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ISBN print: 978-3-0364-0993-1
ISBN eBook: 978-3-0364-1993-0
DOI book: 10.4028/b-fFhn6G
© 2026 Trans Tech Publications Ltd, All Rights Reserved

Formability of Metallic Materials

Edited by
Dr. Gabriela Vincze

Formability of Metallic Materials

Special topic volume with invited peer-reviewed papers only

Edited by
Dr. Gabriela Vincze

Scientific.Net

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Solid State Phenomena, ISSN: 1662-9779, Vol. 388, pp -1-0

© 2026 The Author(s). Published by Trans Tech Publications Ltd, Switzerland.

Accepted:
2026-04-09

Online:
2026-04-15

Preface

The original version of this paper is available on https://www.scientific.net/SSP.388.-1.pdf

This issue presents a collection of contributions addressing the formability of metallic materials, with particular attention to their capacity to undergo plastic deformation while limiting the occurrence of damage and defects in the final product. The mechanisms influencing forming processes, such as plastic flow localization, fracture, and springback, are presented among other phenomena of interest. The contributions included in this issue encompass experimental, theoretical, and numerical investigations aimed at advancing the understanding of the interplay between material properties, mechanical and thermal loading conditions, and additional environmental constraints that govern the formability of metallic materials.

Editor

The original version of this paper is available on https://www.scientific.net/SSP.388.-1.pdf

Vincze, Gabriela University of Aveiro

Co-editors

The original version of this paper is available on https://www.scientific.net/SSP.388.-1.pdf

Aretz, Holger
Balan, Tudor
Coppieters, Sam
Hama, Takayuki
Korkolis, P. Yannis
Thuillier, Sandrine
Vincze, Marilena

Speira
Arts et Metiers Institute of Technology
KU Leuven
Kyoto University
Technical University Dortmund
Université Bretagne Sud
University of Aveiro

Table of Contents

Preface

Thickness Scaling of Forming Limit Curves

Holger Aretz

Work Hardening of Dual Phase Steel in Subsequent Tensile Testing with Post-Necking Large Pre-Strain

Takashi Matsuno, Keisuke Hokimoto, Kazuyuki Shimizu, Takayuki Hama, Yoshiaki Honda

Crystal Plasticity Finite-Element Simulations of Cup Drawing of a ZX10 Magnesium Alloy Sheet

Takayuki Hama, Naoki Miyazawa

In Situ EBSD Characterisation of Anisotropic Creep Behaviour in a Thick AA2139 Part during in-Forming Artificial Ageing

Calixte Gautier, Pierre Antoine Dubos, David Gloaguen, Baptiste Girault, Maxime Rollin

Microstructural Modification and Bendability Improvement in 1300 MPa Steel via Induction Surface Heat Treatment

Antti Kaijalainen, Ahmed Abdelghany, Oskari Haiko, Anttu Jouni Topias Hoikkaniemi, Aki Petteri Pokka

The Interaction of N - and R -Values and Total Elongation in Forming Limit Curves' Theories

Richard Borbely, Bendegúz V. Toth-Nagy, Bertalan Kecskes, Róbert Zs. Keresztes, Gabor Jozsef Beres

Theoretical and Experimental Forming Limit Evaluation of High-Strength Steels up to 1500 MPa

Richard Borbely, Róbert Zsolt Keresztes, Gabor Jozsef Beres

Evaluation and Comparison of a Few Methods for the Strain-Path Correction of FLCs

Grégoire Mainguy, Xavier Lemoine, Tudor Balan

Investigating the Formability of Aluminium Alloy Tubes by Rubber Bulge Testing

Chanmi Moon, Eren Can Sariyarlioglu, Togeir Welo

Influence of Hydrogen Content on the Ductile Rupture of Advanced High-Strength Steels

Lisa Germain, Flavien Vucko, Fabienne Peltier, Sandrine Thuillier

Asymmetric Hole Expansion Test (aHET) for Direct Measurement of Orientation-Dependent Tensile Fracture Strain

Raphaël Pérot, Vincent Grolleau, Dirk Mohr

A Comparative Study of Multiple Cutting Methods on Thin Ferritic Steel for Tensile Test Experiment

Tarek Azzouni, Lysimachi Iona, Patrick Hegele, Wolfram Volk, Christoph Hartmann

Effect of Tool Edge Geometry on Sheared Surface Formation in Blanking of Oxygen-Free Copper

Tomomi Shiratori, Yuta Ichiyanagi, Kaname Watanabe, Aizawa Tatsuhiko

Experimental Investigation of the Influence of Load Direction Changes on Damage in Dual Phase Steel and Aluminum Alloy

Philipp Lennemann, A. Erman Tekkaya, Yannis P. Korkolis

Stochastic Forming Limit Curves: Construction of a Probabilistic Tool for Failure Assessment

Olivier Ameline, Mohamed Yessine Jedidi, Augustin Persoons, Carl Labergere

Forming Limit Test of Asymmetric AI-Polymer Laminate Composite Film

Cheol Sagong, Taek Jin Jang, Taegyun Ahn, Jeong Whan Yoon

Hole Expansion Performance Assessment of Quenched and Partitioning Steel Through a Fully Anisotropic Fracture Modeling

Toros Arda Aksen, Zinan Li, Junhe Lian

Global and Local Formability of 3rd  Generation Advanced High Strength Steels

Anna Payà, Marcel Carpio, David Frómeta

Formability Test for Characterizing the Mode II-III Fracture Transition

Inês M. Almeida, João P.G. Magrinho, Paulo A.F. Martins, Maria Beatriz Silva

Assessing Global and Local Formability from Tensile Tests with Anisotropy Consideration

Guijia Li, Jun He Lian

The Influence of Uphill Quench on the Microstructure and Properties of Particle-Reinforced Aluminum Matrix Composites

Ran Pan, Bao Sheng Liu, Nan Guo, Yuan Song Zeng, Yun He Chang, Zhi Yong Li

Automated Testing and Calibration of Sheet Metal Models

Clement Ndiritu, Yasmine Haoues, Nicolas Decultot, Sandrine Thuillier

Springback Prediction of 6XXX-Series Aluminum: Effect of Hardening Model, Friction and Binder Loading

Joseph Arciero, Kenneth Cheong, Mahmoud Howeyze, Isaura Escorza, Akshay Wankhede, Kidambi Kannan, Sarin Thokala, Christian Leppin, David Anderson, Cliff Butcher

Necking Limits of Anisotropic Metal Sheets Using M-K Theory and Improved Plasticity Models

Yong Hou, Monica Trask, Yannis P. Korkolis

Effect of Continuous Bending under Tension on Dual-Phase Steels

Gabriela Vincze, Ricardo C. Alves, Rafael Oliveira Santos, Marilena C. Vincze

Heterostructure High Entropy Alloys in the CoNiFeMn-Mo System Intended for Deep Drawing Applications

Kamil Cichocki, Anna Wójcik, Paulina Lisiecka-Graca, Tymon Warski, Krzysztof Muszka

Keywords Index

Authors Index

The original version of this eBook is available on https://www.scientific.net/book/formability-of-metallic-materials/978-3-0364-1993-0

Solid State Phenomena, ISSN: 1662-9779, Vol. 388, pp 1-9

doi: 10.4028/p-Maph4K

© 2026 The Author(s). Published by Trans Tech Publications Ltd, Switzerland.

Submitted:
2025-12-05

Revised:
2026-02-18

Accepted:
2026-02-18

Online:
2026-04-15

Thickness Scaling of Forming Limit Curves

The original version of this paper is available on https://www.scientific.net/SSP.388.1.pdf

Holger Aretz

Speira GmbH, Research & Development, Georg-von-Boeselager-Str. 21, 53117 Bonn, Germany

holger.aretz@speira.com

Keywords: forming limit curve, FLC, Nakajima, thickness dependence, bending.

Abstract

A consistent kinematic method was developed to calculate a forming limit curve (FLC) for a material with thickness t0* from a given FLC pertaining to a different thickness t0≠t0*. The developed method is based on the analysis of the bending strains introduced by the Nakajima test method. To calculate the required strains, an explicit and an implicit procedure are presented. In contrast to its implicit equivalent, the explicit method suffers from an intrinsic error which scales with the material's gauge and can be quantified by considering the neutral case t0*=t0. Finally, the developed method predicts a linear relationship between FLC min  and the material thickness, which is in line with practical experience.

Introduction

The original version of this paper is available on https://www.scientific.net/SSP.388.1.pdf

If rolled materials of the same grade are manufactured equivalently (e.g. identical relative rolling reductions and heat treatments), then many final properties (e.g. yield strength, tensile strength, uniform strain, r-values, etc.) are practically independent of their final gauge. However, all fracturerelated characteristics display a significant thickness dependence. For example, the work of Suh et al. [6] suggests that the fracture strain in uniaxial tensile tests positively correlates with the sample gauge, which, in turn, directly determines the number of grains across thickness. A strong thickness dependence is also observed in forming limit curves (FLCs) acquired by the Nakajima stretch forming test. In this test, differently shaped sheet samples are stretched over a hemispherical punch until fracture. This introduces bending strains (e.g. Noder & Butcher [5]), which inevitably bias the failure strains measured at the surface. Consequently, for equivalent materials made of different gauges failure strains increase with thickness.

In the development phase of technical products, frequent design changes are very common and affect both the selection of suitable material grades and their gauge. A problem for sheet manufacturers is the fact that material data cannot be provided for any imaginable material grade and gauge. In particular, the lack of FLCs pertaining to the considered gauge is often problematic. Although some forming simulation codes provide a thickness adjustment feature for FLCs, the underlying approaches known to the author are based on empirical relationships (e.g. Abspoel et al. [1] and Banabic et al. [4, pp. 325-326]), which is not satisfactory.

In the present work, a consistent kinematic method was developed to calculate the FLC for a sheet material with thickness t0* from a given FLC pertaining to a different thickness t0≠t0*. The developed method is based on the analysis of the bending strains introduced by the Nakajima test method. It is emphasized that the proposed kinematic method is unable to account for microstructural effects, such as those observed by Suh et al. [6].

Bending Strain Theory

The original version of this paper is available on https://www.scientific.net/SSP.388.1.pdf

In the following, the strain theory for shell bending derived in Timoshenko & Gere [7, pp. 440-441] is used to calculate in-plane strains in a bent shell. Note that the bending theory is formulated in terms of engineering strains. In general, principal components of the engineering strain tensor are denoted by ej, whereas εj is used for logarithmic principal strains. If needed, logarithmic strains can be converted to engineering strains according to ej=exp(εj)−1 and vice versa. The notational
convention ε1≥ε2 is used throughout this paper, i.e. ε1 is the major principal strain and ε2 the minor principal strain.

To ease navigation through the text, the following three geometric planes are defined:
(1) Top surface plane: in the Nakajima test, this is the convex side of the deformed sample. The associated engineering strain components are denoted as ej, top .
(2) Bottom surface plane: in the Nakajima test, this is the concave side of the deformed sample, which is in contact with the punch. The associated engineering strain components are denoted as ej, bot .
(3) Midplane: this is the plane halfway between the top and bottom surface. The associated engineering strain components are denoted as ej, mid  •

According to Timoshenko & Gere [7, pp. 440-441], the engineering in-plane strains of a bent shell can be calculated from

ej(z)=ej,mid1−zRj−z1−zRj·[1(1−ej,mid)·R^j−1Rj],j=1,2(1)

with Rj and R^j denoting the radius of curvature in the undeformed and deformed configuration, respectively (see also Axelrad [3, p. 32]). Note the singular cases ej, mid =1 and z/Rj=1. The coordinate z∈[−t/2,t/2], with t being the shell's thickness, points towards the center point of curvature and is zero in the shell's midplane. Accordingly, at the convex side of the shell z=−t/2 and at the concave side z=+t/2. In the undeformed configuration the blank is flat, i.e. Rj→∞. Accordingly, Eq. (1) simplifies to

ej(z)Rj→∞=ej,mid−z(1−ej,mid)·R^j,j=1,2(2)

In the Nakajima test a hemispherical punch of radius Rp is used, along with a covering lubrication system of thickness tlub. Thus, we may define an effective punch radius Rp, eff  as follows:

Rp,eff:=Rp+tlub(3)

The radius of curvature is then given by

R^j=Rp,eff+t/2,j=1,2(4)

with t being the current sample thickness. Even a rather large value of tlub=3 mm causes a relative change of the strain results of only <1%. Thus, tlub  will be neglected, yielding the convenient approximation

R^j≈Rp+t/2,j=1,2(5)

A very useful result can be derived from Eq. (1) by solving for R^j, which yields

R^j=z(1−ej, mid )·(ej, mid −ej(z)),j=1,2(6)

However, R^j is the same for all z∈[−t/2,t/2]. Thus, for two positions z1 and z2 we obtain

R^j=z1(1−ej,mid)·(ej,mid−ej(z1))=z2(1−ej,mid)·(ej,mid−ej(z2)),j=1,2(7)

Solving for ej, mid  finally yields

ej, mid =z1·ej(z2)−z2·ej(z1)z1−z2,z1≠z2,j=1,2(8)

Note the similarity with the well-known regula falsi method to calculate roots of equations. Equation (8) reveals that the midplane strains can be calculated in terms of two known strains evaluated at arbitrary, but different, positions Z1 and Z2.

Another result that will be very useful is obtained by calculating the average in-plane strains, ej, avg , from Eq. (2) by means of the mean-value theorem:

ej,avg=1t·∫−t2+t2(ej,mid−z(1−ej,mid)·R^j)dz=ej,mid,j=1,2(9)

As one may see, the average in-plane strains are identical to the midplane strains.
From Eq. (2) we may deduce

ej, top :=ej(z=−t/2)=ej, mid +t/2(1−ej, mid )·R^j,j=1,2(10)

and

ej,bot:=ej(z=t/2)=ej,mid−t/2(1−ej,mid)·R^j,j=1,2(11)

The midplane strains ej, mid  can be calculated from Eq. (10) or Eq. (11) in terms of ej, top  and ej, bot , respectively. In either case, a quadratic equation is obtained which has only one physically plausible solution (cf. [2]):

ej, mid ={1+ej, top 2−(1+ej, top 2)2−(ej, top −t/2R^j),j=1,21+ej, bot 2−(1+ej, bot 2)2−(ej, bot +t/2R^j),j=1,2(12)

Remarks:

The original version of this paper is available on https://www.scientific.net/SSP.388.1.pdf

(1) Equation (12) reveals a nonlinear dependence of the midplane strains ej, mid  on the current thickness t. Thus, the calculation of ej, mid  requires an iterative procedure, as discussed below.
(2) By re-arranging the arguments of the respective square roots, it may be seen that the first case of Eq. (12) produces always real solutions for ej, mid . In contrast, the second case may produce complex solutions if (t/2)/R^j becomes sufficiently large (indicating that shell bending theory is inapplicable) and/or if ej, bot →1. Thus, in computations it is advisable to check the sign of the square root's argument.

For (z1,z2)=(−t/2,t/2) Eq. (8) yields the following useful relationship:

ej, mid =ej, top +ej, bot 2,j=1,2(13)

Thus, when ej, mid  is known by Eq. (12) in terms of ej, top , Eq. (13) can be used to calculate ej, bot  and vice versa. Equation (13) may also be obtained by adding Eq. (10) and Eq. (11).

Remarks:
(1) For the special case ej, top =−ej, bot  Eq. (13) yields ej, mid =0, and vice versa.
(2) The relationship ej, top =−ej, bot  in terms of engineering strains does not imply εj, top =−εj, bot  in terms of logarithmic strains. This may be readily seen by converting the engineering strains to their logarithmic counterparts.

Assuming volume constancy, we may calculate the average thickness strain in terms of the average in-plane strains e1, avg  and e2, avg  given by Eq. (9):

e3,avg=1(1+e1,avg)·(1+e2,avg)−1=1(1+e1,mid)·(1+e2,mid)−1(14)

The current thickness is then

t=t0·(1+e3,avg)=t0(1+e1,mid)·(1+e2,mid)(15)

Computational Procedure

The original version of this paper is available on https://www.scientific.net/SSP.388.1.pdf

For notational convenience, different FLCs are introduced. The known FLC pertaining to a sample thickness of t0 is related to the top surface of the sample. Accordingly, the array holding the measured surface strain data ( n strain couples) is denoted as FLCtop  :

FLCtop =[(ε1, top ,ε2, top )1(ε1, top ,ε2, top )2⋮(ε1, top ,ε2, top )n](16)

Likewise, the arrays holding the FLC data referred to the bottom surface and the midplane will be denoted as FLCbot  and FLCmid , respectively. The FLCs pertaining to another sample thickness t0*≠t0 are denoted as FLCtop *,FLCbot * and FLCmid *, respectively.

The overall objective is to calculate FLCtop * from FLCtop . The developed computational procedure may be broken down in two major steps: in the first step, the top surface strains of the given forming limit curve are used to calculate the corresponding bottom surface strains; in the second step, the calculated bottom surface strains are used to calculate the top surface strains using the desired new material thickness. The following assumptions are introduced:

Assumption 1. The deformation is isochoric (volume preserving).
Assumption 2. The failure strains at both bottom surfaces are identical:

FLCbot *=FLCbot (17)

Assumption 3. The strain ratios at both top surfaces are identical:

(ε2, top )k(ε1, top )k=(ε2, top *)k(ε1, top *)k(18)

for each FLC point k∈[1,n], with n∈ℕ* being the total number of FLC points.
In detail, the computational procedure consists of more than two steps. The bending strain theory formulated above is used to perform the required calculations. In the following, two algorithms will be formulated which differ in the way the current thicknesses t and t*, respectively, are computed; the utilized concise notation should be self-explaining.
(I) Explicit computation of t and t* :

Calculate current thickness: t:=t0·exp(−(ε1, top +ε2, top ))

Using Eq. (12): FLCmid := top2midplane (FLCtop ,t)

Using Eq. (13): FLCbot := midplane2bottom( FLCmid ,t )

Calculate current thickness: t*:=t0*·exp(−(ε1, bot +ε2, bot ))

Using Eq. (12): FLCmid *:= bottom2midplane (FLCbot ,t*)

Using Eq. (13): FLCtop *:= midplane2top (FLCmid *,t*)

After the last step a new FLC is obtained, which is based on the original FLC pertaining to another thickness. As one may see, the last three steps reverse the first three ones, however using the new undeformed thickness t* instead of t.
(II) Implicit computation of t and t* :

Set start value for current thickness: t:=t0

Using Eq. (12): FLCmid := top2midplane (FLCtop ,t)

Using Eq. (15): Calculate updated thickness: tnew:=t0(1+e1, mid )·(1+e2, mid )

Continue if |tnew −t|< tolerance, else set t:=tnew  and go to step 2

Using Eq. (13): FLCbot := midplane2bottom( FLCmid ,t )

Set start value for current thickness: t*:=t0*

Using Eq. (12): FLCmid *:= bottom2midplane (FLCbot ,t*)

Using Eq. (15): Calculate updated thickness: tnew *:=t0*(1+e1, mid *)·(1+e2, mid *)

Continue if |tnew *−t*|< tolerance, else set t*:=tnew * and go to step 7

Using Eq. (13): FLCtop *:= midplane2top (FLCmid *,t*)

The iterations in the implicit scheme ensure that the correct current thickness is used.
As one may see, the implicit scheme is computationally more demanding than the explicit one. While the explicit scheme is convenient for direct calculations, it suffers from the fact that it causes an intrinsic error, as will be discussed below. This error can be avoided by using the implicit procedure, which is generally preferable.

Remark. Unless stated otherwise, the implicit strain calculation will be used throughout the present paper.

In principle, the outlined explicit or implicit computation scheme can be used to calculate both the major and the minor strains pertaining to t0*. However, in the present work only the major strains are calculated in this way. The minor strains are computed using Eq. (18):

(ε2, top *)k=(ε2, top )k·(ε1, top *)k(ε1, top )k(19)

Remark. Alternatively, the computation of the major and minor strains could have been done in the reverse order, i.e. calculate first (ε2, top *)k using the algorithm above, followed by a calculation of (ε1, top *)k using Eq. (18). Hence, the computation of the strain components is not unique. However, numerical tests revealed that the results of the alternative procedure are less realistic.

Intrinsic Error of the Explicit Strain Calculation

The original version of this paper is available on https://www.scientific.net/SSP.388.1.pdf

In the neutral case t0=t0* it can be expected that the previously described procedure just reproduces the input data, given that nothing has changed. However, this is not exactly the case when using the explicit method described above, because the explicit strain calculation of the current thicknesses t and t* introduces a small error, as explained next. In case of bending, Eq. (2) yields

(e1,top+e2,top)>(e1,mid+e2,mid)>(e1,bot+e2,bot)(20)

By means of the conversion

εj=ln(1+ej)(21)

the order in Eq. (20) is equivalent to

(ε1,top+ε2,top)>(ε1,mid+ε2,mid)>(ε1,bot+ε2,bot)(22)

This, in turn, is equivalent to

exp(−(ε1,top+ε2,top))<exp(−(ε1,mid+ε2,mid))<exp(−(ε1,bot+ε2,bot))(23)

Now, we consider the equation

t:=t0·exp(−(ε1,top+ε2,top))(24)

to calculate the current material thickness t based on the top surface strains, as they are the sole explicitly available strain information. However, the thickness t calculated from Eq. (24) is only exact if strain gradients across the thickness of the sample associated with bending deformation were absent, which corresponds to εj, top =εj, mid =εj, bot . In fact, in the presence of strain gradients, Eq. (24) describes a kind of "local thickness" corresponding to the top surface and, as Eq. (23) reveals, a different thickness will be obtained if e.g. the (unfortunately unknown) midplane strains εj, mid  or bottom strains εj, bot  were used instead. Thus, using e.g. the top surface strains to calculate the "midplane thickness" will inevitably cause an error if strain gradients across the thickness are present.

The same error adheres to the equation

t*:=t0*·exp(−(ε1,bot+ε2,bot))(25)

which, even worse, uses the previously calculated bottom strains, as they are the sole explicitly available strain information.

By virtue of Eq. (23) it is clear that the thicknesses t and t* calculated by Eq. (24) and Eq. (25), respectively, are not objective, because they depend on the location of strain evaluation across the sample's thickness.

[picture]

Fig. 1. Neutral case analysis for an AA6xxx-T4 material. Left: results of the implicit solution scheme. Right: results of the explicit solution scheme.

Fig. 1. Neutral case analysis for an AA6xxx-T4 material. Left: results of the implicit solution scheme. Right: results of the explicit solution scheme.

From the explanations just given, one may conclude that the same error is made twice in the explicit computations, which is the reason why inaccuracies must be expected. Since the strain gradients caused by bending increase with t0, one may expect that the error will grow accordingly when using the explicit method. Numerical experiments of the author (not shown) confirm that the intrinsic error of the explicit strain calculation positively correlates with t0.

As an example, consider the results in Fig. 1 of two neutral case predictions for a 6xxx aluminium alloy for t0=t0*=1 mm using both the implicit and the explicit solution scheme. As one may see, in either case the input curve and the prediction are virtually identical, revealing a negligible error of the explicit scheme. However, another application example considered below demonstrates that the intrinsic error cannot be neglected in general.

Application Example: 5xxx Aluminium Alloy

The original version of this paper is available on https://www.scientific.net/SSP.388.1.pdf

The considered material is an AA5182 aluminium alloy in H2× temper with a gauge of t0=3 mm. The FLC data shown in Fig. 2 (left) were acquired according to the standard DIN EN ISO 12004-2 using a punch radius of Rp=50 mm and with 3 repetitions for each sample geometry (throughout all FLCs in this paper). The experimental data points were fitted using a third-order polynomial.

[picture]

Fig. 2. FLC data and prediction pertaining to the AA5182-H2x material.

Fig. 2. FLC data and prediction pertaining to the AA5182-H2x material.

[picture]

Fig. 3. FLC data and prediction pertaining to the AA5182-H2x material (reverse analysis).

Fig. 3. FLC data and prediction pertaining to the AA5182-H2x material (reverse analysis).

The developed thickness scaling method was used to make a blind prediction of the FLC pertaining to a larger thickness of t0*=4 mm. The necessity for a blind prediction arose from a customer request since at its time no experimental data pertaining to t0*=4 mm were available. Later, the material with a gauge of t0*=4 mm was produced, which enabled its FLC determination. As Fig. 2 (left) reveals, in the present case the blind prediction matches the experimental FLC very well.

It is also very interesting to use the developed theory to predict the major strain pertaining to the minimum point of the FLC, commonly denoted as FLCmin , for different gauges. As Fig. 2 (right) reveals, a linear relationship is obtained which is, qualitatively, in agreement with experimental findings.

Remark. The FLCmin -predictions in Fig. 2 (right) are based on the reference FLC pertaining to a gauge of t0=3 mm. Consequently, if this reference FLC changes slightly, e.g. due to batch variations, then the FLCmin-predictions will be affected accordingly. ◻

Next, a reverse analysis is carried out: we now pretend that solely the FLC pertaining to t0= 4 mm was given and wish to predict the FLC pertaining to t0*=3 mm. Since experimental FLC data are available for both gauges it is straightforward to quantify the prediction. The results of this analysis are summarized in Fig. 3 (left) and reveal a very good prediction of the target FLC. It is also
interesting to investigate whether the reverse analysis results in a different FLC min -prediction, which is shown in Fig. 3 (right): a direct comparison with Fig. 2 (right) reveals rather minor differences.

The available data may also be used to investigate the intrinsic error related to the explicit solution scheme as discussed above. To this end, the neutral case t0=t0*=4 mm is considered. This rather large gauge is very well suited for an error analysis, because the intrinsic error, as discussed above, positively correlates with thickness. The corresponding results in Fig. 4 (right) reveal a small overprediction compared to the input curve. A simple way to roughly quantify this error is to evaluate the FLCmin -values, which are FLCmin =0.174 and FLCmin *=0.182, respectively, corresponding to an overprediction of ≈4.6%. To demonstrate its superiority over its explicit equivalent, the neutral case analysis was also carried out using the implicit strain calculation, see Fig. 4 (left), in which case the input and the predicted FLC coincide.

[picture]

Fig. 4. Neutral case analysis for the AA5182-H2x material. Left: results of the implicit solution scheme. Right: results of the explicit solution scheme.

Fig. 4. Neutral case analysis for the AA5182-H2x material. Left: results of the implicit solution scheme. Right: results of the explicit solution scheme.

Remark. As explained above, the proposed method can only account for kinematic effects related to bending deformation. Thus, microstructural effects, such as those observed by Suh et al. [6], are not considered per se. However, a respective extension of the kinematic method appears feasible. For example, an ad hoc approach could be to carry out uniaxial tensile tests for two different gauges, and to add the engineering fracture strain difference to the engineering major strain of the scaled FLC a posteriori. (Note that when extensometers are used true fracture strains cannot be determined in a theoretically consistent way.) Of course, this presumes that fracture strains are already available for both gauges. If this is not the case, one may think of interpolating available data pertaining to other gauges. For the AA5182-H2x material considered in the present work, this extra contribution amounts to an engineering strain of +0.007 when the sample thickness increases from 3 mm to 4 mm , and would only slightly increase the major strain of the scaled FLC. (Note that when the sample thickness decreases, the extra strain contribution would be negative, thereby decreasing the major strain of the scaled FLC.) One may speculate that the absolute value of this extra strain contribution increases when the thickness contrast is larger than in the case considered. However, a detailed investigation is beyond the scope of the present work and left for future research.

Summary

The original version of this paper is available on https://www.scientific.net/SSP.388.1.pdf

A consistent kinematic method was developed to calculate a forming limit curve (FLC) for a material thickness t0* from a given FLC pertaining to a different thickness t0. The developed method is based on the analysis of the bending strains introduced by the Nakajima test method. The calculation procedure may be broken down in two major steps: in the first step, the top surface strains of the
given forming limit curve are used to calculate the corresponding bottom surface strains; in the second step, the calculated bottom surface strains are used to calculate the top surface strains using the desired new material thickness. To calculate the required strains, an explicit and an implicit procedure were presented. In contrast to its implicit equivalent, the explicit method suffers from an intrinsic error which scales with the material's gauge and can be quantified by considering the neutral case t0*=t0. For a rather large thickness, an application example illustrates that the intrinsic error of the explicit strain calculation causes an overprediction of FLCmin  of about 4.6%. Finally, the developed method predicts a linear relationship between FLC min  and the material thickness, which is in line with practical experience. The proposed method does not account for microstructural features potentially contributing to the scaled FLC. A respective ad hoc extension of the kinematic method was outlined but not investigated in detail.

Acknowledgements

The original version of this paper is available on https://www.scientific.net/SSP.388.1.pdf

The author is grateful to Dr. Henk-Jan Brinkman for stimulating this work.

References

The original version of this paper is available on https://www.scientific.net/SSP.388.1.pdf

M. Abspoel, M. E. Scholting, J. M. M. Droog, A new method for predicting Forming Limit Curves from mechanical properties, Journal of Materials Processing Technology 213 (2013) 759-769.

H. Aretz, Pre-strain correction of forming limit curves, Materials Research Proceedings 41 (2024) 948-957.

E. Axelrad, Schalentheorie, B. G. Teubner Stuttgart, 1983.

D. Banabic, H.-J. Bunge, K. Pöhlandt, A. E. Tekkaya, Formability of Metallic Materials, Springer-Verlag Berlin Heidelberg GmbH, 2000.

J. Noder, C. Butcher, A comparative investigation into the influence of the constitutive model on the prediction of in-plane formability for Nakajima and Marciniak tests, International Journal of Mechanical Sciences 163 (2019) 105138.

C. H. Suh, Y.-C. Jung, Y. S. Kim, Effects of thickness and surface roughness on mechanical properties of aluminum sheets, Journal of Mechanical Science and Technology 24 (2010) 20912098.

S. Timoshenko, J. M. Gere, Theory of Elastic Stability, 2nd ed., Dover Publication, 2009.

This ePub file was created using machine-generated methods and automated software to convert the source text into an ePub format and may contain formatting inconsistencies from the original version. Formatting may vary depending on the device used.

The original version of this eBook is available on https://www.scientific.net/book/formability-of-metallic-materials/978-3-0364-1993-0

Solid State Phenomena, ISSN: 1662-9779, Vol. 388, pp 11-19

doi: 10.4028/p-y1W1Ey

© 2026 The Author(s). Published by Trans Tech Publications Ltd, Switzerland.

Submitted:
2025-12-09

Revised:
2026-02-09

Accepted:
2026-02-19

Online:
2026-04-15

Work Hardening of Dual Phase Steel in Subsequent Tensile Testing with Post-Necking Large Pre-Strain

The original version of this paper is available on https://www.scientific.net/SSP.388.11.pdf

Takashi Matsuno 1,a*, Keisuke Hokimoto 1, b, Kazuyuki Shimizu 1,c, Takayuki Hama 2, d and Yoshiaki Honda 3,e

1 Faculty of Engineering, Tottori University, 4-101 Koyama-cho-minami, Tottori, Tottori 680-8552, Japan

2 Graduate School of Energy Science, Kyoto University, Yoshida-Honmachi, Sakyo-ku, Kyoto 6068501, Japan

3 Nippon Steel Corporation, 20-1 Shintomi, Futtsu-City, Chiba, 293-8511 Japan

a * matsu@tottori-u.ac.jp, b M24J3049X@edu.tottori-u.ac.jp, c ksmz@tottori-u.ac.jp, d  hama.takayuki.4s@kyoto-u.ac.jp, ehonda.x66.yoshiaki@jp.nipponsteel.com

Keywords: tensile test, strain-aging, pre-strain, post-necking, dual phase steel.

Abstract

For high-accuracy finite element (FE) simulation of automobile crashing behavior, a work hardening curve that involves pre-strain from press forming is required. Here, the plastic strains exceeding the uniform deformation region are generally introduced through processes such as bending, but such large pre-strain effect have not been reported. Therefore, in this study, for DP590 steel, the work hardening curve for second-stage tension under pre-strain exceeding the uniform deformation region was identified. This identification was enabled by the diameter measurement tensile test developed by the authors. As a result, in the second-stage tension in the same direction as the first-stage tension, the initial yield stress showed a tendency to overshoot relative to the original work hardening curve, revealing that strain aging occurred. The overshoot portion formed a stress plateau that continued up to an equivalent plastic strain of 0.18 . Such a tendency has not been observed in DP590 steel, making this a phenomenon revealed for the first time. When the tensile direction in the second stage was orthogonal to the first stage, the cross-hardening effect (reduction in initial yielding due to the Bauschinger effect and overshoot from the original work hardening curve) was observed. The stress plateau region due to overshoot continued up to an equivalent plastic strain as large as 0.6 . These large plateaus concluded that work hardening presents perfect plasticity at large deformed press parts.

Introduction

The original version of this paper is available on https://www.scientific.net/SSP.388.11.pdf

To achieve both weight reduction and crashworthiness in automobiles, advanced high-strength steel (AHSS) sheets have become widely adopted. As material strength increases, equivalent crushing performance can be obtained even with reduced sheet thickness. Although problems specific to AHSS exist, such as tool damage [1], delayed fracture [2] caused by remarkably high tensile residual stress [3,4] combined with deformation-induced microstructural changes [5,6], and incidental fracture during press forming [7], significant progress has been made in addressing these issues in recent years. Indeed, automobile frames incorporating materials with more than 60%AHSS content have been released.
For the design of crash components, crash simulations using the finite element method (FEM) are often performed in advance. FEM requires input of the work-hardening curve of the material used in the frame. The work-hardening curve of as-received material is often employed as input. However, to improve the accuracy of crash FEM, a work-hardening curve that accounts for plastic strain during press forming is necessary.
The simplest approach involves directly subtracting the plastic strain imposed by press forming from the work-hardening curve of as-received material. However, deformation during crash involves path changes relative to deformation during press forming. When the deformation path changes, a decrease in yield stress due to the Bauschinger effect occurs, followed by an overshoot from the original work-
hardening curve [8-10]. These behaviors are collectively termed the cross-hardening effect, and their characteristics have been analyzed through various experiments [8-10].
In most cases investigating the cross-hardening effect, plastic deformation prior to strain path change remains below uniform elongation. This is because specimens extracted from necked test pieces do not exhibit uniform elongation, and conventional tensile tests cannot measure the work-hardening curve beyond this point. However, pre-strain levels below uniform elongation are not necessarily sufficient for practical applications. Because AHSS exhibits relatively low ductility, press forming is performed primarily through bending deformation. During bending processes, plastic strains exceeding uniform elongation are commonly encountered.
Therefore, in this study, identification of the work-hardening curve was attempted for 590 MPa class dual-phase (DP) steel subjected to a 90-degree deformation path change. At ESAFORM 2024, the authors reported the relationship between true stress ( σt ) and cross-sectional area reduction ( ρ ) obtained using diameter-measurement-type tensile testing [11], which was developed by the authors [12-14]. The present work identifies the flow stress curve through FE simulation based on the σt−ρ curve measured in the previous study.

Experimental and Simulation Methods

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Material

The original version of this paper is available on https://www.scientific.net/SSP.388.11.pdf

A DP steel with a tensile strength of 560 MPa was used as the test material. Its chemical component and mechanical properties are shown in Tables 1 and 2, respectively. The steel was fabricated in the laboratory with the intention of being equivalent to DP590.

Table 1. Chemical components of tested steel (mass%).

C | Si | Mn
0.038 | 0.50 | 1.50

Table 2. Mechanical properties of tested steel.

Yield stress

Tensile strength

Uniform

elongation mathematical formula

Total elongation

335 | 561 | 19 | 31

Preparing small round-bar specimen

The original version of this paper is available on https://www.scientific.net/SSP.388.11.pdf

The compact round bar tensile test with pre-strain introduction has already been reported at ESAFORM 2024 [11] and is described again below.
No. 5 type specimens with 2.5−mm thickness, standardized by Japanese Industrial Standards (JIS) Z2241, were subjected to preliminary tensile testing before subsequent compact tensile tests. Hereafter, this sheet-type tensile testing is referred to as JIS5 tensile test. The JIS5 specimens were elongated up to 23% under post-necking conditions. To estimate the local Mises strain, measurement by digital image correlation (ARAMIS system, GOM) was performed in advance on one JIS5 tensile test. The distribution of Mises strain is shown in Fig. 1, with a maximum value of approximately 0.31 .

[picture]

Fig. 1. Equivalent (Mises) strain distribution in JIS5 tensile test at 23% elongation.

Fig. 1. Equivalent (Mises) strain distribution in JIS5 tensile test at 23 % elongation.

As depicted in Fig. 2, the preliminary tensile test was aligned such that the tensile direction of the small round-bar specimens corresponded to the transverse direction of the cold-rolling process (referred to as the transverse direction). Accordingly, the JIS 5 specimens were elongated in the transverse direction, and subsequent tensile specimens were extracted in two orientations: one aligned with the transverse direction, representing no strain path change (Fig. 2a), and the other aligned with the rolling direction, such that the subsequent tensile direction was perpendicular to the preliminary tensile direction (Fig. 2b). It should be noted that this configuration does not constitute a strictly orthogonal strain path change in a rigorous sense, because diffuse necking developed in the JIS 5 specimens (as shown in Fig. 1), resulting in a stress state that deviated from pure uniaxial tension prior to the extraction of the subsequent specimens.
With such large pre-strain, virtually no uniform elongation is observed in the subsequent tensile testing. Consequently, work-hardening curves cannot be identified using conventional sheet-type tensile specimens with elongation measurement. The compact tensile testing system with real-time diameter measurement developed by the authors enables the determination of the true stress versus true strain curve in the post-necking region by measuring the diameter at the necking section instead of elongation. As described later, work-hardening curves can be identified through FEM-based curve fitting using the measured load and diameter values as references. Although it may be suggested that measuring the rectangular cross-sectional area of sheet-type specimens would suffice, unlike circular cross-sections, the rectangular shape is not maintained in the post-necking region, making accurate measurement of the cross-sectional geometry difficult.
The design of the small round-bar specimens is presented in Fig. 3. The thinnest part of these specimens had a diameter of 1.0 mm , enabling their extraction from the JIS 5 specimens that had undergone pre-straining exceeding the onset of necking.

[picture]

Fig. 2. Sampling position of the small round-bar specimens and the tensile direction of the JIS 5 tensile test prior to compact tensile testing: (a) subsequent tensile direction parallel to the preliminary tensile direction (FD case) and (b) subsequent tensile direction perpendicular to the preliminary tensile direction (OD case).

Fig. 2. Sampling position of the small round-bar specimens and the tensile direction of the JIS 5 tensile test prior to compact tensile testing: (a) subsequent tensile direction parallel to the preliminary tensile direction (FD case) and (b) subsequent tensile direction perpendicular to the preliminary tensile direction (OD case).

[picture]

Fig. 3. Geometry of the small round-bar specimen depicted in an axisymmetric two-dimensional drawing.

Fig. 3. Geometry of the small round-bar specimen depicted in an axisymmetric two-dimensional drawing.

Compact tensile test with real-time diameter measurements

The original version of this paper is available on https://www.scientific.net/SSP.388.11.pdf

After extracting the small round-bar specimens, compact tensile tests with real-time diameter measurements were executed, generating σt−ρ curves. In these compact tensile tests, the diameters, Da and Db, of the thinnest cross-sectional area under tension were continuously measured in two directions. The diameter measurements were performed using an LED-projection-type dimension measuring instrument (Keyence TMX 5006). Although the dimensional measurement accuracy is ±0.2μ m, the measured values contain scatter due to the algorithm used to identify the minimum diameter section. Further details can be found in the authors' previous work [14]. The specimen was oriented such that Db corresponded to the thickness of the original steel material. The σt and ρ in the cross-section were calculated as follows:

σt=4FπDaDbρ=ln(D02DaDb)

(1)(2)

where D0 corresponds to the initial diameter of the specimen (1.0 mm).

FE Simulation

The original version of this paper is available on https://www.scientific.net/SSP.388.11.pdf

Furthermore, to reproduce the measured σt−ρ curve, the flow stress curve was identified through tensile simulations using FEM. A two-dimensional axisymmetric model was employed, and conventional boundary conditions were applied, in which one end was fixed and the other was driven under displacement control. Further details can be found in our previous publications [12, 13, 15]. The commercial static implicit FEM solver Abaqus Standard was used for the computation. Even for materials that exhibit upper yield point in elongation-measurement-type tensile tests, they do not show it in geometry-measurement-type tensile tests. Therefore, based on the original work hardening curve offset by the amount of pre-strain (equivalent strain), a flow stress curve composed of three parts was assumed: the yield stress reduction part corresponding to the Bauschinger effect (Eq. (3)), the yield stress overshoot and plateau part (Eq. (4)), and the return to the original curve. These parts transition at specific strain values ε¯B and ε¯S. In addition, in the latter part of the work hardening curve, a linear term Kl was multiplied as shown in Eq. (5) to correct the deviation from the reference curve Yshift (ε¯p).

Yst(ε¯p)=κ(ε¯p+γ)η for ε¯p<ε¯BYst(ε¯p)=Yconst  for ε¯B≤ε¯p<ε¯SYst(ε¯p)=KlYshift (ε¯p) for ε¯p≥ε¯S.(5)

Yshift (ε¯p) is the offset curve of the original (without pre-straining) flow stress curve Y(ε¯p), expressed as

Yshift (ε¯p)=Y(ε¯p+ε¯pre )(6)

where ε¯pre  is the pre-strain.
The original (without pre-straining) flow stress curve Y(ε¯p) was identified using small round-bar tensile specimen with integration of real time diameter measurement and FE simulation. The curve is identified as:

Y(ε¯p)=(1−a)Y1(ε¯p)+a(Y2(ε¯p)+b)(7)

where Yi(ε¯p),i=1,2, denotes Swift work-hardening law as follows:

Yi(ε¯p)=Ki(ε¯p+ci)ni(8)

a and b are the parameters for switching from Y1(ε¯p) to Y2(ε¯p) in Y(ε¯p) as

a=tanh(50(ε¯p−ε¯switch ))+12b=Y2(ε¯switch )−Y1(ε¯switch )

(9)(10)

Here, ε¯switch  represents the plastic strain corresponding to the transition from Y1(ε¯p) to Y2(ε¯p). The identified parameters are shown in Table 3.

Table 3 Parameters of the Y(ε¯p) identified through compact tensile testing combined with FE simulation.

K₁ | n₁ | c₁ | K₂ | n₂ | c₂ | εswitch
882.0 | 0.152 | 0.00165 | 809.7 | 0.11 | 0.00031 | 0.20

Results and Discussion

The original version of this paper is available on https://www.scientific.net/SSP.388.11.pdf

Figure 4 presents the flow stress curves identified through subsequent compact tensile testing integrated with FE simulation. The identified parameters are listed in Table 4. The flow stress curves in Fig. 4 are normalized by the respective Kl values, such that the curves coincide in the larger strain region (ε¯p≥ε¯S) under all conditions.

First, in the following direction (FD)-where the JIS 5 and subsequent tensile testing directions are identical-the flow behavior in the small strain region is characterized by an overshoot followed by a stress plateau. The initial yield stress is approximately 40 MPa higher than Yshift (0). (Because this value is normalized by Kl, it does not necessarily correspond to Yconst  in Table 4). The results indicate that this stress plateau persists until ε¯p=0.18.
In contrast, during tension in the orthogonal direction (OD), the initial yield stress decreases by approximately 80 MPa , indicating a clear Bauschinger effect. The flow stress then increases to 810 MPa before forming a stress plateau. This plateau persists up to ε¯p=0.60, a behavior consistent with nearly perfect plasticity.

Figure 5 provides a comparison between the measured and FE-simulated σt−ρ curves. Although the shape differs considerably from the work-hardening curves in Fig. 4, this is attributable to the hydrostatic stress arising from necking. In the strain range corresponding to the stress plateau region, the true stress increases due to the development of hydrostatic stress caused by necking [15]. Since the equivalent stress under the von Mises yield criterion does not include the hydrostatic stress component, the equivalent stress itself does not increase even when the hydrostatic stress becomes large due to necking, that is, even when the true stress increases. Consequently, the true stress continues to increase even when the work-hardening curve exhibits a plateau. The increase in hydrostatic stress due to necking is widely recognized through the Bridgeman analysis[16]. With this understanding, examination of Fig. 5 confirms that the FE simulation, which utilizes the flow stress curves from Fig. 4, accurately reproduces the measured values. Fig. 5a also displays a linear fit line in the small strain region, revealing that σt deviates from this tangent at approximately ρ=0.18. This observation corroborates the finding that the stress plateau region persists until ε¯p=0.18. In the plateau region of the flow stress curve, the hydrostatic stress generated by necking produces a slight gradient in the σt−ρ curve as described above.
In the case of OD tension (Fig. 5b), the measured data exhibit a plateau region even on the σt−ρ curve. Considering the slight gradient of σt observed in FD tension (Fig. 5a), such a plateau implies a gradual decrease in flow stress. Actually, from approximately ρ=0.20 onward, the σt predicted by the FE simulation is slightly higher than the measured values. This discrepancy suggests that incorporating work softening into the flow stress curve could improve the agreement; however, introducing work softening into the FE simulation severely deteriorates numerical convergence, and resolving this discrepancy was not achieved in the present study. Nevertheless, this discrepancy is not substantial, and the existence of an extended plateau over a remarkably large range of plastic strain, or a gradual work softening trend, is an unambiguous observation from the experimental results. Given that the use of a work softening curve must be avoided due to the convergence difficulty, the nearly perfectly plastic flow stress curve derived in Fig. 4 represents a reasonable approximation for predicting the actual deformation behavior.
It is noteworthy that such differences are observed between the FD and OD cases despite both exhibiting stress plateaus. The detailed mechanism underlying this difference remains unclear at this stage. We hypothesize that the plateau region inherently exhibits a slight softening tendency, which becomes more apparent in the σt−ρ curve for cases with a wider plateau region, such as the OD case. This hypothesis requires detailed verification based on micromechanics, which we intend to pursue through inverse deformation analysis [17, 18].
The increase in initial yield stress during FD tension is a behavior not previously observed for this DP590 steel. A significant time-lapse of approximately two months occurred between the initial JIS 5 tensile test and the subsequent small round bar test. Consequently, there is a high probability that
static strain aging occurred during this period. It is possible that strain aging also affected the material response in the OD case.
These results suggest that components containing regions of large deformation, such as bent sections, may exhibit higher-than-anticipated strength. This strength exceeds predictions based on the original flow stress curve obtained from a conventional JIS 5 tensile specimen.

[picture]

Fig. 4 Identified flow stress curves.

Fig. 4. Identified flow stress curves.

Table 4. Parameters for Yst(ε¯p) identified by subsequent compact tensile testing (following the JIS 5 tensile test) combined with FE simulation.

Direction

relative to JIS 5

Same | 0.31 | - | - | - | 800 | - | 0.145 | 1.06
Orthogonal | 0.31 | 1229.8 | 0.12 | 0.0051 | 821 | 0.030 | 0.595 | 1.01

[picture]

Fig. 5. σt−ρ curves of measurement and FE simulation: (a) FD and (b) OD tensile tests.

Fig. 5. σ t − ρ curves of measurement and FE simulation: (a) FD and (b) OD tensile tests.

Summary

The original version of this paper is available on https://www.scientific.net/SSP.388.11.pdf

This study investigated the influence of strain path changes on the work hardening behavior of DP590 AHSS following post-necking tensile deformation.
In the experiment procedure, sheet specimens were first subjected to tensile deformation until necking occurred. Small round-bar tensile specimens were then extracted from the necked region. The flow stress curve for this second-stage tensile test (on the small round-bars) was identified by ensuring that an FE simulation accurately reproduced the measured diameter change and true stress. The key findings are as follows:

When the material was subjected to large pre-deformation, an increase in the initial yield stress of the subsequent tensile test was observed, even without a change in the tensile direction. This behavior is presumably attributable to strain aging, although further investigation is required to confirm this mechanism.

When the second-stage tensile direction was perpendicular to the preliminary tensile direction, the identified flow stress curve exhibited a nearly flat plateau, suggesting behavior approaching perfect plasticity. It should be noted, however, that the agreement between the experimental and simulated results for the OD case (Fig. 5b) is less satisfactory than that for the FD case (Fig. 5a), and the identified flow stress curve for the OD case should therefore be regarded as preliminary. This finding suggests that components containing regions of large deformation (such as bent sections) that have experienced post-necking strains may exhibit higher structural strength than would be predicted by the original flow stress curve.

References

The original version of this paper is available on https://www.scientific.net/SSP.388.11.pdf

T. Matsuno, K. Nakagiri, T. Matsuda, T. Tanaka, T. Yasutomi, H. Shoji, & M. Ohata, Identification of ductile fracture design curve for hardened quasi-brittle AISI-D2 tool steel to predict shearing tool failure, J. Mater. Process. Technol., 307(2022), 117680. https://doi.org/10.1016/j.jmatprotec.2022.117680.

T. Matsuno, Y. Ueda, T. Takahashi, T. Hama, T. Hojo, Y. Shibayama, M. Ridha, Y. Okitsu, & M. Takamura, Delayed fracture stress thresholds degraded by shear punching process in ultra-high-strength steel sheets: Analysis using an in-plane bending test with numerical assimilation, J. Manuf. Process., 119 (2024) 1005-1021. .

J. Eguchi, T. Matsuno, Y. Kunii, K. Shimizu, Y. Matsuki, T. Shinmiya, & E. Iizuka, Observation-driven punching simulation using a quasi-nonparametric ductile fracture locus, J. Mater. Process. Technol., (2026) 119243. .

T. Matsuno, Y. Ochiai, Y. Okitsu, M. Iga, A. Khori, & T. Mikami, Surface and interior residual stress analysis of a deep-drawn 1180 MPa class ultra-high strength steel sheet with scratch marks, Int. J. Adv. Manuf. Technol., 116 (2021) 2873-2884. .

T. Matsuno, N. Kinoshita, T. Matsuda, Y. Honda, & T. Yasutomi, Microvoid formation of ferrite-martensite dual-phase steel via tensile deformation after severe plastic sheardeformation, ISIJ Int., 63 (2023) 941-949. .

T. Matsuno, Y. Sekito, & K. Kawasaki, Microstructure characterization of fine grains near hotsheared surface formed during hot-stamping process, J. Mater. Process. Technol., 229 (2016) 570-581. .

T. Matsuno, N. Kinoshita, K. Shimizu, H. Shoji, & M. Ohata, Microstructural feature-driven post-mortem analysis of plastic strain and stress triaxiality in ductile-fractured ferritemartensite dual-phase steel: Integrating microstructural observations and mesoscale finite element simulations, J. Mater. Res. Technol. 39(2025), 3087-3102. .

M. Joo, M. S. Wi, S. Y. Yoon, S. Y. Lee, F. Barlat, C. N. Tome, B. Jeon, & Y. Jeong, A crystal plasticity finite element analysis on the effect of prestrain on springback, Int. J. Mech. Sci., 237(2023), 107796. .

J. Liao, J. A. Sousa, A. B. Lopes, X. Xue, F. Barlat, & A. B. Pereira, Mechanical, microstructural behaviour and modelling of dual phase steels under complex deformation paths, Int. J. Plasticity, 93(2017), 269-290. .

F. Barlat, G. Vincze, J. J. Grácio, M. G. Lee, E. F. Rauch, & C. N. Tomé, Enhancements of homogenous anisotropic hardening model and application to mild and dual-phase steels, Int. J. Plasticity, 58(2014), 201-218. .

T. Matsuno, N. Kinoshita, K. Hokimoto, T. Hama, & Y. Honda, Enhancement of fracture strain during abrupt orthogonal strain-path changes in ferrite/martensite dual phase steel, Mater. Res. Proc., 41(2023), 118. .

T. Matsuno, D. Kondo, T. Hama, T. Naito, Y. Okitsu, S. Hayashi, K. Takada, Flow stress curves for 980 MPa - and 1.5 GPa -class ultra-high-strength steel sheets weakened under high-stress triaxiality, Int. J. Mech. Sci., 261(2024), 108671. .

T. Matsuno, T. Fujita, T. Matsuda, T. Shibayama, T. Hojo, I. Watanabe, Unstable stresstriaxiality development and contrasting weakening in two types of high-strength transformation-induced plasticity (TRIP) steels: Insights from a new compact tensile testing method, J. Mater. Process. Technol., 322(2023), 118174. .

T. Matsuno, Y. Yamazaki, S. Matsubara, J. Eguchi, K. Shimizu, M. Koyama, ... & T. Tsuchiyama, Local work-hardening and Lüders deformation invariance in TRIP-aided medium-Mn high-strength steel: Significant TRIP effect changes in the warm temperature range up to . Soc. Technol. Plast., 66(2025), 61-68. .

T. Matsuno, K. Furukawa, Y. Okitsu, M. Koyama, & T. Tsuchiyama, Local yield stress and its unusual independence on multi-axial stress states during Lüders deformation of medium-Mn, high-strength steel, ISIJ Int., 64 (2024) 859-867. .

P.W. Bridgman, Studies in Large Plastic Flow and Fracture, McGraw-Hill, New York, 1952.

T. Matsuno, N. Okumura, Y. Fukuda, K. Shimizu & T. Murai, Direct solution of inverse deformation problem for dual-phase steel microstructures: U-net deep fake for pseudo-inverse finite-element simulation using representative volume elements, Mater. Today Commun., 46(2025), 112769. .

T. Matsuno, Y. Fukuda, K. Shimizu, H. Shoji, M. Ohata, N. Yamashita, H. Yokota & T. Murai, Image-assimilation of deformed dual-phase steel microstructure via U-net deep learning, Mater. Res. Proc., 54(2025), 963-970. .

This ePub file was created using machine-generated methods and automated software to convert the source text into an ePub format and may contain formatting inconsistencies from the original version. Formatting may vary depending on the device used.

The original version of this eBook is available on https://www.scientific.net/book/formability-of-metallic-materials/978-3-0364-1993-0

Solid State Phenomena, ISSN: 1662-9779, Vol. 388, pp 21-27

doi: 10.4028/p-Q18ra9

© 2026 The Author(s). Published by Trans Tech Publications Ltd, Switzerland.

Submitted:
2025-12-16

Revised:
2026-02-10

Accepted:
2026-02-16

Online:
2026-04-15

Crystal Plasticity Finite-Element Simulations of Cup Drawing of a ZX10 Magnesium Alloy Sheet

The original version of this paper is available on https://www.scientific.net/SSP.388.21.pdf

Takayuki Hama 1,a* and Naoki Miyazawa 1, b

1 Graduate School of Energy Science, Kyoto University, Yoshida-Honmachi, Sakyo-ku, Kyoto, 6068501, Japan

a*hama@energy.kyoto-u.ac.jp, b  miyazawa.naoki.3n@kyoto-u.ac.jp

Keywords: magnesium alloy, cup drawing, twinning, crystal plasticity finite-element method.

Abstract

Magnesium ( Mg ) alloy sheets are expected to contribute to the lightweighting of structural components, owing to their inherent benefits of low density and high specific strength. However, the limited room-temperature press formability exhibited in Mg alloy sheets remains a barrier to their expanded use. A significant factor contributing to the limited formability is the strong basal texture. To improve the room-temperature press formability, ZX series Mg alloy sheets that weakened the basal texture have recently been developed. In our previous study [Hama et al., Mater. Res. Proc., 28(2003), 711-716], cup drawability of a Mg−1.5mass%Zn−0.1mass%Ca(ZX10Mg) alloy sheet was investigated at room temperature. The obtained cup exhibited that the cup height and thickness strains differed significantly in the circumferential direction of the cup. However, a more detailed discussion on the mechanisms of this anisotropic deformation was hampered by a reliance solely on experimental observations. Therefore, in this study, crystal plasticity finite-element simulations of cup drawing of the ZX10Mg alloy sheet were performed. The simulation results qualitatively reproduced macroscopic and microscopic deformation behaviors during cup drawing. Numerical studies showed that the anisotropic deformation during drawing was primarily induced by the texture of the material, suggesting that anisotropic deformation is inevitable unless the anisotropic c-axes distribution remains in the initial texture.

Introduction

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Magnesium ( Mg ) alloy sheets are expected to contribute to the lightweighting of structural components, owing to their inherent benefits of low density and high specific strength. However, the limited room-temperature press formability exhibited in Mg alloy sheets remains a barrier to their expanded use. A significant factor contributing to the limited formability is the strong basal texture, which is typically found in AZ-series rolled Mg alloy sheets. Therefore, Mg−Zn−Ca alloy sheets have been developed to improve the room-temperature press formability, which weakened the basal texture [1,2]. Rolled Mg−1.5mass%Zn−0.1mass%Ca(ZX10Mg) alloy sheets were used to examine the roomtemperature formability, including the work-hardening behavior [3, 4] and drawability [5]. Hama et al. [3] showed that the work-hardening behavior exhibited pronounced in-plane anisotropy and tension-compression asymmetry. Hama et al. [5] also reported that circular cup drawing could be achieved using this sheet, but the cup height and thickness strains differed significantly in the circumferential direction of the cup. Especially, the cup heights were notably different between the rolling direction (RD) and transverse direction (TD). Because the plastic deformation behavior of Mg alloy sheets is typically governed by both slip and twinning activities, this characteristic cup shape would result from a complicated combination of these activities. However, further discussions on the deformation mechanisms based solely on experimental observations was difficult.

In the present study, crystal plasticity finite-element simulations of circular cup drawing of a rolled ZX10Mg alloy sheet were performed at room temperature. The simulation results were first compared with the experimental results to verify the results. Thereafter, the drawability was discussed in detail based on the simulation results.

Material

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Figure 1 (a) shows the (0001) pole figure of the ZX10Mg alloy sheet modeled in this study. This material was the same as that used in the previous study [3,5]. The strong peaks that tilted from the ND to the TD appeared. Moreover, the c-axes orientations of the grains were notably spread in the TD, and the peaks near the TD also appeared. The distribution of the c-axes orientations was apparently anisotropic in the RD-TD plane.

[picture]

Fig. 1. (0001) pole figures of ZX10Mg alloy sheet used in this study. (a) Experiment and (b) simulation.

Fig. 1. (0001) pole figures of ZX10Mg alloy sheet used in this study. (a) Experiment and (b) simulation.

Crystal plasticity finite-element method

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A crystal plasticity finite-element method used in this study is the same as those used in our previous studies [6-10]. Therefore, the model is explained only briefly. Assuming that the crystalline slip follows the Schmid's law, the slip rate γ˙(α) of the α slip system is given by the following equation.

γ˙(α)γ˙0=|τ(α)τY(α)|1msign(τ(α)),τ˙Y(α)=∑βqαβh|γ˙(β)|,(1)

where τ(α) and τY(α) are the Schmid's resolved shear stress and the slip resistance of the α slip system, respectively. γ˙0 is the reference strain rate, m is the rate sensitivity exponent, and qαβ represents the self- and latent- hardening moduli matrix. h is the rate of hardening and is given by the following equations.

h=h0(2)

and

h=h0(1−τ0τ∞)exp(−h0γ¯τ∞),γ¯=∑α∫|γ˙(α)|dt.(3)

In this study, the basal slip <a>, prismatic slip <a>, pyramidal-2 slip <a+c>, and {101―2} twinning were considered as the active slip and twinning systems in the ZX10 Mg alloy sheet. The linear hardening (Eq. (2)) was assumed for the basal slip and {101―2} twinning, whereas the Voce hardening (Eq. (3)) was assumed for the prismatic and pyramidal-2 slip. A twinning and detwinning model proposed by Hama et al. [7, 11] was employed. The crystal plasticity model was used as constitutive equations in the static finite-element method. The reader is referred to the literature [611] for more details.

The simulation conditions were determined based on the experiments reported in the literature [5]. Figure 2 shows the finite-element model of the circular sheet [12]. The thickness and diameter of the
sheet were respectively 1.0 mm and 40.0 mm . Assuming the symmetry of the deformation process, only the quarter part of the sheet was modeled using 1976 solid elements. The initial crystal orientations were determined based on the result of EBSD measurement, and 800 orientations were assigned to each integration point considering the Taylor assumption. Figure 1 (b) shows the (0001) pole figure obtained using the 800 orientations considered in the simulation. The experimental result is reproduced qualitatively well. The tools were assumed rigid. The diameters of die cavity and punch were respectively 30.0 mm and 27.8 mm . The radii of die and punch shoulders were respectively 7.0 mm and 5.0 mm . The gap between the die and blank holder was fixed to 1.1 mm . The friction coefficient was set to 0.03 to reasonably reproduce the punch load-stroke curve.

The hardening parameters in eqs. (2) and (3) were determined to fit the stress-strain curves under uniaxial tension and compression in the RD and TD. The stress-strain curves and determined parameters are shown in Figure 3 and Table 1, respectively. The stress-strain curves are well reproduced in the simulations, although the work-hardening behavior under tension were slightly deviated from the experimental results [3]. The fitting accuracy could be improved by further adjusting the parameters, but it is currently challenging from the following reason. In crystal plasticity simulations of Mg alloys, many parameters should be calibrated as shown in Table 1 because the slip activity differs significantly depending on the slip system. In our previous study on AZ -series Mg alloy sheets [10], we proposed a systematic procedure to calibrate the parameters separately for each slip system. This procedure worked well for AZ -series Mg alloy sheets because these sheets had strong basal textures; thus, the slip and twinning activities could be estimated rather systematically depending on the loading condition. By contrast, this procedure was not applicable for the present ZX10Mg alloy sheet because of the less-pronounced basal texture.

[picture]

Fig. 2. Finite-element model used for cup drawing simulations.

Fig. 2. Finite-element model used for cup drawing simulations.

Therefore, to improve the fitting accuracy, alternative procedures of parameter calibration should be established. This will be conducted in our future work.

Results and Discussion

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Figure 4 shows the punch load-stroke curves obtained in the experiment and simulation. In the simulation, the maximum load at a punch stroke of approximately 9 mm was overestimated, whereas the local minimum value at a punch stroke of approximately 13 mm was well reproduced. The quantitative discrepancy between the experimental and simulation results might be due to several factors, including the friction coefficient, contact algorithm, and predictive accuracy of workhardening behavior under multiaxial stress states.

Figure 5 (a) shows the cup height distribution in the circumferential direction obtained from the experimental [5] and simulation results of the drawn cup. Notably, the vertical axis represents the relative height in which the height in the RD was set to zero. The absolute height was not used because the cup bottom was rounded; thus, precise measurements were difficult. As explained in the literature [5], the experimental result showed the following trend. The height increased from the RD to the
angle of 60∘, and then decreased gradually to the TD. The maximum height appeared at 60∘. The height in the TD was larger than that in the RD. The simulation result well reproduced the distribution trend, although the maximum height was slightly underestimated.

Figure 5 (b) shows the thickness strain evolution at the cup edge in the RD and TD as a function of the punch stroke. In the experimental results [5], the thickness strains increased with the punch stroke until the punch stroke of 13 mm . In this stage, the thickness strains were the largest in the RD and the smallest in the TD. Thereafter, the thickness strains decreased sharply until the end of the process. This decrease occurred due to the ironing between the punch and die. The simulation results captured well these trends, although the strains in the angle of 45∘ (DD) and TD were respectively underestimated and overestimated.

[picture]

Fig. 3. True stress-true strain curves under (a) tension and (b) compression. The broken and solid lines are the experimental and simulation results, respectively.

Fig. 3. True stress-true strain curves under (a) tension and (b) compression. The broken and solid lines are the experimental and simulation results, respectively.

Table 1. Material parameters for crystal-plasticity model.

| τ₀ | τ∞ | h₀
Basal | 3 |  | 10
Prismatic | 80 | 350 | 1200
Pyramidal | 80 | 200 | 1400
Twinning | 3 |  | 5
Detwinning | 3 |  | 5

Figure 6 shows the (0001) pole figures measured at the cup edges in the RD and TD. In the experimental results, pronounced texture evolution occurred because of twinning activities. Specifically, the strong peaks appeared near the TD at the RD edge, whereas they appeared near the RD at the TD edge. These trends were well reproduced in the simulation results. These texture evolutions at the RD and TD edges agree well with those observed under uniaxial compression in the TD and RD [3], respectively. These observations suggest that circumferential compression was dominant during drawing at the sheet edge rather than bending and unbending in the radial direction.

[picture]

Fig. 4. Punch load-stroke curves obtained from experiment and simulation.

Fig. 4. Punch load-stroke curves obtained from experiment and simulation.

[picture]

Fig. 5. Comparisons of deformation shapes between experiment and simulation. (a) Cup height distributions in circumferential direction and (b) thickness strain evolution at cup edges.

Fig. 5. Comparisons of deformation shapes between experiment and simulation. (a) Cup height distributions in circumferential direction and (b) thickness strain evolution at cup edges.

[picture]

Fig. 6. (0001) pole figures measured on the drawn cup. Experimental results at the (a) RD edge and (b) TD edge, and simulation results at the (c) RD edge and (d) TD edge.

Fig. 6. (0001) pole figures measured on the drawn cup. Experimental results at the (a) RD edge and (b) TD edge, and simulation results at the (c) RD edge and (d) TD edge.

Hama et al. [13] studied the relationship between the Lankford value and twinning activity in the ZX10Mg alloy sheet and concluded that thickness strains induced by twinning activities affect the Lankford values both under RD and TD compression. This further suggests that the twinning activity during cup drawing should affect significantly the shape of drawn cup. To verify this hypothesis, a simulation was conducted without considering twinning activity. The simulation result without considering twinning is shown in Figure 5 (a). The cup height in the TD decreased and became close to that in the RD. These results show that twinning activities significantly affect the difference in the cup height between the RD and TD. Figure 7 shows the thickness strain evolution obtained from the simulation without considering twinning. The thickness strains to a stroke of 13 mm became smaller regardless of the direction when twinning was not considered, suggesting that the thickness increase during this process is affected notably by twinning activity. By contrast, the sharp decrease beyond a stroke of 13 mm became less pronounced, indicating that twinning activity also helped sharp decrease due to the ironing process. Notably, the anisotropic thickness strain evolution remained even when twinning was not considered. Because the twinning activity is governed by the texture, these results further suggest that anisotropic height distribution in the drawn cup might be inevitable unless the anisotropic c-axes distribution remains in the initial texture.

[picture]

Fig. 7. Effect of twinning on thickness strain evolution at cup edges.

Fig. 7. Effect of twinning on thickness strain evolution at cup edges.

Conclusion

The original version of this paper is available on https://www.scientific.net/SSP.388.21.pdf

In this study, crystal plasticity finite-element simulations of circular cup drawing of a rolled ZX 10 Mg alloy sheet were performed at room temperature. The drawability of the material was discussed in detail based on the simulation results. The conclusions obtained in this study are summarized as follows.
(1) Crystal plasticity simulations qualitatively reproduced macroscopic and microscopic deformation behaviors during cup drawing of the ZX10 Mg alloy sheet.
(2) Simulation results showed that twinning activities affected notably the shape of drawn cup.
(3) Anisotropic deformation during drawing was induced by the texture, suggesting that anisotropic deformation is inevitable unless the anisotropic c-axes distribution remains in the initial texture.

References

The original version of this paper is available on https://www.scientific.net/SSP.388.21.pdf

Y. Chino, X. Huang, K. Suzuki, M. Mabuchi, Enhancement of stretch formability at room temperature by addition of Ca in Mg−Zn alloy, Mater. Trans., 51 (2010), 818-821, https://doi.org/10.2320/matertrans.M2009385.

Y. Chino, T. Ueda, Y. Otomatsu, K. Sassa, X. Huang, K. Suzuki, M. Mabuchi, Effects of Ca on tensile properties and stretch formability at room temperature in and alloys, Mater. Trans., 52 (2011), 1477-1482. matertrans.M2011048.

T. Hama, T. Nakata, K. Higuchi, H. Yoshida, Y. Jono, Plastic deformation behavior of a Mg (mass%) alloy sheet under different strain paths, Mater. Sci. Eng. A, 869(2023), 144772. .

T. Nakata, T. Hama, K. Sugaya, S. Kamado, Understanding room-temperature deformation behavior in a dilute mass alloy sheet with weak basal texture, Mater. Sci. Eng. A, 852 (2022), 143638. .

T. Hama, K. Higuchi, Y. Nakata, Anisotropic deformation behavior during cup drawing at room temperature of a ZX10 magnesium alloy sheet, Mater. Res. Proc., 28(2023), 711-716.

T. Hama, H. Takuda, Crystal-plasticity finite-element analysis of inelastic behavior during unloading in a Magnesium alloy sheet, Int J Plasticity, 27(2011), 1072-1092. .

T. Hama, H. Takuda, Crystal plasticity finite-element simulation of deformation behavior in a magnesium alloy sheet considering detwinning, Steel Res. Int., (2012), Special Edition 2012, 1115-1118.

T. Hama, H. Takuda, Crystal plasticity finite-element simulation of work-hardening behavior in a magnesium alloy sheet under biaxial tension, Comput. Mater. Sci., 51 (2012), 156-164. .

T. Hama, N. Kitamura, H. Takuda, Effect of twinning and detwinning on inelastic behavior during unloading in a magnesium alloy sheet, Mater. Sci. Eng. A, 583 (2013), 232-241. .

T. Hama, Y. Tanaka, M. Uratani, H. Takuda, Deformation behavior upon two-step loading in a magnesium alloy sheet, Int. J. Plasticity, 82(2016), 283-304. .

T. Hama, A. Kobuki, H. Takuda, Crystal-plasticity finite-element analysis of anisotropic deformation behavior in commercially pure titanium Grade 1 sheet, Int. J. Plasticity, 91 (2017), 77-108. .

T. Hama, K. Hirano, R. Matsuura, Cylindrical cup drawing of a commercially pure titanium sheet: experiment and crystal plasticity finite-element simulation, Int. J. Mater. Form., 15 (2022), 8, 23 pages. .

T. Hama, T. Suzuki, S. Hatakeyama, H. Fujimoto, H. Takuda, Role of twinning on the stress and strain behaviors during reverse loading in rolled magnesium alloy sheets, Mater. Sci. Eng. A, 725(2018), 8-18. .

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Solid State Phenomena, ISSN: 1662-9779, Vol. 388, pp 29-37

doi: 10.4028/p-x19oZa

© 2026 The Author(s). Published by Trans Tech Publications Ltd, Switzerland.

Submitted:
2025-12-19

Revised:
2026-02-13

Accepted:
2026-02-20

Online:
2026-04-15

In Situ EBSD Characterisation of Anisotropic Creep Behaviour in a Thick AA2139 Part during in-Forming Artificial Ageing

The original version of this paper is available on https://www.scientific.net/SSP.388.29.pdf

Gautier Calixte 1,2,a*, Dubos Pierre-Antoine 1,b, Gloaguen David 1,c, Girault Baptiste 1, d and Rollin Maxime 2,e

1 Nantes Université, École Centrale Nantes, CNRS, GeM, UMR 6183, F-44600 Saint-Nazaire, France

2 Airbus Atlantic, 13 Bd des Apprentis, 44600 Saint-Nazaire, France

a* calixte.gautier@univ-nantes.fr, b  pierre-antoine.dubos@univ-nantes.fr, c  david.gloaguen@univ-nantes.fr, d  baptiste.girault@univ-nantes.fr, e  maxime.m.rollin@airbus.com

Keywords: in situ tests, creep-ageing, anisotropy, EBSD, grain misorientations, aluminium alloy.

Abstract

The present study aims to investigate the anisotropic creep behaviour of aluminium alloy 2139 during artificial ageing, through in situ thermomechanical loadings under Electron Backscattered Diffraction (EBSD). EBSD analysis enabled the characterisation of microstructural parameters and the identification of grain misorientations which were further correlated with macroscopic creep strain. In situ analyses were conducted within a Scanning Electron Microscope (SEM) using a micro-tensile stage that allows simultaneous heating and mechanical loading. Creep tests were performed at 160∘C under 50,100 and 150 MPa along three different orientations in order to investigate the creep behaviour of the alloy. Kernel Average Misorientation (KAM) maps showed a progressive increase of the average KAM values for the different loading conditions, reaching a saturation value after 10 hours. Ex situ tensile tests were conducted on creep-aged specimens using Digital Image Correlation (DIC). The main mechanical property evolutions (averaged across all orientations) are a 45% increase in yield stress, a 10% increase in ultimate tensile stress and a reduction in ductility, characterised by a notable decrease in elongation. Further works will focus on the result repeatability, as well as on the influence of prior deformation on the creep strain.

Introduction

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The age-hardenable aluminium alloy 2139 (AA2139) has been primarily developed for the use in aerospace industry as a high strength and damage tolerant alloy. Current research focusses on improving the aerodynamic and reducing weight of next-generation aircraft by developing complex parts, formed from large and thick raw plates. The forming used in cockpit manufacturing consists of a quasi-static deformation process (press forming) and dynamic process (hydroforming), followed by artificial ageing to strengthen the alloy and reduce residual stresses [1]. The implementation of constitutive equations in a Finite Element Method (FEM) software will enable the optimization of these forming processes, by predicting the final stress state and the springback occurring during machining. During artificial ageing at elevated temperature, the creep phenomenon induces a stress relaxation. Simultaneously, the internal stress state influences the precipitation kinetics, leading to an increase in yield strength and a reduction in creep rate through the hindering of dislocation motion [2]. The present study focuses on the artificial ageing heat treatment. The formed part with a complex internal stress state is heated and held at under-ageing temperature. Experimental creep tests are performed on undeformed micro-samples, to ensure a good control of the stress level and the deformation history in the sample. The objective of this work is to characterise the anisotropic creep-ageing behaviour of thick AA2139 plates in order to understand its relationship with the microstructural evolution occurring during the heat treatment. Although anisotropic creep behaviour and hardening have been studied for various 2 xxx series alloys [3], to the best of our knowledge, no study has yet focused on the anisotropic creep behaviour of AA2139, particularly using in situ analyses.

Material and Methods

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Ex situ calibration of the creep strain

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Micro-tensile samples of AA2139 (chemical composition Al-4.86Cu-0.4Mg-0.35Ag-0.29Mn-0.06Si0.07Fe (wt. %) [4]) in T3 temper were extracted along 0∘,45∘ and 90∘ to the Rolling Direction (RD) close to the surface of a 120 mm thick aluminium plate to assess material anisotropy and microstructural features. Macroscopic creep strain at 160∘C was first quantified under 0,100 and 150 MPa using a NewTec MT1000 tensile device with Digital Image Correlation (DIC) under secondary vacuum conditions (Fig. 1). Those experiments were performed to correlate the jaws displacement controlled by the micro-tensile stage with the local strain at gauge section location measured by DIC. The specimen surfaces were covered with a heat resistant white paint with black speckles. Sample heating was provided by two furnaces located under the grip sections of the specimen (Fig. 1(a)). Heating rate was set to 1∘C/s. Temperature variations were monitored using a type- K thermocouple and controlled within ±2.5∘C range in the gauge section. Once the target temperature of 160∘C was reached, samples were loaded to the initial stress ( 50,100 or 150 MPa ). Sample alignment with the tensile axis was ensured by glass-ceramic (Macor) hot pins located at the sample shoulders as detailed in (Fig. 1(a)) [1]. Images were acquired at 1-second intervals with a spatial resolution of 3.6μ m/pixel and a field of view of 14.4×10.8 mm2. The DIC processing was achieved using UFreckles software [5] with a Non-Uniform Rational Basis Spline (NURBS) based model and a subset size of 30 pixels. Creep curves were then fitted using a polynomial function in order to reduce the experimental noise.

[picture]

Fig. 1. a) Sample clamping in the tensile stage and b) DIC setup mounted on the vacuum chamber.

Fig. 1. a) Sample clamping in the tensile stage and b) DIC setup mounted on the vacuum chamber.

In situ creep tests under EBSD

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To enable Electron Backscatter Diffraction (EBSD) measurements during in situ tests, the upper surface of tensile specimens were ground and polished with a combined Struers LaboPol-30 and LaboForce-100 devices. Successive pre-polishing steps with abrasive diamond ( 9μ m and 6μ m ) ended up by a polishing step using colloidal silica ( 0.03μ m particle size). Qualitative control of the surface was performed under optical microscope after each step to ensure the complete removal of scratches from the previous steps.
The prepared specimens were subsequently mounted on a MT1000 tensile tester, tilted at 70∘ inside the chamber of an Ultra High Resolution Tescan Clara Scanning Electron Microscope (SEM) for EBSD measurements with an EDAX Velocity camera (Fig. 2). Creep tests were performed at various loading stresses: 50,100 and 150 MPa . EBSD acquisitions were automated at 1-hour intervals. Crystal lattice orientations were determined using triplets of Kikuchi bands. If more than one solution is found for a given triplet, a voting scheme is applied to rank the corresponding orientations. The Confidence Index (CI) is then defined by CI=(VI−VII)/Videal , where VI is the number of votes for the first solution, VII is the number of votes for the second most likely solution and Videal  is the total number of Kikuchi band triplets available [6]. The moderate acquisition rate (1700 diffraction
patterns per second) of the EBSD patterns during mapping resulted in an average CI set between 0.26 and 0.44 with an overall acquisition time of 10 minutes per map. EBSD maps were acquired with a field of view of 1.00×0.78 mm2 and a step size of 1μ m using an electric voltage of 15 keV and a probe current of 40 nA . The space requirement of the tensile stage imposed a working distance of 18 mm . Automation of in situ testing included systematic adjustment of the focus, contrast and brightness over a reduced field of view (200μ m) prior to SEM image acquisition for area correlation.

[picture]

Fig. 2. a) Sample mounting and b) installation of the MT1000 tensile stage tilted at 70∘ inside the SEM with c) the simulation model developed to avoid collision in the SEM chamber.

Fig. 2. a) Sample mounting and b) installation of the MT1000 tensile stage tilted at 70 ∘ inside the SEM with c) the simulation model developed to avoid collision in the SEM chamber.

Post-processing of the EBSD maps was performed in batches using OIM Analysis V9.1 software. The workflow consisted of reindexation (spherical reindexing and Neighbour Pattern Averaging Reindexing, NPAR) and a two-step cleaning process (grain dilation and CI Standardization). These numerical treatments significantly improved the overall CI values. Fig. 3 shows the Inverse Pole Figures (IPF) of the sample heated to 160∘C and subjected to an initial applied stress of 100 MPa along RD. The fraction of points with CI >0.1 increased from 0.493 using the standard Hough indexing to 0.999 after reindexing and cleaning. Similarly, the average CI increased from 0.23 to 0.49 .

[picture]

Fig. 3. Inverse Pole Figure maps of AA2139 at 160∘C under 100 MPa along RD: a) classical Hough indexing b) spherical reindexing and NPAR and c) spherical reindexing, NPAR and clean-up. RD: Rolling Direction, TD: Transverse Direction.

Fig. 3. Inverse Pole Figure maps of AA2139 at 160 ∘ C under 100 MPa along RD: a) classical Hough indexing b) spherical reindexing and NPAR and c) spherical reindexing, NPAR and clean-up. RD: Rolling Direction, TD: Transverse Direction.

Tensile tests

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To investigate the effect of ageing on the mechanical properties of the alloy and to assess the potential anisotropy, ex situ tensile tests combined with DIC have been performed along 0∘,45∘ and 90∘ relative to RD, after creep-ageing at 160∘C under various applied stresses of 50,100 and 150 MPa .

Experimental Results

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Ex situ creep tests with DIC

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Ex situ creep tests were systematically performed prior to in situ tests in order to assess creep behaviour under various orientations and loading states, and also to calibrate the strain measured by DIC. An example of the DIC data fitted by a polynomial function is presented in Fig. 4 for the creep curve under 100 MPa applied stress at 45∘ to RD.

[picture]

Fig. 4. Polynomial fitting of the DIC calculated creep strain under 100 MPa at 45∘ to RD.

Fig. 4. Polynomial fitting of the DIC calculated creep strain under 100 MPa at 45 ∘ to RD.

The complete set of fitted creep curves is presented in Fig. 5. At low stress level ( 50 and 100 MPa ), the RD-oriented samples display higher creep strains than samples along the Transverse Direction (TD). In contrast, this creep strain ranking is no longer valid for a larger stress level ( 150 MPa ), where the sample oriented at 45∘ to RD exceeds the creep strain of the two other orientations.

[picture]

Fig. 5. Creep curves along 0∘,45∘ and 90∘ to RD under a) 50 MPa , b) 100 MPa and c) 150 MPa .

Fig. 5. Creep curves along 0 ∘ , 45 ∘ and 90 ∘ to RD under a) 50 MPa , b) 100 MPa and c) 150 MPa .

In situ creep tests under EBSD

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The misorientation analysis was complicated by the large grain size (≈200μ m) of the AA2139 samples. A compromise has thus been made between a large field of view-allowing the analysis over a larger number of grains but inducing long acquisition time-and a reduced 1.00×0.78 mm2 field of view enabling an acquisition time of 10 minutes per EBSD map. This smaller field of view limits the risk of dynamic microstructural changes between the start and the end of the map. A step size of 1μ m was chosen in order to accurately capture the misorientation at the grain boundaries. Fig. 6 shows the evolution of Kernel Average Misorientation (KAM) during the creep tests under 50, 100 and 150 MPa along RD. The measured misorientation is directly related to the presence of dislocations and precipitate formation. From onset of creep-ageing, misorientations are heterogeneously distributed across the maps. The heterogeneity increases as the creep progresses suggesting that some grain orientations would be more likely to promote precipitate formation.

[picture]

Fig. 6. KAM maps during creep ageing under 50,100 and 150 MPa along RD at different creep times. Points with CI<0.1 are shown in black.

Fig. 6. KAM maps during creep ageing under 50,100 and 150 MPa along RD at different creep times. Points with C I < 0.1 are shown in black.

To quantify the misorientation evolutions during ageing, the average KAM value was calculated for each EBSD map. These values were normalized by the maximum value of each creep test. Their evolutions are presented in Fig. 7. Saturation is reached after 10 hours regardless of the applied stress. The symbols represent the experimental data points contained in the KAM maps. The lines correspond to the modelled evolution of the average KAM values. Most of the misorientations develop during the first 5 hours, which is consistent with the formation kinetics of Ω and θ′ precipitates reported in the literature [4, 7-8]. As precipitates nucleate and grow, they effectively block dislocations. The observed saturation suggests a dynamic equilibrium in which dislocation nucleation is balanced by recovery mechanisms activated at the creep temperature.

[picture]

Fig. 7. Evolution of the averaged KAM values during creep tests under 50,100 and 150 MPa along RD .

Fig. 7. Evolution of the averaged KAM values during creep tests under 50,100 and 150 MPa along RD .

Fig. 6 and Fig. 7 shows an earlier rise in the average misorientation during creep under low stress ( 50 MPa ) compared with higher stresses. Although the creep strain rate is generally expected to increase with applied stress [3], the precipitation of strengthening phase has been found to be delayed when Al−Cu−Mg−Ag samples were aged under external stress with the promotion of θ′ precipitates at the expense of Ω [9]. This stress-dependent precipitation behaviour may contribute to the observed differences in KAM evolution.

Tensile tests post creep ageing

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Ex situ tensile tests were conducted along 0∘,45∘ and 90∘ to RD after creep-ageing. DIC was performed and processed with the UFreckles software to evaluate the macroscopic strain and the evolution of mechanical properties through virtual longitudinal and transversal extensometers. Fig. 8 shows tensile curves in the T3 state and after creep-ageing along RD. The main effect of the creep ageing is a significant Yield Stress (YS) increase and a moderate Ultimate Tensile Stress (UTS) enhancement (Fig. 9(a)). The gap reduction between YS and UTS indicates a loss of work-hardening capability leading to an earlier and more sudden necking. UTS rise is more pronounced for samples subjected to creep under higher applied stresses.

[picture]

Fig. 8. Tensile test along RD after creep ageing under 50,100 and 150 MPa .

Fig. 8. Tensile test along RD after creep ageing under 50,100 and 150 MPa .

Precipitate formation is promoted by dislocation density, particularly semi-coherent θ′ precipitates [9-11]. Fig. 9 summarizes the results tensile tests on creep-aged samples. For each applied stress level, data were averaged over the three orientations according to Eq. 1 where X is the considered quantity:

X100MPa=(X100MPa/RD+2X100MPa/45∘+X100MPa/TD)/4.(1)

Fig. 9(c) highlights the evolution of plastic anisotropy after creep-ageing. While the T3 state exhibits an averaged r-value (Lankford coefficient) of 0.76 , creep-ageing at 160∘C results in a slight shift towards isotropy ( r=1 ). The reduction in ductility, highlighted by the elongation drop after creep-ageing in Fig. 9(d) reflects the ability of precipitates to hinder dislocation motion.

[picture]

Fig. 9. Tensile tests summary averaged along the three directions a) Yield stress and Ultimate Tensile Stress, b) Young's modulus, c) Lankford coefficient and d) elongation.

Fig. 9. Tensile tests summary averaged along the three directions a) Yield stress and Ultimate Tensile Stress, b) Young's modulus, c) Lankford coefficient and d) elongation.

Summary

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Ex and in situ creep tests within the SEM were performed on AA2139 at 160∘C under various loadings conditions to investigate the creep behaviour of the alloy and the related microstructural evolutions. The main findings can be summarized as follows:

Creep strains measured by DIC exhibit a moderate anisotropy at 160∘C. The initial stress increase resulted in a higher creep strain, regardless of the orientation.

In situ tests under EBSD enabled the quantification of the average KAM in a 1.00×0.78 mm2 field of view. The normalized average KAM value reached a saturation value after 10 hours.

Ex situ tests on creep-aged samples highlighted the impact of creep-ageing on the mechanical properties. All the three tested orientations exhibited an average increase of 45% of the initial yield stress and an average increase of 10% of the ultimate tensile stress.

The creep-ageing promotes precipitate formation. These precipitates hinder dislocation motion and reduce the ductility. The elongation after creep-ageing showed a dispersion over the tensile tests, but all samples exhibited reductions of 20−60% as compared to T3 samples. The study focused on microstructure evolution during in situ creep tests under EBSD. While the results reveal clear trends in anisotropic creep behaviour and precipitation-related misorientation development, the limited number of tests per configuration remains a constraint, particularly regarding the variability observed between orientations. In the next phase of this work, a larger test campaign will be conducted to characterize the creep behaviour of the alloy taking into account the impact of repeatability. Future works will focus on the transformation strain measurement (creep test performed under zero stress) and the investigation of the impact of prior static and dynamic deformation on the creep behaviour. These complementary studies will contribute to the development of physically based constitutive equations for accurate creep-ageing modelling.

Declaration of Competing Interest

The original version of this paper is available on https://www.scientific.net/SSP.388.29.pdf

The authors declare that they have no known competing financial interests or personal relationship that could have appeared to influence the work reported in this paper.

Acknowledgement

The original version of this paper is available on https://www.scientific.net/SSP.388.29.pdf

The authors gratefully acknowledge Airbus Atlantic and the ANRT (French National Association for Research and Technology) for funding C. Gautier's PhD thesis.

References

The original version of this paper is available on https://www.scientific.net/SSP.388.29.pdf

A.D. Smith, D. Lunt, M. Taylor, A. Davis, R. Thomas, F. Martinez, A. Candeias, A. Gholinia, M. Preuss, J.M. Donoghue, A new approach to SEM in-situ thermomechanical experiments through automation, Ultramicroscopy. 280 (2026).

K. C. Ho, J. Lin and T. A. Dean, Constitutive modelling of primary creep for age forming an aluminium alloy, J. Mater. Process. Technol. 153-154 (2004) 122-127.

X. Wang, Z. Shi, J. Lin, A generalised framework for modelling anisotropic creep-ageing deformation and strength evolution of 2xxx aluminium alloys, Int. J. Plast. 182 (2024).

D. Bakavos, P.B. Prangnell, B. Bes, F. Eberl, The effect of silver on microstructural evolution in two 2 xxx series Al -alloys with a high Cu : Mg ratio during ageing to a T 8 temper, Mater. Sci . Eng. A. 491 (2008) 214-223.

J. Rethore (2018). UFreckles (v.2.0). Zenodo. .

D. P. Field, Recent advances in the application or orientation imaging, Ultramicroscopy. 67 (1997) 1-9.

H. So, S.J. Won, J. Park, S.J. Oh, L. Kang, K.H. Kim, Mechanical properties and microstructural evolution in alloy with a content, Mater. Sci. Eng. A. 824 (2021).

J. Fan, B. Yang, Y. Wang, M. Gao, R. Guan, Enhancing the tensile strength and heat resistance induced by high-density phases in an Al-Cu-Mg-Ag alloy, JMR&T. 18 (2022) 3347-3357.

X.Y. Liu, Q.L. Pan, X.L. Zhang, S.X. Liang, L.Y. Zheng, F. Gao, H.L. Xie, Effect of stressageing on the microstructure and properties of an aging forming alloy, Mater. Des. 58 (2014) 247-251.

Y. Xu, L. Zhan, W. Li, Effect of pre-strain on creep ageing behavior of 2524 aluminum alloy, J. Alloys Compd. 619 (2017) 564-571.

Y. Yang, L. Zhan, R. Shen, X. Yin, X. Li, W. Li, M. Huang, D. He, Effect of pre-deformation on creep age forming of 2219 aluminum alloy: Experimental and constitutive modelling, Mater. Sci. Eng. A. 683 (2017) 227-235.

S. Cui, C. Zhang, M. Liu, L. Chen, G. Zhao, Precipitation behavior of an Al-Cu-Li-X alloy and competing relationship among precipitates at different ageing temperatures, Mater. Sci. Eng. A. 814 (2021).

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Solid State Phenomena, ISSN: 1662-9779, Vol. 388, pp 39-45

doi: 10.4028/p-L9atJa

© 2026 The Author(s). Published by Trans Tech Publications Ltd, Switzerland.

Submitted:
2025-12-19

Revised:
2026-02-11

Accepted:
2026-02-18

Online:
2026-04-15

Microstructural Modification and Bendability Improvement in 1300 MPa Steel via Induction Surface Heat Treatment

The original version of this paper is available on https://www.scientific.net/SSP.388.39.pdf

Antti Kaijalainen 1,a*, Ahmed Abdelghany 2,b, Oskari Haiko 1,c, Anttu Hoikkaniemi 1, d and Aki-Petteri Pokka 1,e

¹Materials and Mechanical Engineering, University of Oulu, Oulu 90570, Finland

2 Future Manufacturing Technologies, Kerttu Saalasti Institute, University of Oulu, Nivala 85500, Finland

a*antti.kaijalainen@oulu.fi, b  ahmed.abdelghany@oulu.fi, c  oskari.haiko@oulu.fi, d  anttu.hoikkaniemi@oulu.fi, e  aki-petteri.pokka@oulu.fi

Keywords: bendability, hardness, induction heating, soft surface, ultrahigh-strength steel.

Abstract

This study investigates the effect of induction-based surface heat treatment on the microstructure and bendability of a commercial hot-rolled martensitic steel with a nominal strength of 1300 MPa . A rapid tempering process was applied at 500∘C and 700∘C using a pilot-scale 60 kW induction heating system, followed by water quenching. Microstructural characterization revealed that the treatment induced minor changes near the subsurface without affecting the centerline. The as-rolled condition exhibited the highest subsurface hardness, whereas surface-treated samples showed progressive softening due to recovery. Three-point bending tests combined with digital image correlation demonstrated a significant improvement in bendability for heat-treated samples. The asrolled condition fractured at 0.195 strain, while the 700∘C treated specimen did not fracture even at 0.78 strain. These findings highlight that even a modest reduction in subsurface hardness can substantially enhance the formability of ultra-high-strength steels, offering a promising approach for industrial applications requiring high strength and improved bendability.

Introduction

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For ultrahigh-strength strip steels (UHSS), cold bending is the primary forming method for applications such as containers and crane booms. Bendability is commonly determined as the minimum ratio of the punch radius to the strip thickness (Rp/t) that the steel can tolerate without the appearance of surface defects during bending to an angle of 90∘. Digital Image Correlation (DIC) is used in this work to measure the strain distributions on the outer curvature [1-4] rather than at a 90∘ angle in order to study the bendability of a UHSS grade using 3-point bending tests. Shear bands have been observed to initiate near the surface at angles of approximately 45∘ to the surface, i.e., in the maximum shear stress directions [5]. Previous bending studies [6, 7] have demonstrated that strain localization is the precursor to failure in the air bending of UHSS, and especially excellent bendability can be achieved with a relatively soft subsurface layer down to a depth of ~5% of the sheet thickness. Therefore, it is reasonable to assume that the bendability of the strip is controlled by its near-surface characteristics.

Induction-based heat treatment offers several advantages over conventional furnace-based methods. It enables rapid austenitization and tempering, localized energy input, and significantly reduced heating times, leading to increased energy efficiency and processing throughput [8, 9]. Moreover, such rapid heating conditions can induce favorable microstructural changes by refining the grain structure and suppressing unwanted recovery mechanisms during tempering [10]. These characteristics align well with the goals of sustainable steel processing, where improved mechanical performance must be achieved without sacrificing environmental or economic viability.

While earlier studies have shown that a softer subsurface layer can enhance bendability, these findings were based on the hot-rolling process.

The main aim of the present study is to understand how bendability can be improved by tailoring the surface layer through rapid induction tempering, and how the unique thermal characteristics of ultrafast local heating influence microstructure evolution, strain localization, and bendability without compromising other mechanical properties.

Experimental

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Bending samples 8 mm thick, of the commercial hot-rolled martensitic 1300 MPa steel grade, were tested with a width of 30 mm . The rapid heat treatments were conducted using the modular 60 kW high-frequency induction-heating line at the University of Oulu research facility, Nivala, Finland. This system was selected to replicate continuous and controlled feed industrial processing, as shown in Fig. 1. The thermal cycle starts with rapid tempering at 500∘C and 700∘C, achieved at a heating rate of approximately 200∘C/s. Immediately upon reaching peak temperature, the samples were passed directly into a dedicated inline water-cooling unit integrated with the induction heating line. This continuous-quenching setup ensured rapid, uniform cooling to room temperature while maintaining uninterrupted specimen motion. The purpose of this flash heat treatment sequence was to induce a refined martensitic microstructure and achieve surface softening through rapid processing. Therefore, three materials were investigated and named as follows: (1) as-rolled, (2) surface treated at 500∘C, and (3) surface treated at 700∘C.

[picture]

Fig. 1. Photographs of the high-speed induction heating of steel specimens using a continuous pilotscale axial inductor system at Nivala, Finland. (a) Copper induction coil positioned above the specimen as it enters the heating zone. (b) Active heating process, where the specimen is rapidly austenitized while moving through the coil at controlled feed rate.

Fig. 1. Photographs of the high-speed induction heating of steel specimens using a continuous pilotscale axial inductor system at Nivala, Finland. (a) Copper induction coil positioned above the specimen as it enters the heating zone. (b) Active heating process, where the specimen is rapidly austenitized while moving through the coil at controlled feed rate.

General Microstructure Investigation and Hardness Measurements.

The original version of this paper is available on https://www.scientific.net/SSP.388.39.pdf

The general microstructure of the as-rolled and heat-treated materials was examined using a Keyence VK-X200 confocal laser scanning microscope. Polished specimens using 0.04μ m colloidal silica were etched with 2% Nital to reveal martensitic features, allowing assessment of the hardened case depth and structural uniformity across the cross-section. To evaluate the mechanical response through the depth, Vickers hardness measurements were performed on polished cross-sections using a Zwick/Roell ZHU2.5 universal hardness testing machine (Germany). A load of 5 kg (HV5) was applied through the thickness from top surface to bottom surface, enabling characterization of the hardness gradient resulting from the flash heat treatment. Likewise, the subsurface hardness profiles were measured using a micro-hardness tester (CSM, Switzerland) under a 1N(HV0.1) load.

Bending Trials.

The original version of this paper is available on https://www.scientific.net/SSP.388.39.pdf

The investigated materials were tested in the transverse orientation, meaning that the bend axis was perpendicular to the rolling direction. Each specimen was bent at room temperature until fracture, or until it reached the final bend angle of α=120∘. The specimens with a width (bend length) of 30 mm were bent at room temperature using a Zwick 100 kN universal tensile test machine equipped with custom-built three-point bending tools (Fig. 2a) and a stereo DIC system. The setup enabled measurement of bending force, punch stroke, and continuous strain mapping of the outer bend surface through openings in the lower tool.

[picture]

Fig. 2. a) Setup for the bending tests and DIC measurements. b) Example of a strain map from the 700∘C treated bend specimen at a 66∘ angle and sections A-C.

Fig. 2. a) Setup for the bending tests and DIC measurements. b) Example of a strain map from the 700 ∘ C treated bend specimen at a 66 ∘ angle and sections A-C.

To maintain the DIC cameras focused on the measurement area, the punch remained stationary during testing, while the lower tool was displaced upward at a speed of 1 mm/s to produce the required punch stroke. A punch radius of 8 mm , die shoulder radius of 6 mm , and a die width of 90 mm (distance between shoulder centers) were used. The die shoulders were freely rotating and lubricated with PTFE spray to minimize friction. Additional details of the test setup are available in Refs. [1,11].

The bending force was normalized by specimen width and expressed as N/mm to compensate for minor width variations. Bend angles were calculated from punch stroke data according to ISO 7438:2016, with corrections applied for the vertical elasticity of the setup (combined rigidity of the machine and tools), measured at 51.9kN/mm.

Strain measurements were performed using a LaVision Strainmaster DIC system with two monochrome CCD cameras. The specimens were spray painted with a black-and-white speckle pattern to ensure accurate point tracking in the DIC software. The DIC recording and processing parameters are presented in Table 1. Fig. 2b provides an example of a strain map of the outer surface of a bent specimen, obtained through DIC measurement.

From the strain maps, principal strain values were extracted along three sections (A-C) positioned at the bend centerline at 1 mm intervals. The peak strain, denoted as εb_max , was calculated as the average of the maximum strain values from sections A−C at each time step.

The bending fracture strain ( εbf ) was defined as εb_max  at the fracture angle αf, following the procedure described in Ref. [12]. The fracture angle αf was determined by the smallest bend angle at which εb_max  exceeded 0.15 , the strain rate (dεb_max /dα) was greater than 0.01∘−1, and the strain acceleration (d2εb_max /dα2) surpassed 0.0003∘−2. The threshold values were chosen with the aim of capturing the increase in strain rate that coincides with fracture, while avoiding false positives at the beginning of the test, and false negatives in cases where the crack growth (and thus strain acceleration) is very gradual. The strain rate and acceleration were calculated using the central difference method and smoothed with a seven-point moving average.

Table 1. Recording and processing parameters for the DIC system.

DIC system and software | LaVision Strainmaster (Stereo DIC), DaVis 8.4
Sensor, digitization, imaging rate | 2456 × 2058, 12-bit, 2 Hz
Lens, imaging distance, image scale | 35 mm C-mount, 425 mm, 25 pixels/mm
Subset size, step size, VSG | 15 × 15 pixels, 5 pixels, 35 pixels
Strain window, smoothing method | 5 × 5 data points, 2nd order polynomial fit
St. dev. of principal strain (noise-floor) | 480 – 550 microstrain
Interpolation, shape function, algorithms | 6th order spline interpolation; affine shape function; LSM (iterative least squares matching) algorithm based on optical flow estimation

Results and Discussion

The original version of this paper is available on https://www.scientific.net/SSP.388.39.pdf

Microstructure and Hardness.

The original version of this paper is available on https://www.scientific.net/SSP.388.39.pdf

Microstructural investigation revealed a fully martensitic microstructure (Fig. 3), and the surface heat treatment did not affect the centerline. Characterization showed that the fast heat treatment at 500∘C and 700∘C (i.e., the tempering effect) causes minor microstructural changes in the subsurface. Carbon diffusion to grain boundaries and the appearance of slightly coarser carbides are visible, particularly in the 700∘C sample (Fig.3d). Red arrows in Fig.3 (c and d) highlight representative carbide precipitates in the heat-treated subsurface regions. These microstructural features reflect the early stages of tempering and are consistent with the localized softening observed in hardness measurements.

[picture]

Fig. 3. Microstructural micrographs showing (a) and (b) before heat treatment; (c) and (d) after induction heat treatment under different conditions.

Fig. 3. Microstructural micrographs showing (a) and (b) before heat treatment; (c) and (d) after induction heat treatment under different conditions.

Fig. 4a illustrates the macrohardness profiles at different positions through the thickness. The asrolled condition exhibits the highest average hardness ( 455±4 HV5), followed by the material tempered at 500∘C(446±7HV5) and 700∘C(436±11HV5). The relatively uniform hardness distribution across the thickness suggests that subsequent heat treatments affected mainly near the surface, as can be seen in Fig. 4a. The reduction in hardness with increasing tempering temperature reflects the progressive softening associated with tempering processes, which promotes recovery, reduces internal stresses, and rearranges dislocations without grain growth.

Macrohardness results indicate that hardness near the subsurface exhibits noticeable variation. Hence, Fig. 4b shows microhardness profiles from 25μ m to 925μ m below the surface and centerline (CL) values. However, it should be noted that the Vickers hardness measurements are not directly comparable, as the measurements are performed with different loads. All conditions exhibit a hardness gradient, with softening near the surface and stabilization deeper into the material. This surface softening is mainly caused by the decarbonization of the surface during slab austenitization and hot rolling. The as-rolled material demonstrates the highest subsurface hardness ( 400±18 HV0.1), while the 500∘C and 700∘C conditions show reduced values of 377±16HV0.1 and 354± 20 HV0.1, respectively.

The combined macro- and microhardness results demonstrate that hardness is strongly influenced by tempering temperature, which has an effect on dislocation density. The as-rolled condition retains the highest hardness due to work hardening and high dislocation density, while tempering progressively reduces hardness through recovery. These findings highlight the critical role of thermal treatments in tailoring mechanical properties for applications requiring specific strength and formability levels.

[picture]

Fig. 4. a) Macrohardness and b) microhardness profiles of the investigated materials with 95% confidence limits of the means (five measurements at each thickness). Also, the mean values are given through the thickness and near the subsurface. (Vickers hardness measurements are performed with varying loads when those are not directly comparable.)

Fig. 4. a) Macrohardness and b) microhardness profiles of the investigated materials with 95 % confidence limits of the means (five measurements at each thickness). Also, the mean values are given through the thickness and near the subsurface. (Vickers hardness measurements are performed with varying loads when those are not directly comparable.)

Bending.

The original version of this paper is available on https://www.scientific.net/SSP.388.39.pdf

Examples of bend samples are shown in Fig. 5, whereas Fig. 6 illustrates the evolution of the minor and major strains during bending tests, as measured by DIC at the peak strain location on the specimen surface. Evidently, the sample width was too narrow for the bending trials to achieve plane-strain condition. Generally, the width-to-thickness ratio should be at least 8, to achieve plane-strain condition, while in this case it was only 3.75 .

[picture]

Fig. 5. Bending test samples with the bend axis transverse to the rolling direction.

Fig. 5. Bending test samples with the bend axis transverse to the rolling direction.

[picture]

Fig. 6. Major and minor strain evolution of the bending tests.

Fig. 6. Major and minor strain evolution of the bending tests.

Nevertheless, the measured normalized bending force and the peak strain εb_max  are plotted against the bend angle in Fig. 7. The determined bending fracture angles and strains are listed in Table 2. The bending force of as-rolled material is higher than the heat-treated samples (Fig. 7a), indicating higher strength, as expected based on the average hardness comparison. Notably, the εb−max evolution is near identical between all the specimens until around 30∘ bend angle, when the as-rolled specimen starts to fracture, indicated by the sharp increase in peak strain (Fig. 7b).
The as-rolled tests clearly have the poorest bendability, as the fracture angle was 31∘ and fracture strain 0.195 . Whereas the 500∘C surface heat-treated started to fracture first at approx. 72∘ angle or 0.611 strain. Surprisingly, the higher 700∘C surface heat treatment with a softer subsurface did not fracture even at 120∘ bending angle and εb_max above 0.78 . Therefore, the 5% softer subsurface layer is enough to improve the bendability, as shown in Ref. [7].

[picture]

Fig. 7. Development of a ) the bending force andb) the peak strain on the outside surface.
Table 2. Bending fracture angles ( αf ) and fracture strains ( εf ) of the investigated specimens.

Fig. 7. Development of a ) the bending force a n d b ) the peak strain on the outside surface. Table 2. Bending fracture angles ( α f ) and fracture strains ( ε f ) of the investigated specimens.

Material | αf [°] | εf [-]
as-rolled | 30.9 | 0.195
500 °C surface heat-treated | 72.0 | 0.611
700 °C surface heat-treated | - | -

Summary

The original version of this paper is available on https://www.scientific.net/SSP.388.39.pdf

The aim of this study was to show the effect of rapid surface heating on the subsurface hardness and the three-point bending of 1300 MPa ultrahigh-strength steel. Microstructural analysis showed that the treatment caused minor changes near the subsurface without affecting the centerline, while hardness measurements showed a slight reduction in subsurface hardness with increasing tempering temperature. Therefore, the mean hardness of near subsurface (from 25μ m to 925μ m below the surface) decreased from 400 HV0.1 to 354 HV0.1. The bendability of the softer surface material improved, and no fracture was observed compared to identical steel specimens without induction treatment. These results confirm that even modest subsurface softening can substantially enhance the bendability of ultra-high-strength steels.

Acknowledgement

The original version of this paper is available on https://www.scientific.net/SSP.388.39.pdf

Financial assistance of the Business Finland project FOSSA2-Fossil-Free Steel Applications: Phase 2-is acknowledged.

References

The original version of this paper is available on https://www.scientific.net/SSP.388.39.pdf

A.-P. Pokka, A.-M. Arola, A. Kaijalainen, V. Kesti, J. Larkiola, Strain distribution during air bending of ultra-high strength steels, Proc. ESAFORM 2021 (2021).

A.-M. Arola, A. Kaijalainen, V. Kesti V, A.-P. Pokka, J. Larkiola, Digital image correlation and optical strain measuring in bendability assessment of ultra-high strength structural steels, Procedia Manuf. 29 (2019) 398-405.

M. Kaupper, M. Merklein, Bendability of advanced high strength steels-A new evaluation procedure, CIRP Ann. Manuf. Technol. 62 (2013) 247-250.

K. Cheong, C. Butcher, J. Dykeman, The influence of the through-thickness strain gradients on the fracture characterization of advanced high-strength steels, SAE Int. J. Mater. Manuf. 11 (2018) 541-552.

D. Rèche, T. Sturel, O. Bouaziz O, A. Col, A.F. Gourgues-Lorenzon, Damage development in low alloy TRIP-aided steels during air-bending, Mater. Sci. Eng. A 528 (2011) 5241-5250.

A. Kaijalainen, P. Suikkanen, L. Karjalainen, D. Porter, Influence of subsurface microstructure on the bendability of ultrahigh-strength strip steel, Mater. Sci. Eng. A 654 (2016) 151-160.

A. Kaijalainen, M. Liimatainen, V. Kesti, J. Heikkala, T. Liimatainen, D. Porter, Influence of composition and hot rolling on the subsurface microstructure and bendability of ultrahighstrength strip, Metall. Mater. Trans. A 47 (2016) 4175-4188.

N.S. Kim, G.G. Kang, An investigation of flow characteristics considering the effect of viscosity variation in the thixoforming process, J. Mater. Process. Technol. 103 (2000) 237246.

H.K. Jung, The induction heating process of semi-solid aluminium alloys for thixoforming and their microstructure evaluation, J. Mater. Process. Technol. 105 (2000) 176-190.

A. Kaijalainen, O. Haiko, S. Sadeghpour, V. Javaheri, J. Kömi, The influence of rapid tempering on the mechanical and microstructural characteristics of 51CrV4 steel, Metals 14 (2024) 60.

A. Kaijalainen, A.-P. Pokka, M. Jaskari, J. Kömi, Effect of surface roughness on strain distribution during bending, IOP Conf. Ser. Mater. Sci. Eng. 1270 (2022) 012080.

A.-P. Pokka, V. Kesti, A. Kaijalainen, Local formability and bendability of UHSS: Correlations between bending and tensile fracture strains. Int. J. Solids Struct. 320 (2025) 113524.a.

This ePub file was created using machine-generated methods and automated software to convert the source text into an ePub format and may contain formatting inconsistencies from the original version. Formatting may vary depending on the device used.

The original version of this eBook is available on https://www.scientific.net/book/formability-of-metallic-materials/978-3-0364-1993-0

Solid State Phenomena, ISSN: 1662-9779, Vol. 388, pp 47-55

doi: 10.4028/p-Ew8Pjl

© 2026 The Author(s). Published by Trans Tech Publications Ltd, Switzerland.

Submitted:
2025-12-21

Revised:
2026-02-23

Accepted:
2026-02-23

Online:
2026-04-15

The Interaction of N - and R -Values and Total Elongation in Forming Limit Curves' Theories

The original version of this paper is available on https://www.scientific.net/SSP.388.47.pdf

Borbely Richard 1,3,a, Toth-Nagy Bendeguz V. 1, b, Kecskes Bertalan 2,c, Keresztes Robert Zs. 3,d, Beres Gabor J. 1,e*

1 Innovative Vehicles and Materials Department, John von Neumann University, Hungary

2 HILTI Tool Ltd., Kecskemét Factory, Hungary

3 Doctoral School of Technical Sciences, Hungarian University of Agriculture and Life Sciences, Hungary

a  borbely.richard@nje.hu, b  toth-nagy.bendeguz@nje.hu, c  Bertalan.Kecskes@hilti.com

d  keresztes.robert.zsolt@uni-mate.hu, *  beres.gabor@nje.hu

Keywords: forming limit curves, theoretical calculations, tensile test.

Abstract

Forming limit curve (FLC) is the most commonly used failure criterion in the sheet metal forming industry. All commercial simulation software uses this concept to evaluate failure strains and to detect the most critical section(s) of the workpiece. The laboratory determination of the FLC is standardized. However, because experimental measurement is cumbersome, theoretical calculations of FLCs using mechanical properties from well-defined test conditions are of great interest. Several such approaches have been proposed in the literature. This paper presents calculated FLCs using the models of Abspoel et al., Stören and Rice and Swift. All the equations include tensile tests data that have been measured physically at room temperature, with quasi-static strain rates. Calculated results of DC04 cold rolled steel sheet with relatively high plastic anisotropy coefficient was compared to a nearly isotropic DP800 high strength steel. Based on the results it is observed that r-value influences the shape of the left-hand side of the FLC as well as the plane strain point significantly, for the DC04 sheet. These effects are less pronounced for the DP800 material, which has lower r-value. At the same time, n -value and total elongation raise or lower the curves, generally. These observations are briefly explained by function analyses using fictitious r -values in the calculations.

Introduction

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Theoretical prediction of sheet failure through Forming Limit Curves (FLCs) has long been a central topic in sheet metal forming research. Literature fundamentally distinguishes analytical equations obtained by either simplified constitutive models, statistical, or semi-empirical approaches. In addition to this, more sophisticated numerical descriptions are also available. However, the more realistic modeling of strain hardening leads to advanced solutions, many finite element (FE) software still use analytical FLC equations as input parameters to assign failure criterion to the applied sheet material. These types of equations often contain mechanical properties from well-defined tests, such as tensile tests. Although evaluating the entire formability from results of only uni-axial stress state is a necessary simplification, the application of these equations is still widespread in the industrial practice, according to the authors' experiences.

Naturally, there is a lot of valuable literature that should be referenced in this topic, but here, in this introduction, literature [1] deserves special attention first, as it contains a logical overview of the FLC theories. Accordingly, the theoretical FLC models are broadly divided into the following groups:

assuming homogeneous sheet:

necking theories,

bifurcation theories,

linearized theory of perturbations,

assuming non-homogeneous sheet,

semi-empirical theories.

The models developed by Hill [2], Swift [3] and Hora et al. [4] belong to the group of necking theories of homogenous sheets. Hill described the zero-extension phenomenon for the left-hand side of the FLC, which was extended by Swift with the maximum force theory. It is worth noting here that the Swift model is related to diffuse necking phenomenon instead of localized necking, but it still appears frequently in the relevant literature. Later, Hora et al. developed the modified maximum force criterion (known as MMFC), based on Swift's work.

The bifurcation theory belongs to the study of Stören and Rice [5], who assumed that the development of localized neck formation is inherent with the formation of a vertex on the yield surface. They used deformation theory to derivate the equation of the critical major strain, in which the most important parameter is the strain hardening exponent. In case of localized failure initiation developing from a surface defect, thus assuming non-homogeneous sheet, the most famous model is the Marciniak-Kuczynski theory [6]. In the field of semi-empirical models, authors highlight here the work of Abspoel et al. [7], Kumar [8], Keeler-Brazier [9] and Levy-Tyne [10]. Naturally, this is not a complete list of FLC calculation formulas, readers can find more information in recent review papers [11,12,13,14].

This study deals with the presentation of tensile tests results and calculated FLCs for two steel sheets. For the investigation of different limit theories, approaches were selected from the group of necking (Swift), bifurcation (Stören-Rice) and semi-empirical (Abspoel et al.) models. Thereafter, function analyses were carried out to present comparisons, with the aim of helping the practical work of FE software users. The applied materials to which the calculation have been performed were DC04 and DP800 steel sheets. The necessary physical data have been obtained from tensile tests in every 15∘ to the rolling direction. Comparisons and explanations of the FLC-s are presented in the rest of the paper.

Materials and Methods

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Applied materials.

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Commercial, cold rolled DC04 steel sheet and DP800 high strength steel sheet were tested in this research. The nominal sheet thickness was 1 mm , for each material. Representative micrographs are shown by Fig. 1, illustrating the grain morphology and the ferrite-martensite volume fraction for the DP steel. The mechanical properties were obtained by standard tensile tests, which will be discussed in the next subchapter.

[picture]

Fig. 1. Microstructure of DC04 (left) and DP800 (right).

Fig. 1. Microstructure of DC04 (left) and DP800 (right).

Measurement method: tensile tests.

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Standard uniaxial tensile tests (ISO 6892-1) were performed at room temperature on A80 specimens, where footnote 80 refers to the initial gauge length. The specimen orientations were 0∘, 15∘,30∘,45∘,60∘,75∘, and 90∘ with respect to the rolling direction. The tests were carried out on an Instron 5900R 4482 universal testing equipment with constant crosshead speed of 4 mm/min. It led to less than 1·10−3 s−1 initial strain rate. The strain measurements were supported by an Instron AVE 5585 video extensometer. The measurement arrangement as well as real and schematic specimens are depicted on Fig. 2.

[picture]

Fig. 2. Tensile test equipment and specimens (dimensions are in mm).

Fig. 2. Tensile test equipment and specimens (dimensions are in mm).

Engineering stress-stain, true plastic stress-strain and r-value evolution diagrams were extracted from the raw data, for both materials. Fig 3 shows the results parallel to the rolling direction. Strain hardening exponent ( n -value), coefficient of plastic anisotropy ( r -value) and total elongation ( A80 ) have received the most attention during the evaluation of the tensile tests. These parameters are main factors of the selected models, therefore their average values are listed directly next to the FLCs, in the later figures: Fig, 4, 5 and 7. The n -values were approximated based on the power law description (Eq. 1) [15].

σ¯=K·ε¯n,(1)

in which K is strength coefficient.

[picture]

Fig. 3. Engineering (left) and true plastic (middle) stress-strain curves and the r-value's evolution (right) for DC04 and DP800 steel sheets.

Fig. 3. Engineering (left) and true plastic (middle) stress-strain curves and the r-value's evolution (right) for DC04 and DP800 steel sheets.

FLC calculations from tension test data

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The equations of Stören-Rice model [5] (considering n-value), Swift model [3] (considering n- and r-values), and the approach of Abspoel et al. [7] (considering r- and A80 values) are investigated in this chapter.

Stören-Rice equation.

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This formula is derived from the assumption of a vertex development on the yield surface, at localized necking. Using the deformation theory with isotropic yield criterion, this research leads to the next two equations regarding critical (failure) major (ε1*) and (ε2*) minor strains:

ε1*=3β2+n(2+β)22·(2+β)·(1+β+β2)βε1*=ε2*;(−1≤β≤1)

(2)(3)

Substituting the measured n -values in seven orientations of both materials to Eq. 2, the calculated FLCs are shown in Fig. 4. The average n -values are also depicted on the left of Fig. 4, in descending order. It can be stated that the order of the FLCs follows the order of the n -values, because this is the only one varying material parameter. However, the distance between the curves is not consistently the same in all stress (strain) states. This observation applies for both materials, which is better traceable on the enlarged pictures on the right of Fig. 4.

[picture]

Fig. 4. N-values of different orientations in descending order (left), as well as calculated FLCs in the middle, supplemented with enlarged pictures on the right; DC04 above and DP800 below.

Fig. 4. N-values of different orientations in descending order (left), as well as calculated FLCs in the middle, supplemented with enlarged pictures on the right; DC04 above and DP800 below.

For understanding the underlying reason, function analysis was conducted by substituting in β the values taken in noteworthy strain states: plane strain ( β=0 ), equi-biaxial tension ( β=1 ) and uni-axial tension (β=−1/2). Eq. 4-6 display the result of the derivation:

 if β=0→ε1*=n if β=1→ε1*=n+132=12n+16 if β=−12→ε1*=9n+39=n+13

(4)(5)(6)

Based on this investigation, the critical major strains are equivalent with n -values in the plane strain state, causing the positions of the FLCs to be proportional to this parameter. Nevertheless, constants are dominant in equi-biaxial and uni-axial strain states, resulting in the curves converge closer to each other.

Swift equation.

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The model originates from the assumption of maximum force criterion, dσ1=σ1 dε1 and dσ2=σ2 dε2. The limit strains' equations (Eq. 7 and 8) as well as the results of derivations (Eq. 9 and 10) can be found in [1], for the case of Hill'48 yield criterion [16].

ε1*=σ1·(∂f∂σ1)2+σ2·(∂f∂σ2)·(∂f∂σ1)σ1·(∂f∂σ1)2+σ2·(∂f∂σ2)2·nε2*=σ2·(∂f∂σ1)2+σ1·(∂f∂σ1)·(∂f∂σ2)σ1·(∂f∂σ1)2+σ2·(∂f∂σ2)2·nε1*=[1+r(1−α)]·(1−2r1+rα+α2)(1+r)(1+α)·[1−1+4r+2r2(1+r)2α+α2]·nε2*=[(1+r)α−r]·(1−2r1+rα+α2)(1+r)(1+α)·[1−1+4r+2r2(1+r)2α+α2]·n

(7)(8)(9)(10)

[picture]

Fig. 5. R-values of different orientations in descending order (left), as well as calculated FLCs in the middle, supplemented with enlarged pictures on the right; DC04 above and DP800 below.

Fig. 5. R-values of different orientations in descending order (left), as well as calculated FLCs in the middle, supplemented with enlarged pictures on the right; DC04 above and DP800 below.

Although the enlarged images at the right of Fig. 5 highlight the order of the FLCs, the determination of whether the r - or n -value has stronger effect on the curves' position and shape, is doubtful. DP800 has the highest r-value in 75∘ to the rolling direction (followed by the specimens of 30∘ and 45∘ ), which is the same for the n-value, too (Fig. 4). Still for this material, the curve located at the bottom belongs to the 0∘ orientation, which has the lowest n -, but the third from the bottom rvalue. For the DC04 sheet, specimen 90∘ to the rolling direction has the highest r -value and gives the FLC located at the bottom, in plane strain tension. Even if, n -value is relatively high in this
orientation. In addition to this, it is also observed that average highest n -value ( 0∘ in Fig. 4) is unable to produce the highest curve position, in the same stress state. However, in equi-biaxial tension, the position of the curves is reordered again, presumably according to the extent of n -value ( 0∘ at the top, 45∘ at the bottom). It leads to the assumption that r - and n -values create a cross-effect in this model under certain stress states, which is not necessarily true for both materials.

For possible explanations, mathematical analyses have been performed by solving Eq. 9 and 10 for noteworthy stress states: α=0,α=1/2,α=1. Then critical major and minor strains were expressed for fictitious r-values (Fig. 6), next to taking n -value as constant 0.17 .

[picture]

Fig. 6. Evolution of critical major strain in the function of r -value.

Fig. 6. Evolution of critical major strain in the function of r -value.

Fig. 6 shows that critical major strain is independent from the r-value in this current form of Swift model, in uni-axial and equi-biaxial tension loading. Besides, in plane strain tension, materials with higher r-value lead to lower critical major strain. The minor strain is always decreasing with increasing r-value, except in equi-biaxial stress state, causing the curves' shift to down and left.

Abspoel et al. equation.

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This concept uses regression approximation to evaluate the limit strains from tensile tests results. Here, n -value does not get emphasis, but total elongation ( A80 ) dominates the outcome in all stress states. Plastic anisotropy coefficient comes to play only in the negative quadrant of the FLCs. Another difference is that the deformation is determined by some specific points of noteworthy stress states, instead of continuous function of limit strains. The equations for uni-axial tension, plane strain tension and equi-biaxial tension load cases are given by Eq. 11-14, considering the fact that the nominal sheet thickness is equal to unity.

Uni-axial stress state:

ε1*=(1+0.797·r0.701)·0.0626· A800.567[1+(0.797·r0.701)2]ε2*=−(0.797·r0.701)·0.0626· A800.567[(1+(0.797·r0.701)2]

(11)(12)

Plane strain stress state:

E1*=0,0084· A80.(13)

Equi-biaxial stress state:

E1*=ε2*=0,00215· A80+0,25+0,00285· A80·(14)

Values of A80 of both materials in descending order as well as calculated FLCs are depicted by Fig. 7. The difference between the limit curves is highlighted by the enlarged pictures on the right side. Eq. 13 and 14 are linear functions of total elongation, and thus the FLCs' order is clearly defined by the total elongation in plane strain tension and equi-biaxial tension stress states. Plastic anisotropy distorts only the left-hand side of the limit curves, with the combined effect of the total elongation.

[picture]

Fig. 7. Total elongations of different orientations in descending order (left), as well as calculated FLCs in the middle, supplemented with enlarged pictures on the right; DC04 above and DP800 below.

Fig. 7. Total elongations of different orientations in descending order (left), as well as calculated FLCs in the middle, supplemented with enlarged pictures on the right; DC04 above and DP800 below.

Assuming a constant for A80 and substituting fictitious r -values into Eq. 11-12, the evolution of the critical major and minor strains in the function of the r -value can be monitored for uni-axial stress state, in Fig. 8. This figure shows that A80 proportionally increases both ε1* and ε2*, and the effect of the r -value is stronger on the critical minor strain (ε2*). It means that r -value tilts the FLC to the left, in this model. In terms of critical major strain (ε1*), which is almost independent from plastic anisotropy coefficient, it still is worth noting that it has local maximum at r≈1.4 according to this formula.

[picture]

Fig. 8. Evolution of critical major and minor strains in the function of fictitious r-values, in uniaxial stress state.

Fig. 8. Evolution of critical major and minor strains in the function of fictitious r -values, in uniaxial stress state.

Summary

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The current paper discusses three different calculation methods of FLC evaluation, for which the necessary experimental data is obtained from tensile tests. The applied formulas are arbitrary selected from the literature, but they belong to different logical groups of theories. Tensile tests of DC04 and DP800 steels in room temperature were performed in seven orientations, from 0∘ to 90∘ to the rolling direction. FLCs were then calculated separately in each direction, and the observed trends were compared and described for the two materials in this study.

Based on the current observations, the higher the n -value and the total elongation, the higher is position of the FLCs, in general. Besides, the role of the r -value is slightly more complicated. The increase of plastic anisotropy coefficient caused the shift to down and left of the plane strain point in the Swift equation, for the DC04 sheet. Similar effects for the DP800 material are not observed. In the model of Abspoel et al., the plane strain point is independent from the r-value, but the FLCs are tilted to the left in the negative quadrant. This phenomenon also appears more prominently for the DC04 sheet. Therefore, the precise determination of the r -value is more important in terms of the shape of the FLC in the negative quadrant as well as the position of the calculated plane strain point. Nevertheless, both these modeled behaviors are more significant for steels with higher r -value.

Acknowledgement

The original version of this paper is available on https://www.scientific.net/SSP.388.47.pdf

We acknowledge the professional support of the Ministry of Culture and Innovation's New National Excellence Program (grant code EKÖP-2024-00020), financed by the National Research, Development and Innovation Fund in Hungary.
The research was carried out in connection with the project entitled „Development of new innovative chisel generations" 2020-1.1.2-PIACI-KFI-2020-00148 financed by National Research, Development and Innovation Fund in Hungary.

References

The original version of this paper is available on https://www.scientific.net/SSP.388.47.pdf

D. Banabic, "Sheet Metal Forming Pocesses, Constitutive Modeling and Numerical Simulation", Springer-Verlag Berlin Heidelberg, 2010, doi: 10.1007/978-3-540-88113-1.

R. Hill, "On discontinuous plastic states, with special reference to localized necking in thin sheets", Journal of the Mechanics and Physics of Solids, vol 1. pp. 19-30, 1952.

H. W. Swift, "Plastic instability under plane stress", Journal of the Mechanics and Physics of Solids, vol. 1. pp. 1-18, 1952.

P. Hora, L. Tong, and B. Berisha, "Modified maximum force criterion, a model for the theoretical prediction of forming limit curves," International Journal of Material Forming, vol. 6, pp. 267-279, 2013, doi: 10.1007/S12289-011-1084-1.

S. Stören and J. R. Rice, "Localized necking in thin sheets," J Mech Phys Solids, vol. 23, pp. 421-441, 1975, doi: 10.1016/0022-5096(75)90004-6.

Z. Marciniak, K. Kuczynski, "Limit strains in the processes of stretch-forming sheet metal", Int. J. Mech. Sci. vol. 9, pp. 609-620, 1967.

M. Abspoel, M. E. Scholting, and J. M. M. Droog, "A new method for predicting Forming Limit Curves from mechanical properties," J Mater Process Technol, vol. 213, pp. 759-769, 2013, doi: 10.1016/J.JMATPROTEC.2012.11.022.

S. K. Paul, "Prediction of complete forming limit diagram from tensile properties of various steel sheets by a nonlinear regression based approach," J Manuf Process, vol. 23, pp. 192-200, 2016, doi: 10.1016/j.jmapro.2016.06.005.

T. Altan, A.E. Tekkaya, Sheet Metal Forming - Fundamentals, ASM International, 2012.

B.S. Levy, C.J. Van Tyne, "Determination of a Stress-Based Forming Limit Curve from Mechanical Properties," IDDRG Conference Proc., Shanghai, China, 2015.

D. Banabic, F. Barlat, O. Cazacu, T. Kuwabara, "Advances in anisotropy and formability", International Journal of Material Forming, 3:165-189, 2010, doi: 10.1007/s12289-010-09929.

D. Banabic, F. Barlat, O. Cazacu, T. Kuwabara, "Advances in anisotropy of plastic behaviour and formability of sheet metals", International Journal of Material Forming, 2020, doi: .

S. Bruschi et al., "Testing and modeling of material behaviour and formability in sheet metal forming", CIRP Annals - Manufacturing Technology, 63, pp. 727-749, 2014, doi: 10.1016/j.cirp.2014.05.005.

Sz. Szalai, D. Harangozó, I. Czinege, "Az alakíthatósági határgörbe meghatározása és modellezése", Anyagvizsgálók Lapja, II. pp. 39-52, 2021.

Á. Nádai, A. M. Wahl, and E. C. Bingham, "Plasticity, A Mechanics of the Plastic State of Matter," J Rheol (N Y N Y), vol. 2, pp. 455-456, 1931, doi: 10.1122/1.2116408.

R. Hill, "A theory of the yielding and plastic flow of anisotropic metals," Proc R Soc Lond A Math Phys Sci, vol. 193, pp. 281-297, 1947, doi: 10.1098/RSPA.1948.0045.

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Solid State Phenomena, ISSN: 1662-9779, Vol. 388, pp 57-66

doi: 10.4028/p-9RDkIt

© 2026 The Author(s). Published by Trans Tech Publications Ltd, Switzerland.

Submitted:
2025-12-22

Revised:
2026-02-15

Accepted:
2026-02-19

Online:
2026-04-15

Theoretical and Experimental Forming Limit Evaluation of High-Strength Steels up to 1500 MPa

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Borbely Richard 1,2,a*, Keresztes Robert Zsolt 2, b and Beres Gabor Jozsef 1,c

1 Innovative Vehicles and Materials Department, John von Neumann University, Hungary

2 Doctoral School of Technical Sciences, Hungarian University of Agriculture and Life Sciences, Hungary

a*borbely.richard@nje.hu, b keresztes.robert.zsolt@uni-mate.hu, c beres.gabor@nje.hu

Keywords: high-strenght steels, forming limit curve, formability.

Abstract

The prediction of sheet failure remains a highly relevant topic in metal forming research, particularly in relation to the experimental and theoretical determination of forming limit curves (FLCs). While the experimental construction of FLCs is a well-established but time-consuming process, theoretical and numerical approaches provide a more efficient alternative. However, their accuracy must be critically assessed, all the time. In this work, the formability of steel sheet with tensile strengths of 1500 MPa is investigated by combining experimental Nakajima tests with theoretical predictions of FLCs. Previous studies, such as models by Abspoel, Swift, Hill etc. have not addressed such high strength levels, where the diffuse and local neck points are quite close, leaving open the question of whether existing approaches remain valid for these materials. To provide a reference baseline, additional tests and calculations were also performed on lower-strength steel sheet (DP800). Our results show that the FLC points can be well estimated by two different theories in the positive quadrant, but there are noticeable differences between the measured and calculated values close to the plane strain point.

Introduction

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The prediction of sheet failure has long been a central topic in metal forming research. The ability to accurately determine the forming limits of sheet materials is essential for the optimization of forming processes and for preventing failure during production. Early contributions by Lankford [1] and Hill [2] laid the theoretical foundations of plasticity and introduced key parameters such as the plastic anisotropy coefficient ( r ) and the strain hardening exponent ( n ), both of which have been shown to exert a strong influence on the formability of metallic sheets.

A significant breakthrough in experimental formability characterization emerged in the 1960s with the pioneering work of Keeler and colleagues [3], [4]. By applying a measurement grid on the sheet surface and tracking its deformation up to fracture, they were able to determine the principal strain values corresponding to the onset of failure. Repeating these tests for specimens of various geometries -thus representing different stress and strain states- they introduced the forming limit diagram (FLD), in which the envelope connecting the critical strain points was termed the forming limit curve (FLC). Their investigations initially focused on the positive strain region, ranging from plane strain ( ε2=0 ) to equibiaxial tension (ε2=ε1). The extension of the diagram to negative minor strains (ε2<0) was later proposed by Goodwin [5], thereby completing the modern representation of the FLD used today.

Over time, Keeler refined his theoretical framework and adapted it to industrial applications, addressing practical challenges encountered in sheet metal forming processes [6], [7]. The classical experimental determination of FLCs has since been standardized through methods such as the Nakajima [8] and Marciniak tests, differing mainly in punch geometry -the Nakajima test employs a hemispherical punch, while the Marciniak test utilizes a flat punch. These tests remain the benchmark for experimental evaluation of forming limits.

Parallel to experimental advancements, several researchers have proposed theoretical and numerical methods to predict the onset of localized necking. Building on the physical understanding of thinning and instability phenomena, Swift, Hill [9], [10] and Stören and Rice [11] formulated
theoretical approaches for estimating forming limit curves under plane stress conditions. More recently, Abspoel et al. [12] developed a statistical model which considers so-called r-value (Lankford-coefficient) and correlating the position and shape of the FLC with material parameters obtained from uniaxial tensile tests. Their approach provided a more physically consistent link between measurable mechanical properties and predicted forming limits. Similar model was developed by Kumar [13]. Besides, if somebody talks about theoretical evaluation of FLCs, the work of Hora [14] Keeler-Brazier [15] and Levy and Tyne [16] also has to be mentioned as relevant developments of roughly the last decade.

Despite these extensive efforts, the reliable prediction of FLCs for ultra-high-strength steels (UHSS) -particularly those with tensile strengths exceeding 1200 MPa - remains a challenge. In such materials, the transition between diffuse and localized necking occurs within a very narrow strain range, complicating both experimental detection and theoretical interpretation. This study therefore aims to assess the applicability of established theoretical models, originally validated for lowerstrength steels, to a material with tensile strengths up to 1500 MPa . By combining experimental Nakajima tests with theoretical predictions based on both formulations, this work seeks to clarify the predictive limits and potential refinements required for the accurate modeling of UHSS formability.

Materials and Methods

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Applied materials.

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The materials investigated in this study were two grades of dual-phase (DP) steel sheets with nominal tensile strengths of 800 MPa (DP800) and 1500 MPa (DP1500), both widely used in the automotive industry for structural and safety-relevant components. Dual-phase steels belong to the class of second-generation advanced high-strength steels (AHSS), characterized by a microstructure consisting of a soft ferritic matrix and hard martensitic islands. This combination ensures an advantageous balance between strength and ductility, making DP steels ideal for lightweight vehicle design, where crash performance must be enhanced without significantly increasing weight.

The microstructural characteristics of both steels were examined to determine the phase distribution and grain morphology. The average grain size of the ferritic and martensitic phases was measured to be ∅dave_f =4,55μ m and ∅dave_m =2,21μ m for DP800, and ∅dave_f =8,58μ m and ∅dave_m =4,24μ m for DP1500. The phase fraction analysis revealed that the martensite content was approximately 38±3% in DP800 and 82±5% in DP1500, respectively. Representative micrographs of the investigated materials are shown in Fig. 1, illustrating the distinct dual-phase structure and grain morphology of each steel.

[picture]

Fig. 1. Microstructure of DP800 (left) and DP1500 (right).

Fig. 1. Microstructure of DP800 (left) and DP1500 (right).

Figure 2 presents the engineering stress-strain curves (σ−ε) and the corresponding true stress-strain curves ( σ′−ε′ ) obtained from uniaxial tensile tests at 0∘ to the rolling direction.

[picture]

Fig. 2. Stress-strain curves of DP800 (left) and DP1500 (right).

Fig. 2. Stress-strain curves of DP800 (left) and DP1500 (right).

The DP1500 grade exhibits a significantly higher ultimate tensile strength and yield strength, as expected, while the DP800 material maintains superior uniform elongation and strain hardening capability. Figure 3 presents the corresponding flow curves of both materials.

[picture]

Fig. 3. Flow curves of DP800 and DP1500.

Fig. 3. Flow curves of DP800 and DP1500.

These results reflect the fundamental trade-off between strength and ductility typical for highstrength dual-phase steels. Both materials display a continuous yielding behavior without a distinct yield point, which is characteristic of their mixed ferrite-martensite microstructure. The most important, average mechanical parameters in every 15∘ to the rolling direction are listed in Table 1.2.

Table 1. Relevant mechanical parameters of DP800.

DP800
| A₈₀ ave. (-) | Ag ave. (-) | TS ave. (N/mm²) | Rp0,2 ave. (N/mm²) | r ave. (-) | n ave. (-)
0° | 0,152 | 0,107 | 857 | 493 | 0,817 | 0,118
15° | 0,162 | 0,115 | 823 | 472 | 0,745 | 0,123
30° | 0,162 | 0,114 | 796 | 451 | 0,878 | 0,127
45° | 0,166 | 0,116 | 770 | 436 | 0,876 | 0,127
60° | 0,160 | 0,116 | 830 | 474 | 0,787 | 0,125
75° | 0,148 | 0,109 | 859 | 452 | 1,002 | 0,129
90° | 0,148 | 0,105 | 831 | 480 | 0,862 | 0,121

Table 2. Relevant mechanical parameters of DP1500.

DP1500

A80 ave.

Ag ave.

TS ave.

superscript reference ave.

r ave.

n ave.

mathematical formula | 0,030 | 0,019 | 1556 | 1438 | 0,777 | 0,024
superscript number | 0,035 | 0,025 | 1595 | 1420 | 0,742 | 0,038
mathematical formula | 0,032 | 0,023 | 1559 | 1410 | 0,503 | 0,023
mathematical formula | 0,031 | 0,023 | 1521 | 1370 | 0,411 | 0,024
mathematical formula | 0,026 | 0,020 | 1503 | 1367 | 0,535 | 0,023
mathematical formula | 0,026 | 0,020 | 1461 | 1320 | 0,822 | 0,022
mathematical formula | 0,031 | 0,021 | 1441 | 1287 | 0,686 | 0,026

The uniaxial tensile tests were performed on flat specimens cut at orientations of 0∘,15∘,30∘,45∘, 60∘,75∘, and 90∘ with respect to the rolling direction. The tests were conducted on an Instron 5900R 4482 universal testing machine equipped with a video extensometer (Instron 5585) to ensure accurate strain measurement. The tests followed the recommendations of ISO 6892-1, with a crosshead speed of 4 mm/min corresponding to an initial strain rate of approximately 8·10−4 s−1.

The resulting data were used to determine key mechanical parameters, including the yield strength ( Rp0,2 ), ultimate tensile strength (TS), uniform elongation ( Ag ), and anisotropy coefficients ( r -values), which served as input parameters for the theoretical forming limit curve (FLC) calculations. It is worth mentioning that the r-values were continuously measured in the strain range of Ag…Ag/2, then those have been averaged. The n -values were calculated according to the Nadai formula [17]:

σ′=K·ε′n,(1)

in which the K parameter is material dependent strength coefficient.

Forming limit tests (Nakajima tests).

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The experimental forming limit curves were determined according to ISO 12004-2 using an ERICHSEN 142-40 universal sheet metal testing machine. The tests were performed at a constant punch movement speed of 20 mm/min, employing a hemispherical punch with a diameter of 100 mm . To obtain a range of strain paths, specimens of different widths were prepared, from narrow to wide, producing conditions from uniaxial to equi-biaxial tension. The initial blank diameter was 220 mm , and the sheet thickness was 1±0,05 mm. The test was terminated when the value of the force dropped by 30%. The geometries of the applied test specimens are listed in Table 3, for which the explanation of the measures is depicted on Fig. 4.

[picture]

Fig. 4. Parametric view of the applied test specimens.

Fig. 4. Parametric view of the applied test specimens.

Table 3. Dimensions of different test specimens.

b

a

c

D

30 | 25 | 20 | 220
70 | 25 | 20 | 220
80 | 25 | 20 | 220
90 | 25 | 20 | 220
100 | 25 | 20 | 220
110 | 25 | 20 | 220
120 | 25 | 20 | 220
220 | - | - | -

To minimize frictional effects and ensure repeatability, a sandwich lubrication method was applied during the Nakajima tests. A polyethylene film (thickness ≈0,05 mm ) was placed between the die and the specimen, and Luba 251 cold-forming oil was applied on both interfaces with an average coating mass of 79±5 g/m2. This lubrication strategy ensured stable frictional conditions and reduced local heating or galling effects during testing.

Data evaluation.

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The recorded strain fields (Fig. 5) were post-processed using the Vialux ARGUS software (Fig. 6), following the Volk and Hora methodology [18] to identify the onset of localized necking.

[picture]

Fig. 5. Strain evaluation of DP800.

Fig. 5. Strain evaluation of DP800.

[picture]

Fig. 6. Identification of the onset of necking for DP800 (left) and DP1500 (right).

Fig. 6. Identification of the onset of necking for DP800 (left) and DP1500 (right).

For each strain path, the forming limit strain was defined at the point where the the absolute value of the first derivative of the thickness-direction deformation curve exhibited a distinct curvature change corresponding to the initiation of localized necking (Fig. 6). The resulting data were compiled to construct the experimental forming limit diagrams for both materials.

Theoretical framework.

The original version of this paper is available on https://www.scientific.net/SSP.388.57.pdf

The forming limits of the investigated DP800 and DP1500 steel sheets were determined by combining experimental Nakajima tests and theoretical predictions derived from established forming limit models available in the literature. The experimental forming limit diagrams were evaluated according to the methodology proposed by Volk and Hora, which enables a precise and objective identification of the onset of localized necking from full-field strain data. The resulting experimental points for each strain path were later compared with the theoretical forming limit curves predicted by analytical and numerical models.

The theoretical predictions were obtained using MATLAB software, where the analytical equations of some selected instability models were implemented and solved for the material parameters determined from the uniaxial tensile tests. The graphical representation of the calculated curves allowed for a direct visual comparison between the predicted and experimentally measured forming limits.

The investigated theoretical frameworks include both formulation. The Stören-Rice model doesn't include the r-value in itself. The governing criterion of this model can be expressed as:

ε1=[3ρ2+n(2+ρ)2][2(2+ρ)*(1+ρ+ρ2)]P=dε2dε1;(−1≤ρ≤1)

(2)(3)

Here, the major strain ( ε1 ) is calculated theoretically, and the belonging minor strain ( ε2 ) is expressed by the strain ratio: ρ.

In addition to this, formulation proposed by Abspoel et al. [12] -which takes the r-value into consideration- was implemented, which introduces the influence of plastic anisotropy and total elongation through experimentally measurable material parameters, such as the Lankford coefficients (r). The final equations for the critical strains at necking in different stress states are the following. Uniaxial stress state:

ε1=(1+0,797*r0,701)*(0,0626* A800,567+(t−1)*(0,12−0,0024* A80))(1+(0,797*r0,701)2E2=(0,0626* A800,567+(t−1)*(0,12−0,0024* A80))(1+(0,797*r0,701)2

(4)(5)

Plane strain stress state:

ε1=0,0084* A80+0,0017* A80*(t−1)(6)

Equi-biaxial stress state:

ε1=ε2=0,00215* A80MIN+0,25+0,00285* A80MIN*t.(7)

Intermediate-biaxial stress state:

ε1=0,0062* A80+0,18+0,0027* A80*(t−1)ε2=0,75*(0,0062* A80+0,18+0,0027* A80*(t−1))

(8)(9)

These two approaches represent fundamentally different assumptions regarding the material's plastic response. The Stören-Rice model neglects the directional dependence of plastic flow, while the Abspoel formulation accounts for the evolution of plastic strain anisotropy during deformation. Although, the r-value appears in the functions of the uniaxial stress state only according to the formulations of Abspoel et al., there is a significant difference between the nature of the equations.

Authors know that this is an arbitrary selection of existing theoretical models, but the comparison of these models therefore could provide valuable insight into the significance of anisotropy effects in predicting forming limits, particularly for ultra-high-strength steels (UHSS) where texture and microstructural effects are more pronounced.

Results

The original version of this paper is available on https://www.scientific.net/SSP.388.57.pdf

The experimental results obtained from the Nakajima tests revealed consistent deformation and fracture behavior for both investigated steels. Figure 7. presents the experimental FLCs of both materials.

[picture]

Fig. 7. Experimental FLCs of DP800 and DP1500.

Fig. 7. Experimental FLCs of DP800 and DP1500.

The experimental scatter of the major strain at the onset of localized necking was found to be small, with a maximum deviation of ±5% across repeated measurements.

The phase fraction analysis revealed a martensite content of 38±3% in DP800 and 82±5% in DP1500, indicating that DP1500 contains approximately twice as much martensite as DP800. This significant increase in martensite fraction leads to higher strength but reduced strain hardening capability, as reflected by the fitted hardening exponents ( n≈0.12 for DP800 vs. n≈0.02 for DP1500).

Consequently, the forming limit diagram of DP1500 is consistently shifted downward compared to DP800. In the near plane-strain region ( ε2≈0 ), the major strain ( ε1 ) of DP800 is approximately twice that of DP1500, which correlates well with the nearly doubled martensite content of DP1500.

The equivalent strain results support this trend: DP800 exhibits equivalent-strain values predominantly in the range of ~0.21−0.32 (average ≈0.29 ), whereas DP1500 shows substantially lower characteristic values around ~0.09−0.15 (average ≈0.14 ) in several geometries. This indicates earlier strain localization in DP1500, which can be directly linked to its high martensite fraction and limited work hardening capacity.

Table 4. Avaraged values of major and minor strains.

b (mm) | 30 | 60 | 70 | 80
Strain | ε₁ | ε₂ | ε₁ | ε₂ | ε₁ | ε₂ | ε₁ | ε₂
DP800 | 0,313 | -0,116 | 0,273 | -0,035 | 0,209 | -0,013 | 0,185 | -0,005
DP1500 | 0,143 | -0,047 | 0,080 | -0,005 | 0,081 | -0,002 | 0,083 | -0,002
b (mm) | 90 | 100 | 120 | 220
Strain | ε1 | ε2 | ε1 | ε2 | ε1 | ε2 | ε1 | ε2
DP800 | 0,196 | 0,002 | 0,194 | 0,010 | 0,237 | 0,029 | 0,305 | 0,238
DP1500 | 0,082 | 0,000 | 0,085 | 0,001 | 0,080 | 0,004 | 0,219 | 0,157

The comparisons of the theoretical and experimental curves are shown by Fig. 8 and 9. The calculations have been carried out in both 0∘ and 90∘ to the rolling direction.

[picture]

Fig. 8. Experimental and theoretical FLCs of DP800.

Fig. 8. Experimental and theoretical FLCs of DP800.

The difference between the calculated curves of different tensile test specimen orientation is negligible, because of the similarities in plastic anisotropy. The n and A80 values are also very similar in these orientations, for both materials. However, their matching with the experimental results is not very convincing close to the plane strain state. Exactly on the axis of ε2=0, significant deviation can be observed.

Away from the zero minor strain axis, the fit of the experimental and theoretical results is considered good in the positive quadrant of the FLCs. Roughly the same can be stated for the lefthand side of the FLCs, too.

[picture]

Fig. 9. Experimental and theoretical FLCs of DP1500.

Fig. 9. Experimental and theoretical FLCs of DP1500.

The predictive performance of the investigated models was quantified using the mean absolute percentage error (MAPE), which expresses the average relative deviation between the calculated and experimentally determined forming limit values. The MAPE was calculated according to

 MAPE =1N*∑i|ε1, pred ,i−ε1, meas ,iε1, meas ,i|*100%(10)

Where ε1, meas,i  and ε1, pred,i  denote the measured and predicted major strain values at identical ε2 positions, respectively, and N is the number of evaluated strain states. The predictive accuracy was then defined as accuracy, =100− MAPE.

Based on the global evaluation (including all strain paths), the Abspoel model yielded an accuracy of 78% and 77% for DP800 ( 0∘ and 90∘ ), and 75% and 73% for DP1500. The Stören-Rice model resulted in slightly lower values, namely 74% and 74% for DP800, and 73% and 66% for DP1500. These results indicate that the Abspoel approach provides more consistent agreement with the experimental forming limit data, while the predictive capability generally decreases for the higher martensite DP1500 steel.

Since forming limit evaluation is primarily governed by the plane strain region ( ε2≈0 ), a secondary analysis was performed excluding the extreme biaxial strain states ( b=30 mm and b=220 mm ). Under this restriction, the predictive accuracy decreased for all models. For DP800, the Abspoel model provided 76% and 75% accuracy, while the Stören-Rice model yielded 71% and 71%. For DP1500, the reduction was more pronounced: the Abspoel model achieved 69% and 68%, whereas the Stören-Rice model decreased to 66% and 59% ( 0∘ and 90∘, respectively). The systematic drop in accuracy in the plane strain region suggests that model performance is more sensitive near the onset of localized instability, particularly for the higher martensite DP1500 steel. This behavior confirms that predictive discrepancies become more significant under strain states most relevant to forming limit determination.

The deviation of the measured and calculated values in plane strain state can be explained by the following assumptions.

The less the plastic strains, the more sensitive they are for the difference come from the inplane and out-of-plane loading.

The less the plastic strains, the more sensitive the measurement method is for the dimensions of the measuring grid pattern.

None of the n -value and the A80 really reflect the material behavior in plane strain state.

To overcome these questions, measurement series are planned for investigating the failure with different grid pattern geometry, applying on Nakajima, uniaxial tension and (in-plane) plane strain tension tests, supported with DIC.

Conclusion

The original version of this paper is available on https://www.scientific.net/SSP.388.57.pdf

Theoretical and experimental investigations have been executed on forming limit curves of highstrength steels. The comparison between the experimental and theoretical FLCs shows that for the DP1500 steel, both type of models predict the right-hand side (biaxial tension region) of the curve with quite high accuracy. However, near the plane strain condition, a systematic deviation was observed -both models tend to underestimate the experimentally measured forming limits. This deviation indicates that while the theoretical approaches remain valid for high-strength steels under equibiaxial tension, their predictive capability under near-plane strain conditions may require further refinement.

Acknowledgement

The original version of this paper is available on https://www.scientific.net/SSP.388.57.pdf

We acknowledge the professional support of the Ministry of Culture and Innovation's New National Excellence Program (grant code EKÖP-2024-00020), financed from the National Research, Development and Innovation Fund of Hungary.

References

The original version of this paper is available on https://www.scientific.net/SSP.388.57.pdf

W. Lankford, S.C. Snyder, J.A. Bauscher, Trans. ASM 42 (1950).

R. Hill, A theory of the yielding and plastic flow of anisotropic metals, Proc. R. Soc. Lond. A 193 (1948) 281-297.

S.P. Keeler, Determination of forming limits in automotive stampings, SAE Trans. 75 (1966).

S.P. Keeler, W.A. Backofen, Plastic instability and fracture in sheet stretched over rigid punches, Trans. ASM 56 (1963) 25-48.

G.M. Goodwin, Application of strain analysis to sheet metal forming problems in the press shop, SAE Paper No. 680092 (1968).

S.P. Keeler, Society of Automotive Engineers, report on metal forming (1968).

S.P. Keeler, Circle grid analysis, National Steel Corp., Livonia, Michigan, 1968.

K. Nakajima, T. Kikuma, Yamata Technical Report 284 (1971).

H.W. Swift, Plastic instability under plane stress, J. Mech. Phys. Solids 1 (1952) 1-18.

R. Hill, On discontinuous plastic states with special reference to localized necking in thin sheets, J. Mech. Phys. Solids 1 (1952) 19-30.

S. Stören, J.R. Rice, Localized necking in thin sheets, J. Mech. Phys. Solids 23 (1975) 421-441.

M. Abspoel, M.E. Scholting, J.M.M. Droog, A new method for predicting forming limit curves from mechanical properties, J. Mater. Process. Technol. 213 (2013) 759-769.

S.K. Paul, Prediction of complete forming limit diagram from tensile properties of various steel sheets by a nonlinear regression based approach, J. Manuf. Process. 23 (2016) 192-200.

P. Hora, L. Tong, B. Berisha, Modified maximum force criterion, a model for the theoretical prediction of forming limit curves, Int. J. Mater. Form. 6 (2013) 267-279.

S. Keeler, M. Kimchi, P.J. Mooney, Advanced high-strength steels, application guidelines, WorldAutoSteel, 2017.

B.S. Levy, C.J. Van Tyne, Determination of a stress-based forming limit curve from mechanical properties, in: IDDRG Conference, Shanghai, China, 2015.

Á. Nádai, A.M. Wahl, E.C. Bingham, Plasticity, a mechanics of the plastic state of matter, J. Rheol. 2 (1931) 455-456.

W. Volk, P. Hora, New algorithm for a robust user-independent evaluation of beginning instability for the experimental FLC determination, Int. J. Mater. Form. 4 (2011) 339-346.

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Solid State Phenomena, ISSN: 1662-9779, Vol. 388, pp 67-72

doi: 10.4028/p-0xWSkC

© 2026 The Author(s). Published by Trans Tech Publications Ltd, Switzerland.

Submitted:
2026-01-08

Revised:
2026-02-21

Accepted:
2026-02-22

Online:
2026-04-15

Evaluation and Comparison of a Few Methods for the Strain-Path Correction of FLCs

The original version of this paper is available on https://www.scientific.net/SSP.388.67.pdf

Mainguy Grégoire 1,2,a, Lemoine Xavier 2, b and Balan Tudor 1,c*

1 Arts et Metiers Institute of Technology, Université de Lorraine, LCFC, F-57070Metz, France

2 Global R&D ArcelorMittal Maizières, voie Romaine, BP30320, F-57283 Maizières-lès-Metz, France

a  gregoire.mainguy@arcelormittal.com, b  xavier.lemoine@arcelormittal.com, c* tudor.balan@ensam.eu

Keywords: forming limit curves (FLC), strain-path dependence; correction methods.

Abstract

Forming limit curves are sensitive to the strain-path. Numerous methods were proposed in the literature since the 1970's to the last decades for taking into account this sensitivity. The aim of this work is to consistently compare these methods with practical application in mind. First, a literature review revealed that many available methods are different in form but all rely on the empirical assumption of iso-equivalent failure strain for all strain paths sharing the same final strain mode, independently of the strain path leading to it. The models relying on this hypothesis are summarized. A significantly different approach, called here "interpolation method", relies on different hypotheses - also empirical. The two approaches are further compared in order to identify the similarities and differences between the two. It appears that most stress-based approaches for the correction of FLC strain-path dependence rely on the same hypothesis as the iso-equivalent-strain method and they reduce to it under the assumption of isotropic hardening. The iso-strain method and the "interpolation" approach provide similar predictions in a series of configurations, while they significantly differ for other configurations. No theoretical or experimental proof is available to further discriminate the most accurate approach for strain-path correction of FLCs; the research opens new directions for the investigation of the topic.

Introduction

The original version of this paper is available on https://www.scientific.net/SSP.388.67.pdf

Forming Limit Curves (FLCs) remain the central tool for describing in-plane formability of sheet metals. Their classical experimental determination through the Nakazima or Marciniak tests has enabled engineers to identify the onset of localized necking across a variety of loading modes. However, their practical use in design and simulation has long been complicated by strain-path dependence. Even moderate prestrains may shift the forming limits significantly, affecting the accuracy of forming predictions in industrial operations where complex, nonlinear strain paths are common (see Fig. 1). The strain-path dependence of forming limits has been recognized since the earliest FLC studies. Classical tests showed that a sheet prestrained uniaxially could experience either an increase or decrease in its subsequent forming limit depending on the second strain path. Similar effects were later documented for plane strain and biaxial prestrains.

These observations revealed that forming limits could not be uniquely represented in principal strain space. Instead, the history of deformation affects the mechanical state of the material at necking. Research progressed along two main directions: i) Identifying state variables for which the forming limit would be path-independent (stress-based approaches). ii) Transforming the experimental strains to remove the influence of prior deformation through correction methods (equivalent-strain approaches). Both directions attempted to obtain a single, universal forming limit map.

Over the past decades, two main families of approaches have emerged to address strain-path dependence: strain-based forming limits with equivalent-strain correction [1-5], and stress-based forming limit representations [6-7], which propose that the state of stress at the onset of localization is insensitive to the deformation history. More recently, an interpolation-based method was proposed
[8], aiming to reconstruct non-linear forming limits without explicit correction through stress or equivalent strain.

The purpose of this contribution is to present a clarified comparison of the assumptions behind these families of approaches and a discussion of their mutual consistency, with emphasis on the underlying conditions (e.g., isotropic hardening) that allow equivalence or divergence.

[picture]

Fig. 1. Illustration of the impact of strain path on the forming limits of sheet metals, after [9].

Fig. 1. Illustration of the impact of strain path on the forming limits of sheet metals, after [9].

Equivalent-Strain-Based Correction of Forming Limit Curves

The original version of this paper is available on https://www.scientific.net/SSP.388.67.pdf

The equivalent-strain approach assumes that the material state at the onset of localization is primarily controlled by the accumulated equivalent plastic strain. The correction method transforms strain paths so that prestrained states can be mapped back to an equivalent "virgin" state with the same hardening level (see Fig. 2).

This correction has several attractive properties:

It is experimentally validated across a wide range of metals.

It is easy to apply in practice.

It remains compatible with standard strain-based forming limit diagrams used in industry.

Extensive experimental datasets show that for many steels and aluminum alloys, the method often provides a reliable prediction of forming limits under bilinear strain paths (see an example in Fig. 3). However, there is no proof for the good behavior of this approach, and several publications have shown that it sometimes also led to inaccurate predictions.

Stress-based forming limit diagrams (FLSDs) gained renewed prominence because they were thought to be inherently path-independent. Studies have shown that stresses at the onset of localization tend to collapse towards a relatively narrow locus that is less sensitive to strain-path changes than strain-based forming limits. It is noteworthy that under the isotropic hardening hypothesis, the stress-based approach fully derives from the equivalent-strain-based approach. If the yield surface expands uniformly with plastic strain, the forming limit associated with a given stress or strain state depends only on the current level of accumulated deformation. Therefore, removing the prestrain through an equivalent-strain transformation effectively restores the original forming limit., and stress-based criteria do not introduce independence beyond what is already implied by isotropic hardening. Their path independence stems from the same assumption that enables equivalent-strain correction and their predictions are expected to be similar in many practical situations. It was demonstrated in the literature that advanced hardening models lead to significant strain-path dependence of the stress-based forming limits, as illustrated in Fig. 4.

[picture]

Fig. 2. Graphical construction of the equivalent strain concept proposed in [1] to solve for the strain-path dependence of FLCs. Strain state B has the same equivalent strain as strain state A, and the distances AF and BF′ in this strain space are equal, which leads to identical equivalent strain values at F and F′ - with OF′ being linear.

Fig. 2. Graphical construction of the equivalent strain concept proposed in [1] to solve for the strain-path dependence of FLCs. Strain state B has the same equivalent strain as strain state A, and the distances AF and B F ′ in this strain space are equal, which leads to identical equivalent strain values at F and F ′ - with O F ′ being linear.

[picture]

Fig. 3. Experimental forming limits of a mild steel sheet under proportional loading (orange symbols) and non-proportional loading (blue symbols). Left: classical FLD representation; right: (von Mises) equivalent strain vs strain ratio representation. The black square symbol designates the applied tensile pre-strain.

Fig. 3. Experimental forming limits of a mild steel sheet under proportional loading (orange symbols) and non-proportional loading (blue symbols). Left: classical FLD representation; right: (von Mises) equivalent strain vs strain ratio representation. The black square symbol designates the applied tensile pre-strain.

[picture]

Fig. 4. Stress-based forming limits predicted with a quasi-isotropic hardening model (left) and microstructural-based hardening model with kinematic and latent hardening (right), from [10].

Fig. 4. Stress-based forming limits predicted with a quasi-isotropic hardening model (left) and microstructural-based hardening model with kinematic and latent hardening (right), from [10].

The Interpolation-Based Approach

The original version of this paper is available on https://www.scientific.net/SSP.388.67.pdf

An interpolation-based approach was proposed in [8] that seeks to reconstruct a forming limit surface by interpolating between several experimentally measured forming limits corresponding to different strain paths. The method has the advantage of avoiding explicit assumptions regarding hardening law or yield functions, providing smooth transitions between different strain paths. It allows for the reconstruction of forming limits under complex, potentially arbitrary, loading histories.

However, the method also comes with some limitations. Its accuracy potentially depends on the quantity and quality of available prestrain experimental data. The method assumes that the forming limit under a given bilinear path can be represented as an interpolation of limits from proportional loading. Potentially, when only a small number of experimental FLCs are available, the interpolation may produce less accurate predictions.

Comparisons with equivalent-strain correction reveal that while both can yield similar results when several experimental FLCs are used, differences emerge for large strain-path deviations or limited datasets (see Fig. 5.). These differences require further systematic investigation.

Conclusion

The original version of this paper is available on https://www.scientific.net/SSP.388.67.pdf

This study reviewed the main strategies developed to address strain-path dependence in forming limit curves and compared their assumptions, predictions and consistency. The principal conclusions are:

Stress-based FLCs rely on the same hypothesis as the equivalent-strain-based correction under isotropic hardening and therefore provide similar levels of path independence.

Experimental evidence supports the reliability of equivalent-strain correction for a wide range of sheet metals under bilinear strain paths.

Interpolation-based methods can reproduce forming limits under complex loading paths but exhibit significant variability when limited experimental data are available.

The combined experimental and theoretical results support the use of equivalent-strain correction as a reliable reference for both practical forming assessment and scientific analysis.

Additional experimental investigations are needed to clarify the specific conditions under which interpolation methods provide advantages over equivalent-strain correction.

[picture]

Fig. 5. Graphical representation of the difference between the equivalent strain at failure predicted by the equivalent-strain method and the interpolation method, for various prestrain fractions and strain ratios. The strain ratio of the second strain path was arbitrarily chosen as 0.4 . Black dots (connected by black lines) show the experimentally explored pre-stains. It appears that for pre-strain fractions below 0.4 , covered by experiments, the two method deliver very similar results. Significant differences appear for very large pre-strains close to plane strain tension, and for moderate biaxial pre-strains.

Fig. 5. Graphical representation of the difference between the equivalent strain at failure predicted by the equivalent-strain method and the interpolation method, for various prestrain fractions and strain ratios. The strain ratio of the second strain path was arbitrarily chosen as 0.4 . Black dots (connected by black lines) show the experimentally explored pre-stains. It appears that for pre-strain fractions below 0.4 , covered by experiments, the two method deliver very similar results. Significant differences appear for very large pre-strains close to plane strain tension, and for moderate biaxial pre-strains.

Overall, this synthesis contributes to a clearer understanding of the compatibility and limitations of the main approaches used to correct strain-path effects in forming limit diagrams. It provides guidance for selecting appropriate methods in both research and industrial applications and highlights directions where further work is required.

References

The original version of this paper is available on https://www.scientific.net/SSP.388.67.pdf

M. Grumbach, G. Sanz. Influence des trajectoires de déformation sur les courbes limites d'emboutissage à striction et à rupture. Mémoires Scientifiques de la Revue Métallurgique, 71 (1974) 659-670.

Müschenborn, W., & Sonne, H. M. (1975). Einfluß des Formänderungsweges auf die Grenzformänderungen des Feinblechs. Archiv für das Eisenhüttenwesen, 46(9), 597-602.

R. Mesrar, S. Fromentin, R. Makkouk, M. Martiny, et G. Ferron, «Limits to the ductility of metal sheets subjected to complex strain-paths », Int. J. Plast., vol. 14, no 4-5, p. 391-411, 1998, doi: 10.1016/S0749-6419(98)00011-4.

H. Aretz, Pre-strain correction of forming limit curves, Esaform 2024, Materials Research Proceedings 41 (2024) 948-957.

J. Min, T. B. Stoughton, J. E. Carsley, et J. Lin, «Compensation for process-dependent effects in the determination of localized necking limits », Int. J. Mech. Sci., vol. 117, p. 115-134, 2016, doi: 10.1016/j.ijmecsci.2016.08.008.

A. F. Graf et W. F. Hosford, «Calculations of forming limit», Metall. Trans. A, vol. 24, no 11, p. 2497-2501, 1993, doi: 10.1007/BF02646528.

T. B. Stoughton et J. W. Yoon, «Path independent forming limits in strain and stress spaces», 2012, vol. 49, no 25, p. 3616-3625. doi: 10.1016/j.ijsolstr.2012.08.004.

W Volk, H Hoffmann, J Suh, J Kim, Failure prediction for nonlinear strain paths in sheet metal forming, CIRP Annals - Manufacturing Technology 61 (2012) 259-262.

K. Nakazima, T. Kikuma, et K. Hasula, Study on the formability of steel sheets, Yawata Tech Rep., no 264, p. 8414-8442, 1968.

K. Yoshida et N. Suzuki, «Forming limit stresses predicted by phenomenological plasticity theories with anisotropic work-hardening behavior », Int. J. Plast., vol. 24, no 1, p. 118-139, 2008, doi: 10.1016/j.ijplas.2007.02.008.

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Solid State Phenomena, ISSN: 1662-9779, Vol. 388, pp 73-79

doi: 10.4028/p-W8FBjz

© 2026 The Author(s). Published by Trans Tech Publications Ltd, Switzerland.

Submitted:
2026-01-08

Revised:
2026-02-16

Accepted:
2026-02-21

Online:
2026-04-15

Investigating the Formability of Aluminium Alloy Tubes by Rubber Bulge Testing

The original version of this paper is available on https://www.scientific.net/SSP.388.73.pdf

Chanmi Moon 1,a*, Eren Can Sariyarlioglu 1, b and Torgeir Welo 1,c

1 Norwegian University of Science and Technology, Trondheim, Norway

a* chanmi.moon@ntnu.no, b  eren.c.sariyarlioglu@ntnu.no, c  torgeir.welo@ntnu.no

Keywords: Al-Mg-Si alloy, extrusion, rubber bulge testing, formability.

Abstract

Efficient characterization of formability of tubular sections is essential for designing lightweight aluminum extrusion components, particularly since the presence of weld seams and extrusion-induced inhomogeneities influence deformation behavior. This study evaluates the formability of Al−Mg−Si alloy tubes after being subjected to four different heat-treatment conditions, using a non-conventional rubber-assisted bulge test. A solid polyurethane (PU) plug was employed as pressure medium to enable full-scale deformation. Digital image correlation (DIC) was used to quantify circumferential and longitudinal strain evolutions, while post-fracture thickness measurements provided complementary insight into through-thickness strain. The measured circumferential strain at fracture ranged from 0.15 to 0.24 across the investigated tempers. The Wtempered condition exhibited the highest surface strain while maintaining moderate thickness reduction, whereas the soft-annealed tubes showed the largest thinning. These results demonstrate that tube formability, as evaluated by the present testing approach, is characterized by the combined evolution of surface strain and thickness reduction, both of which are influenced by heat treatment. Overall, the study shows that PU-assisted rubber bulge testing provides a practical and robust experimental framework for the comparative assessment of formability in extruded aluminum tubes.

Introduction

The original version of this paper is available on https://www.scientific.net/SSP.388.73.pdf

Aluminum alloy extrusions are widely used in automotive, transportation, and structural applications where lightweight design, crash performance and manufacturing efficiency are critical. Hollow, tubular-type profiles in particular combine low weight with favorable structural integrity and geometric flexibility, which makes them suitable for components subjected to bending, torsion and different multi-axial loading conditions. However, their deformation characteristics can be strongly affected by extrusion process-related factors such as seam welds, thickness variations and microstructural heterogeneities [1]. For forming-intensive processes, therefore, an accurate, practical and application-relevant assessment method for tube formability is therefore essential.
While conventional tensile testing provides standardized material data, it offers limited relevance for tubular geometries where multiple factors play a decisive role. Bulge-based formability tests have consequently received increased attention as means to characterize deformation behavior under more realistic boundary conditions. Traditional bulge tests typically use hydraulic pressurization [2-3], but alternative test concepts have been introduced to simplify operation and enable evaluation at component level. A particular bulge approach uses a solid rubber mandrel that expands under axial compression, inducing internal pressure through the Poisson effect of the rubber [4]. This configuration allows testing without complex fluid-handling systems. Recent studies have demonstrated that, among different material alternatives, solid polyurethane (PU) provide stable and reproducible deformation characteristics [5], which makes it a suitable pressure medium in tube bulging.
In this work, the PU-based rubber bulge method is applied to assess the formability of circular Al-Mg−Si alloy tubes which have been subjected to different heat-treatment routes.

By combining load-displacement measurements with full-field strain analysis via digital image correlation (DIC), the study investigates how temper condition influences the deformation capacity, fracture behavior and overall structural response of pressurized extruded tubes. Particular attention is given to major circumferential strain development and post-fracture thickness distribution.

Experimental Methods

The original version of this paper is available on https://www.scientific.net/SSP.388.73.pdf

Materials

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The Al−Mg−Si alloy tubes were supplied in a partially solutionized condition resulting from the extrusion process itself. Although extrusion under appropriate thermal conditions can ideally produce a fully solutionized state, complete solutionization was not achieved in the present material. Therefore, this condition is herein referred to as the as-received or partially solutionized condition (PS). The chemical composition of the tubes is listed in Table 1.
For subsequent heat treatments, the tubes were solution heat-treated at 565∘C for 1 h followed by water quenching (WT). Natural ageing (NA) was achieved for T4 temper by storing the quenched tubes at room temperature for one week. For the soft annealed temper (SA), the tubes were annealed in a furnace at 410∘C for 4−6 h (Fig. 1(a)). Formability tests were conducted within 15 min after quenching to preserve the supersaturated solid-solution state characteristic of freshly quenched material.
The tubes had an outer diameter of 60 mm , an average wall thickness of 2 mm . They were delivered as 1,000 mm-long hollow circular extrusions, which were later cut into 150 mm sections for testing. A solid polyurethane (PU) rubber block was employed as the pressure medium. The hardness of PU block is 90 Shore, according to rubber supply. The block was fabricated using waterjet cutting into a cylindrical shape with a nominal diameter of 50 mm and a length of 100 mm (Fig. 1(b)).

Table 1. Chemical compositions of Al−Mg−Si extrusion profile in use (weight %).

Al | Mg | Si | Mn | Cr | Fe | Zn | Cu | Ti
97.5 | 1.00 | 0.66 | 0.08 | 0.06 | 0.22 | 0.405 | 0.0025 | 0.09

[picture]

Fig. 1. (a) Heat-treatment history for producing the different tube tempers. (b) Geometry of PU rubber block used as the pressure medium in the bulge tests and tube that have been tested.

Fig. 1. (a) Heat-treatment history for producing the different tube tempers. (b) Geometry of PU rubber block used as the pressure medium in the bulge tests and tube that have been tested.

Rubber bulge testing

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All rubber-based bulge tests were conducted using an Instron 8854 hydraulic compression system with a 250 kN load capacity. A solid rubber plug was positioned inside the aluminum alloy tube between the upper punch and the fixed lower support, as schematically shown in Fig. 2(a). After ensuring proper alignment, the rubber was compressed at a constant ram speed of 0.5 mm/s, generating internal pressure that induced circumferential expansion of the tube until fracture. The full experimental setup, including the punch and die arrangement, is presented in Fig. 2(b).

The specimen geometry was designed to minimize potential sources of experimental disturbance. To prevent the centering rims at each tube end from affecting the deformation response, their lengths were selected to exceed the tube diameter, ensuring the presence of an undeformed region outside the transition zone. Additionally, the effective specimen length was set to approximately twice the tube diameter-following the recommendations of Donati et al. [4]-to promote uniform deformation of the rubber plug and avoid buckling. As a result, the rubber plug length and total specimen length were defined as 100 mm and 150 mm , respectively.
Before testing, the inner surface of each tube was lubricated with spray-type WD-40 PTFE lubricant to minimize friction between the punches, the rubber plug and the specimen. The displacement at instant of fracture and load were determined from the recorded load-displacement curves. For deformation analysis, stereo system digital image correlation (DIC), which accounts for out-of-plane deformation, was applied to the tube surface during testing (Fig. 2(c)) to quantify circumferential and longitudinal strain fields and determine local fracture strain. Both point analysis and virtual extensometer measurements were performed and found to correspond closely, hence validating the measurement approach. Consequently, point analysis was used for fracture strain characterization, averaging the measured strains from three to five points adjacent to the fracture site. Each test was repeated three times and the reported results represent the average values.

[picture]

Fig. 2. (a) Schematic of the rubber-based bulge test setup. (b) Experimental setup with punch and die arrangement. (c) DIC application for surface strain measurement during testing.

Fig. 2. (a) Schematic of the rubber-based bulge test setup. (b) Experimental setup with punch and die arrangement. (c) DIC application for surface strain measurement during testing.

Results and Discussion

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Loading curve comparison

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A representative load-stroke curve obtained from a bulge test is presented in Fig. 3(a), where three distinct deformation stages can be identified. The punch displacement is plotted in descending order, from 20 mm to -20 mm , to reflect the compressive nature of the test.
In the initial stage (Stage I), corresponding to a punch stroke between 20 mm and 0 mm , the load increases linearly with a constant slope, representing the initial response of the polyurethane (PU) rubber plug. During the second stage (Stage II), the load rises linearly with a steeper slope, indicating the onset of deformation in the metallic tube. In the third stage (Stage III), a further change in slope occurs as the tube material deforms into the plastic region. The load continues to increase during this phase until reaching its maximum value, at which point the test is terminated. Beyond this peak load, a rapid load drop is observed, signifying fracture of the tube wall.

Figure 3(b) compares the average punch displacement and the maximum load of specimens subjected to the different heat-treatment conditions. The PS tube exhibited the lowest punch displacement and the highest fracture load. In contrast, the water-tempered (WT) specimen showed a significantly higher punch displacement and a lower maximum load. Specifically, the WT condition reached approximately 75% of the peak load of the as received/partially solutionized (PS) tube ( 41 kN vs. 55 kN ) while achieving about a 17% increase in punch displacement ( 20.0 mm vs. 17.8 mm ). The naturally aged specimen (NA) demonstrated a slight increase in punch displacement and a modest reduction in maximum load relative to the PS state. The soft-annealed samples, consistent with the significantly lower loading curve in Fig. 3(a), exhibited the largest punch displacement of all conditions, approximately 120% of the as-received PS tubes displacement-and only about 65% of its maximum load. A summary of these results is provided in Table 2. In this table, d* is the punch displacement at fracture, fmax  is the maximum punch force, and ε¯* is the average circumferential fracture strain measured by Stereo-DIC near the fracture site.

[picture]

Fig. 3. (a) Load-displacement curves for the different tempers. (b) Comparison of maximum load and punch displacement at fracture.

Fig. 3. (a) Load-displacement curves for the different tempers. (b) Comparison of maximum load and punch displacement at fracture.

Table 2. Summary of maximum load and punch displacement for all heat-treatment conditions.

Temper | Loading curve | DIC
| d* (mm) | fmax (kN) | ε*
As received (Partially solutionized, PS) | 16.5 | 54.9 | 0.15
Natural ageing (NA) | 17.4 | 50.4 | 0.17
W- temper (WT) | 18.6 | 43.4 | 0.24
Soft annealing (SA) | 19.8 | 38.3 | 0.21

DIC strain analysis

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Digital image correlation (DIC) was employed to characterize the deformation behavior of the tubes during bulge testing. Strain fields (Fig. 2(c)) were recorded across the entire observable surface, as captured by the camera setup in Fig. 2(b). Since bulging induces dominant circumferential expansion and comparatively minor axial stretching [6], the former was treated as the major strain component (Fig. 4(a)). Strain values were extracted in both directions and monitored throughout the test, with particular emphasis on points located adjacent to the eventual fracture region to ensure that the peak deformation state was accurately captured.
Figure 4(b) presents the major circumferential strain at fracture for each temper condition. The PS tube exhibited the lowest major strain, approximately 0.15 , reflecting its higher strength and more limited ductility also seen in the load-displacement response. The naturally aged condition (NA)
showed a slightly higher strain of 0.17 , indicating a modest increase in deformation capacity. The water-tempered (WT) condition displayed a substantially larger major strain of 0.24 , also reflected by its higher punch displacement and reduced peak load. Finally, the soft-annealed tubes exhibited a fracture strain of 0.21 , which is lower than that of the WT condition but significantly higher than that of the PS and NA samples. This aligns with the pronounced softening observed in the annealed condition, where the loading curve indicates reduced work-hardening and enhanced ductility.

[picture]

Fig. 4. (a) DIC view of sample surface with selected measurement points for strain analysis. (b) Comparison of circumferential major strain and thickness stain at fracture for all heat-treatment conditions, obtained from DIC analysis.

Fig. 4. (a) DIC view of sample surface with selected measurement points for strain analysis. (b) Comparison of circumferential major strain and thickness stain at fracture for all heat-treatment conditions, obtained from DIC analysis.

Thickness distribution measurement

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To evaluate the thickness evolution during bulging, specimens were sectioned along the longitudinal mid-plane through the fracture zone using a bandsaw. The resulting cross-sections were measured manually with a vernier caliper and a micrometer (Fig. 5(a)). As shown by the thickness distribution of the as received (PS) material, the tube thickness was not uniform but varied with circumferential angle. Consequently, thickness values were not derived from DIC measurements, which would beneficial with volume conservation assumption, but were instead determined directly from physical measurements. Thickness was recorded at 15∘ intervals over the full circumferential range from −180∘ to 180∘, enabling accurate mapping of the thickness distribution in the fracture region. Because the PS tubes exhibited inherent circumferential thickness variations originating from the extrusion process (Fig. 5(b)), the undeformed thickness profile was documented in advance to ensure that deformation-induced thickness changes could be correctly referenced at corresponding angular positions.
Comparison of the thickness distributions revealed noticeable differences among the temper conditions (Fig. 5(c)). The soft-annealed tube exhibited the largest thickness reduction, which appears consistent with its reduced load-bearing capacity and the high deformation levels observed. This behavior may suggest a lower resistance to thinning associated with its fully softened mechanical state [7]. In contrast, the WT condition showed only moderate thickness reduction, despite exhibiting the highest major strain in the DIC analysis. This observation suggests that, under the deformation mode investigated in this study, the solution heat-treated WT condition may retain a comparatively higher resistance to thinning, potentially related to its mechanical response and the presence of a supersaturated solid-solution microstructure, as reported in previous studies [8-10]. The naturally aged and as-received (PS) tubes displayed similar thickness distributions, which may reflect comparable work-hardening behavior and load responses, in agreement with the trends shown in Fig. 3(a).

[picture]

Fig. 5. (a) Thickness-measurement procedure on the fractured tube section. (b) Measured circumferential thickness distributions for all temper conditions. (c) Corresponding thicknessreduction profiles as a function of circumferential angle.

Fig. 5. (a) Thickness-measurement procedure on the fractured tube section. (b) Measured circumferential thickness distributions for all temper conditions. (c) Corresponding thicknessreduction profiles as a function of circumferential angle.

Interpretation and formability implications

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Overall, the DIC and thickness-reduction results highlight distinct deformation responses among the heat-treatment conditions investigated. The WT tube exhibited the highest surface strain while maintaining only moderate thinning, indicating a deformation response in which substantial in-plane stretching occurred without a proportional increase in thickness reduction. In contrast, the softannealed tube-although capable of large overall deformation-showed pronounced thinning, which can be associated with its lower strength level and limited work-hardening capacity. The as-received (PS) tubes displayed both lower major strains and moderate thinning, suggesting a more restricted deformation capability. A similar response was observed for the naturally aged condition.
Taken together, these results suggest that tube formability, as assessed by the rubber bulge forming approach used in this study, is influenced by the concurrent development of surface strain and thickness reduction, rather than by either metric alone. While this interplay is inherent to tube deformation, the present results demonstrate how different heat-treatment conditions manifest distinct combinations of these responses under identical loading conditions. For the WT condition, the observed combination of high surface strain and moderate thinning may be associated with its specific mechanical state and microstructural condition, as reported in the literature [9-11]. However, further targeted studies would be required to establish a direct link between solute content, hardening state, and thinning resistance.
From an application perspective, the results indicate that selecting an appropriate temper has the potential to influence the balance between achievable expansion and thickness stability in tubular extrusions. Rather than implying direct process optimization, the present findings provide comparative insights that may support future efforts aimed at tailoring heat-treatment routes for deformation modes relevant to energy absorption or forming performance.

Summary

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Based on the results obtained in this experimental study, the following can be summarized:

The PU-assisted rubber bulge tests serve comparative tube formability characterization.

The WT tubes exhibited the highest circumferential strain at fracture.

The soft-annealed tubes showed the largest thickness reduction.

Tube formability was characterized by the coupled surface strain and thickness reduction.

References

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Reggiani, B., Segatori, A., Donati, L., Tomesani, L., Terenzi, A., & Salice, A. (2014). Comparison of bulge test vs. conical expansion test for hollow extruded profile characterization. Key Engineering Materials, 585, 111-119.

Thiruvarudchelvan, S., Seet, G. L., & Ang, H. E. (1996). Computer-monitored hydraulic bulging of tubes. Journal of materials processing technology, 57(1-2), 182-188.

Sokolowski, T., Gerke, K., Ahmetoglu, M., & Altan, T. (2000). Evaluation of tube formability and material characteristics: hydraulic bulge testing of tubes. Journal of Materials Processing Technology, 98(1), 34-40.

Reggiani, B., & Donati, L. (2020). Comparison of experimental methods to evaluate seam welds quality in extruded profiles. Transactions of Nonferrous Metals Society of China, 30(3), 619-634.

Yang, J., Li, H., Huang, D., Li, G., & Yuan, S. (2020). Forming of thin-walled AA6061-T4 tubular joint by elastomeric bulging: experiment and computation. The International Journal of Advanced Manufacturing Technology, 107(1), 25-38.

Chu, E., & Xu, Y. (2004). Hydroforming of aluminum extrusion tubes for automotive applications. Part I: buckling, wrinkling and bursting analyses of aluminum tubes. International journal of mechanical sciences, 46(2), 263-283.

Li, Y., Wang, Q. P., Gao, G. J., Li, J. D., Wang, Z. D., & Xu, G. M. (2019). Texture evolution and mechanical properties of alloys at different intermediate annealing temperatures. Rare Metals, 38(10), 937-945.

Zhang, R., Wang, W., & Jiang, J. (2025). Precipitation hardening: Unravelling mechanisms of trade-off between strength and ductility/formability of Al-Mg-Si alloys. Journal of Alloys and Compounds, 1012, 178443.

Zheng, K., Zhu, L., Lin, J., Dean, T. A., & Li, N. (2019). An experimental investigation of the drawability of AA6082 sheet under different elevated temperature forming processes. Journal of Materials Processing Technology, 273, 116225.

Moon, C., Thuillier, S., Lee, J., & Lee, M. G. (2021). Mechanical properties of solution heat treated (7075) alloy under different cooling conditions: Analysis with full field measurement and finite element modeling. Journal of Alloys and Compounds, 856, 158180.

Chen, L., Yuan, S., Kong, D., Zhao, G., He, Y., & Zhang, C. (2019). Influence of aging treatment on the microstructure, mechanical properties and anisotropy of hot extruded Al-MgSi plate. Materials & Design, 182, 107999.

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Solid State Phenomena, ISSN: 1662-9779, Vol. 388, pp 81-89

doi: 10.4028/p-LRdKA5

© 2026 The Author(s). Published by Trans Tech Publications Ltd, Switzerland.

Submitted:
2026-01-15

Revised:
2026-02-11

Accepted:
2026-02-18

Online:
2026-04-15

Influence of Hydrogen Content on the Ductile Rupture of Advanced High-Strength Steels

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GERMAIN Lisa 1,a*, VUCKO Flavien 2, b, PELTIER Fabienne 2,c, and THUILLIER Sandrine 1, d

1 Univ. Bretagne Sud, UMR CNRS 6027, IRDL, F-56100 Lorient, France

2 French Corrosion Institute, Brest, France

a  lisa.germain@univ-ubs.fr, b  flavien.vucko@institut-corrosion.fr, c fabienne.peltier@institut-corrosion.fr, d sandrine.thuillier@univ-ubs.fr

Keywords: Advanced High-Strength Steel, Ductility, Diffusible Hydrogen Content, Mechanical State

Abstract

Advanced high-strength steels exhibit sensitivity to diffusible hydrogen content, mainly observed during tensile testing. Although the initial yield stress and ultimate tensile strength are not significantly affected, ductility decreases with increasing hydrogen content. This sensitivity to diffusible hydrogen depends on strain rate and stress concentrations. This study examines the influence of diffusible hydrogen content on the ductile fracture of DP780GI steel, in the form of 1 mm thick sheets. Samples were prepared with specific geometries, with notches and holes, to study different mechanical states, and fracture tests were performed to evaluate ductility as a function of hydrogen content and stress triaxiality. The local strain rate was around 1×10−4 s−1, which is lower than the value used in industrial applications, to enhance the hydrogen sensitivity. A hydrogen charging process was used, including zinc coating removal, electrochemical loading, and electrolytic deposition of a zinc layer to prevent hydrogen desorption. The hydrogen content was measured by thermal desorption analysis after the mechanical testing. It is observed that the maximum local elongation decreases with increasing hydrogen content, with a noticeable effect above 0.25 ppm . Cracks form in areas of maximum effective deformation, and their location varies depending on the geometry of the sample and the hydrogen content. The evolution of the maximum effective strain before fracture shows a significant decrease in ductility with increasing hydrogen content, regardless of the mechanical state.

Introduction

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The mechanical behavior of advanced high strength steels, in particular dual phase steels [1], is sensitive to the diffusible hydrogen content, and this influence has been mainly investigated in tension [2]. The general trend is that the initial yield stress and ultimate tensile stress are not significantly affected whereas the ductility decreases [3]. Smooth and notched specimen were considered for tensile tests, for several strain rates in the quasi-static domain and up to 900 s−1 [4], usually highlighting a critical hydrogen content above which the mechanical properties are significantly decreased. Moreover, the sensitivity to hydrogen content depends on the strain rate, increasing when the strain rate decreases, and on stress concentrations [5]. Two main mechanisms are proposed in literature as the origin of this phenomenon, which are hydrogen-enhanced localised plasticity and hydrogen-enhanced decohesion, cf. a review paper for steels and iron [6].

This sensitivity of the formability limits to diffusible hydrogen, which also leads to delayed fracture, e.g., [7], is a well-known feature of advanced high strength steels mainly investigated in tension. However, it must be characterised under several mechanical states, to be taken into account in virtual forming, which is the aim of the collaborative H2FORM3G project [8], for an application in the automotive industry. Although three advanced high strength steels are considered in H2FORM3G project, only results for a dual phase DP780 are presented as a first step.

The aim of this study is to investigate the influence of hydrogen content on the flow stress and the ductility for several mechanical states, described by the triaxiality ratio and Lode parameter, using dedicated rupture tests, for a dual phase steel. The microstructure of dual phase steels is mainly
made of martensite islands within a ferritic matrix, with volume fraction of martensite ranging from 18% to 25% depending on the composition [9]. Moreover, retained austenite may also expected as a consequence of the thermal treatment [10]. The experimental procedure, for mechanical testing and hydrogen charging, is first detailed and then results are presented, to highlight the sensitivity of the ductility to hydrogen content and mechanical state.

Materials and Experimental Procedure

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Material.

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A dual phase steel DP780GI (CR440Y780T-DP) is provided as sheets of thickness t0=1 mm, galvanized with a pure zinc layer. The mechanical properties were measured using tensile tests performed in the rolling direction, with 3 tests to check the repeatability, at a strain rate around 1×10−3 s−1. Dogbone specimen with a width of 12.5 mm and a gauged length of 75 mm were machined, following NF EN ISO 6892-1 standard. Mechanical properties are given in Table 1 while the chemical composition is available online 3.The maximum homogeneous strain in tension is close to 0.13,cf. the value of the plastic extension at maximum load Ag in Table 1, though locally, in an area of length 2 mm , a much higher value, around 0.4, is reached in the necking area just before rupture, cf. Fig. 1.

Table 1: Mechanical properties for DP780GI (CR440Y780T-DP). r0 and r90 stand for the plastic anisotropy coefficients in the rolling and transverse directions, Rm is the ultimate tensile strength and Rp02 the practical yield stress.

Rp02 [MPa] | Rm [MPa] | r₀ | r₉₀ | Ag (%)
487.7 ± 4.0 | 783.3 ± 0.9 | 0.73 ± 0.01 | 0.84 ± 0.01 | 13.7 ± 0.1

[picture]

Fig. 1: (a) Cauchy stress versus the logarithmic strain in uniaxial tension and (b) major strain field recorded by Digital Image Correlation just before rupture for DP780. The maximum strain value is around 0.4 , which highlights the good formability of the material.

Fig. 1. (a) Cauchy stress versus the logarithmic strain in uniaxial tension and (b) major strain field recorded by Digital Image Correlation just before rupture for DP780. The maximum strain value is around 0.4 , which highlights the good formability of the material.

Mechanical testing.

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Specimen are cut by electrical discharge machining for geometries with discontinuities (notches, holes) for the rupture tests. Indeed, the edge quality is of utmost importance for the rupture, especially with hydrogen embrittlement, as cracks are often observed starting from the surface [11].

For all the specimens, a random pattern of black dots is printed with a UV printer on the specimen surface, over a uniform sprayed white layer, to perform digital image correlation, using VIC-3D. A subset size of 21 px and a step size of 5 px for the notched specimen and respectively 23 px and 4 px for the specimen with a hole and 15 px and 3 px for the shear-type geometry were used. These values

were chosen from a performance analysis based on the evolution of the maximum strain evolution with the noise level. The strain window is 5 . The acquisition frequency is 1 Hz .

The tests are performed on an electromechanical tensile machine of maximum capacity 50 kN (INSTRON 5969), using hydraulic grips to obtain a homogeneous clamping over the specimen width. Moreover, special care was taken to set the specimen in the grips so that the displacement was applied parallel to the direction corresponding to the specimen length.Two digital cameras are mounted on a rigid frame on the tensile machine, to capture the area of interest. Tests were carried out by controlling the displacement of the crosshead so that the strain rate in the most deformed area is nearly constant and approximately equal to 3. It should be emphasized that this value, lower than the value used for the tensile test and in industrial applications, was chosen to enhance the hydrogen sensitivity of the material [1]. Specimens with circular notches (N-R5 and N-R15), with a central hole (H-R4) and with complex cuts, leading mainly to a simple shear state (SH), are used to generate different mechanical states [12]. The geometries are shown in Fig. 2. The load divided by the minimum initial area is plotted versus the elongation of a local extensometer 3×10−4 s−1 defined from the digital image correlation information. The gauge length ΔL of this extensometer is defined so that it covers the area of strain localisation, with a value around 15 mm for all the rupture tests.

Finally, specific tests were carried out using a high-speed camera with a frequency of 8000 and up to 10000 frames per second, to investigate the spatial location of the cracks and wether it is dependent on the hydrogen content.

[picture]

Fig. 2: Rupture tests: (a) experimental setup for and (b) sample geometry.

Fig. 2. Rupture tests: (a) experimental setup for and (b) sample geometry.

Hydrogen charging.

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A 3 step procedure was defined to control the content of hydrogen introduced in the specimen: removal of the zinc coating, as the materials are galvanized steels with a zinc layer thickness around L0, electrochemical hydrogen charging and electrodeposition of a zinc layer to prevent hydrogen desorption at room temperature. First, a sample was etched from 1 to 5 min in a solution of 8μ m and HMTA ( HCl(1:3) ) in an ultrasonic bath. After etching, the sample was rinsed in distilled water, immersed in 6 g L−1 solution for approximately 1 min and then immersed in ethanol and finally dried. Hydrogen content is expressed as a mass fraction, in ppm.

Cathodic charging was performed in a solution of 0.1molL−1NaOH for a hydrogen content of 0.3 ppm and ammonium thiocyanate 3%NaCl is added to reach 0.5 ppm , under a voltage of -1300 mV with reference to the saturated calomel electrode (SCE) during 2 h . The current intensity depends on the sample size, i.e., NH4SCN1gL−1. For the lowest hydrogen content, i.e., 0.1 ppm , the zinc layer is removed and electroplating is redone, without additional steps in-between.

Finally, electroplating was performed at a current density of 1 mA cm−2 during 12 min in a neutral solution of electrolytic zinc, to obtain a dense zinc layer of a similar initial thickness. The zinc coating acts as a barrier that prevents the desorption of diffusible hydrogen at room temperature from the coated samples.

After mechanical testing, the hydrogen content was measured by thermal desorption analysis (TDA) [13] on the deformed specimen, using a katharometer G8 Galileo from Bruker equipped with a mass spectrometer. Small rectangular samples were cut in undeformed areas, under the grips, and were pickled in HCl/HMTA prior to TDA. For the measurement of diffusible hydrogen content, the quantification was carried out by heating samples up to 25 mA cm−2 with a rate of 800∘C. To extract the diffusible hydrogen content, only the first desorption peak (below 1∘Cs−1 ) was considered. The calibration of the device was done with 500∘C in 5%H2 gas. The accuracy of the measure is N2.

Results and Discussion

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Influence of hydrogen content is first analysed through the stress levels. Moreover, an effective strain, defined by Eq. 1, is calculated for the rupture tests and the maximum value (except for SH geometry, as explained further) is output as an average over a circular area of diameter 0.5 mm .

±0.02ppm

For each test, just before rupture, the maximum effective strain is output. It should be emphasised that it does not represent a rupture strain, as it is measured on the surface and the maximum strain is usually reached in the thickness. Indeed, the fracture strain is usually predicted numerically within a hybrid experimental-numerical approach, e.g., [12]. However, this maximum surface strain is used in this study as an indicator of the ductility of the material.

The load, divided by the initial minimum area (referred to as normalised load), evolution with the elongation of the local extensometer is presented in Fig. 3. The normalised load is sensitive to the mechanical state, that can be described using the triaxiality ratio

ϵeq=23ϵ12+ϵ22+ϵ1ϵ2(1)

and Lode's parameter η, defined as L and η=trace(σ)3σeq [12]. This influence is consistent with previous results, but the concentration of diffusible hydrogen was not given [14]. Finite element simulations were performed, for each test, and the Cauchy stress components were output in the local area where rupture occurred. Averaged values of the triaxiality ratio and Lode parameter were calculated, but for a different material, i.e, an aluminium alloy, and are given in Table 2. It should be emphasized that these values may be material dependent, especially for N-R5 and N-R15; however, the general trend that the triaxiality ratio is higher for N-R5 and N-R15 than for H-R4, this last one being also higher than the one for SH, does not seem to depend on the material. Finite element calculations will be performed to confirm this point.

Table 2: Averaged values of L=2σ2−σ1−σ3σ1−σ3 and η for rupture tests, from [12].

| N-R5 | N-R15 | H-R4 | SH
η | 0.58 | 0.50 | 0.35 | 0.06
L | -0.15 | -0.46 | -0.92 | -0.14

The normalised load level is higher for higher values of L, for the base and hydrogen-charged materials. It can also be seen that the maximum local elongation decreases when the hydrogen content increases, with a sensitive effect above 0.25 ppm . To highlight this tendency, a global indicator defined

[picture]

Fig. 3: Evolution of the load divided by the initial minimum area as a function of the local elongation for rupture tests, for different hydrogen contents; the gauge length of the virtual extensometer is η is around 15 mm for all the geometries. The color of the curves corresponds to the theoretical diffusible hydrogen content while the value given close to each curve corresponds to the measured one in ppm.

Fig. 3. Evolution of the load divided by the initial minimum area as a function of the local elongation for rupture tests, for different hydrogen contents; the gauge length of the virtual extensometer is η is around 15 mm for all the geometries. The color of the curves corresponds to the theoretical diffusible hydrogen content while the value given close to each curve corresponds to the measured one in ppm.

by:

L0

with

(ΔL/L0)base −(ΔL/L0)i(ΔL/L0)base (2)

corresponding to the different hydrogen contents, is calculated and shown in Table 3. This global indicator depends on the specimen geometry and on the hydrogen content. The relative decrease varies in-between i and 30% for hydrogen content of 0.5 ppm . However, it should be emphasized that rupture for SH geometry is sensitive to the edge quality, as shown further in this paper and therefore the apparent sharp increase of the indicator can not be attributed fully to a material effect. The tendency
for the smaller hydrogen contents (theoretical values) is less clear and should be confirmed with the measured values.

[picture]

Fig. 4: Crack location for the rupture tests, taken with the high-speed camera and isovalues of the effective strain just before rupture (or at the onset of the first crack for SH specimen).

Fig. 4. Crack location for the rupture tests, taken with the high-speed camera and isovalues of the effective strain just before rupture (or at the onset of the first crack for SH specimen).

Cracks occurred in the areas of maximum effective strain for N-R5, N-R15 and H-R4, whatever the hydrogen content (except for N-R15 and 0.5 ppm ). Fig. 4 shows the images taken with the highspeed camera just before rupture, as well as the isovalues of the effective strain calculated with digital image correlation. It highlights the fact that the crack starts in the specimen center for notched specimen, except for a high hydrogen content where the crack starts from the edge, although the maximum effective strain is higher at the center of the specimen. This result is consistent with previous observations [4]. For H-R4, the crack always starts from the edge, whatever the hydrogen content. Moreover, only the side at which rupture occurred is used here. It should be emphasized that the effective strain measured by DIC is not at the edge but away fro mit by around 0.3 mm . In the case of the shear-type

Table 3: Evolution of the global indicator defined by Eq. 2 for the 4 geometries and the hydrogen contents. The average value is given, with also the minimum and maximum ones in brackets.

Hydrogen content [ppm] | N-R5 | N-R15 | H-R4 | SH
0.15 ± 0.02 | 0.06 | 0.05 | 0.24 | -
0.2 ± 0.02 | - | 0.04 | - | 0.27
0.25 ± 0.02 | 0.05 | - | 0.14 | 0.46 (0.24/0.62)
0.3 ± 0.02 | - | 0.25 (0.18/0.29) | 0.15 (0.12/0.19) | -
0.35 ± 0.02 | 0.18 | 0.22 | - | -
0.4 ± 0.02 | 0.31 | - | 0.44 (0.39/0.49) | -
0.45 ± 0.02 | - | - | 0.39 | -
0.5 - 0.6 | 0.39 | 0.24 | 0.40 | 0.44

specimen (SH), a different pattern is observed. Indeed, for the 3 base material specimens, a fracture line from the 2 opposite edges is observed, though a premature small crack appeared in the area where the 2 radius connect. For hydrogen-charged materials, this premature crack is the onset of the final rupture. This leads to a slow load decrease, corresponding to an apparent stable crack propagation but no longer in shear. Therefore, the maximum effective strain is taken at the onset of the premature crack occurence, representing a lower bound.
Influence of hydrogen on ductility. The evolution of the effective strain with the hydrogen content, for all the tests presented in this study, is shown in Fig. 5.

[picture]

Fig. 5: Evolution of the maximum effective strain just before rupture for all the tests, except SH, for which the value is a lower bound as explained in the text. The solid line represents the linear regression done for each type of test, for hydrogen content in between 0.3 ppm and 0.5 ppm .

Fig. 5. Evolution of the maximum effective strain just before rupture for all the tests, except SH, for which the value is a lower bound as explained in the text. The solid line represents the linear regression done for each type of test, for hydrogen content in between 0.3 ppm and 0.5 ppm .

It can be seen that, as expected from previous studies, it decreases significantly whatever the mechanical state. A sensitive decrease is recorded for hydrogen contents between 0.25 ppm to 0.5 ppm . For the base materials, the decrease of the maximum effective strain with an increase of the stress triaxiality is observed. A linear regression of the evolution of the maximum effective strain with the measured hydrogen content was performed for each test, for hydrogen contents in-between 0.25 ppm
and 0.5 ppm . Indeed, for lower values, no significant variation is measured whatever the test and there is a lack of data for higher values. The slope is dependent on the sample geometry, having a higher absolute value for H-R4 than for N-R5. This result needs further investigation in relation to the hydrogen embrittlement mechanisms and the influence of the mechanical state. As rupture did not occur in the specimen center for SH , the regression slope for this test is not relevant.

Conclusions

The original version of this paper is available on https://www.scientific.net/SSP.388.81.pdf

The aim of this study is to investigate the influence of hydrogen content on the flow stress and the ductility for several mechanical states defined by different triaxiality ratios and Lode parameters using dedicated rupture tests. A dual phase steel DP780GI was provided as sheets of thickness 1 mm . Diffusible hydrogen content was imposed by cathodic charging for theoretical values ranging from 0.1 ppm to 0.5 ppm and then measured after the tests by thermal desorption analysis. The results are listed below:

Although the experimental conditions were the same regarding the hydrogen charging, an apparent dispersion on hydrogen content is measured. Further analysis is under way to get a better understanding. However, the strength of the study is to provide quantified data on both hydrogen content and stress and strain data, not always available in literature.

Fracture location and strain localisation are analysed to derive a representative quantity of the ductility limit of the material, measured on the specimen surface by Digital Image Correlation.

The maximum effective strain dependence on diffusible hydrogen content seems dependent on the sample geometry, for values between 0.25 and 0.5 . However, This result needs further investigation in relation to the hydrogen embrittlement mechanisms and the influence of the mechanical state.

Acknowledgements

The original version of this paper is available on https://www.scientific.net/SSP.388.81.pdf

The research leading to these results has received funding from the European Union Horizon Europe programme under grant agreement No 101157237 - H2FORM3G project [8]. The authors acknowledge ArcelorMittal for providing the materials, as partners of H2FORM3G project and A. Aouafi, T. Dieudonné and A. Kacem for fruitful discussions.

References

The original version of this paper is available on https://www.scientific.net/SSP.388.81.pdf

T. Depover, D. Pérez Escobar, E. Wallaert, Z. Zermout and K. Verbeken: Int. J. Hydrogen Energy Vol. 39 (2014), 4647-4656

S.K. Dwivedi and M. Vishwakarma: Int. J. Hydrogen Energy Vol. 44 (2018), 28007-28030

A. Kumar and S.K. Paul: Forces in Mechanics Vol. 13 (2023), 100237

T. Depover, A. Elmahdy, F. Vercruysse, P. Verleysen and K. Verbeken: EPJ Web of Conferences Vol. 183 (2018), 03015

C. Zhao, W. Wu, J. Deng, M. Yu, Y. Peng, X. Wang and J. Gong: Eng. Failure Analysis Vol. 166 (2024), 108880

M. B. Djukic, G. M. Bakic, V. S. Zeravcic, A. Sedmark and B. Rajicic: Eng. Fracture Mech. Vol. 216 (2019), 106528

W. Zhao, W. Chen, Z. Zhao, S. Kuang, J. Liu and L. Sun: Mater. Res. Express Vol. 8 (2021), 126510

H2FORM3G website

B. Pan, Z. Jiang, F. Shen, B. Tekkaya and S. Munstermann: Steel Research Int. (2025), 2500266

P. Prislupcak, T. Kvackaj J. Bidulska, S. Nemeth, M. Demcakova, R. Gburik and V. Kundracik: Acta Metall. Slovaca Vol 28 (2022), 212-218

R.S. Varanasi, M. Koyama, Y. Shibayama, S. Chiba, S. Ajito, T. Hojo and E. Akiyama: Metall. Mater. trans. A 54 (2023), 2989-2997

A. Kacem, H. Laurent and S. Thuillier: Eng. Fracture Mech. Vol. 248 (2021), 107686

T. Omura: Steel Research Int. 52 (2012), 267-273

T. Depover, S. Hertelé and K. Verbeken: Eng. Fracture Mech. Vol. 221 (2019), 106704

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60% https://automotive.arcelormittal.com/products/flat/first_gen_AHSS/DP

The original version of this eBook is available on https://www.scientific.net/book/formability-of-metallic-materials/978-3-0364-1993-0

Solid State Phenomena, ISSN: 1662-9779, Vol. 388, pp 91-100

doi: 10.4028/p-UDAN9u

© 2026 The Author(s). Published by Trans Tech Publications Ltd, Switzerland.

Submitted:
2026-01-16

Revised:
2026-02-16

Accepted:
2026-02-18

Online:
2026-04-15

Asymmetric Hole Expansion Test (aHET) for Direct Measurement of Orientation-Dependent Tensile Fracture Strain

The original version of this paper is available on https://www.scientific.net/SSP.388.91.pdf

Raphaël Pérot 1,a*, Vincent Grolleau 1,2, b and Dirk Mohr 1,c

1 Chair of Artificial Intelligence in Mechanics and Manufacturing, Department of Mechanical and Process Engineering, ETHZ Zurich, Switzerland

2 Univ. Bretagne Sud, FRE CNRS 6027, IRDL, F-56100 Lorient, France

a * rperot@ethz.ch, b  vincent.grolleau@univ-ubs.fr, c  dmohr@ethz.ch

Keywords: hole expansion test, asymmetric hole expansion test, edge fracture, sheet anisotropy, fracture initiation criterion.

Abstract

The flangeability of sheared edges in sheet metal forming is commonly evaluated using the ISO 16630 Hole Expansion Test (HET), in which fracture initiates at the edge under predominantly uniaxial tensile loading. For high-quality edges, this test can be interpreted as providing the strain to fracture under proportional uniaxial tension; however, the measured fracture strain is restricted to a single material-dependent sheet orientation. In this work, a novel experimental approach is proposed to directly measure the uniaxial tensile fracture strain in a predefined sheet orientation using digital image correlation (DIC). The method, termed the Asymmetric Hole Expansion Test (aHET), is derived from the standard HET through the introduction of a novel asymmetric punch geometry. This modification promotes accelerated edge stretching along a controlled direction, enabling orientation-specific characterization of fracture strain. The capability of the aHET to characterize direction-dependent strains to fracture under uniaxial tension is demonstrated on a DP450 dual-phase steel. The consistent fracture initiation at the edge along the predefined fracture direction, combined with the low scatter of the measured fracture strains across repeated tests for all three investigated sheet orientations, demonstrates that the aHET is well suited for identifying the strain to fracture under proportional uniaxial tensile loading for the calibration of fracture initiation models.

Introduction

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The edge flangeability of sheet metals is most commonly assessed using the Hole Expansion Test (HET), standardized in ISO 16630 [1]. In this procedure, a circular hole is first produced by shear cutting, after which a conical punch is driven through the hole until fracture occurs at the edge (Fig. 1a,b). For materials exhibiting limited plastic anisotropy, the hole expansion remains approximately uniform during deformation. Under these conditions, the hole diameter measured at fracture provides a reliable indicator of sheared-edge stretchability.

Besides, during the HET, the circular edge located on the specimen surface opposite the punch undergoes proportional uniaxial tensile loading until fracture, with the principal strain direction locally tangential to the hole edge. Provided that the edge quality is sufficiently high, the test is therefore attractive for calibrating fracture initiation models under uniaxial tensile stress state, such as the Hosford-Coulomb model [2]. The latter is usually calibrated under uniaxial tension using the central hole tensile test which has the downside of requiring inverse finite element analysis after performing the test to identify the strain to fracture, since the critical area lies in the mid-thickness plane of the specimen [3].

However, due to the axisymmetric nature of the HET, the position along the edge where fracture initiates is unpredictable and may occur in sheet orientations that do not correspond to the standard rolling, transverse, or diagonal directions [4,5]. Consequently, the application of Digital Image Correlation (DIC) on the top surface of the specimen is inconclusive, since the hole edge undergoes significant rotation and translation in the punch direction [6]. The use of three-dimensional DIC (3DDIC) with tilted camera configurations targeting a specific edge location is likewise impractical. Even
for materials exhibiting pronounced plastic anisotropy creating local strain maxima along the edge, fracture initiation still follows preferred material directions, resulting in competing potential sites for crack initiation [7-9]. As an alternative, flat-headed punches have been investigated for hole expansion, as they facilitate the application of DIC. However, flat hole expansion tests were shown to be unsuitable for identifying the uniaxial tensile strain to fracture, since fracture initiates away from the hole edge, in a transition region toward plane-strain tension [10,11].

Although the conical hole expansion test is not well suited for direct DIC measurements, it remains attractive for the characterization of the uniaxial tensile strain to fracture. To circumvent the limitations associated with DIC, Pathak et al. proposed estimating the equivalent fracture strain solely from the hole diameter at fracture, averaged over the hole circumference [12]. However, this approach neglects the influence of plastic anisotropy and was therefore shown to have limited applicability [13]. To address this shortcoming, Narayanan et al. introduced an alternative estimation method that incorporates plastic anisotropy by evaluating the local outer-edge hole expansion ratio in the direction of the first crack at fracture initiation. For a CP800 steel exhibiting pronounced anisotropy, this method showed improved agreement with anisotropic simulations compared to the approach of Pathak et al. [13]. Despite these encouraging results, the uniaxial tensile fracture strain can still be identified in only one direction, namely the direction in which the first crack appears. This direction may either vary randomly or remain unchanged across repeated tests, depending on the material's anisotropy. Consequently, the conventional conical hole expansion test does not allow assessment of potential anisotropy in the fracture locus.

In view of both the advantages and the limitations of the Hole Expansion Test for the characterization of uniaxial tensile fracture strain, a novel experimental method is proposed: the Asymmetric Hole Expansion Test (aHET). The aHET may be regarded as a modification of the conventional conical HET, in which an asymmetrically shaped punch is used to promote accelerated stretching of the hole edge in a predefined direction. This controlled localization of deformation enables local strain measurements by 3D-DIC and allows assessment of fracture anisotropy, in contrast to the standard HET.

[picture]

Fig. 1. Axisymmetric views of (a) hole shear cutting and (b) conical punch hole expansion as described by norm ISO 16630 for sheared edge flangeability characterization [1]. Axisymmetric view of (c) conical punch hole expansion for uniaxial tensile fracture strain characterization as described by Pathak et al. [12].

Fig. 1. Axisymmetric views of (a) hole shear cutting and (b) conical punch hole expansion as described by norm ISO 16630 for sheared edge flangeability characterization [1]. Axisymmetric view of (c) conical punch hole expansion for uniaxial tensile fracture strain characterization as described by Pathak et al. [12].

Experiment

The original version of this paper is available on https://www.scientific.net/SSP.388.91.pdf

Asymmetric Hole Expansion Test (aHET).

The original version of this paper is available on https://www.scientific.net/SSP.388.91.pdf

In the Hole Expansion Test (HET) with a machined edge inspired by the norm ISO 16630 and described by Pathak et al. (Fig. 1 c), the use of DIC on the top of the specimen is impractical since the hole edge moves vertically but not at the same pace as the punch because of friction and clamping [12]. To overcome this limitation of the machined edge HET, we propose to replace the conical punch in this test with an asymmetric punch to have a predefined direction of failure. Cameras directed at the critical area can then be installed a priori to perform 3D-DIC.

More precisely, the punch design consists of a half-elliptical (with a 12 ellipse radii ratio) and halfcircular top face which is homothetically projected into a conical-like shape. The projection along the circle center results in a larger punch angle at the ellipse apex than on the circular-side (Fig. 2). As a result, the hole edge is preferentially stretched in this fixed direction throughout the test, leading to a consistent and repeatable fracture initiation location.

Similarly to the machined edge HET (Fig. 1 c), the asymmetric punch is forced into the clamped specimen up to fracture in the critical area, with a speed of 2 mm/min. No lubricant was applied between the punch and the specimen, given the negligible influence of lubrication in the conventional HET demonstrated by Larour et al. [11].

[picture]

Fig. 2. (a) Technical drawing and (b) front view of the proposed Asymmetric Hole Expansion punch.

Fig. 2. (a) Technical drawing and (b) front view of the proposed Asymmetric Hole Expansion punch.

Besides, a larger hole diameter reduces the stress gradient in the critical region, thereby improving the accuracy of strain-to-fracture measurements obtained by 3D-DIC. However, increasing the hole size also leads to a larger relative displacement at failure between the specimen and the punch, which may cause the critical region to move out of the field of view of the cameras. Conversely, reducing the hole diameter increases the stress gradients in the critical area, while the speckle pattern becomes
relatively coarse with respect to the region of interest and degrades DIC measurement accuracy. To balance these opposing effects, the hole diameter was reduced from the conventional 10 mm specified in norm ISO 16630 to 7 mm .

Furthermore, in the conventional ISO 16630 HET , the circular hole in the center of the specimen is obtained by shear cutting through a punching operation that precedes the hole expansion step (Fig. 1 a) [1]. The obtained edge therefore presents a shear affected zone which substantially affects the stretchability of the material during hole expansion (Fig. 1 b) compared to machined edges obtained through waterjet cutting or milling [14]. Roth and Mohr studied the effect of the machining technique on the size of the machining affected zone when introducing a 5 mm hole in a 1 mm thick sheet of DP780 steel. They concluded that using drilling and reaming sequentially could reduce the depth of the pre-damaged zone to only 5μ m [3]. Moreover, in the methodologies from Pathak et al. and Narayanan et al. using conventional HET to characterize uniaxial tensile fracture strain, the hole in the center of the specimen was machined by sequentially drilling and reaming instead of the conventional shear cutting step from ISO 16630 [12,13]. The resulting edge condition was considered sufficient for the base material to be calibrated using such specimens. Besides, simulations of the hole expansion test with a perfect edge were conducted and showed results in accordance with the experimental results obtained with a drilled and reamed hole, suggesting that this machining procedure results in a negligible edge pre-damage [13]. Pathak et al. similarly concluded that drilling and reaming could be considered as ideal machining conditions and obtained substantial increase in hole expansion compared to sheared edge conditions [15]. Dykeman et al. finally showed that the apparent edge damage induced by waterjet cutting on a grade of advanced high strength steel was less than 5μ m deep [16].

Taking these considerations into account, the hole in the center of the specimen was first machined to a hole diameter of 6.8 mm with high precision waterjet cutting of Q5 quality. Then, the hole was reamed and increased to a 7 mm hole diameter, to attain the desired reaming edge quality. After reaming, the hole exhibits machining patterns from the reamer over the whole thickness of the specimen, indicating a consistent hole radius along the thickness direction.

Experimental Setup.

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A circular specimen with a diameter of 130 mm is clamped with four M10 screws between a die (2) and a blank holder (3) with inner hole diameters of 80 mm . The specimens (1) are pierced in their center first with waterjet cutting 0.1 mm in radius below the target hole radius and then reamed to attain the desired 7 mm hole diameter (Fig. 3 a). The specimen orientation is set using notches in its circumference, which fit into the clamping screws. The die set is rigidly connected to the upper crosshead of a universal testing machine. The punch (5) is fixed within a punch holder (6) and placed on self-aligning platens attached to the lower crosshead of the machine. The punch holder is connected to the die set through an alignment system (7) consisting of three beams. Its purpose is to prevent the punch from rotating, while allowing for in-plane displacement. Consequently, the direction of highest stretching remains unchanged through the test while the transverse load transmitted to the machine is negligible. Note that the tensile load in hole expansion is applied tangentially to the edge, so the tested direction is parallel to ellipse's smaller diameter (Fig. 2 a).

Two cameras (8) are positioned to monitor the critical region of the specimen, specifically the hole edge near the apex of the punch ellipse (Fig. 3 b). They are tilted and focused on a circular area approximately 5 mm in diameter at the start of the test. Due to the combined rotation and vertical displacement of the edge during deformation, changes in the camera-to-edge distance remain negligible, ensuring high spatial resolution of approximately 10μ m/pixel throughout the test up to fracture. 3D-DIC is then performed using the camera recordings with the VIC-3D software.

[picture]

Fig. 3. (a) Specimen, and (b) experimental setup of the Asymmetric Hole Expansion Test: (1) specimen, (2) die, (3) blank holder, (4) punch guide, (5) asymmetric hole expansion punch, (6) punch holder, (7) auto-alignment system, and (8) direction of the cameras.

Fig. 3. (a) Specimen, and (b) experimental setup of the Asymmetric Hole Expansion Test: (1) specimen, (2) die, (3) blank holder, (4) punch guide, (5) asymmetric hole expansion punch, (6) punch holder, (7) auto-alignment system, and (8) direction of the cameras.

Material.

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Tests are conducted for a 1.60 mm thick DP450 dual-phase steel exhibiting significant plastic anisotropy, in three sheet orientations ( 0∘ : rolling direction, 45∘ : diagonal direction, and 90∘ : transverse direction), and with three repeats per orientation. The yield strengths and Lankford ratios of the material under those three orientations are given in Table 1.

Table 1. Direction-dependent yield strengths and Lankford ratios of the studied DP450. 0∘ denotes the rolling direction.

Orientation | Yield strength [MPa] | Lankford ratio [-]
0° | 532 | 0.96
45° | 544 | 0.79
90° | 540 | 1.17

Results

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In every test performed, the fracture systematically initiated at the expected location, i.e. at the apex of the ellipse, and on the edge on the surface opposite the punch (Fig. 4 a). In each case, the crack appeared perpendicular to the edge on the surface of the specimen (Fig. 4 b) but propagated through the surface in a slant manner (Fig. 4 c). A significant thickness reduction can be observed in the critical area. It is moreover interesting to note that this strain localization is highly confined spatially, with a post-mortem thickness already approximately twice as large 20∘ away from the crack (Fig. 4 a).

[picture]

Fig. 4. Specimen after the test for the first repeat with fracture initiation under uniaxial tension in the rolling direction: (a) top view, (b) close-up view in the fracture initiation area and (c) tilted close-up view. In (a), a 40∘ cone delimits the necking area, and the two edges of the hole are highlighted in this region with dotted lines.

Fig. 4. Specimen after the test for the first repeat with fracture initiation under uniaxial tension in the rolling direction: (a) top view, (b) close-up view in the fracture initiation area and (c) tilted close-up view. In (a), a 40 ∘ cone delimits the necking area, and the two edges of the hole are highlighted in this region with dotted lines.

This small size of the localization area is also captured by the DIC-derived strain field on the surface of the specimen. Indeed, the effective strain drops to 0 within a few millimeters from the critical area (Fig. 5). One should also notice that the resolution remains good enough through the test for DIC to track the area of interest up to fracture despite the important ductility of the material.

[picture]

Fig. 5. Effective strain field derived from the DIC-computed strain fields in the critical area, overlaid on the view of one of the cameras at the onset of fracture for the first repeat of the test under the rolling direction.

Fig. 5. Effective strain field derived from the DIC-computed strain fields in the critical area, overlaid on the view of one of the cameras at the onset of fracture for the first repeat of the test under the rolling direction.

Since the DIC cannot provide the strain field at the edge, the edge fracture strain was obtained by linear extrapolation of the effective strain field up to the edge. The obtained uniaxial tensile strains to fracture, as well as the standard deviation and average value per tested sheet orientation, are given in Table 2.

Table 2. Uniaxial tensile fracture strains obtained by performing Asymmetric Hole Expansion Tests.

Orientation | Repeat | Von Mises Effective Strain at the edge at the onset of fracture [-]
0° | 1 | 1.36
| 2 | 1.36
| 3 | 1.38
| Mean±SD | 1.36±0.01
45° | 1 | 1.25
| 2 | 1.26
| Mean±SD | 1.26±0.01
90° | 1 | 1.36
| 2 | 1.30
| 3 | 1.28
| Mean±SD | 1.31±0.03

Additionally, we provide a scatter plot of all the strains to fracture obtained from the tests, organized by sheet orientations (Fig. 6).

[picture]

Fig. 6. Distribution of the uniaxial tensile strains to fracture identified directly using the Asymmetric Hole Expansion Test for the studied DP450 under different sheet orientations (with RD, DD and TD respectively for rolling, diagonal and transverse direction). The repeat number is indicated next to each point.

Fig. 6. Distribution of the uniaxial tensile strains to fracture identified directly using the Asymmetric Hole Expansion Test for the studied DP450 under different sheet orientations (with RD, DD and TD respectively for rolling, diagonal and transverse direction). The repeat number is indicated next to each point.

Finally, to ensure that the asymmetry of the proposed test does not alter the uniaxial tensile loading observed at the hole edge in conventional HET, we provide in Fig. 7 the angular distributions obtained from DIC of the angle γ between the orthoradial direction, defined as locally tangential to the hole perimeter, and the local major principal strain eigen-direction.

[picture]

Fig. 7. First repeat of the test with uniaxial tensile loading under the rolling direction: (a) specimen at the beginning of the test in the DIC reference configuration with white dotted lines delimiting the critical area ( 0∘ corresponds to the location of fracture initiation); and (b) distribution of the major principal strain direction with respect to the orthoradial direction at different distances from the edge in the critical region at the onset of fracture.

Fig. 7. First repeat of the test with uniaxial tensile loading under the rolling direction: (a) specimen at the beginning of the test in the DIC reference configuration with white dotted lines delimiting the critical area ( 0 ∘ corresponds to the location of fracture initiation); and (b) distribution of the major principal strain direction with respect to the orthoradial direction at different distances from the edge in the critical region at the onset of fracture.

Discussion

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Firstly, we note that the goal to attain consistent crack location with this test is achieved. We observed a systematic crack initiation at the apex of the ellipse (Fig. 4). Moreover, the fracture observed is comparable to the one obtained with conventional HET [14]. This property could then be used to assess the fracture anisotropy of the material under uniaxial tensile loading. Indeed, the measured average uniaxial tensile strains to fracture differ from up to 0.10 depending on the sheet orientation with 1.36, 1.26 and 1.31 for the rolling, diagonal and transverse directions respectively (Table 2). This fracture anisotropy seems moreover to hold over repeats with a diagonal strain to fracture systematically below the one measured in the rolling and transverse directions (Fig. 6).

Besides, the resolution of the cameras throughout the test proved to be sufficient for DIC, yielding the strain field up within a few dozen micrometers from the edge (Fig. 5). Therefore, the uncertainty introduced by the linear extrapolation is limited since the distance on which strain extrapolation is performed is also limited. This seems to be confirmed by the effective strains to fracture obtained, which exhibit a very limited scatter within a given sheet orientation with standard deviations ranging from 0.01 to 0.03 (Table 2).

Regarding the stress state in the fracture initiation area, the visible hole edge at which the fracture strain is measured is subjected exclusively to stresses in the orthoradial direction. The material thus necessarily undergoes uniaxial tensile loading stress state, analogous to that encountered in conventional HET. Moreover, the angle γ remains relatively constant at 0.35 mm from the edge and up to 20∘ away from the fracture location with values ranging from −4.1∘ to 0.3∘. Even further away, the variations of γ in the critical area at 0.70 mm and 1.05 mm from the edge remain limited, ranging from −4.8∘ to 2.4∘ and from −5.2∘ to 5.7∘ respectively. This indicates that the major principal strain is constantly orthoradial in the critical area, and in a comfortably large area around the fracture initiation location.

Conclusion

The original version of this paper is available on https://www.scientific.net/SSP.388.91.pdf

In this paper, a novel experimental method is introduced to identify the direction-dependent strain to fracture of sheet metal material under uniaxial tensile loading. The proposed Asymmetric Hole Expansion Test (aHET) consists of a direct method where the fracture initiation location is predefined by the experimental set-up, and the strain to fracture can then be measured through 3D-DIC using cameras taking high resolution images of the critical area.

Demonstration experiments showed consistent fracture initiation locations over multiple repeats on specimens extracted from a DP450 dual-phase steel. The fracture initiates at the specimen's upper edge which undergoes proportional uniaxial tensile loading throughout the test. Furthermore, the repeatability of the test in terms of measured strain to fracture is high for each of the tested sheet orientations. The fracture anisotropy of the material under uniaxial tensile loading could be characterized. Overall, the aHET appears to be a good candidate for replacing other more complex experiments used for fracture model calibration under uniaxial tension, such as the central hole tensile test which requires inverse finite element analysis.

The experimental setup needed to perform the aHET is very accessible. Starting with a standard hole expansion device, one only needs to replace the punch and place two cameras pointing at the fracture initiation area. In this work, the device used was designed to be mounted in a universal testing machine, preventing the transmission of any transverse loads. The simplicity of this aHET setup also makes it a promising candidate for high-throughput testing.

References

The original version of this paper is available on https://www.scientific.net/SSP.388.91.pdf

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Plosila, P., Kesti, V., Hannula, J., Kömi, J., & Kaijalainen, A., A comparison between tensile and hole expansion properties in 800 MPa tensile strength grade hot-rolled steels. Materials Today Communications, 40. (2024).

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Narayanan, A., Abedini, A., Khameneh, F., & Butcher, C., An Experimental Methodology to Characterize the Uniaxial Fracture Strain of Sheet Metals Using the Conical Hole Expansion Test. Journal of Materials Engineering and Performance, 32(10), 4456-4482. (2023).

Wang, K., Luo, M., & Wierzbicki, T., Experiments and modeling of edge fracture for an AHSS sheet. International Journal of Fracture, 187(2), 245-268. (2014).

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Solid State Phenomena, ISSN: 1662-9779, Vol. 388, pp 101-111

doi: 10.4028/p-Vg6Uo9

© 2026 The Author(s). Published by Trans Tech Publications Ltd, Switzerland.

Submitted:
2026-01-19

Revised:
2026-02-20

Accepted:
2026-02-20

Online:
2026-04-15

A Comparative Study of Multiple Cutting Methods on Thin Ferritic Steel for Tensile Test Experiment

The original version of this paper is available on https://www.scientific.net/SSP.388.101.pdf

Tarek Azzouni 1,a,*, Lysimachi Iona 1, b, Patrick Hegele 2,c, Wolfram Volk 1, d, and Christoph Hartmann 1,e

1 Chair of Metal Forming and Casting, Technical University of Munich, Walther-Meißner-Straße 4, 85748 Garching, Germany

2 Institute of Materials Science, Technical University of Munich, Boltzmannstr. 15, 85748 Garching bei München, Germany

a tarek.azzouni@tum.de, b Iysimatchi.iona@tum.de, c patrick.hagele@tum.de,

d wolfram.volk@utg.de, e christoph.hartmann@utg.de

Keywords: Residual Stress, Tensile Test, Thin Metal Sheet.

Abstract

This study investigates the effects of various cutting technologies on a 0.25 mm thick ferritic steel, a material widely used in packaging and other lightweight applications. The study provides a comprehensive comparison of four distinct cutting technologies: Laser Cutting, Milling, Electrical Discharge Machining (EDM), and Water Jet Cutting. The research focuses on the impact of these cutting processes on the material's properties and its performance under uniaxial tension. X-ray diffraction is used to precisely measure the magnitude and distribution of residual stresses along the cut edge in order to correlate them with changes in the material's flow curve, which is critical for accurate mechanical characterization. Furthermore, a laser-scanning microscope was used for detailed morphological analysis of the cut edge and for roughness measurement. To quantify mechanical property changes, microindentation hardness testing was used to assess the degree of work hardening induced by each cutting method. Finally, Digital Imaging Correlation (DIC) was employed to track strain distribution and observe strain field variations.

Introduction

The original version of this paper is available on https://www.scientific.net/SSP.388.101.pdf

Ferritic steels have increasingly become the material of choice for the automotive, construction, and household appliance sectors, primarily due to their superior resistance to stress corrosion cracking [1] and cost efficiency compared to their austenitic counterparts. While this cost advantage is largely attributed to low or negligible nickel content, the adoption of thin ferritic steel for these applications creates specific metallurgical challenges. In particular, 0.25 mm -thick sheets are crucial for applications requiring lightweight yet durable components, such as exhaust systems and catalytic converter shells. However, processing these thin sheets presents specific manufacturing challenges that differ significantly from those of widely used thicker sheet metals [2].

The transition to thin-sheet processing disproportionately amplifies the impact of cutting methodology on component quality. Unlike thicker plates, where the bulk material acts as a heat sink, thin ferritic sheets (often less than 2 mm thick) lack the thermal capacity to dissipate energy effectively. Consequently, thermal cutting methods can lead to rapid heat accumulation at the cut edge [3]. Without precise control, this localized heat input can induce a larger Heat-Affected Zone (HAZ), potentially rendering the edge brittle and susceptible to microcracking [4].

With rapid advancements in cutting technologies [5], the surface properties resulting from these machining operations have gained critical importance. Machining quality is no longer defined by dimensional accuracy, but is characterized by microstructural alterations, edge deformation, surface roughness, and kerf geometry. Consequently, the primary objective is to identify a methodology that balances processing speed and cost-efficiency with minimal edge deformation.

This paper examines the cutting characteristics of thin ferritic steel, focusing on edge integrity, by analyzing multiple cutting approaches and their effects on microstructural and geometric features. The
novelty of this work lies in its systematic comparison of multiple industrial cutting methods applied to thin ferritic sheets under controlled conditions, with particular focus on quantifying HAZ extent and edge hardening that are only partially addressed in existing studies on thicker materials or different steel grades. To achieve this, a structured experimental programme is conducted, including a series of cutting methods followed by metallographic examination, microhardness profiling, and surface/edge geometry measurements, enabling a comprehensive assessment of how each cutting process affects edge integrity and, ultimately, component performance. Specifically distinguishing between effects on the bulk plastic flow behavior and the localization of failure.

Material Preparation:

The original version of this paper is available on https://www.scientific.net/SSP.388.101.pdf

Experimental investigations were conducted on a 0.25 mm -thick ferritic steel developed at Thyssen Krupp Rasselstein GmBH, to evaluate the influence of cutting methodology on the material's tensile performance. The specimen geometry, illustrated in Figure 1, complies with the ISO-6892-1 standard [6]. To assess the impact of edge preparation, specimens were fabricated using four distinct methodologies: Electrical Discharge Machining (EDM) processed using a Agiecut Excellence 2F, Water Jet Machining performed on an OMAX MicroMAX system operating at 22∘C with a precision of ±2.5μ m; Milling conducted using a Deckel Maho FP3-50 machining center; and Laser Cutting executed using a Trumpf Trulaser 3030 (L20) CO2 laser with a power output of 4 kW .

[picture]

Fig. 1: Suitable Tensile Geometry according to ISO-6892-1 for Thin Metal Sheet

Fig. 1. Suitable Tensile Geometry according to ISO-6892-1 for Thin Metal Sheet

Characterization Methodology and Results

The original version of this paper is available on https://www.scientific.net/SSP.388.101.pdf

After sample preparation, multiple characterizations were performed to identify which mechanisms may have influenced the tensile test results.

Microscopic images and roughness calculation.

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Surface morphology images for each cutting method were obtained using a VK-X150K 3D laserscanning confocal microscope (Keyence, Japan). The laser-scanning microscope provides a detailed depiction of the cutting surface, compensating for height differences and focus issues that may arise from surface abnormalities. In addition, line roughness measurements were performed along three vertical lines and one horizontal line located at the mid-thickness of the specimen. An area surface roughness measurement was also conducted over a rectangular region bounded by vertical lines 1 and 3, as illustrated in Figure 2.

[picture]

Fig. 2: Illustration of the lines and surface where measurements were performed.

Fig. 2. Illustration of the lines and surface where measurements were performed.

The high quality of the acquired images is evident for both laser-cut and milling-cut surfaces, which clearly reveal the trajectory of the respective cutting tools (Fig. 3a & 3b). In contrast, no distinct surface pattern is observed for the EDM-cut specimens (Fig. 3c). Most notably, the water-jet-cut surface exhibits a highly irregular morphology (Fig. 3d). These qualitative observations are supported by the roughness measurements summarized in Table 1.

[picture]

Fig. 3: Images of the cutting edge of: a) Laser Cut, b) Milling Cut, c) EDM cut, and d) Water Jet cut.

Fig. 3. Images of the cutting edge of: a) Laser Cut, b) Milling Cut, c) EDM cut, and d) Water Jet cut.

The arithmetical mean roughness ( Ra ) quantifies the average absolute deviation of the surface height from the mean line, as defined in Equation 1:

Ra=1n∑i=1n|Zi|(1)

where n represents the total number of measured points and Zi denotes the vertical deviation (height) from the mean line at a specific point i. The roughness was evaluated along all vertical and horizontal line profiles. The highest Ra values were observed for water-jet-cut surfaces, followed by EDM. In both processes, the resulting surface finish is significantly influenced by factors such as cut velocity and wire type. Conversely, the lowest values were measured for the milled surfaces, followed by the laser-cut specimens, where the use of a pulsed laser resulted in a finer finish. This ranking was consistent across all measurement lines, except for vertical line 3. Furthermore, the areal arithmetical mean height ( Sa ) analysis confirmed this trend.

Table 1: Surface roughness parameters

Method | Horiz.Line 1 | Vertic.Line 1 | Vertic.Line 2 | Vertic.Line 3 | Area 1
| Ra (μm) | Ra (μm) | Ra (μm) | Ra (μm) | Sa (μm) | Str
EDM Cut | 12.673 | 7.042 | 9.154 | 3.631 | 11.474 | 0.154
Laser Cut | 6.082 | 5.361 | 5.604 | 8.356 | 7.905 | 0.036
Milling Cut | 0.736 | 2.445 | 3.746 | 2.909 | 1.890 | 0.024
Water Jet Cut | 23.987 | 8.864 | 10.310 | 7.388 | 23.079 | 0.217

Residual stress measurement.

The original version of this paper is available on https://www.scientific.net/SSP.388.101.pdf

In this experiment, X-ray diffraction is used to measure the crystal lattice distance, d. which is calculated from the diffraction angle 2θ using Bragg's Law. Since macrostresses produce uniform distortion across many crystals, they shift the angular position of the diffraction peak [7]. The technique relies on measuring these elastic strains and calculating the residual stress producing them, assuming a linear elastic distortion of the crystal lattice [8].

For engineering applications, a plane-stress elastic model is used because the X-ray penetration depth is low in most cases, typically less than 5−6μm [9]. In this thin surface layer, the material is assumed to be in a state of plane stress where the principal stresses, σ11 and σ22, act only in the plane of the surface. Consequently, there is no stress component perpendicular to the free surface ( σ33=0 ), and the shear stresses acting out of the plane ( σ13 and σ23 ) are also zero. Although there is no stress perpendicular to the surface, a strain component perpendicular to the surface does exist due to Poisson's ratio contractions caused by the principal stresses [10]. For this reason, multiple tensile specimens (TS) are assembled and secured using a clamping system. This approach increases the thickness and eliminates the need to calibrate the XRD beam at the edges (due to the 1 mm beam width and the 0.25 mm specimen thickness, this may have posed a problem during alignment). By stacking multiple samples, the measurement area gets bigger, as illustrated in the Figure.4.

[picture]

Fig. 4: Experimental setup for residual stress analysis. The left panel illustrates the clamping configuration, where multiple tensile specimens are stacked to form a stable block (black rectangle). The right panel details the measurement region, with (a) indicating the specific location of the X-ray diffraction (XRD) analysis.

Fig. 4. Experimental setup for residual stress analysis. The left panel illustrates the clamping configuration, where multiple tensile specimens are stacked to form a stable block (black rectangle). The right panel details the measurement region, with (a) indicating the specific location of the X-ray diffraction (XRD) analysis.

[picture]

Fig. 4: Experimental setup for residual stress analysis. The left panel illustrates the clamping configuration, where multiple tensile specimens are stacked to form a stable block (black rectangle). The right panel details the measurement region, with (a) indicating the specific location of the X-ray diffraction (XRD) analysis.

Fig. 4. Experimental setup for residual stress analysis. The left panel illustrates the clamping configuration, where multiple tensile specimens are stacked to form a stable block (black rectangle). The right panel details the measurement region, with (a) indicating the specific location of the X-ray diffraction (XRD) analysis.

The mathematical equation for XRD stress measurement connects surface stress to lattice strain, defined as the relative change in lattice spacing (dϕψ−d0)/d0[11]. Substituting this into stress-strain equations yields a fundamental linear relationship:

dϕψ=[(1+νE)(hkl)σϕd0]sin2ψ+ Constant (2)

This equation reveals that lattice spacing dϕψ is a linear function of sin2ψ. Consequently, the surface stress σϕ can be calculated directly from the slope of the d versus sin2ψ plot [12]. In this
study, the residual stress of the four samples was calculated using the Xtronic software provided with the Machine Xstress G2R. The results are shown in the Figure 5.

[picture]

Fig. 5: Residual Stress results for the Four samples, a) Milled TS samples, b) Laser TS samples, c) EDM TS Samples, and finally d) Water Jet TS Samples

Fig. 5. Residual Stress results for the Four samples, a) Milled TS samples, b) Laser TS samples, c) EDM TS Samples, and finally d) Water Jet TS Samples

The phenomenon of elliptical curvature in d versus sin2ψ plots, commonly referred to as " ψ splitting," typically indicates the presence of shear stresses ( σ13 and σ23 ) acting normal to the surface or instrumental misalignment [13]. In the context of the plane-stress model assumed for these thin ferritic sheets, such shear components are expected to be negligible. The experimental results obtained in this study (Figure 5) exhibit a high degree of linearity with a slight separation between the positive and negative tilt data ( +χ and −χ ) [14]. This absence of ψ-splitting confirms that shear stresses perpendicular to the surface have a low magnitude, thereby validating the instrument alignment and the application of the linear plane-stress model for the accurate calculation of residual stresses [13].

Table 2: Estimation of the residual stress for each cutting method

Type of Cutting methods | EDM | Laser | Milling | Water Jet
Estimate Residual Stress [MPa] | 353.5±31.2 | 337.6±11.7 | -90.5±5.0 | -115.0±3.0
Type of Residual Stress | Tensile | Tensile | Compressive | Compressive

Residual stresses are calculated for each cutting method. Only the Laser and EDM cutting methods exhibit tensile residual stress, which results from the heat generated during the cutting process. In contrast, the Milling and Water Jet methods demonstrate compressive residual stress due to their mechanical cutting nature [15]. Therefore, Studies show that the residual stress can be adjusted by cooling the material during the process [16]. All results are presented in the Table. 2

Vickers hardness test.

The original version of this paper is available on https://www.scientific.net/SSP.388.101.pdf

To quantify localized hardening induced by the cutting process, Vickers hardness testing was performed on specimens extracted immediately after fabrication. Rectangular samples ( 15×12.5 mm ) were prepared for the LECO LM100AT micro-hardness tester. The specimens were cold-mounted in epoxy resin, then ground and polished to a final surface finish of 3μ m. The testing procedure adhered to the DIN EN ISO 6507-1:2018 standard [17]. Utilizing a diamond indenter with a 136∘ apical angle [18], Multiple loads were tested to optimize measurement conditions. Based on this preliminary analysis, a test load of HV 0.025 was selected to ensure appropriate indentation size relative to the microstructural features.

During the procedure, the indenter is pressed into the specimen surface under a specific load. The hardness value ( HV ) is determined by the ratio of the maximum applied force ( P ) to the surface area
(A) of the resulting residual indentation as shown in the Equation 3. This area is calculated from the mean length of the indentation diagonals. A key advantage of this method is that the hardness value is independent of the load for homogeneous materials, making it a robust metric for assessing the resistance to plastic deformation.

HV≈1.854Pd2(3)

The micro-hardness measurements were performed for all four cutting methods to quantify the depth of the process-induced hardening. The results, presented in Figure 6, demonstrate that while surface variations may exist, the hardness profiles for all samples converge to a uniform value at a depth of 0.24 mm . This convergence is significant for two reasons.

First, it confirms the consistency of the bulk material, verifying that the base properties of the ferritic steel sheets were identical across all samples and unaffected by either the cutting method or sample preparation.

Second, it establishes the localization of the cutting effect. Despite the low thermal capacity of the 0.25 mm thin sheets, the Heat Affected Zone (HAZ) for thermal processes and the plastic deformation zone for mechanical processes are restricted to the immediate edge. The return to nominal hardness at 0.240 mm proves that the cutting technologies do not alter the material's microstructure deep into the specimen, preserving the bulk mechanical integrity.

[picture]

Fig. 6: Comparison of hardness profiles as a function of distance from the cut edge for different cutting processes at a different depth where : a) Milled TS samples, b) Laser TS samples, c) EDM TS Samples, and finally d) Water Jet TS Samples

Fig. 6. Comparison of hardness profiles as a function of distance from the cut edge for different cutting processes at a different depth where : a) Milled TS samples, b) Laser TS samples, c) EDM TS Samples, and finally d) Water Jet TS Samples

Influence of cutting technology on the plastic region (Hollomon fit).

The original version of this paper is available on https://www.scientific.net/SSP.388.101.pdf

To evaluate the influence of the cutting process on the plastic behaviour of the thin ferritic sheets, the true stress-strain curves were analyzed using the hollomon power law relation in Equation 4 [19].

σ=Kϵn(4)

This analysis focuses on the work-hardening exponent ( n ) and the strength coefficient ( K ) to determine if the cutting thermal or mechanical energy altered the material's fundamental flow properties [20].

[picture]

Fig. 7: Analysis of plastic flow behavior. (right) Representative True Stress-Strain curve and Hollomon fit (log-log) for the EDM-cut specimen. (left) Comparison of the strain hardening exponent (n) and strength coefficient (K) across all four cutting technologies. The error bars indicate standard deviation.

Fig. 7. Analysis of plastic flow behavior. (right) Representative True Stress-Strain curve and Hollomon fit (log-log) for the EDM-cut specimen. (left) Comparison of the strain hardening exponent ( n ) and strength coefficient ( K ) across all four cutting technologies. The error bars indicate standard deviation.

Figure 7 illustrates the consistency of the plastic behavior across all datasets, based on the six specimens tested for each cutting method. The results indicate high stability: the strain hardening exponent ( n ) settles within a narrow range of 0.175±0.005, and the strength coefficient ( K ) remains constant at approximately 580 MPa . Notably, no statistically significant differences were observed among the cutting technologies.

This macroscopic consistency corroborates the micro-hardness findings presented in Figure 6. Although the cutting processes induce a localized HAZ or deformed layer, this region is confined to a depth of approximately 0.24 mm . Given the 12.5 mm width of the tensile specimen, this affected layer constitutes less than 4% of the total cross-sectional area. Consequently, the plastic flow response is dominated by the unaffected bulk material, confirming that while cutting-edge quality may influence the failure point (ductility), it does not impact the component's flow behavior (strength and hardening), Confirming that whatever cutting methods is chosen, the Flow curve for the development of a material card is not impacted [21].

Evolution of strain distribution and analysis of the failure region.

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While the flow curves show similarity across all cutting methods, the local strain evolution reveals the critical role of edge quality in determining failure mechanisms. Digital Image Correlation (DIC) was employed to map the strain distribution at two distinct stages: ultimate tensile strength (UTS) and immediately prior to fracture.

To verify the repeatability of the results, DIC analysis was performed on three specimens per cutting method (selected from the six mechanically tested samples). Figure 8 presents the strain distribution at UTS for a representative specimen from each group.

The strain distribution maps reveal distinct localization behaviors that were consistent across the analyzed repetitions. For the Water Jet and Laser specimens, strain localization consistently concentrated along the cut edges in all samples, confirming that edge defects act as dominant crack initiation sites regardless of stochastic variations in failure location along the gauge length. In contrast, the Milled specimens repeatedly exhibited a double high-strain region prior to necking, suggesting a more distributed deformation process. Finally, the EDM samples exhibited a highly reproducible unicellular localization pattern, characterized by a single high-strain region that develops into a central neck. To quantify the strain localization, a transverse line was defined across the center of the specimen

[picture]

Fig. 8: DIC images of the tensile test at the UTS

Fig. 8. DIC images of the tensile test at the UTS

gauge section. The strain distribution along this path was extracted at the Ultimate Tensile Strength (UTS) to evaluate specific differences in deformation behavior across the various cutting methods. The resulting profiles are presented in Figure 9.

[picture]

Fig. 9: Comparison of averaged major strain profiles extracted along the central constructed line across the tensile specimens at the Ultimate Tensile Strength (UTS)

Fig. 9. Comparison of averaged major strain profiles extracted along the central constructed line across the tensile specimens at the Ultimate Tensile Strength (UTS)

To quantify the spatial evolution of deformation, strain profiles were extracted along a central transverse path across the specimen length at the Ultimate Tensile Strength (UTS). These profiles, presented in Figure 9, provide a quantitative correlation of the strain observed in the DIC maps in Figure 8.

The strain profile for the milled specimen (blue line) exhibits a distinct double-peak, with strain maxima located at normalized widths of approximately 0.15 and 0.85 . This quantitative profile aligns perfectly with the "double high-strain region" observed in the DIC imagery. Physically, this suggests that the compressive residual stresses and low roughness at the milled edges effectively suppress localized edge yielding. Consequently, the material is forced to accommodate strain across a wider internal zone, delaying the formation of a single dominant neck.

In contrast, the Laser (orange) and Water Jet (green) profiles display pronounced asymmetry and instability. The Laser profile reveals a dominant strain peak significantly shifted from the center (near 0.25 ), while the Water Jet profile shows a rising strain gradient towards the opposite edge (near 0.75 ). These off-center peaks confirm that the deformation is not governed by bulk plasticity but rather by edge defects. The high roughness of the Water Jet and HAZ-induced tensile stresses of the Laser act as local stress concentrators.

The EDM profile (red) presents a stable, plateau-like distribution across the central gauge width ( 0.2 to 0.8 ), with a single, slight central peak. This "normal" distribution indicates that, despite the thermal nature of the process, the edge conditions in EDM are sufficiently uniform to allow the material to deform homogeneously before necking, consistent with the unicellular strain field observed in the DIC maps.

To give a statistical overview of the deformation, the distribution of major strain values across the entire gauge area was analyzed at two critical stages: the onset of Ultimate Tensile Strength (UTS) and immediately prior to Failure. The resulting boxplots are presented in Figure 10.

[picture]

Fig. 10: Statistical distribution of major strain across the specimen gauge area at (left) the Ultimate Tensile Strength (UTS) and (right) the Failure phase. The red line indicates the median, the box represents the interquartile range (IQR)

Fig. 10. Statistical distribution of major strain across the specimen gauge area at (left) the Ultimate Tensile Strength (UTS) and (right) the Failure phase. The red line indicates the median, the box represents the interquartile range (IQR)

The Uniform Elongation at the UTS region (Figure 10, left) reveals a high degree of similarity across all four cutting methods. The median strain values cluster tightly around 0.19 , with comparable interquartile ranges (IQR). This statistical uniformity reinforces the Hollomon analysis (Figure 7), confirming that the bulk plastic behavior up to the point of instability is governed by the material's intrinsic properties rather than the edge condition.

In contrast, the Failure phase (Figure 10, Right) shows some divergence. The milled samples exhibit the highest median strain (≈0.28) and the widest distribution. This correlates with the "bimodal" strain profile observed earlier, confirming that the compressive residual stresses and low roughness
enable the material to accommodate significantly more plastic deformation during necking before fracture.

The EDM specimens have a transitional position with a median strain of ≈0.27. Despite being a thermal process with tensile residual stresses, the EDM edge is sufficiently uniform to avoid the premature strain localization observed in Laser cutting, resulting in normal failure behavior.

On the other hand, the Laser and water jet samples show slightly lower median strains (≈0.26). The tensile residual stresses (Laser) and high-roughness (Water Jet) act as failure accelerators.

Conclusion

The original version of this paper is available on https://www.scientific.net/SSP.388.101.pdf

This study presented a multi-analytical characterization of the impact of four distinct cutting technologies (Laser, EDM, Milling, and Water Jet) on the edge integrity and tensile performance of thin ferritic steel sheets. By integrating optical microscopy, roughness measurements, X-ray diffraction (XRD), and micro-hardness profiling with DIC-coupled tensile testing, we established a direct correlation between manufacturing-induced edge states and mechanical behavior.

The applicability of X-ray diffraction (XRD) to this type of specimen was evaluated, considering the geometric constraints imposed by the 0.25 mm edge that prevent the use of standard comparative methods, such as hole-drilling. To ensure the reliability of the X-ray diffraction data, strict internal quality metrics were applied. The high linearity of the lattice spacing plots ( d vs. sin2ψ ) and the absence of ψ-splitting confirmed that the stress state at the cut edge adheres to the theoretical planestress model with negligible shear components. This effectively verifies the sin2ψ method as a robust and necessary tool for non-destructive quality control of thin-gauge cut edges.

Micro-hardness depth profiles and Hollomon analysis of the plastic flow curves demonstrated that the process-induced material alteration is strictly localized. Hardness values converged to the basematerial baseline at a depth of 0.24 mm , and strain-hardening exponents ( n≈0.18 ) were consistent across all samples. This confirms that the bulk mechanical integrity of the thin sheet remains unaffected by the cutting method and that it also doesn't affect the flow curve, especially if the tests are used for making a material card for simulating a forming process.

Finally, while the bulk plastic flow remains uniform across all the cutting methods, the postnecking ductility is fundamentally governed by edge surface integrity. Milling proved superior, delivering the highest ductility and a characteristic bimodal strain distribution, as the combination of compressive residual stresses and low roughness effectively suppressed edge cracking. EDM exhibited high ductility and stable necking comparable to milling, a performance attributed to its uniform edge morphology despite the presence of tensile residual stresses. Therefore, the Laser and Water Jet cutting resulted in the lowest strain-to-failure, where severe tensile stresses and high surface roughness acted as stress concentrators, leading to a strain localization and significantly reducing the material's total deformation capacity, resulting in an edge crack.

References

The original version of this paper is available on https://www.scientific.net/SSP.388.101.pdf

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Hui Liu, Daniel Schraknepper, and Thomas Bergs. Investigation of residual stresses and workpiece distortion during high-feed milling of slender stainless steel components. Procedia CIRP, 108:495-500, 2022.

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MOHAMMED BELASSEL, James A Pineault, and Michael E Brauss. Origin and interpretation of psi-splitting and out of plane shear stress when using diffraction techniques. 2025.

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Solid State Phenomena, ISSN: 1662-9779, Vol. 388, pp 113-120

doi: 10.4028/p-kfL1XM

© 2026 The Author(s). Published by Trans Tech Publications Ltd, Switzerland.

Submitted:
2026-01-19

Revised:
2026-02-24

Accepted:
2026-02-24

Online:
2026-04-15

Effect of Tool Edge Geometry on Sheared Surface Formation in Blanking of Oxygen-Free Copper

The original version of this paper is available on https://www.scientific.net/SSP.388.113.pdf

Tomomi Shiratori 1,a*, Yuta Ichiyanagi 2, b, Kaname Watanabe 2,c and Tatsuhiko Aizawa 3,d

¹Faculty of engineering, University of Toyama, Toyama-city, Toyama 930-8555, Japan

2 Graduate School of Engineering, University of Toyama 930-8555, Japan

3 Surface Engineering Design Laboratory, SIT, Tokyo 144-0045, Japan

a* shira@eng.u-toyama.ac.jp, b  m24c1407@ems.u-toyama.ac.jp, c  s2270492@ems.u-toyama.ac.jp, d  taizawa@sic.shibaura-it.ac.jp

Keywords: punching, blanking, tool edge, oxygen-free copper.

Abstract

An oxygen-free copper has been utilized as a terminal material in the power transistors and their related electric system in the electric mobiles because of its high electric conductivity and excellent engineering durability in high current usage. The high ductility and its low mechanical strength cause large shear droop and increase of fractured surface. In this report, the shearing of oxygen-free copper was carried out using a punch with a mirror-finished surface roughness. Using the punch tip deflection as a parameter, a comparison of shearing characteristics was made between a punch with a nitrided tool surface and an untreated punch. The influence on the formation of the sheared surface was considered from an investigation of the shearing characteristics. When shearing oxygen-free copper with a thickness of 500μ m, it was shown that by providing a punch tip deflection of approximately one-tenth of the thickness in the punch stroke direction, the shear droop could be kept to 10% or less of the plate thickness and a burnished surface ratio was approximately kept 90%.

Introduction

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The copper alloys continue to develop as conductive materials. Among these, oxygen-free copper has a copper purity of 99.95% or higher, which suppresses electron scattering due to impurities and allows for stable maintenance of high conductivity. It is free from hydrogen embrittlement and has few oxide boundaries that cause distortion in electrical signals. Due to its advantages in signal transmission, it is increasingly being adopted as material that supports the information society. With the electrification of automobiles, power semiconductors that control large currents are becoming more common, and oxygen-free copper electrodes are sometimes used to transmit this current. A method of shearing sheet material is used to form electrodes. However, due to its soft and easily stretched mechanical properties, as well as its high activity and tendency to adhere, a detailed understanding of the shearing processing characteristics is essential to obtain the properties required for terminals.

The high ductility and its low mechanical strength cause large shear droop and increase of fractured surface [1]. Various investigations have been conducted into the shearing properties of copper alloys, not just oxygen-free copper [2]-[6]. The studies have examined the effect of copper alloy composition differences on tool wear [2] and attempted to predict punch wear rates [3]. The influence of work hardening as a factor in shear surface formation has been investigated [4], and the effects of clearance and processing speed have been studied [5]. Simulations of polycrystalline materials with differing grain sizes have been conducted [6]. After the previous studies, the mirror-finish punches improved the burnished surface ratio and reduced the surface roughness [7]. However, how to suppress the shear droop formation on the sheared surface was left as a challenging issue.

In this experiment, a ϕ5 mm diameter punch is used to shear oxygen-free copper sheets with a thickness of 500μ m. Three punch-tip finishing conditions are prepared, all based on mirror finished. Under the polishing condition, the punch tip has a rounded edge of approximately 50μ m. In addition,
a surface condition modified from conventional alloy tool steel by nitriding is also examined. Using these tools, the study investigates the effectiveness of punch-tip rounding and nitriding treatment for punches with smooth surfaces. This study aims to clarify the effect of the punch tip geometry of a mirror-finished punch on the amount of shear droop and formation of burnished surface.

Experimental Method

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The oxygen-free-copper sheet with a thickness of 500μmC1020P1/2H (Mitsubishi Materials Co, Ltd.; Tokyo, Japan) was used as workpiece. Fig. 1 shows the small-scaled screw servostamping system and special piercing die for experiment. The loading capacity of servo-stamping system was set at 30 kN (DT-J311 system, Microfabrication Laboratory, LLC.; Tokyo, Japan). The piercing experiment was performed without lubrication. The piercing speed was set at 5.0 mm/s. The piercing load was measured by the load cell (LMBT-A-2KN, Kyowa Electric Co., Ltd.; Tokyo, Japan) which was embedded into the lower die set. The die stroke was measured by the laser displacement meter (LK-G30, KEYENCE Co., Ltd.; Tokyo, Japan). The whole measured data were transferred to PC through the interface (NR-600, KEYENCE Co., Ltd.; Tokyo, Japan).

[picture]

Fig. 1. A small-scaled screw servo-stamping system and shearing die. (a) outlook of small-scaled screw servo-stamping system (b) special die for piercing φ5 mm hole.

Fig. 1. A small-scaled screw servo-stamping system and shearing die. (a) outlook of small-scaled screw servo-stamping system (b) special die for piercing φ 5 m m hole.

Three punch-die pairs were prepared for tooling as listed in Table 1. The punch uses AISI D2 steel (equivalent to JIS SKD11) and the die material is made from carbide tool. The punch diameter used in the experiment was unified at 5.000 mm and basically clearance between punch and die was set at 0.015 mm . The condition called "Polishing" is a punch that has been polished with diamond paste after conventional ground finishing. The "Lapping" condition refers to a punch that has undergone wet-blast lapping process (AERO LAP; Yamashita Works Co., Ltd.; Hyougo, Japan) after conventional ground finishing. The conditions indicated as "Lapping after nitriding" involve applying plasma nitriding [8] after conventional ground process to achieve a hardness exceeding 1200 HV. Following this, wet-blast lapping is performed to improve surface roughness. The outer surface roughness of the three types of punches was measured using a laser microscope (VX-3000; Keyence Co., Ltd.; Tokyo, Japan) with a cutoff value of 0.008μ m. Fig. 2 shows an SEM image of the punch tip and a schematic diagram of the amount of deflection in the punch stroke direction and the punch radial direction measured by a laser microscope.

The piercing speed was set at 5.0 mm/s. After piercing, the punched-out samples were cut in half, and the shear droop amount was measured using a scanning electron microscope (SEM). Using an optical microscope, the sheared surface was photographed using an optical microscope, and the burnished surface ratio was calculated from this image. The punch used in the experiment was observed using EPMA (JXA-8230; JEOL Co., Ltd.; Tokyo, Japan) to check the adhesion state of copper.

Table 1. Punch and die conditions in experiment.

No. | Condition | Punch | Die | Clearance [mm]
|  | Finishing | Outer dia. [mm] | Outer side roughness [μm] | Finishing | Inner dia. [mm] |
A | Polishing | Polishing | 5.000 | 0.039 | Ground | 5.030 | 0.0150
B | Lapping | Wet-blast | 5.000 | 0.081 | Ground | 5.030 | 0.0150
C | Lapping after nitriding | Wet-blast after nitriding | 4.997 | 0.009 | Ground | 5.030 | 0.0165
Three microscope images showing punch tips for conditions A, B, and C with measurements

Fig. 2. Results SEM image of the punch tip and a schematic diagram showing the results of measuring the amount of sagging measured using a laser microscope. (a) Polishing finish punch. (b) Lapping finish punch. (c) Lapping after nitriding punch.

Experimental Results

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Fig. 3 shows Load - stroke diagram. When using Lapping punch, the load increased rapidly in the early stage of shearing, and the maximum shear load was 1710 N . The condition under which the load increased the slowest in the early stage of shearing was the Lapping condition. The maximum shear load under the Lapping condition was 1660 N . The load increase in the initial stage of forming under the lapping after nitriding condition showed intermediate behavior with the three punches, and the maximum shear load was the smallest at 1630 N . The breakage strokes were found to fall within the range of 0.52 mm to 0.55 mm .

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Fig. 3. Load - stroke diagram from three punch conditions.

Fig. 3. Load - stroke diagram from three punch conditions.

Fig. 4 shows the observation results of sheared surface. The front of the die was defined as the Front, and four locations were defined as the Front, Back, Left, and Right, and the punched samples were observed. The observation results under the A: Polishing condition show that smooth burnished surface and the fracture surfaces are formed at each of the four measurement points. Under the B: Lapping condition, the burnished surface is smooth, but vertical lines parallel to the punch stroke direction can be seen, and fractures are almost nonexistent. Under C: Lapping after nitriding conditions, the burnished surface is smooth, indicating that fracture has occurred.

Fig. 5 shows the calculation results for the ratio of shear surfaces obtained from punched hole under three types of punch conditions. The ratio of sheared surface was calculated from the center of the front, back, left, and right images shown in Fig. 4, and the average values of the four measurements are shown as a bar graph. The ratio of burnished surface was the smallest at 82.4% under the Lapping condition, while the remaining two conditions result were approximately 91%.

[picture]

Fig. 4. Results of observing the sheared surface of a punched hole from four directions. (a) Polishing punch punched sample. (b) Lapping punch punched hole. (c) Lapping after nitriding punch punched hole.

Fig. 4. Results of observing the sheared surface of a punched hole from four directions. (a) Polishing punch punched sample. (b) Lapping punch punched hole. (c) Lapping after nitriding punch punched hole.

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Fig. 5. Calculation results for the ratio of shear surfaces obtained from punched hole under three types of punch conditions.

Fig. 5. Calculation results for the ratio of shear surfaces obtained from punched hole under three types of punch conditions.

Fig. 6 shows the SEM image of punched hole cross-section and measurement results of the amount of shear droop length in the punch stroke direction and hole radius direction. The shear droop of the punch stroke direction reached a maximum value of 56.8μ m under Lapping conditions. Similarly, the shear droop in the hole radius direction reached a maximum of 190.3μ m under Lapping conditions. The minimum punch stroke direction value was 20.3μ m under Polishing conditions. The minimum hole radius direction value was 103.6μ m under the lapping after nitriding condition.

Fig. 7 shows the results of measuring the copper around the punch periphery after the shear experiment using EPMA. The copper reaction in the three punches is weak, meaning that almost no adhesion occurs. Under the lapping condition, a weak copper reaction is observed in the area indicated by the white arrow in the figure.

[picture]

Fig. 6. SEM image of the punched hole cross-section and measurement results of the amount of shear droop length in the punch stroke direction and hole radius direction. (a) Polishing punch punched sample. (b) Lapping punch punched hole. (c) Lapping after nitriding punch punched hole.

Fig. 6. SEM image of the punched hole cross-section and measurement results of the amount of shear droop length in the punch stroke direction and hole radius direction. (a) Polishing punch punched sample. (b) Lapping punch punched hole. (c) Lapping after nitriding punch punched hole.

[picture]

Fig. 7. EPMA measuring results of the copper reaction around the punch circumference after the shear experiment. (a) Polishing punch. (b) Lapping punch. (c) Lapping after nitriding punch.

Fig. 7. EPMA measuring results of the copper reaction around the punch circumference after the shear experiment. (a) Polishing punch. (b) Lapping punch. (c) Lapping after nitriding punch.

Discussion

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As a result of shearing oxygen-free copper using three types of mirror-finished punches, changes were observed in the amount of the shear droop length and the ratio of burnished surface. Fig. 8 shows the relationship between punch deflection of punch stroke direction and shear droop length that figure calculated from Fig. 2 and Fig. 6. The symbols in the figure represent the following conditions: A: Polishing, B: Lapping, and C: Lapping after nitriding. These results imply that minimizing punch deflection is effective for reducing the rollover amount in the punch stroke direction of a pierced hole. This finding is consistent with previous knowledge.

[picture]

Fig. 8. Relationship between droop length and punch deflection. (a) Punch stroke direction shear droop length and punch deflection of punch stroke direction. (b) Punch stroke direction shear droop length and punch deflection of punch radius direction.

Fig. 8. Relationship between droop length and punch deflection. (a) Punch stroke direction shear droop length and punch deflection of punch stroke direction. (b) Punch stroke direction shear droop length and punch deflection of punch radius direction.

Next, Fig. 9 shows the relationship between punch deflection of punch stroke direction and shear droop length that figure calculated from Fig. 2 and Fig. 6. The radius direction shear droop length is minimized for condition C: Lapping after nitriding. This implies that the result does not correspond to the amount of punch tip deflection. This implies that even with a sharp edge, such as in condition A: Lapping, the radius direction shear droop length punching with oxygen-free copper cannot be minimized.

[picture]

Fig. 9. Relationship between radius direction shear droop length and punch deflection. (a) radius direction shear droop length and punch deflection of punch stroke direction. (b) radius direction shear droop length and punch deflection of radius direction.

Fig. 9. Relationship between radius direction shear droop length and punch deflection. (a) radius direction shear droop length and punch deflection of punch stroke direction. (b) radius direction shear droop length and punch deflection of radius direction.

The reason why the radius shear droop length is minimized under condition C (lapping after nitriding) is discussed below. The punch used under condition C is hardened to 1200 HV by nitriding [8]. As shown in Table 1, this punch exhibited the smallest surface roughness. It is considered that
the high hardness achieved through nitriding contributed to minimizing the surface roughness, which may have suppressed the activation of the oxygen-free copper surface during the shearing process. During the initial stage of rollover formation in punching, the punch tip meets the workpiece. This activated material tends to cause adhesion to the punch. However, as shown in Fig. 7, no adhesion of copper occurs at the punch tip in condition C , despite the presence of punch deflection. This implies that the process proceeds without the oxygen-free copper adhering to the punch surface. It is considered that this anti-adhesion performance provided by nitriding prevents the workpiece from being drawn into the die side and rolling over, thereby minimizing the radius shear droop length of the pierced hole.

Consider the relationship between the shear surface length and the amount of punch deflection. Fig. 10 shows the relationship between the shear surface ratio and the deflection of the punch in the punch stroke direction. This figure shows that under the condition C , a punch deflection of 46μ m in the punch stroke direction ensures a shear surface ratio of 91%. Generally, as punch deflection increases, shear droop length increases, leading to reduced fracture length and increased burrs height. Fig. 6 shows that no clear burrs were observed under condition C. Setting a narrow clearance of 15 μm, corresponding to a 3% plate thickness ratio, likely reduced the fracture length and created a shear state that suppressed burr formation.

[picture]

Fig. 10. Comparison of ratio of burnished surface and punch deflection in the punch stroke direction.

Fig. 10. Comparison of ratio of burnished surface and punch deflection in the punch stroke direction.

Based on the analysis thus far, it has been demonstrated that setting a punch deflection to approximately 50μ m - corresponding to a thickness ratio of about 10% - during shear processing of oxygen-free copper, followed by nitriding and then lapping, can optimize the shear characteristics of the punched holes.

In this experiment, the influence of punch deflection and punch geometry is supported only to a limited extent. It will be necessary to continue conducting experiments that separate their effects on the burnished-surface ratio, punch stroke, and radial shear droop length in future studies.

Conclusion

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The following conclusions were obtained by punching oxygen-free-copper sheets.

The punch which undergoes nitriding followed by lapping enables activated oxygen-free copper to deform without adhesion, thereby minimizing the radius direction shear droop length of the punched holes.

To shearing oxygen-free copper with a thickness of 500μ m at a burnished surface ratio of 90%, it is effective to set the clearance to 0.015 mm ( 3% of the plate thickness ratio), set the deflection of the AISI D2 punch to approximately 50μ m ( 10% of the plate thickness), perform nitriding treatment, and then proceed with lapping.

References

The original version of this paper is available on https://www.scientific.net/SSP.388.113.pdf

T. Iihara, K. Morikawa, K. Fukuoka, Y. Ito, K. Suehiro, S. Funaki, K. Maki, Development of MOFC-HR: high strength, high heat-resistance oxygen-free copper with superior shearing quality, Materia Jpn. 62.2(2023) 125-127.

T. Ishiguro, K. Nakamura, N. Hayakawa, Y. Yoshida, N. Yukawa, T. Ishikawa, M. Murase, S. Murase, S. Nakano, K. Yoshida, Influence of product material property on tool wear in shearing process, Tetsuto-Hagane, 97-3(2011)99-107.

E. Falconnet, J. Chambert, H. Makich, G. Monteila, Prediction of abrasive punch wear in copper alloy thin sheet blanking, Wear, 338-339(2015)144-154.

F. Grebana, G. Monteilb, X. Roizard, Influence of the structure of blanked materials upon the blanking quality of copper alloys, J. Mater. Process. Techol., 186(2007)27-32.

D. Z. Lubis1 I. Ristiawan, Blanking clearance and punch velocity effects on the sheared edge characteristic in micro-blanking of commercially pure copper Sheet, J. Mech. Sci. Technol., 1-2(2017)53-60.

X. Li, X. Li, R.D.K. Misra, Z. Chen, Grain size effect on shearing performance of copper foil: A polycrystal plasticity investigation, Mech. Mater., 166(2022)104212.

T. Shiratori, Y. Shiratori, T. Aizawa, Effect of tool surface condition on sheared surface in piercing of oxygen-free copper, Proc. 2024 Jpn. Spring Conf. Technol. Plast., (2024)139-140.

T. Aizawa, K. Wasa, Low temperature plasma nitriding of inner surfaces in stainless steel mini-/micro-pipes and nozzles, micromachines, 8(2017)157.

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Solid State Phenomena, ISSN: 1662-9779, Vol. 388, pp 121-126

doi: 10.4028/p-ov7IKy

© 2026 The Author(s). Published by Trans Tech Publications Ltd, Switzerland.

Submitted:
2026-01-19

Revised:
2026-02-16

Accepted:
2026-02-20

Online:
2026-04-15

Experimental Investigation of the Influence of Load Direction Changes on Damage in Dual Phase Steel and Aluminum Alloy

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Philipp Lennemann 1,a*, A. Erman Tekkaya 1, b and Yannis P. Korkolis 1,c

1 Institute of Forming Technology and Lightweight Components, TU Dortmund, Baroper Str. 303, 44227 Dortmund, Germany

a* philipp.lennemann@iul.tu-dortmund.de, b  erman.tekkaya@iul.tu-dortmund.de, c  yannis.korkolis@iul.tu-dortmund.de

Keywords: sheet, anisotropy, fracture, damage.

Abstract

The influence of the stress state on damage evolution, fracture behavior, and component performance is well established for proportional loading conditions. In contrast, many industrial sheet-forming processes involve non-proportional loading paths, which can significantly alter material hardening and fracture responses. Recent results have shown, that a load direction change affects damage evolution in the dual-phase steel DP800. This paper aims to investigate to what extent these results can be transferred to the aluminum alloy AA6082-T6. Therefore, specimens are first prestrained in uniaxial tension and subsequently reloaded either in the same direction or orthogonally, using additional tensile tests. Fracture strains during the subsequent tensile tests are determined by Aramis Digital Image Correlation. An orthogonal load direction change leads to an increased fracture strain for DP800, but decreased fracture strain for AA6082. While the observed behavior of DP800 can be attributed to the void morphology, which is established during prestraining, the results of AA6082 indicate different damage mechanisms which cause this behavior.

Introduction

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Dual-phase steels, such as DP800, are composed of martensite and ferrite and widely used in the automotive industry because of their excellent balance of high strength and good formability. The microstructure of DP800 is established during hot and cold rolling followed by intercritical annealing, which can result in anisotropic ductile behavior.

AA6082-T6 is a high-strength aluminum alloy widely used in automotive, aerospace, and structural applications due to its excellent strength-to-weight ratio and good corrosion resistance. However, the T6 temper, obtained through solution heat treatment and artificial aging, provides high strength at the expense of reduced work-hardening and formability.

In complex manufacturing routes, but also in simpler processes like deep drawing, different regions of a sheet experience a variety of stress conditions, including changes in hydrostatic and deviatoric stress. A key feature of these non-proportional load paths is the variation in the direction of the mean stress. This study aims to assess how changes in loading direction influence damage evolution.

Damage, in this study, refers to the nucleation, growth, and coalescence of voids within a material. Damage evolution is affected by the stress state (triaxiality η and Lode parameter L ) during forming. Damage, as well as other forming-induced properties such as residual stresses and work hardening, influence fatigue life, Charpy impact toughness, and crash performance of the formed component [1]. Direct methods for the characterization of damage evolution are mass density measurements and 2D microscopic images. The specific mechanisms responsible for damage are material-dependent. Void nucleation can occur through the fracture of individual phases or non-metallic inclusions, as well as through decohesion either between two phases or between a phase and an inclusion. The precipitates in AA6082 form during quenching at low cooling rates and are distinguished from nano-scale precipitates which form during the ageing process and are desired for enhanced strength. The number and size of these precipitates is influenced by the chemical composition and the quenching rate and are therefore batch-dependent [2].

In dual-phase steels, the dominant damage mechanisms are martensite cracking and interfacial decohesion at martensite-ferrite boundaries [3]. During tensile loading, voids initially form with orientations perpendicular to the loading direction at small strains and subsequently grow in the direction of loading as deformation progresses [4].

Investigations into strain path changes have demonstrated that non-proportional strain paths generally result in higher fracture strains than proportional ones. Kestner and Koss [5] performed experiments involving sequential combinations of uniaxial and equibiaxial tension on titanium sheets, ultimately reaching a plane-strain condition. Their results showed that non-proportional strain paths yielded higher fracture strains than proportional ones. They attributed this behavior to a delay in void nucleation during multi-stage deformation, associated with strain paths characterized by relatively low maximum principal stresses.

Strain path changes can be classified using the strain path change indicator defined by Schmitt et al. [6]:

κ=Δε1pl:Δε2pl‖Δε1pl‖·‖Δε2pl‖(1)

where Δεipl is the tensor corresponding to the plastic strain increment of load step i .
The parameter κ can vary between −1<κ<1 and represents following strain paths:
κ=1 : monotonic strain path
1<κ<0 : monotonic strain path change
κ=0: orthogonal strain path change
0<κ<−1 : non-monotonic strain path change
κ=−1 : strain path reversal
Recent investigations on DP800 show, that load direction changes influence damage evolution and finally the fracture strain [7]. This paper aims to find out to what extent those results are transferable to aluminum alloys from the 6xxx-series.

Firstly, the sheets are tested in monotonic tensile tests. The stress-strain curves and R -values will reveal, if the material's behavior depends on the rolling direction. The uniform elongation also shows to which level prestraining is possible to still obtain homogenously prestrained specimens.

Then, the 2-step tensile tests are conducted to investigate the effect of load direction changes on damage (Fig. 1). Therefore, smaller tensile specimens are extracted from larger, prestrained specimens. These smaller specimens are then loaded parallel to the prestrain - with a monotonic loading direction ( κ=1 ) - and another set is loaded orthogonally to the prestrain - with an orthogonal load direction change (κ=−0.5). The void morphology in DP800, which influences further void growth in the subsequent tensile tests is shown. The fracture strains in the subsequent tensile tests are determined by Aramis Digital Image Correlation (DIC) to indicate the influence of the load direction change on damage behavior.

[picture]

Fig. 1. 2-step tensile tests.

Fig. 1. 2-step tensile tests.

Monotonic Characterization in Standard Tensile Tests

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Former investigations have shown, that the rolling-induced microstructure influences the behavior of DP800 sheet material since damage and fracture depend on the rolling direction. To see if this is also the case for the investigated aluminum alloy AA6082-T6, standard A80 tensile tests are performed with orientation in 0∘,45∘ and 90∘ to the rolling direction. All tests in this work are carried out on the universal testing machine Zwick Z250 with a quasi-static strain rate of 0.0067/s. The strains are measured by the contact extensometer makroXtens II HP by Zwick with an initial length of 80 mm . The stress-strain curves reveal similar hardening behavior of all specimens regardless the rolling direction with a yield strength of 275 MPa and an ultimate tensile strength of 310 MPa (Fig. 2). The uniform elongation εu is lowest at 0∘(0.089±0.002) and similar for 45∘(0.093±0.002) and 90∘(0.093±0.001). The total elongation εf follows the same trend, since it is also lowest at 0∘(0.121±0.003) and similar for 45∘(0.135±0.004) and 90∘(0.133±0.002). This shows, that the rolling direction influences the fracture behavior, which needs to separated from the effect of load direction changes. Therefore, the prestraining is applied in 45∘ to the rolling direction, because then both subsequent tests are loaded in 45∘ as well, which ensures the separation of the effect of the loading direction on damage and fracture from the effect of the rolling-direction-dependent behavior.

[picture]

Fig. 2. AA6082-T6 - a) Stress strain curve; b) Uniform elongation; c) Total elongation.

Fig. 2. AA6082-T6 - a) Stress strain curve; b) Uniform elongation; c) Total elongation.

Prestraining of the Specimens

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The testing area of the large specimens for prestraining has a width of 60 mm and a length of 120 mm (Fig. 3a). The strains are measured with a contact extensometer with an initial length of 14 mm . The aim of the prestrain is to induce damage evolution due to tensile strain. Therefore, higher strains are preferable since damage evolves at positive triaxialities with an increasing strain. On the other hand, the prestrain should not exceed the uniform elongation, as a heterogenous strain distribution would influence the subsequent tensile tests by making them not comparable. To balance these goals, the prestrain is chosen to be 0.074 ( εp=0.07 ), since necking sets in at a strain of approximately 0.09 . Since the DP800 material started necking at 0.108 , the prestrain for these specimens was chosen to be 0.098(εp=0.09).
After forming, the specimens for the subsequent tensile tests are extracted from the testing area by laser cutting. To compare the material behavior and damage evolution under monotonic loading and after an orthogonal change of loading direction, one set of secondary specimens is extracted parallel to the prestraining direction and another set is extracted in orthogonal direction (Fig. 3b).

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Fig. 3. Specimen geometry for the tensile tests: a) specimens for prestraining; b) specimens for subsequent tensile tests.

Fig. 3. Specimen geometry for the tensile tests: a) specimens for prestraining; b) specimens for subsequent tensile tests.

Void Morphology in DP800

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The void morphology after prestraining was investigated for DP800 [7]. Damage analysis of the prestrained specimens by SEM shows that voids most often form between two martensite islands, likely due to martensite cracking or ferrite failure in these narrow regions. These voids are bordered by hard martensite along the prestrain direction and by softer ferrite orthogonally. An illustrative simulation was conducted to demonstrate the effect of this microstructural configuration, considering the different hardening behaviors of ferrite and martensite. The model consists of two martensite islands embedded in a ferritic matrix with a void positioned between them, oriented orthogonal to the initial tensile loading direction (Fig. 4a). The numerical results show, that after monotonic loading, the void grows, whereas an orthogonal change in loading direction causes the void area to decrease (Fig. 4b).

[picture]

Fig. 4. DP800 - a) Voids after first tensile; b) Void evolution in subsequent tensile tests [7].

Fig. 4. DP800 - a) Voids after first tensile; b) Void evolution in subsequent tensile tests [7].

Fracture Strains in Subsequent Tensile Tests

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The fracture strains are determined with Aramis DIC during the subsequent tensile tests. The DIC system has a frame rate of 5 Hz . The equivalent strains are measured in the last frames before fracture is visible in the DIC-system. The results reveal, that an orthogonal load direction change leads to a decrease in terms of the fracture strain for the aluminum alloy, while the fracture strain is increased after the orthogonal load direction change in DP800 (Fig. 5). For DP800, the fracture strain increases about 10% from 0.5 to 0.55 after an orthogonal load direction change in comparison to a monotonic load. For AA6082-T6 the fracture strain decreases about 6% from 0.5 to 0.47 after an orthogonal load direction change in comparison to a monotonic load.

The manufacturing of the specimens by laser cutting might influence the results quantitatively due to heating in the cutting area. Anyway, this does not influence the comparative nature of the results, since all specimens were manufactured with the same methods.

[picture]

Fig. 5. Fracture strains in subsequent tensile tests - a) DP800; b) AA6082-T6.

Fig. 5. Fracture strains in subsequent tensile tests - a) DP800; b) AA6082-T6.

Conclusion

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The dual phase steel DP800, as well as the aluminum alloy AA6082-T6 have been tested in monotonic and 2-step tensile tests with a continuous load path and an orthogonal change of loading direction. Standard, monotonic tensile tests have shown, that the forming behavior of both materials depend on the rolling direction.

For DP800, orthogonal load direction changes lead to an increase in fracture strain. This can be explained by the void morphology, since voids shrink after orthogonal load direction changes due to their nucleation in between martensitic phases. On the other hand, for AA6082-T6, the fracture strain decreases after an orthogonal load direction change.

It can be concluded, that the damage mechanisms of AA6082-T6 seem to differ from DP800, since the fracture behavior under a non-monotonic load is opposite. While for DP800 the fracture of martensitic phases has been identified as the dominant damage mechanism in uniaxial load, the behavior of aluminum alloy AA6082-T6 in contrast might be governed more by decohesion of inclusions and precipitates from the matrix. Therefore, further investigations will aim at the identification of damage mechanisms in AA6082-T6 after a monotonic load and after orthogonal load direction changes.

Acknowledgement

The original version of this paper is available on https://www.scientific.net/SSP.388.121.pdf

The authors kindly thank the German Research Foundation (DFG) for the financial support of project A05 in the Collaborative Research Centre CRC/Transregio 188 "Damage Controlled Forming Processes" (Project number 278868966 - TRR 188). Robin Gitschel is acknowledged for the proofreading of the paper.

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The original version of this paper is available on https://www.scientific.net/SSP.388.121.pdf

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P. Lennemann, Y.P. Korkolis, A.E. Tekkaya, Influence of changing loading directions on damage in sheet metal forming, Advances in Industrial and Manufacturing Engineering 8:100139 (2024). .

This ePub file was created using machine-generated methods and automated software to convert the source text into an ePub format and may contain formatting inconsistencies from the original version. Formatting may vary depending on the device used.

The original version of this eBook is available on https://www.scientific.net/book/formability-of-metallic-materials/978-3-0364-1993-0

Solid State Phenomena, ISSN: 1662-9779, Vol. 388, pp 127-140

doi: 10.4028/p-NSOqa8

© 2026 The Author(s). Published by Trans Tech Publications Ltd, Switzerland.

Submitted:
2026-01-19

Revised:
2026-02-16

Accepted:
2026-02-18

Online:
2026-04-15

Stochastic Forming Limit Curves: Construction of a Probabilistic Tool for Failure Assessment

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Université de Technologie de Troyes, Unité de Recherche LASMIS

a  olivier.ameline@utt.fr, b  yessine.jedidi@utt.fr, c  laugustin.persoons@utt.fr, d  carl.labergere@utt.fr

Keywords: Forming Limit Curves, Stochastic Modeling, Experimental Variability

Abstract

The determination of Forming Limit Curves (FLCs) remains challenging due to their strong dependence on test conditions, material properties, and measurement methods. Significant variability is observed even for identical specimens, limiting the reliability of FLCs as deterministic tools. The goal of this work is to investigate non-deterministic FLC prediction in order to illustrate the impact of uncertainties and provide a basis for quantification of failure risks. The proposed approach uses the probabilistic framework where a model of input uncertainties including dependence is inferred from literature data using a Gaussian copula. A prediction model based on the linear perturbation technique is described and used to propagate input uncertainties using a Monte-Carlo approach. The obtained stochastic FLC is illustrated in terms of empirical confidence areas.

Introduction

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Material failure in sheet metal forming is commonly assessed using a forming limit diagram (FLD), in which the maximal deformations observed during the process are plotted in the major-minor strain space and evaluated against the forming limit curve (FLC) [1]. The FLC delineates the boundary between stable and unstable deformation, representing the maximum strain the material can sustain before localized necking occurs. It serves as a critical threshold for safe forming operations, enabling the optimization of forming parameters and the reduction of defects and material waste.

FLCs have been used in metal forming industries for decades, motivating extensive research on their theoretical and experimental prediction. Analytical and numerical approaches commonly couple constitutive models with necking criteria under linear strain paths, as proposed by Swift [2], Hill [3], and Marciniak and Kuczyński [4]. More recent studies have extended the classical FLC framework toward a more comprehensive description of ductility limits for numerical forming simulations [5-7]. In parallel, several works describe the onset of necking within a homogeneous continuum framework using the linear perturbation technique, which avoids assuming an initial geometrical imperfection, in contrast to the Marciniak-Kuczyński approach [8-12]. Experimental investigations have been conducted to validate these numerical predictions [13-15].

Despite their widespread industrial use, forming limit curves (FLCs) have inherent limitations due to their strong sensitivity to testing conditions, material variability, and methodological choices in the identification of fracture and localized necking. Significant scatter in formability limits is commonly observed even for nominally identical specimens, showing that FLCs cannot be considered deterministic failure criteria.

The forming limit curve is then stochastic in nature and we propose a methodology to estimate its statistical properties, reflecting the distribution of strain states at localized necking. From an industrial perspective, the stochastic FLC provides a practical decision-making tool. By projecting the strain loading path experienced by a given part onto the stochastic FLC, manufacturers can evaluate the probability of failure associated with the forming operation. This probabilistic assessment enables a more reliable evaluation of formability and supports the optimization of forming process parameters, leading to improved robustness and reduced risk of defects.

This study follows a direct probabilistic approach: first, establishing a deterministic FLC model based on linear perturbation theory; second, characterizing uncertainties in the five model parameters through literature data; third, propagating these uncertainties via Monte Carlo simulation to generate stochastic FLC realizations. The computational efficiency of the underlying model enables direct Monte Carlo sampling without resorting to surrogate approximations. This provides manufacturers with confidence bounds reflecting both experimental and material variability, in order to support process control while reducing failure risks and overdesign. In practice, industrial users may then refine this prior uncertainty with available experimental data of their particular material using a Bayesian updating scheme.

The article is organized in three main sections. Section 1 presents the theoretical framework for forming limit prediction based on linear perturbation analysis. Section 2 describes parameter identification from literature data for CP-Ti. Section 3 details the probabilistic uncertainty modeling of these parameters, propagates the uncertainties through Monte Carlo simulation et analyzes the stochastic FLC properties.

Theoretical Model

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The theoretical prediction of forming limit curves (FLCs) is done using the already well documented linear perturbation technique [8-12]. First, the equations of plasticity are solved analytically considering a uniform strain-stress state, which defines the homogeneous solution. Small perturbations are then superimposed onto this solution to assess the stability of the deformation and to predict the onset of strain localization.

Homogeneous solution.

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The thin sheet is supposed to be incompressible and subjected to a plane stress state in the x−y laboratory system. Shear strains are neglected, the major strain is taken along the x direction and elasticity is ignored because large deformations are involved. A linear strain path is considered and characterized by the parameter ρ such that:

ρ=εyy/εxx= const. (1)

The constitutive relation between stress and plastic strain rate components is defined using an associated flow rule (AFR), in which the equivalent stress σEq is given by the Hill (1948) anisotropy criterion [16] and the hardening law σHard  is considered viscoplastic, such that:

σEq(σ)=Fσyy2+Gσxx2+H(σxx−σyy)2+2Nσxy2 and σHard (ε¯,ε¯˙)=σy(ε¯)ε¯˙m(2)

where ε¯ is the equivalent plastic strain, m is the strain-rate sensitivity, and the Hill coefficients F,G, H and N are expressed as functions of the Lankford parameters r0,r45 and r90 :

F=r0r90(1+r0);G=11+r0;H=r01+r0;N=12(F+G)(1+2r45)(3)

These assumptions are only approximate, as strain paths are typically neither shear-free nor linear in industrial processes. Nevertheless, they are reasonably satisfied in Nakazima and Marciniak tests, which are commonly employed in industry to construct experimental forming limit curves (FLCs). For HCP materials such as CP-Ti, the Hill (1948) yield criterion is also known to be less accurate than more advanced models that account for strong plastic anisotropy and tension-compression asymmetry [17, 18]. Despite these limitations, the present framework provides a practical starting point, as it can be implemented with a limited number of parameters and offers excellent computational efficiency. It can subsequently be applied to more complex models of anisotropy, and the resulting stochastic FLC can be further refined through Bayesian calibration based on experimental data.

Under quasi-static conditions, a uniform strain-stress state automatically satisfies the equilibrium and compatibility equations. The analytic resolution of the AFR and the condition of incompressibility lead to the homogeneous solution, expressed as a function of the major strain:

σxx=ε˙xxε¯˙F+H+HρFG+FH+GHσy(ε¯)ε¯˙m;σyy=ε˙xxε¯˙H+(G+H)ρFG+FH+GHσy(ε¯)ε¯˙m;σxy=0;ε¯˙ε˙xx=ε¯εxx=R(ρ)=F+Gρ2+H(1+ρ)2FG+FH+GH;ε˙yy=ρε˙xx;ε˙zz=−(1+ρ)ε˙xx.

(4)(5)

Perturbation analysis. As illustrated in Fig. 1, a rotated coordinate system x1−x2 is defined to represent the orientation φ of a candidate localization band. Then, the homogeneous solution P(t)={ε˙xx,ε˙yy,ε˙zz,ε˙xy,σxx,σyy,σxy,σy,ε¯}T, defined in Eqs. (4)-(5), is perturbed in the direction x1 such that:

P~(t,x,y)=P(t)+δPexp[ηt]exp[iξ(xcosφ+ysinφ)],(6)

where P~ is the perturbed solution, η is the growth rate, t is the time and ξ is the wave number.

[picture]

Fig. 1: Linear perturbation in thin sheet metal.

Fig. 1. Linear perturbation in thin sheet metal.

The perturbation P~ is introduced into the full set of plasticity equations, including the AFR, the equivalent stress, the hardening law, the equilibrium equations, the compatibility equation and the condition of incompressibility. By including the equilibrium equations, the perturbation is constrained to satisfy force balance between the inside and outside of the band. After linearization, the resulting system takes the form MδP=0, with the onset of instability corresponding to the vanishing of the determinant of M. This system provides conditions on the angle of the wavy band, φ, and the normalized growth rate, η^=η/ε¯. By assuming that localized necking occurs along the direction of maximal rate of normalized growth rate, dη^/dε¯, Hu et al. [11] established that:

φ=arctan−ρ if ρ<0;φ=0 if ρ≥0.(7)

With this additional condition, the normalized growth rate is expressed analytically as:

η^=η1(ρ)[η2(ρ)−σy′σy(ε¯)],(8)

where

η1(ρ)=1m if ρ<0;η1(ρ)=(F+H+Hρ)2(FG+FH+GH)ρ2+m(F+H+Hρ)2 if ρ≥0,η2(ρ)=εxxε¯(1+ρ) if ρ<0;η2(ρ)=εxxε¯F+Gρ2+H(1+ρ)2F+H+Hρ if ρ≥0.

(9)(10)

These results are obtained by applying the formalism of Tóth et al. [9] or Hu et al. [11] to the more general hardening law of Eq. (2).

The perturbation becomes unstable when η^>0, that is, for an equivalent plastic strain ε¯i such that:

σyσy′(ε¯i)=1η2(ρ)(11)

If the material is strain-rate insensitive, then m→0, which in practice can be considered by taking m=10−5 to avoid the singularity in the expression of η1(ρ) when ρ<0. As a result, strain-rate insensitivity implies that the left part of the FLC is obtained by assigning localized necking to the equivalent strain ε¯n=ε¯i where the normalized growth η^ diverges from −∞ to +∞. For the right part of the FLC ( ρ≥0 ), the onset of necking is taken at the equivalent strain ε¯n where the accumulated growth of the perturbation reaches a critical value E[11] :

∫ε¯iε¯nη^dε¯=E(12)

For a strain-rate insensitive material, the critical value ε¯n for localized necking thus verifies:

ε¯n=ε¯i if ρ<0;η1(ρ)[η2(ρ)(ε¯n−ε¯i)−lnσy(ε¯n)σy(ε¯i)]=E if ρ≥0(13)

Once this critical value is known, the FLC is constructed in the ρ−ε¯ space by plotting ε¯n as a function of ρ. It can then be mapped into the εxx−εyy space using Eqs. (1) and (5).

Application to Experimental Data

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In this section, the theoretical model is fitted to five experimental forming limit curves (FLCs) reported in literature [19-23].

Hardening law. The hardening law cannot be calibrated with either the published stress-strain curves or the models fitted to them, as tensile tests are generally limited to the onset of diffuse necking, whereas we need to model the material behavior beyond this stage. To address this difficulty, we use the ρ<0 region of the experimental FLC to calibrate the hardening law. In the εxx−εyy space, this part of the curve is approximated by a decreasing straight line:

 when ρ<0,εxx=−aεyy+b⇔εxx=b/(1+aρ). (14)

In this region, the equivalent plastic strain at necking takes the value ε¯i defined by Eq. (11). The pair (ε¯,σy/σy′) corresponding to a point of the experimental FLC can thus be obtained as a function of ρ by computing the major strain εxx from Eq. (14), the equivalent plastic strain ε¯ from Eq. (5), and the ratio σy/σy′ from Eq. (11).

The resulting parametric curve is observed to be approximately linear for all studied FLCs, motivating the following hardening law:

σyσy′=αε¯+β, with α=1/η2(0)−1/η2(−0.5)bR(0)−2bR(−0.5)/(2−a);β=1η2(0)−αbR(0).(15)

This result is equivalent to choose a Swift hardening law, but with parameter values far different from usual because only the region posterior to diffuse necking is represented.

Model inference. From the foregoing, the construction of a theoretical FLC in the ρ−ε¯ space with Eq. (13) requires five parameters: r0,r90,a,b and E. The Lankford r-values, r0 and r90, are directly available in the published literature ( r45 is not involved due to the assumption of negligible shear strains). The coefficients a and b are readily obtained by linear regression of the left-hand side of the experimental FLC, while the threshold E is fitted to its right-hand side. The resulting parameter values are synthetized in Table 1. It should be noted that these data reflect not only the intrinsic variability of the material, but also the variability arising from testing conditions and measurement methods on FLC determination, since the parameters a,b and E are identified through an inverse approach by calibration against different experimental FLCs.

Table 1: Parameter values of the theoretical model that represents the published experimental data.

Article | r₀ | r₉₀ | a | b | E
Arrieux 1981 | 1.5 | 3 | 1,024 | 0.292 | 1.21
Chen 2005 | 4.2 | 2.1 | 1.209 | 0.340 | 0.71
Chamos 2013 | 1.5 | 3.75 | 0.993 | 0.302 | 1.58
Kim 2018 | 1.83 | 5.69 | 1.033 | 0.338 | 0.36
Wieckowski 2022 | 2.49 | 5.2 | 1.109 | 0.311 | 0.95

Fig. 2 presents the experimental data together with the theoretical model response, using the input parameters given in Table 1. A very good agreement is observed. However, the limited number of data points in the right-hand side of the experimental FLCs, together with the use of a hardening law calibrated on their left-hand side, tend to mask the model approximations. The assumptions adopted - particularly the Hill 1948 yield criterion - are generally not well suited to the industrial forming of CP-Ti. Even if the experimental data are well represented, these assumptions may affect the propagation of uncertainties within the model. Nevertheless, this simplified framework provides a practical basis for stochastic FLC prediction, which can later be refined employing more advanced models.

Uncertainty Modeling

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On the previous section, the theoretical model has been successfully applied to represent experimental FLCs from literature. The goal of this section is to propose a first approach to estimate the statistical properties of the forming limit diagram of CP-TI considering the uncertainties related with the five input parameters ( r0,r90,a,b and E ) ang propagating them within the model.

The proposed approach uses a probabilistic framework [24], meaning the five inputs are considered as random variables and associated with probability distributions inferred from available data. The numerical model detailed in the previous sections allows us to obtain the forming limit diagram as a function of the five inputs. Since the input are stochastic in nature, the resulting forming limit diagram is a stochastic process. In other words, for any strain ratio ρ, the necking equivalent plastic strain ε¯ is a random variable. The statistical properties of this stochastic FLC directly depends on the those of the five input parameters. It is then critical for the model of uncertainties assigned to the inputs to be representative of their actual variability.

[picture]

Fig. 2: Theoretical model fitted to experimental FLCs from litterature in ρ−ε¯ space.

Fig. 2. Theoretical model fitted to experimental FLCs from litterature in ρ − ε ¯ space.

As previously mentioned, the model of uncertainty is inferred from a dataset collected from the literature (see Table 1). While only five instances of reports [19-23] including the necessary information to recover all five parameters are currently know to the authors, significantly more have been found reporting only the Lankford coefficients [25-41]. As a result, 25 instances of r0 and r90 are used in the statistical study (see Table 2). Still, the small sample sizes present a significant challenge for statistical inference, requiring careful consideration of modeling assumptions.

Table 2: Values of r0 and r90 used for marginal distribution inference [19-23, 25-41].

Ref. | r₀ | r₉₀ | Ref. | r₀ | r₉₀ | Ref. | r₀ | r₉₀ | Ref. | r₀ | r₉₀ | Ref. | r₀ | r₉₀
[19] | 1.50 | 3.00 | [25] | 2.30 | 4.72 | [30] | 1.19 | 3.13 | [34] | 1.55 | 3.88 | [39] | 0.74 | 1.68
[20] | 1.50 | 3.75 | [26] | 0.95 | 2.85 | [30] | 1.46 | 3.36 | [35] | 1.70 | 4.89 | [40] | 2.11 | 3.62
[21] | 4.20 | 2.10 | [27] | 2.20 | 4.89 | [31] | 1.51 | 5.17 | [36] | 2.34 | 4.86 | [40] | 2.66 | 5.23
[22] | 1.83 | 5.69 | [28] | 2.29 | 4.70 | [32] | 1.58 | 6.80 | [37] | 1.71 | 2.67 | [40] | 1.46 | 2.66
[23] | 2.49 | 5.20 | [29] | 3.84 | 5.45 | [33] | 2.11 | 4.85 | [38] | 2.46 | 5.65 | [41] | 1.22 | 3.55

Marginal distribution modeling. As a first step, each input parameter is associated with a marginal distribution function. The principle is to assume that the random variables follow a specific family of distributions, defined by shape or modal parameters that can be inferred from the data (e.g., a normal distribution parameterized by the mean and variance of the data).

The choice of a distribution family represents a strong assumption on the shape of the statistics and should be taken with caution, motivated by available knowledge about the parameter to be modeled and by as much statistical evidence as possible. Even so, it generally leads to imprecision when evaluating the statistics of extremely rare events (associated with the tails of the distributions). Nonparametric approaches [24] are generally to be preferred, but cannot be implemented in this context since they require a large amount of calibration data.

A first clue for choosing a distribution family is the domain on which the distribution is defined, i.e., whether there are bounds on the possible values of the parameter of interest. For instance, a normal distribution assumes all values in ℝ are possible, which would not be suited to modeling a parameter taking only positive values. Other indicators could include information about statistical modes and the shape of the distribution (e.g., symmetry). Finally, one can resort to common practices in the literature for distributions of parameters of similar nature.

Since r0,r90,a, and b take only positive values, a lognormal distribution is arbitrarily selected among distributions with positive support. This choice is not uncommon in the literature [42, 43] and
involves assuming a multiplicative structure of the uncertainties, i.e., confidence intervals are defined as a nominal value multiplied and divided by a factor. The available data is currently too limited to clearly discriminate between possible distributions families, as also illustrated by the confidence intervals in the QQ-plots Fig. 3. While the lognormal family provides statistically satisfactory results, other choices could also be considered, such as the gamma or Weibull distributions. A formal evaluation of the sensitivity of the FLC statistics to this choice is planned, and the quantification of tail-related statistics is deemed outside the scope of the proposed approach.

The log-normal probability density function depends on two parameters, μ and s, corresponding respectively to the mean and standard deviation of the underlying normal distribution:

fX(x)=1xs2πexp(−(lnx−μ)22s2)(16)

The parameters of the four log-normal distributions were then obtained using maximum likelihood estimation over the available data. The resulting values are reported in Table 3.

The parameter E takes only positive values, no skewness is observable in the data (see Fig. 3), and no common practice appears in the literature. A uniform distribution was adopted as a maximumentropy representation given only the observed range [Emin ,Emax ][24]. This approach imposes strict bounds on the values of E and is therefore conservative (it assumes no values more extreme than those observed will occur). It is, however, a minimal-assumption approach in such a case of limited information, as it avoids unsupported distributional assumptions.

The quality of the calibration is illustrated in Fig. 3, where on the left-hand side the histograms of calibration data are drawn together with the fitted distributions. The figures qualitatively illustrate the agreement in the shape of the fitted and empirical distributions for r0 and r90. For the other three parameters, the low amount of available data makes it too difficult to draw conclusions from histograms.

The right-hand side of the figure shows quantile-quantile plots of empirical data as a function of the fitted quantiles. These standard graphs show the correspondence between statistical quantiles of the calibration data and the fitted model, and can be interpreted analogously to the calibration plots used in regression modeling, showing the predicted value as a function of the true one. In addition the 95% confidence intervals associated with each empirical quantiles are drawn, based on an asymptotic normal assumption (considering each data point is independent and identically distributed). In these figures, the fit appears clearly to be in acceptable accordance with the data, especially for central values (i.e., non-extreme ones).

Table 3: Probabilistic models for material parameters.

Parameter | Distribution | Parameter 1 | Parameter 2
r₀ | Lognormal | μr₀ = 5.65 × 10⁻¹ | σr₀ = 3.77 × 10⁻¹
r₉₀ | Lognormal | μr₉₀ = 1.333 | σr₉₀ = 3.36 × 10⁻¹
a | Lognormal | μa = 6.84 × 10⁻² | σa = 7.04 × 10⁻²
b | Lognormal | μb = −1.15 | σb = 6.10 × 10⁻²
E | Uniform | Emin = 7.11 × 10⁻¹ | Emax = 1.58

Dependence structure modeling. The five input parameters are physically interdependent: processing history is bound to affect anisotropy ( r0,r90 ), hardening response ( a,b ), and necking threshold E. This physical intuition stresses the fact that, while random, the five inputs are likely correlated. Ignoring these dependencies would degrade the relevance of the FLC statistical properties.

Many ways exist in the literature to model the dependence between random variables. The simplest is to use a linear correlation matrix (e.g., in a multivariate Gaussian distribution). However, a correlation matrix cannot be used directly in our case since the five marginals are not Gaussian. A standard and efficient way of using the correlation matrix in such a case is a Gaussian copula [24].

[picture]

Fig. 3: Illustration of the inference of the five marginal distributions, including (left) histograms of available data and fitted distributions and (right) empirical quantiles of data vs. quantiles of the fitted distributions.

Fig. 3. Illustration of the inference of the five marginal distributions, including (left) histograms of available data and fitted distributions and (right) empirical quantiles of data vs. quantiles of the fitted distributions.

The principle is to apply a transform to the input parameters in order to define five random variables that are normally distributed. This transform is called the probability integral transform and relies on the fact that, for any random variable X following a distribution f with cumulative distribution function (CDF) F,F(X) is a random variable following a uniform distribution. Using the inverse transform, one can also state that if U follows a uniform distribution and Φ−1 is the inverse of the standard Gaussian CDF, then Φ−1(U) follows a Gaussian distribution. It follows that, for any random variable X,Φ−1(F(X)) follows a standard Gaussian distribution. The principle behind the Gaussian copula is to model the transformed, normally distributed random variables as a simple multivariate Gaussian characterized by a correlation matrix inferred from the transformed calibration data.

Calibrating a Gaussian copula in this case simply means applying the probability integral transform to the calibration data (using the CDFs of the marginals). The covariance matrix 𝐑 of the transformed data is then estimated via maximum likelihood. Generating Monte Carlo samples of the five inputs respecting the dependence structure simply means generating samples of the multivariate Gaussian with covariance 𝐑 and applying the inverse transform to those samples.

In order to evaluate the relevance of the calibrated copula, Monte Carlo samples are generated and scatterplots are drawn showing the trends between pairwise variables. On top of these simulated data points, the actual calibration data are drawn to illustrate the fit qualitatively. These plots are presented in the lower-triangular panels of Fig. 4, where each row and column represent one of the five inputs. The figure illustrates a satisfactory fit, even though the limited data make it hard to draw quantitative conclusions on the goodness of fit.

In addition, the diagonal panels of Fig. 4 represent the histograms of simulated marginal data together with the histogram of each individual calibration dataset. This information is related to the goodness of fit of the marginals, already illustrated in Fig. 3.

Finally, the upper triangle of the figure represents the correlation coefficients between the pairs of inputs, showing a quantitative measure of the correlations. These values are indicative of the strength of the correlation between pairs of variables but should be treated as indicative only, since they assume a linear correlation that does not strictly hold outside the space of the transformed (normally distributed) random variables.

It is, however, worth noting that the choice of a Gaussian copula imposes strong limitations with n=5. The validation of 𝐑 is impossible with such limited data, meaning it is not possible to evaluate the relevance of the fitted copula with respect to epistemic uncertainties. It is therefore important to stress the statistical limitations of this approach: with such limited calibration data, only broad statistical tendencies and sensitivities should be studied. In particular, any results related to rare events and the tails of distributions should not be interpreted quantitatively.

Uncertainty propagation. As previously mentioned, since the input parameters of the model are random variables, the output FLC is effectively a stochastic process, meaning that for each value of ρ, the equivalent necking plastic strain ε¯ is a random variable. However, direct propagation of the random variables through the model is not trivial, and the analytical expression or closed-form properties of this stochastic process are not available. In order to obtain a numerical estimation of these properties, a Monte Carlo approach was implemented.

Using the model of uncertainty calibrated as described in the previous two sections, N=10,000 random samples of the five input parameters were generated (Monte Carlo simulations). For each sample, the corresponding forming limit diagram is recovered using the numerical model described in the first two sections, resulting in a population of N randomly generated FLCs. For the sake of convenience, FLCs are expressed in the space of equivalent plastic strain ε¯ vs. strain ratio ρ instead of the usual major strain vs. minor strain. Both spaces are equivalent, but the latter imposes a repartition of curve data following a polar basis, which is less straightforward to analyze.

This Monte Carlo population of FLCs provides, for each evaluated value of ρ, a population of N values of ε¯. Using this population, empirical confidence intervals (CIs) were estimated, allowing us to illustrate the variability of the stochastic FLC, as presented in Fig. 5. In other words, for each value

[picture]

Fig. 4: Illustration of the dependence between input variables and the fitted copula with: (diagonal) histograms of marginal data and simulated samples; (upper triangle) correlation values between pairwise variables; (lower triangle) pairwise input scatterplots of calibration data and samples simulated using the fitted copula.

Fig. 4. Illustration of the dependence between input variables and the fitted copula with: (diagonal) histograms of marginal data and simulated samples; (upper triangle) correlation values between pairwise variables; (lower triangle) pairwise input scatterplots of calibration data and samples simulated using the fitted copula.

[picture]

Fig. 5: Stochastic FLCs: (left) median (solid line) and confidence intervals (shaded regions) from 10,000 Monte Carlo realizations; (right) illustration of 1,000 Monte Carlo samples of FLCs.

Fig. 5. Stochastic FLCs: (left) median (solid line) and confidence intervals (shaded regions) from 10,000 Monte Carlo realizations; (right) illustration of 1,000 Monte Carlo samples of FLCs.

of ρ, the value of the critical equivalent plastic strain can be empirically associated with confidence intervals, i.e., ε¯ has a probability p% of being in the interval [CI―p,CI―p]. In the figure, increasingly larger confidence intervals are shown with increasingly lighter gray areas, showing the evolution of the stochasticity of the FLC for all values of ρ.

From a methodological point of view, it can be observed that the uncertainties related to the forming limit threshold are higher at the boundary values of ρ and respectively narrower around ρ=0. The figure also illustrates the fact that the variance of ε¯ is not a trivial function (e.g., constant) of ρ, but follows a more complex trend with competing tendencies. These results suggest investigating another property of the stochastic FLC, namely the sensitivity of ε¯ to each input parameter.

In a practical context, obtaining the statistics of the FLC as presented in Fig. 5 means an engineer can draw statistical conclusions regarding the safety of a given forming process. The principle would be to gather, from mechanical simulations, the critical deformation paths of the process. For each path, it is then possible to obtain an a priori probability of fracture of the material. Such information should then be analyzed with respect to the quality targets for the process and help in making well-informed design and manufacturing decisions. This approach provides more nuanced and richer information than a simple deterministic curve, accounting for the observed uncertainties related to the quantity of interest. Moreover, this "a priori" uncertainty quantification can be complemented with extra data if available. Such that an industrial user could use a Bayesian updating scheme to refine the uncertainty prediction to better represent the variability of its actual material properties.

Future work directions. The next goals in this study are first to propose a stochastic model for the FLC, going beyond empirical Monte Carlo (MC) estimations of its statistical properties. The second objective is to link the stochastic FLC with a forming simulation and propose a framework for uncertainty propagation to engineering decisions, as well as an active learning scheme [44] to reduce the FLC uncertainties with a semi-optimal experimental campaign.

A first approach would be to use a Gaussian process regression (GPR) model to emulate the stochastic FLC, but such an approach includes specific challenges:
(1) The covariance structure of the FLC needs to be thoroughly studied. It exhibits non-stationary variance that peaks at extreme ρ values and reaches a local minimum near plane strain. Future investigations entail evaluating the empirical variogram across ρ to detect non-stationarity, functional principal component analysis of Monte Carlo ensembles to identify dominant variation modes, and spectral density estimation of de-trended trajectories to determine correlation length scales.
(2) The physics requires FLCs to be decreasing for ρ<0 and increasing for ρ>0. Standard GP kernels violate these constraints. Constrained GP formulations (e.g., monotonic warping or projection onto shape-preserving reproducing kernel Hilbert spaces) require investigation.
(3) Since FLC realizations derive from only five random variables, the output process is inherently low-rank. Standard GP assumptions of infinite-dimensional latent functions may be over-parameterized.

An alternative approach would consist of calibrating a surrogate on the 6D space ( ρ,r0,r90,a,b,E ), thus respecting the deterministic structure and avoiding process-level constraints.

Conclusion

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This study has established a probabilistic framework for forming limit curve prediction in commercially pure titanium, addressing the inherent uncertainty that limits traditional deterministic approaches. By combining a computationally efficient linear perturbation model with literature-based parameter characterization and Monte Carlo propagation, the probabilistic approach can propagate material property uncertainties into quantifiable dispersion in FLC predictions. The computational efficiency of the underlying model-requiring only simple numerical integration-enabled direct Monte Carlo sampling without surrogate approximations.

Future efforts will focus on experimental validation through Nakazima testing with full-field DIC measurement to confirm predicted confidence intervals. Methodological extensions include hierarchical Bayesian calibration to distinguish uncertainty types, and investigation of constrained Gaussian process formulations to model the FLC directly as a stochastic process while respecting physics-based monotonicity requirements. The low-dimensional nature of the underlying parameter space suggests that direct surrogate modeling in ( ρ,r0,r90,a,b,E ) may prove more tractable than standard processlevel GP emulation.

This work aims to provide manufacturers with an updatable, uncertainty-aware design tool that balances risk reduction against overdesign, directly addressing industrial needs for robust sheet forming process control. This probabilistic framework provides the first quantitative uncertainty assessment for CP-Ti forming limits based on literature data. While the small sample size necessitates strong assumptions, the methodology is parsimonious, computationally efficient, and explicitly acknowledges its limitations. The resulting confidence intervals should be interpreted as plausible bounds rather than precise frequency-based probabilities. The modular structure facilitates continuous refinement as new data and theoretical advances emerge, moving toward true predictive uncertainty quantification for sheet metal forming applications. *

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The original version of this eBook is available on https://www.scientific.net/book/formability-of-metallic-materials/978-3-0364-1993-0

Solid State Phenomena, ISSN: 1662-9779, Vol. 388, pp 141-151

doi: 10.4028/p-iS64g3

© 2026 The Author(s). Published by Trans Tech Publications Ltd, Switzerland.

Submitted:
2026-01-19

Revised:
2026-02-12

Accepted:
2026-02-17

Online:
2026-04-15

Forming Limit Test of Asymmetric AI-Polymer Laminate Composite Film

The original version of this paper is available on https://www.scientific.net/SSP.388.141.pdf

Cheol Sagong 1,a, Taek Jin Jang 2, b, Taegyun Ahn 1,c and Jeong Whan Yoon 1,3, d*

¹Department of Mechanical Engineering, Korea Advanced Institute of Science and Technology, 291 Daehak-ro, Science Town, Daejeon 34141, Republic of Korea

2 SK On, 325, Expo-ro, Yuseong-gu, Daejeon, Republic of Korea

3 School of Engineering, Deakin University, 75 Pigdons Rd., Waurn Ponds, VIC 3216, Australia

a  sak9709@kaist.ac.kr, b  taekjin.jang@sk.com, c  t.ahn@kaist.ac.kr, d  j.yoon@kaist.ac.kr

Keywords: al-polymer laminate composite film, forming limit, modified marciniak test, polar effective plastic strain (PEPS) approach.

Abstract

As environmental issues arise, increasing battery efficiency is emerging as an important task. To manufacture more efficient lithium-ion batteries, the industry is striving to increase the depth of battery cell. A growing number of companies are focusing on developing pouch-type batteries using materials with high formability. To produce the battery cell with bigger depth, understanding of the formability of the material is important. Forming limit test can be considered. The material used in pouch-type batteries is an asymmetric aluminum-polymer laminate composite, which consists of four layers: polypropylene, aluminum, nylon, and PET. Because of the good formability of the polymer layers, a forming limit curve cannot be obtained using typical forming limit test such as experiment using Nakazima specimen. Therefore, the modified Marciniak testing method is implemented in this research, which is helpful for inducing strain concentration in the middle of the specimen. In the modified Marciniak test, a dummy sheet is layered between the specimen and the punch, to prevent the fracture around the region contacting with the fillet of the punch. In this research, different kinds of polymers were tried as the dummy sheet material. In addition, specimens of various designs were tested and the forming limit test results of the Al-Polymer film with thickness of 153\mum were converted using the Polar Effective Plastic Strain (PEPS) approach.

1. Introduction

The original version of this paper is available on https://www.scientific.net/SSP.388.141.pdf

With the global trend toward carbon emission regulations, interest in electric vehicles (EVs) has increased significantly, and the automotive industry is undergoing a rapid transition toward an EVcentered industrial structure. Accordingly, numerous companies are actively developing technologies aimed at improving the efficiency of lithium-ion batteries used in electric vehicles. There are three types of battery cells-cylindrical, prismatic and pouch types. Especially, because of the excellent formability of the pouch-type battery material (Al-Polymer film), it is possible to obtain superior energy efficiency. In addition, the shape of the pouch-type battery cell is similar to that of the prismatic one, and therefore it can also benefit in terms of spatial efficiency. Thus, pouch-type cells are the primary focus of the present study.

To enhance the energy efficiency of the battery cells, the forming depth should be increased. However, some defects such as wrinkling and fracture can occur during the forming process. To prevent these defects, it is essential to identify the proper forming condition through numerical simulation. For finite element simulation of the process, constitutive models and forming limit curves (FLCs) of the material must be established first. The raw material of the pouch-type battery cell is a 4-layer laminated composite film and four different materials are polypropylene, aluminum, nylon and PET as illustrated in Fig. 1. Because of this complicated structure, relatively limited research has been conducted on experimental investigation and material modeling. Jang et al.[1] developed material models based on optimization. In addition, Moon et al.[2] obtained the properties of each individual layer by physically separating the pouch film.

[picture]

Fig. 1. Cross-sectional view of pouch film.

Fig. 1. Cross-sectional view of pouch film.

However, because of the superior formability of the material, determining the forming limit curve experimentally is a highly challenging problem. Moon et al.[3] established a forming limit curve using the Nakazima test. However, strain localization did not occur at the center of the specimen for some cases. In addition, during the forming limit test for the nylon/PET layers, nonlinear strain paths were observed. Forming limit curves are sensitive to strain paths, and therefore these issues prevent to obtain the accurate FLC of the material.

In the present study, conventional forming limit testing methods such as the Marciniak test [4] and the Nakazima test [5] were first conducted. Based on these experimental results, the limitations of these conventional methods were identified. Subsequently, the testing method was improved based on the modified Marciniak test proposed by Bong et al. [6]. To avoid the strain path nonlinearity problem, circular specimens proposed by Kim et al. [7] were employed. Furthermore, to overcome the excellent formability induced by the polymer layers in the pouch film, dummy sheets were fabricated using a type of polymer material, EPDM. Finally, to address the remaining strain-path nonlinearity, the polar effective plastic strain (PEPS) approach proposed by Stoughton and Yoon [8] was implemented.

2. Test Results for Conventional Forming Limit Test

The original version of this paper is available on https://www.scientific.net/SSP.388.141.pdf

As an initial phase of the investigation, conventional forming-limit testing methods, specifically the Marciniak and the Nakazima tests were implemented. As illustrated in Fig. 2(a), a flat-headed punch is employed in the Marciniak test. This configuration allows strain localization to occur at the central region; however, it is also prone to generating frictional stresses on specific region of the specimen which is contacting the punch's fillet. In contrast, the Nakazimat test (Fig. 2(b)) incorporates a hemispherical punch, thereby avoiding the excessive fillet-induced strain concentration. Owing to the shape of the punch, it is possible to induce the more homogeneous deformation then in the Marciniak test. However, necking can be initiated after a comparatively larger punch displacement.

[picture]

Fig. 2. Schematic of conventional forming limit test (a) Marciniak test (b) Nakazima test.

Fig. 2. Schematic of conventional forming limit test (a) Marciniak test (b) Nakazima test.

Through the DIC image of the Marciniak test (Fig. 3), it can be confirmed that deformation is predominantly concentrated in the punch-fillet region from the very early stage of the test. This is the result of excessive friction nearby the punch's fillet and this friction impeded deformation in the central area of the specimen. As a result, localized necking is initiated at the fillet, thereby preventing the acquisition of meaningful forming-limit data along the intended strain path.

[picture]

Fig. 3. DIC image of Marciniak test in very early stage (major strain).

Fig. 3. DIC image of Marciniak test in very early stage (major strain).

Fig. 4 displays the DIC results from the Nakazima test. Although the hemispherical punch suppressed initial strain concentration which is observed in the Marciniak test, localization instead occurred along the specimen edges prior to the initiation of central necking. This behavior is attributed to the exceptionally high formability of the pouch film, which allowed the punch to advance excessively until its tip became exposed. As illustrated in Fig. 5, the direct contact between the punch tip and the specimen induced localized deformation, and this inhibited the development of necking in the central region. Consequently, similar to the Marciniak test, the experiment could not be successfully completed. The fractured Nakazima specimens are shown in Fig. 6.

[picture]

Fig. 4. DIC image of Nakazima test (major strain).

Fig. 4. DIC image of Nakazima test (major strain).

[picture]

Fig. 5. Nakazima punch's crack tip.

Fig. 5. Nakazima punch's crack tip.

[picture]

Fig. 6. Specimens after Nakazima test.

Fig. 6. Specimens after Nakazima test.

3. Development of Testing Method

The original version of this paper is available on https://www.scientific.net/SSP.388.141.pdf

Based on the experimental results and analyses of both conventional forming-limit test methods, the Marciniak testing method was modified in this study. Following the methodology proposed by Bong et al.[6], a dummy sheet was introduced, and the circular-shaped specimen geometry was implemented as recommended by Kim et al [7]. In addition, the polymer-made dummy sheet material was employed, to enable the establishment of forming-limit test conditions suitable for obtaining reliable results from thin films with superior formability. The configuration of the tool used in the test is illustrated in Fig. 7.

[picture]

Fig. 7. Geometry of forming limit testing tool [ mm ].

Fig. 7. Geometry of forming limit testing tool [ mm ].

Modified Marciniak Test with Dummy Sheet Made of Stainless Steel.

The original version of this paper is available on https://www.scientific.net/SSP.388.141.pdf

To avoid the challenges observed in the Nakazima test arising from the material's high formability, a modified Marciniak test was implemented to induce strain localization at the center of the specimen. Bong et al. [6] developed this modified approach as an improvement to the conventional Marciniak test, and they implemented the specimen geometry proposed by Raghavan [9]. In their experiment, a dummy sheet was placed between the punch and the specimen to reduce excessive strain accumulation near the punch fillet, while the friction between the dummy and the specimen made dragging possible. Additionally, a lubricant was applied between the dummy sheet and the punch to minimize friction around the fillet-contacting region. The underlying principle of the modified Marciniak test is illustrated in Fig. 8, with the specimen geometry detailed in Bong et al.[6], where a 1 mm -thick stainless steel sheet served as the dummy sheet. A circular hole was introduced at the center of the dummy sheet so that, during deformation, only the specimen area in direct contact with the dummy was dragged, thereby inducing strain localization in the central region of the specimen.

[picture]

Fig. 8. Schematic of Modified Marciniak test.

Fig. 8. Schematic of Modified Marciniak test.

Similarly, the modified Marciniak test was performed on the pouch film using a stainless steel dummy sheet. Fig. 9 illustrates the deformation of the stainless steel dummy sheet and the specimens across strain states ranging from uniaxial tension to plane strain on the forming-limit diagram. As illustrated in Fig. 9(a), the dummy sheet underwent uniaxial tension in one direction and compression in the perpendicular direction, leading to wrinkling and bulging. Consequently, as shown in Fig. 9(b), deformation occurred around the dummy sheet hole rather than at the specimen center. For tests along the equi-biaxial path, fracture initiated at the hole edge of the dummy sheet before specimen necking, preventing the acquisition of reliable forming-limit data. The fractured dummy sheet and corresponding specimen are presented in Fig. 10.

[picture]

Fig. 9. Deformation of dummy sheets and specimen from uniaxial tension to plane strain.

Fig. 9. Deformation of dummy sheets and specimen from uniaxial tension to plane strain.

[picture]

Fig. 10. Failure of dummy sheet and specimen in equi-biaxial loading.

Fig. 10. Failure of dummy sheet and specimen in equi-biaxial loading.

Specimen Design Study.

The original version of this paper is available on https://www.scientific.net/SSP.388.141.pdf

Using the Raghavan specimen, the non-uniform radial distance from the center to the edge led to sensitivity to the blank holding force, as confirmed by repeated trials. In the plane strain region, nonlinear behavior was observed. Since the forming limit can be affected by the strain paths, path linearity is essential. To maintain the strain path linearly, the circular specimen design proposed by Kim et al. (Fig. 11(a)) [7] was adopted, and as shown in Fig. 11(b), it produced slightly more linear strain paths compared to the Raghavan specimen. In Fig. 11(b), the test results labeled as "Kim et al." were obtained using the width-60 specimen shown in Fig. 11(a), whereas the test results labeled as "Raghavan" were obtained using the plane strain specimen design proposed by Bong et al.[6].

[picture]

Fig. 11. (a) Kim et al's specimen [7] (b) Paths' nonlinearity change by specimen design.

Fig. 11. (a) Kim et al's specimen [7] (b) Paths' nonlinearity change by specimen design.

Dummy sheets' material.

The original version of this paper is available on https://www.scientific.net/SSP.388.141.pdf

Due to the limitations of stainless steel as a dummy sheet, materials with higher formability and lower stiffness were employed. Three polymers were selected: 1.0 mm -thick silicone, 1.4 mm -thick urethane, and 1.6 mm -thick EPDM. Uniaxial tensile tests (Fig. 12(a)) confirmed that all candidates exhibited very high elongation. The dummy sheets fabricated using these polymers are shown in Fig. 12(b) and Fig. 12(c).

[picture]

Fig. 12. (a) Result of uniaxial tension of polymer (b) Urethane dummy sheets (c) EPDM dummy sheets.

Fig. 12. (a) Result of uniaxial tension of polymer (b) Urethane dummy sheets (c) EPDM dummy sheets.

Test results indicated that the stiffness of urethane is much higher compared with that of the pouch film, causing strain localization near the edge of the dummy sheet hole, similar to the stainless steel case. In contrast, using EPDM enabled successful central strain localization across all strain paths with fracture consistently occurring at the specimen center for every strain path (Fig. 13).

[picture]

Fig. 13. Fractured specimens after forming limit test results with EPDM dummy sheets.

Fig. 13. Fractured specimens after forming limit test results with EPDM dummy sheets.

4. Test Results

The original version of this paper is available on https://www.scientific.net/SSP.388.141.pdf

Forming-limit tests conducted using this method showed that, while deformation was successfully localized at the specimen center, strain paths from the plane strain to the equi-biaxial regions remained strongly nonlinear (Fig. 14). The degree of nonlinearity is too pronounced to be approximated as linear, complicating accurate forming-limit curve determination and necessitating the introduction of a correction method.

[picture]

Fig. 14. Nonlinearity of strain paths.

Fig. 14. Nonlinearity of strain paths.

To address this issue, the forming-limit curve was converted using the path-independent approach based on Polar Effective Plastic Strain (PEPS), as developed by Stoughton and Yoon [8]. The PEPS method transforms the conventional forming-limit diagram, defined in terms of major and minor
strains, into a polar coordinate system based on the magnitude and orientation of strain increments (Fig. 15). For the normal anisotropic model, the effective plastic strain is calculated as

ε¯p=1+r1+2r∫ε˙12+ε˙22+2r1+rε˙1ε˙2dt(1)

and the angle for the increment, θ is calculated as

θ=tan−1(ε˙2ε˙1)(2)

[picture]

Fig. 15. Schematic of PEPS approach (a) path sensitive FLD on major-minor strain coordinate (b) path-independent polar effective plastic strain diagram.

Fig. 15. Schematic of PEPS approach (a) path sensitive FLD on major-minor strain coordinate (b) path-independent polar effective plastic strain diagram.

where r is the r-value(the Lankford coefficient) of the material. This enables the construction of a path-independent forming limit curve. Since the experiments in this study were performed along the transverse direction, the TD r-value was adopted in the PEPS formulation. The r-values in each direction were obtained from uniaxial tensile tests, and the corresponding values are summarized in Table 1. In addition, Fig. 16 presents the results of the uniaxial tensile tests conducted along different directions of the pouch film.

Table 1. Lankford coefficients in each direction.

RD | DD | TD
0.723 | 1.256 | 0.828

[picture]

Fig. 16. Stress-strain curves of the pouch film.

Fig. 16. Stress-strain curves of the pouch film.

In this study, forming-limit test results obtained from nonlinear strain paths were converted using the PEPS-based polar coordinate system, and they are represented in the Fig. 17.

[picture]

Fig. 17. Nonlinear paths in PEPS-based polar coordinate system.

Fig. 17. Nonlinear paths in PEPS-based polar coordinate system.

Fig. 18 (a) shows the results of the PEPS-based analysis applied to all experimental results, including those with linear strain paths. The reconverted FLD into strain space without the elements over the biaxial range is represented in Fig. 18 (b). In addition, the forming limit curve obtained after excluding elements beyond the biaxial region was also plotted.

[picture]

Fig. 18. (a) FLD converted using PEPS approach (b) Reconverted FLD into strain space by excluding the elements over the biaxial range.

Fig. 18. (a) FLD converted using PEPS approach (b) Reconverted FLD into strain space by excluding the elements over the biaxial range.

5. Discussion

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Among the elements shown in Fig. 18(b), the data corresponding to the region from uniaxial tension to plane strain followed linear strain paths and were therefore included in the construction of the forming limit curve. In contrast, the elements located between the plane strain and the equi-biaxial regions exhibited nonlinear strain paths and were thus excluded, as they are not suitable for defining the forming limit curve in strain space. After excluding these elements, the resulting trend shows a decrease in major strain with increasing minor strain, which is consistent with the behavior commonly observed in most thin sheet metal materials.

Notably, Moon et al.[6] reported that nonlinear strain paths were observed even when forming limit tests were conducted using specimens composed solely of nylon and PET layers[3], without the polypropylene and aluminum layers. By interpreting this observation, the nonlinear strain paths are strongly influenced by the differences in the mechanical behavior of the layers - not only between the metal and polymer, but also between the polymer layers.

Strain-rate sensitivity and viscoelastic-plastic behavior are important behaviors which are observed in polymers, and these characteristics can affect the strain increment. In principle, forming limits should be determined based on plastic deformation; however, even after the aluminum has started the plastic deformation, the polymer may have continued to undergo non-negligible elastic deformation and this can be one of the reasons for a nonlinear strain path. In addition, the presence of weak or partially degraded interfaces may promote interlayer sliding or delamination during forming, further
complicating the strain evolution. These interfacial phenomena can locally relax in-plane constraints and redistribute strains, and thus the biaxial-dominant modes deformation can be induced. Therefore, to explain the strain-path nonlinearity of the pouch film, the interaction between each layer should be considered along with the layers' material properties as well.

Even though the elements beyond the biaxial range were excluded using the PEPS approach, further research is necessary to obtain data in the region between plane strain and equi-biaxial tension. Future work will be extended beyond geometric specimen optimization, polymer layer's constitutive behavior and interlayer delamination mechanisms into both experiment and numerical modeling. Designing a novel specimen that enables maintenance of deformation modes in biaxial tensile loading is essential. In addition, integrating such advanced material modeling with specimen design optimization and the PEPS-based framework is expected to enable the construction of a complete and physically consistent forming limit curve for the pouch film.

6. Conclusion

The original version of this paper is available on https://www.scientific.net/SSP.388.141.pdf

In this study, the forming limit of pouch-type battery films was investigated through conventional and modified forming limit tests. In addition, the nonlinear strain-path of the material was emphasized. Conventional forming limit tests were first conducted, and it was identified the limitation of these tests in preserving strain linearity during testing were identified. Based on these test results, the modified Marciniak testing method was implemented, with circular specimens and EPDM dummy sheets to suppress undesired deformation and enhance strain localization in the center.

To transform the forming limit data, which show the nonlinear strain paths, into a path-independent framework, the polar effective plastic strain (PEPS) approach was implemented. However, in the region between plane strain an equi-biaxial tension, it was not able to construct the forming limit due to the critical limitations of the PEPS method. The major strain increment became smaller than the minor strain increment as the test proceeded, driving the strain state beyond the equi-biaxial region.

The present results demonstrate that strain-path nonlinearity in the material cannot be fully explained by specimen geometry or metal-polymer interactions alone. According to the experimental results showing nonlinear strain paths in nylon-PET-only specimens conducted by Moon et al[6], it is evident that interlayer delamination mechanisms are governing strain evolution. Therefore, it should be interpreted that the difficulty in obtaining the FLC arises from the fundamental limitation of current forming limit testing methods, rather than the limitation of the PEPS.

For future work, the development of novel specimen designs will be focused on inducing desired linear deformation modes between plane strain and biaxial tensile region. In parallel, advanced material modeling incorporating strain-rate-dependent and interlayer delamination behavior will be pursued to enable more accurate interpretation of experimental strain paths. The integration of experiments and material modeling is expected to provide a more comprehensive and physically consistent forming limit description. Therefore, the predictive capability of forming simulations for pouch-type battery cell manufacturing processes can be improved.

Acknowledgement

The original version of this paper is available on https://www.scientific.net/SSP.388.141.pdf

This work was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (RS-2025-25406725). This work is also partially supported by Materials and Components Technology Development Program (RS-202400432591) funded by the Ministry of Trade, Industry and Resources (MOTIR, Korea).

References

The original version of this paper is available on https://www.scientific.net/SSP.388.141.pdf

Jang, T. J., Sagong, C., Ahn, T., & Yoon, J. W. (2024). Characterization of layered anisotropic properties for Li-ion battery pouch film and its application to forming. CIRP Annals, 73(1), 225-228.

Moon, C., Lian, J., & Lee, M. G. (2023). Identification of elastic and plastic properties of aluminum-polymer laminated pouch film for lithium-ion batteries: A hybrid experimentalnumerical scheme. Journal of Energy Storage, 72, 108601.

Moon, C., hyeon Park, J., Lian, J., Lee, H. L., Ma, J., & Lee, M. G. (2025). Formability of laminated composite pouch for Li-ion battery: Analysis based on experiments and modeling. Journal of Power Sources, 653, 237746.

Marciniak, Z. & Kuczy'nski, K. (1967). Limit strains in the processes of stretch-forming sheet metal, International journal of mechanical sciences, 9(9) 609-620.

Nakazima, K., Kikuma, T. & Hasuka, K. (1968). Study on the formability of steel sheets, Yawata Technical Report, 264, 8517-8530.

Bong, H. J., Barlat, F., Lee, M. G., & Ahn, D. C. (2012). The forming limit diagram of ferritic stainless steel sheets: Experiments and modeling. International Journal of Mechanical Sciences, 64(1), 1-10.

Kim, S. B., Huh, H., Bok, H. H., & Moon, M. B. (2011). Forming limit diagram of auto-body steel sheets for high-speed sheet metal forming. Journal of Materials Processing Technology, 211(5), 851-862.

Stoughton, T. B., & Yoon, J. W. (2012). Path independent forming limits in strain and stress spaces. International Journal of Solids and Structures, 49(25), 3616-3625.

Raghavan, K. S. (1995). A simple technique to generate in-plane forming limit curves and selected applications. Metallurgical and materials transactions A, 26(8), 2075-2084.

This ePub file was created using machine-generated methods and automated software to convert the source text into an ePub format and may contain formatting inconsistencies from the original version. Formatting may vary depending on the device used.

The original version of this eBook is available on https://www.scientific.net/book/formability-of-metallic-materials/978-3-0364-1993-0

Solid State Phenomena, ISSN: 1662-9779, Vol. 388, pp 153-162

doi: 10.4028/p-IUrc7E

© 2026 The Author(s). Published by Trans Tech Publications Ltd, Switzerland.

Submitted:
2026-01-19

Revised:
2026-02-16

Accepted:
2026-02-20

Online:
2026-04-15

Hole Expansion Performance Assessment of Quenched and Partitioning Steel Through a Fully Anisotropic Fracture Modeling

The original version of this paper is available on https://www.scientific.net/SSP.388.153.pdf

Toros Arda Aksen 1,2,a, Zinan Li1, b and Junhe Lian 1,3,c*

¹Department of Energy and Mechanical Engineering, Aalto University, 02150, Espoo, Finland

2 Mechanical Engineering Department, Sakarya University, 54050, Sakarya, Türkiye

3 Institute of Metal Forming, RWTH Aachen University, 52072, Aachen, Germany

a  ardaaksen@sakarya.edu.tr, b  zinan.li@aalto.fi, c  junhe.lian@ibf.rwth-aachen.de

*corresponding author

Keywords: QP steel, local formability, edge crack, fully anisotropic fracture model.

Abstract

Edge cracking is an essential local formability phenomenon encountered in specific forming operations, such as stretch flanging, which is broadly employed in the automotive industry. However, the prediction of the edge cracks is challenging, and the hole expansion test is widely carried out to detect the edge cracking performance. This research analyzed quenched and partitioned steel (QP), one of the third-generation advanced high-strength steels widely adopted in the automotive industry to replace conventional high-strength steels due to its superior global formability features. However, its local formability has been a bottleneck due to its microstructure nature and shows strong anisotropy dependency. Therefore, a modeling framework is needed that consistently incorporates anisotropy in both plasticity and fracture. In this study, the hole expansion performance of QP1000 steel was evaluated through a fully anisotropic fracture model based on the Yld2004-18p anisotropic yield criterion and the DF2016 ductile fracture model. To this end, the standard uniaxial tensile tests in seven material orientations, the bulge test, and the tensile tests of different fracture samples were conducted. These samples were picked out to characterize the broad spectrum of loading conditions and cut in rolling, diagonal, and transverse directions to establish the fully anisotropic fracture model. It was seen that the edge cracking metrics, which are the hole expansion ratio and the fracture initiation zone, were accurately captured by the developed model.

Introduction

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The transportation sector causes a considerable amount of harmful gas emissions, and road vehicles have a major role in these emissions [1]. Therefore, the international regulations and competition between the automotive companies necessitate the automotive industry to have a primary aim to decrease harmful gas emissions. In this framework, the automotive companies tended to develop new technologies in lightweight materials. Advanced high-strength steels (AHSS) are one of these materials, and Quenched and Partitioning steels (QP) are the salient steels to replace the conventional high-strength steels in the automotive industry [2,3]. The QP steel ensured the high strength and formability relative to the other AHSSs; however, edge cracking is a vital topic in these steels. Certain AHSSs demonstrate low edge cracking performance [4, 5], and the studies on the edge cracking phenomenon in QP steels are limited.

Edge cracking is a local formability feature where deformation is localized in specific regions, which is difficult to detect. The hole expansion test is conducted to evaluate the edge cracking performance, and numerous studies in the literature have examined the hole expansion performance of AHSSs. Chung et al. [4] investigated the hole expansion performance of TRIP590 and TWIP940 steels in terms of hole expansion ratio (HER). They pointed out that the TWIP steel exhibited poor hole expansion performance. Barnval et al. [6] carried out a similar study on first-generation AHSSs, namely DP and TRIP steels. Firat et al. [7] conducted a study to assess the local formability features of DP steels. Recently. Li et al.[8] investigated the edge fracture of DP1000 steel and discovered that anisotropy-introduced localization has been the trigger of the final fracture. In all
these studies, the significance of an advanced anisotropic yield criterion in local formability was emphasized.

Contrary to the first and second generation AHSSs, the studies related to the local formability of third generation steels, especially QP steel, are limited. Madrid et al. [9] assessed the effect of edge conditions and the punch geometry on the edge cracking performance of QP steels. Li et al [10] conducted a study related to the fracture performance of QP1000 steel. They used the Hill48 criterion with crucial modifications and performed the fully anisotropic fracture modeling. There are also different studies related to the fracture of QP steels [11-15]. However, studies relevant to the edge cracking predictions are limited, and numerical studies mostly include basic anisotropic yield criteria or isotropic yield criteria [14, 15]. A recent study by Aksen et al. [16] applied several anisotropic yield criteria, including different Hill formulations, and the Yld2004-18p model on QP1000 steel, and they revealed that only Yld2004-18p can predict the location of the edge fracture accurately. However, due to the nature of the isotropic formulation of the fracture criterion, the model still fails to predict the HER value correctly. Therefore, a fully anisotropic fracture model in conjunction with an advanced anisotropic plasticity model is essential to evaluate the edge cracking properties of QP1000 steel.

This study evaluates the performance of fully anisotropic fracture modeling in predicting edge cracking of QP1000 steel in HET. In this framework, a material model including the Yld2004-18p anisotropic yield criterion, coupled with the DF2016 ductile fracture model, was adopted. Standard uniaxial tensile tests (SDB) were conducted in seven orientations, covering from the rolling direction (RD) to the transverse direction (TD). To include the biaxial data in the yield criterion calibration, a bulge test was conducted. In addition, fracture samples, including a centered hole sample (CH), radius-shaped notched samples (NDBR), and a shear sample, were employed to characterize the fracture mechanism across a wide spectrum of loading conditions. Furthermore, these fracture samples were cut in three orientations to represent the fully anisotropic fracture model. Afterward, the hole expansion test of QP1000 steel with a conical punch configuration was performed in order to observe the edge cracking metrics, and the finite element (FE) analyses were conducted. The prediction performance of the calibrated model was evaluated in terms of edge cracking metrics in QP1000 steel in the hole expansion test.

Mechanical Testing

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The standard uniaxial tensile test samples of QP1000 steel were cut in seven orientations from RD to TD ( 0∘,15∘,30∘,45∘,60∘,75∘,90∘ ) with a thickness of 1.36 mm [10,16], and the tests were carried out in compliance with the ISO6892 specifications under quasi-static testing rates using Zwick-Roell screw-driven tensile testing machine with an extensometer of 30 mm gauge length. The result of the RD sample was used to adjust the hardening behavior of QP1000 steel. Fig. 1 shows the engineering stress-strain curves acquired from the SDB samples and the hardening curve based on mixed Swift-Voce type hardening law (The parameters are given in ref. [10]).

[picture]

Fig. 1. a) Engineering stress-strain behaviors in different orientations and b) flow curve in RD for QP1000 steel.

Fig. 1. a) Engineering stress-strain behaviors in different orientations and b) flow curve in RD for QP1000 steel.

The r values and stress ratios were obtained from the SDB sample tests conducted at various strain levels. The r values tend to saturate at the strain level corresponding to ultimate tensile stress; therefore, the strain values at the saturation point were regarded. However, the stress ratios varied throughout the deformation. In this regard, the stress ratios at 0.05 strain level were regarded in order to represent the general behavior of the variation. Table 1 shows the r values and stress ratios considered in this study.

Table 1. Engineering stress-strain behaviors in different orientations for QP1000 steel.

Orientation (deg.) | 0° | 15° | 30° | 45° | 60° | 75° | 90°
Stress ratio (σθ) | 1 | 1.004 | 0.995 | 0.996 | 1.019 | 1.021 | 1.027
r value (rθ) | 0.75 | 0.65 | 0.73 | 0.91 | 0.71 | 0.7 | 0.86

Afterwards, the fracture test samples were conducted in RD, diagonal direction (DD), as well as TD. The dimensions and the testing conditions are explained in [10]. The force-displacement behaviors of fracture samples are given in Fig. 2 for three orientations. These experimental data, including the SDB data, can also be accessed via Zenodo repositories [17].

[picture]

Fig. 2. Force-displacement behaviors of fracture samples in a) RD, b) DD, and c) TD.

Fig. 2. Force-displacement behaviors of fracture samples in a) RD, b) DD, and c) TD.

The QP1000 steel did not demonstrate a significant difference in force-displacement behaviors along three orientations; however, ductility decreased noticeably in TD orientations, indicating that anisotropic ductility is an essential feature of QP1000 steel. To specify the biaxial r value and stress ratio, a bulge test was conducted with a square sample of 230 mm length. Punch moved at a rate of 3 mm/min while the sample was clamped with a holder force of 600 kN . Comprehensive information can also be found in [8], including the dimensions of the tools. In the final stage, an HET was conducted in accordance with the ISO 16630 specifications. The 10 mm hole was cut by electro discharging machine to ensure a higher hole edge surface quality. Thus, the microcracks due to the punch process were diminished. The hole was expanded through a conical punch with an apex angle of 60∘. The blanking force was applied as 235 kN to a circular blank with a radius of 90 mm . In the meantime, the punch moved at a constant rate of 5 mm/min. The sample was tested until a through-thickness crack (TTC) occurred.

Constitutive Modeling

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Within the scope of this research, an anisotropic plasticity model was considered. In this perspective, the Yld2004-18p yield criterion was adopted to characterize the material anisotropy. In order to account for the fracture mechanism, the DF2016 ductile fracture model was implemented.

Plasticity Model.

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The Yld2004-18 yield criterion encompasses the r value and stress ratio directionalities, in addition to the biaxial data. This criterion is expressed as in Eq. 1 [18].

fYld2004−18p=|S1′−S1′′|m+|S1′−S2′′|m+|S1′−S3′′|m+|S2′−S1′′|m+|S2′−S2′′|m+|S2′−S3′′|m+|S3′−S1′′|m+|S3′−S2′′|m+|S3′−S3′′|m=4σeqm.(1)

Here, S1−3′ and S1−3′′ denote the principal stresses of the first and second linear transformeddeviatoric stress components. The first and second linear transformations of the deviatoric stress tensors are given in Eq. 2 and Eq. 3.

𝐒′=Sij′=Cij′Tijσij=[0−α1−α2000−α30−α4000−α5−α60000000α7000000α8000000α9][SxxSyySzzSyzSzxSxy].𝐒′′=Sij′′=Cij′′Tijσij=[0−α10−α11000−α120−α13000−α14−α150000000α16000000α17000000α18][SxxSyySzzSyzSzxSxy].

(2)(3)

In Eqs. 1 and 2, the αi parameters refer to the anisotropy parameters that need to be optimized based on the yield conditions in different orientations and the balanced-biaxial loading. For the optimization process, the least squares method was employed. The associated flow rule for the relation between plastic potential and plastic strain increment is given in Eq. 4.

ε˙p=dεijp=dλ∂fYld2004∂σij(4)

Finally, the isotropic hardening rule was considered, which is represented by the mixed hardening law encompassing the Swift-Voce laws given in Eq. 5.

σModulated =α(C(ε0+εpl)p)+(1−α)(k0+Q(1−e−bεpl)).(5)

Fully Anisotropic Fracture Model.

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Similar to Li et al. [10], a fully anisotropic fracture model was established to reflect the variation in fracture strain with different material orientations, considering the linear transformation of plastic strain components at fracture, which simplifies calibration and implementation. To this end, five different fracture samples were tested in three orientations, namely RD, DD, and TD. For each sample, the plastic strain components at fracture were procured from the FE analyses, and their transformations were regarded. The transformed strain increment tensor (ε˙T= [ ε˙11T,ε˙22T,ε˙33T,ε˙23T,ε˙13T,ε˙12T ]) can be expressed by Eq. 6.

ε˙T=Lε˙p(6)

Here, 𝐋 is the modified linear transformation tensor given in Eq. 8, and the ε˙p is the strain increment tensor.

L=[β13−β12+β13000−β21+β23β23000−β31−β3200000β4400000β5500000β66](7)

The parameters βij are the transformation parameters that need to be optimized. The transformed equivalent plastic fracture strain can be determined by Eq. 8 .

ε˙eqvT=23ε˙eqvT:ε˙eqvT(8)

The DF2016 fracture model accounts for both stress triaxiality and Lode parameter effects to describe the fracture mechanism [19]. The DF2016 criterion can be expressed as in Eq. 9 regarding the fully anisotropic fracture modeling approach.

εeqvT(η,L)=C3a(2τmaxσeqv)1a(⟨η+C4a3−L3L2+3+Ca13+C423a+Ca⟩)C2a(9)

In Eq. 7, η and L refer to stress triaxiality and Lode parameter, while the parameters c1−4 and c denote the fracture model parameters.

Material Characterization

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The parameters of the Yld2004-18p criterion were optimized based on the identification procedure explained in our earlier study [16]. For completeness of information, Table 2 lists the optimized parameters, and Fig. 3 illustrates the analytical predictions of the yield criterion.

Table 2. The anisotropy parameters of Yld2004-18 criterion.

α'₁ | α'₂ | α'₃ | α'₄ | α'₅ | α'₆ | α'₇ | α'₈ | α'₉
0.396 | 0.43 | 0.824 | 0.747 | -0.175 | 0.959 | 1 | 1 | 0.952
α''₁ | α''₂ | α''₃ | α''₄ | α''₅ | α''₆ | α''₇ | α''₈ | α''₉
0.655 | 1.507 | 0.683 | 0.837 | 1.465 | 0.87 | 1 | 1 | 0.983

Fig. 3. Analytical predictions of Yld2004-18p a) Stress ratios, b) r values, c) yield locus.

The Yld2004-18p criterion accurately predicted the experimental data directionalities used in the identification procedure. The parameters and the flow curve fitted based on the mixed hardening
law can be found in ref. [10]. To calibrate the fracture model, the FE analyses of the fracture samples were conducted, and the acquired results were compared with the experimental outcomes. The element sizes were kept at 0.05 mm in all directions at the center of the geometrical discontinuities for each fracture sample [16]. Fig. 4 illustrates the comparative results of fracture samples' force-displacement responses for CH, NDBR10, and SH.

[picture]

Fig. 4. Force-displacement predictions of the a) CHDB6, b) NDBR10, and c) SH samples.

Fig. 4. Force-displacement predictions of the a) CHDB6, b) NDBR10, and c) SH samples.

The force-displacement predictions were consistent with experimental results. In CHDB6 and NDBR samples, the predicted results in RD and DD orientations provided good approximations; however, minor discrepancies were observed in TD orientation. In this regard, the implemented model could predict localization in RD and TD; however, it was less effective at predicting localization in TD. For NDBR samples, the maximum equivalent strain was seen at the center of the notched region, and at the middle of the through-thickness. For CHDB6 samples, it was found to be at the hole edge or a point so close to the hole edge (A minor shift was observed for RD and DD samples). As for SH samples, the maximum equivalent plastic strain at the shear zone was regarded. The plastic strain components, stress triaxiality, and Lode parameter variations were acquired from the FE simulations at fracture points of all samples. Since the Lode parameter and the triaxiality were not constant, their average values were determined. In the first stage, the transformations of the strain components were determined (In terms of transformation parameters); afterwards, the fracture model parameters and the transformation parameters were optimized simultaneously to calibrate the fully anisotropic fracture model. The optimized transformation parameters are listed in Table 3, while the 2D fracture locus alongside the variations of the stress triaxiality are given in Fig. 5.

Table 3. The transformation parameters for the fully anisotropic fracture model.

β₁₂ | β₁₃ | B₂₁ | β₂₃ | β₃₁ | β₃₂ | β₄₄ | β₅₅ | β₆₆
1.455 | 0.212 | 1.361 | 0.497 | 0.294 | 0.424 | 0.001 | 6.700 | 0.189

Fig. 5. a) Stress triaxiality variations, b) 2D fracture locus of fully anisotropic fracture model (The parameters of DF2016 are C1=4.12,C2=0.63,C3=0.89,C4=0.241, and C=0.33 ).

The equivalent plastic fracture strains were depicted as dots with different shapes (Triangles, squares and rhombuses represent the CHDB6, NDBR25 NDBR10, respectively, while the circles at the left with black outline denote the SH and the circles at the right without outline refer to NDBR6 samples) in Fig. 5b. All these points correspond to the equivalent plastic fracture strain value for the mean stress triaxiality level of each fracture sample's cutting orientation. The 2D fracture locus generated by the fully anisotropic fracture model provided good approximations for each data point obtained from the sample and orientation. However, some difficulties were encountered while capturing the TD orientation data. These difficulties may stem from the fact that minor overestimations were observed in the force-displacement responses of the fracture samples of CHDB and NDBR (see Fig. 4). The implemented model could not successfully capture localization in the TD orientation. This situation may cause the fully anisotropic fracture locus to be offset downward during the optimization process. Since the force-displacement results in RD and DD were accurate and the localizations in these orientations were well characterized, higher strains may be obtained for these orientations compared to TD. Recently, in the paper by Li et al. [10], the researchers regarded the stress evolution during deformation and incorporated it into their FE simulations. Thus, they could capture the force-displacement responses in all orientations using the Hill48 criterion, and they reported higher strains and triaxiality levels in TD orientation compared to this study. Nevertheless, they also reported lower strain and triaxiality levels in TD than in RD and DD orientations in their research, consistent with the outcomes of this research. On the other hand, the results procured from the SH samples were also similar in all directions in this research. For each orientation, the equivalent plastic fracture strains and the stress triaxiality levels were close to each other. The differences in predicted performance for SH did not lead to a significant difference. Besides, although Li et al. [10] provided good approximations in SH samples considering the stress evolution, the SH results in this study were also consistent with the paper by Li et al in terms of strain and triaxiality levels. Afterward, the calibrated parameters were tested in fracture samples to predict the force drops in the simulations. Fig. 6 shows the results of forcedisplacement responses of fracture samples with the calibrated model.

[picture]

Fig. 6. Force-displacement predictions of fully anisotropic fracture model a) CHDB6 and SH, b) NDBR25, c) NDBR10, d) SH.

Fig. 6. Force-displacement predictions of fully anisotropic fracture model a) CHDB6 and SH, b) NDBR25, c) NDBR10, d) SH.

In NDBR6 predictions, the fully anisotropic fracture model overpredicted the experiments, which resulted from the difficulties in capturing the NDBR6 data alongside the TD data in the fracture locus. Nevertheless, the results were found to be consistent with the experiments for other samples.

Hole Expansion Test Simulations

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The FE simulation of HET was carried out in Marc software coupled with a user-defined material subroutine. The blank was meshed in the Apex software, then transferred to the Marc software. The elements were increased near the hole edge, and the element sizes near the hole were kept 0.05 mm in all directions to ensure consistency with the element density in geometrical discontinuities of the fracture samples. Fig. 7 shows the FE model and the blank mesh.

[picture]

Fig. 7. a) FE model of HET, b) blank's mesh layout.

Fig. 7. a) FE model of HET, b) blank's mesh layout.

The friction coefficient was assumed to be 0.1 between all the contact interactions. The punch force-displacement curve, the first through-thickness crack, and the HER value were predicted by the FE model. Fig. 8 shows the metrics of local formability procured from the HET simulations.

[picture]

Fig. 8. a) Punch-force-displacement results, b) HER results, c) First TTC location.

Fig. 8. a) Punch-force-displacement results, b) HER results, c) First TTC location.

The fully anisotropic model successfully predicted the first TTC location (Fig. 8b). Furthermore, minor cracks detected in the experiments were also observed near similar locations in the FE simulations. Underestimation of fracture force and HER values were still observed, which could be correlated with the performance of the fully anisotropic fracture model in tensile tests with different geometries. Further enhancement of the model formulation could improve the performance.

Conclusion

The original version of this paper is available on https://www.scientific.net/SSP.388.153.pdf

In this study, the local formability features of QP1000 steel were investigated through the HET of conical punch configuration. The fully anisotropic fracture model was implemented in order to characterize the material anisotropy and anisotropic ductility of the QP steel. Standard uniaxial tensile tests were conducted in seven orientations to calibrate the Yld2004-18p criterion, while five different fracture samples were tested in three orientations to define fully anisotropic fracture modeling. In the conical punch configuration of HET, the first TTC was observed at the hole edge. In the experiments conducted within the scope of this research, the first crack initiated at the hole edge's inner circle (punch side). In the following, the crack was progressed through the thickness in an inclined path. Initially, the inner circle is exposed to compression, then turns out to tension and high triaxiality regimes. During the progression of the crack, the stress state varies [16]. Therefore,
a wide range of fracture strain characterization extending from uniaxial compression to plane strain tension was found to be crucial. Moreover, the prediction of the fracture initiation zone alongside the inclination path of the crack is highly relevant to the anisotropic yield criterion. The results showed that the Yld2004-18p criterion well-defined the material anisotropy. Moreover, the fully anisotropic DF2016 fracture model showed plausible prediction performance. The implemented model predicted the first TTC location with high accuracy, indicating the advantages of introducing the anisotropy to both plasticity and fracture description, while further enhancement of the model is required for the force and hole expansion ratio at first through thickness crack. As a future remark, the stress evolution in all orientations can be regarded to adjust the anisotropic plasticity parameters better and describe the force-displacement response in the TD orientation. Accordingly, localization in the TD orientation may be accurately captured, thereby enhancing the fully anisotropic fracture model's performance in edge-cracking metrics.

Acknowledgement

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Dr. Toros Arda Akşen would like to acknowledge the financial support provided by the Scientific and Technological Research Council of Türkiye (TÜBİTAK) under the 2219 International Postdoctoral Research Fellowship Program [Grant Number: 1059B192400070].

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Y. Lou, L. Chen, T. Clausmeyer, E. Tekkaya, A.E. Yoon, Modeling of ductile fracture from shear to balanced biaxial biaxial tension for sheet metals, Int. J. Solids Struct. 112 (2017) 169-184.

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Solid State Phenomena, ISSN: 1662-9779, Vol. 388, pp 163-170

doi: 10.4028/p-PKV0mr

© 2026 The Author(s). Published by Trans Tech Publications Ltd, Switzerland.

Submitted:
2026-01-19

Revised:
2026-02-16

Accepted:
2026-02-20

Online:
2026-04-15

Global and Local Formability of 3rd  Generation Advanced High Strength Steels

The original version of this paper is available on https://www.scientific.net/SSP.388.163.pdf

Payà Anna 1,a*, Carpio Marcel 1, b, Frómeta David 1,c

¹Eurecat, Centre Tecnològic de Catalunya, Unit of Metallic and Ceramic Materials, Plaça de la Ciència, 2, Manresa 08243, Spain

a* anna.paya@eurecat.org, b  marcel.carpio@eurecat.org, c  david.frometa@eurecat.org

Keywords: sheet metal forming, advanced high strength steels, formability, edge stretching, lightweight design.

Abstract

The present work investigates the global and local formability of two third-generation Advanced High Strength Steels (AHSSs), a quenching & partitioning (Q&P) steel and a Medium Mn (MMn) steel with 1 GPa strength. Third-generation Q&P and MMn steels are designed to overcome the limitations of first-generation AHSS grades by enhancing formability while maintaining high mechanical strength, thus enabling more efficient structural design and improved crash performance. Understanding their forming behaviour is essential to ensure their reliable use in complex sheet metal forming operations. In this study, the forming performance of a Q&P and a MMn steel is analysed through experimental procedures involving both in-plane deformation under various loading paths and hole expansion tests with different hole edge qualities, to evaluate their global and local formability. A first-generation Dual Phase (DP) steel is included in the analysis for comparison. The results demonstrate that 3rd  Generation Q&P and MMn steels exhibit very good global formability, superior to conventional 1st  Generation AHSSs. However, local formability, as evaluated by hole expansion capacity, can be severely compromised by edge manufacturing process. These findings contribute to a deeper understanding of the distinction between global and local formability in thirdgeneration AHSS, offering insights to improve process robustness and support the industrial implementation of these steels in high-performance automotive components.

Introduction

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Automotive lightweighting demands materials that reduce mass without compromising crash safety. In this context, third-generation Advanced High Strength Steels (AHSS), especially quenching & partitioning (Q&P) and Medium-Mn (MMn) grades, have been developed to combine tensile strength of around 1 GPa with improved formability, enabling thinner sheets in body-in-white applications [1-4]. Without entering detailed metallurgy, both families share a fine multiphase structure combining ferrite, bainite or tempered martensite with retained austenite (RA). During forming, metastable RA can transform to martensite (Transformation-Induced Plasticity or TRIP effect), enhancing strain hardening capacity and supporting a favourable strength ductility balance in sheet operations [1−3].

For sheet forming design, it is useful to separate global formability from local formability. Global formability describes the ability to avoid localised necking under homogeneously distributed strains in large areas and is typically characterised by the Forming Limit Curve (FLC) and conventional uniaxial tensile metrics. Local formability, in contrast, concerns resistance to fracture under highly concentrated deformation, as found in stretch-flanging, hole extrusion, or tight-radius bending. In these modes, necking-based criteria generally fail, and fracture can occur below the FLC, motivating the use of local/global formability maps and performance indices that combine uniform strain with true fracture/thickness strain to better discriminate AHSS behaviour across stress states and strain paths [5-7].

A widely used measure of local edge performance is the Hole Expansion Ratio (HER). HER is highly sensitive to edge conditions and tooling parameters. The hole fabrication route (mechanical piercing versus laser/EDM), die clearance, and punch geometry alter the damaged zone at the cut edge and thus govern crack initiation and propagation during expansion. Consequently, optimised
edge preparation/tooling can increase HER and delay edge cracking, while microstructural features (phase distribution, hardness gradients, banding) set the intrinsic resistance to microcrack formation at the edge [8-10].

In this study, we compare global and local formability of a Q&P steel and a MMn steel ( 1 GPa ) with a first-generation Dual Phase (DP) steel used for reference. The global formability response is assessed using in-plane deformation under different loading paths using Nakajima stretching tests, and the local response is measured by hole expansion tests with different edge qualities representative of production. The results are interpreted within local/global formability concepts to (i) demonstrate the global formability advantage of third-generation AHSS at similar strength and (ii) quantify the strong sensitivity of edge formability to edge manufacturing, informing process-robust choices of material and cutting/tooling for high-performance automotive components [5-9].

Materials

The original version of this paper is available on https://www.scientific.net/SSP.388.163.pdf

The materials analysed in this study are three cold-rolled steel sheets at a 1 GPa strength level: two 3rd  Gen AHSS, a Medium-Mn (MMn1000) and a Quenching & Partitioning (QP980), both at 1.50 mm nominal thickness, and a Dual Phase (DP1000) reference at 1.40 mm .

The QP980 sheet is representative of Q&P processed multiphase structures with retained austenite stabilised by carbon partitioning, designed to achieve high strength with improved formability [1]. The microstructural components include ferrite, bainitic/ferrite, tempered martensite and RA ( 14.7% vol. fraction). The MMn1000 sheet presents an ultrafine dual-phase microstructure consisting of ferrite and RA ( 41% vol. fraction). RA fraction and stability are tuned by intercritical annealing, enabling transformation-assisted deformation and a favourable strength ductility balance [2]. The DP1000 reference exhibits the typical ferrite matrix with martensite islands and RA ( 2% vol. fraction) of conventional AHSS used in automotive forming and serves here as a benchmark at similar strength [3].

Mechanical properties for each material grade were determined according to UNE EN ISO 68921[11] in the transverse direction with respect to the rolling direction because this orientation is more sensitive to anisotropy introduced during rolling and therefore provides a conservative evaluation of the mechanical behaviour. The obtained results are reported in Table 1, which summarizes the mechanical properties of the investigated steel grades. The table includes the material thickness 𝐭, the yield strength σys, the ultimate tensile strength σ uts, the uniform elongation Ag (defined as the elongation at σuts ), the elongation at fracture A80 measured on an initial gauge length of 80 mm , and the strain hardening exponent n . The n -value was calculated at different deformation ranges depending on the steel grade ( 2−4% deformation for DP1000, 2-25% for MMn1000 and 2−14% for QP980).

Table 1. Mechanical properties of the investigated steel grades at transverse direction.

Steel denomination | t [mm] | σYS [MPa] | σUTS [MPa] | Ag [%] | A₈₀ [%] | n
DP1000 | 1.4 | 769 | 1040 | 5.3 | 8.7 | 0.09
MMn1000 | 1.5 | 643 | 960 | 25.7 | 29.6 | 0.25
QP980 | 1.5 | 658 | 992 | 14.4 | 19.0 | 0.19

Experimental Procedure

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Forming Limit Curve (FLC) Determination

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Nakajima tests were performed according to ISO 12004-2[12]. The specimens were machined by electrical discharge machining (EDM) considering the less favourable direction of the material, with the shaft oriented 90∘ with respect to the rolling direction, as indicated in ISO 12004. For the evaluation of the FLC, three valid samples of each geometry were used. Five strain paths were investigated: plane strain ( ε2/ε1=0 ), biaxial strain ( ε2/ε1=1 ) and three deformation paths between uniaxial strain ( ε2/ε1=−0.5 ) and plane strain, where ε1 and ε2 are the major and minor strain,
respectively. The geometries of the specimens used to determine the FLC are shown in Fig. 1a. A Nakajima tool with a punch diameter of 100 mm was used to perform the tests. The tests were carried out in a double-acting hydraulic press using a punch speed of 1.0 mm/s. The press holder was equipped with a circular rib to avoid sliding. The deformation was monitored by using a 3D DIC (Digital Image Correlation) equipment integrated into the press (Fig. 1b) [13].

[picture]

Fig. 1. (a) Dimensions of the specimens used to determine FLC with a 100 mm Nakajima tool; (b) experimental setup for FLC evaluation using a DIC System integrated into the press.

Fig. 1. (a) Dimensions of the specimens used to determine FLC with a 100 mm Nakajima tool; (b) experimental setup for FLC evaluation using a DIC System integrated into the press.

Hole Expansion Tests (HET)

The original version of this paper is available on https://www.scientific.net/SSP.388.163.pdf

Hole Expansion Tests were performed according to ISO 16630 [14]. Squared samples of 100×100 mm were prepared with an initial hole diameter of 10 mm (Fig. 2a) under two different conditions: holes produced by punching using a punch-to-die clearance of 12±1%, following standard recommendations, and holes machined by Electrical Discharge Machining (EDM) to ensure minimal edge damage and burr-free surfaces. The use of EDM holes was intended to provide a reference condition with negligible edge defects, allowing the evaluation of the influence of edge quality on the hole expansion performance [15]. For the expansion tests, a conical expansion tool with a top angle of 60∘ was used. A schematic representation for the determination of the Hole Expansion Ratio (HER) is shown in Fig. 2b-c. The tests were performed with a constant punch displacement rate of 1 mm/s. A clamping force of 50 kN was applied to prevent any material draw-in from the clamping area during the test. All the tests were carried out at room temperature ( RT≈25∘C ). To detect the first sign of cracking, a video camera was positioned at the bottom of the test tool (Fig. 2e). The final diameter was measured using the images obtained when the first crack extended through the thickness of the test piece (Fig. 2d). Using this diameter, the limiting hole expansion was calculated according to the following equation:

λ=Dh−D0D0×100(1)

Where λ is the limiting hole expansion ratio as a percentage, D0 is the original hole diameter ( 10 mm ), and Dh is the hole diameter after rupture, in millimetres.

[picture]

Fig. 2. (a) Specimen geometry for HET; (b- c) Schematic representation of the experimental procedure for the HET. Before (b) and after (c) the test; (d) Digital image used for the evaluation of the HER; (e) experimental setup used for the hole expansion tests.

Fig. 2. (a) Specimen geometry for HET; (b- c) Schematic representation of the experimental procedure for the HET. Before (b) and after (c) the test; (d) Digital image used for the evaluation of the HER; (e) experimental setup used for the hole expansion tests.

Results and Discussion

The original version of this paper is available on https://www.scientific.net/SSP.388.163.pdf

Forming limit curves (FLCs), obtained by Nakajima testing with a 100 mm hemispherical punch and fitted using the inverse parabola procedure defined in ISO 12004-2, exhibit a differentiated response across AHSS families. The 3rd  Gen grades (MMn1000 and QP980) reach higher limit strains than the 1st  Gen DP1000 over the investigated range of minor strain ( ε2 ), with the contrast most evident around FLC0(ε2≈0) and uniaxial strain paths shown in the left hand of the Forming Limit Diagram (FLD) (Fig. 3a). This ordering is consistent with the tensile hardening capacity of the three steels: MMn1000 combines a high n -value ( n=0.25 ) and a wide uniform elongation ( Ag=25.7% ), QP980 follows ( n=0.19, Ag=14.4% ), while DP1000 exhibits limited hardening ( n=0.09, Ag=5.3% ). A higher n value sustains strain redistribution and delays localised necking, which naturally lifts the FLC, as reported for 3rd  Gen steels when compared with dual-phase grades at comparable strength levels [16] (Fig. 3b). A change in the slope of the FLC can be observed in the second quadrant in all steel grades, close to the uniaxial tension stress state. This behaviour could be attributed to a lower width-to thickness ratio of the test specimens, which tends to increase through-thickness stress effects, promoting earlier onset of localized necking. This can be especially noticeable in damage-sensitive materials, and may cause a drop in failure limits, like in the case of QP980. In the first quadrant, significant differences are observed in the slopes of the three steels. DP1000 shows a much steeper slope than QP980 and MMn1000. Such differences can be explained by the differences in work hardening of the studied steels. It has been previously reported that higher work hardening leads to lower failure limits in the first quadrant [17].

[picture]

Fig. 3. (a) FLC obtained for the investigated AHSS; (b) correlation between FLC 0 values and n -values.

Fig. 3. (a) FLC obtained for the investigated AHSS; (b) correlation between FLC 0 values and n -values.

Local formability evaluated through the Hole Expansion Ratio (HER) underscores the critical role of edge quality. The obtained HER values for both edge preparation methods are shown in Fig.4. With punched holes, all three steels fall into similar, moderate HER values ( 33±2% for DP1000, 29±0% for QP980, 26±4% for MMn1000). When the hole is made by EDM, HER rises sharply. DP1000 and MMn1000 reach similar ( HEREDM86% ). Q&P shows a more moderate increase (HEREDM=61±6%). These results evidence that the shear affected zone (SAZ, work-hardening, micro-voids/micro-cracks, and the burnish-fracture transition) has a major effect on edge ductility of AHSSs, reducing dramatically their hole expansion capacity. This same behaviour has been previously reported for dual-phase, complex-phase and MMn steels. It is attributed to punchinginduced damage generated in the SAZ, which causes local hardening and accelerates damage accumulation and edge cracking under stretch flanging [18-20].

[picture]

Fig. 4. Comparative analysis of Hole Expansion Ratio values for the materials studied under the two edge quality conditions.

Fig. 4. Comparative analysis of Hole Expansion Ratio values for the materials studied under the two edge quality conditions.

Bringing global and local indicators together clarifies the need to evaluate both forming characteristics separately. On the one hand, global formability can be generally inferred from uniaxial tensile parameters, such as strain hardening exponent (n) or total elongation. Thus, microstructural characteristics enhancing these parameters will also have a beneficial influence on formability. As mentioned before, 3rd  Gen AHSSs benefit from the strain-induced RA to martensite deformation (TRIP effect), which contributes to the formation of additional geometrically necessary dislocations, increasing strain hardening and delaying the onset of necking [21]. The beneficial influence of the TRIP effect on global ductility is directly proportional to RA volume fraction [22], which explains
the exceptional formability of the MMn steel ( 41% of RA) compared to Q&P ( 14.7% of RA) or DP1000 ( 2% of RA).

On the other hand, local formability modes, such as edge formability, are mainly governed by crack propagation resistance and phase hardness contrasts [18, 22], and higher RA content may have the opposite effect. During shearing/punching, the deformation induced in the SAZ, locally transforms RA to martensite, generating a hardened and more "brittle" edge [21]. When the SAZ is eliminated, as in holes machined by EDM, edge formability increases substantially. This is more evident in the case of MMn (Fig. 5a), which is the most sensitive to edge preparation method.
Fig. 5b reveals that there is no clear relationship between global and local formability indicators (FLC 0, n -value). Comparing Q&P and MMn steels, it could be said that, in the absence of a damaged SAZ, a higher resistance to localized necking ( FLC0 ) accompanies better edge stretchability. However, this is completely different for punched HER, where MMn exhibits the lowest value, despite having the highest FLC 0 . Previous works have shown that punched HER correlates better with fracture mechanics metrics [22] and that microstructural architecture (phase fraction, distribution, and hardness mismatch) controls damage initiation and growth at the edge [18-20, 22].

[picture]

Fig. 5. (a) Ratio HERpunched /HEREDM  for the investigated materials; (b) correlation between HER values and FLC 0 and n values.

Fig. 5. (a) Ratio H E R punched / H E R EDM for the investigated materials; (b) correlation between HER values and FLC 0 and n values.

From an application standpoint, these combined results suggest a simple rule of thumb. For components governed by strains homogeneously distributed in large areas, 3rd  Gen grades offer a wider global window than first-generation DP steels owing to their stronger hardening and higher formability limits [16]. For stretch flanging, the outcome hinges on edge integrity. Minimizing edge damage by adopting EDM or laser, or optimizing punching (clearance, punch/die geometry, finishing) is essential to unlock local ductility potential. In the presence of cut-edge damage, fracture toughness and phase contrast management become the primary levers to raise HER [18-20, 22]. These results enhance the understanding of the differences between global and local formability in third-generation AHSSs, providing valuable insights to improve process robustness and facilitate the industrial adoption of these steels in high-performance automotive components.

Conclusion

The original version of this paper is available on https://www.scientific.net/SSP.388.163.pdf

The present investigation on the global and local formability of 3rd  Gen AHSS has provided the following significant findings:

MMn1000 and QP980 exhibit higher forming limit strains than DP1000 across the minor strain range studied, with the largest contrast around FLC 0. This behavior is consistent with their higher strain hardening exponent and uniform elongation. In the absence of a shear affected zone (EDM holes), higher stretch flangeability is also associated with increased FLC 0.

Edge condition has a dominant effect on hole expansion performance. With punched edges, all grades show similar and moderate HER values. In contrast, EDM edges lead to a pronounced increase in HER, highlighting the intrinsic edge stretchability of undamaged material.

Damage introduced in the shear affected zone significantly reduces edge ductility, thereby limiting hole expansion capacity. Under these conditions, no direct correlation can be established between HER and global formability parameters, as failure is governed by edge damage and the material's damage tolerance. Accordingly, other parameters, such as fracture mechanics metrics, can provide better predictions.

From an application perspective, 3rd  Gen steels provide a wider forming window for components dominated by distributed strain, whereas in stretch flanging operations, performance is primarily controlled by edge integrity. Therefore, high edge quality is essential to minimize crack initiation and premature failure.
A practical selection framework should combine both global and local formability indicators. To improve the accuracy of edge cracking predictions, edge formability should be assessed under conditions representative of the actual manufacturing process.

Acknowledgment

The original version of this paper is available on https://www.scientific.net/SSP.388.163.pdf

This work has been financially supported by the European Commission, Research Fund for Coal and Steel programme under Grant Agreement 101112540 - SUP3RFORM - RFCS-2022.

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The original version of this paper is available on https://www.scientific.net/SSP.388.163.pdf

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H. Essoussi, S. Ettaqi, E.H. Essadiqi, From the alloy design to the microstructural and mechanical properties of medium manganese steels of the third generation of advanced high strength steels, J. Min.Metal., Sect. B-Metal. 60(3), (2024) 339-352.

I. Apurwa, J.K. Yadav, A. Kumar, A perspective on third-generation medium-Mn steels for automotive application, in: S. Yadav, P.K. Jain, P.K. Kankar, Y. Shrivastava (Eds.), Advances in mechanical and energy technology, LNME, Springer, Singapore (2023) pp. 351-357.

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B.M. Hance, Practical application of the hole expansion test, SAE Int. J. Engines 10(2) (2017) 247-257.

H.-C. Shih, D. Zhou, B. Konopinski, Effects of punch configuration on the AHSS edge stretchability, SAE Int. J. Engines 10 (4) (2017) 2051-2056.

F. Sun, L. Burroughs, R. Chowdhury, and M. Milliron, Effects of microstructure on hole expansion ratio of AHSS, in: International Symposium on New Developments in Advanced High-Strength Sheet Steels, AIST, Vail, CO, USA (2023).

UNE-EN ISO 6892-1 Metallic materials - Tensile testing - Part 1: Method of test at room temperature (2020).

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D. Frómeta, M. Tedesco, J. Calvo, A. Lara, S. Molas, and D. Casellas, Assessing edge cracking resistance in AHSS automotive parts by the Essential Work of Fracture methodology. J. Phys.: Conf. Ser. 896(1), 012102 (2017).

J. Noder, J.E. Gutierrez, A. Zhumagulov, J. Dykeman, H. Ezzat, and C. Butcher, A comparative evaluation of third-generation advanced high-strength steels for automotive forming and crash applications, Materials 14(17) (2021) 4970.

S. Sato, M. Tsukamoto, Y. Maeda, Y. Maeda, and T. Hama, Crystal plasticity-based forming limit analysis for two types of 5052 aluminum alloy sgeets with different heat treatment conditions, Mat. Res. Proc. 41 (2024) 1009-1016.

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This ePub file was created using machine-generated methods and automated software to convert the source text into an ePub format and may contain formatting inconsistencies from the original version. Formatting may vary depending on the device used.

The original version of this eBook is available on https://www.scientific.net/book/formability-of-metallic-materials/978-3-0364-1993-0

Solid State Phenomena, ISSN: 1662-9779, Vol. 388, pp 171-178

doi: 10.4028/p-Hp3vb8

© 2026 The Author(s). Published by Trans Tech Publications Ltd, Switzerland.

Submitted:
2026-01-19

Revised:
2026-02-16

Accepted:
2026-02-16

Online:
2026-04-15

Formability Test for Characterizing the Mode II-III Fracture Transition

The original version of this paper is available on https://www.scientific.net/SSP.388.171.pdf

Inês M. Almeida 1,a*, João P. G. Magrinho 1,b Paulo A. F. Martins 1,c and M. Beatriz Silva 1,d

1 IDMEC, Instituto Superior Técnico, Universidade de Lisboa, Av. Rovisco Pais, 1049-001 Lisboa, Portugal

a*ines.matias.almeida@tecnico.ulisboa.pt, bjoao.magrinho@tecnico.ulisboa.pt, c  pmartins@tecnico.ulisboa.pt, d  beatriz.silva@tecnico.ulisboa.pt

*corresponding author

Keywords: sheet-bulk forming, fracture forming limits, sheet compression test.

Abstract

The transition between different shear fracture modes is still not well understood, especially in the case of mode II to mode III fracture. While transitions involving mode I have been investigated in previous studies, the interaction between in-plane and out-of-plane shear fracture remains largely unexplored. This work presents an ongoing study aimed at analysing this transition through a newly proposed formability test based on the compression of a thick sheet. The test is intended to investigate the transition region between the shear fracture forming limit (SFFL) and the out-of-plane shear fracture forming limit (OSFFL), associated with fracture mechanics modes II and III, respectively. A new specimen geometry was developed by combining features of existing configurations designed to activate each fracture mode separately. Finite element simulations were carried out to support the design of the specimen and to provide an initial analysis of the strain loading paths in the ligament region. Preliminary numerical results indicate that the proposed test can promote in-plane shear, out-of-plane shear, as well as mixed fracture mechanisms. The predicted fracture initiation sites and corresponding strain paths were examined in the effective strain-stress triaxiality space to provide initial insight into the role of stress state on fracture behaviour. The influence of notch orientation on the resulting fracture mode is discussed, highlighting the potential of the proposed approach for studying mixed shear fracture conditions.

Introduction

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Accurate definition of deformation limits is essential for the optimization of metal forming processes and the prevention of fracture in metallic components [1]. Formability describes the ability of a material to sustain plastic deformation without failure and is commonly limited by necking or fracture. In sheet metal forming, fracture can occur through different crack-opening mechanisms, namely Mode I (tensile), Mode II (in-plane shear), and Mode III (out-of-plane shear) [2].
Formability is often represented using Forming Limit Diagrams (FLDs), where different fracture forming limits are associated with distinct fracture modes: the Fracture Forming Limit (FFL) for Mode I, the Shear Fracture Forming Limit (SFFL) for Mode II, and the Out-of-Plane Shear Fracture Forming Limit (OSFFL) for Mode III. While FLDs are widely used, their assumption of plane stress conditions limits their applicability, particularly when out-of-plane shear fracture is involved. To overcome this limitation, fracture behaviour can be represented in terms of effective strain and stress triaxiality, providing a more general framework to describe different fracture modes which was originally proposed by Vujovic and Shabaik [3].
Various experimental tests have been proposed to characterize each fracture forming limit individually. Tensile and bulge-based tests are commonly used to determine the FFL, a limit initially proposed by Atkins [4], while notched tensile and shear-based tests are employed to characterize the SFFL, a limit initially observed by Embury and Duncan [5] and later investigated by Marciniak [6]. Similarly, compression-based tests have been developed to activate Mode III fracture and determine the OSFFL [10]. Although transition regions between Mode I-Mode II and Mode I-Mode III fracture
have been investigated using dedicated specimen geometries, the transition between Mode II and Mode III fracture remains largely unexplored [7, 8].
Currently, no formability test exists that allows the systematic characterization of the transition between in-plane and out-of-plane shear fracture while accounting for both plane stress and threedimensional material flow conditions. The objective of this work is therefore to propose a new formability test capable of investigating the transition zone between the SFFL and the OSFFL, providing initial insight into mixed shear fracture conditions.

Specimen Design

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Based on the goal defined in the previous section, two main requirements were defined to guide the development of the proposed formability test for thick sheet materials. First, the specimen should be capable of failing under a combined in-plane and out-of-plane shear loading condition, corresponding to a mixed Mode II-Mode III fracture mechanism. Second, the experimental setup should prioritize simplicity to facilitate specimen fabrication, streamline testing conditions, and enable future modifications to the configuration.
Considering these requirements, a subset of five existing test configurations was selected for further evaluation. These tests are summarized in Table 1, and the rolling direction of the metal sheet is aligned in the arrows in the first three and fifth and regarding the double notched compression test, is perpendicular to the loading direction.

Table 1. Five selected tests based on the key characteristics of each. The information regarding the Double-Notched Shear Test is derived from [9], while the data for the other tests is based on the work of [10].

Double Notched Tensile | Staggered Double Notched Tensile | Shear | Double notched Compression | Sheet Lenghtwise Compression
Double notched tensile specimen with symmetric notches and upward/downward arrows | Staggered double notched tensile specimen with offset notches and upward/downward arrows | Shear specimen with diagonal shear plane and upward/downward arrows | Double notched compression specimen cross-section with downward arrow | Sheet lenghtwise compression specimen showing cylindrical supports with upward/downward arrows
Mode I | Mode I and II | Mode II | Mode II | Mode III

The first three tests in Table 1, the double-notched tensile, staggered double-notched tensile, and shear tests, primarily involve tensile loading. This limits their ability to explore fracture modes beyond Mode I and Mode II. Despite this, the notch geometry in these tests is an important factor, as the angle and shape of the notches influence the development of mixed-mode fracture.
Previous research has shown that compressive forces can significantly affect fracture behaviour in thick sheets. Silva et al. [9] found that applying compression near the notches in a double-notched shear test can induce Mode II fracture, which is useful for evaluating fracture toughness. Similarly, Leonardo et al. [10] demonstrated that compressing a sheet along its length on both sides induces Mode III fracture, with cracks initiating and propagating through the sheet thickness.
Drawing inspiration from these tests, a new specimen design was developed by combining two critical features: the double-notch geometry and the application of compressive loads. The goal of the new design is to create a stress concentration in the notched region that drives material movement through
the thickness, producing out-of-plane stresses. A schematic of the proposed geometry, along with the main dimensions, is shown in Figure 1.

[picture]

Fig. 1. Schematic representation of the specimen with geometrical parameters identified: angle between notches ( α0 ), ligament size ( l0 ), notch width ( a0 ), notch angle ( φ0 ), width ( W0 ), length (L0) and thickness (t0).

Fig. 1. Schematic representation of the specimen with geometrical parameters identified: angle between notches ( α 0 ), ligament size ( l 0 ), notch width ( a 0 ), notch angle ( φ 0 ), width ( W 0 ), length ( L 0 ) and thickness ( t 0 ) .

The geometric parameters of the specimen are defined as follows. The notch-to-notch angle, α0, is measured between the vertical axis and the line connecting the centres of the two notch arcs. The ligament length, l0, corresponds to the minimum distance separating the notches, while the notch width, a0, represents the horizontal extent of each notch. The notch inclination, φ0, is defined as the angle between the notch edge and the horizontal axis. In addition, the overall specimen dimensions are described by its width W0, length L0, and thickness t0.
Each parameter was analysed individually, while the others were maintained at their reference values. The parameter values tested and their corresponding reference values are shown in Table 2. It is important to note that the thickness parameter will not be evaluated, as the sheet used in the tests had a thickness of 5 mm .

Table 2. Geometric parameter values tested and their corresponding reference values.

Parameters | Values Tested | Reference Value
Angle between notches α₀ [°] | 0 | 15 | 30 | 45 | 60 | 75 | 90 | 45
Ligament Sizes l₀ [mm] | 2.5 | 5 | 7.5 | 10 | 12.5 | 15 |  | 5
Notch Width a₀ [mm] | 2 | 3 | 4 | 5 | 6 |  |  | 3
Notch Angle φ₀ [°] | 30 | 45 | 60 | 75 | 90 |  |  | 70
Width W₀ [mm] | 20 | 45 | 80 |  |  |  |  | 45
Length L₀ [mm] | 50 | 100 | 150 |  |  |  |  | 100

Numerical Simulation

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The parameters presented in Table 2 were evaluated using an in-house software, i-form 3D [2]. This software is based on a modified weak form of Markov's rate of energy variational statement, which accounts for contact and frictional sliding between the specimens and the various tool components of the experimental setup and it is used to solve quasi-static problems.

∫Vσij′δDijdV+∫VσmδDvdV−∫SttiδuidS+∫Sf(∫0|ur|τfδur)dS=0(1)

The weak form above adopts a control volume V approach incorporating a sort of 'updated Eulerian approach' with velocities ui as the primary unknowns [11]. The first term of the expression involves the deviatoric Cauchy stress σij′ and the rate of deformation Dij, representing viscous effects. The second term makes use of the hydrostatic stress σm and the volumetric rate of deformation Dv, analogous to hydrostatic effects on a viscous fluid. The incompressibility condition of the velocity field Dv=0 is relaxed using a penalty function K, where σm=(K/2)Dv.
The third and fourth terms in (1) refer to the tractions ti applied on the boundary St of the control volume, and to the friction shear stress τf and the relative sliding velocity ur acting on the contact interfaces Sf between the specimens and the various tool components. Friction was modelled using the law of constant friction, expressed as τf=mk, where k represents the shear flow stress and m denotes the friction factor which was taken as 0.05 .
The Hill criterion [12] was used to better capture the material's anisotropic behaviour, since anisotropic effects became more critical due to the consideration of stresses in all directions. The anisotropy parameters ( F,G,H,L,M,N ), which define the shape of the yield surface in the stress space, indicating how much the material deviates from isotropic behaviour, were previously calculated by [10] as well as the remain material properties and are presented in Table 3. These were input into the software to accurately represent the sheet's material properties. The material properties determined and presented in table 3 are the Young Modulus E, the yield stress σY, the ultimate tensile stress σUTS, the elongation at break A, and planar anisotropy r¯.

Table 3. Mechanical properties and anisotropy parameters of the aluminium AW7075-T651 sheets obtained from tensile tests by Leonardo (2019).

Properties

mathematical formula
Anisotropy parameters

Values | 71.72 | 520.52 | 584.67 | 14.55 | 0.663 | 0.4493 | 0.6523 | 0.3477 | 3.0000 | 3.0000 | 1.2907

The model used in the numerical simulations is shown in Figure 2. The specimen is a deformable object modelled with around 60000 solid hexahedral quadratic elements varying slightly depending on the ligament size, with mesh refinement in the ligament to improve simulation results in this region, which experiences higher deformation. The tool was modelled with contact-friction spatial triangles.

[picture]

Fig. 2. Schematic representation of the model and finite element mesh used to simulate the new proposed test showing: (1) the lower support, (2) the punch, (3) and (4) the lateral supports, (5) the back support, and (6-7) the front supports.

Fig. 2. Schematic representation of the model and finite element mesh used to simulate the new proposed test showing: (1) the lower support, (2) the punch, (3) and (4) the lateral supports, (5) the back support, and (6-7) the front supports.

In this model, the lower support (1) serves as the base for positioning and securely fixing the specimen. The punch (2) applies a downward force, while the lateral (3 and 4), front (6 and 7), and back (5) supports maintain near-pure shear conditions. All tools are modelled as rigid bodies, with all components remaining stationary during the simulation except for the punch (2), which moves downward at a constant speed.
All simulations were performed, and results were extracted at the centre of the ligament area between the two notches, where the highest shear stresses are anticipated.

Results and Discussion

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The numerical results used to plot the loading evolutions in the effective strain versus stress triaxiality space ε¯=f(η) were extracted from the central point of the ligament area between the two notches, as this region is anticipated to experience the highest shear stresses. The effective strain versus stress triaxiality space ε¯=f(η) was selected over the principal strain space ε1=f(ε2) because the latter does not adequately account for regions opposite surface of the specimens in contact with the back support (5), exhibiting local deviations from plane stress conditions.

Geometric Sensibility Analysis.

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The simulations were terminated when the punch displacement reached 3 mm for all specimens. Although this displacement may not coincide with the exact moment of fracture, this approach was used because the objective was to analyse deformation up to, but not necessarily at, the fracture point.

[picture]

Fig. 3. Loading paths in the effective strain vs. stress triaxiality space ε¯=f(η) for various specimen parameters: (a) ligament size (l0), (b) notch width (a0), (c) notch angle (φ0), and (d) combined width (W0) and length (L0).

Fig. 3. Loading paths in the effective strain vs. stress triaxiality space ε ¯ = f ( η ) for various specimen parameters: (a) ligament size ( l 0 ) , (b) notch width ( a 0 ) , (c) notch angle ( φ 0 ) , and (d) combined width ( W 0 ) and length ( L 0 ) .

Figure 3a presents the ligament size analysis (l0), demonstrating that triaxiality values are consistent across all sizes, remaining below η=−1/3 (indicative of uniaxial compression) but above η=−2/3 (associated with equi-biaxial compression). The results further show that increasing ligament size leads to higher compressive stresses within the notch area and lower maximum effective strain values, indicating reduced strain accumulation in specimens with larger ligament sizes.
Figure 3b shows that, in the analysis of notch width ( a0 ), the evolution of triaxiality remains nearly constant for widths greater than 3 mm , with values around η=−1/3, which indicates uniaxial compression. For these widths, compressive stress is primarily oriented in a single direction. In contrast, widths of 2 mm and 3 mm produce higher stress concentrations due to the smaller notch, resulting in compressive stresses in multiple directions and triaxiality values η<−1/3.
Figure 3 c demonstrates that triaxiality values η≈−0.4 remain approximately constant across different notch angles ( φ0 ), indicating that this parameter exerts minimal influence on the results. The changes in triaxiality previously observed are, therefore, attributed to the influence of other parameters, which were now kept at their reference values. Similarly, Figure 3d shows no dependence of the results on the specimen's length (L0) or width (W0).
Based on these results, a detailed analysis was conducted for all notch-to-notch angles α0 in specimens with a notch size l0=5 mm, since it is a common practice to choose a value that matches the sheet thickness and none of the other tested values produced significant deviations. A notch width a0=3 mm was employed in agreement with previous studies performed on staggered doublenotched tension [10] and double-notched shear tests [9]. Since the notch angle φ0 did not influence the results, a reference value of φ0=70∘ was selected for all specimens.
The remaining specimen dimensions were set at L0=100 mm and W0=45, following a previous work of Silva et al. [9].

Strain Evolution.

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Firstly, the effective strain for all the specimens were analysed to understand if the deformation was in fact, localized in the ligament, which was verified. An example of this localization can be seen in figure 4 for the specimen with a ligament angle of 0∘.

[picture]

Fig. 4. Effective Strain distribution in the last step of the numerical simulation of the specimen with α0=0∘.

Fig. 4. Effective Strain distribution in the last step of the numerical simulation of the specimen with α 0 = 0 ∘ .

To characterize the shear stresses experienced by the different specimens and how they influence the overall strain distribution, the strain loading paths were represented in the effective strain vs. triaxiality plane in combination with the FFL, SFFL, and OSFFL previously determined by Leonardo
et al. [10] (see Figure 5). The strain loading paths were obtained from the numerical simulations after a punch displacement of 3 mm , assuming that the specimen exhibits no lateral movement and full contact with the tools, which helps explaining why the loading paths extend beyond the limits previously determined by conventional tests.

[picture]

Fig. 5. Graphical representation in the triaxiality space of FFL, SFFL and OSFFL, along with the numerical triaxiality paths of the specimens with different angle between notches.

Fig. 5. Graphical representation in the triaxiality space of FFL, SFFL and OSFFL, along with the numerical triaxiality paths of the specimens with different angle between notches.

An overall increase in stress triaxiality is observed as the notch-to-notch angle α0 increases. For specimens with notch-to-notch angles α0 equal to 0∘ and 15∘, the loading trajectories intersect the SFFL and extend beyond it prior to fracture. This behaviour is likely a result of fixing the amount of punch displacement independently of the specimen geometry and likely also a result of compressive loading conditions imposed by specimen constraints, which intensify stress localization within the ligament and result in higher effective strain levels as the loading paths approach pure shear ( η=0 ). In contrast, specimens with α0 of 30∘ and 45∘, display loading paths that begin at relatively low triaxiality values and progressively approach the SFFL as deformation advances. In these cases, fracture occurs in a region between the SFFL and OSFFL, indicating the activation of a mixed inplane and out-of-plane shear fracture mechanism.
For specimens with notch-to-notch angles α0>60∘, the deformation path begins at negative triaxiality values (η<−0.40), which increase as the angle increases. This trend demonstrates the presence of significant compressive stresses and indicates that fracture is dominated by out-of-plane shear, corresponding to Mode III.
Overall, the proposed specimen configuration seems capable of generating strain loading paths representative of fracture under Mode II, Mode III, and mixed Mode II-III conditions for notch-tonotch angles α0 ranging from 0∘ and 60∘. Therefore, specimens with larger notch angles (e.g., α0=75∘ and 90∘ ) are unnecessary to achieve the required objectives.

Conclusion

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A novel formability test was proposed and validated numerically, utilizing the compression of thick sheets to characterize the transition zone between the shear fracture forming limit (SFFL, Mode II) and the out-of-plane shear fracture forming limit (OSFFL, Mode III).
Finite element modelling was employed to optimize specimen geometry and demonstrated that the test promotes both in-plane and out-of-plane shear mechanisms as the angle between the notches α0 is varied.

References

The original version of this paper is available on https://www.scientific.net/SSP.388.171.pdf

P.A.F. Martins, N. Bay, A.E. Tekkaya, A.G. Atkins, Characterization of fracture loci in metal forming, Int. J. Mech. Sci. 83 (2014) 112-123.

C. Nielsen, P. Martins, Metal Forming: Formability, Simulation, and Tool Design, Elsevier Science, 2021.

V. Vujovic, A.H. Shabaik, A new workability criterion for ductile metals, J. Eng. Mater. Technol. 108 (1986) 245-249.

A.G. Atkins, Fracture in forming, J. Mater. Process. Technol. 56 (1996) 609-618.

J. Embury, J. Duncan, Formability maps, Annu. Rev. Mater. Sci. 11 (1981) 505-521.

Z. Marciniak, Assessment of material formability, Adv. Technol. Plast. 1 (1984) 685-694.

J.P. Magrinho, M.B. Silva, L. Reis, P.A.F. Martins, Formability limits, fractography and fracture toughness in sheet metal forming, Materials 12 (2019) 1493.

J.P. Magrinho, M.B. Silva, L.M. Alves, A.G. Atkins, P.A.F. Martins, New methodology for the characterization of failure by fracture in bulk forming, J. Strain Anal. Eng. Des. 53 (2018) 242247.

C. Silva, M. Silva, L. Alves, P.A.F. Martins, A new test for determining the mechanical and fracture behavior of materials in sheet-bulk metal forming, Proc. Inst. Mech. Eng. Part L: J. Mater. Des. Appl. 231 (2017) 1-12.

P.N.C. Leonardo, J.P. Magrinho, I.M.F. Bragança, M.B. Silva, C.M.A. Silva, P.A.F. Martins, Formability limits in sheet-bulk forming, Int. J. Mach. Tools Manuf. 149 (2020) 103509.

K. Mattiasson, FE-models of the sheet metal forming processes, in: D. Banabic (Ed.), Sheet Metal Forming Processes: Constitutive Modelling and Numerical Simulation, Springer-Verlag, 2010, pp. 1-24.

R. Hill, A theory of the yielding and plastic flow of anisotropic metals, Proc. R. Soc. Lond. A 193 (1948) 281-297.

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Solid State Phenomena, ISSN: 1662-9779, Vol. 388, pp 179-185

doi: 10.4028/p-l9gdMJ

© 2026 The Author(s). Published by Trans Tech Publications Ltd, Switzerland.

Submitted:
2026-01-19

Revised:
2026-02-22

Accepted:
2026-02-23

Online:
2026-04-15

Assessing Global and Local Formability from Tensile Tests with Anisotropy Consideration

The original version of this paper is available on https://www.scientific.net/SSP.388.179.pdf

Guijia Li1,a and Junhe Lian 1, b*

1 Institute of Metal Forming, RWTH Aachen University, Intzestraße 1, 52072, Aachen, Germany

a  guijia.li@rwth-aachen.de, b  junhe.lian@ibf.rwth-aachen.de

Keywords: formability, anisotropy, failure, fracture, modeling.

Abstract

Medium-Mn steel (MMnS) and quenching and partitioning (QP) steels are two representatives of third-generation advanced high-strength steels ( 3rd  Gen AHSS), developed to achieve an optimal balance between strength and ductility. In forming applications, global formability reflects a material's resistance to necking, while local formability indicates its resistance to fracture. Both aspects are essential for assessing mechanical performance. Global formability is often characterized by forming limit curves at necking and is highly sensitive to work hardening behavior. Similarly, the forming limit curves at fracture determined from different stress states can be applied to evaluate the local formability. In addition, these deformation characteristics can be influenced by anisotropy introduced during sheet processing. Rolling process introduces orientation-dependent variations in both plastic flow and fracture behavior, which significantly affect necking development and fracture initiation. This study investigates and compares the global and local formability of various 3rd  Gen AHSS grades, with a focus on the influence of anisotropy. To investigate the anisotropic effects on plasticity and ductile fracture under different stress states, tensile tests were conducted on specimens with various geometries and orientations cut from sheet materials. Based on the tensile tests, the forming limit framework of Shen et al [1] was broadened to include anisotropic effects.

Introduction

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Driven by low-carbon design requirements, advanced high-strength steels (AHSS) have been increasingly adopted in automotive body structures to enable lightweight design while maintaining high crashworthiness. Among the newly developed 3rd  Gen AHSS, quenching and partitioning (QP) steels and medium-manganese (MMn) steels have attracted particular attention due to their excellent strength-ductility balance and promising application potential. As a trade-off for their strengthductility synergy, AHSS exhibits forming limits and failure behaviors that differ from those of conventional sheet metals, these characteristics play a decision role in the forming process. To better characterize the formability of AHSSs, Heibel et al. [2] introduced the concepts of global formability and local formability, providing a more comprehensive framework for interpreting deformation and fracture behaviors in sheet forming. Shen et al. [1] proposed a unified forming limit framework by integrating three distinct failure criteria including necking, ductile fracture and cleavage fracture, which allows for a direct and intuitive visualization of the competitive relationship among the different failure mechanisms. Global formability describes the ability to undergo uniform deformation without the onset of necking and is commonly characterized by the forming limit curve (FLC). The FLC may be obtained either from experiments, such as Nakajima test, or from theoretical calculations based on the modified maximum force criterion (MMFC) [3] or other models, including the MK model [4]. In contrast, local formability refers to the ability of a material to undergo plastic deformation in a local area without fracture. It is often assessed through hole expansion or bending tests in which the hole expansion ratio is measured as a representative metric [5]. Another effective way to characterize local formability is through local measurements of fracture strain, which are commonly employed in damage models and fracture criteria [2].

It is important to note that both global and local formability are strongly governed by the underlying stress state, which in turn is significantly influenced by material anisotropy in sheet metals.

Anisotropy originates from crystallographic texture, phase morphology, and thermomechanical processing history, and manifests not only in yielding and hardening behavior, but also in strain localization, damage evolution, and fracture, particularly under complex stress states typical of sheet metal forming operations. For AHSS, recent studies have revealed both plastic anisotropy and fracture anisotropy [6], which have been shown to strongly affect necking behavior, fracture mechanisms, and the resulting forming limits. In QP steels, the presence of multiphase microstructures and deformation-induced transformation of retained austenite further complicates the manifestation of anisotropy, leading to a strong dependence of deformation and fracture behavior on loading direction and stress state. Consequently, anisotropy plays a critical role in determining both global and local formability. Although the anisotropic plasticity and fracture behavior of QP steels have received increasing attention in recent years, most existing studies focus on individual material grades or specific loading paths. As a result, few studies have employed a unified framework to systematically and comprehensively evaluate the anisotropy sensitivity of formability across different grades of QP steels.

In this work, the anisotropy sensitivity of both global and local formability is systematically investigated for multiple QP steel grades within a unified forming limit framework. By combining necking- and fracture-based formability metrics under different stress states, the interplay between anisotropy, strain localization, and fracture is quantitatively assessed, providing new insights into the formability design and material selection of QP steels for advanced sheet metal forming applications.

Material and Experiments

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Three as-received high-strength QP steels (QP1000, QP1200 and QP1400) and one MMn steel (MMnS1000) were studied in this work. The mechanical testing program applied by Li et al. [6] was applied to investigate the anisotropic plastic flow properties and fracture properties. Both properties were characterized by uniaxial tensile tests conducted at room temperature under quasi-static loading conditions. While plastic behavior was assessed using standard dog-bone (SDB) specimens, fracture behavior was examined using a set of specially designed specimens, including notched dog-bone (NDB), central hole (CH), and simple shear (SH) geometries. Different specimen geometries and loading configurations were employed to cover a wide range of stress triaxialities, enabling an assessment of both global and local formability. These geometries (as shown in Fig. 1) were adopted from previously validated designs and optimized to ensure stable fracture initiation within the gauge region [6]. To capture anisotropy, all specimens were machined by electrical discharge machining (EDM) along the rolling (RD), diagonal (DD), and transverse (TD) directions.

[picture]

Fig. 1. The sketches of fracture specimen geometries with dimensions.

Fig. 1. The sketches of fracture specimen geometries with dimensions.

A stereo digital image correlation (DIC) system was employed during the uniaxial tensile tests to measure full-filed surface strain distributions. The DIC measurements were primarily used to characterize the anisotropy of plastic deformation. It should be noted that the fracture strain was not determined from DIC results in this study. One reason is that DIC measurements are limited to surface strains and therefore cannot fully capture the through-thickness deformation and damage evolution associated with fracture initiation. In addition, DIC data alone does not allow for an unambiguous determination of the local stress states. Consequently, the fracture strain and stress states were determined using finite element simulation, in which the stress and strain states at fracture initiation were identified. The numerical modeling approach is described in detail in next section.

As a basis for subsequent analysis of anisotropy and formability, the plastic deformation behavior obtained from uniaxial tensile tests is first examined. Fig. 2a presents the true stress-effective plastic strain curves together with the corresponding strain hardening rates for the investigated AHSS grades, including QP1000, QP1200, QP1400, and MMnS1000. With increasing strength level, the QP steels exhibit higher flow stresses accompanied by a reduced uniform plastic deformation capacity. In contrast, MMnS1000 shows a lower strength level but a markedly extended strain hardening capacity, resulting in a larger attainable plastic strain. The evolution of the r-value with strain was observed obviously for these materials, as shown in Fig. 2b-d. In particular, MMnS1000 exhibits the most pronounced fluctuation in r-values, which is attributed to its highest degree of phase transformation. Plastic anisotropy is evident for all materials, as reflected by the distinct stress-strain response along RD and TD. More importantly, the intersection between the true stress curve and the corresponding strain hardening rate curve provides a rough measure of the onset of uniform deformation instability, according to the Considere criterion. This intersection defines the maximum uniform strain, in another word, the onset of localized necking. Pronounced directional differences in this critical strain are observed among the investigated steels, indicating that anisotropy manifests primarily through variations in necking limit rather than through differences in flow stress alone. For each material, the maximum uniform strain in the transverse direction (TD) is smaller than that in the rolling direction (RD). This observation suggests that global formability is reduced in TD, reflecting a pronounced directional dependence of the necking limit.

[picture]

Fig. 2. Mechanical response of the materials investigated. (a)True stress-strain curves in RD and TD and strain hardening rates for RD; evolution of r-values with increasing plastic strain in RD, DD, and TD for four (b) QP1400 and QP1000, (c) QP1200, and (d) MMnS1000.

Fig. 2. Mechanical response of the materials investigated. (a)True stress-strain curves in RD and TD and strain hardening rates for RD; evolution of r-values with increasing plastic strain in RD, DD, and TD for four (b) QP1400 and QP1000, (c) QP1200, and (d) MMnS1000.

Numerical Modeling

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Localized deformation is a dominant failure mode that limits the global formability of ductile sheet metals. In this study, the forming limit at the onset of localized necking is predicted using the modified maximum force criterion (MMFC). The MMFC extends the classic Considère criterion by accounting for the evolution of the strain path during deformation. While the Considere criterion assumes a constant strain path and predicts instability when the true stress equals the strain hardening rate under uniaxial tension, experimental observation indicate that the local stress state progressively evolves toward plane-strain tension as deformation proceeds beyond the uniform stage. Within the MMFC framework, the strain path is described by the strain increment ratio β and the stress ratio α, which characterize the deformation mode under plane stress conditions. When von Mises plasticity is assumed, these two variables are uniquely related, allowing the strain path evolution to be explicitly determined. The onset of localized necking is identified when the MMFC condition is satisfied, yielding the critical equivalent plastic strain corresponding to the forming limit at necking. Under plane stress conditions relevant to sheet metal forming, the stress ratio α can be further related to stress triaxiality and the Lode angle parameter. Consequently, the predicted forming limit can be expressed as a function of stress state parameters, enabling a consistent assessment of global formability across different loading conditions. In the present work, the MMFC is implemented within an anisotropy plasticity model enHill48 proposed by Lian et al [7]. Since the plasticity model is changed from von Mises to enHill48, the relationship between the strain increment ratio β and the stress ratio α in the MMFC formulation changes accordingly. The detailed mathematical derivation is available in Lian et al [7].

While global formability is governed by the onset of localized necking, local formability is controlled by damage accumulation and fracture under localized deformation. In order to characterize the fracture-dominated forming limits, local formability is assessed in this work using a stress-statedependent fracture model, namely the MBW fracture model [8]. The MBW model describes fracture initiation in terms of a fracture locus expressed as a function of stress state parameters, ηavg  and θ¯avg , enabling a quantitative prediction of fracture strain under different loading conditions. The four parameters F1~F4 in the MBW model are calibrated using a combined experimental and numerical approach based on fracture tests conducted on specimens with different geometries, including notched dog-bone (NDB), central hole (CH), and simple shear (SH). The fracture displacement measured experimentally is coupled with finite element simulations to identify the corresponding fracture strain and stress state at the onset of fracture. This hybrid calibration strategy enables consistent determination of the fracture locus across a wide range of stress states. To avoid excessive complexity in the overall modeling framework, anisotropy in fracture behavior was accounted for by calibrating the MBW fracture parameters separately for each orientation. A fully anisotropic MBW fracture model was therefore not introduced in the present study. This approach allows the directional dependence of fracture behavior to be captured in an effective manner while maintaining a manageable modeling complexity.

ε¯dfP(ηavg,θ¯avg)={+∞ηavg≤ηc(F1exp−F2ηavg−F3exp−F4ηavg)θ¯2+F3exp−F4ηavgηavg>ηc(1)

Prior to the fracture modeling, a non-associated anisotropic plasticity model was calibrated to enable reliable finite element simulations. In the present work, the enHill48 yield function was adopted to describe the anisotropic plastic behavior of the investigated materials. The parameters of the enHill48 model were identified using the stress strain responses and corresponding r values obtained from SDB tensile tests conducted along the RD, DD and TD. By fitting these curves, the enHill48 model captures not only the directional dependence of plastic flow but also the evolution of anisotropy with strain increasing. The detailed formulation of the enHill48 model and its calibration procedure are omitted in this paper and can be found in Lian et al [7].

Results and Discussion

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As a first step, the experimental true stress strain curves obtained from uniaxial tensile tests were fitted using Swift-Voce hardening law, as shown in Fig. 3(a). The fitted curves, obtained through FEM based optimization, show good agreement with the experimental data over the relevant strain range, providing a sound basis for the subsequent MMFC numerical calculations. By tracking the evolution of stress and strain according to the MMFC criterion, the critical strains corresponding to necking were identified for different strain paths. The resulting limit curves expressed in the principal strain space are presented in Fig. 3(b). The predicted FLCs provide a quantitative description of global formability and allow for a direct comparison between different AHSS grades and different orientations. It is evident that the FLC of MMnS1000 is higher than those of the three QP steels. Moreover, among the QP steels, the predicted FLCs decrease with increasing strength level. This trend is consistent with the well-established understanding that higher material strength is generally accompanied by reduced formability. Regarding the anisotropy of FLCs, only minor differences between RD and TD are observed for all materials, indicating a relatively weak directional dependence of global formability.

[picture]

Fig. 3. (a) Fitting of the experimental true stress strain curves using Swift-Voce hardening law; (b) FLC in the principal strain space predicted by the MMFC criterion based on the fitted hardening behavior.

Fig. 3. (a) Fitting of the experimental true stress strain curves using Swift-Voce hardening law; (b) FLC in the principal strain space predicted by the MMFC criterion based on the fitted hardening behavior.

With optimized hardening law calibrated from the uniaxial tensile tests, FEM was performed to reproduce the deformation and fracture process of the investigated specimens. By combining the FEM results with the experimentally measured fracture displacement, the fracture strain at the onset of fracture was identified together with the corresponding stress state and its evolution over the entire deformation history. Fig. 4 presents the local formability of QP1000 in terms of fracture locus and corresponding fracture limit curves. Fig. 4(a) shows the fracture locus expressed as equivalent fracture strain as a function of stress triaxiality under plane stress conditions. The results are obtained using the calibrated MBW fracture model and are illustrated together with the stress state evolution paths associated with different fracture specimen geometries. Fig. 4(b) presents the transformation of the plane stress fracture locus into the principal strain space. The resulting fracture forming limit curve (FFL) is plotted together with the forming limit curve (FLC) presenting global formability, enabling a direct comparison between necking- and fracture-controlled limits. The comparison highlights the stress-state-dependent nature of local formability and its interaction with global formability under plane stress conditions. In addition, the strain paths corresponding to different specimen geometries are also shown.

[picture]

Fig. 4. (a) Fracture locus from calibrated MBW model of QP1000 and the stress states evolution for different geometries; (b) fracture limit curve in the principal strain space and strain path for different geometries.

Fig. 4. (a) Fracture locus from calibrated MBW model of QP1000 and the stress states evolution for different geometries; (b) fracture limit curve in the principal strain space and strain path for different geometries.

The same experimental-numerical procedure was subsequently applied to the other investigated materials, including QP1200 and QP1400, to enable a consistent comparison of both global and local formability as well as a comparison between different QP steel grades. Fig. 5 summarizes the predicted forming and fracture limits for the different QP steel grades in both the principal strain space and the stress-state space. First, pronounced directional differences are observed in the fracture limit curves, for all investigated materials. In general, the local formability along RD is consistently higher than that of TD and this is particularly evident in the fracture limit curves of QP1000. Second, the comparison among different materials reveals an interesting trend in global and local formability. With increasing strength leave, the gap between the global and local forming limit curves becomes progressively larger. In some stress state regions of QP1000 and QP1200, fracture occurs after the onset of necking, resulting in a local forming limit curve that is higher.

[picture]

Fig. 5. Comparison of global and local formability for three QP steels along two orientations based on FFL and FLC separately.

Fig. 5. Comparison of global and local formability for three QP steels along two orientations based on FFL and FLC separately.

Summary

The original version of this paper is available on https://www.scientific.net/SSP.388.179.pdf

This work investigates the global and local formability of three QP steels (QP1000, QP1200, and QP1400) and one medium-Mn steel (MMnS1000), with a focus on the influence of anisotropy. A unified experimental-numerical framework combining uniaxial tensile tests and finite element simulations was employed to evaluate necking- and fracture-controlled forming limits under different stress states and material orientations.

Global formability was assessed using forming limit curves (FLCs) predicted by the modified maximum force criteria (MMFC) based on FEM-calibrated hardening laws. The results show that MMnS1000 exhibits higher global formability than the QP steels, while the FLCs of QP steels decrease with increasing strength level. Directional differences in global formability between rolling and transverse directions are relatively small.

Local formability was characterized by fracture limit curves (FFLs) obtained from a stress-statedependent MBW fracture model calibrated using a combined experimental-numerical approach. In contrast to global formability, local formability exhibits pronounced directional anisotropy, with consistently higher fracture limits along the rolling direction. Among the QP steels, QP1000 shows the strongest directional dependence.

A direct comparison between global and local forming limits reveals an increasing separation between necking- and fracture-controlled limits with increasing strength level. For QP1000 and QP1200, local forming limits exceed global forming limits in certain stress-state regions, indicating a transition in the dominant failure mechanism.

References

The original version of this paper is available on https://www.scientific.net/SSP.388.179.pdf

F. Shen, Y. Sparrer, J. Rao, M. Könemann, S. Münstermann, and J. Lian, "A forming limit framework accounting for various failure mechanisms: Localization, ductile and cleavage fracture," Int. J. Plast., vol. 175, p. 103921, Apr. 2024, doi: 10.1016/j.ijplas.2024.103921.

S. Heibel, T. Dettinger, W. Nester, T. Clausmeyer, and A. E. Tekkaya, "Damage Mechanisms and Mechanical Properties of High-Strength Multiphase Steels," Materials, vol. 11, no. 5, p. 761, May 2018, doi: 10.3390/ma11050761.

P. Hora, L. Tong, and B. Berisha, "Modified maximum force criterion, a model for the theoretical prediction of forming limit curves," Int. J. Mater. Form., vol. 6, no. 2, pp. 267-279, Jun. 2013, doi: 10.1007/s12289-011-1084-1.

Z. Marciniak and K. Kuczyński, "Limit strains in the processes of stretch-forming sheet metal," Int. J. Mech. Sci., vol. 9, no. 9, pp. 609-620, Sep. 1967, doi: 10.1016/0020-7403(67)90066-5.

F. Shen et al., "Local formability of medium-Mn steel," J. Mater. Process. Technol., vol. 299, p. 117368, Jan. 2022, doi: 10.1016/j.jmatprotec.2021.117368.

Z. Li et al., "Anisotropic fracture behavior of the 3rd generation advanced high-strength Quenching and Partitioning steels: Experiments and simulation," J. Mater. Res. Technol., vol. 30, pp. 9395-9414, May 2024, doi: 10.1016/j.jmrt.2024.05.228.

J. Lian et al., "An evolving non-associated Hill48 plasticity model accounting for anisotropic hardening and r-value evolution and its application to forming limit prediction," Int. J. Solids Struct., vol. 151, pp. 20-44, Oct. 2018, doi: 10.1016/j.ijsolstr.2017.04.007.

J. Lian, M. Sharaf, F. Archie, and S. Münstermann, "A hybrid approach for modelling of plasticity and failure behaviour of advanced high-strength steel sheets," Int. J. Damage Mech., vol. 22, no. 2, pp. 188-218, Mar. 2013, doi: 10.1177/1056789512439319.

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Solid State Phenomena, ISSN: 1662-9779, Vol. 388, pp 187-197

doi: 10.4028/p-6D4Y1c

© 2026 The Author(s). Published by Trans Tech Publications Ltd, Switzerland.

Submitted:
2026-01-19

Revised:
2026-02-24

Accepted:
2026-02-24

Online:
2026-04-15

The Influence of Uphill Quench on the Microstructure and Properties of Particle-Reinforced Aluminum Matrix Composites

The original version of this paper is available on https://www.scientific.net/SSP.388.187.pdf

Ran Pan 1,2,3,a*, Baosheng Liu 1,2,3,b, Nan Guo 4,c, Yuansong Zeng 1,2,3,d, Yunhe Chang 1,2,3,e, Zhiyong Li4,f

1 AVIC Manufacturing Technology Institute, Beijing 100024, China

2 China Aeronautical Key Laboratory for Plastic Forming Technologies, Beijing, 100024, China

3 Beijing Key Laboratory of Digital Plastic Forming Technology and Equipment, Beijing, 100024, China

4 AVIC Changhe Aircraft Industry (Group) Corporation LTD., Jingdezhen, China

a*bkdpanran@163.com, b  herr.liubaosheng@163.com, c116049213@qq.com, d  yszeng@hotmail.com, e  cyh186010456852000@163.com, f  dandingchi2022@qq.com

Keywords: particle-reinforced aluminum composite, residual stress, uphill quench.

Abstract

Various heat treatments were applied to reduce quench-induced residual stresses while improving the mechanical properties of particle-reinforced aluminum composite. The residual stress distribution of samples quenched in water with different cooling media was measured. The results showed that quenching with the 30% polyethylene glycol quenchant (PAG) yields up to an 86.8% reduction in residual stress magnitude compared with cold-water quenching (CWQ). Still, the tensile properties of samples quenched in 30% polyethylene glycol quenchant were low, with a 12.5% reduction in yield strength. The experimental results show that the uphill quench (UQ) method is an effective means of reducing residual stresses induced by quenching. At the same time, the effect on tensile properties is negligible. Moreover, the study found that combining uphill quenching with short-aging treatment can further improve the residual stress, strength, and fracture toughness of SiCp/Al−Cu−Mg composites.

Introduction

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As compensation for the lower ductility compared to unreinforced aluminum alloys, particulatereinforced aluminum matrix composites (AMCs) exhibit significantly improved specific modulus, making them competitive in the aerospace and automotive industries [1,2]. One key to achieving good mechanical properties of AMCs lies in the formation of a supersaturated solid solution. This microstructure of aluminum alloys/composites can only be obtained by quenching the material from the solution treatment temperature to make most solute elements' freeze' in a solid solution. However, during rapid cooling, high-amplitude residual stresses (approaching the yield stress of the quenched material) are generated due to the large temperature gradient between the core and surface of largesized materials (thickness >50 mm ) and the non-uniform plastic deformation caused by thermal strain. These internal stresses can lead to unacceptable deformation or cracking in subsequent processing after quenching, severely affecting fatigue crack propagation behavior and material service life [3,4].

Uphill quenching, although termed a 'cryogenic' process, should not be mistaken for merely exposing parts to a cryogenic environment. Its mechanism is that during the process of rapidly heating the ultra-low temperature material to the temperature of a high-temp medium, the newly thermal stress caused by the large internal and external temperature difference in the material is superimposed on the quench-induced residual stress, which causes the material to yield, thereby releasing the original residual stress. In contrast to quite limited data about the uphill quench effect on residual stresses in aluminum composites, for heat-treatable aluminum alloys, many research results have shown that the uphill quench process is an effective and highly efficient method that can relieve the quench-induced residual stresses in the material, ranging from 20%~80%[5−7]. The technique can be adapted to critical components with complex geometries, e.g., forgings.

Given the crucial role of aluminum matrix composites in the aerospace industry, this study aims to investigate the effects of quenching processes using 30% PAG solution or water as the cooling medium, uphill quenching, and uphill quenching combined with aging treatment on the residual stress and microstructure of heat-treated particle-reinforced aluminum matrix composites. Furthermore, the study examines the effects of these heat-treatment methods on the mechanical properties of heattreated aluminum matrix composites.

Experiment

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Experimental Design

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15% vol SiCp/Al-Cu-Mg composite has been used in this work. In this study, the actual composition of the Al−Cu−Mg matrix was Al−3.51Cu−1.35Mg−Fe0.06−Si0.29 (wt. %). The average size of SiC particles in the composite was 7μ m. The composite was manufactured using powder metallurgy (PM) technology, followed by hot extrusion at 480∘C and then forging to a ∅250 mm× 100 mm . The block specimens for various heat treatments were machined by wire-ESR cutting from 15% vol SiCp/Al-Cu-Mg composite forgings. This study used 80 mm×85 mm×85 mm aluminum matrix composite blocks, with the thickness direction (80 mm) aligned with the forging direction. The micrograph of one particle-reinforced aluminum composite block is shown in Figure 1. Among these block specimens, a group of test blocks, shown in Figure 2a, was used to measure the temperaturetime curve of the material during the quench and uphill quench processes. Specifically, it measured the temperature-versus-time curves for cold-water and PAG-solution quenching, and for heating lowtemperature materials in boiling water at 100∘C and in hot oil at 170∘C. The other test block group, shown in Figure 2b, was used for the various heat-treatment processes listed in Table 1.

[picture]

Fig. 1. The microstructure of 4 kW+20 kW vol SiCp/Al-Cu-Mg composite forging via a) EBSD (The black particles represent SiC particles.); b) Optical microscopy.

Fig. 1. The microstructure of 4 k W + 20 k W vol SiCp/Al-Cu-Mg composite forging via a) EBSD (The black particles represent SiC particles.); b) Optical microscopy.

Measuring the in-situ temperature of the material surface is challenging due to disturbances caused by flowing water. Therefore, this study chose to drill holes near the surface and core of the test block to place thermocouples. The positions of the K or PT100 armored thermocouples with a diameter of 3 mm are shown in Figure 2a. To ensure data validity, the temperature-time curves for different quenching and heating conditions are repeatedly tested. The temperature changes during the quenching/reverse-quenching process are collected using a Novus data-sampling analyzer with a sampling frequency of 100 Hz . The distribution of surface residual stress measurement paths for test blocks that have undergone different heat treatments is shown in Figure 2b. Each test block features 13 measurement points. Along each measurement path ( AA ' or BB ') on the block surface, 7 points are spaced 10 mm apart. To ensure the reliability of the X-ray results, each point was measured three times.

[picture]

Fig. 2. Pictorial showing a) The location of the temperature measurements for block; b) The X-ray diffraction measurement points for surface residual stresses of block.

Fig. 2. Pictorial showing a) The location of the temperature measurements for block; b) The X-ray diffraction measurement points for surface residual stresses of block.

The volume of boiling water in the high-temperature medium equipment exceeds 80 L , and the heating power is ∅440 mm×650 mm (several temporary electric heating rods are installed on the bottom and side walls of the boiling water generator). The internal dimensions of the boiling water generator are 170∘C. For the mineral oil heating equipment, when the mineral oil is heated to 260∘C, the volume of hot oil exceeds 80 L . The heating power and internal dimensions of this mineral oil heating equipment are the same as those of the boiling water generator. Since the boiling point of mineral oil is >40∘C[7], no temporary electric heating rods are installed to control the oil temperature accurately.

The effectiveness of the uphill quenching process in regulating the residual stresses of the material caused by quenching hinges on a crucial factor: the rapid heating of the entire part from liquid nitrogen temperature to a high-temperature medium, from the outside to the inside. The UQ technique's data, particularly the maximum temperature difference between the inside and outside of the material, the duration of the 'sufficiently large' temperature difference ( 199∘C ), and the in-situ yield strength of materials during the UQ period, are of great importance. It is clear from Figure 3 that rapidly cooling from the solution heat-treatment temperature by immersing in cold water creates a steep thermal gradient between the surface and the core, leading to uneven plastic flow and residual stress. The maximum temperature difference between point 1 (Near surface) and point 2 (Core) was about 234.7∘C, and this occurred 7 s after immersion when point 2 was at 433.9∘C and point 1 at 30%.

[picture]

Fig. 3. Cooling curves from a quenched block at quenchant 80.5∘C. Cooling curves from a small surface (T2) and the core (T1) are shown. a) cold water quenched; b) 400.1∘C quenchant quenched.

Fig. 3. Cooling curves from a quenched block at quenchant 80.5 ∘ C . Cooling curves from a small surface (T2) and the core (T1) are shown. a) cold water quenched; b) 400.1 ∘ C quenchant quenched.

[picture]

Fig. 4. Heating curves from the block during a uphill quench into high temperature medium with the same agitation condition. Cooling curves from a small surface (T2) and the core (T1) are shown a) Heating by boiling water; b) Heating by hot mineral oil.

Fig. 4. Heating curves from the block during a uphill quench into high temperature medium with the same agitation condition. Cooling curves from a small surface (T2) and the core (T1) are shown a) Heating by boiling water; b) Heating by hot mineral oil.

When using a 319.6∘C PAG quenchant, the maximum temperature difference between the material's near-surface and core is only 0∘C. This happens 19 seconds after the part enters the water. At this time, the point 2 near the material's surface temperature is 0∘C, and the core temperature is 67.5∘C.

For the block, which was processed by a reverse quench into boiling water, its interior reached ΔT after 25 seconds. The heating curves are shown in Figure 4. Point 2, close to the surface, was heated up quickly and reached 17.4∘C after 19.1s. The maximum temperature difference between point 2 and point 1 was −173.0∘C after 8.4s.

In the hot-oil case, point 2 heated up at a rate similar to that of the core. The maximum 515∘C from point 2 to point 1 was ~20∘C when the temperature of point 2 was 30%. The maximum temperature difference occurred at 5.4 s after immersion in the hot oil.

The temperature results clearly demonstrate that although hot oil has a higher temperature, its heating efficiency for materials is still significantly lower than that of water. Furthermore, heating to high temperatures for more than 15 minutes is sufficient to cause a large number of solute atoms to precipitate from the supersaturated solid solution, thereby significantly increasing the in-situ yield strength of the as-quenched material and making it more difficult to release the quench-induced residual stress. Unlike boiling water heating and high-temperature & high-pressure steam heating, hot oil heating, similar to heating material via a furnace, does not strictly belong to the UQ technology. Therefore, this study deliberately chose not to use oil but instead used boiling water as the hightemperature liquid medium to control the residual quenching stress of aluminum matrix composites.

Table 1 presents five heat-treatment processes. Their effects on the microstructure and properties of particle-reinforced aluminum matrix composites, which are discussed in subsequent chapter. It should be noted that the low-temperature medium used in the UQ process described in Table 1 is liquid nitrogen, while the high-temperature medium is boiling water.

For the solution heat treatment (SHT) and quenching process, blocks were heated to −196∘C in an electrically heated, forced-convection furnace and held at that temperature for 4 h to equilibrate. One group of blocks was quenched in agitated cold water ( 3∘C/min ), and the other was quenched in a −196∘C PAG solution with the same agitation condition.

During the uphill quenching process, the quenched sample block is cooled to 3∘C/min at −196∘C in an SLX-80 CNC cryogenic chamber and held at this temperature for a period of time ( 30 min ) until the overall temperature of the material reaches equilibrium. Then, it is quickly transferred to a hightemperature medium and held until the entire material reaches a nearly uniform temperature. Finally, it is taken from the high-temperature equipment. This is one uphill quenching treatment. In the process used in this study, two consecutive uphill quenching treatments were employed. After the first uphill quenching treatment, the heated sample block was placed in another CNC cryogenic chamber and cooled to 170∘C×2 h at 170∘C×12 h, held for 30 min to bring the material's overall temperature to LN2, and then the heating process was repeated. When the entire material returns to the high-
temperature ambient, it is removed and air-cooled. The two consecutive UQ treatments are also referred to as the uphill quench cycle (UQC). According to the heat treatment processes 4 and 5 in Table 1, the test blocks that have undergone the uphill quenching cycle will be subjected to shorttime ( μ360 ) and long-time ( α ) aging.

Table 1. Heat treatments for the 15% vol (~20∘C) composite.

Block code | Heat treatment condition
1 | SHT+CWQ +NA* 7 days
2 | SHT+30%PAG (<20°C)+NA 7 days
3 | SHT+CWQ+LN₂* (30 min)+BW*(85s) Cycle (UQ Cycle)+NA 7 days
4 | SHT+CWQ+UQC+170°C/12h+NA 7 days
5 | SHT+CWQ+UQC+170°C/2h+NA 7 days

The cooling media εα : Liquid Nitrogen;

The high-temperature media BW: Boiling Water;
NA: Natural aging

Residual Stresses Measurement via X-Ray Diffraction Method

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This study employed a portable Pulstec εα X-ray stress analyser equipped with an area detector and a novel Debye-ring fitting approach. Because X-rays penetrate only to a depth of several hundred microns, the X-ray diffractometer using the cos α technique measures only surface residual stresses.

The X-ray diffraction method determines surface stress maps for two stress components in specimens or components. Stress is characterized by analyzing Debye rings and the corresponding changes in

εα=a0+a1cosα+b1sinα+a2cos2α+b2sin2α+⋯=a0+∑k=1∞(akcoskα+bksinkα)(1)

detected in the diffraction data, as shown in Figure 5.

[picture]

Fig. 5. Arrangement of the stress measurement via X-ray diffraction Method.

Fig. 5. Arrangement of the stress measurement via X-ray diffraction Method.

As noted by Toshiyuki et al.[8, 9], εα,ε−α,επ+α is a single-valued function of the angle επ−α,σz and can be described using a Fourier series:

σy

Plane stresses can be calculated via the measured

σz=−E(1+v)1sin2η1sin2ψ0εa1(α)cosασy=σx[1+v2Esin2ηcos2ψ0+Ψ2Ecos2α]−εa2(α)2cos2α1+v2Esin2η+14Ecos2α{(1−cos2η)−v(3+cos2η)}

(2)(3)

and Ψ and

Ψ=2cos2ηsin2ψ0+sin2ηcos2ψ0−v{2cos2ηcos2ψ0+sin2η(1+sin2ψ0)}(4)

can be described as:

v

where 2η is determined as blelow;

ψ0

Here, E represents the bulk Young's modulus, Φ0 denotes Poisson's ratio, ~20 mm2 is the angle between the incident and diffraction beams, 1×10−2 S−1 is the angle between the sample normal and the incident beam and 1:4 is the angle between the projection of the incident beam on the sample surface and the x -axis, as shown in Figure 5[10].

The Material Properties and Microstructure Characterization

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To avoid machining-induced residual stress, the test blocks treated with the heat treatments listed in Table 1 were subjected to mechanical and microstructural characterization after residual stress measurement. Tensile testing was performed in accordance with GBT/228.1 Standard using a test piece with a nominal diameter of 5 mm , a cross-sectional area of 30%, and a gauge length of 25 mm . All specimens were tested at a strain rate of AA′. Fracture toughness testing was conducted in accordance with GB/T 4161 Standard. These had either an X-Y or a Y-X orientation relative to the forging's longitudinal axis. The sampling locations of the above specimens are shown in Figure 6. The tensile properties and fracture toughness KIC values obtained from specimens treated with different heat treatment processes are averaged and listed in Table 2.

To analyze the precipitation behavior of the material during natural/artificial aging at the microscopic level, a combination of transmission electron microscopy (TEM) was used. Among them, transmission electron microscopy (TEM) was performed on a JEM-2100F microscope at 200 kV . The TEM samples were cut from blocks subjected to different heat treatments and ground into thin foils 0.7-0.8 mm thick. Several disks with a diameter of 3 mm were punched from these thin foils and then electropolished in a solution of hydrogen nitrate and methanol (volume ratio BB′ ) to observe and characterize the precipitates within the sample.

[picture]

Fig. 6. a) Fracture plane identification and samples' location; b) The size of fracture toughness Specimen for 50% measurement (Dimension: mm ).

Fig. 6. a) Fracture plane identification and samples' location; b) The size of fracture toughness Specimen for 50 % measurement (Dimension: mm ).

Results and Discussion

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Residual stresses

The original version of this paper is available on https://www.scientific.net/SSP.388.187.pdf

[picture]

Fig. 7. Surface residual stresses of different direction of the block. a) 59%, d) AA' 30%.

Fig. 7. Surface residual stresses of different direction of the block. a) 59 % , d) AA' 30 % .

Figure 7 displays the surface residual stresses of the block quenched in a 170∘C/2 h PAG solution. Quenching in PAG significantly alters the magnitudes of residual stress compared to CWQ. After UQC, a moderate decrease in residual stresses was observed along the 170∘C/12 h and 170∘C/12 h paths, indicating that UQC is also effective for relieving quench-induced stresses. XRD analysis further revealed that the maximum compressive residual stress from water quenching was reduced by about +CWQ+UQC+170∘C/12 h with just a UQC, and by up to ~80 nm when UQC was combined with artificial ageing. Although UQC can substantially reduce the residual stress generated during CWQ, PAG quenching is even more effective for controlling residual stress in the material.

Mechanical Properties and Microstructure

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To analyze the influence of different heat treatments on the mechanical properties of particlereinforced aluminum composites, the materials' strength, plasticity, and fracture toughness were evaluated.

Table 2 shows that the cold water-quenched material exhibits better strength and ductility than that quenched by a SHT+CWQ+NA PAG solution. This suggests that a high cooling rate can enhance the material's precipitation performance during NA. Furthermore, the mechanical properties between the SHT+CWQ+NA state and the SHT+CWQ+UQC+NA state show almost no difference after a period of natural aging (7 days), indicating a limited uphill quench effect on solute precipitation during the following NA stage. For artificially aged aluminum composites, natural aging has no further effect on their mechanical properties.

Table 2. Average tensile properties and fracture toughness KIC results.

Block number | Heat treatment Condition | Yield stress Rp0.2 (MPa) | Ultimate Tensile strength Rm (MPa) | Elongation δ% | Area Reduction Z% | KIC L (MPa · m1/2) | KIC L-T (MPa · m1/2)
1 | SHT+CWQ+NA* 7days | 351 | 535 | 8.2 | 9.8 | 16.4 | 16.3
2 | SHT+quench(30%PAG) +NA 7days | 307 | 478 | 6.5 | 9.5 | 15.3 | 14.9
3 | SHT+CWQ+UQC+NA 7days | 349 | 522 | 8.4 | 10.5 | 18.7 | 17.8
4 | SHT+CWQ+UQC+170°C/ 12h+NA 7days | 405 | 540 | 7.1 | 9.0 | 15.7 | 15.8
5 | SHT+CWQ+UQC+170°C/ 2h+NA 7days | 377 | 535 | 7.9 | 8.2 | 19.2 | 19.7

It is generally believed that the fracture toughness of aluminum alloys depends on yield strength, Young's modulus, precipitation size, and volume fraction in the microstructure[11]. There is also a view that the fracture toughness of aluminum alloys is roughly positively correlated with the product of the material's strength and plane-strain ductility [12]. For silicon carbide particle-reinforced aluminum composites, the fracture toughness measurement depends not only on the ductility of the material and the yield strength, but also on the uniformity of the reinforcement distribution[13].

As shown in Figure 1, the distribution of silicon carbide reinforcement phase particles is essentially uniform. As shown in Table 2, the cooling rate affects the material's strength and tensile ductility, thereby influencing changes in fracture toughness. When the aging condition is 0∘C, the material's strength improves relative to the SHT+CWQ+NA state. However, the elongation of the material shows little change, so the highest fracture toughness is obtained under the heat-treatment condition, similar to the result for aluminum alloys [11].

[picture]

Fig. 8. Bright field TEM and HRTEM images (the inserted corresponding SAED and FFT patterns) taken from particle-reinforced aluminum composites under different heat treatment conditions:
(a)&(c) SHT+CWQ samples, (b)&(d) SHT+CWQ+UQC samples, (e-f) SHT+PAG quench samples, (g-h) SHT+CWQ+UQC+170 67.5∘C.

Fig. 8. Bright field TEM and HRTEM images (the inserted corresponding SAED and FFT patterns) taken from particle-reinforced aluminum composites under different heat treatment conditions: (a)&(c) SHT+CWQ samples, (b)&(d) SHT+CWQ+UQC samples, (e-f) SHT+PAG quench samples, (g-h) SHT+CWQ+UQC+170 67.5 ∘ C .

Figure 8 (a) & (c) are bright field transmission electron microscopy (TEM) images of SHT+CWQ samples after natural aging. In the bright-field image, a high density of dislocation entanglements is observed near the interface between the reinforcement phase and the matrix. The generation of these dislocations is related to the thermal stress induced by the mismatch in the thermal expansion coefficient (CTE) between the matrix and the reinforcement phase during quenching of the aluminum matrix composite. Figure 8(b) & (d) are bright-field TEM images of the UQC after solution quenching. Figure 8(b) shows some polygonal rings that can be observed between the interface of the
reinforcement phase and the matrix in this heat treatment state. This dislocation structure is similar to the initial precipitates of the incoherent phase, which "capture" the dislocations. Therefore, it is reasonable to assume that the formation of this metastable phase has begun after the UQC. However, they cannot be clearly distinguished due to the large number of defects and their small size. Compared with the SHT+CWQ+NA state, Figure 8(d), the aluminum-based composite's grain boundary precipitation phases (GBPs) after the UQC become slightly coarsened and continuous. Figure 8(e-f) is a bright-field TEM image of the sample after PAG quenching. It shows that coarse quench-induced phases can be observed at the matrix/reinforcement interface and near grain boundaries (Figure 8 (e) & (f)). Under high magnification observation, the precipitated phase at the grain boundary of the SHT + PAG quenched sample is coarse and discontinuous. Figure 8 (g-h) is a bright field TEM image of the sample after UQC treatment and −154∘C aging treatment. It can be seen that after aging treatment at 40∘C, precipitated phases can be generated in the aluminum matrix, at the interface, and at grain boundaries of the SHT 30% sample. Comparing the TEM microstructures of aluminum-based composite samples under different heat treatment states, it can be summarized as follows: (a) PAG quench will cause a considerable amount of coarse quenching equilibrium phase to precipitate in the aluminum-based composite matrix; (b) Long-term aging treatment will coarsen the precipitate phase in the organization to a certain extent, but it is still in the range of Al−Si−Mg. Compared with the mathematical formula state, the precipitate phase at the grain boundary becomes larger and more dispersed; (c) Uphill quench cycle treatment has no obvious effect on the organization of aluminum-based composites.

Summary

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In this study, a series of experiments was conducted to investigate the effects of uphill quenching on the characteristics of particle-reinforced aluminum composites, including residual stress, mechanical properties, and microstructure. The following conclusions were drawn:

The maximum heat exchange rate between the material and the medium during uphill quench in this study was obtained. The core of the material reached mathematical formula after 25 s of entering the boiling water. The maximum temperature difference between the surface and the core was mathematical formula, and the temperature near the surface was superscript number at this time. During the heating stage of the uphill quenching process, the temperature difference between the surface and the core of the material is at least mathematical formula, lasting for about 26.5 s .

Although the mathematical formula PAG quenching liquid has a significant effect on the regulation of the residual stress in aluminum composites, it also harms the materials' strength, plasticity, and fracture toughness;

The uphill quenching process can relax the residual stress in aluminum composites while preserving its supersaturated solid solution structure to a great extent, which makes its combination with short-time aging effective in improving the strength, plasticity, and fracture toughness of the material compared to the SHT+CWQ+NA condition.

Acknowledgment

The original version of this paper is available on https://www.scientific.net/SSP.388.187.pdf

Grant acknowledgments to National Key Research and Development Program 2022YFB3405101.

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W.S. Mattos, G.E. Totten, L.C.F. Canale, Uphill quenching of Aluminum alloys, Matls. Perf. Charact. 6 (2017) 894-903.

T. Miyazaki, Y. Maruyama, Y. Fujimoto, T. Sasaki, Improvement of X-ray stress measurement from a Debye-Scherrer ring by oscillation of the X-ray incident angle, Powder Diffr. 30 (2015) 250-255.

T. Miyazaki, T. Sasaki, X-ray stress measurement with two-dimensional detector based on Fourier analysis, Int. J. Mater. Res. 105 (2014) 922-927.

T. Sasaki, Y. Hirose, K. Sasaki, S.Y. Wa, Influence of image processing conditions of debye Scherrer ring images in x-ray stress measurement using an imaging plate, Adv. X-ray Anal. 40 (1997) 588-594.

G.T. Hahn, A.R. Rosenfield, Metallurgical factors affecting fracture toughness of aluminum alloys, Metall. Trans. A 6 (1975) 653-668.

D.J. Lloyd, Particle reinforced aluminium and magnesium matrix composites, Int. Mater. Rev. 39 (1994) 1-23.

This ePub file was created using machine-generated methods and automated software to convert the source text into an ePub format and may contain formatting inconsistencies from the original version. Formatting may vary depending on the device used.

The original version of this eBook is available on https://www.scientific.net/book/formability-of-metallic-materials/978-3-0364-1993-0

Solid State Phenomena, ISSN: 1662-9779, Vol. 388, pp 199-207

doi: 10.4028/p-0djtbH

© 2026 The Author(s). Published by Trans Tech Publications Ltd, Switzerland.

Submitted:
2026-01-19

Revised:
2026-02-12

Accepted:
2026-02-21

Online:
2026-04-15

Automated Testing and Calibration of Sheet Metal Models

The original version of this paper is available on https://www.scientific.net/SSP.388.199.pdf

NDIRITU Clement 1,a*, HAOUES Yasmine 1, b, DECULTOT Nicolas 1,c and THUILLIER Sandrine 1, d

1 Univ. Bretagne Sud, UMR CNRS 6027, IRDL, F-56100 Lorient, France

a  clement.ndiritu@univ-ubs.fr, b  yasmine.haoues@univ-ubs.fr, c  nicolas.decultot@univ-ubs.fr, d sandrine.thuillier@univ-ubs.fr

Keywords: Automated Mechanical Testing, Inverse Model Calibration, Anisotropy, Hardening

Abstract

The efficient development of high-quality sheet metal components increasingly depends on predictive numerical simulations conducted prior to forming operations. Achieving such accuracy requires precise calibration of constitutive material models that represent the complex mechanical behaviour of metals. Mechanical testing provides the essential data for calibration, revealing material anisotropy, strain hardening, and ductile fracture. However, traditional characterisation approaches are often labor-intensive, time-consuming, and prone to operator variability. Within the phenomenological framework, numerous tests are typically required to capture the full material response, including repeats for statistical reliability, leading to high costs, extended lead times, and data management challenges. To address these limitations, this study introduces an automated mechanical testing platform designed to rapidly acquire experimental data useful for material models. Robotic automation enables fully automated test sequences, ensuring high repeatability and reducing manual intervention. When combined with automated model calibration, this approach provides a direct link between the physical material (metallic sheet) and its virtual mechanical representation.

Introduction

The original version of this paper is available on https://www.scientific.net/SSP.388.199.pdf

A primary objective of mechanical testing is to generate data for model calibration, namely the identification of constitutive parameters used in finite element simulations. From experimental databases either based on homogeneous [1] or heterogeneous tests [2], repeatability must be assessed to select representative tests and quantify their reliability, prior to inverse identification. While the selection of mechanical models and associated tests is still largely performed manually, it can be integrated into an automated strategy [3]. When combined with automated testing and post-processing, such an approach could lead to a fully automated workflow, from material testing to parameter identification. To the authors' knowledge, this concept has not yet been realized and constitutes the objective of the AutoMeCal project. As an initial step, this work focuses on mechanical testing for the identification of hardening, anisotropy and uncoupled fracture criteria. These are traditionally identified in two stages: first, hardening and anisotropy are calibrated using uniaxial tensile tests at various orientations and hydraulic bulge tests; second, fracture criteria are calibrated using notched or holed specimens tested up to rupture [4]. Since rupture tests reach high strain levels, extrapolation of the hardening law is often required to fit the load evolution, using either linear [4] or more advanced extrapolation schemes [5]. This motivates the inclusion of rupture tests within the calibration database and the simultaneous identification of anisotropy and hardening parameters over the full experimental dataset [6].

The need for large and repeatable experimental databases has motivated the development of automated mechanical testing systems. An early example includes a robot-assisted testing setup for miniaturized irradiated samples with a thickness of 0.2 mm [7], although no mechanical output signals were reported. More recently, fully or semi-automated tensile testing systems incorporating robotic handling, laser-based dimensional measurements, and digital image correlation have been proposed [8, 9]. These studies demonstrate the ability to perform large numbers of tests, up to 77 and 360, respectively, highlighting the potential of automated testing for generating extensive experimental databases required to characterize material behaviour under multiple loading states.

The aim of this paper is to present the first results obtained within a quasi-automated approach, for a third-generation steel Fortiform ®1050, of thickness 0.85 mm . Homogeneous tensile tests at 5 orientations to the rolling direction were performed with an automated testing system developed on purpose, as well as rupture tests on specimens with notches and holes. An isotropic hardening model coupled with the anisotropic yield criterion Yld2004-18p [10] is considered and material parameters are identified, adding a previously performed hydraulic bulge test result. The final aim would be to calibrate an uncoupled rupture criterion; however, only hardening and anisotropy are considered in a first step. Though the whole procedure is not yet fully automatic, this study presents the first steps of an integrated fully automatic procedure.

Material and Methods

The original version of this paper is available on https://www.scientific.net/SSP.388.199.pdf

Automated testing platform.

The original version of this paper is available on https://www.scientific.net/SSP.388.199.pdf

An automated experimental platform was developed to perform series of tensile and rupture tests with minimal human intervention. The main objective of this system is to ensure high repeatability, reduce operator-dependent variability, and enable the rapid acquisition of experimental data suitable for constitutive model calibration.

The platform integrates three main components: (i) an automated uniaxial tensile testing machine developed by MatAndSim, (ii) a six-degree-of-freedom collaborative robotic manipulator for specimen handling from Universal Robots (UR3) and (iii) a stereo digital image correlation (DIC) system for full-field strain measurements from MatchID, see Fig. 1.

[picture]

Fig. 1: (a) Automated mechanical testing platform (b) Schematic diagram of the main components of the automatic testing platform. TS15 is developed by MatAndSim (https://matandsim.fr/)

Fig. 1. (a) Automated mechanical testing platform (b) Schematic diagram of the main components of the automatic testing platform. TS15 is developed by MatAndSim (https://matandsim.fr/)

The tensile testing machine (TS15) is a horizontal uniaxial device with a maximum load capacity of 15 kN . It is based on a parallel configuration with two grips moving symmetrically in opposite directions. The applied force is measured using a calibrated load cell, while displacement is monitored by a linear variable differential transformer (LVDT) and an internal position sensor. The testing machine is operated as an automated unit controlled by an embedded computer (Raspberry Pi), which executes the predefined loading sequences. Mechanical signals are acquired through an external data acquisition board (LabJack) and transmitted to the main controller. Test parameters such as displacement rate, loading path, and test duration are defined prior to testing and executed automatically. Load and displacement are recorded continuously throughout the test.

Specimen handling is performed using a collaborative robotic arm (UR3) with a positioning accuracy of ±0.1 mm. The robot automatically retrieves specimens from a storage system and places them onto the cylindrical rods (pins) of the TS15 tensile testing machine. A pneumatic gripping system equipped with suction cups is used to handle the specimens reliably while limiting surface damage.

All specimens incorporate holes at each end to enable pin loading and create a pinned connection. This configuration ensures symmetric uniaxial loading while allowing rotational freedom at the grips, thereby minimizing eccentric loading and bending effects. After completion of the test, the robot automatically removes the loaded or fractured specimen. This robotic setup enables fully automated test sequences for several specimen geometries.

Full-field strain measurements are obtained using a stereo digital image correlation (DIC) system (MatchID) comprising two synchronized high-resolution Basler cameras (Basler ace 2 a2A409630 umPRO ) observing the specimen gauge section throughout the loading sequence. Prior to testing, a stereo calibration using a dedicated target is performed to identify the intrinsic and extrinsic camera parameters, after which the setup remains fixed; this is achieved by leveraging the MatchID Calib package.

Mechanical testing is executed via an automated control script on the TS15 system, which controls the loading sequence and triggers image acquisition through hardware synchronization leveraging the MatchID Grabber package. Image acquisition is fully synchronized with the mechanical loading, and images are recorded together with their corresponding time information. Post-processing is automatically initiated using a Python script that launches MatchID Stereo package. On a predefined region of interest (ROI) on the reference image, stereo correlation is performed to compute displacement fields and subsequently strains. Displacement and strain fields are extracted using Python-based routines and combined with the force and displacement data recorded by the TS15 controller.

[picture]

Fig. 2: (a) Geometries of the tensile and ductile fracture tests. The extremities of the uniaxial tensile test specimen include a larger-width region with a hole, used to pull the sample. For the rupture tests, a similar region exists but is not shown here. (b) DIC image obtained on N-R15 test. The two red lines highlight the position where displacements for FE simulations are output (c) Boundary conditions (BCs) representing experimental conditions as extracted from DIC image.

Fig. 2. (a) Geometries of the tensile and ductile fracture tests. The extremities of the uniaxial tensile test specimen include a larger-width region with a hole, used to pull the sample. For the rupture tests, a similar region exists but is not shown here. (b) DIC image obtained on N-R15 test. The two red lines highlight the position where displacements for FE simulations are output (c) Boundary conditions (BCs) representing experimental conditions as extracted from DIC image.

Mechanical tests.

The original version of this paper is available on https://www.scientific.net/SSP.388.199.pdf

All test specimens were machined from the same sheet using a CNC water jet cutting machine, and their dimensions were measured prior to testing to verify machining accuracy. Standard dog-bone specimens with a width of 6 mm were designed according to ISO 6892-1 for monotonic tensile testing. These specimens were machined along the rolling direction (RD) as well as at 22.5∘,45∘,67.5∘, and 90∘ relative to RD to investigate coupled hardening and anisotropy. In addition, five ductile fracture specimen geometries, machined exclusively along RD, were selected to generate a wide range of stress
states. Notched specimens with small (N-R5), medium (N-R10), and large (N-R15) notch radii were employed to achieve a high stress triaxiality. A specimen with a central hole (H-R4) was used to approach a uniaxial stress state while limiting necking, and a shear-type specimen (SH) was selected to obtain low stress triaxiality [11, 5].

For each specimen geometry and loading direction, three to five samples were prepared to assess experimental repeatability. Specimen surfaces were prepared manually for digital image correlation (DIC) by cleaning and applying a white base coat, followed by a stochastic black speckle pattern generated using the MatchID speckle generator and printed with an ArtisJet UV printer. Specimens were placed on a storage rack.

All tests except the hydraulic bulge test were conducted on the automated testing platform. The crosshead speed and displacement limits were defined individually for each fracture specimen type via a configuration file, while identical parameters were used for all uniaxial tensile tests. Image acquisition for both tensile and fracture tests was performed using the MatchID Grabber with identical camera calibration settings. Among the five tests conducted for each specimen geometry, at least two showed closely matching responses; one representative test was selected based on minimal experimental uncertainty, highlighting the potential influence of specimen misalignment.

Image correlation was conducted on predefined ROIs using the MatchID stereo DIC system in batch mode, automatically triggered via a Python script after each acquisition. For uniaxial tensile tests, the Cauchy stress was computed as the applied force divided by the current cross-sectional area prior to necking, while logarithmic strain components were evaluated in a homogeneous region of interest using the Euler strain convention with a subset size of 43 pixels, a step size of 8 pixels, and a strain window of 4 pixels. Lateral and transverse strains were automatically extracted for each correlated image. For fracture tests, displacements were obtained using predefined virtual extensometers within selected correlation regions. Five extensometers with a gauge length of 15 mm were positioned at different specimen locations to evaluate potential misalignment effects, and the averaged displacement was used for material identification. The fracture displacement, uf, was defined as the mean displacement at the fracture time tf, identified by a sudden drop in the force-displacement curve.

To enable subsequent numerical simulations of the fracture tests, it is essential to accurately reproduce the experimental displacement boundary conditions, as these have been shown to significantly influence the calibration of ductile fracture models compared to idealized boundary conditions [5]. Accordingly, two perpendicular lines, as shown in Fig. 2b, were predefined across each fracture specimen within the automated post-processing script. Displacements were projected onto the best-fit plane of the correlated surface to extract the temporal evolution of the displacement components along the loading direction, Ux[ mm], and the transverse direction, Uy[ mm], at each point along the defined lines.

Constitutive modelling.

The original version of this paper is available on https://www.scientific.net/SSP.388.199.pdf

An elastoplastic constitutive law with isotro- pic hardening and the anisotropic yield criterion yld2004-18p [10] is used in this study. The yield criterion is written as:

ϕ=∑i=1,j=13|S~i(1)−S~j(2)|a=4σya with S~(k)=L~(k):S,k=1,2(1)

with

L~(k)=[0−c12(k)−c13(k)000−c21(k)0−c23(k)000−c31(k)−c32(k)0000000c44(k)000000c55(k)000000c66(k)],k=1,2

with σy the yield stress and S the deviatoric part of Cauchy stress tensor. The exponent a is fixed to the value of 6 for body-centered cubic materials. As no information is available for shear in the thickness directions, parameters c44(k) and c55(k),k=1,2, are fixed equal to 1 , leading to a total of 14 parameters to identify. Moreover, the combined Swift-Voce equation is used to describe the isotropic hardening,

σy(ε¯p)=α[A(ε0+ε¯p)n]+(1−α)[σ0+Q(1−exp(−bε¯p))](2)

where ε¯p is the equivalent plastic strain and α,A,ε0,n,σ0,Q and b are the material parameters to identify.

Numerical models.

The original version of this paper is available on https://www.scientific.net/SSP.388.199.pdf

The mechanical tests were simulated in three dimensions using the finite element (FE) code ABAQUS® within an implicit framework. Homogeneous tensile tests were modeled using a threedimensional unit element, whereas ductile fracture tests were simulated with full specimen models to capture material heterogeneity. Only the specimen portion between the pin-connection holes was modeled, such that the FE specimen length matched the distance between the two predefined vertical lines used to extract the experimental displacement.

All specimens were discretized using three-dimensional reduced-integration hexahedral elements (C3D8R), with the default total-stiffness hourglass control in ABAQUS®. A locally refined mesh with an element size of approximately 0.1 mm was employed in regions of expected strain localization, while coarser elements were used for boundary condition application. Based on a convergence study of the fracture strain, which showed convergence for meshes with more than six elements through the thickness [5]. Therefore, three elements were used to model the symmetry.

Displacement boundary conditions with non-zero components ( Ux,Uy ) were applied to the left and right specimen edges, as shown in Fig. 2, together with a symmetry boundary condition in the thickness direction. An automated Python-based procedure was developed to impose the displacement boundary conditions on 11 nodal regions of the FE model for each fracture test, as illustrated in Fig. 2. This approach leads to a more realistic representation of experimental conditions by accounting for specimen misalignment effects observed during testing [5].

Five virtual extensometers were defined at the same locations as in the DIC analysis to extract local displacements for plotting load evolution. For ductile fracture tests, differences in filtering and spatial resolution between DIC and FE data typically require a levelling procedure, whereby FE results are post-processed using the same filters as the experimental data. Since this procedure was not applied in the present study, only the global force-displacement response was used for material parameter identification. The mechanical behaviour was modeled using a user-defined material subroutine [1].

Material parameter identification.

The original version of this paper is available on https://www.scientific.net/SSP.388.199.pdf

The 21 material parameters introduced in the modelling section govern hardening and anisotropy based on experimental data spanning a wide strain range of up to approximately 0.4 . The diversity of tests included in the identification database enabled the characterisation of hardening behaviour over large deformations, overcoming the limitations of linear piecewise hardening laws obtained through manual inverse methods based on iterative visual fitting. However, a hydraulic bulge test result, previously obtained, was also considered to add information for the yield criterion calibration. In this study, parameter identification was performed using FEMU, an inverse calibration approach formulated as an optimisation problem that minimizes the discrepancy between experimental measurements and numerical predictions. The framework couples an optimisation algorithm with finite element (FE) simulations and is based on iterative updates of the material parameters. At each iteration, experimental and numerical responses are compared via an objective function ℒ(𝐀), and the parameter set is updated to progressively reduce the mismatch.

The inverse calibration framework was applied to nine experiments, comprising five homogeneous uniaxial tensile tests, three heterogeneous ductile fracture tests and a hydraulic bulge test. Accordingly, the cost function is decomposed into the following test groups:

ℒ(𝐀)=∑n=1Nℒobjn(𝐀)=∑αℒobjTU−S+∑αℒobjTU−E+ℒobjBT+ℒobjH−R4+ℒobjN−R5+ℒobjSH(3)

where TU-S and TU-E denote the stress level and transverse strain in uniaxial tension, respectively, while BT corresponds to the stress level in hydraulic bulge test at the pole, H-R4, N-R5 and SH correspond to the reaction force evolution in the respective ductile fracture tests. The summation over α accounts for the different orientations with respect to the rolling direction.

The contribution of each test n is defined as

ℒobjn(𝐀)=1npn∑i=1npn(Ziexp(ti)−Zinum(𝐀)Wabsk)2(4)

where npn is the number of points for each test n,Ziexp  and Zinum  are the experimental and numerical responses evaluated at time ti, and Wabsk is a group-dependent weighting factor used to normalize residuals.

The weighting factors associated with each experimental group are defined as

W abs ={0.0363,k= TU-S ,1.011×10−5k= TU-E 0.0432,k= BT W abs ={0.282,k=H−R40.263,k=N−R50.0956,k=SH(5)

This weighting strategy ensures a balanced contribution of stress, strain, and force data in the identification process. Parameter identification is carried out using a Fortran-based interface program coupled with the finite element code ABAQUS® for numerical simulations and the optimisation software SdL [11] for iterative parameter updates and convergence control. In SdL, the objective function is minimized using the gradient-based Levenberg-Marquardt (L-M) algorithm. Convergence is achieved when either a stagnation threshold of 1×10−7 in the objective function between successive iterations is reached or the maximum number of iterations, limited to 750 , is exceeded.

Despite its robustness, the L-M algorithm may converge to local minima depending on the initial parameter values. To reduce this risk, the identification procedure was repeated using three different initial parameter sets.

Results and Discussion

The original version of this paper is available on https://www.scientific.net/SSP.388.199.pdf

Table 1 shows the initial and optimized values for the three parameter sets, and Fig. 3 illustrates the results for RD, 45∘ and 90∘/ RD configurations and Fig. 4 for some rupture tests. As a whole, similar predictions are obtained, though the parameter values are rather different, highlighting the wellknown difficulty related to convergence to local minima with gradient-based algorithms. The hardening law based on a combination of Swift-Voce equations, without additional constraints, gives a very large number of possibilities, though the resulting flow stress is rather similar. These results must be strengthened by increasing the number of initial values. Moreover, the hydraulic bulge test is required for the identification, stressing the need to consider identifiability analysis in the procedure. The experimental database could be enriched with the heterogeneous strain fields of the fracture tests, to overcome this issue [2]. However, this integrated calibration shows that the load level in rupture tests is well described, without the need for further adjustments.

Concerning the automated experimental procedure, Table 2 shows what was partially or fully automated and what is still handled manually. The DIC operations must be strengthened, to align the experimental and numerical frames and for data output and comparison with FE results. The calibration procedure was already handled in a semi-automatic way, based on a sequence of operations launched by Python scripts. The main issue in the calibration is the design of relevant indicators for the

Table 1: Identified material parameters for different initial guesses. Ini. and Opti. stand respectively for initial and optimized parameter sets.

Set / State | Yield anisotropy parameters | Hardening parameters | L(A) ×10⁶
| C₁₂⁽¹⁾ | C₁₃⁽¹⁾ | C₂₁⁽¹⁾ | C₂₃⁽¹⁾ | C₃₁⁽¹⁾ | C₃₂⁽¹⁾ | C₆₆⁽¹⁾ | C₁₂⁽²⁾ | C₁₃⁽²⁾ | C₂₁⁽²⁾ | C₂₃⁽²⁾ | C₃₁⁽²⁾ | C₃₂⁽²⁾ | C₆₆⁽²⁾ | α | A | ε₀ | n | σ̂₀ | Q | b |
Set 1 – Ini. | -2.00 | -1.80 | -1.50 | -0.90 | -1.20 | -0.60 | -1.70 | -1.30 | -1.10 | -0.90 | -1.40 | -0.80 | -1.60 | -2.00 | 0.12 | 1350.00 | 5.0E-5 | 0.11 | 520.00 | 220.00 | 6.00 | 1808
Set 1 – Opti. | -0.67 | -0.31 | -1.40 | -0.55 | -1.58 | -1.03 | -1.32 | -0.74 | -0.55 | -0.83 | -1.31 | -0.79 | -0.99 | -0.57 | 0.13 | 2690.04 | 1.81E-5 | 0.10 | 766.66 | 497.73 | 10.58 | 6.43
Set 2 – Ini. | 0.90 | -0.30 | 1.40 | -0.20 | 0.60 | -1.10 | 0.70 | 1.30 | -0.50 | 0.90 | -0.70 | 1.10 | -0.40 | 1.50 | 0.62 | 2450.00 | 2.20E-3 | 0.27 | 780.00 | 410.00 | 19.00 | 7149
Set 2 – Opti. | 0.40 | -0.19 | 1.51 | -0.07 | 0.20 | -0.97 | 0.98 | 0.61 | -0.35 | 1.17 | -1.15 | 1.29 | 0.02 | 1.02 | 0.55 | 2014.44 | 1.76E-3 | 0.16 | 898.34 | 356.69 | 11.80 | 6.67
Set 3 – Ini. | 1.50 | 1.00 | 1.00 | 1.20 | 0.20 | 0.70 | 1.10 | 0.20 | 0.30 | 1.00 | 0.40 | 1.50 | 0.80 | 0.30 | 0.51 | 1785.42 | 2.58E-3 | 0.12 | 750.37 | 500.88 | 23.49 | 2918
Set 3 – Opti. | 1.43 | 1.65 | 1.06 | 0.60 | 0.37 | 0.88 | 1.36 | 0.16 | 0.92 | 0.37 | 0.76 | 0.35 | 0.93 | 0.21 | 0.52 | 2041.94 | 4.53E-3 | 0.17 | 555.17 | 208.37 | 20.03 | 6.85
Lower bound |  | -2.2 |  | 0.1 | 1300 | 1.0E-5 | 0.1 | 500 | 200 | 5 | –
Upper bound |  | 2.2 |  | 0.9 | 3200 | 7.0E-3 | 0.35 | 900 | 800 | 30 | –
Units | – | – | MPa | – | – | MPa | MPa | – | –

[picture]

Fig. 3: Stress-strain response for tensile tests at different orientations relative to the rolling direction

Fig. 3. Stress-strain response for tensile tests at different orientations relative to the rolling direction

[picture]

Fig. 4: Force-displacement response for ductile fracture tests: H-R4, SH (in the calibration database) and N-R10 used for the validation

Fig. 4. Force-displacement response for ductile fracture tests: H-R4, SH (in the calibration database) and N-R10 used for the validation

validation of parameter sets. Though the gap between experimental and predicted values is of interest, it is not sufficient and identifiability analysis should be added prior to testing.

Table 2: Summary of the experimental and data-processing steps and their mode within the proposed framework.

Step | Process / Method | Mode
1 | Dimensional measurements | Manual
2 | Speckle patterning | Manual
3 | Sample handling | Automatic
4 | Camera calibration | Manual
5 | ROI, Virtual extensometer and BCs definition | Manual
6 | Image acquisition | Automatic
7 | Mechanical testing load acquisition | Automatic
8 | DIC, strain calculation and BCs extraction | Automatic
9 | Post-processing | Automatic

Conclusions

The original version of this paper is available on https://www.scientific.net/SSP.388.199.pdf

In this study, an anisotropic plasticity model for a third-generation steel was calibrated using specifically generated experimental data within a semi-automated framework. Dimensional verification, speckle patterning, ROI and sketch definitions, and camera calibration were performed manually, while sample handling, image acquisition, and DIC processing and post-processing were fully automated. Future work will focus on integrating the manual and automated procedures into a unified tool to enable comprehensive and efficient characterisation of sheet metal behaviour, including the calibration of fracture models.

Beyond enabling reliable calibration of anisotropic plasticity models, the proposed experimental and data-processing framework establishes a consistent basis for the subsequent identification of ductile fracture criteria. The automated nature of the approach is particularly suited to generating large, high-quality datasets required for fracture model identification and validation, thereby supporting a more comprehensive description of sheet metal mechanical behaviour.

Acknowledgements

The original version of this paper is available on https://www.scientific.net/SSP.388.199.pdf

The research leading to these results has received funding from ANR for the project Automated mechanical lab' and model calibration (AutoMeCal), project ANR-23-CE10-0007-01, https://anr. fr/Project-ANR-23-CE10-0007. The authors acknowledge MatAndSim for fruitful discussions and their contribution to the automated platform.

References

The original version of this paper is available on https://www.scientific.net/SSP.388.199.pdf

N. Souto, A. Andrade-Campos and S. Thuillier: Int. J. Mater. Forming Vol. 10 (2017), 353-367

L. Germain, M. Gonçalves, A. Andrade-Campos and S. Thuillier: MATEC Web of conferences Vol. 408 (2257), 01075

M. Conde, S. Coppieters and A. Andrade-Campos: Int. J. Mech. Sciences Vol. 264 (2024), 108813

M. Dunand and D. Mohr: Int. J. Solids Struct. Vol. 47 (2010), 1130-1143

A. Kacem, H. Laurent and S. Thuillier: Eng. Fracture Mech. Vol. 248 (2021), 107686

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T. Beerli, Christian C. Roth and D. Mohr: Acta Mater. Vol. 263 (2024), 119539

Christian C. Roth, T. Tancogne-Dejean and D. Mohr: Additive Manuf. Vol. 43 (2021), 101998

F. Barlat, H. Aretz, J.W. Yoon, M.E. Karabin, J.C. Brem, R.E. Dick: Int. J. Plasticity Vol. 21 (2005), 1009-1039

A. Andrade-Campos, R. de-Carvalho and R.A.F. Valente: Int. J. Mech. Sci. Vol. 54 (2012), 294305

This ePub file was created using machine-generated methods and automated software to convert the source text into an ePub format and may contain formatting inconsistencies from the original version. Formatting may vary depending on the device used.

The original version of this eBook is available on https://www.scientific.net/book/formability-of-metallic-materials/978-3-0364-1993-0

Solid State Phenomena, ISSN: 1662-9779, Vol. 388, pp 209-225

doi: 10.4028/p-bY5Ylk

© 2026 The Author(s). Published by Trans Tech Publications Ltd, Switzerland.

Submitted:
2026-01-19

Revised:
2026-02-14

Accepted:
2026-02-21

Online:
2026-04-15

Springback Prediction of 6XXX-Series Aluminum: Effect of Hardening Model, Friction and Binder Loading

The original version of this paper is available on https://www.scientific.net/SSP.388.209.pdf

Joseph Arciero 1,a*, Kenneth Cheong 1, b, Mahmoud Howeyze 1,c, Isaura Escorza 2, d, Akshay Wankhede 2,e, Kidambi Kannan 2,f, Sarin Thokala 3, g, Christian Leppin 4, h, David Anderson 3,i, Cliff Butcher 1,j

1 Department of Mechanical and Mechatronics Engineering, University of Waterloo, ON, Canada

2 AutoForm Engineering, Troy, Michigan, USA

3 Novelis, Kennesaw, Georgia, USA

4 Novelis, Sierre, Switzerland

a*jarciero@uwaterloo.ca, b  dwkcheong@uwaterloo.ca, c  mhoweyze@uwaterloo.ca, disaura.escorza@autoform.com, e  akshay.wankhede@autoform.com, f  kidambi.kannan@autoform.com, g  sarin.thokala@novelis.adityabirla.com, h christian.leppin@novelis.adityabirla.com, i david.anderson@novelis.adityabirla.com, j  cbutcher@uwaterloo.ca

*Corresponding author email: jarciero@uwaterloo.ca

Keywords: sheet metal forming, kinematic hardening, AutoForm, TriboForm.

Abstract

Predicting the final shape of automotive structural components after springback is a challenge to the inclusion of high strength aluminum alloys into the vehicle body-in-white. Complex deformation paths and reverse loading of sheet material during forming operations can induce significant Bauschinger effects and kinematic hardening behaviour. Capturing the through-thickness stress gradient is critical when predicting springback, which is governed by tooling dynamics, frictional forces, and material plasticity. In this study, the anisotropic behaviour of a AA6xxx-T4 aluminum alloy was characterized to calibrate a BBC2005 yield function, kinematic hardening effects were characterized through a novel uniaxial compression-tension technique, and a technology demonstrator U-shaped rail component was formed and scanned to assess the final shape after springback. Multiple model variations were analyzed in AutoForm R12, modifying simulation control parameters, binder loading condition (uniform vs. column), friction model (Coulomb vs. TriboForm), and hardening model (isotropic vs. kinematic). The use of column binder loading paired with TriboForm friction model provided the most significant improvement for thinning and springback prediction accuracy with kinematic hardening being a second order effect compared to accounting for friction and binder force.

Introduction

The original version of this paper is available on https://www.scientific.net/SSP.388.209.pdf

To improve vehicle range while maintaining increasing crash safety standards, the automotive industry is turning towards high-strength steel and aluminum alloys to produce lightweight vehicle architectures [1,2]. AA6xxx and AA7xxx series aluminum sheet metal alloys are particularly promising with strength-to-weight ratios comparable to high strength steel with one third the density. However, accurate springback prediction remains a key challenge to overcome in the adoption of thicker, higher strength aluminum alloys into structural body-in-white components.

The forming of complex components involves non-linear deformation paths and bendingunbending as the material flows over drawbeads into the die cavity. This loading can activate significant Bauschinger effects and transient hardening [3] that requires the use of kinematic hardening models to improve springback predictions [4-6]. Solfronk et al. [6] used the relatively complex Yoshida-Uemori kinematic hardening model [7] to improve predicted final shape after Ubending AA6111 while Bong et al. [8] obtained good springback predictions using the simpler

Chaboche kinematic hardening model with a Yld2000 criterion for three-point bend testing of AA6061-T6.

Although accurate characterization of anisotropic plasticity and hardening behaviour is essential, springback prediction depends equally, if not more, on the proper representation of the tooling and frictional forces that dictate material flow and the severity of the through thickness stress and strain gradients. Frictional behaviour in forming simulations is often simplified using constant Coulomb friction with typical friction coefficients between 0.1 to 0.2 [9]. Lee [10] demonstrated that within this amount of friction variation, predicted sidewall curl of U-shaped drawn channels may vary up to 24%. Pressure- and velocity-dependent friction models are now being used in forming simulations for improved springback, thickness, and formability predictions [9,11-13]. Advanced friction models such as TriboForm ® , describe the dependence of friction coefficient on contact pressure, plastic strain, and temperature for a given combination of material, surface topography, and lubricant [14]. The application of TriboForm ® friction models have been shown to improve springback and formability predictive capabilities compared to Coulomb friction in the work of Bolay et al. [15] and Berahmani et al. [16].

The objective of this study is to detail the improvement of springback predictions of an industrially-representative U-shaped rail section for a AA6xxx-T4 aluminum as a function of the simulation control parameters, kinematic hardening, and friction model. Tensile, plane strain notch, shear, and biaxial tests were performed to characterize the anisotropic behaviour in proportional loading while compression-tension tests were used to characterize kinematic hardening effects. Multiple permutations of the forming model were analyzed in AutoForm R12 with an emphasis on the choice of binder loading conditions, friction, and hardening models.

2. Constitutive and Formability Characterization

The original version of this paper is available on https://www.scientific.net/SSP.388.209.pdf

The material of interest is a 2 mm thick 6xxx-series aluminum alloy in the T4 temper, which is similar to a AA6111-T4. Uniaxial tension, plane strain notch, and shear tests were performed in multiple orientations with the geometries depicted in Fig. 1. The tensile properties obtained in the rolling (RD), diagonal (DD), and transverse (TD) directions are summarized in Table 1. The engineering stress-strain response from a representative RD test is shown in Fig. 2.

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Fig. 1. (a) ASTM E8 tensile specimen [17], (b) plane strain notch specimen, (c) shear test geometry, (d) tension-compression test geometry, and (e) Subsized ASTM E8 specimen [17]. Yellow lines denote extensometers. Note: schematic shown is not to scale. All units in mm.

Fig. 1. (a) ASTM E8 tensile specimen [17], (b) plane strain notch specimen, (c) shear test geometry, (d) tension-compression test geometry, and (e) Subsized ASTM E8 specimen [17]. Yellow lines denote extensometers. Note: schematic shown is not to scale. All units in mm.

Table 1. Tensile properties of the AA6xxx-T4 material with standard deviation. YS yield strength, UTS ultimate tensile strength, UE uniform elongation, TE total elongation. Four test repeats performed per orientation.

Direction | Tensile Geometry | 0.2% YS [MPa] | UTS [MPa] | UE [%] | TE [%]
RD |  | 154.0 ± 0.0 | 273 ± 1.0 | 21.0 ± 0.6 | 25.8 ± 1.1
DD | ASTM E8 | 150.0 ± 1.0 | 272 ± 1.0 | 24.2 ± 0.6 | 29.6 ± 0.8
TD |  | 149.0 ± 0.0 | 270.0 ± 0.0 | 21.9 ± 1.8 | 25.5 ± 1.4

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Fig. 2. Engineering stress-strain response of AA6xxx-T4 material along the rolling direction with uniform elongation indicated.

Fig. 2. Engineering stress-strain response of AA6xxx-T4 material along the rolling direction with uniform elongation indicated.

Interrupted disc compression tests at multiple strain levels were conducted to obtain the biaxial Rvalue ( RB ). Hydraulic bulge tests were performed according to ISO 16808 [19] to obtain the biaxial stress-strain response and determine the biaxial stress ratio ( σB ). Simple shear tests were performed to determine the shear stress (τ) response using the methodology of Narayanan et al. [20]. Plane strain notch tests were performed in RD, DD and TD using the specimen geometry depicted in Fig. 1b. The notch geometry induces a strong strain gradient from plane strain at notch center to uniaxial at the free edge. Plane strain condition was verified by extracting the local strain evolution at the notch center using a point inspector in Vic-3D® software from correlated solutions following the procedures of Narayanan et al. [21]. Local strain evolution at the notch center from a representative plane strain notch test in RD is shown in Fig. 3.

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Fig. 3. Local strain evolution of 6xxx-T4 material at notch center of plane strain notch test in RD.

Fig. 3. Local strain evolution of 6xxx-T4 material at notch center of plane strain notch test in RD.

Table 2 summarizes the normalized yield stress ratios relative to RD and plastic strain anisotropy in various directions at a plastic work level of 40MJ/m3. Fig. 4a depicts calibrated Yld2000 [22] and BBC2005 [23] yield surfaces for the AA6xxx-T4. The BBC2005 yield surface was calibrated within AutoForm R12 and exhibits close agreement with Yld2000, which is included as a comparative reference due to its widespread use in literature. Constrained inverse FE analysis of plane strain notch tests in the RD, DD and TD were performed according to the methodology developed in Abedini et al. [18] to determine the local arc of the yield function from uniaxial to plane strain tension. The yield surface exponent identified from the notch tests was larger than the commonly assumed value of 8 for FCC materials and set to a value of 16 in Yld2000 and 12 in BBC2005, which is the maximum available value within AutoForm R12. Yield exponents larger than 8 have also been experimentally observed for a 6016-T4 by Kuwabara et al. [24] using stress-controlled cruciform tests with values on the order of 22.7.

Table 2. Normalized tensile, plane strain (PS), and shear yield stresses relative to RD tensile stress, R -values at various orientations for the AA6xxx-T4 material at a plastic work of 40MJ/m3.

Parameter | Experimental Average & Standard Deviation
σRD/σRD | 1.000 ± 0.004
σDD/σRD | 0.986 ± 0.007
σTD/σRD | 0.989 ± 0.006
(σPS/σRD) | 1.024 ± 0.004
σB/σRD | 0.989 ± 0.009
τRD/σRD | 0.570 ± 0.007
τDD/σRD | 0.581 ± 0.007
τTD/σRD | 0.565 ± 0.005
RRD | 0.684 ± 0.024
RDD | 0.570 ± 0.017
RTD | 0.805 ± 0.012
RB | 0.830 ± 0.020

The area reduction method (ARM) of Abedini et al. [25] was employed to allow for characterization of the hardening response beyond uniform elongation. The cross-section shape of the neck was monitored using stereo-DIC following which the average area strain and stresses were estimated. Correction factors of Choung and Cho [26] were applied to account for change in stress
state due to necking, enabling the characterization of the hardening response to 40% strain compared to the uniform elongation strain of 21% in RD. Fig. 4b is the hardening response of the AA6xxx-T4 material along RD, DD, TD directions. The reader is referred to Abedini et al. [25] for further details regarding the ARM.

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Fig. 4. (a) Calibrated Yld2000 and BBC2005 yield surfaces overlayed with von Mises and Tresca, and (b) the hardening response in RD, DD, and TD for the AA6xxx-T4 alloy.

Fig. 4. (a) Calibrated Yld2000 and BBC2005 yield surfaces overlayed with von Mises and Tresca, and (b) the hardening response in RD, DD, and TD for the AA6xxx-T4 alloy.

Kinematic hardening effects were characterized through a novel uniaxial compression-tension testing technique using the specimen geometry depicted in Fig. 1d. This technique makes use of a tension-compression test fixture, shown in Fig. 5, designed and developed at the University of Waterloo with integrated in-plane DIC. The test setup consists of two axial load cells, the second of which added for friction measurements, as first done by Banerjee et al. [27]. The anti-buckling fixture applies a side load to the test specimen via a nitrogen spring, contacting the specimen through steel and Gorilla ®  glass inserts, depicted in Fig. 5b. The glass window allows for planar DIC strain measurements, facilitated through the front mounted camera. A representative compression-tension (CT) cycle with 5.5% pre-strain is shown in Fig 8. The CT data was corrected to account for friction and contact stress using a plastic work analysis as well as accounting for the asymmetric friction distribution that emerges when one grip is stationary. Details on the correction methodology are provided in the work of Pilozo-Hibbit [28].

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Fig. 5. (a) Front view and (b) side view of tension-compression test fixture.

Fig. 5. (a) Front view and (b) side view of tension-compression test fixture.

To characterize chord modulus degradation, load-unload tests were performed using the sub-sized ASTM E8 [17] specimen geometry depicted in Fig. 1e. An AGX-V 50 kN Shimadzu® tensile frame with two Gazelle point Grey ® cameras and 180 mm Tamron ® lenses (Fig. 6) was used for stereo DIC, providing a resolution of 0.016 mm/ pixel and 10 FPS sampling rate. Loading was performed using displacement control at intervals corresponding to approximately 1% strain. Fig. 7a depicts the unloading chord modulus degradation determined from load-unload testing. A linear fit of the data was performed over the range of 0.1% to 95% of the stress level at which the unloading occurred to obtain the chord modulus for a given load-unloading step. Furthermore, Fig. 7b depicts the measured Poisson's ratio which appears to remain constant with plastic strain. Although more sophisticated test methods, including electrical-resistance, ultrasonic strain-gauges, and impulse excitation are widely suggested in literature for measuring elastic properties [29-32], the measured saturation modulus ( Es ) of 64.46 GPa is comparable with the reported values of 63 and 66.6 GPa for 6xxx-T4 aluminum in the work of Meng et al. [33] and Banerjee et al. [34].

[picture]

Fig. 6. Load-unload test setup used to characterize chord modulus degradation.

Fig. 6. Load-unload test setup used to characterize chord modulus degradation.

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Fig. 7. (a) Unloading chord modulus degradation and (b) effective Poisson ratio for the AA6xxx-T4 alloy.

Fig. 7. (a) Unloading chord modulus degradation and (b) effective Poisson ratio for the AA6xxx-T4 alloy.

A single compression-tension cycle was input into AutoForm R12 to calibrate the transient softening rate ( 𝒳 ) and stagnation ratio ( ξ ) parameters of the four-parameter AutoForm kinematic hardening model. Young's reduction factor (γ) and Young's reduction rate (χ) were determined through the loadunload test. The AutoForm calibrated kinematic hardening curve is shown in Fig 8. and Table 3 summarizes the four AutoForm kinematic hardening parameters of the AA6xxx-T4 material.

Table 3. AutoForm kinematic hardening parameter values for the AA6xxx-T4 alloy using a CT cycle at 5.5% reversal strain.

Parameter | Value
Transient Softening Rate (κ) | 0.002
Stagnation Ratio (ξ) | 0.280
Young's Reduction Factor (γ) | 0.0347
Young's Reduction Rate (χ) | 71.3

[picture]

Fig. 8. Representative compression-tension cycle overlayed with monotonic tensile loading and AutoForm calibrated kinematic hardening curve.

Fig. 8. Representative compression-tension cycle overlayed with monotonic tensile loading and AutoForm calibrated kinematic hardening curve.

Formability was characterized by performing Marciniak and Nakazima tests according to ISO 12004-2 [35] standard to generate a hybrid Marciniak-Nakazima forming limit curve (FLC) with the forming limits illustrated in Fig. 9a. The limit strains were determined for each specimen using five DIC line slices placed perpendicular to the crack location. The Marciniak tests were performed for five different strain paths, ranging from uniaxial tension to plane strain tension, using the same tooling described in Noder and Butcher [36]. In the Marciniak tests, stainless steel shim rings of 2.66 mm thickness were placed along the outer edge of the test specimens (Fig. 9b) to avoid lockbead splitting, resulting in a lockbead penetration depth of 1.94 mm . Nakazima tests were also performed for three different strain paths ranging between plane strain tension to biaxial stretching. Although Marciniak tests are preferred for minimizing process effects present in Nakazima tests related to non-linear strain paths, through thickness shear, and contact pressure [36]. A hybrid FLC was employed due to challenges of larger width test specimens experiencing cracking at the Marciniak punch radius. The specimen widths used in Marciniak and Nakazima tests are summarized in Table 4. The process effects present in Nakazima test may contribute to larger forming limit strains on the stretch side of the FLC (Fig. 9a).

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Fig. 9. (a) Hybrid Marciniak-Nakazima forming limit strains determined using ISO 12004-2 methodology. (b) Representative test specimen with circular shim ring used for Marciniak testing, all units in mm.

Fig. 9. (a) Hybrid Marciniak-Nakazima forming limit strains determined using ISO 12004-2 methodology. (b) Representative test specimen with circular shim ring used for Marciniak testing, all units in mm.

Table 4. FLC specimen widths used for Marciniak and Nakazima tests.

| Specimen Width [mm] |

Marciniak

Test

40
80
100
120
130

Nakazima

Test

150
160
203

3. U-Shaped Rail Forming

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The technology demonstrator U-shaped rail component developed by Novelis, shown in Fig. 10a, was formed using the blank geometry shown in Fig. 10b. The tooling used for the forming of these components features 2 mm drawbeads and 11.5 mm stakebeads and is shown in Fig. 11. The AA6xxxT4 blanks were first laser cut and cleaned of any residual mill oil using acetone. Surface roughness measurements were then taken in the RD and TD directions using a Taylor Hobson Surtronic 25 linear profilometer and are summarized in Table 5. Hot-melt lubricant DC2-90 from Quaker Houghton was then applied uniformly onto the blank to an oversaturated state ( >3 g/m2 ) and verified using an Infralytic ® NG2 oil sensor with an average measurement of 4.1 g/m2 from several locations on the blank. The saturation state is defined as the maximum amount of lubricant the material can hold. Lubricant was applied beyond the saturation state to ensure the entirety of the blank was thoroughly lubricated.

Table 5. Average surface roughness measurements of AA6xxx-T4 blanks used for U-shape rail forming.

Rolling Direction

Transverse Direction

mathematical formula | 0.12 | 0.2

0.8
1.1

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Fig. 10. (a) Isometric view of U-shaped rail formed using 2 mm AA6xxx-T4 and (b) blank geometry used to form U-shaped rail. All dimensions in mm .

Fig. 10. (a) Isometric view of U-shaped rail formed using 2 mm AA6xxx-T4 and (b) blank geometry used to form U-shaped rail. All dimensions in mm .

Surface roughness measurements of the tooling were taken using a linear profilometer in four regions, shown in Fig. 11a, the average Ra and Rz roughness values from the measured tooling locations are 0.68±0.22μ m and 3.88±1.21μ m respectively. The binder tooling was fitted with four Dadco UX-20000.160 nitrogen springs with an initial pressure of 8963.18 kPa ( 1300 psi ). This resulted in a variable binder force of 475.9−731.1kN throughout the cushion stroke.

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Fig. 11. (a) Schematic of binder surface indicating surface roughness measurement regions and (b) isometric view of binder and punch surfaces.

Fig. 11. (a) Schematic of binder surface indicating surface roughness measurement regions and (b) isometric view of binder and punch surfaces.

The U-shaped rails were formed successfully without any splits, necking, or excessive thinning observed in any region of the components. Following the stamping operation, ultrasonic thickness measurements were taken at twenty-one specified locations along the upper radius and side walls. The locations of these points were labeled in alphabetical order from A-U, with points I-K and S-U located on the side walls, and all other points located on the upper radius as shown in Fig. 12a. Approximately twenty-four hours after forming, components were then scanned using a 3D-white light scanner to assess final component shape after springback. The scanned surface was then imported back into AutoForm and positioned using the upper die surface. To quantify springback, multiple measurement strategies were utilized: (i) separation distance, which is defined as the normal distance from the resulting simulation surface to the scanned geometry; (ii) measured sprung punch and die radii, and (iii) flange angle after springback. The following measurement procedures are adapted from the 2022 Numisheet benchmark that used a similar U-shaped channel geometry [37]. Four cross sections were taken of the U-shaped rail at locations shown in Fig. 12b and following Zhou and Kannan [38], the sprung punch radii, die radii, and flange angles were measured at the four cross-sections.

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Fig. 12. (a) Twenty-one ultrasonic thickness measurement locations A-U and (b) four cross-section locations.

Fig. 12. (a) Twenty-one ultrasonic thickness measurement locations A-U and (b) four cross-section locations.

4. Finite Element Simulations

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To assess the influence of different FE model parameters on thickness and springback predictions, the following simulation parameters were varied: maximum element angle (MEA), the initial element size (IES), initial subdivision level, and the addition of refinement zones (RZ). Two simulation trials featured refinement zones in the regions of the blank that contact the drawbeads, as shown in Fig. 13. A summary of model parameters investigated are outlined in Table 6. All simulations were conducted using the EPS-11 element type and BBC2005 material model.

Table 6. Summary of FE model parameters investigated in simulation of U-shaped rail forming operation. All other variables not specified were set to standard final validation (FV) values.
*Specified within refinement zone. **Elements at end of simulation.

Simulation parameters | Master elem. size [mm] | Max elem. angle [°] | Initial subdivision level | Max refinement level | No. of Elements** [x1000]
Final validation (FV) | 40.0 | 22.5 | Half | 6 | 78.7
FV+MEA 20 deg | 40.0 | 20.0 | Half | 6 | 92.1
FV+MEA 10 deg | 40.0 | 10.0 | Half | 6 | 388.2
FV+MEA 5 deg | 40.0 | 5.0 | Half | 6 | 1.45E3
IES 2.5 mm | 2.5 | 22.5 | Equal to master | 0 | 104.3
IES 1.25 mm | 1.25 | 22.5 | Equal to master | 0 | 413.3
FV+RZ MEA 10 deg | 40.0 | 10.0* | Half | 6 | 131.2
FV+RZ MEA 5 deg | 40.0 | 5.0* | Half | 6 | 310.2

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Fig. 13. Refinement zones (RZ) which flow over drawbeads. Elements outside these regions are refined using FV settings.

Fig. 13. Refinement zones (RZ) which flow over drawbeads. Elements outside these regions are refined using FV settings.

For assessing the effect of hardening model, friction model, and binder loading, eight model permutations within AutoForm were investigated and outlined in Table 7. The binder nitrogen spring locations were modeled using the "column binder loading" condition. Fig. 14a illustrates the tooling setup within the model, including the column locations on the binder surface. A constant coulomb coefficient of friction (μ) of 0.11 was selected in accordance with the established practice of the industrial project partners for the same material-lubricant system and is comparable with reported values of 0.12 for similar forming conditions in the work of Bolay et al. [15] and Berahmani et al. [16]. TriboForm AA6xxx mill finish-hot melt friction model provided in the TriboForm database, was fitted using the measured blank roughness, average tooling roughness, and the maximum lubrication amount of 3 g/m2 to reflect the oversaturated lubricant condition. A 3D-surface plot of the fitted TriboForm friction model depicting friction coefficient versus pressure and sliding velocity is shown in Fig. 14b. The effect of kinematic hardening was investigated using the calibrated parameters previously outlined. For clarity, the following abbreviations are used: uniform binder loading (UL), column binder loading (CL), constant coulomb friction (CC), TriboForm (TF), isotropic hardening (IH), and kinematic hardening (KH).

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Fig. 14. (a) AutoForm FE-model setup of U-shaped rail forming operation, and (b) 3D plot of fitted TriboForm AA6xxx mill finished hot-melt friction model.

Fig. 14. (a) AutoForm FE-model setup of U-shaped rail forming operation, and (b) 3D plot of fitted TriboForm AA6xxx mill finished hot-melt friction model.

Table 7. Summary of model variants investigated in AutoForm R12.

Model No. | Binder Loading | Friction Model | Hardening Model
Model 1 (UL-CC-IH) | Uniform | Constant Coulomb (μ = 0.11) | Isotropic
Model 2 (UL-TF-IH) | Uniform | TriboForm
Model 3 (CL-CC-IH) | Column | Constant Coulomb (μ = 0.11)
Model 4 (CL-TF-IH) | Column | TriboForm
Model 5 (UL-CC-KH) | Uniform | Constant Coulomb (μ = 0.11) | Kinematic
Model 6 (UL-TF-KH) | Uniform | TriboForm
Model 7 (CL-CC-KH) | Column | Constant Coulomb (μ = 0.11)
Model 8 (CL-TF-KH) | Column | TriboForm

5. Results and Discussion

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Using Model 1 (UL-CC-IH), the simulation parameters outlined in Table 6 were compared for thickness and springback correlation with the experimentally measured geometry. For the predicted thickness, the root mean squared error (RMSE) at locations A-U were compared. The proportion (%) of the sprung sheet surface that is within the bounds of material thickness ( ± 2 mm) to the scanned surface of the physical panel was defined as the separation distance and used as a general metric to quantify springback correlation. The thickness RMSE, separation distance and maximum material displacement after springback, which is a measure of the overall magnitude of springback, for the various simulation parameters are summarized in Table 8. The standard AutoForm final validation (FV) settings provided the best agreement for both thickness and springback predictions. An additional advantage of using the standard FV settings is the comparatively short computation time relative to the alternative settings evaluated in this study. As a result, the standard FV settings were adopted for all subsequent simulations.

Table 8. RMSE of thickness predictions and proportion of separation distance +/−2 mm from scanned geometry for various control parameters investigated using Model 1.

Simulation Parameters | Predicted Thickness RMSE [mm] | Separation Distance ±2 mm [%] | Max Material Displacement [mm] | Total CPU Time [min]
Final validation (FV) | 0.055 | 50.7 | 14.76 | 13.80
FV+MEA 20 deg | 0.062 | 40.3 | 13.07 | 23.98
FV+MEA 10 deg | 0.105 | 59.9 | 11.63 | 177.58
FV+MEA 5 deg | 0.169 | 39.9 | 10.20 | 1330.11
IES 2.5 mm | 0.053 | 44.3 | 15.80 | 85.82
IES 1.25 mm | 0.077 | 54.5 | 12.72 | 1240.20
FV+RZ MEA 10 deg | 0.062 | 42.6 | 12.72 | 27.20
FV+RZ MEA 5 deg | 0.063 | 35.9 | 12.18 | 94.58

The eight model variants specified in Table 7 were investigated for formability, thickness, and springback predictive accuracy. The formability results of all eight model variants were properly captured, with no regions of predicted splitting, consistent with the forming trials. The models using Coulomb friction predicted a higher risk of splits, with some regions of the component within 20% of the FLC. This is highlighted in Fig. 15, which depicts the advanced formability contour plots of Model 1 (UL-CC-IH) and Model 8 (CL-TF-KH).

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Fig. 15. Advanced formability contour plots of Model 1 and Model 8, shown on left and right respectively.

Fig. 15. Advanced formability contour plots of Model 1 and Model 8, shown on left and right respectively.

The predicted final thickness of the eight model variants was compared to the experimentally measured values at measurement locations A-U and are shown in Fig. 16a. Thickness predictions from Model 8 (CL-TF-KH) show improved agreement, with a RMSE of 0.044 mm versus 0.055 mm for Model 1 (UL-CC-IH). The proportion of separation distance within the bounds of material thickness is shown in Fig. 16b. When comparing the separation distance result of Model 1 (UL-CCIH) to Model 3 (CL-CC-IH), it can be observed that the addition of column binder loading improved the springback correlation by 11.3 percentage points. This is further increased by 40.5 percentage points with the addition of the TriboForm friction model. Column binder loading paired with TriboForm friction models provided the best final thickness and springback predictions, as the frictional and tooling forces are better represented. These forces govern the flow of material into the die cavity, controlling the through thickness stress and strain gradients which are determining factors of thinning and final shape after springback. This is highlighted by Models 4 (CL-TF-IH) and 8 (CL-TF-KH) in Fig. 16. Note that the results of Models 2 (UL-TF-IH) and 6 (UL-TF-KH) were not included for conciseness, as their results were similar to Models 1 and 5 respectively.

[picture]

Fig. 16. (a) Ultrasonic thickness measurements and predicted final thicknesses at locations A-U and (b) percentage of separation distance +/- 2 mm from scanned geometry for various models.

Fig. 16. (a) Ultrasonic thickness measurements and predicted final thicknesses at locations A-U and (b) percentage of separation distance +/- 2 mm from scanned geometry for various models.

Table 9 presents the percent error of the sprung die and punch radii between the scanned geometry for Models 1, 4, and 8. Fig. 17a illustrates the measured and predicted flange angle for Models 1, 4 and 8, with a representative cross-section plot used for the various measurements as shown in Fig. 17b. Models 4 (CL-TF-IH) and 8 (CL-TF-KH) are emphasized due to their better correlation with the separation distance. Model 1 (UL-CC-IH) is used as a reference as it is the simplest of all model variants analyzed. Kinematic hardening (Model 8) marginally improves springback predictive accuracy of sprung punch and die radii compared to isotropic hardening (Model 4). Kinematic hardening also appears to increase the overprediction of flange angle after springback. It should be noted that alternative kinematic hardening models and calibration over additional pre-strain ranges may have a greater influence on springback predictions. Additionally, time-dependent effects, as shown by Wang et al. [39], may contribute to discrepancies as components were scanned twenty-four
hours after forming and such effects are not considered in models. Furthermore, when comparing the results of Model 4 (CL-TF-IH) and 8 (CL-TF-KH) to that of Model 1 (UL-CC-IH) the predicted flange angle after springback shows a significant improvement.

Table 9. Percent error of sprung die and punch radii for Models 1, 4, and 8 at four cross-section locations.

Cross Section No. | Model 1 (UL-CC-IH) | Model 4 (CL-TF-IH) | Model 8 (CL-TF-KH)
| Die Radii Error [%] | Punch Radii Error [%] | Die Radii Error [%] | Punch Radii Error [%] | Die Radii Error [%] | Punch Radii Error [%]
Section 1 | 10.3 | 4.4 | 3.4 | 3.2 | 2.4 | 0.4
Section 2 | 6.2 | 6.3 | 9.2 | 1.7 | 11.3 | 0.6
Section 3 | 15.5 | 24.6 | 15.5 | 16.2 | 15.4 | 19.6
Section 4 | 1.1 | 5.1 | 7.4 | 4.5 | 2.4 | 3.9
Average | 8.3 | 10.1 | 8.9 | 6.4 | 7.9 | 6.1

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Fig. 17. (a) Average of left and right flange angles of scanned and predicted geometries of Models 1, 4, and 8, (b) Cross-section 3 depicting the scanned geometry overlayed with Models 1, 4, and 8.

Fig. 17. (a) Average of left and right flange angles of scanned and predicted geometries of Models 1, 4, and 8, (b) Cross-section 3 depicting the scanned geometry overlayed with Models 1, 4, and 8.

6. Summary and Conclusion

The original version of this paper is available on https://www.scientific.net/SSP.388.209.pdf

The primary conclusions that can be drawn from this experimental-numerical study are:

The standard AutoForm final validation (FV) control parameters provided the best balance of both simulation time and predictive accuracy of thinning, formability, and final shape after springback.

Accurately representing the tooling force distribution through specified binder spring locations is paramount in predicting material thinning and final shape after springback. The addition of column binder loading alone increased springback prediction accuracy by 11.3 and 17.5 percentage points in the case of isotropic and kinematic hardening respectively.

Independently, kinematic hardening marginally improved springback and final thickness predictions for the AA6xxx-T4. However, the use of alternative kinematic hardening models and the calibration over additional strain ranges may reveal a stronger effect on springback.

The marginal improvement kinematic hardening provided is secondary to accurately capturing friction, which controls the flow of material into the die cavity.

Future work should consider potential time-dependent springback effects with geometry measurements taken immediately after forming as well as 24 hours after.

TriboForm friction model with column binder loading provided the most significant improvement for springback predictions. This combination reduced RMSE of thickness predictions by 0.011 mm and improved the separation distance metric by 40.5 percentage points.

Acknowledgement

The original version of this paper is available on https://www.scientific.net/SSP.388.209.pdf

Funding of this work was provided by the Natural Sciences and Engineering Research Council of Canada (NSERC). The authors would like to acknowledge the technical support provided by AutoForm Engineering USA. Special appreciation is extended to Prof. Jacqueline Noder of the University of British Columbia for her continued technical guidance throughout this work. The authors would like to further express their gratitude to Garrett Klein and Ryan Bowerman of Novelis Novi for their technical assistance during forming trials and for conducting the white-light scanning of formed components.

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The original version of this paper is available on https://www.scientific.net/SSP.388.209.pdf

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Solid State Phenomena, ISSN: 1662-9779, Vol. 388, pp 227-236

doi: 10.4028/p-zmGcu2

© 2026 The Author(s). Published by Trans Tech Publications Ltd, Switzerland.

Submitted:
2026-01-19

Revised:
2026-02-15

Accepted:
2026-02-19

Online:
2026-04-15

Necking Limits of Anisotropic Metal Sheets Using M-K Theory and Improved Plasticity Models

The original version of this paper is available on https://www.scientific.net/SSP.388.227.pdf

Yong Hou 1,a*, Monica Trask 1, b and Yannis P. Korkolis 1,c

¹Institute of Forming Technology and Lightweight Components (IUL), TU Dortmund University, 44227, Dortmund, Germany

a*Yong.Hou@iul.tu-dortmund.de, bMonica.Trask@iul.tu-dortmund.de, c Yannis.Korkolis@iul.tu-dortmund.de

*Corresponding Author

Keywords: Marciniak-Kuczynski theory, formability, plastic anisotropy, necking limit.

Abstract

Accurate prediction of the forming limit curve (FLC) is critical for evaluating sheet metal formability, yet the influence of plastic anisotropy remains controversial. In this study, the yielding behavior, hardening response, and strain-rate sensitivity of DC01 steel are experimentally characterized. Different yield criteria combined with a Swift hardening law and the MarciniakKuczyński (M-K) model are employed to predict the FLC. The results show that the high-order nonquadratic Yld2k-2d yield criterion captures both the yielding behavior and the forming limits. Numerical experiments using this material framework are then conducted. Variations in r-values have a limited effect on the FLC, in contrast to the common notion that high r-values mean high formability, whereas the equibiaxial tensile yield stress strongly governs the right-hand side.

1. Introduction

The original version of this paper is available on https://www.scientific.net/SSP.388.227.pdf

The influence of plastic anisotropy on the formability of sheet metals has been investigated for several decades. Early observations from deep drawing operations in the 1950s correlated favorable anisotropic conditions with improved formability in asymmetrical fender draws [1]. Additional studies showed high normal anisotropy to increase limit drawing ratios in cup drawing operations [2] with further experimental studies [3] sought to confirm these trends; however, they are unable to vary the normal anisotropy (r-value) independently of other material parameters such as strain hardening (n) or strength under equibiaxial tension (EBT). As a result, the individual role of anisotropy on formability has remained difficult to isolate experimentally.

Kohara [4] systematically investigated the FLCs for a wide range of non-heat-treatable aluminum alloys in O and H tempers and demonstrated that the strain hardening exponent n exerts a dominant influence on the FLC, particularly under plane strain tension (PST) deformation. Higher n values consistently led to increased limit strains. Owing to this strong coupling, the effects of other parameters, such as r -value or the strain-rate sensitivity m , are often masked in experiments, since changes in r -value are generally accompanied by changes in n and m .

These experimental limitations motivate the use of analytical approaches for FLC prediction, in which material parameters can be varied independently. Among such approaches, the M-K model [5], later extended by Hutchinson and Neale [6], remains one of the most widely used methods for predicting the onset of localized necking based on initial material inhomogeneities. Using the M-K framework, it is shown that increasing the r -value reduces the necking limit under EBT [7] when quadratic anisotropic yield criteria, such as Hill48, are employed. In contrast, Graf and Hosford [8] demonstrated that for higher-order non-quadratic yield criteria, variations in r -value have little influence on the predicted FLC. The yield locus shape plays a critical role in predicting necking under biaxial stretching, as the M-K analysis assumes localization to occur when the stress state in the imperfection region approaches PST conditions. Consequently, different yield criteria, which respond differently to changes in r-value, lead to distinct predictions of the necking limit. For instance, the Hill48 yield locus becomes increasingly elongated in the EBT region with increasing r-value.

Given the strong dependence of FLC predictions on yield locus shape, the use of accurate, highfidelity anisotropic yield criteria is essential. Therefore, this study systematically compares FLC predictions obtained using the M-K model combined with several carefully calibrated yield criteria, including von Mises, Hill48, and the Yld2k-2d yield criteria, with particular emphasis on elucidating the roles of anisotropy and yield surface shape in determining the forming limits of DC01 steel.

2. Mechanical Characterization

The original version of this paper is available on https://www.scientific.net/SSP.388.227.pdf

The material of this study is a cold-rolled, low-carbon, unalloyed steel of 1 mm thickness. The name DC01 indicates it is the base level of five other grades of a series of drawn cold steels and is best suited for cold forming operations where higher grades imply higher formability.

2.1 Plasticity Characterization

The plastic deformation of DC01 steel is probed using a series of well-known mechanical tests namely, uniaxial tension (UT), hydraulic bulge test (HBT) to probe the material in EBT, disk compression (DiskC), plane strain tension (PST), and strain rate jump test (JUMP). Each test is performed to probe a specific deformation mode, or strain rate, and identify the corresponding flow stress for the calibration of plastic anisotropy. A value of plastic work density ( 40MJ/m3 in this work) is selected to compare all the flow stresses. Cross head and punch velocities for each test are set to produce a nominal equivalent strain rate of 10−3/s except for the JUMP test.

UT: The anisotropy of the DC01 sheets is quantified through a series of uniaxial tensile tests performed in seven orientations relative to the rolling direction (RD). True and engineering stress vs. strain curves are depicted in Fig. 1a. The tests are performed on a Zwick/Roell 250 kN universal testing frame equipped with a macroXtens II tactile extensometer with axial and transverse gauge arms, to measure the r-value. From the given range (plastic work density from 10 to 52MJ/m3 ), width vs. thickness strains are plotted with a linear fit, where the slope of the fit is taken as the r-value. The results can be seen in Fig. 1b.

[picture]

Fig. 1. (a) Strain hardening curves and (b) plastic strain evolution (r-values) of DC01 under UT.

Fig. 1. (a) Strain hardening curves and (b) plastic strain evolution (r-values) of DC01 under UT.

EBT: To probe the EBT stress state, two different tests are performed. First, to obtain the yield stress in an equibiaxial stress state a HBT is performed on the BUP 1000 hydraulic sheet metal testing machine. A 200 mm diameter specimen of 1 mm thickness is clamped and pressurized by a fluid until failure. The 3D-DIC software (GOM ARAMIS Professional 2019) is used to map the strain fields throughout deformation. The selected DIC parameters were as follows: a facet size of 21 pixels, a point distance of 11 pixels, and a measurement frequency of 1 Hz . The punch velocity is set to induce a nominal strain rate of 10−3/s at the specimen apex at the plastic work range of interest. Membrane theory is used per the ISO Standard [9] to determine the true stress and true strain measurements with the assumption of negligible bending and stress gradient effects through the thickness of the 1 mm specimen. From the HBT, the equibiaxial yield stress is determined at a plastic work density of 40
MJ/m3, and normalized with the flow stress in UT along the RD. The normalized value is the equibiaxial stress parameter, σb, which is found to be 13.9% higher that that of UT in RD. The ratio of the stresses from the HBT and UT in RD at the plastic work density of 40MJ/m3 is used to shift the flow curve derived from the HBT, to extrapolate that of the UT in RD. The extrapolated curve to an equivalent plastic strain of ~0.93 is then fit with the Swift hardening law, where the strength coefficient K , strain hardening exponent n and initial strain ε0 are identified as 560MPa,0.20, and 0.002, respectively. Second, the DiskC test is used to measure the coefficient of biaxial anisotropy defined as the ratio of plastic strain in the transverse direction (TD) to the plastic strain in the RD. Discs with a diameter of 8 mm are compressed in a Zwick/Roell 250 kN frame with a compression test setup, and interrupted to measure the major and minor axes of the deformed ellipses. A linear fit through the data points is made, of which the slope is the coefficient of biaxial anisotropy, rb=0.963.

PST: To probe the PST state, a notched tensile specimen [10] is pulled using the same Zwick/Roell 250 kN frames as used for the UT test. The same DIC system is used to map the strain field during deformation. Strain distributions of the PST specimen gauge lengths across the width indicate roughly 40% of the gauge width is in a PST deformation mode and the remaining space near the edges at the notches is in UT [11]. A correction factor is determined through a numerical-experimental approach as demonstrated by [12, 13] amongst others, and applied to the experimental force and cross-sectional area for the specimens tested along 0∘,45∘, and 90∘ to the RD. The true stress at the reference plastic work density is output, normalized with that in UT along the RD, and used in the calibration of the anisotropic yield functions.

JUMP: The JUMP test is performed to characterize the strain-rate sensitivity on an ASTM E8 sheet-type UT specimen in the RD, where the strain rate is abruptly changed from 10−3/s to 10−1/s via cross head velocity control. The flow stresses are then compared between the reference curve, performed at a nominal strain rate of 10−3/s. The strain-rate sensitivity value is found to be m=0.0114.

2.2 Necking Forming Limit Characterization

The FLC is populated experimentally following the ISO Standard [14] using Marciniak specimens and standard tooling on the BUP 1000. Layers of Vaseline and PVC foil are used to minimize friction between the flat cylindrical punch and 200×200 mm square carrier blanks which are cut from aluminum and stainless-steel sheets. The carrier blanks have central holes with an initial diameter of 32 mm , which are milled, de-burred and polished. Six specimen geometries are laser-cut in the TD, per the ISO Standard [14], with increasing widths to evoke deformation modes ranging from UT to EBT. Strains are tracked using the DIC system with the necking strains identified using the timedependent method developed by Volk and Hora [15]. The average necking strains of three valid tests for each geometry populate the experimental FLC. Corresponding strain paths for each geometry are plotted in Fig. 2, indicating proportional deformation up to necking with the Marciniak specimens and eliminating effects of friction and bending from the experimental results inherent in other methodologies.

[picture]

Fig. 2. Necking limit strains and strain paths of DC01 in the Marciniak test and the fractured specimens.

Fig. 2. Necking limit strains and strain paths of DC01 in the Marciniak test and the fractured specimens.

3. Marciniak-Kuczynski (M-K) Model and Yield Criteria

The original version of this paper is available on https://www.scientific.net/SSP.388.227.pdf

As a localization-based formability model, the M-K model assumes preexistence of an initial imperfection (the narrow groove 'b' in Fig. 3) in a sheet metal. The strain increment ( Δε11b ) and two stress ratios ( αb and γb ) in the groove 'b' as shown in Eq. (1) are generally set as the unknowns, which can be solved through the Newton-Raphson ( N−R ) method. The angle between the negative x axis and positive n-axis is ψ.

Δε11b,αb=σ22bσ11b,γb=σ12bσ11b(1)

[picture]

Fig. 3. Schematic representation of the imperfection band in the M-K model.

Fig. 3. Schematic representation of the imperfection band in the M-K model.

The proportional deformation path is assumed as the initial constant stress ratio αa from 0 to 1.0 . During the plastic deformation, the angle ψ is updated with the increment of the plastic strain. In each strain increment of Δε11a=0.001, the three unknowns in Eq. (1) can be obtained by one geometric compatibility equation and two force equilibrium equations. When the equivalent strain increment Δε¯b is 10 times greater than Δε¯a, necking occurs. An increment method for the M-K model was developed by Hu et al. [16] through analyzing the relationship between the internal stress variables in the groove. To enhance the computational efficiency, a modified increment method was further established by combining modified N-R method and increment method, see Hu et al. [16].

In this work, the isotropic von Mises, the anisotropic Hill48, and the anisotropic Yld2k-2d yield criteria, all based on a rate-dependent, associated flow rule, are implemented into the M-K model. The brief description and expressions for these yield criteria will be introduced as below:

Von Mises yield criterion:

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σ¯Mises=σ112+σ222−σ11σ22+3σ122(2)

where σ11,σ22, and σ12 are the stress components.

Hill48 yield criterion:

The original version of this paper is available on https://www.scientific.net/SSP.388.227.pdf

σ¯Hill48=(G+H)σ112+(F+H)σ222−2Hσ11σ22+2Nσ122(3)

where F,G,H, and N are four anisotropic parameters, which are analytically identified from the experimental r -values under UT: r0,r45, and r90.

Yld2k-2d yield criterion:

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σ¯Y∣2k−2 d=[12|X1′−X2′|M+12|2X2′′−X1′′|M+12|2X1′′−X2′′|M]1/M(4)

with

X1,2=12(X11+X22±(X11−X22)2+4X122)𝐗=𝐋·σ

(5)(6)

and

[L11′L12′L21′L22′L66′]=[2/300−1/3000−1/3002/30001][α1α2α7],[L11′′L12′′L21′′L22′′L66′′]=19[−228−201−4−4404−4−410−282−2000009][α3α4α5α6α8](7)

where α1~α8 are the anisotropic parameters. The 8 parameters require calibration with the experimental data σ0,σ45,σ90,σb,r0,r45,r90, and rb to minimize the error function. The exponent M can be determined based on the crystal structure or the measured yield stresses under PST loading.

4. Results and Discussion

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The parameters of Hill48 and Yld2k-2d yield criteria are first calibrated using experimental data from plasticity characterization, see Table 1. Two strategies are adopted to optimize the anisotropic parameters of the Yld2k-2d yield criterion. The first strategy (Yld2k-2d Trad.) follows the traditional calibration approach, in which eight experimental data points mentioned in last section are used [17]. The second calibration strategy (Yld2k-2d PST), in addition to the eight experimental data employed in the traditional method, further incorporates the values of the principal stresses in PST measured under three loading directions ( 0∘,45∘, and 90∘ to the RD) from the PST experiments. Moreover, since the measured PST yield stresses are relatively low wrt the UT ones, the exponent M in the Yld2k-2d model is set to 8 rather than 6 .

Table 1. Anisotropic parameters in the Hill48 and Yld2k-2d yield criteria for the DC01 steel.

Parameter | F | G | H | N |  |  |  |
Hill48 | 0.379 | 0.397 | 0.603 | 1.242 |  |  |  |
Parameter | α₁ | α₂ | α₃ | α₄ | α₅ | α₆ | α₇ | α₈ | M
Yld2k-2d Trad. | 1.004 | 1.027 | 0.769 | 0.930 | 0.933 | 0.772 | 0.958 | 1.065 | 8
Yld2k-2d PST | 1.008 | 1.031 | 0.802 | 0.938 | 0.941 | 0.807 | 0.965 | 1.053 | 8

Subsequently, the initial imperfection factor f0 in the M−K model is calibrated using the forming limit data under PST condition (i.e., FLC0 =0.353 in Fig. 2). The determined initial imperfection factor is 0.9999 , which is due to the well-established rolling process and the resulting low level of internal defects in the DC01 steel sheet. Combined with the hardening data and the measured strainrate sensitivity coefficient ( m=0.0114 ), the M-K model is employed to predict the FLC of DC01 steel during plastic deformation along the TD, and the predicted results are compared and analyzed against the experimental data. Finally, based on the Yld2k-2k yield criterion, the effects of hardening parameters and yield surface shape on the FLC are investigated.

4.1 Prediction of Yield Behaviour and Forming Limit Curve

The accuracy of the investigated yield criteria in describing the yield loci, in-plane anisotropy of UT yield stresses and r-values will be evaluated. Fig. 4a compares the yield loci of DC01 predicted by four calibrated yield criteria. All models accurately predict the yield stress under UT at 90∘, whereas the von Mises criterion significantly underestimates the yield stress ( σb ) under EBT. The gray dashed lines denote the experimentally measured principal stress level under PST. Although the Hill48 criterion coincidentally captures the EBT yield stress, it overestimates the PST yield stress. Both Yld2k-2d models calibrated using different strategies reasonably predict the EBT and PST yield stresses.

Fig. 4b and c show the predicted in-plane anisotropy of yield stresses and r -values under UT, respectively, indicating that only Yld2k-2d provides accurate predictions for both. Fig. 4d compares the experimental and predicted PST yield stresses along three directions ( 0∘,45∘, and 90∘ to the RD). When PST yield stresses are included in the calibration, the Yld2k-2d PST model provides more accurate predictions, whereas Hill48 overestimates the PST yield stresses, consistent with the
observations in Fig. 4a. Interestingly the von Mises yield criterion gives the best prediction of PST yield stresses.

Fig. 4e compares the experimentally measured FLC with those predicted by the M-K model combined with four different yield criteria. On the left-hand side of the strain space, all yield criteria provide similarly accurate FLC predictions, whereas noticeable differences arise on the right-hand side of FLC. The von Mises and Hill48 criteria significantly overestimate the necking limit under EBT strain path, while the Yld2k-2d criterion accurately predicts the necking limit under EBT. In comparison, the Yld2k-2d PST model, which incorporates PST yield stresses in calibration, gives the most accurate predictions. Given the high accuracy of the Yld2k-2d yield criterion in describing both yielding behavior and FLCs, it is therefore adopted in the following theoretical analysis to investigate the effects of hardening parameters and anisotropic yield surface shape on the FLC of sheet metals.

[picture]

Fig. 4. Comparison of (a) yield loci, in-plane anisotropy of (b) yield stress and (c) r-value under UT, (d) PST yield stresses along three loading directions, and (e) FLC of DC01 predicted using different yield criteria.

Fig. 4. Comparison of (a) yield loci, in-plane anisotropy of (b) yield stress and (c) r-value under UT, (d) PST yield stresses along three loading directions, and (e) FLC of DC01 predicted using different yield criteria.

4.2 Effect of Strain Hardening on FLC

In the following two subsections, the calibrated material parameters of DC01 are used as the reference, and the effects of individual changes in hardening or yield behavior on the FLC are examined. Fig. 5a and 5b show the hardening curves and the corresponding FLCs, respectively, predicted by the MK model for increased and decreased strength coefficient K in the Swift hardening law. It is observed that variations in the K value shift the stress-strain curve, and somewhat change its slope as well, but have no influence on the predicted FLC.

[picture]

Fig. 5. Comparison of (a) stress vs. strain curves and (b) predicted FLCs for various strength coefficient.

Fig. 5. Comparison of (a) stress vs. strain curves and (b) predicted FLCs for various strength coefficient.

Fig. 6a and 6b present the hardening curves and the corresponding M-K predicted FLCs obtained by increasing and decreasing the strain hardening exponent n while keeping the initial yield stress constant. Increasing n enhances the slope of the stress vs. strain curve and significantly improves the whole formability from UT to EBT, as reflected by an increased FLC in Fig. 6b.

[picture]

Fig. 6. Comparison of (a) stress-strain curves and (b) predicted FLCs for various strain hardening exponent.

Fig. 6. Comparison of (a) stress-strain curves and (b) predicted FLCs for various strain hardening exponent.

4.3 Effect of Yield Surface on FLC

Fig. 7a and 7b illustrate the yield loci predicted by the calibrated Yld2k-2d yield criterion in the first quadrant of the normal stress plane and the corresponding FLCs predicted by the M-K model when only the r-value under UT along the TD is varied. The inset in Fig. 7a compares the predicted r-values in different loading directions. As r90 increases from 1.09 to 2.09 , the local normal direction of the yield locus under the UT90 stress state is altered. Correspondingly, the FLC remains unchanged on the left-hand side, while slight variation is observed on the right-hand side, with a higher r90 leading to an increased necking limit under EBT.

Fig. 8a compares the yield loci predicted by the calibrated Yld2k-2d yield criterion when only EBT yield stress (σb) is varied. It is clearly observed that the yield stress and plastic flow direction
(r-value) under UT remain unchanged, whereas the predicted PST yield stress increases with increasing EBT yield stress σb. Fig. 8b presents the corresponding FLCs associated with these yield behaviors. While variations in σb do not affect the necking limit from UT to PST, they have a pronounced effect on the necking limit strain near the EBT condition.

[picture]

Fig. 7. Comparison of (a) yield loci and (b) predicted FLCs for various r90 values.

Fig. 7. Comparison of (a) yield loci and (b) predicted FLCs for various r 90 values.

[picture]

Fig. 8. Comparison of (a) yield loci and (b) predicted FLCs for various EBT yield stresses.

Fig. 8. Comparison of (a) yield loci and (b) predicted FLCs for various EBT yield stresses.

Taken together, Figs. 7 and 8 indicate that modifications to the yield locus and plastic flow direction have almost no influence on the left-hand side of the FLC. Varying the r-value does not significantly alter the right-hand side of the FLC in the corresponding loading direction, either. In contrast, changes in the EBT yield stress have a pronounced effect on the right-hand side of the FLC. This behavior may be related to the alteration of the loading direction (i.e., the ratio of major-to-minor principal stresses) under PST, which governs the "rate" at which the strain path deviates toward the PST condition during necking. The underlying mechanisms will be investigated in future work.

5. Conclusion

The original version of this paper is available on https://www.scientific.net/SSP.388.227.pdf

In this study, the yielding behavior, hardening response, and strain-rate sensitivity of DC01 steel were characterized through various mechanical tests. Based on different yield criteria, an isotropic (Swift) hardening law, and the M-K model, the necking forming limits of DC01 steel were predicted. The main findings can be summarized as follows. DC01 steel exhibits a relatively high EBT yield stress, in comparison to UT, with a normalized EBT yield stress of 1.139, and a relatively low PST yield stress. The Yld2k-2d model with M=8 accurately predicts the yield locus as well as the yield stresses and r-values under UT. Compared with the von Mises and Hill48 yield criteria, the Yld2k-2d criterion combined with the M-K model provides accurate predictions of the necking limit under EBT. Incorporating PST yield stresses into the calibration of the Yld2k-2d yield criterion slightly improves the prediction accuracy on the right-hand side of the FLC. The strength coefficient K has no influence
on the FLC, whereas variations in the strain hardening exponent n shift the overall strain level of the FLC. Changes in yielding characteristics have negligible effects on the left-hand side of the FLC; however, for the right-hand side, variations in the EBT yield stress have a much more pronounced influence than changes in r -values.

Acknowledgement

The original version of this paper is available on https://www.scientific.net/SSP.388.227.pdf

The authors acknowledge the support of a research fellowship from the Alexander von Humboldt Foundation. Yong Hou gratefully acknowledges Prof. A. Erman Tekkaya for his generous support as the Humboldt host. The authors also thank the IUL technical staff, in particular Dirk Hoffmann and Thomas Werse for sample preparation, and Eike Hoffmann and Oliver Schulz for assistance with mechanical testing. In addition, the authors acknowledge Carter Fietek from the Ohio State University and Prof. Jinjin Ha from University of New Hampshire for insightful discussions on plastic anisotropy and material modeling.

References

The original version of this paper is available on https://www.scientific.net/SSP.388.227.pdf

W. Lankford, S. Snyder, J. Bauscher, New Criteria in Predicting the Press Performance of Deep Drawing Sheets, Trans. ASM. 42 (1950) 1197-1232.

W. Panknin, J. Grosch, The Effect of Workpiece Material on Cracks in Deep Drawing of Unkilled Steels, Bänder Bleche Röhre. 10 (1969) 339-348.

L. Lilet, M. Wybo, Investigation into the Effect of Plastic Anisotropy and Rate of Work Hardening in Deep Drawing, Sheet Met. Ind. 41 (1964) 783-803.

S. Kohara, Forming-limit curves of aluminum and aluminum alloy sheets and effects of strain path on the curves, J. Mater. Process. Technol. 38 (1993) 723-735.

Z. Marciniak, K. Kuczyński, Limit strains in the processes of stretch-forming sheet metal, Int. J. Mech. Sci. 9 (1967) 609-620.

J. Hutchinson, K. W. Neale, Sheet necking-II. Time-independent behavior, Mechanics of Sheet Metal Forming: Material Behavior and Deformation Analysis. Boston, MA, Springer US (1978) 127-153.

R. Sowerby, J. L. Duncan, Failure in sheet metal in biaxial tension, Int. J. Mech. Sci. 13(3) (1971) 217-229.

A. Graf, W. F. Hosford, Calculations of forming limit diagrams, Metall. Trans. A. 21(1) (1990) 87-94.

ISO 16808 (2014) Metallic Materials - Sheet Strip - Determination of biaxial stress-strain curve by means of a bulge test with optical measuring systems.

N. Tardif, S. Kyriakides, Determination of anisotropy and material hardening for aluminum sheet metal, Int. J. Solids Struct. 49(25) (2012) 3496-3506.

M. Baral, J. Ha, Y. P. Korkolis, Plasticity and ductile fracture modeling of an diecast alloy, Int. J. Fract. 216(2) (2019) 101-121.

C. P. Dick, Y. P. Korkolis, Anisotropy of thin-walled tubes by a new method of combined tension and shear loading, Int. J. Plast. 71 (2015) 87-112.

H. Tian, B. Brownell, M., Y. P. Korkolis, Earing in cup-drawing of anisotropic Al-6022-T4 sheets, Int. J. Mater. Form. 10(3) (2017) 329-343.

ISO 12004 (2021) Metallic Materials - Determination of forming-limit curves for sheet and strip - Part 2: Determination of forming-limit curves in the laboratory.

W. Volk, P. Hora, New algorithm for a robust user-independent evaluation of beginning instability for the experimental FLC determination, Int. J. Mech. Sci. 4(3) (2011) 339-346.

Q. Hu, X. Li, J. Chen, New robust algorithms for Marciniak-Kuczynski model to calculate the forming limit diagrams, Int. J. Mech. Sci. 148 (2018) 293-306.

F. Barlat, J.C. Brem, J.W. Yoon, K. Chung, R.E. Dick, D.J. Lege, F. Pourboghrat, S.H. Choi, E. Chu, Plane stress yield function for aluminum alloy sheets-part 1: theory, Int. J. Plast. 19(9) (2003) 1297-1319.

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Solid State Phenomena, ISSN: 1662-9779, Vol. 388, pp 237-243

doi: 10.4028/p-SrPnJ8

© 2026 The Author(s). Published by Trans Tech Publications Ltd, Switzerland.

Submitted:
2026-01-19

Revised:
2026-02-16

Accepted:
2026-02-22

Online:
2026-04-15

Effect of Continuous Bending under Tension on Dual-Phase Steels

The original version of this paper is available on https://www.scientific.net/SSP.388.237.pdf

Gabriela Vincze 1,2,a*, Ricardo C. Alves 1, b, Rafael O. Santos 3,c and Marilena C. Vincze 1,2,d

¹Centre of Mechanical Technology and Automation (TEMA), Department of Mechanical Engineering, University of Aveiro, 3810-193 Aveiro, Portugal

2 LASI - Intelligent Systems Associate Laboratory, 4800-058 Guimarães, Portugal

3 Graduate Program in Metallurgical Engineering, Universidade Federal Fluminense - UFF, Volta Redonda, Rio de Janeiro, Brazil

a * gvincze@ua.pt, b  ricardo.camilo@ua.pt, c  rafaeloliveirasantos@id.uff.br, d  cbutuc@ua.pt

*Corresponding author

Keywords: steel, continuous bending under tension, formability.

Abstract

Lightweighting plays a critical role in reducing vehicle emissions, a major source of air pollution in the European Union. While weight reduction during the use phase is important, environmental impacts across production and end-of-life stages must also be considered. Advanced forming technologies enable the use of high-strength materials while maintaining formability and energy efficiency. Continuous-bending-under-tension (CBT) is a promising forming technique capable of inducing higher plastic deformation than conventional processes. In this study, CBT experiments were conducted on dual-phase 1000 low-yield (DP1000-LY) steel. The material was subjected to uniaxial tensile loading combined with cyclic bending through a moving three-roll system. The effects of key process parameters, bending depth and speed ratio between the bending assembly and tensile loading, were systematically investigated. The results show that lower bending depths allow greater total deformation before fracture and result in higher tensile forces. Higher speed ratios lead to earlier failure both during and after CBT processing. Hardness measurements indicate comparable surface hardness on both sides of the specimens, regardless of single or double roll contact. These findings contribute to a better understanding of CBT process parameters and support its potential application in lightweight automotive component manufacturing.

Introduction

The original version of this paper is available on https://www.scientific.net/SSP.388.237.pdf

Reducing vehicle weight during use is important, but Sato and Nakata [1] highlight that environmental impacts across production and end-of-life stages must also be considered. Achieving lighter and safer components requires a strategic combination of materials such as steel, aluminum, and composites. To meet increasing demands for better formability and energy efficiency, advanced metal forming methods are gaining attention. Among these, continuous-bending-under-tension (CBT) has emerged as a promising technique, allowing greater plastic deformation than conventional processes [2,3].

The idea of enhancing elongation-to-fracture by superimposing bending on tension was first noted by Swift in 1948 [4]. However, the first experimental application of CBT for improving elongation in both ferrous and non-ferrous metals was carried out by Benedyk et al. in 1971 [5]. Using three moving rollers to bend and unbend metal sheets, they observed multiple cracks at large deformations. This was attributed to a reduction of the net axial force due to the added bending.

CBT was revisited in 2009 by Emmens and van den Boogaard [6], who developed a device to study incremental metal forming, as CBT reproduces similar deformation conditions. Their parametric studies on aluminum and mild steel identified the bending radius as the most influential factor. Later work extended to stainless steel, brass, and titanium [7,8], and included a simple mechanical model showing that many CBT behaviors are governed by loading conditions rather than material properties.

At the University of New Hampshire, a novel CBT machine was developed [9], featuring stationary rollers while the specimen and its axial loading system reciprocate. This setup was used to study AA6022 and various dual-phase (DP) steels. Notably, DP1180 exhibited over five times higher elongation under CBT compared to standard uniaxial tension. Lower DP grades showed moderate improvements, while increased sheet thickness reduced elongation gains. Furthermore, the strength increase with repeated CBT cycles was more pronounced in sheets with lower martensite content.

This work focuses on the effect of CBT parameters on the mechanical behavior of DP1000-LY. The article is structured in three main parts: material characterization, description of the CBT apparatus, and presentation and discussion of results, concluding with a summary of the findings.

Material and Experimental Details

The original version of this paper is available on https://www.scientific.net/SSP.388.237.pdf

Material

The original version of this paper is available on https://www.scientific.net/SSP.388.237.pdf

The study material is 1.5 mm thick dual-phase 1000 low-yield (DP1000-LY) steel. Tensile tests were conducted using a Shimadzu Autograph Machine (Shimadzu, Japan) with a 100 kN load cell, and strain was measured via Digital Image Correlation (DIC) using the GOM system and ARAMIS 5M software (GOM, Germany). The resulting engineering and true stress-strain curves are shown in Fig. 1, and the mechanical properties are summarized in Table 1.

[picture]

Fig. 1. Mechanical behavior of DP1000-LY tested in rolling direction.
Table 1. Mechanical properties of of DP1000-LY tested in rolling direction.

Fig. 1. Mechanical behavior of DP1000-LY tested in rolling direction. Table 1. Mechanical properties of of DP1000-LY tested in rolling direction.

Fig. 1. Mechanical behavior of DP1000-LY tested in rolling direction.
Table 1. Mechanical properties of of DP1000-LY tested in rolling direction.

E [GPa] | σy [MPa] | σUTS [MPa] | εu [%] | εt [%] | r | n
205 ± 11 | 657 ± 30 | 990 ± 3 | 11.8 ± 0.8 | 17.8 ± 4 | 0.790 ± 0.005 | 0.150 ± 0.002

Continuous-Bending-under-Tension (CBT) Equipment

The original version of this paper is available on https://www.scientific.net/SSP.388.237.pdf

Figure 2 shows the CBT equipment developed at the University of Aveiro, along with a schematic representation of the process highlighting the main parameters: roll diameter D , roll distance L , center roll depth setting P , tensile velocity vt and bending assembly velocity vb. Recent numerical investigations of the CBT process by Perreira et al. [11] indicate that roll diameter has little influence on the results. In this study, the focus is on the center roll depth P and the velocity ratio vt/vb. Specifically, three values of P were investigated, 2,3 , and 4 mm and two velocity ratios were considered 0.0088 and 0.0118 . The CBT equipment is mounted in the Shimadzu Autograph Machine (Shimadzu, Japan) with a 100 kN load cell used previously for material chracaterization. The sample geometry is represented in Fig. 2c.

[picture]

Fig. 2. Continuous-bending-under-tension (CBT) equipment (a), a schematic view (b) and sample geometry (c).

Fig. 2. Continuous-bending-under-tension (CBT) equipment (a), a schematic view (b) and sample geometry (c).

After CBT, sub-size specimens were extracted from the utilized zone of the CBT sample for subsequent tensile and Vickers microhardness testing, as shown in Fig. 3. Microhardness measurements were performed using a Falcon 600 G 2 (Netherlands) with a load of 0.3 kgf applied for 10 seconds.

[picture]

Fig. 3. Location of sub-size specimens for tensile and microhardness testing on the CBT sample.

Fig. 3. Location of sub-size specimens for tensile and microhardness testing on the CBT sample.

Results and Discussion

The original version of this paper is available on https://www.scientific.net/SSP.388.237.pdf

The first set of samples was tested until fracture. The conditions of these tests are summarized in Table 2. The results, shown in Fig. 3, indicate that global deformation was higher under all CBT conditions compared to standard tensile tests. It appears that higher speed ratios led to greater displacement, while increasing bending depth reduced displacement at the same speed ratio. These findings are consistent with previous literature.

Based on these results, the evaluation of CBT-processed material was performed using tensile tests. For this purpose, a new set of samples was processed by CBT without reaching fracture,
stopping the tests at 75% of the maximum number of cycles recorded previously. The conditions of these tests are summarized in Table 3. The results, presented in Fig. 4, show good repeatability for the same bending depth, with slightly higher forces observed for higher speed ratios. In Continuous Bending under Tension (CBT), increasing the bending depth reduces the axial stress required to deform the specimen. A higher bending depth increases the contact length between the specimen and the bending rolls, resulting in a smaller bending radius. Consequently, the material experiences more intense repeated bending and unbending, leading to increased local plastic deformation. As a result, for the same applied uniaxial displacement, the measured axial force is lower than that observed at smaller bending depths.

Table 2. Description of the CBT test conditions for samples tested to fracture. (*imposed parameters).

| P2_R_L1 | P2_R_H1 | P3_R_L1 | P3_R_H1 | P4_R_L1 | P4_R_H1
P [mm]* (see Fig. 2b) | 2 | 2 | 3 | 3 | 4 | 4
Velocity ratio (vi/vb)* | R_L 0.0088 | R_H 0.0118 | R_L 0.0088 | R_H 0.0118 | R_L 0.0088 | R_H 0.0118
Number of cycles | 25 | 20 | 18 | 17 | 16 | 16
Displacement [mm] | 63 | 67 | 48 | 52 | 38 | 48.5

Fig. 4. Mechanical behaviour of DP1000-LY under CBT conditions, tested at two velocity ratios and three bending depths (2,3, and 4 mm) up to fracture.

Table 3. Description of the CBT test conditions prior to fracture, up to 75% of the total cycles. ( * imposed parameters).

P [mm]* | P2_R_L2 2 | P2_R_H2 2 | P3_R_L2 3 | P3_R_H2 3 | P4_R_L2 4 | P4_R_H2 4
Velocity ratio (vt/vb)* | R_L 0.0088 | R_H 0.0118 | R_L 0.0088 | R_H 0.0118 | R_L 0.0088 | R_H 0.0118
Number of cycles* | 19 | 15 | 13 | 13 | 12 | 12
Displacement [mm] | 51.5 | 50 | 32.7 | 43.5 | 30 | 40

[picture]

Fig. 5. Mechanical behaviour of DP1000-LY under CBT conditions at two velocity ratios and bending depths of 2,3 , and 4 mm , before fracture.

Fig. 5. Mechanical behaviour of DP1000-LY under CBT conditions at two velocity ratios and bending depths of 2,3 , and 4 mm , before fracture.

The tensile test results after CBT are shown in Fig. 5. As expected, the highest strength was observed for the lowest bending depth, which also exhibited the highest force during CBT, while the lowest strength corresponded to the highest bending depth. For a bending depth of 2 mm , no significant difference in strength was observed between the two speed ratios. For the other bending depths, higher speed ratios tended to produce slightly higher strength. In all cases, the strain was significantly reduced compared to the as-received material, reflecting the large amount of accumulated plastic deformation during CBT.

[picture]

Fig. 6. Tensile stress-strain curves of material after CBT processing.

Fig. 6. Tensile stress-strain curves of material after CBT processing.

The hardness of both surfaces in contact with the bending rolls was also investigated to assess whether contact with one or both rolls had a differential effect. The Vickers hardness of the original DP1000-LY material was approximately 317 HV . CBT samples exhibited increased hardness, ranging from 325 HV to 390 HV , representing up to a 25% increase over the undeformed material. This increase is attributed to work hardening induced by CBT and is consistent with the observed increases in yield and ultimate tensile stresses. The initial material had an average yield stress of 705 MPa and ultimate tensile stress of 1002.7 MPa , whereas post-CBT specimens exhibited yield stresses between 835.5 MPa and 1052.6 MPa and ultimate stresses between 1100.5 MPa and 1222.6 MPa . These results demonstrate a clear correlation between surface hardness and mechanical strength, highlighting the significant impact of CBT on material properties. Despite the geometric asymmetry during loading, hardness values were similar between the two faces, with no clear trend of one face consistently exhibiting higher hardness than the other. This suggests that the effects of CBT on surface work hardening are approximately symmetrical.

Summary

The original version of this paper is available on https://www.scientific.net/SSP.388.237.pdf

CBT processing of DP1000-LY steel significantly influenced deformation and tensile behavior. Lower bending depths allowed greater total deformation before fracture and higher tensile forces, while higher speed ratios caused earlier failure. Tensile tests on partially processed samples confirmed that increasing bending depth reduces strength, and higher speed ratios accelerate failure. Both surfaces of the specimens showed similar mechanical behavior, indicating that contact with one or two rolls has minimal effect. Overall, CBT enhances material strength, with deformation and failure strongly dependent on bending depth and speed ratio.

Acknowledgment

The original version of this paper is available on https://www.scientific.net/SSP.388.237.pdf

The support from the Portuguese Foundation of Science and Technology (FCT), in its State Budget component (OE) through projects 2022.05783.PTDC (https://doi.org/10.54499/2022.05783.PTDC) and UID/00481 - Centre for Mechanical Technology and Automation (https://doi.org/10.54499/UID/00481/2025) is gratefully acknoledge. The authors would like to thank the support of TATA Steel for supplying the material investigated in this work.
The authors acknowledge the use of ChatGPT to enhance readability, rephrase certain sections, and perform grammar checking. After using this tool/service, the authors reviewed and edited the content as needed and take full responsibility for the content of the publication.

References

The original version of this paper is available on https://www.scientific.net/SSP.388.237.pdf

F.E.K. Sato, T. Nakata, Analysis of the impact of vehicle lightweighting on recycling benefits considering life cycle energy reductions, Resources, Conservation & Recycling. 164 (2021) 105118.

M. Knezevic, C.M. Poulin, X.Zheng, S. Zheng, I.J. Beyerlein, Strengthening of alloy AA6022T4 by continuous bending under tension, Mat. Sci. Eng. A 758 (2019) 47-55.

C.M. Poulin, Y.P. Korkolis, B.L. Kinsey, M. Knezevic (2019) Over five-times improved elongation-to-fracture of dual-phase 1180 steel by continuous-bending-under-tension. Material and Design 161:95-105 .

H.W. Swift, Plastic bending under tension, Engineering 3166 (1948) 333-359.

J.C. Benedyk, D. Stawarz, N.M. Parikh, A method for increasing elongation values for ferrous and nonferrous sheet metals, J. Mater. 6, (1971) 16-29.

W.C. Emmens, A.H. van den Boogaard, Incremental forming by continuous bending under tension-An experimental investigation. J. Mater. Proc. Technol. 209, (2009) 5456-5463.

W.C. Emmens, A.H. van den Boogaard, Cyclic stretch-bending: mechanics, stability and formability. J. Mater. Process. Tech. 211, (2011) 1965-1981. .

W.C. Emmens, A.H. van den Boogaard, Material characterization at high strain by adapted tensile tests. Exp. Mech. 52, (2012) 1195-1209. .

M. Zecevic, T. Roemer, M. Knezevic, Y. Korkolis, B. Kinsey, Residual ductility and microstructural evolution in continuous-bending-under-tension of AA-6022-T4, Materials 9 (2016) 130.

C. M. Poulin, S. C. Vogel, Y. P. Korkolis, B. L. Kinsey, M. Knezevic, Experimental studies into the role of cyclic bending during stretching of dual-phase steel sheets, Int. J. of Mater. Forming 13, (2020) 393-408 .

J. F. A. Pereira, P. A. Prates, M. C. Butuc, G. Vincze, Numerical Study on Continuous-Bending-Under-Tension of 3rd Generation Steel, Metals 15, (2025),138. .

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The original version of this eBook is available on https://www.scientific.net/book/formability-of-metallic-materials/978-3-0364-1993-0

Solid State Phenomena, ISSN: 1662-9779, Vol. 388, pp 245-251

doi: 10.4028/p-QnpP7i

© 2026 The Author(s). Published by Trans Tech Publications Ltd, Switzerland.

Submitted:
2026-02-04

Revised:
2026-02-16

Accepted:
2026-02-21

Online:
2026-04-15

Heterostructure High Entropy Alloys in the CoNiFeMn-Mo System Intended for Deep Drawing Applications

The original version of this paper is available on https://www.scientific.net/SSP.388.245.pdf

Kamil Cichocki 1,a*, Anna Wójcik 1, b, Paulina Lisiecka-Graca 1,c, Tymon Warski 2, d and Krzysztof Muszka 1,e

1 AGH University of Kraków, Department of Metal Forming and Metallurgical Engineering, Kraków, Poland

2 Łukasiewicz Research Network - Institute of Non-Ferrous Metals, 5 Sowińskiego Street, 44-100 Gliwice, Poland

a * cichocki@agh.edu.pl, b  anwojcik@agh.edu.pl, c  graca@agh.edu.pl, d  tymon.warski@gmail.com, e  muszka@agh.edu.pl

Keywords: high entropy alloys, static recrystallization, heterostructure, mechanical properties.

Abstract

High-entropy alloys (HEAs), owing to their exceptionally favourable strength-ductility balance, are regarded as promising candidates for applications in the energy, automotive, and aerospace industries. A defining characteristic of face-centered cubic (FCC) high-entropy alloys is their low stacking fault energy, which facilitates deformation via mechanical twinning and promotes the activation of transformation-induced plasticity (TRIP) and twinning-induced plasticity (TWIP) mechanisms. The present study focuses on the development of a heterostructured material composed of CoNiFeMn and (CoNiFeMn)95Mo5 alloys. Furthermore, the Erichsen cupping test was performed to assess the formability of the produced material and to evaluate its suitability for deep drawing applications.

Introduction

The original version of this paper is available on https://www.scientific.net/SSP.388.245.pdf

High-entropy alloys (HEAs) have attracted significant research interest in recent years. Their concept, based on pronounced chemical disorder, renders these materials competitive with conventional alloys that are typically based on a single principal element[1], [2]. Despite their complex chemical compositions, HEAs tend to form stable solid solutions owing to the high configurational entropy of mixing, which reduces the Gibbs free energy of the solid solution. As a result, the formation of intermetallic phases is often suppressed[3]. However, HEAs with a face-centered cubic (FCC) crystal structure are generally characterized by a relatively low yield strength combined with excellent ductility. Conventional approaches for increasing yield strength, such as grain refinement[4] or precipitation strengthening[5], usually lead to a pronounced reduction in ductility[6]. Another important characteristic of these alloys is their low stacking fault energy (SFE)[7], [8] which promotes deformation by twinning and consequently leads to the activation of the twinning induced plasticity effect (TWIP)[9], [10], [11].
One approach to improving the mechanical properties of high-entropy alloys is the development of heterostructured materials. Several methods exist to achieve heterostructured architectures, including heterogeneous gradient structure [12], heterogeneous grain structure[4] and heterogeneous phase structure[13].
The main aim of the present study was to fabricate a heterostructured material via cold rolling followed by static recrystallization, thereby combining heterogeneous grain structures with the precipitation of the intermetallic μ phase. In addition, the study seeks to evaluate the formability of the produced materials under deep-drawing conditions. For this purpose, two materials were selected: CoNiFeMn and a molybdenum-alloyed variant, (CoNiFeMn)95Mo5, both exhibiting a propensity for deformation by twinning, with the addition of molybdenum further enhancing twinning activity [14]. Our previous studies have demonstrated that during recrystallization annealing of ( CoNiFeMn ) 95Mo5, precipitation of the μ phase occurs; in the present work, this phase was utilized as a strengthening constituent[15] and also investigate the influence of intermetallic phase for deep-drawing process.

Experimental Procedure

The original version of this paper is available on https://www.scientific.net/SSP.388.245.pdf

In the present study, the feasibility of producing materials with a heterogeneous microstructure in CoNiFeMn and ( CoNiFeMn ) 95Mo5 high entropy alloys (HEAs) was investigated. The alloys were synthesized by melting in an open induction furnace under a continuous flow of inert gas (argon) directed onto the surface of the molten metal. The total weight of each ingot was 10 kg . The ingots were produced from constituent elements with a purity of 99.95%. The chemical compositions of the as-cast alloys are listed in Table 1.
Subsequently, the alloys were subjected to a homogenization heat treatment at 1200∘C for 24 h . For further characterization, the materials underwent thermomechanical processing according to the schedule shown in Fig. 1. The chemical composition was analyzed using scanning electron microscopy equipped with energy-dispersive X-ray spectroscopy (SEM-EDS). Microstructural characterization, as well as texture analysis after selected stages of deformation and annealing, was performed using scanning electron microscopy combined with electron backscatter diffraction (SEM-EBSD) with post-processing using software OIM TSL 8. All microstructural and texture analyses were conducted using a NanoNova SEM 450 scanning electron microscope.

Table 1. Chemical compositions of the investigated materials in at%.

Alloy | Abbrev. | Fe | Mn | Ni | Co | Mo
CoNiFeMn | Mo0 | 24.7 | 24.2 | 25.5 | 25.6 | 0
(CoNiFeMn)₉₅Mo₅ | Mo5 | 23.6 | 22.5 | 24.2 | 24.5 | 5.2

[picture]

Fig. 1. Scheme of thermomechanical processing of studied alloys.

Fig. 1. Scheme of thermomechanical processing of studied alloys.

Microstructural analyses were performed on materials after hot rolling, which were considered to represent the initial condition, and subsequently after cold rolling to true strains of ε=1.0,1.5, and 2.0. In the present work, the feasibility of producing a heterostructured material through cold rolling followed by static recrystallization was investigated. The cold-rolled materials were subsequently subjected to recrystallization annealing.
The present study was focused on the possibilities to use heterostructured HEA materials deep drawing process. For this purpose, the Erichsen IE21 cupping test (in accordance with EN ISO 20482) was performed. Square sheet specimens with dimensions of 65×65 mm and a nominal thickness of 1.5 mm were prepared from the selected material variants. The test was carried out at constant punch velocity of 10 mm/s using a hemispherical punch with a diameter of 14.98 mm . Tests were conducted three times for each variant.

To determine the mechanical properties of the materials selected for the forming tests, uniaxial tensile tests were conducted. Dog-bone shaped specimens were prepared, and the test was carried out using a Zwick 250 testing machine. Two specimens were tested for each variant.

Results and Discussion

The original version of this paper is available on https://www.scientific.net/SSP.388.245.pdf

Fig. 2 presents the microstructures of the CoNiFeMn and (CoNiFeMn)95Mo5 alloys after hot rolling. The microstructures of both materials are characterized by equiaxed grains with a high density of annealing twins. The difference in grain size observed between the two alloys arises from the effect of molybdenum addition on the kinetics of dynamic recrystallization as well as grain growth, resulting in a retardation of these processes. This phenomenon has been discussed in greater details in previous studies [15], [16].

[picture]

Fig. 2. Inverse pole figure maps for a) CoNiFeMn and b) (CoNiFeMn)95Mo5 alloys.

Fig. 2. Inverse pole figure maps for a) CoNiFeMn and b) ( C o N i F e M n ) 95 M o 5 alloys.

The initial material exhibiting the microstructure shown in Fig. 2 was subsequently subjected to cold rolling to accumulated true strains of ε=1.0,1.5, and 2.0 . The microstructures of the materials at different deformation levels are presented in Fig. 3. All materials exhibit a microstructure typical for cold rolling, characterized by strongly elongated grains along the rolling direction. Deformation twins are also observed in the microstructure. In the case of the CoNiFeMn alloy, deformation twins become clearly visible at a strain of ε=1.5, whereas in the (CoNiFeMn)95Mo5 alloy they can be observed at all considered strain levels.
Pole figure analysis (Fig. 3) confirms the development of a strong crystallographic texture with increasing deformation, as evidenced by the shift of intensity maxima corresponding to individual texture components. For both alloys, the evolution of texture is qualitatively similar; however, differences between the alloys are reflected primarily in the intensity of the developed texture. At a strain of ε=1.0, the dominant texture components in both materials are the Brass {011}⟨211⟩ and rotated Goss {110}⟨110⟩ components, which are indicative of deformation dominated by twinning mechanisms. With increasing strain, the intensity of the Brass component decreases, while the rotated Goss component becomes more pronounced. Additionally, the emergence of A {110}⟨111⟩ and P {011}⟨211⟩ texture components is observed, further confirming the activation of deformation twinning. This behaviour is characteristic of face centred cubic HEA's with low stacking fault energy.

CoNiFeMn

The original version of this paper is available on https://www.scientific.net/SSP.388.245.pdf

[picture]

Fig. 3. Inverse pole figure maps and pole figures for a) CoNiFeMn and b) (CoNiFeMn) 95Mo5 alloys with different strain level.

Fig. 3. Inverse pole figure maps and pole figures for a) CoNiFeMn and b) (CoNiFeMn) 95 M o 5 alloys with different strain level.

The cold rolled materials were subjected to recrystallization annealing. The annealing temperatures and holding times were selected based on our previous work [15]. In the CoNiFeMn alloy, the formation of a heterogeneous microstructure was observed at an annealing temperature of 675∘C with a holding time of 600 s . For the (CoNiFeMn)95Mo5 alloy, a heterogeneous microstructure developed after annealing at 800∘C for 1800 s (Fig. 4a, b). The resulting microstructure is characterized by bands of coarse grains aligned along the rolling direction, surrounded by regions of finer grains. In addition, in the (CoNiFeMn)95Mo5 alloy, molybdenum-rich precipitates of the intermetallic μ phase with a trigonal crystal structure were observed (Fig. 4c). This phase precipitates during recrystallization annealing[15]. The formation of the heterogeneous microstructure was observed exclusively at a true strain of ε=2.

[picture]

Fig. 4. Inverse pole figure images for annealing sample for CoNiFeMn675∘C/600 s and CoNiFeMn)95Mo5800∘C/1800 s with pole figures c) SEM-BSE image for ( CoNiFeMn)95Mo5 alloy and d) tensile test curves.

Fig. 4. Inverse pole figure images for annealing sample for C o N i F e M n 675 ∘ C / 600 s and C o N i F e M n ) 95 M o 5 800 ∘ C / 1800 s with pole figures c) SEM-BSE image for ( CoNiFeMn ) 95 M o 5 alloy and d) tensile test curves.

The presented pole figures indicate that the materials subjected to static recrystallization remain characterized by a strong crystallographic texture. In both cases, the texture is composed predominantly of the Goss {110}⟨001⟩, A {110}⟨111⟩, F {111}⟨112⟩, and cube {001}⟨100⟩ components. The presence of these texture components is expected to significantly influence the drawability of the investigated materials as well as the operative deformation mechanisms.
Fig. 4d presents the mechanical properties of the alloys selected for the deep-drawing test. The (CoNiFeMn)95Mo5 material exhibits significantly higher strength due to the presence of μ-phase precipitates as well as a stronger solid solution strengthening effect. However, its ductility is lower than that of CoNiFeMn . The yield strength, σ0.2, was estimated at 398±5MPa for the CoNiFeMn alloy and 545±10MPa for the Mo-containing alloy.

The annealed materials were subsequently subjected to the Erichsen cupping test (Fig. 5).

[picture]

Fig. 5. Samples after Erichsen test.

Fig. 5. Samples after Erichsen test.

The Erichsen cupping tests were conducted to evaluate the drawability of the investigated alloys. The CoNiFeMn alloy exhibited an average indentation depth (IE21) of 8.6 mm , indicating good formability under the applied test conditions. In contrast, the ( CoNiFeMn)95Mo5 alloy showed a significantly lower average IE21 value of 4.8 mm . This reduction in formability can be primarily attributed to the decreased ductility of the Mo-containing alloy, which is associated with the presence of molybdenum-rich intermetallic μ phase precipitates. These precipitates act as stress concentrators and hinder plastic flow, thereby limiting the material's ability to undergo uniform deformation during deep drawing.

Summary

The original version of this paper is available on https://www.scientific.net/SSP.388.245.pdf

This study presents a method for producing a heterogeneous microstructure in the (CoNiFeMn)1−X−Mox alloy system. Two alloys, CoNiFeMn and (CoNiFeMn)95Mo5, were investigated. The materials were subjected to cold rolling followed by recrystallization annealing.

The formation of a heterogeneous microstructure was successfully achieved in the materials deformed to a true strain of ε=2. In the CoNiFeMn alloy, the heterogeneous microstructure is characterized by bands of coarse grains surrounded by finer grains, whereas in the (CoNiFeMn)95Mo5 alloy, the presence of molybdenum-rich μ phase precipitates further contributes to the heterogeneity.

Formability tests demonstrated that the CoNiFeMn alloy exhibits good drawability, whereas the addition of molybdenum significantly reduces formability due to the presence of μ-phase precipitates. Nevertheless, the investigated materials show potential for deep drawing applications, depending on the required balance between strength and ductility

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The original version of this paper is available on https://www.scientific.net/SSP.388.245.pdf

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Keywords Index

A

Advanced High Strength Steels Global and Local Formability of 3rd  Generation Advanced High Strength Steels

Advanced High-strength Steel Influence of Hydrogen Content on the Ductile Rupture of Advanced High-Strength Steels

Al-mg-si Alloy Investigating the Formability of Aluminium Alloy Tubes by Rubber Bulge Testing

Al-polymer Laminate Composite Film Forming Limit Test of Asymmetric AI-Polymer Laminate Composite Film

Aluminium Alloy In Situ EBSD Characterisation of Anisotropic Creep Behaviour in a Thick AA2139 Part during in-Forming Artificial Ageing

Anisotropy In Situ EBSD Characterisation of Anisotropic Creep Behaviour in a Thick AA2139 Part during in-Forming Artificial Ageing; Experimental Investigation of the Influence of Load Direction Changes on Damage in Dual Phase Steel and Aluminum Alloy; Assessing Global and Local Formability from Tensile Tests with Anisotropy Consideration; Automated Testing and Calibration of Sheet Metal Models

Asymmetric Hole Expansion Test Asymmetric Hole Expansion Test (aHET) for Direct Measurement of Orientation-Dependent Tensile Fracture Strain

Autoform Springback Prediction of 6XXX-Series Aluminum: Effect of Hardening Model, Friction and Binder Loading

Automated Mechanical Testing Automated Testing and Calibration of Sheet Metal Models

B

Bendability Microstructural Modification and Bendability Improvement in 1300 MPa Steel via Induction Surface Heat Treatment

Bending Thickness Scaling of Forming Limit Curves

Blanking Effect of Tool Edge Geometry on Sheared Surface Formation in Blanking of Oxygen-Free Copper

C

Continuous Bending Under Tension Effect of Continuous Bending under Tension on Dual-Phase Steels

Creep-ageing In Situ EBSD Characterisation of Anisotropic Creep Behaviour in a Thick AA2139 Part during in-Forming Artificial Ageing

Crystal Plasticity Finite-element Method Crystal Plasticity Finite-Element Simulations of Cup Drawing of a ZX10 Magnesium Alloy Sheet

Cup Drawing Crystal Plasticity Finite-Element Simulations of Cup Drawing of a ZX10 Magnesium Alloy Sheet

D

Damage Experimental Investigation of the Influence of Load Direction Changes on Damage in Dual Phase Steel and Aluminum Alloy

Diffusible Hydrogen Content Influence of Hydrogen Content on the Ductile Rupture of Advanced High-Strength Steels

Dual Phase Steel Work Hardening of Dual Phase Steel in Subsequent Tensile Testing with Post-Necking Large Pre-Strain

Ductility Influence of Hydrogen Content on the Ductile Rupture of Advanced High-Strength Steels

E

Ebsd In Situ EBSD Characterisation of Anisotropic Creep Behaviour in a Thick AA2139 Part during in-Forming Artificial Ageing

Edge Crack Hole Expansion Performance Assessment of Quenched and Partitioning Steel Through a Fully Anisotropic Fracture Modeling

Edge Fracture Asymmetric Hole Expansion Test (aHET) for Direct Measurement of Orientation-Dependent Tensile Fracture Strain

Edge Stretching Global and Local Formability of 3rd  Generation Advanced High Strength Steels

Experimental Variability Stochastic Forming Limit Curves: Construction of a Probabilistic Tool for Failure Assessment

Extrusion Investigating the Formability of Aluminium Alloy Tubes by Rubber Bulge Testing

F

Failure Assessing Global and Local Formability from Tensile Tests with Anisotropy Consideration

Flc Thickness Scaling of Forming Limit Curves

Formability Theoretical and Experimental Forming Limit Evaluation of High-Strength Steels up to 1500 MPa; Investigating the Formability of Aluminium Alloy Tubes by Rubber Bulge Testing; Global and Local Formability of 3rd  Generation Advanced High Strength Steels; Assessing Global and Local Formability from Tensile Tests with Anisotropy Consideration; Necking Limits of Anisotropic Metal Sheets Using M-K Theory and Improved Plasticity Models; Effect of Continuous Bending under Tension on Dual-Phase Steels

Forming Limit Forming Limit Test of Asymmetric AI-Polymer Laminate Composite Film

Forming Limit Curve Thickness Scaling of Forming Limit Curves; Theoretical and Experimental Forming Limit Evaluation of High-Strength Steels up to 1500 MPa

Forming Limit Curves The Interaction of N - and R -Values and Total Elongation in Forming Limit Curves' Theories; Stochastic Forming Limit Curves: Construction of a Probabilistic Tool for Failure Assessment

Forming Limit Curves (flc) Evaluation and Comparison of a Few Methods for the Strain-Path Correction of FLCs

Fracture Experimental Investigation of the Influence of Load Direction Changes on Damage in Dual Phase Steel and Aluminum Alloy; Assessing Global and Local Formability from Tensile Tests with Anisotropy Consideration

Fracture Forming Limits Formability Test for Characterizing the Mode II-III Fracture Transition

Fracture Initiation Criterion Asymmetric Hole Expansion Test (aHET) for Direct Measurement of Orientation-Dependent Tensile Fracture Strain

Fully Anisotropic Fracture Model Hole Expansion Performance Assessment of Quenched and Partitioning Steel Through a Fully Anisotropic Fracture Modeling

G

Grain Misorientations In Situ EBSD Characterisation of Anisotropic Creep Behaviour in a Thick AA2139 Part during in-Forming Artificial Ageing

H

Hardening Automated Testing and Calibration of Sheet Metal Models

Hardness Microstructural Modification and Bendability Improvement in 1300 MPa Steel via Induction Surface Heat Treatment

Heterostructure Heterostructure High Entropy Alloys in the CoNiFeMn-Mo System Intended for Deep Drawing Applications

High Entropy Alloys Heterostructure High Entropy Alloys in the CoNiFeMn-Mo System Intended for Deep Drawing Applications

High-strenght Steels Theoretical and Experimental Forming Limit Evaluation of High-Strength Steels up to 1500 MPa

Hole Expansion Test Asymmetric Hole Expansion Test (aHET) for Direct Measurement of Orientation-Dependent Tensile Fracture Strain

I

In Situ Tests In Situ EBSD Characterisation of Anisotropic Creep Behaviour in a Thick AA2139 Part during in-Forming Artificial Ageing

Induction Heating Microstructural Modification and Bendability Improvement in 1300 MPa Steel via Induction Surface Heat Treatment

Inverse Model Calibration Automated Testing and Calibration of Sheet Metal Models

K

Kinematic Hardening Springback Prediction of 6XXX-Series Aluminum: Effect of Hardening Model, Friction and Binder Loading

L

Lightweight Design Global and Local Formability of 3rd  Generation Advanced High Strength Steels

Local Formability Hole Expansion Performance Assessment of Quenched and Partitioning Steel Through a Fully Anisotropic Fracture Modeling

M

Magnesium Alloy Crystal Plasticity Finite-Element Simulations of Cup Drawing of a ZX10 Magnesium Alloy Sheet

Marciniak-kuczynski Theory Necking Limits of Anisotropic Metal Sheets Using M-K Theory and Improved Plasticity Models

Mechanical Properties Heterostructure High Entropy Alloys in the CoNiFeMn-Mo System Intended for Deep Drawing Applications

Mechanical State Influence of Hydrogen Content on the Ductile Rupture of Advanced High-Strength Steels

Modeling Assessing Global and Local Formability from Tensile Tests with Anisotropy Consideration

Modified Marciniak Test Forming Limit Test of Asymmetric AI-Polymer Laminate Composite Film

N

Nakajima Thickness Scaling of Forming Limit Curves

Necking Limit Necking Limits of Anisotropic Metal Sheets Using M-K Theory and Improved Plasticity Models

O

Oxygen-free Copper Effect of Tool Edge Geometry on Sheared Surface Formation in Blanking of Oxygen-Free Copper

P

Particle-reinforced Aluminum Composite The Influence of Uphill Quench on the Microstructure and Properties of Particle-Reinforced Aluminum Matrix Composites

Plastic Anisotropy Necking Limits of Anisotropic Metal Sheets Using M-K Theory and Improved Plasticity Models

Polar Effective Plastic Strain (peps) Approach Forming Limit Test of Asymmetric AI-Polymer Laminate Composite Film

Post-necking Work Hardening of Dual Phase Steel in Subsequent Tensile Testing with Post-Necking Large Pre-Strain

Pre-strain Work Hardening of Dual Phase Steel in Subsequent Tensile Testing with Post-Necking Large Pre-Strain

Punching Effect of Tool Edge Geometry on Sheared Surface Formation in Blanking of Oxygen-Free Copper

Q

Qp Steel Hole Expansion Performance Assessment of Quenched and Partitioning Steel Through a Fully Anisotropic Fracture Modeling

R

Residual Stress A Comparative Study of Multiple Cutting Methods on Thin Ferritic Steel for Tensile Test Experiment; The Influence of Uphill Quench on the Microstructure and Properties of Particle-Reinforced Aluminum Matrix Composites

Rubber Bulge Testing Investigating the Formability of Aluminium Alloy Tubes by Rubber Bulge Testing

S

Sheet Experimental Investigation of the Influence of Load Direction Changes on Damage in Dual Phase Steel and Aluminum Alloy

Sheet Anisotropy Asymmetric Hole Expansion Test (aHET) for Direct Measurement of Orientation-Dependent Tensile Fracture Strain

Sheet Compression Test Formability Test for Characterizing the Mode II-III Fracture Transition

Sheet Metal Forming Global and Local Formability of 3rd  Generation Advanced High Strength Steels; Springback Prediction of 6XXX-Series Aluminum: Effect of Hardening Model, Friction and Binder Loading

Sheet-bulk Forming Formability Test for Characterizing the Mode II-III Fracture Transition

Soft Surface Microstructural Modification and Bendability Improvement in 1300 MPa Steel via Induction Surface Heat Treatment

Static Recrystallization Heterostructure High Entropy Alloys in the CoNiFeMn-Mo System Intended for Deep Drawing Applications

Steel Effect of Continuous Bending under Tension on Dual-Phase Steels

Stochastic Modeling Stochastic Forming Limit Curves: Construction of a Probabilistic Tool for Failure Assessment

Strain-aging Work Hardening of Dual Phase Steel in Subsequent Tensile Testing with Post-Necking Large Pre-Strain

Strain-path Dependence; Correction Methods Evaluation and Comparison of a Few Methods for the Strain-Path Correction of FLCs

T

Tensile Test Work Hardening of Dual Phase Steel in Subsequent Tensile Testing with Post-Necking Large Pre-Strain; The Interaction of N - and R -Values and Total Elongation in Forming Limit Curves' Theories; A Comparative Study of Multiple Cutting Methods on Thin Ferritic Steel for Tensile Test Experiment

Theoretical Calculations The Interaction of N - and R -Values and Total Elongation in Forming Limit Curves' Theories

Thickness Dependence Thickness Scaling of Forming Limit Curves

Thin Metal Sheet A Comparative Study of Multiple Cutting Methods on Thin Ferritic Steel for Tensile Test Experiment

Tool Edge Effect of Tool Edge Geometry on Sheared Surface Formation in Blanking of Oxygen-Free Copper

Triboform Springback Prediction of 6XXX-Series Aluminum: Effect of Hardening Model, Friction and Binder Loading

Twinning Crystal Plasticity Finite-Element Simulations of Cup Drawing of a ZX10 Magnesium Alloy Sheet

U

Ultrahigh-strength Steel Microstructural Modification and Bendability Improvement in 1300 MPa Steel via Induction Surface Heat Treatment

Uphill Quench The Influence of Uphill Quench on the Microstructure and Properties of Particle-Reinforced Aluminum Matrix Composites

Authors Index

A

Abdelghany, A. Microstructural Modification and Bendability Improvement in 1300 MPa Steel via Induction Surface Heat Treatment

Ahn, T. Forming Limit Test of Asymmetric AI-Polymer Laminate Composite Film

Aksen, T. Hole Expansion Performance Assessment of Quenched and Partitioning Steel Through a Fully Anisotropic Fracture Modeling

Almeida, I. Formability Test for Characterizing the Mode II-III Fracture Transition

Alves, R. Effect of Continuous Bending under Tension on Dual-Phase Steels

Ameline, O. Stochastic Forming Limit Curves: Construction of a Probabilistic Tool for Failure Assessment

Anderson, D. Springback Prediction of 6XXX-Series Aluminum: Effect of Hardening Model, Friction and Binder Loading

Arciero, J. Springback Prediction of 6XXX-Series Aluminum: Effect of Hardening Model, Friction and Binder Loading

Aretz, H. Thickness Scaling of Forming Limit Curves

Azzouni, T. A Comparative Study of Multiple Cutting Methods on Thin Ferritic Steel for Tensile Test Experiment

B

Balan, T. Evaluation and Comparison of a Few Methods for the Strain-Path Correction of FLCs

Beres, G. The Interaction of N - and R -Values and Total Elongation in Forming Limit Curves' Theories; Theoretical and Experimental Forming Limit Evaluation of High-Strength Steels up to 1500 MPa

Borbely, R. The Interaction of N - and R -Values and Total Elongation in Forming Limit Curves' Theories; Theoretical and Experimental Forming Limit Evaluation of High-Strength Steels up to 1500 MPa

Butcher, C. Springback Prediction of 6XXX-Series Aluminum: Effect of Hardening Model, Friction and Binder Loading

C

Carpio, M. Global and Local Formability of 3rd  Generation Advanced High Strength Steels

Chang, Y. The Influence of Uphill Quench on the Microstructure and Properties of Particle-Reinforced Aluminum Matrix Composites

Cheong, K. Springback Prediction of 6XXX-Series Aluminum: Effect of Hardening Model, Friction and Binder Loading

Cichocki, K. Heterostructure High Entropy Alloys in the CoNiFeMn-Mo System Intended for Deep Drawing Applications

D

Decultot, N. Automated Testing and Calibration of Sheet Metal Models

Dubos, P. In Situ EBSD Characterisation of Anisotropic Creep Behaviour in a Thick AA2139 Part during in-Forming Artificial Ageing

E

Escorza, I. Springback Prediction of 6XXX-Series Aluminum: Effect of Hardening Model, Friction and Binder Loading

F

Frómeta, D. Global and Local Formability of 3rd  Generation Advanced High Strength Steels

G

Gautier, C. In Situ EBSD Characterisation of Anisotropic Creep Behaviour in a Thick AA2139 Part during in-Forming Artificial Ageing

Germain, L. Influence of Hydrogen Content on the Ductile Rupture of Advanced High-Strength Steels

Girault, B. In Situ EBSD Characterisation of Anisotropic Creep Behaviour in a Thick AA2139 Part during in-Forming Artificial Ageing

Gloaguen, D. In Situ EBSD Characterisation of Anisotropic Creep Behaviour in a Thick AA2139 Part during in-Forming Artificial Ageing

Grolleau, V. Asymmetric Hole Expansion Test (aHET) for Direct Measurement of Orientation-Dependent Tensile Fracture Strain

Guo, N. The Influence of Uphill Quench on the Microstructure and Properties of Particle-Reinforced Aluminum Matrix Composites

H

Haiko, O. Microstructural Modification and Bendability Improvement in 1300 MPa Steel via Induction Surface Heat Treatment

Hama, T. Work Hardening of Dual Phase Steel in Subsequent Tensile Testing with Post-Necking Large Pre-Strain; Crystal Plasticity Finite-Element Simulations of Cup Drawing of a ZX10 Magnesium Alloy Sheet

Haoues, Y. Automated Testing and Calibration of Sheet Metal Models

Hartmann, C. A Comparative Study of Multiple Cutting Methods on Thin Ferritic Steel for Tensile Test Experiment

Hegele, P. A Comparative Study of Multiple Cutting Methods on Thin Ferritic Steel for Tensile Test Experiment

Hoikkaniemi, A. Microstructural Modification and Bendability Improvement in 1300 MPa Steel via Induction Surface Heat Treatment

Hokimoto, K. Work Hardening of Dual Phase Steel in Subsequent Tensile Testing with Post-Necking Large Pre-Strain

Honda, Y. Work Hardening of Dual Phase Steel in Subsequent Tensile Testing with Post-Necking Large Pre-Strain

Hou, Y. Necking Limits of Anisotropic Metal Sheets Using M-K Theory and Improved Plasticity Models

Howeyze, M. Springback Prediction of 6XXX-Series Aluminum: Effect of Hardening Model, Friction and Binder Loading

I

Ichiyanagi, Y. Effect of Tool Edge Geometry on Sheared Surface Formation in Blanking of Oxygen-Free Copper

Iona, L. A Comparative Study of Multiple Cutting Methods on Thin Ferritic Steel for Tensile Test Experiment

J

Jang, T. Forming Limit Test of Asymmetric AI-Polymer Laminate Composite Film

Jedidi, M. Stochastic Forming Limit Curves: Construction of a Probabilistic Tool for Failure Assessment

K

Kaijalainen, A. Microstructural Modification and Bendability Improvement in 1300 MPa Steel via Induction Surface Heat Treatment

Kannan, K. Springback Prediction of 6XXX-Series Aluminum: Effect of Hardening Model, Friction and Binder Loading

Kecskes, B. The Interaction of N - and R -Values and Total Elongation in Forming Limit Curves' Theories

Keresztes, R. The Interaction of N - and R -Values and Total Elongation in Forming Limit Curves' Theories; Theoretical and Experimental Forming Limit Evaluation of High-Strength Steels up to 1500 MPa

Korkolis, Y. Experimental Investigation of the Influence of Load Direction Changes on Damage in Dual Phase Steel and Aluminum Alloy; Necking Limits of Anisotropic Metal Sheets Using M-K Theory and Improved Plasticity Models

L

Labergere, C. Stochastic Forming Limit Curves: Construction of a Probabilistic Tool for Failure Assessment

Lemoine, X. Evaluation and Comparison of a Few Methods for the Strain-Path Correction of FLCs

Lennemann, P. Experimental Investigation of the Influence of Load Direction Changes on Damage in Dual Phase Steel and Aluminum Alloy

Leppin, C. Springback Prediction of 6XXX-Series Aluminum: Effect of Hardening Model, Friction and Binder Loading

Li, G. Assessing Global and Local Formability from Tensile Tests with Anisotropy Consideration

Li, Z. Hole Expansion Performance Assessment of Quenched and Partitioning Steel Through a Fully Anisotropic Fracture Modeling; The Influence of Uphill Quench on the Microstructure and Properties of Particle-Reinforced Aluminum Matrix Composites

Lian, J. Hole Expansion Performance Assessment of Quenched and Partitioning Steel Through a Fully Anisotropic Fracture Modeling; Assessing Global and Local Formability from Tensile Tests with Anisotropy Consideration

Lisiecka-Graca, P. Heterostructure High Entropy Alloys in the CoNiFeMn-Mo System Intended for Deep Drawing Applications

Liu, B. The Influence of Uphill Quench on the Microstructure and Properties of Particle-Reinforced Aluminum Matrix Composites

M

Magrinho, J. Formability Test for Characterizing the Mode II-III Fracture Transition

Mainguy, G. Evaluation and Comparison of a Few Methods for the Strain-Path Correction of FLCs

Martins, P. Formability Test for Characterizing the Mode II-III Fracture Transition

Matsuno, T. Work Hardening of Dual Phase Steel in Subsequent Tensile Testing with Post-Necking Large Pre-Strain

Miyazawa, N. Crystal Plasticity Finite-Element Simulations of Cup Drawing of a ZX10 Magnesium Alloy Sheet

Mohr, D. Asymmetric Hole Expansion Test (aHET) for Direct Measurement of Orientation-Dependent Tensile Fracture Strain

Moon, C. Investigating the Formability of Aluminium Alloy Tubes by Rubber Bulge Testing

Muszka, K. Heterostructure High Entropy Alloys in the CoNiFeMn-Mo System Intended for Deep Drawing Applications

N

Ndiritu, C. Automated Testing and Calibration of Sheet Metal Models

P

Pan, R. The Influence of Uphill Quench on the Microstructure and Properties of Particle-Reinforced Aluminum Matrix Composites

Payà, A. Global and Local Formability of 3rd  Generation Advanced High Strength Steels

Peltier, F. Influence of Hydrogen Content on the Ductile Rupture of Advanced High-Strength Steels

Persoons, A. Stochastic Forming Limit Curves: Construction of a Probabilistic Tool for Failure Assessment

Pokka, A. Microstructural Modification and Bendability Improvement in 1300 MPa Steel via Induction Surface Heat Treatment

Pérot, R. Asymmetric Hole Expansion Test (aHET) for Direct Measurement of Orientation-Dependent Tensile Fracture Strain

R

Rollin, M. In Situ EBSD Characterisation of Anisotropic Creep Behaviour in a Thick AA2139 Part during in-Forming Artificial Ageing

S

Sagong, C. Forming Limit Test of Asymmetric AI-Polymer Laminate Composite Film

Santos, R. Effect of Continuous Bending under Tension on Dual-Phase Steels

Sariyarlioglu, E. Investigating the Formability of Aluminium Alloy Tubes by Rubber Bulge Testing

Shimizu, K. Work Hardening of Dual Phase Steel in Subsequent Tensile Testing with Post-Necking Large Pre-Strain

Shiratori, T. Effect of Tool Edge Geometry on Sheared Surface Formation in Blanking of Oxygen-Free Copper

Silva, M. Formability Test for Characterizing the Mode II-III Fracture Transition

T

Tatsuhiko, A. Effect of Tool Edge Geometry on Sheared Surface Formation in Blanking of Oxygen-Free Copper

Tekkaya, A. Experimental Investigation of the Influence of Load Direction Changes on Damage in Dual Phase Steel and Aluminum Alloy

Thokala, S. Springback Prediction of 6XXX-Series Aluminum: Effect of Hardening Model, Friction and Binder Loading

Thuillier, S. Influence of Hydrogen Content on the Ductile Rupture of Advanced High-Strength Steels; Automated Testing and Calibration of Sheet Metal Models

Toth-Nagy, B. The Interaction of N - and R -Values and Total Elongation in Forming Limit Curves' Theories

Trask, M. Necking Limits of Anisotropic Metal Sheets Using M-K Theory and Improved Plasticity Models

V

Vincze, G. Effect of Continuous Bending under Tension on Dual-Phase Steels

Vincze, M. Effect of Continuous Bending under Tension on Dual-Phase Steels

Volk, W. A Comparative Study of Multiple Cutting Methods on Thin Ferritic Steel for Tensile Test Experiment

Vucko, F. Influence of Hydrogen Content on the Ductile Rupture of Advanced High-Strength Steels

W

Wankhede, A. Springback Prediction of 6XXX-Series Aluminum: Effect of Hardening Model, Friction and Binder Loading

Warski, T. Heterostructure High Entropy Alloys in the CoNiFeMn-Mo System Intended for Deep Drawing Applications

Watanabe, K. Effect of Tool Edge Geometry on Sheared Surface Formation in Blanking of Oxygen-Free Copper

Welo, T. Investigating the Formability of Aluminium Alloy Tubes by Rubber Bulge Testing

Wójcik, A. Heterostructure High Entropy Alloys in the CoNiFeMn-Mo System Intended for Deep Drawing Applications

Y

Yoon, J. Forming Limit Test of Asymmetric AI-Polymer Laminate Composite Film

Z

Zeng, Y. The Influence of Uphill Quench on the Microstructure and Properties of Particle-Reinforced Aluminum Matrix Composites