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ISBN print: 978-3-0364-0987-0
ISBN eBook: 978-3-0364-1987-9
DOI book: 10.4028/b-r53Jgv
© 2026 Trans Tech Publications Ltd, All Rights Reserved

Material Behaviour Modelling

Edited by
Prof. Anne-Marie Habraken
Dr. Emin Semih Perdahcioglu

Material Behaviour Modelling

Special topic volume with invited peer-reviewed papers only

Edited by

Prof. Anne-Marie Habraken
and Dr. Emin Semih Perdahcioglu

Scientific.Net

Copyright © 2026 Trans Tech Publications Ltd, Switzerland

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Solid State Phenomena, ISSN: 1662-9779, Vol. 390, pp -1-0

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

Accepted:
2026-04-15

Online:
2026-04-17

Preface

The original version of this paper is available on https://www.scientific.net/SSP.390.-1.pdf

Constitutive Modelling for any Material: metals like Al, steel, Ti, Mg... but also glass, wood, food, textile, composite..., covering complex loading paths, high strain rates, high temperatures, cyclic loading, etc. are described and applied or identified, if it targets forming processes.
The focus is:

APPROACHES BRIDGING THE SCALES: from atomistic to macroscopic scale able to explain formability mechanism; hardening behaviour, rate sensitivity, anisotropy, toughness, phase transformation, etc.

FRACTURE PREDICTION: by continuum damage mechanics, fracture criteria, crack propagation etc.

EXPERIMENTAL PROCEDURE of model parameter identification, or physical experimental method applied for model validation.
Modeling work with experimental inspiration, validation, and verification are presented in this issue.

Editors

The original version of this paper is available on https://www.scientific.net/SSP.390.-1.pdf

Anne Marie Habraken
University of Liege
Semih Perdahcioglu
University of Twente

Co-editors

The original version of this paper is available on https://www.scientific.net/SSP.390.-1.pdf

Lionel Leotoing
INSA Rennes
Sandra Monteiro
University of Coimbra
Lian Junhe
RWTH Aachen University

Table of Contents

Preface

Improved Mapping of Yield Locus Distortion through a Strain-Dependent Calibration of the Barlat YId2000-2d Yield Criterion

Marcel Rentz, Marion Merklein

A Novel Approach for Determining the Bauschinger Effect in Dual-Phase Steel Sheets with Tensile-Compression Tests

Adam Häckel, David Naumann, Marion Merklein

Numerical Quantification of Structure-Property Relations in DP800 - Influence of Phase Material Models in RVE

Niklas C. Fehlemann, Sebastian Münstermann

Empirical Modelling of the Cold Crystallization of PLA

Ghinwa Julie El Hajj Sleiman, Ewen Labalec, Gaël Colomines

Laser-Induced Solid-State Foaming of HIPed Ti6Al4V-ELI Alloy

Pasquale Guglielmi, Angela Cusanno, Antonio Piccininni, Emanuele Fulco, Donato Sorgente, Gianfranco Palumbo

1D Stress Evolution Prediction via Thermodynamically-informed Neural Networks

Alireza Fallahnejad, Emin Semih Perdahcioglu, Ton van den Boogaard

Deriving Macroscopic Damage Properties by Micromechanical Modeling of DP1000

Maximilian Neite, Sebastian Münstermann

Texture Data Preparation for Finite Element Simulations of Puncture Tests: Insights from the VPSC Approach

Alois C. Ott, Leo Schwarzmeier, Johannes Kronsteiner, Nikolaus P. Papenberg, Thomas Antretter

Modeling Transient Temperature-Driven Microstructure Evolution in Inconel 718 during Hot Forming

Nadine Elekyabi, Holger Brüggemann, Emad Scharifi, Junhe Lian

Quantifying Grain Boundary and Orientation Effects on Copper under Compression: A Crystal Plasticity Finite Element Approach

Muhammet Batuhan Paçacı, Haluk Darendeliler

Experimental Study on the Relationship between Cooling Rate, Secondary Dendrite Arm Spacing and Tensile Properties in Aluminum Casting

Taegyun Ahn, Juwon Lee, Janghoon Lee, Jeong Whan Yoon

Predictive Modeling of 42CrMo4 Hardness after Two-Step Heat Treatment Using Symbolic Regression

Johann Albers, Fabian Dittrich, Sebastian Härtel

How to Take the Lüders Plateau into Account in the Formability Prediction of Steel Sheets

Xavier Lemoine

Microscale to Macroscale Analyses of Superplasticity of Ti-6242S Titanium Alloy

Muritala Arowolo, Vincent Velay, Vanessa Vidal, Moukrane Dehmas

Study on the Effects of Grain Shape, Size and Size Distribution on the Mechanical Behavior of Metals

Ruxandra Bocan, Emin Semih Perdahcioglu, Ton van den Boogard

Investigation of Friction Coefficient at Stack Compression Tests

Martin László Kölüs, Imre Nagy-Gombai, Gábor Kalácska, Gábor Jozsef Beres

Mechanical Performance of a Steel-Polymer Biphasic Lattice Structure

Edoardo Mancini, Anastasia Ciccarella, Andrea Di Giuliano, Lino Micarelli, Giuseppe Dell'Avvocato, Katia Gallucci

Further Developments of Hill-Gotoh Theory of Anisotropic Sheet Metal Plasticity

Wei Tong

A Compact Micromechanical Model for the Elastic-Viscoplastic Deformability of PLA-Hemp Biocomposites

Ndeye Niang, Sami Holopainen, Thierry Barrière, Vincent Placet, Xavier Gabrion

Modeling the Influence of Material Variabilities on the Forming and Failure Behavior of PP/PE Cores in Metal-Polymer-Metal Sandwich Sheets

Johannes Pusicha, Johannes Buhl

Keywords Index

Authors Index

The original version of this eBook is available on https://www.scientific.net/book/material-behaviour-modelling/978-3-0364-1987-9

Solid State Phenomena, ISSN: 1662-9779, Vol. 390, pp 1-9

doi: 10.4028/p-v2sNIf

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

Submitted:
2025-12-15

Revised:
2026-02-13

Accepted:
2026-02-21

Online:
2026-04-17

Improved Mapping of Yield Locus Distortion through a Strain-Dependent Calibration of the Barlat YId2000-2d Yield Criterion

The original version of this paper is available on https://www.scientific.net/SSP.390.1.pdf

Marcel Rentz 1,a*, Marion Merklein 1, b

¹Institute of Manufacturing Technology, Friedrich-Alexander-Universität Erlangen-Nürnberg, Egerlandstraße 13, 91058 Erlangen, Germany

a* marcel.rentz@fau.de, b  marion.merklein@fau.de

Keywords: material modeling, distortional hardening, plane strain, shear stress.

Abstract

Accurate modeling of the real material behavior is fundamental to improve the accuracy of the finite element analysis (FEA) of sheet metal forming processes. Classical material models such as Hill'48 or Barlat Yld2000-2d do not consider the material behavior under plane strain and shear, even though these states are the primary cause of failures observed in sheet metal forming. Moreover, yield criteria are conventionally calibrated at the onset of plastic deformation to determine the initial yield locus. Isotropic hardening is subsequently assumed, based on the flow curve under uniaxial tension. However, some modern sheet metals exhibit a pronounced distortional hardening behavior, which cannot be sufficiently mapped by the conventional modeling strategy. Hence, this contribution aims to improve the mapping of the yield locus distortion by considering the plane strain and shear stress states and by performing the parameter calibration at higher plastic strains. Hereby, the yield locus exponent of the Barlat Yld2000-2d is adapted in order to accurately map the material behavior under plane strain or shear. Moreover, the influence of a strain-dependent calibration of the yield locus on the mapping accuracy is investigated. Two materials, AA5182 and DP600, are being examined. It is observed that the consideration of the plane strain state leads to a reduction of the yield locus exponent while the consideration of the shear stress state is accompanied with an increase of the yield locus exponent.

Introduction

The original version of this paper is available on https://www.scientific.net/SSP.390.1.pdf

Over the past few decades, FEA has become an indispensable tool for designing and improving sheet metal forming processes. Numerical simulations significantly reduce development time and costs in industrial manufacturing by predicting stresses, sheet thickness distributions and potential failure zones. Therefore, the mapping accuracy of forming simulations is essential and depends on the boundary and contact conditions, as well as the material model. In particular, the chosen yield criterion has a significant impact on numerical prediction accuracy. Increasing computing power and improved capabilities in material characterization enable the use of advanced material models. Compared to classical yield criteria, advanced models provide a better mapping of real material behavior. Nevertheless, modern yield criteria require a large number of input parameters, which leads to an increase in characterization effort and longer computing times. For this reason, classical material models, such as Hill'48 or Barlat Yld2000-2d [1], are still widely used. These yield criteria have the disadvantage of not considering complex load conditions, such as plane strain or shear stress.

In sheet metal forming, both mentioned states can result in failure. Under plane strain conditions, the forming limit potential is at minimum. This can be explained by the material flowing solely from the thickness direction, resulting in early necking of the material [2]. Particularly in deep drawing, plane strain conditions lead to localized necking in the transition area between the bottom and skirt of a deep drawn part, which can cause bottom fractures. According to Xavier et al. [3], more than 80% of all failures in formed automotive components occur under near-plane strain conditions Although pure shear does not cause thinning of the sheet, component failure can occur under shear conditions, notably for dual-phase steels [4]. Shear-induced failure can be observed in bending operations [5] and in deep drawing processes in the transition region between the flange and skirt [6]. In summary, the plane strain and shear stress states both account for a significant proportion of
material failure in sheet metal forming processes. Consequently, accurate consideration of the material behavior in these states is highly relevant. However, in the classical yield criterion Barlat Yld2000-2d [1], the material behavior under plane strain or shear is not considered in the calibration process. Rather, it is defined by two grid points and the shape of the yield locus respectively. As a result, the mapping of the aforementioned states depends on the eight input parameters of the Yld2000-2d and the choice of the yield locus exponent m , which is heuristically selected based on values found in the literature. In the case of face-centered cubic (fcc) materials, a value of m=8 is recommended, while for body-centered cubic (bcc) metals, a value of m=6 is usually chosen [7].

Another challenge for the conventional Yld2000-2d material model is the hardening behavior of certain modern materials. Conventionally, yield criteria are calibrated at the onset of plastic deformation to determine the initial yield locus. Isotropic hardening is then assumed, based on the flow curve under uniaxial tension. However, it is not possible to accurately map the anisotropic or distortional hardening behavior of modern materials with this conventional modeling strategy. To overcome this challenge, Wang et al. [8] used an approach in which the anisotropy coefficients α1 to α8 of the Yld2000-2d are varied as a function of plastic strain using sixth-degree polynomial functions. By comparing the resulting yield locus with data points, determined in a biaxial tensile test with cruciform specimen and based on a deep drawing simulation of a cylindrical cup, it was possible to achieve a higher mapping accuracy compared to the conventional Yld2000-2d. To consider the evolution of the Lankford coefficient, Küsters and Brosius [9] replaced the yield locus exponent with a strain-dependent Bézier function. Furthermore, Rentz and Merklein [10] adapted the yield locus exponent of the Yld2000-2d yield criterion to accurately map the material behavior under plane strain or shear. This approach improved the mapping accuracy of a deep drawing process. However, this method is limited to a fixed plastic strain.

Accurate mapping of the material behavior under plane strain and shear is important for improving failure prediction. Thus, this study investigates a strain-dependent calibration of the Yld2000-2d yield criterion, whereby either the plane strain or shear stress state is considered by adapting the yield locus exponent. Furthermore, the dependence of the yield locus exponent of the plastic strain is examined.

Methodical Approach

The original version of this paper is available on https://www.scientific.net/SSP.390.1.pdf

In this contribution, the two commonly used materials AA5182-O and DP600, with a nominal sheet thickness of t0=1.0 mm are used. The naturally hard aluminum alloy AA5182 offers a good corrosion resistance and weldability. The combination of a high strength and good formability makes the aluminum alloy suited for cold forming. AA5182 is often used for car body parts and bumpers. The dual-phase steel DP600 shows a very high strength combined with a good formability due to its specially developed microstructure. These properties make this steel grade particularly suitable for manufacturing of car body parts that require high energy absorption in the event of an accident. DP600 is consequently ideal for door reinforcements and safety cells.

Identifying the parameters of the conventional Barlat Yld2000-2d yield criterion requires the characterization of two stress states. The hardening behavior under uniaxial tension and the Lankford coefficients are characterized in the tensile test according to DIN EN ISO 6892-1. Thereby, tensile specimens are tested in, diagonal to and transverse to the rolling direction. The flow behavior under equi-biaxial tension is identified in the hydraulic bulge test according to DIN EN ISO 16808. Moreover, the strain-dependent evolution of the biaxial anisotropy rb is determined in the layer compression test according to Lenzen and Merklein [11]. The specimen stack is compressed by 50% of its initial height. To adapt the Yld2000-2d yield criterion, the principal stress components under plane strain and shear need to be classified. Therefore, hydraulic bulge tests with an elliptical die are conducted according to [12] to characterize the material behavior under near plane strain conditions. Additionally, shear tests according to [13] are carried out. For all characterization tests, the equivalent strain rate of 0.00401 s for the dual-phase steel according to SEP 1240 and the equivalent strain rate of 0.00671 s for the aluminum alloy according to VDA 239-300 is selected.

One challenge of the commonly used Yld2000-2d yield criterion is that the material behavior under plane strain or shear condition cannot be accurately mapped, as these are not directly implemented in
the calibration process. Rather, these states are obtained by interpolation between two grid points respectively, which causes a deviation from the actual material behavior under plane strain or shear. However, the yield locus exponent m affects the shape of the modeled yield locus. The plane strain area is particularly sensitive to this, whereby a change in the geometry of the yield locus also occurs in the shear area. Consequently, the yield locus exponent is adapted in order to integrate the stress point of the plane strain state or the shear stress point respectively. The extension of the Yld2000-2d material model is conducted in Matlab 2023 according to [10]. However, the identification of the yield loci is not limited to a fixed equivalent plastic strain of εpl,eq=0.05. In this contribution, the parameter calibration is performed within the range covered by the experimental data with a step size of Δεpl,eq=0.025. Figure 1 shows the approach to consider further material data, such as the shear stress, in the Yld2000-2d yield criterion.

[picture]

Fig. 1. Approach to the extension of the Yld2000-2d with further material information by varying the yield locus exponent m .

Fig. 1. Approach to the extension of the Yld2000-2d with further material information by varying the yield locus exponent m .

Material Characterization

The original version of this paper is available on https://www.scientific.net/SSP.390.1.pdf

As described above, different characterization tests are conducted to investigate the hardening behavior of the two materials AA5182 and DP600. In Figure 2, the flow behavior under uniaxial tension in dependence of the rolling direction can be observed.

[picture]

Fig. 2. Flow curves under uniaxial tension for AA5182 and DP600.

Fig. 2. Flow curves under uniaxial tension for AA5182 and DP600.

To guarantee statistical reliability, three specimens ( N=3 ) are tested for each rolling direction. For AA5182, the flow curves are on a similar level. Therefore, the aluminum alloy exhibits a pronounced isotropic material behavior under uniaxial tension. For DP600, the yield stresses under a load diagonal and transverse to the rolling direction, are comparable, amounting to 419.9 MPa and 427.7 MPa , whereas the yield stress in rolling direction is significantly lower at 390.9 MPa . However, the dual-phase steel shows a higher work hardening potential at 45∘ to the rolling direction compared to 0∘ and 90∘. Hence, anisotropic hardening can be identified for DP600 in case of uniaxial tensile loading.

Moreover, to ensure the comparison of the flow behavior under different strain and stress states, the calculation of the von Mises equivalent plastic strain and stresses are necessary. In Figure 3, the stress state-dependent equivalent flow curves for the aluminum alloy AA5182 and dual-phase steel DP600 are depicted.

[picture]

Fig. 3. Equivalent flow curves of the aluminum alloy AA5182 and the dual-phase steel DP600 in dependence of the stress state.

Fig. 3. Equivalent flow curves of the aluminum alloy AA5182 and the dual-phase steel DP600 in dependence of the stress state.

In case of both materials, it can be observed that the failure of the material occurs at lower equivalent plastic strains for uniaxial tension and plane strain compared to equi-biaxial tension and shear. For AA5182, isotropic hardening behavior is evident at the onset of plastic deformation. This can be observed by a similar flow behavior independent of the stress or strain state. Though, the hardening potential under shear deviates from the other strain and stress states, starting from an equivalent plastic strain of 0.1 . As a consequence, the aluminum alloy exhibits anisotropic hardening behavior at higher plastic strains. In contrary, the dual-phase steel DP600 shows nearly isotropic material behavior at lower and higher strains.

It can be summarized that the aluminum alloy AA5182 shows isotropic behavior under uniaxial tension, while a slight anisotropic, stress-state dependent hardening behavior can be observed. For the dual-phase steel DP600, a pronounced anisotropic material behavior under uniaxial tension is determined. However, a stress-state dependent hardening behavior cannot be identified.

The resulting equivalent flow curves can subsequently be used in order to calibrate the conventional Yld2000-2d yield criterion. Table 1 shows the necessary parameters in order to calculate the yield locus using the conventional modeling strategy. Hereby, a yield locus exponent of m=8 is selected for the aluminum alloy, while m=6 is chosen for the dual-phase steel. For the parameter identification, an equivalent plastic strain of εpl,eq=0.05 is selected. This can be explained due to the incorrect determination of the radii of curvature in both hydraulic bulge tests, resulting in high standard deviations of the flow stress at low equivalent plastic strains. Consequently, the onset of plastic deformation under equi-biaxial and plane strain conditions cannot be accurately identified.

Table 1. Parameters for the conventional identification of the Yld2000-2d at an equivalent plastic strain of εpl,eq=0.05.

Yld2000-2d | σ₀ | σ₄₅ | σ₉₀ | σb | r₀ | r₄₅ | r₉₀ | rb | mconv
AA5182 | 214.98 | 210.73 | 205.65 | 210.30 | 0.762 | 0.610 | 0.605 | 1.108 | 8
DP600 | 560.05 | 632.55 | 572.85 | 586.19 | 0.775 | 1.032 | 1.076 | 0.720 | 6

Yield Locus Exponent Evolution and Yield Locus Distortion

The original version of this paper is available on https://www.scientific.net/SSP.390.1.pdf

Conventionally, the parameter identification for yield criteria is performed at the onset of plastic deformation. The yield locus is then expanded isotropically based on the flow curve under uniaxial tension. However, distortionally hardening materials pose a challenge, as they cannot be accurately modeled assuming isotropic hardening. Therefore, a strain-dependent calibration of the Yld2000-2d yield criterion is investigated. Hereby, the conventional Yld2000-2d is adapted by integrating either the stress point of the plane strain state or the shear stress point respectively according to [10]. This is being achieved by adjusting the yield locus exponent m . The parameter calibration is conducted, starting from an equivalent plastic strain of εpl,eq=0.05 until the upper limit of the available experimental data, with a step size of Δεpl,eq=0.025.

In Figure 4, the strain-dependent evolution of the yield locus exponent for the investigated aluminum alloy AA5182 and dual-phase steel DP600 is depicted.

[picture]

Fig. 4. Strain-dependent evolution of the yield locus exponent.

Fig. 4. Strain-dependent evolution of the yield locus exponent.

It should be noted that the parameter identification for both materials is limited to the achieved equivalent plastic strains under uniaxial tension. Due to that, the upper limit for the aluminum alloy is at an equivalent plastic strain Epl,eq =0.20, while the identification is limited to Epl,eq =0.15 in case of the dual-phase steel. For both materials, it can be identified that a strain-dependent consideration of the plane strain state leads to a reduction of the yield locus exponent m , which leads to a rounder yield locus geometry. This is due to a comparable hardening potential under equi-biaxial and plane strain conditions, which results in a rounder shape of the yield locus in the biaxial tension area. Considering the shear stress state in the Yld2000-2d yield criterion leads to an increase of the yield locus exponent in dependence of the equivalent plastic strain, which is accompanied with an increasing flattening of the corresponding yield locus. This can be explained by the lower work hardening potential under shear conditions, which is more pronounced for the aluminum alloy. Consequently, the yield locus exponent increases to a greater extent for the aluminum alloy than for the dual-phase steel.

The adapted strain-dependent yield loci can be calculated, using the determined material parameters in Figure 3 and the respective yield locus exponents. For a better overview, Figure 5 only considers the yield loci with a step size of Δεpl,eq=0.05.

[picture]

Fig. 5. Yield locus distortion for the aluminum alloy AA5182.

Fig. 5. Yield locus distortion for the aluminum alloy AA5182.

The black yield locus represents the common modeling strategy with a conventional choice of the yield locus exponent of m=8 for aluminum alloys. Assuming isotropic hardening of the material, the normalized yield locus remains unchanged in case of plastic deformation. In the left diagram of Figure 5 , the conventional yield locus and the adapted yield locus determined at an equivalent plastic strain of εpl,eq=0.05 are in good agreement. However, it can be observed that an increasing plastic strain leads to a distortion of the yield locus when considering the plane strain state in the Yld2000-2d yield criterion. Using the conventional approach leads to premature reaching of the forming limit due to the deviation between numerically calculated and real material behavior. Hence, the conventional modeling strategy is only able to accurately map the beginning of plastic deformation under plane strain conditions in case of the aluminum alloy. A contrary result is observed in the right diagram of Figure 5. Hereby, the onset of plastic deformation under shear conditions cannot be mapped precisely with the conventional modeling strategy. Nevertheless, the shape of the conventionally identified yield locus and the yield loci determined at higher plastic strains ( εpl,eq≥0.10 ) are similar. Therefore, the conventional modeling strategy is only suitable for considering the shear stress state at higher plastic strains for AA5182.

Similar observations can be concluded for the dual-phase steel DP600. As can be seen in the left diagram of Figure 6, the conventional modeling strategy overestimates the onset of yielding under plane strain conditions. This influences the failure prediction accuracy once more. Though, the yield loci calculated at higher plastic strains ( εpl,eq≥0.10 ) align well with the conventionally identified yield locus. As a result, the conventional modeling strategy cannot map the onset of yielding but can map the hardening behavior under plane strain conditions for the dual-phase steel. Considering the shear stress state, the conventionally identified yield locus has a similar curve shape compared to all of the adapted yield loci. Thus, the Yld2000-2d yield criterion accurately maps the overall material behavior of the dual-phase steel under shear conditions.

[picture]

Fig. 6. Yield locus distortion for the dual-phase steel DP600.

Fig. 6. Yield locus distortion for the dual-phase steel DP600.

Consequently, the following Table 2 summarizes the suitability of the conventionally identified Yld2000-2d yield criterion for the two investigated materials. With exception of the shear stress state in case of the DP600, the distortional hardening behavior is not mapped accurately with the conventional modeling strategy. Using the conventional yield criterion leads to a deviation between the modeled and real material behavior because the overall material behavior under plane strain or shear is not mapped precisely. Due to that, enhanced modeling strategies are necessary to implement the shown dependence of the yield locus exponent of the plastic strain to improve numerical mapping accuracy. Therefore, future works will focus on this aspect.

Table 2. Suitability of the conventionally identified Yld2000-2d for modeling the material behavior.

Material | Consideration of plane strain | Consideration of shear
Onset of yielding | High plastic strains | Onset of yielding | High plastic strains
AA5182 | + | - | - | +
DP600 | - | + | + | +

Summary and Outlook

The original version of this paper is available on https://www.scientific.net/SSP.390.1.pdf

Due to the relevance of the plane strain and shear stress states in sheet metal forming processes, a consideration of the mentioned states in the yield criterion is desirable. The commonly used Barlat Yld2000-2d yield criterion offers the possibility of modeling the anisotropic material behavior of modern materials. Adjusting the yield locus exponent allows the stress point of the plane strain or shear stress state to be directly integrated into the yield locus, respectively. Within this contribution, a strain dependent calibration of the Yld2000-2d yield criterion is investigated in order to improve the mapping of the distortional hardening behavior of modern materials. For both materials, it is observed that a strain-dependent consideration of the plane strain state leads to a reduction of the
yield locus exponent, while a strain-dependent integration of the shear stress state is accompanied with an increase of the yield locus exponent. Additionally, the use of a conventional modeling strategy results in a deviation of the modeled and real material behavior with the exception of the shear stress state in case of the DP600. The reason for this is, that the overall material behavior under plane strain or shear is not sufficiently mapped. In future works, the optimal equivalent plastic strain should be identified, at which the parameter identification leads to the most accurate numerical results. Moreover, the strain-dependent calibration of the Yld2000-2d yield criterion should be implemented into a user-defined subroutine and subsequently validated in a deep drawing process of cross-die samples in comparison to the conventional and adapted modeling strategy.

Acknowledgement

The original version of this paper is available on https://www.scientific.net/SSP.390.1.pdf

The authors would like to thank the German Research Foundation (DFG) for supporting the present investigations, which were performed within the scope of the research project "Improvement of the mapping accuracy in material modelling with the integration of the yield locus at plane strain" (project number 412137480).

References

The original version of this paper is available on https://www.scientific.net/SSP.390.1.pdf

F. Barlat, J.C. Brem, J.W. Yoon, K. Chung, R.E. Dick, D.J. Lege, F. Pourboghrat, S.-H. Choi and E. Chu, Plane stress yield function for aluminum alloy sheets-part 1: theory, International Journal of Plasticity 19(2003)9, p. 1297-1319.

Y. Hou, K. Du, A.A. El-Aty, M.-G. Lee, J. Min, Plastic anisotropy of sheet metals under plane strain loading: A novel non-associated constitutive model based on fourth-order polynomial functions, Materials & Design 223(2022), 111187.

M.D. Xavier, R.L. Plaut and C.G. Schön, Uniaxial Near Plane Strain Tensile Tests Applied to the Determination of the FLC Formabillity Parameter, Materials Research 17(2014)4, p. 982986.

J.H. Kim, J.H. Sung, K. Piao and R.H. Wagoner, The shear fracture of dual-phase steel, International Journal of Plasticity 27(2011)10, p. 1658-1676.

A.W. Hudgins, D.K. Matlock, J.G. Speer and C.J. Van Tyne, Predicting instability at die radii in advanced high strength steels, Journal of Materials Processing Technology 210(2010)5, p. 741-750.

B.A. Behrens, C. Bonk and I. Peshekhodov, On modelling of shear fracture in deep drawing of a high-strength dual-phase sheet steel, Journal of Physics: Conference Series 896(2017), 012125.

J.W. Yoon, F. Barlat, R.E. Dick, K. Chung and T.J. Kang, Plane stress yield function for aluminum alloy sheets-part II: FE formulation and its implementation, International Journal of Plasticity 20(2004)3, p. 495-522.

H. Wang, M. Wan, X. Wu and Y. Yan, The equivalent plastic strain-dependent Yld2000-2d yield function and the experimental verification. Computational Materials Science 47(2009)1, p.12-22.

N. Küsters and A. Brosius, An adapted yield criterion for the evolution of subsequent yield surfaces. Journal of Physics: Conference Series 896(2017), 012020.

M. Rentz and M. Merklein, Improvement of material modeling by considering the plane strain and shear stress state in the Yld2000-2d yield criterion, Journal of Physics: Conference Series 3104(2025), 012004.

M. Lenzen and M. Merklein, Determination of strain dependent anisotropy in layer compression tests and resulting influence on the yield locus modelling, In NUMIFORM 2019: The 13th International Conference on Numerical Methods in Industrial Forming Processes, 2019.

M. Lenzen and M. Merklein, Improvement of Numerical Modelling Considering Plane Strain Material Characterization with an Elliptic Hydraulic Bulge Test, Journal of Manufacturing and Materials Processing 2(2018)1, 6.

M. Merklein and M. Biasutti, Forward and Reverse Simple Shear Test Experiments for Material Modeling in Forming Simulations, in: G. Hirt and A.E. Tekkaya, Proceedings of the 10th International Conference on Technology of Plasticity, ICTP 2011, p. 702-707.

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The original version of this eBook is available on https://www.scientific.net/book/material-behaviour-modelling/978-3-0364-1987-9

Solid State Phenomena, ISSN: 1662-9779, Vol. 390, pp 11-18

doi: 10.4028/p-4ggPVB

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

Submitted:
2025-12-17

Revised:
2026-02-16

Accepted:
2026-02-21

Online:
2026-04-17

A Novel Approach for Determining the Bauschinger Effect in Dual-Phase Steel Sheets with Tensile-Compression Tests

The original version of this paper is available on https://www.scientific.net/SSP.390.11.pdf

Adam Häckel 1,a*, David Naumann 1, b, Marion Merklein 1,c

1 Institute of Manufacturing Technology, Friedrich-Alexander-Universität Egerlandstraße 13, 91058

Erlangen, Germany

a  adam.haeckel@fau.de, b  david.naumann@fau.de, c  marion.merklein@fau.de

*corresponding author

Keywords: material characterization, dual-phase steels, bauschinger effect.

Abstract

In order to reliably predict a material's behavior during the forming process, robust calculations with precisely calibrated material models are required. Especially when it comes to mapping phenomena depending on complex interactions of different effects, sophisticated measuring techniques have to be used in order to capture them sufficiently. Springback of sheet metal components is governed by elastic behavior, determined by geometry and current material properties. While well understood for most materials, dual-phase steels are exceptional due to their non-linear elasticity and pronounced kinematic hardening, which strongly affect elastic response. Kinematic hardening is characterized via the Bauschinger coefficient from tension-compression tests. As the Bauschinger effect depends on pre-strain and strain rate, precise crosshead control is essential. Therefore, state-of-the art characterization techniques control the process speed by calculating the crosshead velocity from the pre-set clamping length and strain rate. This method, however, does not account for setup-related influences such as machine stiffness or specimen slippage. Therefore, to improve the characterization accuracy of the Bauschinger effect, an alternative method for crosshead control during the tensile-compression test is introduced and analyzed in this study. To compare this innovative approach with the conventional one, both methods are used to capture the effect of relaxation on the Bauschinger effect with different dual-phase and mild steel. The mentioned novel method is based on the optical strain rate control during tensile tests by Naumann, using Digital Image Correlation with an Aramis setup by ZEISS. The intended pre-strain before load reversal is actively controlled by measuring the strain in situ. After characterizing the material cards for each setup, the resulting Chaboche-Rousselier curves are compared to the experimental ones. The results demonstrate that the applied method provides a reliable proof of concept and achieves precision comparable to, as well as exceeding, the conventional displacement strain rate control method.

Introduction

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Dual-phase steels (DP steels) were developed in the 1970s based on low-carbon microalloyed sheet steels for the automobile industry. Due to their excellent balance of strength, ductility and energy absorption, they are one of the most widely produced advanced high strength steels (AHSS), leading to their standardization in the 2000s. To date, however, the elastic-plastic material behavior of DP steel is not fully understood, which manifests as inaccuracies during numerical predictions, especially during springback prognosis simulations. The main difficulty hereby lies in describing the complex interactions between several material and forming process related effects, ranging from the micro- to the macroscale. These effects are predominantly the Bauschinger effect [1], non-linear elasticity [2], pre-strain [3] as well as strain rate [4]. Various sophisticated material models have been developed in the last decades tackling this challenge and providing the tools for highly accurate numerical simulations. Specially for predicting the springback of DP steels after sheet forming processes, the hardening models developed by Chaboche and Rousselier (CR) [5] as well as Yoshida and Uemori [6] are widely adopted. Calibrating these models however, requires extensive material characterization of high precision, such as tensile-compression tests (TCT), compression-tensile tests (CTT) along with regular tensile and compression tests. While associated challenges, such as buckling
during compression [7] or precise strain measurements despite localized necking [8] have been addressed thoroughly, strain rate dependency remains an issue, especially for springback related predictions of DP-steels.
The Bauschinger effect describes the phenomenon whereby a metal that has been plastically deformed in one direction exhibits a reduced yield stress when the loading direction is reversed, compared to the forward direction [9]. At its core, the effect arises from the internal stress state created during forward plastic deformation. Dislocations accumulate and interact with obstacles such as grain boundaries, precipitates or phase interfaces, leading to heterogeneous microstructures with regions of high and low dislocation density [10]. These heterogeneous dislocation structures give rise to back-stresses, which are categorized as long-range internal stresses that oppose further forward deformation but assist the onset of plasticity upon load reversal. Under reversed stress, dislocations encounter reduced resistance and can remobilize at lower applied stress, resulting in the lowered reverse yield strength characteristic of the Bauschinger effect [11]. The magnitude of the effect depends on microstructural factors such as grain size, dislocation density, phase distribution, and precipitate content, as well as on loading conditions including pre-strain level, strain rate, and temperature. In alloys and steels with multiple phases, differences in plastic incompatibility between phases enhance residual stress gradients and thus accentuate the Bauschinger effect [10]. During cyclic deformation, the transient component of the effect leads to an initial rapid reduction in yield stress upon load reversal, followed by a more stabilized response as the reversed deformation progresses. Thermally activated processes, such as dislocation climb or recovery, can mitigate back-stresses if hold times or temperatures are sufficient, thereby modifying the Bauschinger response. The Bauschinger effect is most commonly described using equation (1) [9], determining the Bauschinger coefficient (BC) as the absolute ratio between the stress before load reversal ( σ1 ) and re-yielding after (σ2).

BC=|σ2σ1|(1)

To investigate relaxation effects on elastic-plastic material behavior with industrial relevance, relaxation mechanisms on a timescale of seconds must be considered. The dominating effects in the first seconds during relaxation are rearrangements and gliding of dislocations, cross-slip as well as microplastic redistribution between phases, such as ferrite and martensite [12]. These effects however, can already be activated during forming, depending on the applied strain rate [10]. Therefore, the influence of a strain controlled holding time may be diminished due to slow strain rates [13].
If conventional tensile tests are carried out by applying displacement strain rate control (DSRC), strain and strain rate are calculated by the crosshead position during testing. This negatively affects the accuracy of the strain rate through influences like machine compliance and specimen clamping. With the conventional DSRC setup, a compensation of machine influences is necessary. Because of the specimen's slippage between the clamps, the initial clamping length cannot be taken into account accurately when it comes to reaching the intended pre-strain before load reversal. To compensate the discrepancy between the intended and the actually achieved strain, a preliminary investigation with separate tensile and compression tests is conducted. By measuring the achieved true strain with a digital correlation system (DIC), the traveled distance of the crossbeam can be matched with the corresponding true strain. This workaround is required for every testing configuration (material, pre-strain, strain rate, etc.) and poses an inefficient use of resources. To overcome this challenge, different approaches have been presented in the past and also were converted into standards like the ISO 6892-1 method A1. Thereby, a closed loop control is applied with a strain measurement system, which directly determines the strain at the gauge length of the specimen. Tactile and optical measuring systems can be used for this purpose. Further development was done by Naumann and Merklein by utilizing local optical strain rate control (OSRC) with DIC to account for local strain phenomena like diffuse and localized necking [14]. This not only makes it possible to specifically influence and thus control the strain rate on the sample, but also the strain itself. As an attempt to enhance the experimental characterization of the Bauschinger effect and consequently improving the calibration
of state-of-the-art hardening models such as the Chaboche-Rousselier and Yoshida-Uemori models, this paper introduces a method of conducting strain and strain rate controlled tensile-compression tests utilizing a DIC setup.

Setup and Methodology

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All tests were carried out on universal testing machines from ZwickRoell with a nominal force of 100 kN and a maximum crosshead speed of 600 mm/min and 1500 mm/min. A special tool design along with the miniaturized specimen geometry is presented by Staud and Merklein [7]. They were used to apply defined tension and compressive loads to the specimens. The specimens with an initial gauge length and width of 2 mm were laser cut and subsequently wire cut by electrical discharge machining in the area of free span between the clamping jaws to reduce the influence of edge quality. Clamping was done by hydraulic pressure and force closure. The individual components are displayed in Fig. 1. The relatively short gauge length enables compression strains to a certain level without the risk of buckling and also without friction influence as it would appear with an anti-buckling device.

[picture]

Fig. 1. Test setup for the tensile-compression-tests with 2D-DIC.

Fig. 1. Test setup for the tensile-compression-tests with 2D-DIC.

The strains were measured contactless by a two-dimensional digital image correlation system from Zeiss ARAMIS with a 12 Mpx camera setup. The closed control loop was built up between the electronic control unit (ECU) of the universal testing machine (UTM) and the ARAMIS DIC-system. Strains were live calculated over a gauge length of 1 mm by a virtual extensometer on the specimen surface and fed back to the ECU with a 0 to 10 V analog signal. The control of the crosshead movement was calculated by an embedded PID-controller of the ECU. Post processing of the strains was done by an analyzed area of 1.5×1.5 mm2, where strains were averaged to reduce measurement noise and measurement errors. Three different steel grades with a nominal thickness of 1 mm were investigated in the scope of this contribution. Table 1 shows the different configurations of the investigation, varying material, type of strain rate control, relaxation holding time and pre-strain.

Table 1. Varied parameters for tensile-compression tests.

Material | Strain rate control | Holding time [s] | Strain [%]
DC04 | DSRC | 0 | 1
DP600 | 1.5
|  |  | 2
DP1000 | OSRC | 10 | 3
| 5

A mild steel DC04, and two AHSS with a dual-phase composition of martensitic phase embedded in a ferritic matrix DP600 and DP1000 are used in the TCTs in rolling direction. A constant strain rate of ε˙=0.1%/s is intended to achieve, providing a control loop designed to be as robust as possible. As a final step, the experimental stress-strain-curves of the TCT are used to characterize the kinematic hardening behavior and to calibrate the Chaboche-Rousselier constitutive model. The material response is described using a nonlinear kinematic hardening formulation, in which the total back stress is defined as the superposition of individual back-stress components αi, represented by equation
(2). Considering the uniaxial nature of the TCT, the Armstrong-Frederick evolution equation, used by the CR constitutive equations, simplifies to equation (3) [5]. ε˙p is the plastic strain rate tensor. Ci and γi are the material parameters. As literature suggests, two CR-terms ( N=2 ) are used [15]. The material parameters are identified with the non-linear least-square Fit method (Isqnonlin) by fitting the tensile and compression segments of the stress-strain curve, minimizing the error in stress between data and model [16].The final difference between the experimental and modelled stress-strain curve is presented as the root mean square error (RMSE).

α=∑i=1Nαi.α˙i=Ciε˙p−γiαi|ε˙p|.

(2)(3)

Results

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In the following, the results of the TCTs are presented. The stress-strain curves as well as the strain rate over the true strain with the target strain of ε=0.05 and nominal strain rate of ε˙=0.1%/s are used to compare the OSRC with the DSRC method. The influence of relaxation with a holding time of 10 s(t10) compared to 0 s(t0) for the materials DC04, DP600 and DP1000 is investigated. Additionally, the averaged Bauschinger coefficients (BC) are plotted against the achieved true strain for every test configuration. In Fig. 2, the reference measurements with DC04 are shown. While an influence of relaxation is not discernible between t0 and t10 for both control methods, the stress-strain curves, however, show a distinct difference in the measured true strain between DSRC and OSRC.

[picture]

Fig. 2. a) Stress-strain curve and b) strain rate over true strain for DC04.

Fig. 2. a) Stress-strain curve and b) strain rate over true strain for DC04.

While the strains for OSRC deviate from the target strain with ε=0.047, the DSRC method overshoots with ε=0.054. The actually measured strain rate of the observed area highlights the success of the OSRC method in regulating the strain rate. After an initial overshoot, the strain rate stabilizes at 0.092±0.006%/s during the tensile segment and at −0.097±0.006%/s during compression. DSRC also provides a near constant strain rate for DC04 with an average tensile strain rate of 0.059±0.007%/s and an irregularly erratic compression strain rate. The deviating strain rates of DSRC show the distinct difference between the strain rate control methods. As DSRC provides a constant tensile strain rate, it misses the targeted rate due to the imprecisely set clamping length. After load reversal, the DSRC strain rate closes in on the target rate. It diverts, however into a highly irregular strain rate inside the measured area. OSRC on the other hand provides flexible adjustment of the crossbeam, controlled by the in-situ measured true strain, a highly reliable strain rate across the whole testing cycle.
Fig. 3 shows the typical stress-strain curves for DP600 with less pronounced elastic-plastic transitions after load reversal. The total tensile and compression strains behave similarly to the DC04 specimens with DSRC overshooting (ε=0.052) and OSRC undershooting (ε=0.048) the target value. For DP600 no influence of relaxation is noticeable in the stress-strain curves. The DSRC strain rate curve
of the observed area is severely erratic, while OSRC provides a steady tensile strain rate of 0.095±0.005%/s and a compression strain rate of −0.097±0.007%/s. In Fig. 4, the total true strains of DP1000, in contrast to DC04 and DP600, overlap for DSRC and OSRC mostly, with OSRC being closer to the target strain with 0.048 than DSRC with 0.047 .

[picture]

Fig. 3. a) Stress-strain curve and b) strain rate over true strain for DP600.

Fig. 3. a) Stress-strain curve and b) strain rate over true strain for DP600.

Apart from the longer overshoot observed in Fig. 4b) which results from the higher strength and consequently larger elastic strain of DP1000 compared to DC04 and DP600, the strain rate of the OSRC setup proves to be reliably consistent for all three materials. The tensile strain rate of DP1000 was 0.095±0.006%/s, without prior mapping of each configuration's elastic-plastic behavior. The DSRC strain rates, however, do not reach the intended strain rate for all configurations and also differ for each material. This highlights the DSRC's dependency on the material's strength and therefore the need of subsequent, often iteratively conducted pre-examinations for each particular setup. The absence of an influence due to relaxation in the stress-strain curve is attributed to the small strain rate of 0.1%/s used in the TCTs, activating most relaxation mechanisms during plastification.

[picture]

Fig. 4. a) Stress-strain curve and b) strain rate over true strain for DP1000.

Fig. 4. a) Stress-strain curve and b) strain rate over true strain for DP1000.

The determined and averaged Bauschinger coefficients for the steels DC04, DP600 and DP1000 across engineering pre-strains of 1%,1.5%,2%,3% and 5% are presented in Fig. 5. The resulting plastic strains differ between materials because the target values were defined in terms of engineering strain, not considering elastic strain. Consequently, materials with higher elastic fractions reached lower plastic strains at the same nominal pre-strain. The measured evolution of BC follows a trend that can be approximated by a power-law relationship. Nevertheless, some data points slightly deviate from this behavior and fall outside the corresponding fit. The overall small standard deviation of the Bauschinger coefficient demonstrates the repeatability of the conducted experiments, especially for the OSRC method.

The comparison between DSRC and OSRC reveals that the Bauschinger coefficients are slightly but consistently higher under OSRC than under DSRC. This difference could be attributed to the constant strain rate loading at 0.1%/s under OSRC, in contrast to the first lower but accelerating strain rate with higher fluctuations under DSRC. The constant strain rate enables a first relaxation of the initially induced back stresses by the overshoot during elastic strain, while an accelerating strain rate causes back-stresses to build up instead, resulting in a stronger Bauschinger effect. In addition, transient effects immediately following load reversal may contribute to a rapid, short-lived drop in yield stress, which decays within a small strain interval. Under OSRC, these transient contributions, together with the pre-relaxed back-stress field, may slightly modify the early reversal response, whereas under DSRC they appear less pronounced. Relaxation holds have little influence on the coefficient for DSRC, while for OSRC a slight reduction in BC is observed, following the hold. This behavior could be interpreted as the hold activating the remaining relaxation potential after the relaxation inducing slow loading process. The exact mechanisms responsible for these subtle trends are not yet fully clarified, and further targeted experiments would be required to achieve a comprehensive understanding.

[picture]

Fig. 5. Bauschinger coefficients over plastic strain with approximated trendlines following a power law for a) DSRC tests and b) OSRC tests.

Fig. 5. Bauschinger coefficients over plastic strain with approximated trendlines following a power law for a) DSRC tests and b) OSRC tests.

The Chaboche-Rousselier parameters were identified by fitting the model response to the experimentally measured stress-strain curves of the TCT, focusing on the accurate reproduction of the transition from forward plastic deformation to re-yielding in compression. The fitting procedure minimized the deviation between experimental and model curves over the relevant strain range, with emphasis on capturing the curvature and saturation behavior of the kinematic hardening response. As presented in Fig. 6, the fitted CR-model curves yield consistent results. The OSRC curves achieve equally low RMSE compared to the DSRC results, rising respectively for higher strength materials. This degree of deviation from the experimental stress-strain-curves aligns with previous investigations on the Chaboche-Rousselier hardening model [17], validating the suitability of the OSRC method for material card calibration.

[picture]

Fig. 6. Comparison of fitted curve deviation a) single curve of DP1000 OSRC t0 b) mean value and standard deviation of all specimens RMSE values compared by configuration.

Fig. 6. Comparison of fitted curve deviation a) single curve of DP1000 OSRC t0 b) mean value and standard deviation of all specimens RMSE values compared by configuration.

Summary and Outlook

The original version of this paper is available on https://www.scientific.net/SSP.390.11.pdf

The study reaffirms the proof of concept for the optical strain rate control methodology by Naumann and Merklein [14] by demonstrating its applicability to reverse loading experiments, a domain in which reliable local strain rate control has so far remained insufficiently addressed. By overcoming limitations of conventional displacement-based control strategies, the presented results highlight the potential of optical strain rate control to close an important methodological gap in the experimental characterization of cyclic and reverse plasticity. The results show that OSRC attains a level of measurement precision comparable to, and in some cases exceeding, that of displacementcontrolled approaches. Mean deviations of the Bauschinger coefficient remain similarly low, and the RMSE values of the fitted Chaboche-Rousselier curves are consistently as low as those obtained with DSRC. Moreover, the RMSE of the CR-parameter identification demonstrates that OSRC is fully suitable for material parameter calibration, providing a level of fidelity that aligns with established standards. A central advantage of OSRC is the increased control over the experimental setup. The strain as well as the strain rate can be regulated directly within the optically tracked region of interest. It ensures a more accurate representation of the local deformation. In addition, the method requires no mechanical workarounds and can be rapidly implemented within existing optical measurement infrastructures.
Future work should investigate how OSRC affects the stability and accuracy of calibrated material parameters, particularly in numerical and experimental sheet-metal forming applications. Further development of the software environment could enhance usability and streamline data processing. Extending the methodology to higher strain rates and elevated temperatures will help assess its robustness under more demanding conditions.

Acknowledgement

The original version of this paper is available on https://www.scientific.net/SSP.390.11.pdf

The authors would like to thank the German Research Foundation (DFG) for funding the project "Analysis of the elastic-plastic material behavior of higher strength steel materials under cyclical and swelling load in dependence of the relaxation behavior" (Project number 519030020).

References

The original version of this paper is available on https://www.scientific.net/SSP.390.11.pdf

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J. L. Chaboche and G. Rousselier, "On the Plastic and Viscoplastic Constitutive EquationsPart II: Application of Internal Variable Concepts to the 316 Stainless Steel," Journal of Pressure Vessel Technology, vol. 105, no. 2, S. 159-164, 1983.

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Solid State Phenomena, ISSN: 1662-9779, Vol. 390, pp 19-29

doi: 10.4028/p-MDl8Q0

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

Submitted:
2025-12-18

Revised:
2026-02-17

Accepted:
2026-02-21

Online:
2026-04-17

Numerical Quantification of Structure-Property Relations in DP800 - Influence of Phase Material Models in RVE

The original version of this paper is available on https://www.scientific.net/SSP.390.19.pdf

Fehlemann Niklas C. 1,a*, Münstermann Sebastian 1,b

1 Institute of Metal Forming, RWTH Aachen University

a*niklas.fehlemann@ibf.rwth−aachen.de,b Sebastian.muenstermann@ibf.rwth-aachen.de

*Corresponding author

Keywords: dual phase steel, representative volume elements, damage, microstructure, sensitivity analysis, crystal plasticity.

Abstract

Understanding the relationships between (micro) structure and (mechanical) properties is inevitable for the design of modern structural metallic materials. A crucial property for most highstrength steels is ductile damage tolerance. Ductile damage in form of voids or microcracks can accumulate during cold forming, which either leads to failure in the forming process or subsequently affects the performance. Structure-property relations are often investigated using numerical methods, e.g. crystal plasticity (CP) modeling with statistically representative volume elements (sRVE). In a previous study, CP-simulations on 3D-sRVE were coupled with surrogate modeling techniques performing a variance-based sensitivity analysis. This analysis enables quantitative descriptions of the relationships between microstructure features with the damage tolerance, quantified by different indicators for individual damage mechanisms. To investigate the effect of the individual material models and the corresponding phase properties, 500 sRVE simulations were carried out with different CP-parameter sets and the damage tolerance was investigated in the present paper. All sets stem from the same DP800 but were calibrated with different approaches. Surrogate models were trained on the simulative database to calculate Sobol Indices (SI), which are a measure of how strong damage tolerance is affected by a particular microstructure feature. The SI are compared for the individual material models and damage indicators. The structure-property quantification is heavily influenced by the different material models, resulting in different values for the SI and a different order of the individual microstructure features. The main factor for the pronounced differences is the differently evolving mechanical phase contrast between ferrite and martensite.

Introduction

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Establishing a mapping between process, structure, property and performance relationships is of crucial importance for the precise understanding of process chains and associated materials [1]. Since experimental investigations can consume a significant amount of resources to explore a larger space of different process parameters or microstructures, simulative and data-based methods are often used in this context, especially for structure-property mappings [2, 3]. An important advantage of numerical approaches is that individual parameters can be varied independently from the others, e.g. to show the singular effect of martensite banding on mechanical properties [4]. When numerical models are employed, a core piece is the material model used. In the case of microstructure-property relations, crystal plasticity (CP) models in conjunction with statistically representative volume elements (sRVE) are often used to map microstructure and mechanical properties. One of the most important mechanical property is the tolerance against initiation and evolution of ductile damage, since ductile damage can lead to failure during cold forming or can influence the final product properties [5].

In a previous study by the authors [6], a combined numerical and data-based structure-property quantification was carried out on a dual-phase steel DP800. Microstructural features like grain sizes and shapes were linked with damage tolerance using CP-simulations with a newly calibrated material model [7]. Based on a dataset of 250 virtual microstructure-damage pairs, surrogate modeling was employed to quantify the effect of the individual microstructure features on various damage
indicators. However, the question of how the material models of the individual phases influences this structure-property quantification remained unanswered. Therefore, the aim of this study is to conduct a second structure-property quantification with a different material model calibrated for the same steel in order to reveal the effect of the material model on structure-property quantification.

The general procedure is as follows: first, a dataset of sRVE of a DP800 with virtual microstructure concepts, i.e., for example, varying phase ratios or smaller or larger ferrite or martensite grains are generated. These sRVE are simulated with a second CP model for the same base DP800 and evaluated for their damage tolerance using different damage indicators for cleavage fracture of martensite or phase boundary decohesion. Based on this database of sRVE simulations, variance-based sensitivity analysis is performed with surrogate modeling to quantify the effect of the individual microstructure variation on the damage tolerance. Finally, the results for both CP-parameter sets are compared, and the differences are presented and discussed.

Materials & Methods

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The methodology for the numerical and data-based structure-property quantification employed in this study closely resembles the approach presented in Fehlemann et al. [6], and as such, it will only be briefly summarized here.

Microstructure model.

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The base material is a standard DP800 steel with a strongly heterogeneous microstructure made up of ferrite and martensite. The martensite is either present as small, irregular-shaped islands or larger, elongated bands and makes up about 30% of the material. For the generation of the sRVE, the open source sRVE generator DRAGen [8] is employed. DRAGen is especially good at capturing the heterogeneous, irregular microstructure of the DP800. In this study, seven different microstructural features and their respective effect on the damage are investigated, with the subsequently used abbreviations in brackets:

The phase ratio between ferrite and martensite (P−R)

The grain size of ferrite (F_GrSize)

The aspect ratio of the ferrite grains ( F_Ar )

The island size of martensite (M_GrSize)

The aspect ratio of the martensite islands ( M_Ar )

Binary variable indicating presence of martensite bands ( N_bands)

The hardening of the ferrite phase (f_hard)

Details on how the variations are implemented in DRAGen and the material model can be found in [6]. It is important to note that the modifications on the microstructure are made in relation to the base microstructure. This means that a higher value for F_GrSize results in larger grains compared to the base material and vice versa. Example sRVE with different virtual microstructures are presented in Fig. 1.

Material model.

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The aim of this study is the comparison of material model parameter sets and their effect on the structure-property quantification. Two sets of CP-parameters were utilized to investigate the influence of the material model and the associated mechanical (phase) properties on the effect of the microstructure features on damage tolerance. The first set (henceforth denoted as S1 ), calibrated by Fehlemann et al. [7] and utilized for the study in [6] is based on in situ micropillar compression tests. The fundamental principle underlying this methodology for CP-calibration is based on the fact that the strain hardening measured in the in situ micropillar compression test cannot be directly transferred to the polycrystal, given that the strain hardening due to free surfaces in the micropillar differs from that in the polycrystal [9]. Consequently, the micropillar tests were milled from pre-strained macroscopic tensile specimens to estimate the hardening of the polycrystal via the change in CRSS at different pre-strain levels. The parameters for the martensite can be fitted directly from the micropillar compression tests, since the martensite contains enough interfaces to closely resemble
bulk behavior [10]. S1 also considers grain size effects with a Hall-Petch type scaling law as well as the effect of the carbon content on the martensite hardness.

The second set ( S2 ) is derived from Pütz et al. [11]. In this case, the calibration of the CPparameters for the ferrite was achieved using nanoindentations, whereby the force-indentation curves for individual grains were fitted. Subsequently, the model parameters for the martensite were inversely fitted to the homogenized true stress-strain curve of the bulk material. An optimal fit for the martensite is reached when the homogenized stress-strain curve from the DP800 sRVE closely resembles the homogenized true stress-strain curve of the bulk material. Since the structure-property pairs for the first set are taken from Fehlemann et al. [6], only simulations with S2 are carried out in this study. All simulations in this paper are carried out with the open-source multiphysics framework DAMASK [12].

[picture]

Fig. 1. Example sRVE with virtual microstructures. a.) Base microstructure, b.) fine grained ferrite and martensite, c.) Coarse ferrite and martensite, d.) ferrite elongated in RD plus finegrained martensite.

Fig. 1. Example sRVE with virtual microstructures. a.) Base microstructure, b.) fine grained ferrite and martensite, c.) Coarse ferrite and martensite, d.) ferrite elongated in RD plus finegrained martensite.

Damage indicators

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As in [6], three damage indicators are used for the most common ductile damage mechanism in DP-steels. It should be noted that the indicators are only for damage initiation, since there is no damage model to predict the evolution of damage during plastic deformations. All indicators are calculated solely from the stress- and strain fields after the simulation has finished. The first indicator for cleavage fracture of martensite (denoted as Ind C, indicator for cleavage fracture) is determined by the percentage of elements in an sRVE that exceed a specified threshold of the maximum principal stress. The value for the principal stress is derived using the methodology by Steinbrunner et al. [13]. According to their findings, the fracture of martensite occurs at approximately 5% strain, which corresponds to a stress threshold of 3500 MPa for S1 and 1350 MPa for S2. The difference stems from the different material parameters for martensite in the two sets. To assess the damage initiation by phase boundary decohesion, a novel indicator was developed. This indicator is based on the magnitude and scatter of the equivalent plastic strains at the interfaces between ferrite and martensite. The indicator is denoted as IndPBD (Indicator for phase boundary decohesion), and further details can be found in [6]. As an additional measure of the overall damage tolerance, an equally weighted normalized average of IndC and IndPBD is calculated, which is referred to as IndH (Homogenized indicator). Furthermore, it was decided not to use an indicator for damage initiation in ferrite grains, as this rarely occurs in the DP800 under consideration [14, 15].

Structure-property quantification.

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In order to calculate the aforementioned Sobol indices [16], surrogate modeling approaches are required, since SI cannot be calculated directly from the numerically generated database itself. SI are a measure of the fraction of the variance of the output of a given system caused by a specific input feature. Thus, they are a measure of how important a specific input feature is [17]. To mitigate the effect of the surrogate modeling strategy, two different model classes are used with different hyperparameters. Gaussian Process Regression (GPR) [18] is used with three different kernels, while Polynomial Chaos Expansion (PCE) [19] is used with two different forms of the underlying distributions. All five surrogate models are trained on the simulative database, and the SI are calculated from the trained models. It is noteworthy that the calculation process of the SI is different for both PCE and GPR. For PCE, the SI can be directly derived from the model parameters [19]; for GPR, they are calculated by sampling from the model in a specific manner [17]. Therefore, the SI shown in this paper are the average over the five models. The database consists of 250 virtual microstructure concepts from the aforementioned 7d design space. To reduce the scatter, 2 sRVE are simulated for each microstructure concept, yielding 500 sRVE simulations in total. The load case is uniaxial tension along the rolling direction (RD) until 12% global strain, shortly before the uniform elongation of the base material.

Results

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Material model comparison.

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The global, homogenized true stress-strain curves for both parameter sets are presented in Fig. 2, alongside the experimental true stress-strain curve from the uniaxial tensile tests. It is evident that both parameter sets demonstrate a strong agreement with the experimental data. For S1, the prediction quality for low strains, up to 0.06 , is particularly excellent, with subsequent minor deviations. For S2, the trend is opposite, with small overestimations at the beginning and a good fit for higher strains. It can be stated that, in general, both fits are adequate, particularly when taking into consideration the complexity of micromechanical modeling of multiphase materials.

[picture]

Fig. 2. Comparison of simulative (solid blue) and experimental (dashed black) true stress strain curves for the parameter sets S1 (a) and S2 (b).

Fig. 2. Comparison of simulative (solid blue) and experimental (dashed black) true stress strain curves for the parameter sets S 1 (a) and S 2 (b).

In addition to the homogenized true stress - strain curves, the individual strain hardening curves for both phases are shown in Fig. 3. Fig. 3a depicts the CRSS of the ferrite, while Fig. 3b shows the true stress-strain curves for the martensite. A pronounced strength difference is observable for martensite: while the yield stress for S1 is around 3000 MPa , it is only around 800 MPa for S2. On the other hand, S2 shows a more pronounced strain hardening of martensite. For the ferrite, the trend is vice versa, since the strain hardening is more pronounced for S1. A direct comparison between S1 and S2 in terms of the yield strength is not straightforward, as the values for S1 are skewed by the incorporated grain size scaling effects.

[picture]

Fig. 3. Comparison of material models S1 and S2 in terms of phase properties. a.) scaled ferrite hardening, b.) martensite true stress - strain curves.

Fig. 3. Comparison of material models S 1 and S 2 in terms of phase properties. a.) scaled ferrite hardening, b.) martensite true stress - strain curves.

Best and worst microstructures.

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Before evaluating the surrogate models and computing the structure-property quantification by means of the SI, the optimal and critical microstructures are derived directly from the numerical dataset of virtual microstructure-damage pairs. To circumvent the overestimation of potential outliers, the 250 data points for each indicator are sorted, and the 95th and 5th percentiles are calculated. The mean value of the individual microstructure features is subsequently determined and designated as the "optimal" or "critical" microstructure. These are represented in the radar charts in Fig. 4, alongside the respective microstructures for S1, taken from [6]. The microstructure of the base material is also
plotted for comparison. The rows show the three damage indicators, with the optimal microstructure configurations on the left side and the critical on the right side. All arms are normalized in the interval I∈[0,1].

[picture]

Fig. 4. Radarcharts showing the optimal (left) and critical (right) microstructures for all three indicators. Upper row: IndC, middle row: IndPBD, lower row: IndH.

Fig. 4. Radarcharts showing the optimal (left) and critical (right) microstructures for all three indicators. Upper row: IndC, middle row: IndPBD, lower row: IndH.

The discrepancies between S1 and S2 are comparatively small for a same load case. For Ind B in the initial row, the absence of bands and the elongation of martensite perpendicular to the loading direction (low values for M−ar ) are advantageous, which strongly deviates from the base microstructure, which exhibits more elongated martensite and pronounced banding. A visible discrepancy between S1 and S2 is observed for the phase ratio, where S2 predicts more martensite to be critical, whereas S1 predicts a lesser amount of martensite to be critical (Fig. 4b). For Ind PBD, particularly the critical microstructures are highly analogous (Fig. 4d). For the optimal microstructure, deviations are evident for the ferrite grain size and the bands. The radar charts for IndH (Fig. 4e) reveal that both models suggest relatively small values for all microstructure features to be optimal. A notable discrepancy is once again evident for the phase ratio for the critical microstructure configurations (Fig. 4f), where a medium amount of martensite is predicted to be critical in S1, compared to a high amount of martensite for S2.

Surrogate Model quality.

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The set of 250 virtual microstructure-damage pairs was utilized to train the five surrogate models (2x PCE, 3x GPR). Prior to the calculation of the SI from the models, it is imperative to evaluate their prediction quality, as a bad model will most likely produce wrong or at least imprecise SI. The coefficient of determination ( R2 ) serves as the primary metric for model quality. This measure is preferable to the mean squared error (MSE), as it would overestimate errors for higher values of the damage indicators. The results are presented in Fig. 5, where the predictions and ground truth values are compared. The GPR predictions are located on the left side, while the PCE predictions are depicted on the right. It is evident that the model quality is generally very good, with R2-values ranging between 0.75-0.9. Graphically, the good model prediction is underlined by the fact that all values lie closely along the blue ideal line. The 95%-confidence interval (shaded) computed directly from the residuals ytrue −ypred  is relatively narrow, which is also an indicator of good model quality. Overall, the GPR predictions are superior to the PCE predictions, a trend that holds true for all three damage indicators. In contrast, no significant differences are observed among the indicators. If substantial deviations between the prediction and the ground truth occur, they are predominantly underestimations of large ground truth values, i.e. they occur on the right-hand side of the graphic and right to the ideal line. Therefore, residuals that fall out of the confidence interval predominantly occur on this side of the ideal line.

Structure property quantification.

The original version of this paper is available on https://www.scientific.net/SSP.390.19.pdf

In Fig. 6, the SI are displayed. The original SI from [6], determined using the material model S1, are slightly shaded, while the new SI, as determined in this study based on the S2 CP model, are darker blue. For the brittle indicator IndB, the phase ratio ( P−R ) shows a significantly increased effect (0.1→0.55), while the influence of all microstructural morphology features, particularly the martensite aspect ratio, is reduced. For the ductile indicator IndPBD, the impact of the martensite aspect ratio is increased, while both grain size effects are reduced. For the ferritic phase, this can be attributed to the absence of Hall-Petch type grain size scaling effects in the S2 model, which should clearly diminish the effect of the grain size. In contrast to IndB, the phase ratio's effect is slightly decreased rather than increased. Finally, the effects of both aspect ratios are interchanged for the homogenized indicator IndH : the effect of the ferrite aspect ratio is increased, and the effect of the martensite aspect ratio is decreased. All other features remain largely unaffected by the material model change.

[picture]

Fig. 5. Model quality estimation. (a), (c), (e) show the GPR-predictions, (b), (d), (f) the PCE predictions. The shaded area denotes the confidence interval based on the residuals. (a) and (b) Cleavage indicator, (c) and (d) PBD-indicator, (e) and (f) Homogenized indicator.

Fig. 5. Model quality estimation. (a), (c), (e) show the GPR-predictions, (b), (d), (f) the PCE predictions. The shaded area denotes the confidence interval based on the residuals. (a) and (b) Cleavage indicator, (c) and (d) PBD-indicator, (e) and (f) Homogenized indicator.

[picture]

Fig. 6. Comparison of the SI based on the two different material models.
a.) IndC, b.) IndPBD, c.) IndH.

Fig. 6. Comparison of the SI based on the two different material models. a.) Ind C , b.) Ind P B D , c.) Ind H .

Discussion

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Comparing the structure-property quantifications with two different material models reveal pronounced differences between optimal and critical microstructures as well as the SI. Since all other parameters of the studies, such as the feature space and RVE size, are equal, the differences must stem from the parameter sets. The differences in the calibration approach, especially the difference in the micromechanical data used, result in significant discrepancies. First, only the S1 model is really validated, since the S2 model uses the macroscopic true stress-strain curve for inverse calibration of the martensite phase. Consequently, potential errors in calibrating the CP parameters of the ferrite can be offset in the martensite calibration and still result in a good overall fit of the homogenized true stress-strain curve from the uniaxial tensile tests, as shown in Fig. 2. Furthermore, calibrating the ferrite using nanoindentations depends on the target value [20]. In literature, the force-indentation curve, post-mortem imprint, or pile-up topography is used, all with their own (dis)advantages. Fig. 3 shows that the S2 model would likely predict the micropillar compression for ferrite and martensite incorrectly, since it underestimates the ferrite hardening, while simultaneously overestimating the martensite hardening, although the yield stress of the martensite is also strongly underestimated by S2. Therefore, it can be concluded that the S2 parameter set models the phase contrast between ferrite and martensite smaller than the S1 model, caused by weaker martensite and presumably stronger ferrite without the grain size effects. The fact that the homogenized stress-strain curve is nevertheless predicted correctly is due to the inverse calibration of the martensite and the lack of validation.

The exact reasons behind the differences in the SI between S1 and S2 are beyond the scope of this study. It is assumed that due to the presumably higher fidelity of the S1 model (as discussed before), the S1SI are closer to reality. Since it can be concluded that the S2 model is at least inaccurate, if not
downright wrong (since it will fail on predicting the micropillar compression experiments), the SI are also not trustworthy. This therefore underscores the need for a highly accurate material model if a structure-property quantification or a computational microstructure optimization is planned. Not only are the levels of the SI nearly completely different, but also the order of the different indices is perturbed. This underlines the importance of a correct modeling of the individual phase properties and the associated evolution of the mechanical phase contrast for a precise damage assessment.

Summary

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In this study, a numerical and data-based structure-property quantification approach was employed to establish a correlation between microstructural features and ductile damage tolerance of a DP800 steel. The results were then compared to those of a similar study that employed a differently calibrated material model. This comparison was undertaken to investigate the influence of the material model parameter set on the structure-property quantification. A pronounced effect was observed, with both the values and the order of the SI being significantly different. The underlying reason for this discrepancy is presumed to be the differently modelled evolution of the mechanical properties in both models S1 and S2, which consequently leads to diverging predictions of ductile damage behavior. This highlights the necessity for precise calibration and validation of the material models and the right choice of the experimental database for calibration. Furthermore, the extent to which the material model affects other variables, apart from damage initiation, such as global formability or effective stiffness, should be investigated.

Acknowledgement

The original version of this paper is available on https://www.scientific.net/SSP.390.19.pdf

This research was funded by Deutsche Forschungsgemeinschaft (DFG, German Research Foundation; Projectnumber 278868966 - TRR 188; Damage Controlled Forming Processes, subproject B05). The simulations were carried out with computing resources granted by RWTH Aachen University under project rwth1645.

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The original version of this paper is available on https://www.scientific.net/SSP.390.19.pdf

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Solid State Phenomena, ISSN: 1662-9779, Vol. 390, pp 31-38

doi: 10.4028/p-LwDCV7

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

Submitted:
2025-12-19

Revised:
2026-02-24

Accepted:
2026-02-25

Online:
2026-04-17

Empirical Modelling of the Cold Crystallization of PLA

The original version of this paper is available on https://www.scientific.net/SSP.390.31.pdf

EL HAJJ SLEIMAN Ghinwa 1,2,a*, LABALEC Ewen 2,b, COLOMINES Gaël 2,c

1 Icam School of Engineering, 35 av. Champ de Manœuvres, 44470 Carquefou, France

2 Nantes Université, UMR CNRS 6144, Laboratoire GEPEA, F-44000 Nantes, France

a  ghinwa.el-hajj-sleiman@icam.fr, b  ewen.labalec@gmail.com, c  gael.colomines@univ-nantes.fr,

(*corresponding author)

Keywords: Cold crystallization, PLA, empirical modelling.

Abstract

Polylactic acid (PLA) has emerged as a promising alternative to conventional petroleum based plastics due to its biodegradability, renewable sourcing, and lower environmental impact. However, PLA exhibits a slow crystallization kinetics compared to other semi-crystalline polymers, such as polyethylene (PE) or polypropylene (PP), resulting in an amorphous structure after processing. This amorphous morphology can adversely influence the mechanical properties and overall performance of PLA components. The present study investigates the cold crystallization behavior of PLA using Differential Scanning Calorimetry (DSC) with the objective of developing an empirical model capable of describing crystallinity as joint function of holding temperature and time. The resulting model is intended to serve as a practical reference for industrial applications, facilitating improved control of PLA's microstructure and mechanical performance.

Introduction

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Polylactic Acid (PLA) is a widely used biodegradable polymer derived from renewable resources, making it an attractive alternative to petroleum-based plastics. PLA has garnered significant attention in various applications, such as packaging, biomedical devices, and 3D printing, due to its biocompatibility and eco-friendly nature. However, despite its promising potential, PLA exhibits relatively poor crystallization behavior, which limits its thermal mechanical properties, such as strength, rigidity, and thermal stability. One of the key challenges in optimizing PLA's properties for practical applications lies in understanding and controlling its crystallization behavior.

Cold crystallization is a phenomenon that occurs when an amorphous polymer, like PLA, is heated above its glass transition temperature ( Tg ) but below its melting temperature ( Tm ), causing crystallization to take place without prior nucleation. This process is particularly relevant in PLA, as it can significantly enhance its mechanical and thermal properties. The rate and extent of cold crystallization in PLA are influenced by various factors, including molecular weight, cooling rate, processing conditions, and the presence of additives and fillers [1].

Long Yu et al. [2] have studied the cold crystallization and the postmelting crystallization of PLA using a high-pressure differential scanning calorimetry (DSC). They have shown that cold crystallization of PLA happened around 70∘C. However, this temperature (Tcc) as well as the size of the cold crystallization exotherms decreased with increasing pressure. Omaima Alhaddad et al. [3] have shown in their study that crystallization behaviors of neat PLA depend on the heating rate. The cold crystallization temperature (Tcc) for all values shifted to higher temperatures as the heating rate increased and the peak height simultaneously increased. Ingrid et al. [4] investigated non-isothermal cold crystallization kinetics of PLA compounds. In their study, they applied Ozawa & Mo model [5] which described successfully the experimental data.

Several studies have reported the used of modified Avrami model [6] to describe the nonisothermal crystallization kinetics of PLA samples [7], [8], [9], [10]. Jeziorny [11] proposed that the kt parameter in the Avrami model should be corrected to account for the heating rate.

In this study, we examine the cold crystallization behavior of PLA and develop an empirical model to describe its crystallization kinetics, with the objective of enhancing the attainable crystallinity. The model couples a sigmoidal arctangent temperature dependence with a Gaussian time-efficiency
factor. While the formulation is empirical rather than mechanistic, it accurately reproduces measured trends and generates a practical response surface suitable for processing optimization and design.

Materials and Methods

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Materials

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The PLA used in this study is PLI 005 supplied by NATUREPLAST [12] in the form of granulates. The properties of the PLA are presented in the following Table 1:

Table 1. Properties of PLI 005 (NATUREPLAST)

Density [kg/m3] | 1250
MFI [190°C; 2.16 kg] | 25-35
Shrinkage during molding [%] | 0.3-0.4
Glass transition temperature [°C] | 60 ± 1.2
Melting temperature [°C] | 170-180
Young's modulus [MPa] | 3500
Drying time [h] | 3
Drying temperature [°C] | 60

Characterization methods

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Differential Scanning Calorimetry (DSC) measurements were carried out using a Mettler Toledo instruments DSC1 STARe System under a nitrogen atmosphere ( 50 mL/min ). The PLA samples were equilibrated isothermally at 5∘C for 10 minutes to ensure thermal stability, followed by heating to 220∘C at a rate of 5∘C/min, subsequent cooling at a rate of 10∘C/min to 5∘C, and a second heating cycle to 220∘C at a rate of 5∘C/min to evaluate thermal transitions associated with melting and recrystallization.

Empirical modelling of cold crystallization

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Initial measurements performed at a range of isothermal holding temperatures were used to determine the onset of cold crystallization and to identify the holding times required to achieve complete crystallinity. Although the expected temperature range for pure PLA is typically 80−85∘C, our results indicate deviations from this behavior.
As shown in Fig. 1, a pronounced increase in crystallinity occurs around a threshold temperature of approximately 75∘C. Above this point, a gradual rise in the crystalline fraction is observed, followed by a plateau beyond 90∘C.
Base on the experimental results presented in Fig. 1, the crystallinity as a function of temperature exhibits behavior which can be described by Eq. 1. In fact, the function arctan(x) naturally generates a smooth S -shaped curve because it is monotonically increasing. As x→−∞,arctan(x)→−π2, and as x→+∞,arctan(x)→π2, with an inflection point located at x=0. [13]

τt∞( T)=τ∞−τ0πarctan(k−1( T−TT))+τ∞+τ02(1)

However, in Eq. 1, the lower/ upper asymptotes, the inflection point as well as the steepness factor are represented below:
τt∞(T) : Crystallization rate (%) for a given temperature ( ∘C )
τ∞ : Crystallization rate for an infinite temperature (upper asymptotic value)
τ0 : Crystallization rate for a non-crystallized sample at room temperature, or residual crystallization rate (%) (lower asymptotic value)
TT : The transition temperature from a quasi-amorphous state to a crystalline state ( ∘C ) (inflection point)
k : a parameter that represents the rapidity of a transition from one to another ( ∘C ) (controls the steepness of the transition)

[picture]

Fig. 1. Measurement of the degree of crystallinity

Fig. 1. Measurement of the degree of crystallinity

The model parameters of Eq. 1 were estimated using the Solver tool in Microsoft Excel by minimizing the sum of squared residuals. It is important to note that τ∞ does not account for the melting process occurring around 160−170∘C.
Although this model has no direct physical meaning and serves only as a mathematical description of the phenomenon, the lower and upper plateaus it exhibits are physically consistent. Below the glass transition temperature, the crystallinity remains unchanged because the material is in a glassy state. Likewise, once a certain temperature is exceeded, the crystallinity no longer increases, indicating that the maximum achievable crystalline fraction has been reached. It is also expected that above a specific temperature cold crystallization ceases altogether; however, this threshold could not be identified within the scope of this study due to time limitations.
Within the framework of modeling crystallinity as a function of temperature and time, the minimum holding time required to achieve complete crystallization was determined from DSC data, based on the crystallinity curves obtained at different isothermal temperatures (Fig. 2).

[picture]

Fig. 1. Holding time required for complete crystallization

Fig. 1. Holding time required for complete crystallization

It is observed that the holding time required for the transformation to reach a chosen completion level ' p ' is inversely proportional to the temperature. Eq. 2 describes the temperature dependence of the time required to achieve complete crystallization at a given holding temperature. In the equation, " r " is a kinetic fitting parameter related to the crystallization rate, and " TT ", represents a characteristic
temperature associated with the onset of crystallization. The inverse dependence on ( T−TT ) reflects the strong slowdown of crystallization kinetics as the isothermal temperature approaches TT′. The model parameters of Eq. 2 were also estimated using the Solver tool in Microsoft Excel by minimizing the sum of squared residuals.

tτp%( T)=r T−TT,(2)

Eq. 1 and Eq. 2, which describe the crystallinity as a function of temperature at infinite time and the minimum holding time required for complete crystallization as a function of holding temperature, respectively, enable the construction of an analytical model describing the degree of crystallization of PLA as a function of both holding temperature and holding time.
The first in constructing the model consists of characterizing the melting transformation, which reflects the crystalline fraction of the material, with crystallinity being quantified through the melting enthalpy. The heat flow associated with melting can be described by a Gaussian function, which reproduces the bell-shaped peak observed in the DSC thermograms (Fig. 3).

[picture]

Fig. 2. Modelling of the transformation

Fig. 2. Modelling of the transformation

This Gaussian function can be represented by a peak of amplitude A , width σ, and position v , as shown in Eq. 3.

α(t)=Ae−σ(t−v)2(3)

α(t) : the heat flow released during the transformation, as a function of time (W)
A: the peak amplitude of the signal (W)
σ : a parameter related to the kinetics of transformation. It controls the width of the distribution (s−2)
t : the time elapsed since the start of the transformation (s)
v: the time at which the released power is maximum, in other words the peak position (s)
Due to the properties of Gaussian curves, the released power is positive over an infinite time interval. In reality, however, the transformation occurs over a finite time. To address this, an arbitrary transformation percentage p,98%, is defined to represent the point at which the transformation is considered complete (Fig. 4).

[picture]

Fig. 3. Arbitrary truncation of the curve representing the transformation

Fig. 3. Arbitrary truncation of the curve representing the transformation

Consequently, at time t=0 s, a very small percentage, 1−p, of the transformation has already occurred. We therefore seek the value of v such that: ∫−∞0α(t)dt∫−∞∞α(t)dt=1−p
The integral of a Gaussian over the entire domain is:

∫−∞∞α(t)dt=∫−∞∞Ae−σ(t−v)2dt=Aπσ

Using the error function, which is a mathematical function that calculated the area under the curve of the Gaussian function from 0 to x , we get:

∫−∞0α(t)dt=A2πσ(1+erf(−vσ))=A2πσ(1−erf(−vσ))

Thus, the value of v is (Eq. 4):

∫−∞0α(t)dt∫−∞∞α(t)dt=A2πσ(1−erf(−vσ))Aπσ=1−p12(erf(vσ)−1)=p−1v=erf−1(2p−1)σ(4)

We therefore choose to analyze the transformation over the time interval t∈[0,2v]. Considering that the holding time t(p%) required to reach a transformation of p% corresponds to this interval, we then obtain the Eq. 5:

tτp%=2v=2erf−1(2p−1)σ=r T−TT′(5)

It is therefore possible to express the concentration of the transformation, σ, as a function of temperature T, and we obtain (Eq. 6):

σ=(2( T−TT′)erf−1(2p−1)r)2(6)

Now that the transformation α is properly characterized as a function of temperature and of the running time, we define an efficiency factor η, which represents the normalized cumulative Gaussian starting at t=0. This factor is defined by Eq. 7:

η(t)=∫0tα(t)dt∫0∞α(t)dt(7)

By integrating ∫0tα(t)dt, we get:

∫0tα(t)dt=A2πσ[erf((t−v)σ)+erf(vσ)]

Integrating ∫0∞α(t)dt, we get:

∫0∞α(t)dt=A2πσ[erf(vσ)+1]

Hence, the efficiency factor becomes (Eq. 8):

η(t)=erf((t−v)σ)+erf(vσ)erf(vσ)+1(8)

To simplify the calculation, erf(x) is approximated by tanh(x/a), with a=π2. However, tanh(x) can be converted to an exponential form, tanh(x)=e2x−1e2x+1. The factor, η, becomes (Eq. 9):

η(t)=exp(2(t−v)a−1σ)−exp(−2va−1σ)exp(2(t−v)a−1σ)+1(9)

Now, to express the crystallinity in relative terms, Eq. 1 was normalized between its two asymptotic limits τ0 and τ∞. The relative crystallinity rate was defined as in Eq. 10:

τr(T)=τt∞(T)−τ0τ∞−τ0(10)

Which ensures that τr=0 when τt∞=τ0 and τr=1 when τt∞=τ∞. By substituting Eq. 1 in Eq. 10 , the relative crystallinity rate equation becomes (Eq. 11):

τr( T)=1πarctan(k−1( T−TT))+12(11)

By multiplying the relative crystallinity rate (Eq. 11) by the efficiency factor (Eq. 9), we therefore obtain a measurement between 0 and 1 of the crystallinity rates of PLA as a function of the heating temperature and the holding time at this heating temperature. To transform the scale from 0 to 1 on the interval [0;∞], we multiply by τ∞−τ0 and we add τ0. We find (Eq. 12):

τ(T,t)=Δτ·ητr+τ0(12)

In order to model a complete crystallization, we consider p=98%, then we substitute v and σ with their corresponding equations. Thus Eq. 12 becomes (Eq. 13):

τ(T,t)=Δτ2arctan(k−1( T−TT))+π2πexp(γc1)−1exp(γc1)+c2+τ0(13)

Where: γ=(T−TT′)t, while c 1 and c 2 includes all the constants present in the previous equations. The parameters listed below were determined using Microsoft Excel Solver by minimizing the sum of squared residuals.

Δτ=48.88%k=0.9477(∘C)TT=72.89(∘C)TT′=70.88(∘C)c1=0.07093(∘C−1.min−1)c2=33.1890( unitless )τ0=5.67(%)

Based on this model, we obtain iso-holding time lines and iso-crystallinity lines in function of temperature and holding time.

[picture]

Fig. 4. Representative surface of the crystallinity model

Fig. 4. Representative surface of the crystallinity model

In Fig. 5, three experimental data points (shown in white) were added to assess the validity of the developed analytical model in describing the crystallization behavior. The comparison indicates good agreement, suggesting that the model provides a reliable prediction of the crystallization process.

Conclusion

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Differential Scanning Calorimetry (DSC) measurements were performed to characterize cold crystallization behavior of PLA under different heating temperatures. Based on the DSC results, an empirical model was developed to describe the evolution of cold crystallinity of PLA as a function of both the heating temperature and the isothermal holding time. From this model, a representative crystallinity response surface was established, enabling the prediction of the crystallization behavior over a range of processing conditions.

References

The original version of this paper is available on https://www.scientific.net/SSP.390.31.pdf

E. Tarani et al., "Cold crystallization kinetics and thermal degradation of pla composites with metal oxide nanofillers," Applied Sciences (Switzerland), vol. 11, no. 7, p. 3004, Apr. 2021.

L. Yu, H. Liu, K. Dean, and L. Chen, "Cold crystallization and postmelting crystallization of PLA plasticized by compressed carbon dioxide," J. Polym. Sci. B Polym. Phys., vol. 46, no. 23, pp. 2630-2636, Dec. 2008.

O. Alhaddad, S. H. El-Taweel, and Y. Elbahloul, "Nonisothermal cold crystallization kinetics of poly(lactic acid)/bacterial poly(hydroxyoctanoate) (PHO)/talc," Open Chem., vol. 17, no. 1, pp. 1266-1278, Jan. 2019.

I. D. dos Santos Silva et al., "An investigation of PLA/Babassu cold crystallization kinetics," J. Therm. Anal. Calorim., vol. 141, no. 4, pp. 1389-1397, Aug. 2020.

R. M. R. Wellen and E. L. Canedo, "Nonisothermal melt and cold crystallization kinetics of poly(3-hydroxybutyrate) and poly(3-hydroxybutyrate)/carbon black compounds. Evaluation of Pseudo-Avrami, Ozawa, and Mo models," J. Mater. Res., vol. 31, no. 6, pp. 729-739, Mar. 2016.

M. Avrami, "Kinetics of Phase Change. II Transformation-Time Relations for Random Distribution of Nuclei," J. Chem. Phys., vol. 8, no. 2, pp. 212-224, Feb. 1940.

Z. Su, W. Guo, Y. Liu, Q. Li, and C. Wu, "Non-isothermal crystallization kinetics of poly (lactic acid)/modified carbon black composite," Polymer Bulletin, vol. 62, no. 5, pp. 629-642, May 2009.

P. N. Tri, S. Domenek, A. Guinault, and C. Sollogoub, "Crystallization behavior of poly(lactide)/poly(β-hydroxybutyrate)/talc composites," J. Appl. Polym. Sci., vol. 129, no. 6, pp. 3355-3365, Sep. 2013.

M. Li, D. Hu, Y. Wang, and C. Shen, "Nonisothermal crystallization kinetics of poly(lactic acid) formulations comprising talc with poly(ethylene glycol)," Polym. Eng. Sci., vol. 50, no. 12, pp. 2298-2305, Dec. 2010.

V. H. Sangeetha, R. B. Valapa, S. K. Nayak, and T. O. Varghese, "Investigation on the Influence of EVA Content on the Mechanical and Thermal Characteristics of Poly(lactic acid) Blends," J. Polym. Environ., vol. 26, no. 1, pp. 1-14, Jan. 2018.

A. Jeziorny, "Parameters characterizing the kinetics of the non-isothermal crystallization of poly(ethylene terephthalate) determined by d.s.c.," Polymer (Guildf)., vol. 19, no. 10, pp. 1142-1144, Oct. 1978.

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Solid State Phenomena, ISSN: 1662-9779, Vol. 390, pp 39-50

doi: 10.4028/p-qK6j3v

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

Submitted:
2025-12-19

Revised:
2026-02-17

Accepted:
2026-02-24

Online:
2026-04-17

Laser-Induced Solid-State Foaming of HIPed Ti6Al4V-ELI Alloy

The original version of this paper is available on https://www.scientific.net/SSP.390.39.pdf

Guglielmi Pasquale 1,a*, Cusanno Angela 2,b, Piccininni Antonio 2,c, Fulco Emanuele 1, d, Sorgente Donato 2,e and Palumbo Gianfranco 2,f

1 Department of Engineering, University of Basilicata, via Ateneo Lucano 10, 85100, Potenza, Italy

2 DMMM - Politecnico di Bari, Viale Japigia 182, Bari 70126, Italy

a * pasquale.guglielmi@unibas.it, b angela.cusanno@poliba.it, c antonio.piccininni@poliba.it,

d  emanuele.fulco@unibas.it, e  donato.sorgente@poliba.it, f  gianfranco.palumbo@poliba.it

Keywords: Ti6Al4V-ELI alloy, laser-induced solid-state foaming, hot isostatic pressing, porosity evolution, material modelling.

Abstract

The present work proposes a novel strategy that significantly shortens Solid-State Foaming (SSF) times, delivering a substantial improvement in process efficiency and paving the way for faster production of customized and functionalized prosthetic components. In particular, the evolution of porosity was evaluated in terms of both volume fraction and mean pore diameter and its subsequent effect on microhardness in a Ti6Al4V-ELI alloy that was initially densified via Hot Isostatic Pressing (HIP) and then subjected to Laser-Induced Solid-State Foaming (LISSF). This acronym was introduced by the authors to underline the originality of this approach, which is not reported in the existing literature. Localized heat treatments were performed using a CO2 laser source at a target temperature of 1020∘C, with three distinct dwell times (120, 240, and 360 seconds). To predict density variations and the resulting mechanical properties, two analytical models were implemented and validated: (i) the Johnson-Mehl-Avrami-Kolmogorov (JMAK) kinetic model, which effectively described the time-dependent evolution of porosity and pore growth under different thermal regimes (based on conduction phenomena vs. direct laser exposure) and (ii) the Ryshkewitch-Duckworth (RD) model, which was used to correlate the exponential decay of microhardness with increasing porosity. The experimental results and regression analyses confirm the high predictive accuracy of both models ( R2 greater than 0.95 ), demonstrating the feasibility of the LISSF process for fabricating titanium components with locally controlled porosity for biomedical applications with reference to the manufacturing of customized and functionalized prosthetic components, ensuring both structural reliability and enhanced performance. On the other side, experimental results demonstrated that process parameters play a critical role in the microstructural evolution: specifically, increasing the dwell time to 360 s under direct laser exposure (1020∘C) led to a maximum porosity fraction of approximately 30% and a growth in mean pore diameter up to about 35μ m.

Introduction

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The growing demand for metallic materials with customized properties is fueling the development of innovative strategies for creating porous structures, particularly for biomedical and aerospace applications [1]. Among these, porous titanium alloys stand out for their unique combination of high specific strength and exceptional biocompatibility, making them the material of choice for orthopedic implants and lightweight structural components. In the biomedical field, controlled porosity promotes osseointegration and mitigates stiffness mismatch with bone, while in aerospace engineering, it enables significant weight reduction without sacrificing mechanical integrity [2].

Several techniques have traditionally been employed to introduce porosity into Titanium (Ti) alloys. Powder metallurgy and space-holder methods rely on removable agents during sintering, but often struggle to achieve uniform pore distribution and connectivity [3]. Additive manufacturing (AM) technologies, such as selective laser melting (SLM) and electron beam melting (EBM), have revolutionized the design of complex porous architectures with unparalleled precision; however, these processes remain energy-intensive and require advanced equipment [4]. Alternative approaches, including chemical foaming and gas-phase reactions, have also been explored, though they typically
involve intricate chemistries and may introduce contamination or compromise mechanical performance [5].

In this scenario, Solid-State Foaming (SSF) stands out as a key enabler for next-generation prosthetic manufacturing, combining structural reliability with tailored porosity for enhanced osseointegration [6]. In fact, by avoiding melting, SSF minimizes residual stresses, distortion, and microstructural degradation, ensuring the integrity of the original component. In particular, LaserInduced Solid-State Foaming (LISSF) provides a means to introduce controlled porosity into Ti alloys while drastically reducing foaming time. Unlike AM, which constructs parts layer by layer, LISSF acts directly on fully dense components, locally inducing porosity through thermally activated mechanisms such as creep and diffusion, without melting the material. [7]. Moreover, its localized nature enables selective foaming in predefined regions, making it compatible with pre-existing parts and reducing processing costs compared to AM. A further advantage lies in its capability to create tailored porosity gradients within a single component, a feature particularly valuable for biomedical implants that require a dense core for load bearing and a porous surface to enhance osseointegration [8].

Supported by experimental findings, numerical modeling based on analytical formulations is essential for the design of manufacturing processes aimed at producing components with variable density through solid-state foaming. Such models enable the prediction of porosity evolution and resulting mechanical properties as functions of key process parameters, including temperature and dwell time [7]. This predictive capability allows engineers to optimize thermal cycles and foaming conditions to achieve tailored porosity gradients, which are critical for aerospace structures requiring weight reduction without compromising integrity. By minimizing trial-and-error and guiding process optimization, these models accelerate the development of tailored, high-performance components.

According to this, the present study investigates the application of LISSF to Ti-6Al-4V-ELI alloy previously densified by HIP. The research focuses on understanding porosity evolution - quantified by fraction (P) and average diameter (d) and its influence on mechanical properties (Vickers microhardness). To interpret and predict material behavior, two analytical models were employed: the Johnson-Mehl-Avrami-Kolmogorov (JMAK) model [9] with Arrhenius dependence, describing the kinetics of porosity growth and the Ryshkewitch-Duckworth (RD) model [10,11], correlating hardness with porosity. These models provide a physically meaningful framework for process optimization and pave the way for advanced applications where controlled porosity is critical.

Material and Methods

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Material.

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The proposed approach was focused on a Ti6Al4V-ELI produced by means of HIP. The chemical composition of the investigated material is presented in Table 1.

Table 1. Chemical composition (weight %) of the investigated Ti6Al4V-ELI.

Al% | V% | Fe% | C% | N% | H% | O% | Ti
6.21 | 3.80 | 0.16 | 0.008 | 0.011 | 0.001 | 0.008 | Bal.

The material was produced through a HIP process carried out at 1000∘C under an applied pressure of 120 MPa , using titanium powders with an average particle size ranging from 70 to 100μ m and an argon content corresponding to a pressure of 0.4 MPa . According to this, the HIPed material is affected by the presence of reduced and pressurized microporosities (about 0.01 and 1μ m as P and d, respectively) [12].

Experimental Activity.

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The LISSF experiments were carried out using a CO2 laser source with a maximum power of 2.5 kW , combined with a LASCON® temperature control system which continuously controls the laser power to keep the temperature constant using data acquired by a pyrometer. As shown in

Fig. la, the experimental setup features a square laser spot with a side length of 10 mm , kept fixed at the center of the specimen.

The sample is supported only at its ends, leaving the remaining part suspended and free from direct supports. This configuration ensures localized heating while minimizing mechanical and thermal stresses during each experimental test.

[picture]

Fig. 1. LISSF experimental setup (a) and main dimension ( mm ) of the sample (b) with schematic of section extraction: central slices at 0 mm(1),±10 mm(2),±20 mm(3) and ±30 mm(4) from specimen center.

Fig. 1. LISSF experimental setup (a) and main dimension ( mm ) of the sample (b) with schematic of section extraction: central slices at 0 m m ( 1 ) , ± 10 m m ( 2 ) , ± 20 m m ( 3 ) and ± 30 m m ( 4 ) from specimen center.

Each experimental laser treatment was carried out three times to ensure repeatability and reliability of the results. A sacrificial sample was used to monitor the temperature (T) along the longitudinal direction during the LISSF using three K-type thermocouples, as illustrated in Fig. lb. This allowed the derivation of a fourth-order polynomial (Equation 1) describing the stationary temperature T(∘C), after the initial heating step, as a function of longitudinal position ( x ) within the specimen, enabling direct correlation of measured porosity values after LISSF to process temperature.

T=0.0004x4−1.1898x2+1020(1)

The provided Table 2 outlines the experimental plan designed to analyze the thermal effects on the specimens. This details 12 specific measurement points (ID) characterized by varying dwell times and temperatures distributed along the longitudinal direction of each sample.

The experimental conditions can be categorized into two distinct regions: (i) the zone directly exposed to laser radiation (IDs 1, 5, and 9) and (ii) the zones not exposed (all remaining IDs).

Table 2. Experimental conditions investigated (T: Temperature; DT: Dwell Time).

ID | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12
T, °C | 1020 | 933 | 672 | 238 | 1020 | 933 | 672 | 238 | 1020 | 933 | 672 | 238
DT, sec | 120 | 120 | 120 | 120 | 240 | 240 | 240 | 240 | 360 | 360 | 360 | 360

At the end of each test, every specimen was cut to extract transversal slices starting from the center and extending up to 30 mm outward, with a step of 10 mm . The main dimensions of the samples and a schematic of the section extraction is reported in Fig. 1b.

These sections were then prepared metallographically through grinding and polishing to obtain surfaces suitable for microstructural observation.

Subsequently, the prepared sections were analyzed using optical microscopy supported by quantitative image analysis performed using the ImageJ (R) software which allowed accurate measurement of P and d values according to both dwell time and temperature. Microhardness tests allowed to determine the mechanical characteristics of the heat-treated material according to the different process conditions. In this regard, measurements were performed using a fully automated Qness Q10 tester in compliance with the Vickers standard (DIN EN ISO 6507).

For each experimental condition, the average Vickers microhardness was calculated from 6 random indentations within the extracted transverse sections, applying a 0.5 kg load by maintaining the load for 15 seconds.

Material Modeling.

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As outlined in the introduction, two distinct analytical models were employed to predict the actual material behavior after LISSF: (i) JMAK model with Arrhenius dependence, for describing porosity evolution and (ii) RD model, for correlating microhardness with porosity.

The JMAK approach was adapted to describe the kinetics of porosity growth during solid-state foaming. This model interprets porosity evolution as a sigmoidal process: an initial slow stage, a rapid intermediate growth and a final plateau before pore opening. The Arrhenius dependence accounts for thermal activation, where higher temperatures accelerate creep and superplastic flow, promoting faster pore expansion [7]. The adopted formulations are reported below (Equation 2 and 3).

P(t,T)=Pmax[1−exp(−(k(T)t)m]k(T)=k0exp(−Q/(RT))

(2)(3)

where P is the porosity fraction, Pmax  is representative of the maximum porosity, m is Avrami exponent, k0 is kinetic factor, Q is activation energy and R is the gas constant ( 8.314[ J/(mol·K)] ). Temperature values are consistently expressed in Kelvin. In more detail, Pmax represents the structural saturation limit dictated by the alloy's microstructural stability. This parameter is critical for predicting the onset of pore coalescence and potential collapse under prolonged exposure [13]. In agreement with previous findings on high-strength porous Ti−6Al−4 V foams synthesized by solidstate powder processing [14], a porosity fraction of approximately 0.50 was considered a suitable compromise between structural integrity and functional performance; therefore, this value was adopted as Pmax in the present study.

Once P is predicted, pore size can be estimated through simple scaling relationships, supporting a comprehensive understanding of foaming behavior.

Finally, assuming that the three-dimensional development of porosity can reasonably be approximated as spherical, the prediction of d (expressed in micrometers) can be directly related to P through a first-order relationship (Equation 4), which is established based on the correlation between the experimental results obtained for the corresponding d and P values:

d=91.964·P+7.6611(4)

The RD model explains the exponential decrease in microhardness as porosity increases. According to this approach, the hardness of a porous material (HV) is related (Equation 5) to the hardness of the fully dense material ( H0 ) through an exponential function, where the sensitivity coefficient (b) quantifies how strongly hardness responds to porosity. For the material investigated in this work, the microhardness of the HIPed condition is equal to 302.5±1.8μHV0.5; however, to better assess the predictive capability of the model, this value was not directly imposed as a constraint for determining b .

H(P)=H0eb·(P)(5)

In addition to porosity fraction, the model can be adapted to use pore size as a variable (Equation 6), maintaining the same exponential trend. The modified formulation is proposed to account for the average pore diameter, expressed as:

H(P)=H0eb′·(d/100)(6)

where b ' is the modified sensitivity coefficient. This approach assumes that pore size can serve as a proxy for porosity, maintaining the exponential decay trend observed in the original RD model. Such a variant is particularly useful when porosity fraction is difficult to measure directly, while pore size distribution is readily available from microstructural analysis.

All constants for both the JMAK and RD models were obtained through direct regression against the full experimental dataset using the Root Mean Square Error (RMSE) minimization, expressed as summarized in the Equation (7).

RMSE=1n∑i=1n(yexp−ymod)2(7)

where n represents the number of data points, yexp  and ymod respectively are the experimental and the predicted (model) values.

Results and Discussion

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The results analyze the evolution of P,d and microhardness ( μHV0.5 ) as a function of temperature and dwell time.

Porosity Evolution and Pore Size

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The morphological characteristics of the LISSF specimens; specifically, P and d , are presented in Fig. 3.

The material exhibits markedly different responses under the two thermal exposure conditions. At 1020∘C, which corresponds to the center of the specimen directly exposed to the laser radiation, the material exhibits a significant foaming effect. P (Fig. 3a) at this temperature ranges from 0.263 (dwell time equal to 120 s ) to 0.30(360 s).

Conversely, the regions not directly exposed to laser irradiation ( 933∘C,672∘C, and 238∘C ) exhibit a marked reduction in porosity, stabilizing below 0.05 regardless of the dwell time, as also supported by the micrographs shown in Fig. 2.

[picture]

Fig. 2. Micrographs referring to a LISSF in which a temperature of 1020∘C was set at the laser position for a total dwell time of 120 s .

Fig. 2. Micrographs referring to a LISSF in which a temperature of 1020 ∘ C was set at the laser position for a total dwell time of 120 s .

This observation suggests that the activation energy - or the thermal threshold required for significant pore expansion - is primarily achieved under direct laser exposure. Indeed, this assumption is supported by comparing the porosity obtained at the same temperature when using a conventional furnace heating process, already documented in a previous work [12], with that obtained using laser-based heating. When the two conditions reported below (Table 3) are compared, despite the SSF temperature being the same ( 1020∘C ), a markedly greater capability for pores to grow in size can be observed in the case of LISSF, despite the much longer exposure ( 3600 s ) at high temperature experienced by the furnace-heated sample.

Table 3. Comparison between SSF in furnace and LISSF at 1020∘C.

Temperature, °C | Time, s | P | d, μm
Furnace | 1020 | 3600 | 0.17 | 11.00
LISSF | 1020 | 120 | 0.26 | 33.17

Regarding d, shown in Fig. 3b, a similar trend was recorded. The area subject directly to the laser irradiation result in significantly larger pores, with diameters ranging between 33.17 (dwell time equal to 120 s ) and 34.89μ m(360 s) at 1020∘C.

A slight time-dependence is observed in the direct interaction zone. As the dwell time increases from 120 s to 360 s , the average pore diameter shows a moderate increase. This is likely attributable to the coalescence of pores (Ostwald ripening) [15].

At lower temperatures ( 238∘C−933∘C ), the pore diameter remains relatively constant and small (12−13μ m), indicating that the only conduction heating alone is insufficient to drive further pore growth or coalescence.

[picture]

Fig. 3. Experimental results in terms of (a) P and (b) d according to the different temperature and dwell time investigated.

Fig. 3. Experimental results in terms of (a) P and (b) d according to the different temperature and dwell time investigated.

Microhardness Analysis.

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The mechanical response of the investigated Ti alloy was evaluated through Vickers microhardness testing, as illustrated in Fig. 4. The microhardness exhibits an inverse relationship with the processing temperature and the resulting porosity.

[picture]

Fig. 4. Experimental results in terms of Vickers microhardnesses according to the different temperature and dwell time investigated.

Fig. 4. Experimental results in terms of Vickers microhardnesses according to the different temperature and dwell time investigated.

The lowest hardness values, ranging between 165 (dwell time equal to 360 s ) and 175HV(360 s), are recorded at 1020∘C. This softening can be attributed to the high-volume fraction of porosity which reduces the effective load-bearing area. As the distance from the laser spot increases (temperature decreases), the microhardness recovers significantly. At 933∘C, values range from 252.6 to 261.6 HV and they reach a maximum of nearly 290 HV at 238∘C. When compared to the effect of
temperature, the dwell time appears to have a minor influence on microhardness within the examined range, particularly in the peripherical area of each specimen.

In general, the data confirms that the LISSF is highly localized. Effective foaming, characterized by high porosity (P greater than 0.25 ) and large pores (d greater than 30μ m ), is restricted to the direct laser irradiation zone ( 1020∘C ). The surrounding material retains a denser structure and higher mechanical hardness, resembling the AR properties of the alloy.

Material Modeling Results.

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The analysis compares the experimental data (Exp) with the numerical predictions (Model) for P,d and the resulting microhardness ( μHV0.5 ).

Porosity Modeling.

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As outlined earlier, P was modeled using the JMAK (Johnson-Mehl-Avrami-Kolmogorov) approach. The accuracy of the model was evaluated separately for two distinct regions: (i) the zone directly exposed to laser radiation (IDs 1, 5, and 9) and (ii) the zones not exposed (all remaining IDs). The identification of separate model parameters for the exposed and unexposed zones is required due to the fundamentally different thermal profiles and kinetic mechanisms governing pore evolution in the two regions. Direct laser irradiation produces extremely high heating rates and steep thermal gradients, reaching temperatures near or above the β-transus and activating diffusion- and creep-based expansion mechanisms that are not observed under conduction-based heating [7,16,17]. Consequently, porosity evolution follows distinct kinetic paths and a single parameter set would not be able to simultaneously reproduce the behavior of both regimes while maintaining physical consistency and predictive accuracy.

[picture]

Fig. 5. JMAK model (Equations 2 and 3 ) prediction for P with reference to the zones not exposed to the laser radiation.

Fig. 5. JMAK model (Equations 2 and 3 ) prediction for P with reference to the zones not exposed to the laser radiation.

The model predictions for the zones not exposed show a strong correlation with the experimental findings. The bar chart comparison indicates that the modeled values (Model) closely track the experimental porosity (Exp) across the identified samples. The linear regression analysis confirms this agreement, yielding a RMSE of 0.959 .
Table 4 summarizes the values of the constants determined considering Equations (2 and 3).

Table 4. Constant of JMAK model for P with reference to the zones not exposed to the laser radiation.

Pmax | K₀, s⁻¹ | m | Q, J/mol | RMSE
0.5 | 0.04434 | 0.213 | 6686.9 | 0.959

For the samples directly exposed to the laser spot (1020∘C), the model demonstrates excellent predictive capability. The comparison between experimental and modeled P reveals negligible deviation. The linear regression analysis for this regime exhibits a perfect fit with an RMSE of 1 , suggesting that the JMAK model captures the porosity evolution under direct laser exposure with high precision. Although the laser-exposed region provides only three experimental porosity values, these points span a broad temporal range (120, 240 and 360 s ) and appear to be sufficient to identify the two unknown parameters of the JMAK law at constant temperature. Indeed, at fixed value of temperature the model reduces to a two-parameter problem ( k and m ).

[picture]

Fig. 6. JMAK model (Equations 2 and 3) prediction for P with reference to the zones directly exposed to the laser radiation.

Fig. 6. JMAK model (Equations 2 and 3) prediction for P with reference to the zones directly exposed to the laser radiation.

Table 5 summarizes the values of the constants determined according to the laser regime.

Table 5. Constant of JMAK model for P with reference to the zones directly exposed to the laser radiation.

Pmax | K0, s⁻¹ | m | Q, J/mol | RMSE
0.5 | 2.383 | 0.195 | 114937.2 | 1.000

Laser heating produced localized high temperatures near the β-transus (about 980∘C ), resulting in greater deformability and significantly faster kinetics compared to conduction heating [16]. This effect stems from rapid thermal cycling and high peak temperatures that activate more dynamic nucleation and growth processes in the β-phase. As a result, laser-treated samples exhibit significantly higher porosity under identical process durations [17]. Consequently, laser-treated regions reached higher porosity levels within the same time frame. The model is effective because it captures the observed trend with few parameters that have clear physical meaning, such as the Avrami exponent (growth mechanism), activation energy (thermal sensitivity) and maximum porosity (structural limit).

The modeling of d also displays a high degree of accuracy when compared to the experimental measurements (Exp). The bar chart comparison shows that the model correctly predicts the significant variation in pore sizes, particularly distinguishing between smaller pores (about 12μ m ) and larger pores (33−35μ m) found in specific samples (e.g. IDs 1,5,9) directly exposed to the spot laser action.

The linear regression analysis for p yields an RMSE of 0.997 , confirming that the model effectively replicates the experimental trends regarding the dimensional growth of the porosity.

[picture]

Fig. 7. Empirical model (Equation 4) prediction model for d.

Fig. 7. Empirical model (Equation 4) prediction model for d.

Microhardness Modeling.

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The comparison between experimental Vickers microhardness (Exp) and the model predictions (Model) shows consistent agreement across all tested conditions, with values generally ranging between 170 and 280 HV . As porosity grows, the effective load bearing area is reduced and stress effects become more prominent, leading to a loss in hardness even for small increments of porosity. This behavior aligns with the principles of cellular solids described by Gibson and Ashby [13].

[picture]

Fig. 8. RD model (Equation 5) prediction for P .

Fig. 8. RD model (Equation 5) prediction for P .

[picture]

Fig. 9. Modified RD model (Equation 6) prediction for d.

Fig. 9. Modified RD model (Equation 6) prediction for d.

The regression analysis for the microhardness data yields an RMSE of 0.962 as regard P (Fig. 8) and similarly equal to 0.9631 in the subsequent analysis for d (Fig. 9), indicating that the material modeling approach successfully predicts the mechanical property changes induced by the thermal cycles.

Table 6 summarizes the values of the constants determined with reference to both P and d considering Equations (5 and 6). Although the calibrated values of b and b′ are very close to each other, the value of H0 obtained for the model associated with P is almost identical to the experimental one ( 302.5 vs 305.6 ), demonstrating a strong predictive capability.

Table 6. Constant of RD model for both P and d.

Variable | H0, μHV0.5 | b | b', μ⁻¹
P | 305.6 | 2.1 | -
d | 362.1 | - | 2.2

Conclusion

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In the present work, the feasibility of a LISSF process for HIPed Ti6Al4V-ELI alloy was successfully demonstrated. The study investigated the effects of laser processing parameters on the reactivation and growth of argon-filled pores, leading to the following assessments:

The LISSF technique allows for the generation of porous structures with tunable characteristics. A direct correlation was established between the input energy (governed by controlled laser power to reach a constant temperature) and the resulting porosity. Higher temperatures and prolonged exposure times promoted significant pore expansion, transforming the material from a nearly fully dense state to a porous structure suitable for biomedical applications.

Two distinct heating regimes were identified according to two different regions of the LISSFed specimens: (i) the region directly exposed to laser irradiation and (ii) the adjacent region outside this exposure. The first region exhibited the highest porosity fractions (up to 0.30) and larger pore diameters, whereas the other one showed finer and more uniformly distributed porosity, underscoring the potential for designing functionally graded materials.

A kinetic model based on the JMAK equation was successfully applied to predict both the porosity fraction (P) and the average pore diameter (d). The model showed excellent agreement with experimental data (RMSE greater than 0.95 ) for both variables and laser heating regimes. Furthermore, a linear relationship was validated to predict the microhardness evolution as a function of the thermal history.

Complementarily, R-D model provided a robust description (RMSE greater than 0.96 ) of the mechanical response, confirming the exponential decay of microhardness with increasing porosity and pore size.
In conclusion, LISSF of HIPed titanium offers a promising, flexible route to manufacture components with locally tailored porosity, bridging the gap between solid, structural parts and porous, osseointegrative surfaces. Future developments will focus on calibrating models capable of predicting additional mechanical properties - such as elastic modulus, yield strength, ultimate tensile strength (UTS) and elongation at fracture - based on the porosity observed after LISSF, as well as on optimizing scanning strategies for the fabrication of complex 3D-graded lattice structures.

References

The original version of this paper is available on https://www.scientific.net/SSP.390.39.pdf

Miko T, Petho D, Gergely G, et al. A Novel Process to Produce Ti Parts from Powder Metallurgy with Advanced Properties for Aeronautical Applications. Aerospace. 2023;10(4).

Yang XL, Du XF, Xu ZL, et al. Progress in processing of porous titanium: a review. Rare Metals. Springer; 2024. p. 1932-1955.

Yadav MK, Yarlapati A, Aditya YN, et al. Processing and Development of Porous Titanium for Biomedical Applications: A Comprehensive Review. Journal of Manufacturing and Materials Processing [Internet]. 2025;9(12):401.

Wu D, Li Y, Zhu W, et al. Fabrication and biocompatibility of a 3D-printed porous Ti-25Ta alloy scaffold. Discover Applied Sciences. 2025;7(7).

Tian JH, Li YH, Wu XJ. Research Progress on Biocompatible Sintered Porous Titanium Alloys for Hard Tissue Implants. Transactions of the Indian Institute of Metals. Springer; 2025.

Guglielmi P, Piccininni A, Cusanno A, et al. Manufacturing of a hybrid component in Ti6Al4V-ELI alloy by combining diffusion bonding and superplastic forming. Materials Research Proceedings. Association of American Publishers; 2023. p. 37-44.

Oppenheimer S, Dunand DC. Solid-state foaming of Ti-6A1-4V by creep or superplastic expansion of argon-filled pores. Acta Mater. 2010;58(13):4387-4397.

Murray NGD, Dunand DC. Effect of initial preform porosity on solid-state foaming of titanium. J Mater Res. 2006;21(5):1175-1188.

Duckworth W. Discussion of Ryshkewitch Paper by Winston Duckworth. Journal of The American Ceramic Society-Ryshkewitch. Vol. 36, No. 2.

Ryshkewitch E. Compression Strength of Porous Sintered Alumina and Zirconia. 1953.

Fanfoni M, Tomellini M. The Johnson-Mehl- Avrami-Kohnogorov model: A brief review. Il Nuovo Cimento D [Internet]. 1998;20(7):1171-1182.

Guglielmi P, Palumbo G. Production of porous titanium structures by combining hot isostatic pressing and solid-state foaming. CIRP J Manuf Sci Technol. 2024;52:58-72.

Gibson LJ, Ashby MF. Cellular Solids: Structure and Properties [Internet]. 2nd ed. Cambridge Solid State Science Series. Cambridge: Cambridge University Press; 1997. Available from: .

Lee MH, Kim KB, Han JH, et al. High strength porous Ti-6Al-4V foams synthesized by solid state powder processing. J Phys D Appl Phys [Internet]. 2008;41(10):105404.

Yu W, Zhou X, Kanj MY. Microfluidic Investigation of Foam Coarsening Dynamics in Porous Media at High-Pressure and High-Temperature Conditions. Langmuir. 2022;38(9):28952905.

de Freitas FE, da Silva RAL, Takahashi RJ, et al. Influence of Laser Remelting on Creep Resistance in Ti-6Al-4V Alloy with Thermal Barrier Coating. Coatings. 2025;15(2).

Cerri E, Ghio E. On Strain-Hardening Behavior and Ductility of Laser Powder Bed-Fused Ti6Al4V Alloy Heat-Treated above and below the -Transus. Materials. 2024;17(14).

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/material-behaviour-modelling/978-3-0364-1987-9

Solid State Phenomena, ISSN: 1662-9779, Vol. 390, pp 51-61

doi: 10.4028/p-Vj1faR

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

Submitted:
2025-12-23

Revised:
2026-02-21

Accepted:
2026-02-22

Online:
2026-04-17

1D Stress Evolution Prediction via Thermodynamically-informed Neural Networks

The original version of this paper is available on https://www.scientific.net/SSP.390.51.pdf

Alireza FALLAHNEJAD 1,a,⋆, Emin Semih PERDAHCIOĞLU 1, b, and Ton VAN DEN BOOGAARD 1,c

1 Chair of Nonlinear Solid Mechanics, University of Twente, The Netherlands

a  a.fallahnejad@utwente.nl, b  e.s.perdahcioglu@utwente.nl, c  a.h.vandenboogaard@utwente.nl

Keywords: Constitutive Modeling, Elastoplasticity, Machine Learning, Thermodynamic Consistency

Abstract

Accurate modeling of elastoplastic behavior is crucial for forming simulations, yet conventional constitutive laws require extensive calibration and often fail to generalize across diverse loading paths. To address this limitation, a thermodynamically informed neural-network framework is proposed for predicting one-dimensional stress evolution. The model integrates physical consistency into a data-driven formulation by coupling two neural components: one learns the state evolution, predicting increments of the internal variable, while the other approximates the Helmholtz free-energy potential, from which stresses are obtained via automatic differentiation. Synthetic datasets generated from randomized strain paths with power-law hardening were used for training, ensuring broad coverage of nonlinear responses. The model successfully reproduces monotonic, unloading, reverse, and random loading behaviors with minimal error accumulation and stable recursive inference. Owing to its incremental formulation, the framework maintains predictive accuracy beyond the trained strain range, offering a physically interpretable and data-efficient alternative to conventional constitutive models.

Introduction

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The accurate representation of material behavior under complex loading conditions is essential for predictive forming simulations. Conventional constitutive models are generally categorized as empirical and physically based. Empirical models rely on phenomenological observations and parameter fitting to experimental data [1], which limits their generalization and reliability when extrapolated beyond the calibrated range. Physically based or mechanistic models, such as crystal plasticity, improve predictive capability by linking macroscopic response to microstructural mechanisms. However, their application remains computationally demanding and requires detailed material characterization, which restricts their use in large-scale forming simulations.

Recent advances in machine learning (ML) have motivated the development of data-driven constitutive models that can infer complex stress-strain relationships directly from data. Model-free datadriven frameworks, first introduced by Kirchdoerfer and Ortiz [2], bypass empirical laws by formulating mechanics directly from experimental data and conservation principles. Although conceptually appealing, these approaches require extensive datasets and face challenges related to dimensionality and data sparsity [3-9]. Neural-network-based constitutive models have therefore gained attention for their ability to approximate highly nonlinear mappings between strain and stress. Early works by Ghaboussi et al. [10] and Lefik and Schrefler [11] demonstrated the feasibility of such black-box models, while subsequent studies extended them to various materials and loading conditions [12-18]. Despite their success, these purely data-driven models often lack physical interpretability and may violate thermodynamic laws, leading to poor extrapolation outside the training domain.

To overcome these limitations, recent research has focused on embedding physical knowledge into machine-learning models. Physics-informed neural networks (PINNs) [19] and constrained or physically enhanced neural networks [20,21] incorporate mechanical or thermodynamic constraints directly into the learning process. By integrating physical consistency into the architecture or loss function, these approaches ensure that model predictions comply with energy conservation, material stability,
and dissipation principles. A particularly promising development is the class of thermodynamicsbased artificial neural networks (TANNs) [22,23], which define stresses and dissipation rates through derivatives of learned free-energy and dissipation potentials. These models combine the flexibility of neural networks with the rigor of continuum thermodynamics, resulting in improved generalization and interpretability.

Motivated by these recent developments, the present study proposes a thermodynamically informed neural-network framework for modeling one-dimensional elastoplastic stress evolution. The model couples a learned state-evolution law with a free-energy potential, from which stresses are derived via automatic differentiation, ensuring thermodynamic admissibility. Training is performed on synthetic data generated from random strain paths to capture nonlinear and path-dependent behavior. The model's performance is assessed against unseen loading histories to evaluate its generalization capability. This work serves as a first step toward developing thermodynamically consistent data-driven constitutive surrogates for forming applications, bridging the gap between classical physics-based models and modern machine-learning approaches.

Methodology

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Thermodynamics backbone. The proposed constitutive framework is developed under the smallstrain assumption, ensuring infinitesimal deformations. The thermodynamic notation and structure adopted in this work follow the classical treatment by Asaro and Lubarda [24]. The first law of thermodynamics in local form is

ρe˙=σ:ε˙+ρr−∇·𝐪,(1)

where ρ is the mass density, e the internal energy per unit mass, σ the Cauchy stress, ε the infinitesimal strain tensor, r the internal heat source per unit mass, and 𝐪 the heat flux.

Moreover, the second law of thermodynamics, expressed through the Clausius-Duhem inequality, imposes non-negative entropy productions

Ps=ρs˙−ρrT+1T∇·𝐪−𝐪T2·∇T≥0(2)

where s is the specific entropy and T the absolute temperature. Introducing the Helmholtz free energy, ψ, and using the Legendre transformation in terms of strain and temperature as the primary variables, and combining the first and second laws following standard manipulations, one obtains the dissipation inequality in the form

𝒟=𝒟m+𝒟q,𝒟m=σ:ε˙−ρ(ψ˙+T˙s),𝒟q=−𝐪T·∇T(3)

with both contributions assumed, in a strong form, non-negative ( 𝒟m,𝒟q≥0 )
In addition, to describe the material behavior, internal state variables ζ=(ζ1,…,ζn) are introduced, leading to the free energy function and its rate

ψ=ψ^(T,ε,ζ),ψ˙=∂ψ^∂ε:ε˙+∂ψ^∂TT˙+∑i∂ψ^∂ζiζ˙i.(4)

Substituting into the dissipation inequality and applying the Coleman-Noll procedure yields the constitutive relations:

σ=ρ∂ψ^∂εs=−∂ψ^∂T𝒟m=−ρ∑i∂ψ^∂ζiζ˙i≥0

(5)(6)(7)

These relations ensure thermodynamic consistency of the material model, linking stresses, temperature, and internal variables through a potential-based formulation.

Model formulation. Building upon the thermodynamic foundations established in the previous section, the present framework formulates a constitutive model that integrates thermodynamic principles into a neural representation of material behavior. The approach retains the structure of classical continuum thermodynamics, in which the Helmholtz free energy density ψ characterizes the recoverable part of the internal energy, and the material dissipation 𝒟m accounts for irreversible processes. According to the Clausius-Duhem inequality, the total dissipation must remain non-negative, 𝒟m≥0. Instead of prescribing explicit analytical forms for ψ and the evolution of internal variables, the present framework approximates these quantities through neural representations trained on representative data. This formulation allows the model to capture nonlinear and path-dependent responses while preserving full thermodynamic admissibility.

As shown in Fig. 1, the constitutive framework consists of two complementary neural components. The first, denoted NNζ, represents the state-evolution law.It predicts the increment of the internal variable, expressed as the equivalent plastic strain increment Δεpl, from the current mechanical state. This quantity defines the magnitude of irreversible deformation within each incremental step and thus governs the dissipative behavior. The second network, NNψ represents the energetic potential. Given the current strain and the updated internal variable, it approximates the Helmholtz free-energy density ψ, from which the corresponding stress and energy state are consistently inferred.

[picture]

Figure 1: Schematic of the interaction between the model's components in the prediction mode.

Figure 1. Schematic of the interaction between the model's components in the prediction mode.

To enhance the physical fidelity of the model, a binary classifier is integrated into the inference pipeline alongside NNζ. The classifier determines whether plastic flow is expected to occur during a given increment and effectively acts as a data-driven yield criterion. It receives the same mechanical inputs as NNζ and outputs a probability 𝒫∈[0,1] indicating the likelihood of plastic deformation. This probability is then combined with the raw prediction of NNζ through a hybrid gating rule, which ensures exact zero increments in elastic regimes while allowing smooth activation of plastic flow when the stress state approaches the yield surface. In this manner, the classifier enforces physical consistency by suppressing spurious plastic updates and reducing error accumulation in sequential predictions.

In incremental form, the model operates according to the recursive update procedure detailed in Algorithm 1, sequentially receiving the total strain increment together with the current stress and internal state variable to compute the updated internal variable, free energy, and resulting stress via automatic
differentiation. The initial state is defined by zero stress and equivalent plastic strain ( σt,0=0,ε¯t,0pl= 0 ). This recursive update ensures that the stress evolution is derived consistently from the underlying energy potential while accommodating the history dependence of inelastic deformation. The state computed at each increment becomes the input for the next, enabling the model to reproduce monotonic, unloading, and reverse-loading behaviors within a single thermodynamic framework. When no plastic evolution occurs, the formulation yields purely elastic and energy-conservative behavior; when plastic flow is active, the predicted increments of the internal variable produce positive dissipation, ensuring 𝒟m≥0. Conceptually the two networks play complementary thermodynamic roles: NNζ governs the irreversible mechanisms contributing to dissipation, while NNψ determines the recoverable energy response. Their coupling forms a closed constitutive mapping, which unites data-driven learning with continuum thermodynamics. By structuring the material behavior around the interplay of energy storage and dissipation, this model formulation provides a general and physically consistent foundation for developing neural constitutive laws.

Algorithm 1 Recursive inference procedure of the model.
    Input: Test data \(\left\{\Delta \varepsilon_{i}\right\}_{i=1}^{N}\)
    Initialize: \(\sigma_{t, 0}, \bar{\varepsilon}_{t, 0}^{\mathrm{pl}}\)
    for \(i=1\) to \(N\) do
        Prepare classifier input: \(\mathbf{x}_{\mathrm{cls}}=\left[\Delta \varepsilon_{i}, \sigma_{t, i-1}, \bar{\varepsilon}_{t, i-1}^{\mathrm{pl}}\right]\)
        Predict probability: \(\mathcal{P}_{i}=\) Classifier \(\left(\mathbf{x}_{\text {cls }}\right)\)
        Prepare \(\mathrm{NN}_{\zeta}\) input: \(\mathbf{x}_{\mathrm{NN}_{\zeta}}=\left[\Delta \varepsilon_{i}, \sigma_{t, i-1}, \bar{\varepsilon}_{t, i-1}^{\mathrm{pl}}\right]\)
        Predict raw increment: \(\Delta \bar{\varepsilon}_{\text {raw }, i}^{\mathrm{pl}}=\mathrm{NN}_{\zeta}\left(\mathbf{x}_{\mathrm{NN}_{\zeta}}\right)\)
        Apply correction: \(\Delta \bar{\varepsilon}_{i}^{\mathrm{pl}}=\operatorname{ApplyMethod}\left(\Delta \bar{\varepsilon}_{\text {raw }, i}^{\mathrm{pl}}, \mathcal{P}_{i}\right)\)
        Update internal variable: \(\bar{\varepsilon}_{t, i}^{\mathrm{pl}}=\bar{\varepsilon}_{t, i-1}^{\mathrm{pl}}+\Delta \bar{\varepsilon}_{i}^{\mathrm{pl}}\)
        Prepare \(\mathrm{NN}_{\psi}\) input: \(\mathbf{x}_{\mathrm{NN}_{\psi}}=\left[\Delta \varepsilon_{i}, \sigma_{t, i-1}, \Delta \bar{\varepsilon}_{i}^{\mathrm{pl}}\right]\)
        Predict free energy: \(\psi_{i}=\mathrm{NN}_{\psi}\left(\mathbf{x}_{\mathrm{NN}_{\psi}}\right)\)
        Compute stress: \(\sigma_{t, i}=\frac{\partial \psi_{i}}{\partial \Delta \varepsilon_{i}}\)
    end for
    Return: \(\left\{\Delta \bar{\varepsilon}_{i}^{\mathrm{pl}}, \psi_{i}, \sigma_{t, i}\right\}_{i=1}^{N}\)

Data Generation and Training Procedure

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Training data. The training datasets were produced using a fully automated numerical framework developed for the generation of thermodynamically consistent data under general three-dimensional finite-strain loading. The framework couples a constitutive solver, implementing the thermodynamic backbone described in the last section, with a data-management module for randomized strain-path generation and result extraction. The framework implements the von Mises yield criterion with isotropic hardening and power-law flow and hardening functions. At each integration step, the framework evaluates stress and internal variables by incrementally solving the constitutive equations through an explicit return-mapping scheme that guarantees satisfaction of the dissipation inequality.

Although the framework supports general 3D loading paths, in the present work, only the normal stress-strain response was used to train the neural networks, making the generated data effectively onedimensional. Randomized loading paths were imposed by prescribing sequences of strain increments with varying sign and amplitude, enabling the reproduction of monotonic, cyclic, and non-proportional behaviors within a single dataset. For each increment, the algorithm recorded the strain, stress, and internal variable.

The material parameters correspond to a typical metallic behavior with Young's modulus {Δεi}i=1N 200 GPa , Poisson's ratio σt,0,ε¯t,0pl, initial yield stress i=1, the hardening parameters of N and 𝐱cls=[Δεi,σt,i−1,ε¯t,i−1pl]. Using the mentioned framework, a total of 1500 and 750 samples were
generated as the training and validation datasets, respectively. Each data instance contains the tuple 𝒫i=, where the first three quantities serve as the inputs of (𝐱cls ), the fourth one is the output of NNζ and at the same time the input of 𝐱NNζ=[Δεi,σt,i−1,ε¯t,i−1pl], and the last one is the output of Δε¯raw ,ipl=NNζ(𝐱NNζ). Fig. 2a illustrates the histogram distributions of the training and validation datasets for each input component, as well as the outputs for both networks. Additionally, for a more comprehensive overview, Fig. 2b includes the overall ranges of variation in total strain and stress for the training data.

It is worth noting that the model does not require explicit specification of elastic constants such as Young's modulus or Poisson's ratio. Owing to the thermodynamic structure and the incremental formulation, the model infers the elastic response directly from the stress-strain couples provided in the training set. Thus, all necessary elastic properties are implicitly identified from the data itself, eliminating the need for prior calibration and allowing the framework to operate solely on raw stressstrain histories.

[picture]

Figure 2: (a) Statistical distribution for the networks' components. Training and validation datasets are distinguished. (b) The overall ranges of variation in total strain and stress for the training data.

Figure 2. (a) Statistical distribution for the networks' components. Training and validation datasets are distinguished. (b) The overall ranges of variation in total strain and stress for the training data.

Network architectures and training procedure. Two independent feed-forward neural networks (FFNNs) were employed to represent the energetic and dissipative mechanisms of the constitutive model: Δε¯ipl=ApplyMethod(Δε¯raw ,ipl,𝒫i) for state evolution and ε¯t,ipl=ε¯t,i−1pl+Δε¯ipl for the free-energy potential. Training them separately al-
lows each network to specialize in its respective thermodynamic role, thereby improving convergence stability and overall training efficiency. The network architectures and hyperparameters, including the number of layers, neurons, activation functions, and learning-rate schedules, were selected through a combined grid-search procedure and fine-tuned using the Keras Tuner framework to achieve an optimal balance between accuracy and computational cost.

The first network, NNψ, comprises two hidden layers with 48 neurons each, activated by Leaky 𝐱NNψ=[Δεi,σt,i−1,Δε¯ipl] functions. The output layer employs a linear activation without bias. A weighted mean absolute error (MAE) was used as the loss function to mitigate the dominance of near-zero targets, while optimization was performed using the Nadam algorithm (Adam with Nesterov's accelerated gradient) combined with a cosine-decay-with-restarts learning-rate schedule, initialized at ψi=NNψ(𝐱NNψ). To prevent overfitting, early stopping was applied with a patience of 500 epochs, meaning that training ceased once the validation loss failed to improve over 500 consecutive epochs. All inputs and outputs were normalized to the range σt,i=∂ψi∂Δεi, and a batch size of 20 was used. The training was completed after 3500 epochs.

To enhance the accuracy of {Δε¯ipl,ψi,σt,i}i=1N in elastic regimes, a binary classifier was introduced as a complementary network. It was implemented as an FFNN with two hidden layers containing 64 and 32 neurons, respectively, each followed by a dropout layer with a rate of 0.3 to improve generalization. E= activations were used for all hidden layers, while the output layer employed a sigmoid activation to produce a probability value ν=0.3. The network was trained through 500 epochs using the binary cross-entropy loss and optimized via the Adam algorithm with a learning rate of σy=500MPa. Early stopping with a patience of 50 epochs was applied based on validation loss. The optimal decision threshold was determined through F1-score maximization over the validation set. During inference, the classifier's output probability modulates the prediction of k=1GPa according to a hybrid gating rule, combining hard thresholding and magnitude-based scaling to ensure smooth transitions near the elastic-plastic boundary.

The second network, m=0.15, consists of a single hidden layer with 48 neurons activated by the (Δεt,εtpl,σt;Δε¯pl,ψt) function. The output layer uses a linear activation without bias. The MAE was again adopted as the loss function, and the same optimizer was used as for NNζ, but with an exponential-decay learning-rate schedule initialized at NNζ. The normalization range, batch size, early-stopping parameters, and total number of epochs were identical to those of the first network.

The simpler architecture of NNψ was sufficient to capture the smooth energetic response, whereas NNψ, supported by the classifier, required a deeper representation due to the localized and discontinuous nature of plastic evolution. As shown in Fig. 3, the validation loss (MAE) at the end of training falls below NNζ for NNψ and reaches approximately NNζ for ReLU. The classifier further reduced error accumulation in sequential predictions, particularly for zero-target, i.e., elastic samples, thereby improving the overall thermodynamic consistency of the coupled model.

[picture]

Figure 3: The evolution of loss values through the training iterations for a) 3×10−4, and b) [−1,1]

Figure 3. The evolution of loss values through the training iterations for a) 3 × 10 − 4 , and b) [ − 1 , 1 ]

Results and Dicussion

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The performance of the model was evaluated in inference mode following the recursive procedure outlined in Algorithm 1. Starting from an initial stress-free state, the networks were supplied with prescribed strain increments to predict the corresponding internal variable, free energy, and stress values through automatic differentiation. This setup was applied to various unseen loading paths, including monotonic, cyclic, and random sequences, to examine the model's accuracy and generalization capability. To begin, the model's response under monotonic loading is analyzed to verify its ability to reproduce the elastic-plastic transition and establish a baseline for subsequent cyclic and random tests.
Test Set 1 - Monotonic loading. The first test case consists of 100 monotonic loading increments for the material characterized earlier in the section "Training Data." Fig. 4 shows the evolution of the model outputs, namely the increment of the internal state variable and the updated free energy, along with the corresponding reference data. The predictions exhibit a reasonable match with the target values across the entire loading path. Quantitatively, the average discrepancy in the predicted internal state variable increment is on the order of NNζ. The absolute average and maximum errors in the free energy predictions are ReLU and 𝒫, respectively. The derived stress values show absolute average and maximum errors of 0.72 and 7.94 MPa , respectively. These results confirm that the model accurately captures the linear elastic-to-plastic transition without any sharp drift or systematic bias.

[picture]

Figure 4: Model predictions for test set 1 (monotonic loading): increments of the internal variable and free energy, and the corresponding stress response.

Figure 4. Model predictions for test set 1 (monotonic loading): increments of the internal variable and free energy, and the corresponding stress response.

Test Set 2 - Cyclic loading. The second evaluation scenario consists of 240 loading increments, including loading, unloading, and reverse loading phases. The model successfully tracks the nonlinear hysteretic response without any evidence of error accumulation over increments. As depicted in Fig. 5, the predictions for both the internal state variable increment and stress evolution closely follow the reference data. The average errors in the predicted internal variable increment and free energy are 10−3 and NNζ, respectively. Based on these, the derived stress values exhibit an average absolute error of 0.58 MPa and a maximum deviation of 5.16 MPa . The consistent agreement across loading reversals confirms the model's capability to represent history-dependent behavior and maintain numerical stability during recursive inference.

[picture]

Figure 5: Model predictions for test set 2 (cyclic loading): accurate reproduction of loading, unloading, and reverse loading branches.

Figure 5. Model predictions for test set 2 (cyclic loading): accurate reproduction of loading, unloading, and reverse loading branches.

Test Set 3 - Random loading. The final validation examines the model's performance under randomly varying loading paths, representing the most complex condition considered. This test evaluates the model's generalization capability to arbitrary loading sequences not included in the training data of NNψ and ELUx2. Fig. 6a shows the evolution of the total strain during loading. At several stages, the strain clearly exceeds the lower limit of the training range, providing a stringent test for extrapolation. Fig. 6b presents the corresponding predictions of NNζ, and the derived stress values. The network predictions closely follow the reference results, accurately reproducing the complex nonmonotonic stress evolution. The maximum deviation in the predicted free energy is 3×10−4, while the computed stress values exhibit an average absolute error of 0.86 MPa and a maximum error of 4.18 MPa . These results highlight the robustness of the proposed framework in handling irregular, unseen strain paths. It's worth mentioning that each input component in this test set remains within the range of its respective training values. However, the cumulative total strain exceeds the loading paths seen during training. This highlights an important advantage of the model: because it operates on strain increments rather than total strain, it maintains high predictive accuracy even when the overall loading path extends beyond the training domain.
Discussion. The results presented above demonstrate that the proposed thermodynamically informed neural network achieves accurate and stable predictions of stress and free-energy evolution under various unseen loading paths. These results can be attributed to the physically guided design of the model, in which the stress is not learned directly but obtained as the derivative of the free-energy potential predicted by the energetic network. This formulation ensures that the model inherently satisfies the thermodynamic relations introduced in the backbone derivation, i.e., NNψ. Consequently, the predicted stress fields are fully consistent with an underlying energy landscape learned by the network, thereby preventing unphysical artifacts such as negative dissipation or spurious energy generation. In other words, by embedding the constitutive relationship within the network's structure, the model enforces physics-driven coherence between the state variables, promoting smoothness, stability, and extrapolative reliability.

The model additionally showed no noticeable dependency on the strain-increment size: repeating the same loading path with much coarser or finer increments produced nearly identical predictions. This suggests that the recursive formulation is robust with respect to time-step discretization.

[picture]

(a) Random loading path for test set 3 , plotted alongside the upper and lower strain limits of the training range. A zoomed-in inset on the right highlights the behavior near the lower training bound.

Figure 6. (a) Random loading path for test set 3 , plotted alongside the upper and lower strain limits of the training range. A zoomed-in inset on the right highlights the behavior near the lower training bound.

[picture]

(b) Output NNζ, and computed 10−7 compared with reference data.

Figure 7. (b) Output N N ζ , and computed 10 − 7 compared with reference data.

Figure 6: Model predictions for test set 3 (random loading). (a) Applied strain path showing segments beyond the training limit; (b) comparison of model predictions and target values.

The accuracy of the predictions beyond the trained strain range further demonstrates the advantage of the incremental formulation. Since the model operates on strain increments rather than total strain values, each inference step represents a small, locally learned material update. This structure allows the model to accumulate physically consistent predictions over extended loading paths, even when the total strain exceeds the limits observed during training. As a result, the framework exhibits a remarkable level of generalizability in inference mode despite being trained on a relatively compact dataset.

Nevertheless, the current formulation also has notable limitations. The present study is restricted to one-dimensional loading, where both the state and energy functions depend on a single strain component. Extending the model to a fully three-dimensional constitutive representation would, in principle, require the free-energy network to learn a potential defined over the full strain-state space. This would demand an extremely large and diverse dataset that densely samples the multidimensional domain, making data generation and training computationally prohibitive. In high-dimensional settings, the number of required samples grows exponentially, and the recursive inference mechanism could become numerically unstable due to the compounded propagation of small prediction errors. Although dimensionality-reduction or tensor-decomposition strategies could alleviate these challenges, they would require additional methodological developments beyond the current scope.

Conclusions

The original version of this paper is available on https://www.scientific.net/SSP.390.51.pdf

This study presented a thermodynamically informed neural network for one-dimensional elastoplasticity, coupling separate networks for dissipative evolution and free-energy representation. The model accurately reproduces stress evolution under monotonic, cyclic, and random loading paths while maintaining thermodynamic consistency. Its incremental formulation enables stable predictions beyond the strain range used for training, demonstrating generalization capability.

Although the present work is restricted to one-dimensional loading, the framework is in principle extendable to three-dimensional multi-axial stress states by learning a tensorial free-energy function from which stresses are derived via differentiation. In such a setting, invariant-based formulations can ensure objectivity for isotropic materials, while anisotropy can be incorporated through structural tensors or orientation-dependent features. However, extending the approach to 3D significantly increases data and computational requirements, which currently limits scalability. A systematic comparison with purely black-box neural networks remains an important direction for future work.

Acknowledgment

The original version of this paper is available on https://www.scientific.net/SSP.390.51.pdf

This research is part of the DEPMAT project (with project number P20-22 / N21022b) of the research programme Perspectief which is partly financed by the Dutch Research Council (NWO). It is also part of the Partnership Program of the Materials innovation institute M2i (www.m2i.nl).

References

The original version of this paper is available on https://www.scientific.net/SSP.390.51.pdf

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Solid State Phenomena, ISSN: 1662-9779, Vol. 390, pp 63-74

doi: 10.4028/p-4FOjpQ

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

Submitted:
2026-01-07

Revised:
2026-02-18

Accepted:
2026-02-21

Online:
2026-04-17

Deriving Macroscopic Damage Properties by Micromechanical Modeling of DP1000

The original version of this paper is available on https://www.scientific.net/SSP.390.63.pdf

Maximilian Neite 1,a*, Sebastian Münstermann 1, b

1 Institute of Metal Forming, RWTH Aachen University, Germany

a* maximilian.neite@ibf.rwth-aachen.de, b sebastian.muenstermann@ibf.rwth-aachen.de, (*corresponding author)

Keywords: Crystal Plasticity, Representative Volume Element, Micromechanical Modeling, Damage, Dual Phase Steel.

Abstract

Macroscopic damage models can describe the toughness behavior and formability of metals in terms of limit strains. However, it requires time-, cost-, and material-intensive calibration. In this work, a simulation framework is proposed to derive macroscopic damage model parameters and related properties directly from the microstructure. For this purpose, statistically Representative Volume Elements of the investigated DP1000 steel were generated utilizing the Python framework DRAGen. This was based on quantitative characterization of EBSD measurements of the present microstructure. Mechanical properties were assigned to the geometrical microstructure model by calibrating a phenomenological Crystal Plasticity model for distinct phases. Martensite cracking was identified as the predominant damage mechanism. This behavior on the microscale was represented by an isotropic brittle damage model in DAMASK, using a fracture mechanical literature value as the critical energy release rate parameter. The presented modeling approach enables stress statedependent prediction of macroscopic damage properties out of the present microstructure.

Introduction

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Dual Phase (DP) steel offers an excellent combination of high strength and good formability, which is why this class of Advanced High Strength Steels (AHSS) is highly attractive to the automotive industry [1,2]. It enables lightweight design in car body construction, one of the main challenges current car manufacturing faces [3]. Different types of DP steel can be achieved by tailoring the processing route, which results in different microstructures [4]. The produced microstructures provoke different macroscopic properties, like strength or damage tolerance, which are crucial for the design and manufacturing process [5,6]. This is part of the classical forward consideration of the process - microstructure - property relationship [7].

Macroscopic material properties, such as the toughness of steel, can be described using macroscopic damage models. These models also enable the prediction of forming limit curves, which are essential for assessing a material's formability [8]. However, applying these phenomenological models in practice is often limited by the significant experimental and numerical effort required to calibrate their parameters. The Modified Bai-Wierzbicki (MBW) damage model uses limit strains to separately describe ductile damage initiation and ductile failure based on the stress state, expressed by the Lode angle parameter and stress triaxiality. To cover a broad range of stress state conditions, various sample geometries are typically tested in experiments. Additionally, the effects of strain rate and temperature on material flow must be calibrated. [9,10]

For a backward, reverse engineering approach in microstructure design for tailored materials, it is essential to predict macroscopic material properties from the current microstructure directly [4,11]. To do this, micromechanical simulations of geometrical microstructure models can be conducted using Crystal Plasticity (CP) models [12]. Here, a Representative Volume Element (RVE) of the microstructure is often employed [13,14]. In the past, several RVE generators were developed to enhance accuracy by accounting for more microstructural features. The recently released Python framework Discrete RVE Automation and Generation (DRAGen) [15] can generate RVEs with banded structures, which are often necessary when studying DP steel [16,17]. Additionally, it creates input files for the simulation framework Düsseldorf Advanced Material Simulation Kit (DAMASK)
[18]. This framework allows for multi-physics CP simulations in a phase-field approach, incorporating mechanical, thermal, and damage effects simultaneously and interdependently.

The damage behavior in DP steels has been widely studied in recent years. Tasan et al. [4] reviewed experimental observations in the literature and summarized that damage typically initiates due to cracking of martensite, interface decohesion between ferritic and martensitic constituents, or a combination of both processes. These different damage mechanisms were observed to depend on microstructural characteristics, like phase fractions, ferrite grain size, or distribution and morphology of martensite. For DP steels with banded martensitic structures, martensite cracking was found to be the main damage initiation mechanism [19]. Also, numerical investigations were performed to study the damage behavior of DP steel micromechanically, considering both mechanisms [11,20-24] or specifically for the mechanism of interface decohesion [25,26] or martensite cracking [27].

In this paper, we present a workflow to derive macroscopic damage model parameters from the present microstructure. For this purpose, RVEs were generated with DRAGen based on a quantitative analysis of Electron Backscatter Diffraction (EBSD) measurements. The phenomenological CP model in DAMASK was calibrated to assign mechanical properties to the geometrical microstructure model. Finally, the isotropic brittle damage model implemented in DAMASK was applied to obtain stress state-dependent predictions of damage on the macroscale due to martensite cracking for a coldrolled DP1000.

Microstructure Model

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The ferritic-martensitic microstructure of DP1000 was investigated by EBSD measurements on a metallographically prepared sample of the Normal Direction (ND)-Transverse Direction (TD) plane. The captured area was about 100×100 mm2 in size and measured with a step size of 50 nm . These EBSD data were evaluated quantitatively with the help of the software OIM Analysis 2022 EDAX. After the predefined Confidence Index standardization, grains were calculated based on a threshold angle of 5∘, and grains smaller than five data points were excluded. Founded on the histogram of grain-average Image Quality, the ferritic and martensitic phases were distinguished. Through this separation, the phase ratios were determined by the number of data points per phase.

The grain morphology in the ferritic phase was described by fitted ellipses in terms of the lengths of major and minor semi-axes. Additionally, the grain's slope was expressed by the inclination angle between the ellipse's major axis and the horizontal axis of the EBSD image. Together with this geometrical description of the grains, their crystallographic orientation, defined by the Euler angles φ1,ϕ, and φ2, is stored in a .csv file.

A binary image of the EBSD measurements, colored by phase, was processed with functions of the OpenCV library in Python to describe the geometrical appearance of martensite quantitatively with ellipses. Large, band-like martensitic structures and smaller martensitic islands were analyzed individually. In this study, the focus is on larger microstructure volumes instead of a precise spatial resolution of fine hierarchical substructure constituents of the martensite, like laths or (sub-)blocks [28]. Those constituents were agglomerated to the quantitatively described martensite islands. The crystallographic orientation of martensitic constituents was not analyzed, since their orientation was later assumed to be random due to isotropic mechanical modeling of martensite.

The obtained results of the EBSD analysis are presented in Fig. 1 by Inverse Pole Figure (IPF) illustrations for the [001] direction. Ferritic grains of the investigated DP1000 steel exhibit an elongation along the TD. This observation is typical for so-called pancake grain morphology in coldrolled materials, leading to the assumption that a similar elongation could be observed along the Rolling Direction (RD) in the RD-ND plane [29]. This results in an assumed grain morphology, where two ellipsoid's axes have a similar length, larger compared to the third ellipsoid's dimension. The second IPF depicts pixels assigned to the martensitic phase in black. From this, the phase fraction of martensite in the investigated DP1000 is determined to be about 28%. Here, martensitic band-like formations in TD are visible, and are assumed to be spread in the RD-TD plane accordingly.

[picture]

Fig. 1. EBSD analysis of investigated DP1000: a) IPF[001] for the ND-TD plane, and accordingly, in b) martensite is highlighted in black.

Fig. 1. EBSD analysis of investigated DP1000: a) IPF[001] for the ND-TD plane, and accordingly, in b) martensite is highlighted in black.

The gathered microstructure data were then used to train a synthetic input data generator for the following generation of RVEs. The trained generator is built on a Wasserstein Generative Adversarial Network with Gradient Penalty (WGAN-GP) [30]. It is capable of reproducing the statistical distribution as well as interdependencies of parameters, which quantitatively describe the investigated microstructure.

Afterwards, three-dimensional RVEs were generated utilizing DRAGen. Here, the synthetic microstructural data produced by the trained WGAN-GP were used as input to create ellipsoids depicting grains within the discretized volume element. For this work, ten statistically equivalent RVES of the investigated DP1000 steel were generated. Each one has an edge length of 50μ m with a resolution of 1.28, resulting in RVEs containing 64×64×64 voxels. One martensite band was inserted into each RVE, following the aforementioned quantitative microstructure characterization of the investigated material.

As an example, one generated RVE of DP1000 is depicted in Fig. 2, showing the grains in individual colors and the martensitic band horizontally in red.

[picture]

Fig. 2. Exemplary RVE of the DP1000 generated with DRAGen, color-coded by grains.

Fig. 2. Exemplary RVE of the DP1000 generated with DRAGen, color-coded by grains.

Here, the assumed pancake morphology of the ferrite grains can be observed as well. Most of the generated RVEs contain a martensite phase fraction of around 30%, which is higher compared to the experimental findings. This inaccuracy is caused by the algorithm implemented in DRAGen to create banded structures, making it more difficult to reach the targeted phase ratios during RVE generation.

Crystal Plasticity

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DAMASK was used in this study to assign mechanical properties to the geometrical microstructure model by employing a phenomenological CP model (phenopowerlaw) for ferrite and the isotropic CP model for martensite [18]. The model for ferrite relies on anisotropic dislocation reactions and shearing in specifically activated slip systems, whereas the mechanical behavior of martensite was modeled to be direction independent. However, this model depicts not only the elastic mechanical behavior of martensite. Its hardening law, defined by the hardening exponent a and the initial hardening factor h0, allows the evolution of the resistance to plastic deformation from an initial ξ0 to a saturated value ξ∞, analogous to the model used for ferrite. Both models obey a rate dependency implemented as a power-law relationship with the reference shear rate γ˙0 and the rate sensitivity n.

The CP parameter sets for both phases were calibrated separately in this study based on an inverse procedure. The CP parameters were adjusted iteratively to match the macroscopic flow behavior of the investigated DP1000, studied in a previous work [31]. The literature suggests more sophisticated calibration procedures involving micropillar compression tests [32] or nanoindentations [13]. Noteworthy, these approaches also require considerable experimental effort.

Empirical correlations from literature connecting local chemical composition with flow behavior were used to determine a solely martensitic flow curve. This approach [33] defines the flow properties of martensitic microstructures as a composition of Peierls stress, strengthening due to carbon located at interstitial lattice positions, and strengthening due to dislocation effects. Segregation of alloying elements arising from partitioning and restricted solubility leads to distinct chemical compositions of ferrite and martensite. In previous studies [13], it was observed that particularly manganese and carbon accumulate within the martensitic phase of a DP steel. To characterize these local variations in chemical composition, specifically carbon and manganese content, multiple line scans of Electron Probe Microanalysis (EPMA) were performed over the microstructure investigated. Also, the global chemical composition of homogenously distributed alloying elements defining the material's strength was examined with the help of Optical Emission Spectroscopy (OES). Flow curve partitioning, according to the determined phase fractions, was applied to obtain an individual curve for ferrite [34]. The phases' flow curves, calculated based on chemistry and flow curve partitioning, and the experimental (total) flow curve of the DP1000 are represented in Fig. 3 as dashed lines.

An RVE comprised of only one martensite grain was loaded with quasi-static, uniaxial tension, and the homogenized values of equivalent stress over plastic equivalent strain were extracted. The curve was compared to the martensitic flow curve obtained from the local chemistry, and the CP parameter set for martensite was adjusted until both curves were in good approximation. Afterwards, the ten generated RVEs of DP1000 were subjected to the same loading condition as the martensitic RVE to define the CP parameter set for ferrite in a way that the calculated ferritic flow curve is represented.

The resulting curves from the RVE simulations are depicted in Fig. 3. The overall mechanical responses of the RVEs indicate higher equivalent stresses compared to the experimental flow curve, regardless of the observed scatter. This can be explained by the higher amount of martensite within the created RVEs in relation to the experimentally determined phase fraction. In general, both calibrated CP parameter sets describe the mechanical behavior of the individual phases well, especially at larger strains. The deflections in hardening behavior at small strains can be neglected because of the study's focus on the prediction of ductile damage at usually elevated strains. This results in a sufficient representation of the mechanical properties of DP1000 in micromechanical simulations utilizing the generated RVEs and the calibrated CP models. Table 1 concludes the calibrated values for the important CP parameters governing plastic material behavior.

Table 1. Calibrated CP parameter sets of martensite and ferrite to represent the mechanical behavior of DP1000.

γ̇₀ (s⁻¹) | n (−) | ξ₀ (MPa) | h₀ (MPa) | ξ∞ (MPa) | a (−)
Martensite | 0.001 | 100 | 685 | 1000 | 685 | 1.0
Ferrite | 0.001 | 100 | 210 | 1000 | 520 | 2.0

[picture]

Fig. 3. Results of the CP Calibration in terms of equivalent stress over plastic equivalent strain curves of ten simulated RVEs; comparison with reference (dashed) curves for ferrite and martensite phases, as well as the overall experimental (total) curve.

Fig. 3. Results of the CP Calibration in terms of equivalent stress over plastic equivalent strain curves of ten simulated RVEs; comparison with reference (dashed) curves for ferrite and martensite phases, as well as the overall experimental (total) curve.

Damage Modeling

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Scanning Electron Microscopy (SEM) was used to study the damage mechanism of the investigated DP1000 on the microscale. For this purpose, macroscopically tested samples from a previous study [31] were metallographically prepared in the vicinity of the fracture surface. Martensite cracking was determined as the predominant damage mechanism for distinct stress states at quasi-static conditions. Additionally, only a small number of pores were observed near the fracture surfaces of the ex-situ investigated samples, indicating rapid ductile damage evolution in DP1000.

In this work, martensite cracking was modeled micromechanically within the DAMASK framework [18]. Its phase field approach contains implemented damage models to describe damage initiation and evolution non-locally. The damage field variable φ can vary from an undamaged ( φ=1 ) to a fully damaged ( φ=0 ) condition. A sharp crack is assumed as a diffuse crack interface with gradually degrading material properties by introducing a length scale parameter lc. The spectral solver, based on the Fast Fourier Transform (FFT) method, numerically solves the following differential equation describing damage evolution

μφφ˙=fφ−Div𝐟φ.(1)

Here, damage viscosity is represented by μφ, damage driving force by fφ and damage flux by 𝐟φ. Following Griffith's criterion, brittle damage is created when sufficient energy required to build new internal surfaces, related to the surface tension G, is provided by the release of stored elastic energy. So, the damage flux is defined as

𝐟φ=−Glc∇φ(2)

and the damage driving force, derived by minimization of the total free energy density, as

fφ=Glc−φS:E.(3)

In this model, the damage influence on the stored elastic energy is implemented by a modified Hooke's law, with S being the second Piola-Kirchhoff stress tensor, E the elastic Green-Lagrange strain tensor, and C the elastic stiffness tensor

S=φ2C:E(4)

In this study, the isobrittle damage model [35] implemented in DAMASK was applied to model damage in martensite. For this purpose, μφ=0.001 and lc=2μ m was set. The governing model parameter, critical surface energy, Gc was defined as 423 J m−2. This value was approximated from literature [36], which reveals the mean fracture toughness of martensite in DP800 with microcantilever bending tests to Ji,2%=423 J m−2. This value was adopted and approximated as a characteristic energy-equivalent damage parameter for martensite in the RVE simulations of this study, because high experimental effort is required, and more precise data for DP1000 are not available.

Damage initiation in martensite was assumed in the micromechanical simulations as soon as any material point assigned to the martensite phase leaves the undamaged state. Typically, multiple material points reach the damaged state simultaneously within an increment, leading to numerical instabilities and subsequent loss of convergence under the default solver settings. A detailed study of crack propagation through the 3D RVE would therefore require significantly increased computational resources. At this instant, where φ<1, homogenized values in terms of average values over all material points containing the RVE were calculated for plastic equivalent strain εp, stress triaxiality η, and Lode angle parameter θ¯. By doing this homogenization, the results obtained with periodic RVEs in virtual experiments were bridged to macroscopic properties. This way, each RVE simulation results in one data point in the εp−η−θ¯-space representing damage initiation in martensite on the macroscale.

Distinct prescribed deformation gradient rates were applied to each RVE to obtain stress statedependent results. Quasi-static conditions were simulated for Uniaxial, Plane Strain (PS), EquiBiaxial (Biaxial), and Non-Equi-Biaxial (NE-Biaxial) tension. Those loading conditions are illustrated in Fig. 4, which shows a deformed RVE for each stress state, together with the reference cubic geometry of the RVE as a black wireframe. The deformed RVEs are colored by equivalent stress (vM stress) values to indicate the stress distribution. Regardless of the stress state, a big range in stress values can be observed due to the distinct phases of ferrite and martensite. Martensite shows significantly higher and uniform stress values due to the flat stress-strain curve without hardening, set during CP calibration (cf. Fig. 3). Since the calibrated CP parameter set of the ferritic phase allows for strain hardening, the stress distribution follows the most favorable grain orientations for dislocation slip. These favorable orientations depend on the direction of loading and thus the stress state.

The ten generated RVEs of DP1000 were subjected to each prescribed stress state to consider the scatter of damage. The mean damage initiation strain and its standard deviation obtained from these micromechanical simulations were calculated per stress state. In Fig. 5, the resulting data points are compared to the macroscopically and experimentally-numerically calibrated MBW damage locus of the investigated DP1000 derived in a previous study [31]. This ductile failure locus is assumed to be θ¯-symmetric and can be mathematically described with D1=0.12,D2=0.12,D3=0.18, and D4=1.52 by

εp(η,θ¯)=(D1e−D2η−D3e−D4η)θ¯2+D3e−D4η(5)

[picture]

Fig. 4. Simulation of an RVE under different stress states due to distinct boundary conditions: a) uniaxial, b) plane strain, c) equi-biaxial, and d) non-equi biaxial; the black wireframe shows initial cubic RVE geometry, and the distribution of equivalent stress (vM stress) is illustrated by color code.

Fig. 4. Simulation of an RVE under different stress states due to distinct boundary conditions: a) uniaxial, b) plane strain, c) equi-biaxial, and d) non-equi biaxial; the black wireframe shows initial cubic RVE geometry, and the distribution of equivalent stress (vM stress) is illustrated by color code.

Overall, the micromechanically derived data points are located in the vicinity of the conventionally calibrated MBW damage locus. Here, a macroscopic MBW ductile failure locus is compared to derived data points representing ductile damage initiation in martensite. This can be justified, since damage on the macroscopic scale was modeled in an uncoupled manner due to the observed material behavior. Additionally, the results of the SEM investigations have shown rapid damage evolution in the investigated DP1000, resulting in prompt ductile failure after damage initiation. At the edges of the illustrated θ¯-range, specifically uniaxial and biaxial stress states, the predicted damage strain is closer to the reference MBW locus compared to predictions at θ¯ around 0 . Even though the reference values on the MBW locus are located slightly outside the standard deviation range of the predicted damage strains for most stress states, the results demonstrate the capability of the presented simulation framework. The reference MBW locus is obtained by fitting the locus to multiple macroscopically determined supporting points. Therefore, even the reference locus can contain small deflections at certain stress states. Further, approximations were made when adopting an experimental value of a

DP800 as the governing CP model parameter Gc for DP1000. Noteworthy, in both cases, martensite cracking was described and investigated. In this study, the robustness of the simulation framework to predict macroscopic damage properties was evaluated by simulating multiple statistically equivalent RVEs. However, sensitivity studies on the influence of the key CP model parameter Gc could also be incorporated in the future to demonstrate the model's capabilities and quantify its predictive limits. For DP steels exhibiting a damage mechanism other than martensite cracking, the suitability of the applied model is questionable and needs to be investigated. Finally, inaccuracies created in the microstructure model during RVE generation and following CP calibration are propagated and influence the predictions.

[picture]

Fig. 5.Plastic equivalent strains for damage and their standard deviations derived by RVE simulations compared with the macroscopic MBW locus in dependence on the stress state.

Fig. 5. Plastic equivalent strains for damage and their standard deviations derived by RVE simulations compared with the macroscopic MBW locus in dependence on the stress state.

Summary

The original version of this paper is available on https://www.scientific.net/SSP.390.63.pdf

This work presents a simulation framework to micromechanically derive macroscopic damage properties directly from the present microstructure. To do so, RVEs were generated utilizing DRAGen based on quantitative data gathered from EBSD measurements on DP1000 steel. The chemical composition was captured to empirically calculate individual flow curves for ferrite and martensite. CP parameter sets of a phenomenological CP model implemented in DAMASK were calibrated by inverse, iterative fitting to the flow curves for both phases. The generated RVEs contain more martensite than targeted, hence the total flow curves of CP simulations are slightly above the experimental one. The isobrittle damage model implemented in DAMASK was employed to micromechanically model martensite cracking, which was observed by SEM as the predominant damage mechanism in the investigated material. An experimentally measured toughness of martensite was adopted from literature as the model's governing parameter, the critical surface energy Gc. Ten RVEs were micromechanically simulated under distinct prescribed loading conditions to account for
scatter and the stress state dependency. The procedure was performed on a DP1000 steel and validated with existing macroscopically examined damage properties of this material. The predicted damage strains are close to the reference damage locus described by the MBW model. Considering the assumptions made for the Gc parameter and the deviations resulting from small inaccuracies during RVE generation and consecutive CP calibration, the proposed simulation framework is capable of predicting damage properties of DP1000 from its microstructure in good approximation. The resource-efficient way of determining macroscopic damage properties can also be used in the future for reverse engineering in material design to optimize microstructures.

Further studies are planned to validate the results based on macroscopic simulations using the micromechanically calibrated damage parameters of the MBW model. Additionally, studies on the subsequent damage evolution within the RVE after damage initiation or on incorporating multiple damage mechanisms simultaneously in micromechanical simulations could give further insights into the damage behavior, meaningful for optimizing materials by tailoring resistance microstructures.

Acknowledgements

The original version of this paper is available on https://www.scientific.net/SSP.390.63.pdf

This work was funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) - Project number 391544620. Simulations were performed with computing resources granted by RWTH Aachen University under project rwth1670. The authors gratefully acknowledge material supply from SSAB AB .

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Solid State Phenomena, ISSN: 1662-9779, Vol. 390, pp 75-84

doi: 10.4028/p-FSxD8K

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

Submitted:
2026-01-16

Revised:
2026-02-18

Accepted:
2026-02-20

Online:
2026-04-17

Texture Data Preparation for Finite Element Simulations of Puncture Tests: Insights from the VPSC Approach

The original version of this paper is available on https://www.scientific.net/SSP.390.75.pdf

Alois C. Ott 1,2,a*, Leo Schwarzmeier 1,b, Johannes Kronsteiner 1,c, Nikolaus P. Papenberg 1, d and Thomas Antretter 2,e

1 LKR Light Metals Technologies, AIT Austrian Institute of Technology, Lamprechtshausener Straße 61, Braunau am Inn / Ranshofen, 5282 Austria

2 Montanuniversität Leoben, Chair of Mechanics, Franz-Josef Straße 18, Leoben, 8700 Austria

a alois.ott@ait.ac.at (*corresponding author), b leo.schwarzmeier@ait.ac.at, cjohannes.kronsteiner@ait.ac.at, dnikolaus.papenberg@ait.ac.at, ethomas.antretter@unileoben.ac.at 

Keywords: Forming, Aluminum Alloy, Anisotropy, Alignment, Clustering, Puncture Test

Abstract

Predicting the deformation behavior of rolled and extruded light metal alloys is a challenging task. Due to the high cost of experimental analysis, finite element simulations are often required. A variety of material models at different scales are available for practical use. In this work, the viscoplastic self-consistent (VPSC) approach is employed to consider microstructural effects. These can be incorporated by using measured crystal sizes and orientations - called texture - of the alloy under consideration. For each integration point in the FE mesh, a corresponding texture is assigned and individually deformed in LS-DYNA ® , where VPSC is implemented as a user-defined material model - referred to as FE-VPSC. This study focuses on preprocessing of texture data as well as their compression for accurate and faster FE simulations. For verifying the simulations, a comparison with digital image correlation (DIC) of experimental puncture tests was conducted.

Introduction

The original version of this paper is available on https://www.scientific.net/SSP.390.75.pdf

The usage of sheet metal alloys in industry is very versatile. A single car is made up of hundreds of individual sheet-metal parts. However, due to construction requirements and the intricacies of the material behavior, complex forming processes with multiple deformation steps are often necessary [1,2,3]. Furthermore, anisotropic behavior of the sheet metal can lead to thinning, cracks or uncontrolled shape deviations during forming [4,5,6]. To understand and thereby mitigate these effects, the simulation of forming processes has gained increasing importance over the past decades. When, in addition, microstructural information is incorporated into the material model, further improvements can be achieved. However, simulations taking into account such a degree of detail are computationally expensive. The viscoplastic self-consistent (VPSC) approach represents a mesoscale material model that provides a viable trade-off between accurately capturing the micromechanical response - including as much information as necessary - and computational cost for a given component size. The microstructural input of the material in our work is derived from electron backscatter diffraction (EBSD) measurements. However, before being used in microstructure simulations [7], this information must be carefully preprocessed.
Sample preparation as well as measurement setup for EBSD analysis often leads to texture misalignment and an excessively large number of grain orientations, which can complicate material simulations. For every sheet metal a physically motivated coordinate system - from here on termed the 'reference system' - can be defined, representing the three process directions (rolling-, transversaland normal direction). The measured grain orientations, i.e. the texture, are expressed by rotation matrices relative to the corresponding measurement coordinate system. To provide optimal material data for simulation, the deviations of the reference system to the measurement system are evaluated and corrected in the preprocessing step. In a second step, grains with almost identical crystallographic orientations are combined by a clustering algorithm specifically adapted for this purpose [8,9]. It has
already been shown, that for sheet metals with a single phase of cubic crystals, 100 grains are more than sufficient to represent the texture [8,9,10].
A practical approach to characterize the material behavior involves puncture tests [11]. Using digital image correlation (DIC) during puncture tests, deformations and cracks of a sheet metal specimen can be detected [12,13]. Moreover, these effects can be replicated through an implementation of VPSC in the finite element framework LS-DYNA ® (FE-VPSC). The goal is to use optimized EBSD data to achieve a reasonable agreement with simulation results [14,15,16] at acceptable computational effort.
In the present work, we address these issues by investigating an EN AW-6016-T4 aluminum alloy. For comparison of experiments with simulated puncture tests, the resulting deformation maps are evaluated.

Experimental

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For the investigations in this work, an EN AW-6016-T4 sheet metal alloy with a thickness of 1.2 mm was analyzed via EBSD measurements as well as puncture tests.

Texture. For EBSD measurements, a Velocity Super camera of AMETEK/EDAX was employed. The accelerating voltage amounts to 20 kV with a beam intensity of 18, i.e. a unitless Tescan-specific value ranging from 1 to 20 . The viewing direction on the sample surface was the plane spanned by the transversal direction (TD) and rolling direction (RD). A working distance of 15 mm was chosen, with a view field of 900μ m and a scanning step size of 1μ m. For data preparation and analysis, the software OIM v8.6 was used. The NPAR clean-up method was employed, using a confidence index (CI) threshold of 0.05 . The measured inverse-pole figure (IPF) map as well as the corresponding Pole Figures are provided in Fig. 1.

[picture]

Fig. 1. Texture measurements of EN AW-6016-T4, showing (a) Inverse Pole Figure map in the [001] crystal direction and (b) the corresponding Pole Figures in the [001], [110], and [111] directions.

Fig. 1. Texture measurements of EN AW-6016-T4, showing (a) Inverse Pole Figure map in the [001] crystal direction and (b) the corresponding Pole Figures in the [001], [110], and [111] directions.

Puncture Tests. An electrodynamical testing machine, Linovis (4a Engineering), was employed to perform puncture tests on the chosen sheet metal. In Fig. 2(a) the corresponding test chamber is illustrated. Sheets with dimensions of 80×80 mm were manufactured and painted with white base coat and black speckles. To reduce friction, two layers of Teflon foil coated with bearing grease were applied to the backside. Subsequently, the prepared samples are placed between the two clamping plates and fixed using the two hydraulic cylinders. In a separate chamber on the left side (not visible),
a high-speed camera, FASTCAM NOVA S9, was oriented orthogonal to the sheet surface. The resolution was set to 1024×1024px with a corresponding view field of 35.84×35.84 mm. The samples were deformed with a speed 0.2 mm/s and ~15 mm deformation path. The movement of the impactor ( 20 mm diameter) from the right/back side in the chamber led to fracture of the samples, see Fig. 2(b). For determining the optical measured displacements, the software MercuryRT v3.1. was used. Based on the chosen parameter settings, a frame rate of 60 fps was sufficient to ensure accurate DIC measurements.

[picture]

Fig. 2. Puncture test setup: (a) the Linovis test chamber and (b) samples of four experiments conducted up to fracture.

Fig. 2. Puncture test setup: (a) the Linovis test chamber and (b) samples of four experiments conducted up to fracture.

Methods

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To ensure meaningful FE-VPSC simulations, an adequate preprocessing of the texture input data was necessary. Due to sample preparation and positioning of the EBSD detector, the process directions RD, TD and ND did not coincidence with the principal axis of the measurement system. This issue was handled by using an alignment algorithm.
Subsequently, the number of different grain orientations was reduced in order to provide a balanced trade-off between simulation runtime and output quality. For this purpose, a clustering method was applied on the previously aligned texture data.
Finally, the FE setup for conducting puncture simulations is discussed. It includes the VPSC material parameters, the deformation path, the tooling geometry, as well as their combined application in the FE mesh.

Texture Alignment. The texture input for the VPSC material model was derived from EN AW-6016 sheet metal alloy in T4 temper, see IPF map in Fig. 1(a). It was composed almost entirely of grains exhibiting cube orientation, as shown in Fig. 1(b). This is a result of static recrystallization during heat treatment [17,18]. Due to the FCC crystal symmetry as well as orthotropy introduced during the forming process, the grains are symmetrically oriented.
It is known that a symmetrical alignment of the pole figures is equivalent for aligning RD and TD in 2D plane with the two principal axes x and y , see Fig. 3. For the 2D alignment of pole figures we use the division of the upper hemisphere with a grid as shown in Ref. [7]. Following this procedure, the azimuth angle [0∘,360∘] and altitude [ 0,90∘ ] are divided by 10∘, resulting in a grid with 36×9=324 bins. Each bin is assigned the number of poles contained in it. Stereographic projection of the grid onto the equatorial plane yields bins symmetrically distributed along the x - and y -axes, as illustrated in the pole figures of Fig. 3. Since the {111}<110> slip systems are the main contributors for deformation of aluminum at room temperature, the {111} pole figure is afforded special attention
in the alignment process. Through the optimization process described in Ref. [7], the texture gets realigned until the bins in the {111} pole figure are symmetrically aligned with the RD- and TD-axes. This corresponds to the minimal difference of poles in opposing bins. The resulting alignment amounts to an (Euler) angle of 1.91∘. This value is comparatively low, as the texture is sharp and well defined and was therefore already manually aligned with reasonable accuracy during measurement. For less pronounced textures, deviations introduced by the measurement process can reach up to 5∘ [7].

[picture]

Fig. 3. Pole figures of EN AW-6016-T4 after symmetrical alignment about the RD- and TD-axis of the {111} pole figure.

Fig. 3. Pole figures of EN AW-6016-T4 after symmetrical alignment about the RD- and TD-axis of the { 111 } pole figure.

Texture Clustering. The initial texture of the investigated aluminum sheet consists of 660 grains. From Ref. [8,9,10], it is known that 100 grains are sufficient to describe the orientation distribution function (ODF) of FCC alloys. Since the runtime of VPSC shows a linear relationship with the number of grains used [15], the simulation time of FE-VPSC can be reduced significantly by texture clustering. For this purpose, the clustering algorithm described in Ref. [8,9] is used. It is based on a quaternion representation of each grain and the Euclidean metric defined within this space.

[picture]

Fig. 4. Comparison of original texture (black) and clustered texture (red) with rising cluster radii ( rCl ), i.e. increasing texture compression, and reducing number of grains. The cube texture component is preserved through the clustering.

Fig. 4. Comparison of original texture (black) and clustered texture (red) with rising cluster radii ( r C l ), i.e. increasing texture compression, and reducing number of grains. The cube texture component is preserved through the clustering.

The algorithm starts by assigning each grain to a cluster. The size of a cluster is defined by a radius ( rCl ) determined from the Euclidean distance to its center. Clusters are then iteratively realigned (with occasional re-creation) until each cluster provides a good average of the grains assigned to it. The dominant texture component, in this case 'cube', should be retained as well. This is achieved through the weighted arithmetic mean of the assigned grains, followed by normalization. As shown in Fig. 4 the cube texture component is well preserved for the textures clustered with a rCl of 0.2,0.4, 0.6 and 0.8 . For both tension and compression test simulations, a cluster radius of 0.6 has been shown to provide a good balance between simulation quality and runtime [8,9].

FE-VPSC.

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The fundamental principle of the VPSC approach assumes that each individual grain of the considered polycrystalline aggregate possesses an ellipsoidal shape and is embedded in a homogeneous effective medium (HEM). Within a self-consistent framework, the response of all grains is determined such that stress equilibrium and strain compatibility are satisfied in an average sense. This interaction is established through an appropriate linearization scheme. For each individual grain in the polycrystalline aggregate, the continuum relation for the local strain rate ϵ˙(Γ) is given by

ϵ˙(Γ)=γ0˙∑s=1Nms(Γ)(ms(Γ):σc(Γ)τ^s(Γ))n(1)

Here, ms denotes the geometric Schmid tensor, σc the Cauchy stress tensor, and τ^s the critical resolved shear stress after accumulated shear Γ in each grain. The index s=1,…,N accounts for the different slip systems active in the crystal lattice, while γ˙0 represents a reference shear rate.

Furthermore, VPSC is implemented within the LS-DYNA ® finite element framework via a userdefined material subroutine (UMAT). Each finite element of the metal sheet is assigned a single integration point governed by the FE-VPSC formulation [19]. At each implicit time step, the deformation increment, and the current texture are passed to the VPSC UMAT, which computes the stress response and updates the local texture accordingly. Further details on the implementation, which is based on an approach developed for the FE solver ABAQUS®  [20], can be found in [14,15]. The puncture simulations used the same setup and hardware as described in Ref. [16]. Both the impactor and the die are represented by rigid bodies. Boundary nodes, created to simulate the blank holder, secure the 1.2 mm thick metal sheet. Deformation was realized via the impactor, which moved with a constant velocity of 0.1 mm/s. A standard friction coefficient for aluminum-steel contact at room temperature was applied to define the interactions between the impactor and the metal sheet, as well as the sheet and the die, see Fig. 5. It should be noted that this definition of interaction differs from the experimental setup, where Teflon and lubrication were employed to minimize friction between the impactor and the sheet metal.

[picture]

Fig. 5. Overview of the FE meshes used in the LS-DYNA ® puncture test simulations. VPSC was implemented as a UMAT for the part representing the metal sheet specimen, which comprised approximately 4.4 k elements. The impactor, with a diameter of 20 mm , and the die featuring a 40 mm opening were modelled as rigid bodies, representing the steel components of the experimental setup.

Fig. 5. Overview of the FE meshes used in the LS-DYNA ® puncture test simulations. VPSC was implemented as a UMAT for the part representing the metal sheet specimen, which comprised approximately 4.4 k elements. The impactor, with a diameter of 20 mm , and the die featuring a 40 mm opening were modelled as rigid bodies, representing the steel components of the experimental setup.

The employed VPSC material model was restricted to the {111}<110> slip system, as face-centered cubic (fcc) materials deform predominantly by slip on this system at room temperature [4]. The associated self-hardening behavior was described using the (extended) Voce hardening law, whereby the evolution of the critical resolved shear stress (CRSS) of the slip system is given by

τ^s(Γ)=τ0+(τ1+θ1Γ)[1−exp(−Γ|θ0τ1|)].(2)

Here, Γ denotes the accumulated total shear stress within the grain, while τ0,τ1,θ0 and θ1 represent the Voce hardening parameters. Although VPSC enables the incorporation of latent hardening effects, these were neglected in the present study, as only a single slip system was considered.
The Voce hardening parameters in Equation (2) were taken from Ref. [16], which investigated the same material. Specifically, the flow curve obtained from an experimental tensile test of EN AW-6016-T4 in the rolling direction (RD) was calibrated using LS-OPT (v. 7.0.2). The calibration was performed by systematically varying the hardening parameters within an FE-VPSC simulation of the tensile specimen geometry until agreement with the experimental response was achieved.
The resulting Voce parameters τ0=66.18MPa,τ1=68.56MPa,θ0=423.28MPa,θ1=0MPa, are adopted throughout this study. The N -effective linearization scheme was applied with an interaction parameter neff =10.

Results and Discussion

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This section examines the effects of texture alignment and varying degrees of texture clustering on the FE-VPSC simulations of puncture tests. The simulation results show good agreement with the experimental observations.

Texture Alignment. The measured texture of EN AW-6016-T4, with and without automated alignment, is shown in Fig. 1(b) and Fig. 3, respectively. Both textures were subsequently clustered using a cluster radius of 0.4 . The resulting clustered textures serve as input for the FE-VPSC puncture simulations, see Fig. 6.

[picture]

Fig. 6. Results of FE-VPSC puncture simulations, comparing XY-displacement of (a) measured, i.e. manually aligned, texture and (b) and automatically aligned texture.

Fig. 6. Results of FE-VPSC puncture simulations, comparing XY-displacement of (a) measured, i.e. manually aligned, texture and (b) and automatically aligned texture.

Small differences are visible by comparing the resulting displacement maps after performing both simulations. The angular position of the two resulting maxima obtained from the manually aligned texture input deviates from the results obtained from the automatically aligned texture input by an offset angle of about 10∘, even though the misalignment between manually and automatically aligned texture only amounts to ~2∘. It appears that even minor variations in the texture can strongly affect the puncture test simulations. The maximum value of deformation in the manual case is 1.413 mm in comparison to 1.427 mm for the automatically aligned case.

Texture Clustering. The influence of four texture clustering levels on the FE-VPSC puncture simulations is investigated. Fig. 7(a)-(d) shows the corresponding XY-displacement maps obtained using input textures of EN AW-6016-T4 with cluster radii 0.2,0.4,0.6 and 0.8 , respectively. An overview of the results is given in Table 1.

[picture]

Fig. 7. Resulting XY-displacement of simulated puncture tests via FE-VPSC, using four clustered textures, cluster radii ( rCl)0.2 to 0.8 , as input.

Fig. 7. Resulting XY-displacement of simulated puncture tests via FE-VPSC, using four clustered textures, cluster radii ( r C l ) 0.2 to 0.8 , as input.

Between the input textures with cluster radii of 0.2 and 0.4 only minor deviations in the locations of the two resulting maxima fields can be observed. Accordingly, the maximum deformation changes marginally, from 1.426 mm to 1.427 mm . For a cluster radius of 0.6 more pronounced deviations in the localization of the deformation maxima become apparent, and the maximum displacement increases to 1.442 mm . The deformation map obtained with a cluster radius 0.8 exhibits clear visual differences, with peak values reaching 1.517 mm .

Table 1. Characteristics of simulated puncture tests with varying degrees of input texture clustering.

Cluster Radius | Number of Clusters | XY-Displacement Maximum | Runtime FE-VPSC
0.2 | 167 | 1.426 mm | 37h 11min 35s
0.4 | 57 | 1.427 mm | 19h 55min 11s
0.6 | 30 | 1.442 mm | 13h 44min 50s
0.8 | 13 | 1.517 mm | 10h 18min 44s

Regarding computational runtime, the simulation using a cluster radius of 0.2 required 37 h 11 min 35s. Compared to the original texture consisting of 660 grains, the reduced texture obtained with a cluster radius of 0.2 consists of 167 clusters. Increasing rCl from 0.2 to 0.4 results in an approximate reduction in the number of grains by two thirds i.e. to 57 grains, which corresponds to a runtime reduction of roughly 50% down to 19 h 55 min 11 s . In contrast, clustering with a radius of 0.6 reduces the texture further to 30 grains and yields a runtime as low as 13 h 44 min 50 s . However, this level of data reduction already leads to considerable differences in the simulation results, i.e. in the maximum XY-displacement. The cluster radius of 0.8 reduces the texture to 13 grains but results in only a limited further reduction of the runtime to 10 h 18 min 44 s . It can also be observed that the reduction in computation time is no longer proportional to the number of grains, indicating that solver overhead becomes significant at this level of clustering.

Experimental validation. Based on the presented results, a cluster radius of 0.4 appears to represent the optimal trade-off between accuracy and computational efficiency. For tensile- and compression simulations with FE-VPSC [8,9] even a higher clustering radius of 0.6 was adequate. To verify the FE-VPSC puncture test simulations, a comparison is made with the experimental deformation maps from DIC measurements. To obviate the influence of static friction on the metal sheet surface, the deformation maps are evaluated with an applied puncture force of approximately 1300 N . The resulting deformation maps of the four conducted experiments are shown in Fig. 8. For enhanced visualization, the upper bound of the color scale was capped at 0.20 mm , the maxima exceeding this value are displayed in white. Consequently, it is now evident that the maxima develop with an offset of approximately 45∘ relative to the principal axes. This behavior was expected due to the high fraction of cube texture component and the presence of in-plane shear [21]. This effect is even more pronounced in the puncture simulations at higher forces, which shows good agreement with the experimental observations.

[picture]

Fig. 8. Displacement maps from four experimentally conducted puncture tests performed at deformation forces of approximately 1300 N are presented. For improved visualization, the upper limit of the color scale was capped at 0.20 mm ; deformation values exceeding this threshold are displayed in white. This adjustment highlights the consistent formation of two maxima regions, offset by 45∘, across all tests. In Figures 8(b) and 8(d), localized voids appear in the displacement field. These result from oversized speckles on the specimen surface that could not be reliably captured during digital image correlation (DIC) analysis.

Fig. 8. Displacement maps from four experimentally conducted puncture tests performed at deformation forces of approximately 1300 N are presented. For improved visualization, the upper limit of the color scale was capped at 0.20 mm ; deformation values exceeding this threshold are displayed in white. This adjustment highlights the consistent formation of two maxima regions, offset by 45 ∘ , across all tests. In Figures 8(b) and 8(d), localized voids appear in the displacement field. These result from oversized speckles on the specimen surface that could not be reliably captured during digital image correlation (DIC) analysis.

Summary

The original version of this paper is available on https://www.scientific.net/SSP.390.75.pdf

The puncture test simulations performed with FE-VPSC exhibit good agreement with the experimental DIC results regarding the (RD,TD)-displacement. Anisotropic effects, particularly at 45∘ between the rolling direction (RD) and transverse direction (TD), are clearly visible in the (RD,TD)-displacement field.
These results were achieved by processing the original texture to align the measurement coordinate system with the manufacturing directions. This involves an adjustment of approximately 2∘ which already induced non-negligible changes in the simulation results. In a second step, an efficient data reduction was applied using a clustering algorithm tailored for grain orientation data. A cluster radius of 0.4 provided a favorable balance between accuracy and computational efficiency.

Acknowledgement

The original version of this paper is available on https://www.scientific.net/SSP.390.75.pdf

This research has been funded by the Fonds Zukunft Österreich within the framework "Industrienahe Dissertationen 2025" in the project "MaLTe" administered by FFG under the grant number FO999926074 and by the project "Data-T-Rex" (Wi-2021-305676/13-Au) which is cofinanced by research subsidies granted by the government of Upper Austria. The authors would also like to thank Los Alamos National Laboratory, and especially Miroslav Zecevic, for providing the FE-VPSC code.

References

The original version of this paper is available on https://www.scientific.net/SSP.390.75.pdf

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Solid State Phenomena, ISSN: 1662-9779, Vol. 390, pp 85-96

doi: 10.4028/p-S75TgF

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

Submitted:
2026-01-16

Revised:
2026-02-13

Accepted:
2026-02-21

Online:
2026-04-17

Modeling Transient Temperature-Driven Microstructure Evolution in Inconel 718 during Hot Forming

The original version of this paper is available on https://www.scientific.net/SSP.390.85.pdf

Nadine ELEKYABI 1,a, Holger BRÜGGEMANN 1, b, Emad SCHARIFI 1,c* and Junhe LIAN 1,c

1 Institute of Metal Forming (IBF), RWTH Aachen University, Germany

a  nadine.elekyabi@ibf.rwth-aachen.de, b holger. brueggemann@ibf.rwth-aachen.de, c  emad.scharifi@ibf.rwth-aachen.de, djunhe.lian@ibf.rwth-aachen.de

*corresponding author

Keywords: Superalloys, Non-isothermal Deformation, Microstructure Modeling, Hot Forming

Abstract

Industrial hot forming of nickel-based superalloys is typically carried out under nonisothermal conditions, where rapid temperature changes between forming steps not only affect the thermo-mechanical response but also the microstructural evolution, including processes such as recrystallization. Most microstructure models are developed and calibrated for idealized isothermal conditions, and their applicability to realistic transient temperature paths is still unclear. Therefore, this study investigates the microstructure evolution of Inconel 718 under non-isothermal hot deformation by combining dilatometer tests with full-field simulations using DIGIMU® in order to provide detailed insight into the underlying microstructural mechanisms and to assess the capability of a Full-Field approach for modelling such non-isothermal forming conditions. For this aim, compression tests with temperature increases ( 1020∘C to 1070∘C ) and decreases ( 1120∘C to 1070∘C ) were performed, with the temperature change applied at different strains. The results reveal a path dependence, heating at low strain to 1070∘C leads to higher DRX-fractions and finer, more homogeneous grain structures, whereas later heating at higher strains produces coarser, partially recrystallized microstructures due to reduced strain at the higher temperature. For the temperature decrease, DRX occurs predominantly at 1120∘C, after a late temperature change, no additional DRX takes place at 1070∘C. While an earlier change still allows additional DRX. The Full-Field simulations reproduce these trends in dislocation density, recrystallized fraction and grain size distributions with good qualitative agreement and moderate quantitative deviations.

Overall, the study demonstrates that the timing and direction of temperature changes affect the final microstructural state, and that DIGIMU®, a Full-Field approach, can capture path-dependent microstructure evolution in Inconel 718 and provides a useful digital tool for analyzing and designing transient temperature hot forming processes.

Introduction

The original version of this paper is available on https://www.scientific.net/SSP.390.85.pdf

Hot forming is a central manufacturing route for producing high-performance components with tailored mechanical properties. During such processes, the microstructure evolves continuously through mechanisms such as work hardening, dynamic recovery, dynamic recrystallization (DRX), post-dynamic recrystallization (PDRX) and grain growth [1,2]. Because these mechanisms control grain structure and, in turn, mechanical properties, it is essential to study the microstructural evolution during hot deformation. The resulting microstructure evolution and the kinetics of these mechanisms are closely linked to the process parameters such as strain rate, deformation temperature and deformation degree [3,4].

Nickel-base superalloys, such as Inconel 718, which are widely used in aircraft engines and energy applications because of their excellent high temperature properties [5], are a prominent example where this microstructure sensitivity is critical. As a result, their hot deformation behavior and microstructural evolution have been investigated in numerous works [6-8].
In these works, the temperature histories are typically idealized as constant values. In real hot forming processes, however, the thermal conditions are often non-isothermal. In open-die forging, for
example, after each form passes, an increase and decrease of the core temperature of the material is observed [9]. As a result, the material experiences transient heating and cooling during ongoing deformation, which can modify the competition between hardening, recovery and recrystallization compared with idealized isothermal processes. Lin et al. [10] investigated non-isothermal two-stage hot deformation of a Ni-Fe-Cr base superalloy and showed that with different final deformation temperatures, strain rates and first-stage strains lead to different DRX fractions. However, their study is purely experimental and limited to global quantities in the final states after the experiment, without access to intermediate microstructural states.

To analyze such conditions in more detail, Full-Field models are required that enable the detailed tracking of microstructural states throughout the deformation process. This approach improves the link between process parameters and microstructural changes, since every intermediate state can be analyzed without additional experimental effort [11]. In this work, the Full-Field framework DIGIMU® is used for that purpose. It is designed to simulate multistage hot forming processes within reasonable computation times and to model microstructural evolution at the grain level on a meshed representative volume element (RVE) containing multiple grains. The framework uses an explicit grain structure and uses experimentally derived boundary conditions, such as temperature-time histories and strain rate, to reproduce realistic deformation conditions. Grain boundary migration and recrystallization are described by a finite element formulation coupled with a level-set approach, which allows grain boundaries to move in response to curvature and differences in stored energy between neighboring grains. The underlying kinetic models account for the coupled evolution of dislocation density, grain boundary migration and recrystallization, including dynamic recrystallization (DRX) and post-dynamic recrystallization (PDRX) [12-14].

This study therefore addresses the mentioned gap by combining transient hot compression tests on Inconel 718 with Full-Field simulations in DIGIMU®, using the experimentally measured timetemperature histories as boundary conditions to assess the model's ability to reproduce and analyze the microstructural evolution under transient temperatures. In this way the microstructural behavior is studied under process-relevant thermal conditions that are close to those encountered in industrial hot forming, while allowing the microstructure to be analyzed at arbitrary points in time during the deformation.

Materials and Methods

The original version of this paper is available on https://www.scientific.net/SSP.390.85.pdf

Material. The investigated material is an industrially produced wrought bar of the nickel-based superalloy Inconel 718. The initial stock with a diameter of 160 mm was forged in ten passes to a rectangular cross-section of 87 mm×87 mm. After forging, the material was solution annealed at 1030∘C for 1 h . The chemical composition of used Inconel 718 is given in Table 1.

Table 1. Chemical composition of the investigated Inconel 718 alloy [wt.%]

C | Si | Mn | P | S | Cr | Ni | Mo
0.022 | 0.06 | 0.08 | 0.007 | 0.0004 | 17.63 | 53.9 | 2.97
Al | Ti | Co | Cu | Nb | Fe | B | Ta
0.52 | 0.94 | 0.32 | 0.04 | 4.98 | 18.49 | 0.0024 | <0.02

Experimental Procedure. Non-isothermal hot compression tests were carried out to determine the material behavior under changing temperatures. All tests were conducted on cylindrical Inconel 718 specimens with dimensions 5 mm in diameter and 10 mm in height, using a quenching and deformation dilatometer DIL 805 (TA Instruments).

In this device, the specimens are inductively heated and the temperature is controlled by a thermocouple welded directly to the specimen surface. The dilatometer allows testing in a temperature range from room temperature up to 1500∘C. It is equipped for compression tests with a maximum forming force of 20 kN , enabling strain rates between 0.001 s−1 and 20 s−1 and strains up to 1.2, which covers the deformation conditions investigated in the present study. The schematic experimental procedure for the non-isothermal hot compression tests is illustrated in Figure 1 for the
case of temperature decrease. For each transient temperature test, the specimen was first heated at a rate of 10 K/s to the desired initial deformation temperature T1 and held for 30 s to minimize temperature gradients. The first deformation was carried out at T1 up to a predefined strain ε1. After this the temperature was changed to a second deformation temperature T2 and the deformation was continued to the final strain ε2.

Two types of non-isothermal tests were performed. Tests with a temperature increase from 1020∘C to 1070∘C and tests with a temperature decrease from 1120∘C to 1070∘C. In both series, the temperature change was imposed at different true strain levels ( ε=0.2,0.4 and 0.6 ) in order to cover both the hardening dominated regime and the softening regime of the flow curve. The complete experimental matrix, including the combinations of initial temperature, temperature path and strain level at which the temperature change was applied, is summarized in Table 2. In addition, isothermal tests at constant temperatures of 1020∘C,1070∘C and 1120∘C were also carried out as reference experiments up to a strain of 0.8 . After deformation, all specimens were rapidly quenched with argon gas in order to freeze the microstructure. Subsequently, all specimens were metallographically prepared (grinding, polishing and etching). Microstructural images were taken with optical microscopy and the grain size was evaluated using the line-intercept method in accordance with DIN EN ISO 643 [15].

[picture]

Fig.1. Schematic procedure of the transient temperature hot compression tests

Fig.1. Schematic procedure of the transient temperature hot compression tests

Table 2. Experimental matrix of hot compression tests

Temperature T₁ [°C] | Temperature T₂ [°C] | Strain rate ε̇ [s⁻¹] | Changing rate [°C/s] | Switching strain ε
1020 | 1070 | 0.1 | 100 | 0.2
1020 | 1070 | 0.1 | 100 | 0.4
1020 | 1070 | 0.1 | 100 | 0.6
1120 | 1070 | 0.1 | 100 | 0.2
1120 | 1070 | 0.1 | 100 | 0.4
1120 | 1070 | 0.1 | 100 | 0.6

During all transient tests, the temperature-time history of the specimen was recorded continuously by the welded thermocouple. As an example, Figure 2 shows one measured temperature evolution for the heating path from 1020∘C to 1070∘C and one for the cooling path from 1120∘C to 1070∘C. From these curves, effective heating and cooling rates of about 100 K/s were determined in the temperature change stage, demonstrating that the dilatometer is capable of achieving such rapid transient temperature changes during hot compression. These experimentally measured temperature histories were used directly as thermal boundary conditions in the DIGIMU® simulations described in the following section in order to reproduce the non-isothermal deformation conditions as realistically as possible.

[picture]

Fig. 2. Representative measured temperature-time histories of the specimen during transient temperature tests for a) a temperature increase from 1020∘C to 1070∘C at ε=0.2 and b) a temperature decrease from 1120∘C to 1070∘C at ε=0.2

Fig. 2. Representative measured temperature-time histories of the specimen during transient temperature tests for a) a temperature increase from 1020 ∘ C to 1070 ∘ C at ε = 0.2 and b) a temperature decrease from 1120 ∘ C to 1070 ∘ C at ε = 0.2

Simulation Setup. To generate a representative polycrystalline RVE with a lognormal grain size distribution in DIGIMU®, statistical grain size information from the initial microstructure was used. The starting microstructure of the solution-annealed material was characterized by light optical microscopy, and the grain size distribution was quantified using the line-intercept method [15]. From these measurements, the grain size distribution within a given section was determined. Based on these parameters, a polycrystalline RVE containing 50 grains and following a lognormal grain size distribution was generated. The initial microstructure is shown in Figure 3 and the corresponding grain size distribution is listed in Table 3. The material parameters used in the Full-Field model were based on the parametrization in [16] investigations, in which the hot deformation response of Inconel 718 through four key mechanisms is captured: grain growth, dynamic hardening and recovery, dynamic recrystallization (DRX), and post-dynamic recrystallization (PDRX). This parameter set was adapted for use in the framework of DIGIMU® 5.0. The boundary conditions were defined in analogy to the experimental tests. The temperature evolutions were directly taken from the experimentally measured temperature-time histories T(t) as shown in Figure 2 and Table 2. The simulations were carried out under a constant strain rate of 0.1 s−1, and up to a strain of ε=1, which corresponds to the local strain levels in the center of the specimen when accounting for barreling [17].

[picture]

Fig. 3. Initial microstructure of Inconel 718
Table 3. Grain size distribution of initial RVE of Inconel 718

Fig. 3. Initial microstructure of Inconel 718 Table 3. Grain size distribution of initial RVE of Inconel 718

Mean grain size [μm] | Standard deviation | Min. grain size [μm] | Max. grain size [μm]
89.9 | 65.0 | 10.2 | 435.4

Results and Discussion

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Evolution of Dislocation Density under transient temperatures with DIGIMU®. Figure 4 illustrates the evolution of the mean dislocation density ρ¯ together with the critical dislocation density ρcr used in the DIGIMU® model. DRX is triggered once ρ¯ exceeds ρcr. Panel (a) shows the results for temperature increase from 1020∘C to 1070∘C with switches at equivalent strains in the specimen center εeq=0.2,0.4 and 0.6 , while panel (b) presents the corresponding temperature decreases from 1120∘C to 1070∘C. In the heating cases (Figure 4 (a)) the changing temperature curves (black), initially follow the isothermal reference T1=1020∘C up to the switch strain. For the earliest switch at εeq=0.2, the temperature change takes place before ρ¯ reaches the critical value of the 1020∘C reference ρcr=4.8E+08 mm−2 (upper dashed line). At the moment of the change, the material has therefore not entered the DRX regime corresponding to 1020∘C. The microstructure is still dominated by dynamic hardening. When the temperature is raised to 1070∘C,ρcr drops to 3.5E+08 mm−2 (lower dashed line), so that the current ρ¯ lies above the new critical value and DRX is activated immediately after the change at εeq=0.25. The dislocation density curve gradually adapts to the new thermal condition, and approaches the isothermal 1070∘C reference at εeq=0.55, and finally ends at almost the same level. The system thus has enough strain at the higher temperature for DRX to adapt to the new condition.
For the later switches at εeq=0.4 and 0.6 , the situation is different. Here, the temperature change occurs after the isothermal 1020∘C curve has already crossed its critical value. This means that DRX has already been activated. As a result, when the temperature is increased to 1070∘C,ρ¯ is well above ρcr of 1070∘C and DRX can still proceed, but the remaining strain interval is relatively short and part of the stored energy has already been consumed by DRX at 1020∘C. The non-isothermal curves bend downwards after the change, indicating that recovery and DRX reduce the stored dislocation density, but they do not converge to the isothermal 1070∘C reference. Instead at the final strain ρ¯ remains slightly higher, with values of ρ¯=3.51E+08 mm−2 for the temperature increase at εeq=0.4 and ρ¯=3.66E+08 mm−2 for the temperature increase at ε=0.6, compared to ρ¯=3.35E+08 mm−2 for the isothermal 1070∘C case. This corresponds to an increase of about 4% (temperature increase at εeq=0.4 ) and 6% (temperature increase at εeq=0.6 ), quantifying the slower and incomplete adaptation when the temperature increase is applied in the softening regime.

In the cooling paths in Figure 4 (b) the system starts from a high temperature DRX regime and is then transferred to a cooler state with reduced recovery and grain-boundary mobility. All three nonisothermal curves initially follow the isothermal 1120∘C reference. In all three cases ρ¯ crosses the critical level ρcr of 1120∘C already at ε=0.15, so that DRX is active at 1120∘C before the temperature change. When the temperature is reduced to 1070∘C,ρcr jumps to the higher value of ρcr=3.5E+08 mm−2, and the further evolution of ρ¯ depends on the switching strain. For a change at εeq=0.2,ρ¯ is 2.4E+08 mm−2 at the onset of cooling. During the subsequent deformation, ρ¯ increases gradually and reaches ρ¯=3.7E+08 mm−2, clearly above ρcr, implying an interval during which nucleation is suppressed before it resumes once ρ¯ exceeds ρcr. This triggers a second DRX episode at the lower temperature, after which ρ¯ decreases slightly and approaches a plateau close to the isothermal 1070∘C curve. If the change is delayed to εeq=0.4, cooling occurs in the softening regime of the 1120∘C curve with ρ¯=2.4E+08 mm−2<ρcr of 1070∘C. Immediately after cooling no abrupt increase in DRX activity occurs and the curve evolves into a flat plateau. This indicates that most of the DRX has already taken place at 1120∘C, and the second stage mainly adjusts the microstructure towards a steady-state at 1070∘C without strongly modifying the overall DRX fraction. For the latest change at εeq=0.6, the temperature decrease is applied at a late deformation stage at 1120∘C, where DRX has already reduced ρ¯ to 2.1E+08 mm−2, slightly below ρcr of 1120∘C. After the temperature change the non-isothermal curve rises and reaches ρ¯=3.3E+08 mm−2, but remains below ρcr of 1070∘C over the entire remaining strain range. It runs parallel to, but under, the isothermal curve of the new temperature of 1070∘C, ending at ρ¯=3.14E+08 mm−2 compared to ρ¯=3.35E+08 mm−2.

This behavior suggests that, after the late temperature change, no further DRX is triggered at 1070∘C, the microstructure is already recrystallized and never fully adapts to the new conditions.

[picture]

Fig. 4. Evolution of the critical and mean dislocation density during (a) temperature increase from 1020∘C to 1070∘C at equivalent strains 0.2,0.4 and 0.6 and (b) temperature decrease at 0.2,0.4 and 0.6

Fig. 4. Evolution of the critical and mean dislocation density during (a) temperature increase from 1020 ∘ C to 1070 ∘ C at equivalent strains 0.2 , 0.4 and 0.6 and (b) temperature decrease at 0.2 , 0.4 and 0.6

Microstructure Evolution. Figure 5 combines the evolution of the critical and mean dislocation density with the simulated microstructure for a temperature decrease at εeq=0.6. The points a-d make four selected strain levels along the non-isothermal curve with the corresponding microstructures. At an equivalent strain in the specimen center of εeq=0.4DRX is active and the microstructure consists of deformed grains with few RX nuclei. At the change point εeq=0.6 the microstructure is largely recrystallized. After the change εeq=0.8 the microstructure shows further straining and growth of existing RX nuclei, rather than new DRX grains. In the final state εeq=1.0 the microstructure is similar to that at εeq=0.8. DRX occurs only at 1120∘C and during the stage at 1070∘C mainly the existing grains grow without a second recrystallization event.

[picture]

Fig. 5. Simulated intermediate microstructure states for the temperature decrease from 1120∘C to 1070∘C at ε=0.6. Points a-d mark selected strains along the non-isothermal path at (a) εeq=0.4, (b) εeq=0.6, (c) εeq=0.8 and (d) εeq=1.0

Fig. 5. Simulated intermediate microstructure states for the temperature decrease from 1120 ∘ C to 1070 ∘ C at ε = 0.6 . Points a-d mark selected strains along the non-isothermal path at (a) ε e q = 0.4 , (b) ε e q = 0.6 , (c) ε e q = 0.8 and (d) ε e q = 1.0

Comparison of Experimental and Simulated Microstructures and Grain Size Distributions.

The original version of this paper is available on https://www.scientific.net/SSP.390.85.pdf

The final microstructures for the temperature increase cases from 1020∘C to 1070∘C are shown in Figure 6 for (a) an early temperature change at ε=0.2, (b) a change at ε=0.4 and (c) a later change at ε=0.6. The left image shows the experimental microstructure, the middle the corresponding DIGIMU® microstructure, and the right panel the experimental and simulated grain size distribution. For ε=0.2 (Figure 6 (a)), the experimental microstructure (left) is dominated by fine recrystallized grains, with only a weak tail to larger diameters. The simulated microstructure (middle) exhibits a very similar microstructure consisting of deformed stretched grains and RX grains. The experimental grain size distribution exhibits a peak in the lowest grain size classes, with a large fraction of grains located at 10μ m. The simulated grain size distribution peaks at 14.5μ m. The corresponding grain size distributions reflect reduced refinement when more deformation occurs at 1020∘C before a temperature increase. For the change at ε=0.4, both experiment and simulation display a coarser more heterogeneous grain structure compared to ε=0.2. For the latest change at ε=0.6 (Figure 6 (c)) the trend continues, both experiment and simulation display larger grains and the corresponding grain size distributions show a shift towards larger grain sizes.

The final microstructures for the temperature decrease from 1120∘C to 1070∘C are shown in Figure 7 for a change at (a) ε=0.2, (b) ε=0.4, and (c) ε=0.6. For an early temperature change at ε=0.2 (Figure 7 (a)), both experiment and simulation are filled with fine RX grains and smaller stretched deformed grains. The grain size distributions peak in the smallest classes at 20μ m in the experiment and 14.5μ m in the simulation. This reflects the underlying mechanism, DRX starts at 1120∘C and after the temperature decrease continues at 1070∘C. The additional DRX at 1070∘C leads to further grain refinement, which is consistently captured by the simulation and experiment. For the temperature decrease at ε=0.4 (Figure 7 (b)), the experimental and simulated microstructure remains largely recrystallized. The peak of the grain size distributions still lies in the smallest class, but the peak surface fraction decreases to about 0.45 , indicating that DRX at 1070∘C still contributes to refinement, yet less effectively than in the temperature decrease case at ε=0.2. For the later temperature decrease at ε=0.6 (Figure 7 (c)) the microstructure at the end of deformation shows larger deformed stretched grains with small RX grains. The grain size distribution still peaks at the
smaller classes, but the surface fraction is reduced to 0.45 . In this case DRX proceeds entirely at 1120∘C and no significant additional DRX occurs at 1070∘C consistent with the dislocation density evolution discussed above.

Taken together Figure 6 and Figure 7 demonstrate that the timing and direction of the temperature change provide a microstructural control, while the DIGIMU® simulations reproduce these trends in good agreement with the experiments.

[picture]

Fig. 6. Comparison of experimental (left) and simulated (middle) microstructures and grain size distribution (right) for a temperature increase at (a) ε=0.2, (b) ε=0.4 and (c) ε=0.6

Fig. 6. Comparison of experimental (left) and simulated (middle) microstructures and grain size distribution (right) for a temperature increase at (a) ε = 0.2 , (b) ε = 0.4 and (c) ε = 0.6

[picture]

Fig. 7. Comparison of experimental (left) and simulated (middle) microstructures and grain size distribution (right) for a temperature decrease at (a) ε=0.2, (b) ε=0.4 and (c) ε=0.6

Fig. 7. Comparison of experimental (left) and simulated (middle) microstructures and grain size distribution (right) for a temperature decrease at (a) ε = 0.2 , (b) ε = 0.4 and (c) ε = 0.6

Comparison of Experimental and Simulated DRX-Fractions. The final DRX-fractions for the transient temperature tests are compared for experiment and DIGIMU® simulations in Figure 8. For the temperature increase (Figure 8 (a)) both experiment and simulation indicate that an earlier temperature increase promotes a higher recrystallized fraction. At ε=0.2, the experimental DRX is about 46%, while the simulation predicts 55%. When the temperature change is delayed to ε=0.4, the DRX-fraction decreases 32% in the experiment and 44% in the simulation, and for ε=0.6 it drops further to 23% (experiment) and 37% (simulation). An earlier temperature increase gives the material more strain at 1070∘C, where DRX proceeds more rapidly. In contrast, a later change allocates more deformation to 1020∘C, where DRX is less pronounced, so that a smaller fraction of the microstructure can recrystallize.

For the temperature decrease (Figure 8 (b)), experiment and simulation show similar behavior. At ε=0.2, the experimental DRX-fraction is 50% while the simulation predicts 54%. At ε=0.4, the DRX fractions remain at a similar level, 49% in the experiment and 51% in the simulation. When the temperature decrease is applied at ε=0.6, the corresponding values are 55% (experimental) and 66% (simulative). A later temperature decrease produces a somewhat higher DRX-fraction, since more deformation is applied at 1120∘C, where DRX is effective, before the material is transferred to 1070∘C. This behavior is consistent with the dislocation density and microstructure analyses, which
showed that DRX is already well advanced during high temperature stage and that the deformation at 1070∘C mainly preserves and slightly adjusts the existing recrystallized structure.

Overall the comparison in Figure 8 shows that the DIGIMU® simulations captures the main features of the experimental DRX behavior under both temperature increase and decrease conditions. The deviations are typically on the order of 5−10%, with tendency to overestimate DRX . This can partly be attributed to the experimental measurement by the line-intercept method on microstructure pictures that were taken with optical microscope, where small recrystallized grains may be missed, while the simulation counts all recrystallized regions in the RVE.

[picture]

Fig. 8. Comparison of final DRX-fraction between experiment and simulation for (a) temperature increase from 1020∘C to 1070∘C at strains ε=0.2,0.4 and 0.6 ; and (b) temperature decrease from 1120∘C to 1070∘C at strains ε=0.2,0.4 and 0.6

Fig. 8. Comparison of final DRX-fraction between experiment and simulation for (a) temperature increase from 1020 ∘ C to 1070 ∘ C at strains ε = 0.2 , 0.4 and 0.6 ; and (b) temperature decrease from 1120 ∘ C to 1070 ∘ C at strains ε = 0.2 , 0.4 and 0.6

Conclusions.

The original version of this paper is available on https://www.scientific.net/SSP.390.85.pdf

In this work, the influence of transient temperature paths during hot deformation on the microstructure evolution of Inconel 718 was investigated by combining dilatometer tests with Full-Field simulations in DIGIMU®. Non-isothermal compression tests with temperature increases ( 1020∘C to 1070∘C ) and decreases ( 1120∘C to 1070∘C ) were performed, with the temperature change imposed at different strains. The measured temperature-time curves were directly imposed in the simulations. Isothermal tests at 1020∘C,1070∘C and 1120∘C served as reference. The results show a pronounced path dependence. Early heating to 1070∘C yields higher DRX fractions and finer, more homogeneous grain structures, while later heating leaves less strain at the higher temperature and produces coarser, less recrystallized microstructures. For cooling, DRX takes place mainly at 1120∘C. After an early temperature change, DRX can continue at 1070∘C, whereas a late change transfers an almost fully recrystallized state into a regime where further DRX is suppressed. These mechanisms are consistently reflected in dislocation density evolution, final DRX fractions and grain size distributions.

References

The original version of this paper is available on https://www.scientific.net/SSP.390.85.pdf

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The original version of this eBook is available on https://www.scientific.net/book/material-behaviour-modelling/978-3-0364-1987-9

Solid State Phenomena, ISSN: 1662-9779, Vol. 390, pp 97-109

doi: 10.4028/p-cF5aLT

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

Submitted:
2026-01-18

Revised:
2026-02-18

Accepted:
2026-02-21

Online:
2026-04-17

Quantifying Grain Boundary and Orientation Effects on Copper under Compression: A Crystal Plasticity Finite Element Approach

The original version of this paper is available on https://www.scientific.net/SSP.390.97.pdf

PAÇACI, M. Batuhan 1,a*, DARENDELILER Haluk 1, b

¹Department of Mechanical Engineering, Middle East Technical University, 06800 Ankara Türkiye

a  pacaci.batuhan@metu.edu.tr, b  hdarende@metu.edu.tr

Keywords: Crystal Plasticity, Grain Size, Grain Boundary, Simple Compression

Abstract

The plastic deformation behavior of polycrystalline copper is strongly governed by microstructural attributes such as grain boundaries and crystallographic orientation. While previous studies have examined microstructural features such as grain size and crystallographic orientation, these factors are often treated as isolated [1,2]. In this study, the individual and coupled effects of grain size and orientation distribution on the mechanical response of copper are investigated using the crystal plasticity finite element method. Uniaxial compression simulations are performed employing an Abaqus user-material subroutine (UMAT) subroutine, and the resulting stress-strain response, slip system activity, and strain localization are examined. Polycrystalline representative volume elements (RVEs) with varying grain sizes and textures are generated to quantify their influence on flow behavior and dislocation density. The results indicate that grain refinement diminishes orientation sensitivity and the influence of individual grains, while enhancing material strength. Moreover, grain boundaries promote heterogeneous slip, localized deformation near grain edges, and increased stress-strain heterogeneity.

Introduction

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The plastic deformation of polycrystalline metals arises from crystallographic slip and its interaction with microstructural features such as grain size, grain boundaries, and orientation distribution. Classical continuum plasticity models, although computationally efficient, rely on homogenized material descriptions and therefore fail to capture orientation-dependent anisotropy, strain localization, and intergranular heterogeneity observed experimentally [3].

Crystal Plasticity theory provides a physically motivated framework by linking macroscopic deformation to slip-system-level kinematics and hardening mechanisms [4]. Early constitutive formulations established the theoretical foundation for crystal plasticity by explicitly incorporating resolved shear stress-based slip activation and finite deformation kinematics [5]. Subsequent numerical implementations enabled the integration of crystal plasticity into finite element frameworks, allowing grain-scale stress-strain heterogeneity and texture evolution to be resolved explicitly [6].

Dislocation-based theories first introduced the concept of geometrically necessary dislocations (GNDs) as a mechanistic link between lattice incompatibility and continuum-scale fields, providing a theoretical basis for capturing deformation-induced lattice curvature [7]. Subsequent experimental observations revealed pronounced strain localization and lattice curvature accumulation in the vicinity of grain boundaries, confirming their active role in governing deformation heterogeneity [8, 9]. Building on this theoretical-experimental foundation, crystal plasticity finite element formulations enabled a more comprehensive representation of grain-scale plastic deformation [10].

The increasing demand for micro- and nano-scale material characterization, together with the steady growth in computational power and simulation capability over recent decades, has enabled the investigation of material behavior at progressively smaller length scales. Within this context, the Crystal Plasticity Finite Element Method (CPFEM) has emerged as a robust framework for linking microstructural features to macroscopic mechanical response in polycrystalline metals. In this study, the effects of grain size, crystallographic orientation variation, and their coupled outcomes on the plastic deformation of polycrystalline copper are systematically investigated using CPFEM.

Representative Volume Elements (RVEs) with prescribed microstructural characteristics are generated, and uniaxial compression simulations are performed to evaluate macroscopic stress-strain response, slip activity, and strain localization. The results provide micromechanical insight into how microstructural topology governs deformation behavior in Face-Centered-Cubic (FCC) metals, contributing to improved microstructure-sensitive predictive modeling.

Governing Equations and Modeling

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Crystal plasticity is formulated within a finite deformation framework, where the deformation gradient, F, is multiplicatively decomposed into elastic and plastic parts as F* and Fp, respectively. The velocity gradient L is defined as

L=F˙*·F*−1+F*·F˙P·FP−1·F*−1(1)

The plastic part of velocity gradient is expressed as a sum over all slip systems

Lp=∑α=1nγ˙(α)s(α)⊗m(α)(2)

where γ˙(α) is the slip rate on system α, and s(α) and m(α) denote the slip direction and slip plane normal, respectively, in the reference configuration. In crystal plasticity, it is assumed that slip does not affect crystal elasticity, and the elastic constitutive law can be described as [11]

τ∇*=ℂ:D*(3)

where τ∇* is the Jaumann rate of elastic Kirchhoff stress, D* is the elastic rate of deformation tensor, and ℂ is the elastic stiffness tensor with tensor symmetry. To express the elastic response in the global frame, each grain's lattice orientation must be mapped from its local crystal coordinate system to the macroscopic reference frame. This coordinate transformation is performed using the rotation matrix R. The elastic stiffness tensor in the sample frame is obtained as follows [12]:

ℂ=Rℂ0RT(4)

where coordinate rotation matrix R can be expressed in terms of transformation matrix T as follows

R=[(T11T)2(T12T)2(T13T)22T11TT12T2T13TT11T2T12TT13T(T21T)2(T22T)2(T23T)22T21TT22T2T23TT21T2T22TT23T(T31T)2(T32T)2(T33T)22T31TT32T2T33TT31T2T32TT33TT11TT21TT12TT22TT13TT23TT11TT22T+T12TT21TT13TT21T+T11TT23TT12TT23T+T13TT22TT31TT11TT32TT12TT33TT13TT11TT32T+T12TT31TT13TT31T+T11TT33TT12TT33T+T13TT32TT21TT31TT22TT32TT23TT33TT22TT31T+T21TT32TT21TT33T+T23TT31TT22TT33T+T23TT32T].(5)

Here transformation matrix composed of Euler angles in Bunge's convention is as follows

T=[cosϕ1cosϕ2−sinϕ1sinϕ2cosΦsinϕ1cosϕ2+cosϕ1sinϕ2cosΦsinϕ2sinΦ−cosϕ1sinϕ2−sinϕ1cosϕ2cosΦ−sinϕ1sinϕ2+cosϕ1cosϕ2cosΦcosϕ2sinΦsinϕ1sinΦ−cosϕ1sinΦcosΦ].((6)

The critical resolved shear stress τ(α) on the slip system, α, can be expressed as follows

τ(α)=m*(α)·ρ0ρτ·s*(α)(7)

where τ is the Kirchhoff stress tensor, ρ0 and ρ are the mass densities in undeformed and deformed configurations, and s*(α) and m*(α) are the slip direction and the slip plane normal of system α,
respectively, in the deformed configuration. Rate-independent plasticity is the limiting form of a ratedependent viscoplastic model [13]. In rate-sensitive crystals, the slip rate γ˙(α) depends on τ(α) as follows:

γ˙(α)=γ˙0(α)(τ(α)g(α))|τ(α)g(α)(1m)−1(8)

where γ˙0(α) is the reference slip rate, g(α) the current slip resistance. The slip resistance incorporates lattice friction g0, Voce-type hardening gv, and Taylor hardening

g(α)=g0+gv+Gb(α)ξ2ρtot(9)

where G is shear modulus, b(α) Burger's vector, ξ geometric factor, and ρtot  total dislocation density. The total dislocation density, ρtot α, is computed through projection operations that account for geometrically necessary dislocations (GNDs) of edge and screw types, with their respective line directions denoted as ne(β) and ss(β)[11]

ρtotα=∑β|m(α)·ne(β)||ρGND,eβ|+|m(α)·ss(β)||ρGND,sβ|(10)

Subscripts e and s denote the edge and screw components of the geometrically necessary dislocation density, respectively. Decomposition of the incompatibility tensor into edge, ρGNDe(α), and screw, ρGNDS(α), dislocation densities can be shown as follows

Λ=∑αρGNDe(α)b(α)s(α)⊗n(α)+ρGNDs(α)b(α)s(α)⊗s(α)(11)

where Λ is the incompatibility tensor, b(α) is the magnitude of the Burger vector and n(α) is the edge dislocation line direction [12, 14]:

Λ=−(∇×Fp)T(12)

The constitutive behavior of each grain was introduced through a user-material subroutine (UMAT). The formulation implemented in this work is primarily based on the classical single-crystal plasticity model developed by Huang, a widely referenced framework in CPFEM studies [6]. Nevertheless, several numerical components were adapted from the Oxford UMAT implementation [12].

Throughout the simulations, Abaqus software is used for solving the constitutive updates. The constitutive updates within the UMAT rely on a forward-gradient ( θ-method) time integration scheme. The slip activity γ˙(α) on a system α can be modeled as a function of the applied shear stress τ(α) and the present strength g(α).

Δγ(α)=Δt[γ˙t(α)+θ∂γ˙(α)∂τ(α)Δτ(α)+θ∂γ˙(α)∂g(α)Δg(α)](13)

Here γ˙t(α) describes the evaluated slipping rate at time t. θ represents the Euler time integration scheme. Hence, the increments of shear strain, Δγ(α), can be obtained as follows

Representative polycrystalline geometries were constructed using Neper. The computational domain was defined as a cylindrical representative volume element (RVE) with a 1:1 height-todiameter ratio. An example of created RVE is presented in Fig. 1.

[picture]

Fig. 1. RVE created by Neper to be used in CPFEM simulation.

Fig. 1. RVE created by Neper to be used in CPFEM simulation.

Crystallographic orientations were defined for each grain using Bunge Euler angles ( φ1,Φ,φ2 ) within the crystal plasticity framework. For each grain, all twelve {111}<110> slip systems of the FCC crystal structure were evaluated, and the maximum Schmid factor among these systems was used as the primary indicator for slip system activity.

[picture]

Fig. 2. Boundary conditions and related Reference Points (RP) are represented.

Fig. 2. Boundary conditions and related Reference Points (RP) are represented.

Polycrystalline geometries were meshed with C3D10 quadratic tetrahedral elements (four integration points) to capture strain gradients and rotation fields accurately. Mesh refinement was controlled through the Neper user defined dimensionless scaling factor which defines the characteristic element length, rcl, relative to the average grain size. Mesh length for each grain, cl, can be written as

cl=rcllcell(14)

where lcell  is the characteristic length of a grain. In Neper, the characteristic length of a grain is computed approximately in terms of its volume Vcell  as lcell =(Vcell )1/3.

Meshes were refined by adjusting cell volume to study resolution effects on deformation and stability. The top surface imposed 20% axial compression by applying an axial force (Fig. 2). Boundary conditions allowed free lateral strain, and frictionless contact was assumed to prevent barreling and isolate grain-scale plasticity as shown in Fig. 2.

The material parameters employed in this study are summarized in Table 1. All values were directly adopted from Demir et al. [12]. This reference provides a validated parameter set specifically calibrated for FCC copper, ensuring consistency with the crystal plasticity constitutive framework used in the present work.

Table 1. Selected Material Parameters of Cu for CPFEM.

Model Parameters | Value | Units
C₁₁ | 170 | [GPa]
C₁₂ | 124 | [GPa]
C₄₄ | 75 | [GPa]
b | 2.56(10⁻⁴) | [μm]
g₀ | 16 | [MPa]
h₀ | 250 | [MPa]
n | 2.5 |
q | 1.4 |

Results and Discussions

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Grain Size Effects. Grain-size effects were studied using a single-crystal RVE and polycrystalline RVEs with 10, 100, and 500 grains, all under identical conditions. Models were generated in Neper with random seed placement to create unbiased 3D Voronoi tessellations, ensuring statistically uniform microstructures for reliable evaluation of grain-size influence (Fig. 3).

Grain orientations were randomly assigned to avoid texture bias, with mesh and boundary settings kept consistent with the baseline. After applying compressive load, the true stress -true strain curves were obtained. Stress-strain curves (Fig. 4) show clear grain-size strengthening; smaller grains (higher grain numbers) yield higher flow stress, while the single-grain model exhibits the lowest stress.

Table 2 summarizes flow stresses and stresses at 10% strain. As grain size decreases from single crystal to a RVE of 500 grains, yield stress rises from 41.86 MPa to 47.25 MPa , and stress at 10% strain from 125.0 MPa to 168.8 MPa . This corresponds to ~13% higher flow stress and ~35% higher strength, confirming classical grain-size strengthening due to increased grain-boundary density and GND accumulation [11].

[picture]

Fig. 3. RVE models for a single crystal and polycrystals with 10,100 , and 500 grains respectively.

Fig. 3. RVE models for a single crystal and polycrystals with 10,100 , and 500 grains respectively.

To validate the grain-size dependence of the simulated flow stress, the results were fitted to the classical Hall-Petch relation:

σy=σ0+kyd−1/2(15)

where ky is the Hall-Petch constant, d is the average grain size, σy is the yield strength, and σ0 is the lattice friction. For Hall-Petch validation, 0.2% offset method is used to determine the yield stresses ensuring consistent identification of plastic initiation across microstructures (Fig. 5).

[picture]

Fig. 4. True stress-strain curves for different grain numbers in RVE.

Fig. 4. True stress-strain curves for different grain numbers in RVE.

Table 2. Grain number, flow stresses and stress levels at 10% strain for different grain size.

Grain Number | Flow Stress [MPa] | Stress at 0.1 Strain [MPa]
500 | 47.25 | 168.79
100 | 45.52 | 158.37
10 | 44.06 | 153.04
Single crystal | 41.86 | 125.00

Fig. 5. Closer look at the 0.2% strain for different number of grains.

The corresponding plot of flow stress versus d−1/2 exhibited a good linear correlation (R2≈0.97), confirming that the simulated data follow Hall-Petch scaling. The fitted parameters were found to be σ0=39.40MPa and ky=120.79MPa·μm1/2 and the related curves can be seen in the following Fig. 6. Results show good agreement with the experimentally reported values for pure copper presented by Cordero et al., who reported σ0≈40MPa and ky≈110MPa·μm1/2[15].

[picture]

Fig. 6. Hall-Petch plot with curve fitting for 0∘,45∘,45∘.

Fig. 6. Hall-Petch plot with curve fitting for 0 ∘ , 45 ∘ , 45 ∘ .

Grain Orientation Effects on Single Crystal. To investigate the effect of grain orientation, two distinct sets of Euler angles are considered. The first orientation is arbitrarily defined by the Euler angles ( γ ). Schmid factor analysis (Table 3) indicates dominant activation of slip systems 4, 6, 7, and 8, with values ranging from 0.29 to 0.35 , moderate activation of systems 2 and 3 (approximately 0.14 ), and negligible contributions from the remaining systems (<0.10). The maximum factor of 0.35 makes this orientation a suitable reference for studying crystallographic effects under uniaxial compression.

Accumulated slip ( 0∘,45∘,45∘ ) for all twelve FCC systems under uniaxial compression shows dominant activity in systems 4, 6, 7, and 8, consistent with their high Schmid factors. Systems 2, 3, 11, and 12 exhibit moderate activation, while others remain largely inactive. This reflects the partially aligned slip character of the ( ~0.5 ) orientation, producing a plastic response governed by multiple active systems rather than a single mechanism. Fig. 7 illustrates this distribution, with intense slip in systems (d), (f), (g), and (h).

To identify the orientation with the lowest yield and flow stress, the Euler angles of the second set were chosen to maximize the Schmid factor ( 324∘,108∘,98∘ ). This condition was met for ( 324∘,108∘,98∘ ), where slip system 10 reached 0.49 (Table 3). Systems 4 and 10 show the highest Schmid factors among all slip systems.

Fig. 8 shows accumulated slip contours for the ( 324∘,108∘,98∘ ) orientation, with activity concentrated in systems (d) and (j) which exhibit the most favorable geometric arrangement between the loading axis, the slip plane normal, and the slip direction. These systems accommodate most of the deformation, confirming this orientation corresponds to the lowest yield and flow stress among all cases.

Fig. 9 compares true stress-strain curves for ( 0∘,45∘,45∘ ) and ( 20% ) orientations. The latter, with a maximum Schmid factor of 0.35 , shows higher flow stress and stronger hardening. At 39.7% strain, stresses are approximately 186 MPa and 260 MPa respectively, a difference of about 0∘,45∘,45∘ in plastic strength.

Table 3. Schmid factors for single crystal with two different Euler angles.

Euler angles
0°, 45°, 45° | 324°, 108°, 98°
Slip System | Slip Plane Normal | Slip Direction | Schmid Factor | Schmid Factor
1 | (1,1,1) | [1, -1,0] | 0.00 | 0.22
2 | (1,1,1) | [0,1, -1] | 0.14 | 0.04
3 | (1,1,1) | [1,0, -1] | 0.14 | 0.26
4 | (-1,1,1) | [1,1,0] | 0.29 | 0.46
5 | (-1,1,1) | [0,1, -1] | 0.06 | 0.10
6 | (-1,1,1) | [1,0,1] | 0.35 | 0.36
7 | (1, -1,1) | [1,1,0] | 0.29 | 0.25
8 | (1, -1,1) | [0,1,1] | 0.35 | 0.14
9 | (1, -1,1) | [1,0, -1] | 0.06 | 0.39
10 | (1,1, -1) | [1, -1,0] | 0.00 | 0.49
11 | (1,1, -1) | [0,1,1] | 0.14 | 0.20
12 | (1,1, -1) | [1,0,1] | 0.14 | 0.29

[picture]

Fig. 7. For the single crystal with the Euler angles ( 0∘,75∘,45∘ ), contours of the total accumulated slip ( 5∘ ) on the twelve 10% slip systems: (a) 1, (b) 2, (c) 3, (d) 4, (e) 5, (f) 6, (g) 7, (h) 8, (i) 9, (j) 10, (k) 11, (l) 12.

Fig. 7. For the single crystal with the Euler angles ( 0 ∘ , 75 ∘ , 45 ∘ ), contours of the total accumulated slip ( 5 ∘ ) on the twelve 10 % slip systems: (a) 1, (b) 2, (c) 3, (d) 4, (e) 5, (f) 6, (g) 7, (h) 8, (i) 9, (j) 10, (k) 11, (l) 12.

[picture]

Fig. 8. For the single crystal with the Euler angles 324∘,108∘,98∘ contours of the total accumulated slip ( 0∘,45∘,45∘ ) on twelve 0∘,55∘,45∘ slip systems: (a) 1, (b) 2, (c) 3, (d) 4, (e) 5, (f) 6, (g) 7, (h) 8 , (i) 9 , (j) 10 , (k) 11, (l) 12.

Fig. 8. For the single crystal with the Euler angles 324 ∘ , 108 ∘ , 98 ∘ contours of the total accumulated slip ( 0 ∘ , 45 ∘ , 45 ∘ ) on twelve 0 ∘ , 55 ∘ , 45 ∘ slip systems: (a) 1, (b) 2, (c) 3, (d) 4, (e) 5, (f) 6, (g) 7, (h) 8 , (i) 9 , (j) 10 , (k) 11, (l) 12.

[picture]

Fig.9. True stress-true strain comparison for single crystals with Euler angles of ( 0∘,75∘,45∘ ) and ±4o.

Fig.9. True stress-true strain comparison for single crystals with Euler angles of ( 0 ∘ , 75 ∘ , 45 ∘ ) and ± 4 o .

Comparison of Different Orientations for Single Crystal: To assess the effect of crystallographic alignment, the crystal was rotated from ( Φ ) to ( φ2 ) in φ1 increments. Table 4 shows that the maximum Schmid factor rises from 0.28 to 0.43 , while true stress at 324∘ strain drops from 243.56 MPa to 147.41 MPa , confirming an inverse relation between slip alignment and flow strength. Fig. 10 illustrates this trend, with yield stress and hardening rate systematically decreasing as the Schmid factor increases.

Orientation Effects in Polycrystals. To investigate the effect of crystallographic orientation on the polycrystalline scale, RVEs with 500 grains were analyzed under identical boundary and loading conditions. For the 500 -grain RVE, four orientations ( 108∘,98∘ ), ( 0∘,55∘,45∘ ), ( 10% ), and ( (0∘ ) were selected based on single-crystal results. To maintain these as mean values, 45∘,45∘ deviation was applied to 10% and 324∘,108∘,98∘, while Φ remained fixed since rotation about the loading axis does not affect uniaxial response. This introduces realistic variability without making the RVE behave like a single crystal. Fig. 11 shows stress-strain curves, confirming the same trend: ( φ2, ±10o ) yields the lowest strength, ( ±50o ) the highest, and others fall in between. Table 5 summarizes stresses at φ1 strain for direct comparison. Fig. 12 illustrates slip localization for ≪0.025, ±10o ), with systems (d), (f), (g), and (h) most active, confirming multi-system slip behaviour.

Table 4. Max Schmid Factors for single crystals with different Euler Bunge angles.

Euler Angle | Max Schmid Factor | True Stress at %10 Strain [MPa]
0-55-45 | 0.28 | 243.56
0-50-45 | 0.31 | 226.92
0-60-45 | 0.33 | 209.96
0-45-45 | 0.35 | 202.66
0-65-45 | 0.37 | 179.56
0-70-45 | 0.41 | 158.43
0-75-45 | 0.43 | 147.41

[picture]

Fig. 10. True stress-true strain curves for single crystal at different orientations.

Fig. 10. True stress-true strain curves for single crystal at different orientations.

In the polycrystalline simulations, higher overall stress levels are observed compared to their single-crystal counterparts, reflecting the additional strengthening introduced by intergranular constraints and grain boundary interactions.

[picture]

Fig. 11. True stress-true strain curves for a 500 -grain polycrystal with different Euler angles.

Fig. 11. True stress-true strain curves for a 500 -grain polycrystal with different Euler angles.

Table 5. True stress-strain at ±50o strain for different Euler Bunge Angles.

Euler Angle | Max Schmid Factor | True Stress at 10% Strain [MPa]
0-55-45 | 0.28 | 235.53
0-45-45 | 0.35 | 215.55
0-75-45 | 0.43 | 160.16
324-108-98 | 0.49 | 131.80

Coupled Effect of Orientation and Grain Size Variation. Two orientation distributions (Sets A and B ) were generated around ( p=2.84(10−5) ), which corresponds to the orientation exhibiting the lowest stress among the selected cases. Using the same procedure as before, the p=0.0040 and ±10o angles were varied with magnitudes of ±50o for Set A and 20% for Set B while 8.35% was kept fixed. Welch's t-tests confirmed statistically significant differences ( p -values 4.88% ) between 8.35% and 4.88% cases; specifically, ±10o for 500 grains and ±50o for 50 grains.

Fig. 13 and Table 6 show that as the orientation variation increases from superscript number to mathematical formula, the flow stress increases. For the 50 -grain RVEs, the stress at superscript number strain increases by 17.62 MPa ( mathematical formula ), whereas for the 500-grain RVEs the increase is 10.92 MPa ( mathematical formula ). Grain number moderates this effect: RVEs with fewer grains or larger grain sizes reflect orientation deviations more strongly, while RVEs with larger numbers of grains with smaller grain sizes tend to average them out.

A comparison between the 50 -grain and 500 -grain RVEs demonstrates that grain number and grain size control the macroscopic impact of orientation variation, as shown in Table 6. In the 50-grain RVEs, the smaller number of grains and their larger size limit averaging, so the stress reflects orientation deviations more directly. In contrast, in the 500 -grain RVEs, the larger number of smaller grains increases averaging, reducing the sensitivity to orientation variation. This is evident in the reduction of the stress variation from mathematical formula to mathematical formula.

Overall, the superscript number versus mathematical formula comparison confirms that orientation variation affects the aggregate response, and that its impact depends strongly on the number of grains within the RVE.

Table 6. Comparative analysis of 50 and 500 Grains for σy=σ0+kyd−1/2 and γ standard deviations.

| 50 Gram | 500 Grain
Set | σ = 10 | σ = 50 | σ = 10 | σ = 50
Stress at %20 Strain [MPa] | 210.93 | 228.55 | 223.84 | 234.76
Difference [MPa] | 17.62 | 10.92
Variation [%] | 8.35 | 4.88

[picture]

Fig. 12. For the Euler angles ( mathematical formula ) contours of the total accumulated slip ( mathematical formula ) for the twelve superscript number slip systems in 500 grain polycrystal: (a) 1, (b) 2, (c) 3, (d) 4, (e) 5, (f) 6, (g) 7, (h) 8, (i) 9, (j) 10, (k) 11, (l) 12.

Fig. 12. For the Euler angles ( ) contours of the total accumulated slip ( ) for the twelve slip systems in 500 grain polycrystal: (a) 1, (b) 2, (c) 3, (d) 4, (e) 5, (f) 6, (g) 7, (h) 8, (i) 9, (j) 10, (k) 11, (l) 12.

[picture]

Fig.13. True stress-true strain curves for superscript number versus mathematical formula variations.

Fig.13. True stress-true strain curves for versus variations.

Conclusions

The original version of this paper is available on https://www.scientific.net/SSP.390.97.pdf

The combined findings elucidate the governing mechanisms in crystal plasticity:

Orientation alignment dictates local slip activity and controls the onset and nature of hardening.

Polycrystalline interactions increase overall material strength.

Grain size reduction increases yield stress and flow strength in accordance with Hall-Petch scaling.

Orientation variation and neighborhood constraints govern the transition from local anisotropy to global homogeneity.
In conclusion, the mechanical response of polycrystalline copper is shown to arise from a delicate balance between crystallographic anisotropy and intergranular compatibility. The present CPFEM framework provides a numerically stable and physically consistent basis for analyzing texture effects, hardening behavior, and strain localization in FCC materials.

References

The original version of this paper is available on https://www.scientific.net/SSP.390.97.pdf

P. Wang, L. Ma, X. Cheng, and X. Li, "Effect of grain size and crystallographic orientation on the corrosion behaviors of low alloy steel," Journal of Alloys and Compounds, vol. 857, 158258, 2021; https://doi.org/10.1016/j.jallcom.2020.158258

A. Bahmani, S. Moradi, M. Lotfpour, H. T. Jeong, and W. J. Kim, "Effect of grain size on the corrosion resistance of the Fe41Mn25Ni24Co8Cr2 high entropy alloy," Corrosion Science, vol. 230, 111892, 2024;

R. Hill, Continuum micro-mechanics of elastoplastic polycrystals, Journal of the Mechanics and Physics of Solids, vol.13, 1965,pp.89-101, .

R. J. Asaro and J. R. Rice, "Strain localization in ductile single crystals," Journal of the Mechanics and Physics of Solids, vol. 25, no. 5, pp. 309-338, 1977,

R. J. Asaro, "Micromechanics of crystals and polycrystals," Advances in Applied Mechanics, vol. 23, pp. 1-115, 1983;

Y. Huang, A User-Material Subroutine Incorporating Single Crystal Plasticity in the ABAQUS Finite Element Program. Cambridge, MA: Harvard University, 1991, pp. 1-21

J. F. Nye, "Some geometrical relations in dislocated crystals," Acta Metallurgica, vol. 1, no. 2, pp. 153-162, 1953

A. Vattré, "HDR: Interfaces in crystalline materials," arXiv preprint arXiv:2307.14569, pp. 4050, 2023;

Raabe, D., Zhao, Z., Park, S. J., and Roters, F. (2002). Theory of orientation gradients in plastically deformed crystals. Acta Materialia, 50, 421-440.

A. Arsenlis, D. M. Parks, R. Becker, and V. V. Bulatov, "On the evolution of crystallographic dislocation density in non-homogeneously deforming crystals," Journal of the Mechanics and Physics of Solids, vol. 52, no. 6, pp. 1213-1246, 2004;

R. Hill and J. R. Rice, "Constitutive analysis of elastic-plastic crystals at arbitrary strain," Journal of the Mechanics and Physics of Solids, vol. 20, no. 6, pp. 401-413, 1972;

E. Demir, A. M. Pecchero, C. Hardie, and E. Tarleton, "OXFORD-UMAT: An efficient and versatile crystal plasticity framework International Journal of Solids and Structures, vol. 307, 2025,

D. Peirce, R. I. Asaro, and A. Needleman, "Material rate dependence and localized deformation in crystalline solids," Acta Metallurgica, vol. 31, no. 12, pp. 1951-1976, 1983;

S. Das, A. Tarleton, and E. Tarleton, "Consistent determination of geometrically necessary dislocation density from simulations and experiments," International Journal of Plasticity, vol. 109, pp. 18-42, 2018;

Z.C. Cordero, B.E. Knight, and C.A. Schuh, "Six decades of the Hall-Petch effect - a survey of grain-size strengthening studies on pure metals," International Materials Reviews, vol. 61, pp. 495-512, 2016; .

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Solid State Phenomena, ISSN: 1662-9779, Vol. 390, pp 111-119

doi: 10.4028/p-os8U1C

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

Submitted:
2026-01-19

Revised:
2026-02-20

Accepted:
2026-02-21

Online:
2026-04-17

Experimental Study on the Relationship between Cooling Rate, Secondary Dendrite Arm Spacing and Tensile Properties in Aluminum Casting

The original version of this paper is available on https://www.scientific.net/SSP.390.111.pdf

Taegyun Ahn 1,a, Juwon Lee 1, b, Janghoon Lee 2,c and Jeong Whan Yoon 1,3, d*

1 Department of Mechanical Engineering, Korea Advanced Institute of Science and Technology (KAIST), 291 Daehak-ro, Yuseong-gu, Daejeon 34141, Republic of Korea

2 Manufacturing Solution Division, Hyundai Motor Group, 37 Cheoldobangmulgwan-ro, Uiwang-si, Gyeonggi-do 06797, Republic of Korea

3 School of Engineering, Deakin University, 75 Pigdons Rd., Waurn Ponds, VIC 3216, Australia

at.ahn@kaist.ac.kr, bjuwonlee@kaist.ac.kr, cjhlee0212@kia.com, dj.yoon@kaist.ac.kr

Keywords: Aluminum casting, Cooling rate, Secondary dendrite arm spacing (SDAS), Mechanical properties, Thermal analysis

Abstract

The dendritic microstructure formed during solidification plays a critical role in determining the mechanical properties of aluminum castings. In particular, secondary dendrite arm spacing (SDAS) is strongly influenced by the cooling rate and is closely related to yield strength, ultimate tensile strength, and elongation. However, experimental validation of these relationships requires a consistent methodology for defining cooling rate and linking it to microstructural and mechanical measurements. In this study, an experimental framework was established to investigate the relationships among cooling rate, SDAS, and mechanical properties in aluminum castings. Casting blocks with different thicknesses were fabricated to obtain a wide range of cooling rates. Cooling curves were measured during solidification, and cooling rates were determined using the second derivatives of the cooling curves. SDAS measurements and tensile tests were conducted on specimens extracted from symmetric positions within the casting blocks to ensure equivalent thermal histories. The results showed that the cooling rate-SDAS relationship exhibited a linear trend on a logarithmic scale, consistent with previously reported correlations. Smaller SDAS values were associated with increased yield strength, ultimate tensile strength, and elongation. The agreement between the present results and literature data confirms the validity of the proposed experimental framework for correlating solidification conditions, microstructure, and mechanical properties of aluminum castings.

1. Introduction

The original version of this paper is available on https://www.scientific.net/SSP.390.111.pdf

Aluminum castings are widely used in automotive, aerospace, and structural applications due to their low density, good castability, and favorable mechanical performance. The mechanical properties of aluminum castings are strongly governed by the microstructure developed during solidification [13]. Among various microstructural parameters, secondary dendrite arm spacing (SDAS) has been recognized as a key indicator of solidification conditions and final material performance [4-6].
It is well established that SDAS decreases with increasing cooling rate, leading to finer microstructures [7,8] and improved mechanical properties such as yield strength (YS), ultimate tensile strength (UTS), and elongation (El) [9, 10]. Numerous studies have proposed empirical relationships linking cooling rate and SDAS, as well as correlations between SDAS and mechanical properties [11, 12]. Despite these efforts, experimental verification of such relationships remains challenging due to ambiguities in defining cooling rate and difficulties in directly correlating microstructural observations with mechanical test results obtained from the same thermal conditions. One major experimental challenge arises from the fact that cooling curves obtained during casting are often non-ideal, making the determination of representative cooling rates ambiguous. Additionally, it is not always feasible to extract specimens for microstructural analysis and mechanical testing from the exact same location within a casting, which can lead to inconsistencies
in thermal history and data interpretation. These limitations highlight the need for a robust experimental framework that enables consistent evaluation of cooling rate, SDAS, and mechanical properties.
The objective of this study is to establish and validate an experimental methodology for correlating cooling rate, SDAS, and mechanical properties in aluminum casting. By employing casting blocks with different thicknesses, cooling rates were systematically varied and quantified using the second derivatives of cooling curves. SDAS measurements and tensile tests were conducted on specimens extracted from symmetric locations to ensure equivalent thermal histories. The experimental results were analyzed and compared with literature-reported correlations to assess the validity of the proposed framework.

2. Specimen Preparation

The original version of this paper is available on https://www.scientific.net/SSP.390.111.pdf

Casting Design and Solidification Conditions. To achieve various cooling rates, the casting was designed to have six steps with different thicknesses of 4, 5, 7, 10, 14 and 18 mm , where thinner steps were intended to produce higher cooling rates. A total of seven castings were solidified to ensure repeatability of the experimental data.

[picture]

Fig. 1. Aluminum castings with multiple thicknesses

Fig. 1. Aluminum castings with multiple thicknesses

Temperature Measurement and Specimen Location Strategy. To establish correlations between cooling rate, SDAS, and mechanical properties, it is ideally required that thermocouple measurements, SDAS observations, and tensile tests be conducted at the same location. However, this was not feasible for several reasons.
First, tensile specimens could not be extracted from the thermocouple location due to the presence of the thermocouple hole. Second, SDAS observation requires grinding and polishing, which would damage tensile specimens if performed prior to testing. Conversely, performing tensile tests before SDAS observation would deform the dendritic structure, preventing accurate SDAS measurement. While most of the prior studies do not mention about this problem, a study has addressed this issue by extracting specimens near the thermocouple location or measuring SDAS at the grip region of tensile specimens [11]. However, these approaches may result in discrepancies in thermal history between the gauge region and the thermocouple measurement location.
In this study, thermal symmetry about the center line of the casting was assumed. Thermocouples were placed at a distance of 50 mm from the center line, and SDAS specimens were extracted at the same location. Tensile specimens were extracted from another location 50 mm from the center line on the opposite side, where an identical thermal history was assumed. The validity of this assumption was verified by comparing SDAS values measured at both symmetric locations, which showed good agreement.

Specimen Geometry and Preparation. For SDAS measurement, coin-shaped specimens were fabricated. From the top view, the center of the specimen corresponded to the thermocouple location.

The specimen diameter was 25 mm , which was selected to accommodate grinding and polishing procedures. From the side view, grinding and polishing were performed until the mid-plane along the thickness direction of the casting block was reached, allowing observation of the microstructure at the thickness center.
Tensile specimens were extracted in a similar manner. From the top view, the center of the gauge section coincided with the symmetric position relative to the thermocouple location. From the side view, specimens were extracted from the mid-thickness of the casting blocks.

[picture]

Fig. 2. Locations of thermocouples, specimen extraction positions, and extracted specimens

Fig. 2. Locations of thermocouples, specimen extraction positions, and extracted specimens

Tensile Specimen Type Selection. For tensile testing of cast metals, round specimens are commonly used to minimize the influence of non-uniform pore distribution. However, for casting blocks with thicknesses of 4, 5, and 7 mm , the limited thickness made it impractical to fabricate round specimens. Therefore, flat specimens were employed for these conditions.
Although the combined use of round and flat specimens may raise concerns regarding result comparability, thin casting blocks were required in this study to achieve high cooling rates, making the use of flat specimens unavoidable. While round specimens with reduced gauge diameters could have been extracted, such specimens were expected to be excessively sensitive to porosity, resulting in large scatter and poor repeatability in tensile test results. Consequently, flat specimens were selected to ensure more consistent and reliable mechanical property measurements under high cooling rate conditions.

3. Thermal Analysis

The original version of this paper is available on https://www.scientific.net/SSP.390.111.pdf

Challenges in Analyzing Non-Ideal Cooling Curves. In an ideal cooling curve, a plateau with zero slope appears during phase transformation, where the temperature remains constant due to the release of latent heat. In practice, however, this plateau often exhibits a finite slope as a result of an imbalance between the rate of latent heat release and the rate of heat loss to the surrounding environment. Furthermore, experimental noise and extremely high cooling rates can obscure subtle slope changes, making it difficult to clearly identify the phase transformation region directly from the cooling curve.

Identification of Phase Transformation using Derivatives. To identify phase transformation under non-ideal cooling conditions, numerical differentiation of the cooling curve was employed. While phase transformation can be inferred from changes in the slope (or first derivative) of the cooling curve, gradual slope variations make it difficult to determine a distinct transformation point. Since the second derivative captures the onset of slope change, it provides a clearer mathematical criterion for detecting phase transformation. Moreover, from a physical standpoint, the release of latent heat during phase transformation reduces the cooling rate, resulting in a pronounced response in the second derivative. Previous studies have also reported that using the second derivative provides a more reliable criterion for identifying the onset of phase transformation [13, 14]. Accordingly, the second derivative of the cooling curve was used to identify phase transformation regions in this study.
The material investigated was an Al−Si alloy, in which primary α−Al solidifies first, followed by eutectic Si. Therefore, the first peak in the second derivative was defined as the onset of primary α Al solidification, and the second peak as the onset of eutectic silicon solidification.

[picture]

Fig. 3. Identification of the dendritic growth region based on the second derivative of the cooling curve

Fig. 3. Identification of the dendritic growth region based on the second derivative of the cooling curve

Average Cooling Rate over Dendritic Growth Region. Once the phase transformation interval was identified, the average cooling rate relevant to SDAS formation was calculated. Previous studies have shown that dendritic growth becomes negligible after the onset of eutectic Si solidification, and thus SDAS is not expected to change beyond this point [15, 16]. Based on this assumption, the dendritic growth interval was defined as the temperature range between the onset of primary α-Al solidification and the onset of eutectic Si solidification [8, 17]. The average cooling rate was calculated over this interval and used for correlation with SDAS.

Average Cooling Rate =Tα−Al−Teutectic teutectic −tα−Al

For the 4 mm thick casting block, solidification of primary α-Al began before the molten metal fully filled the mold, making it impossible to directly identify the onset of α−Al solidification from the cooling curve. Consequently, the average cooling rates obtained from other block thicknesses were fitted using an exponential function, and the extrapolated value was adopted as the average cooling rate for the 4 mm block. This exponential fitting was used solely for estimating the cooling rate of the 4 mm block. For blocks with well-defined average cooling rates, the experimentally determined values were directly used, together with the measured SDAS, to establish the relationship between SDAS and average cooling rate.

[picture]

Fig. 4. Measured average cooling rates according to the block thicknesses

Fig. 4. Measured average cooling rates according to the block thicknesses

4. Results and Discussion

The original version of this paper is available on https://www.scientific.net/SSP.390.111.pdf

Relationship between SDAS and Average Cooling Rate. The dendritic microstructures were observed near the thermocouple locations using optical microscopy, as shown in Fig. 5. A clear difference in microstructural scale was observed depending on the casting block thickness, and consequently on the cooling rate. SDAS was measured using the linear intercept method [18], which is widely adopted for SDAS evaluation. The measured SDAS values are plotted as a function of average cooling rate in Fig. 6.

[picture]

Fig. 5. Optical micrographs showing dendritic microstructures in 4 mm and 18 mm casting blocks near thermocouple location

Fig. 5. Optical micrographs showing dendritic microstructures in 4 mm and 18 mm casting blocks near thermocouple location

[picture]

Fig. 6. SDAS versus average cooling rate with fitted curve

Fig. 6. SDAS versus average cooling rate with fitted curve

The relationship between SDAS and average cooling rate is commonly described by a linear fit on a log-log scale. Accordingly, in the present study, the experimental data were fitted using the powerlaw:

SDAS=b(AC)−n(2)

where b=37.303μ m·(∘C/s)0.385,n=0.385 and AC denotes the average cooling rate. The obtained exponent n falls within the range of 1/3 to 1/2 reported by Flemings [7], indicating that the experimental procedure and data analysis employed in this study are consistent with established literature.
It should be noted that some scatter is present in the SDAS measurements, which can be attributed to local variations in dendritic morphology and inherent uncertainties associated with the linear intercept method. Nevertheless, the overall trend of decreasing SDAS with increasing cooling rate is clearly observed, and the fitted relationship captures the experimental data well. This result suggests that the SDAS measurements obtained in this study are sufficiently reliable for establishing quantitative correlations with cooling rate and mechanical properties.

SDAS Agreement at Symmetric Location. To verify the assumption that symmetric locations with respect to the casting centerline experience identical thermal histories, SDAS measurements were performed at both locations. One location corresponded to the thermocouple position, from which specimens for SDAS observation were extracted, while the other was the symmetric position used for tensile specimen extraction.

Table 1. Comparison of SDAS measured at symmetric locations

Thickness of the block [mm] | SDAS at thermocouple location [μm] | SDAS at the symmetric location [μm]
18 | 21.4 | 22.3
14 | 25.3 | 25.7
10 | 28.8 | 27.5
7 | 31.4 | 32.0
5 | 37.6 | 34.3
4 | 46.2 | 44.5

The measured SDAS values at the two symmetric locations are summarized in Table 1. For all casting block thicknesses examined, the SDAS values measured at the symmetric locations showed good agreement.
These results confirm that the assumption of equivalent thermal histories at symmetric locations is reasonable for the present casting geometry. Accordingly, the tensile specimens extracted from the symmetric positions can be reliably correlated with the SDAS values measured at the thermocouple locations, enabling consistent analysis of the relationships among cooling rate, microstructure, and mechanical properties.

Relationship Between Mechanical Properties and SDAS. Since SDAS measured at symmetric locations showed good agreement, the SDAS obtained at the thermocouple locations was correlated with the mechanical properties measured from tensile specimens extracted from the corresponding symmetric positions. As discussed previously, this approach was established to overcome the experimental limitation that microstructure observation and tensile testing cannot be performed simultaneously on a single specimen. The experimental results and fitted relationships are given in Fig. 7 and Eq. 3 to Eq. 5.

[picture]

Fig. 7. Mechanical properties versus SDAS with fitted line

Fig. 7. Mechanical properties versus SDAS with fitted line

YS[MPa]=−1.51(SDAS[μ m])+258.27(3)

UTS[MPa]=−2.31(SDAS[μ m])+346.64(4)

El[%]=−0.128(SDAS[μ m])+10.04(5)

One limitation of the present study is related to the quality of the casting process. The casting experiments were conducted by a non-specialized casting facility, and as a result, the overall casting quality was not optimal. In particular, a relatively high level of porosity was observed in the castings, which contributed to significant scatter in the measured mechanical properties. Since tensile properties are highly sensitive to porosity, especially elongation, this variability limited the consistency of the experimental data. It is expected that improved control of the casting process, such as optimized melt handling, degassing, and mold design, would reduce porosity and lead to more reliable mechanical property measurements. Therefore, further improvement in casting quality would enable more quantitative and refined correlations between SDAS, and mechanical properties.
Although some scatter was observed in the experimental data, a clear decrease in yield strength (YS) and ultimate tensile strength (UTS) with increasing SDAS was evident. The scatter in elongation (El) to fracture was larger than that in YS and UTS, which can be attributed to the higher sensitivity of elongation to casting porosity and specimen geometry. In particular, the use of both flat and round specimens may complicate direct comparison of elongation. To minimize this effect, the crosssectional areas of the flat and round specimens were matched as closely as possible. Despite the increased scatter, elongation also exhibited a decreasing trend with increasing SDAS. Overall, all three mechanical properties-YS, UTS, and elongation to fracture-were found to decrease with increasing SDAS. This trend can be explained in terms of microstructure refinement. A smaller SDAS corresponds to a finer dendritic structure, which reduces the effective interdendritic spacing and
restricts dislocation motion, thereby increasing strength. From a ductility standpoint, in fine dendritic structures, interdendritic regions are fragmented, which limits the growth and coalescence of localized strain zones and delays crack initiation, leading to improved elongation.
While similar trends between microstructure and mechanical properties have been reported for Al−Si alloys [4, 17], different behaviors have been observed in other alloy systems. A more in-depth discussion on the underlying mechanisms governing microstructure-mechanical property relationships can be found in [19].

5. Conclusion

The original version of this paper is available on https://www.scientific.net/SSP.390.111.pdf

In this study, an experimental framework was applied to investigate the relationships among cooling rate, secondary dendrite arm spacing (SDAS), and mechanical properties in aluminum castings. Casting blocks with different thicknesses for obtaining wide range of cooling rates, and the use of the second derivative of cooling curves for identifying phase transformation regions were adopted based on established approaches reported in previous studies [13, 14]. Building upon these methods, this work focused on improving the experimental linkage between thermal history, microstructure, and mechanical properties.
The primary contribution of this study lies in the specimen extraction strategy. Due to inherent experimental constraints, it was not possible to extract both SDAS and tensile specimens from the exact thermocouple location. To address this issue, SDAS observation specimens were extracted from the thermocouple locations, while tensile specimens were extracted from symmetric positions with respect to the casting centerline, where identical thermal histories were assumed. This assumption was validated by comparing SDAS values measured at the symmetric locations, which showed good agreement. This approach enabled a consistent correlation between cooling rate, SDAS, and mechanical properties.
The relationship between SDAS and average cooling rate followed a power-law form, with a fitted exponent consistent with values reported in the literature [7], confirming the reliability of the thermal analysis. Tensile test results demonstrated that yield strength, ultimate tensile strength, and elongation increased with decreasing SDAS. Although some scatter was observed, particularly in elongation due to porosity and specimen geometry effects, consistent trends were obtained across all specimens.
Overall, the results confirm that SDAS is an effective microstructural parameter for describing the mechanical behavior of aluminum castings. The specimen extraction framework proposed in this study provides a practical and reliable methodology for experimentally correlating cooling conditions, dendritic microstructure, and mechanical properties, and can be readily extended to other casting alloys and solidification studies.

Acknowledgments

The original version of this paper is available on https://www.scientific.net/SSP.390.111.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) and the InnoCORE program of the Ministry of Science and ICT (N10250154) and. This work is also partially supported by ARC Training Center of IC220100028.

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The original version of this paper is available on https://www.scientific.net/SSP.390.111.pdf

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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/material-behaviour-modelling/978-3-0364-1987-9

Solid State Phenomena, ISSN: 1662-9779, Vol. 390, pp 121-135

doi: 10.4028/p-oDeJ6s

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

Submitted:
2026-01-19

Revised:
2026-02-24

Accepted:
2026-02-25

Online:
2026-04-17

Predictive Modeling of 42CrMo4 Hardness after Two-Step Heat Treatment Using Symbolic Regression

The original version of this paper is available on https://www.scientific.net/SSP.390.121.pdf

ALBERS Johann 1,a*, DITTRICH Fabian 1, b, and HÄRTEL Sebastian 1,c

Fachgebiet Hybride Fertigung, BTU, Cottbus-Senftenberg 03046, Germany

a  johann.albers@b-tu.de, b  fabian.dittrich@b-tu.de, c  sebastian.haertel@b-tu.de

Keywords: 42 CrMo 4 , Heat Treatment, Material Hardness, Symbolic Regression, Predictive Mode

Abstract

During a heat treatment, a material undergoes microstructural changes that result in an alteration of its hardness. In a two-step heat treatment, the material is first adjusted to an initial hardness via a specified cooling rate. Subsequently, the hardness is reduced through a tempering process, while its ductility is increased. Depending on the tempering duration, tempering temperature, and initial hardness, different resulting hardness values are obtained. The resulting hardness after a chosen heat treatment is thus far been difficult to predict. This work employs symbolic regression to develop a model that predicts the hardness evolution of 42 CrMo 4 steel as a function of cooling rate, tempering duration, and tempering temperature. By describing the model with few parameters, it has also been demonstrated that cooling rates and tempering temperatures leading to a target hardness can be determined. The overall model achieves a coefficient of determination of R2=98.50% for known experimental data and a combined coefficient of determination of R2=93.13% for previously unknown cooling rates (forward) and previously unattained resulting hardness values (inverse). Our work shows that the resulting hardness of 42 CrMo 4 can be predicted using a small number of parameters. This work is anticipated to establish a foundation for further research endeavors. For instance, the approach using symbolic regression can be further adapted to identify physically interpretable constants. Furthermore, the model description offers the possibility of coupling with a simulation model to accurately predict the hardness of a component.

Introduction

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Objective

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This study aims to model the hardness of 42 CrMo 4 (see Table 1) steel following a two-step heat treatment comprising hardening and tempering. Although various models exist [1,2,3,4], predicting final hardness remains challenging due to the complexity of the underlying hardening mechanisms [2,3]. Recent advancements in machine learning offer new methodologies to address such problems, even with limited datasets [5,6].

Table 1: Chemical Composition of 42 CrMo 4 (wt. %).

| Specified | Actual
C | 0.38 − 0.45 | 0.42
Mn | 0.60 − 0.90 | 0.72
Si | < 0.40 | 0.40
Cr | 0.90 − 1.20 | 1.05
Mo | 0.15 − 0.30 | 0.29
Fe | bal. | bal.

Experimental data were obtained in two stages: first, hardening by austenitizing the samples at 850∘C for 30 min and cooling at four rates ( 0.6 K s,4.0 K s,12.5 K s, and 76.0 K s ), and second, tempering at different temperatures ( 400∘C,525∘C,650∘C ) and durations ( 30 min,60 min,120 min ) (see Figure 1 for an example of the heat treatment procedure). Symbolic Regression [6] was employed to model the hardening and tempering behavior, fitting the experimental data while enforcing parsimony to minimize model complexity and mitigate overfitting. First, a model was developed correlating the untreated hardness and cooling rate with the hardness after the hardening step. A subsequent model was then derived to predict the final hardness based the tempering duration, tempering temperature and this initial hardness. These models were combined into a single, consolidated model consistent with the experimental data. The model's predictive capability was validated by forecasting hardness values for unseen data. Furthermore, an objective function was formulated to optimize the process parameters to achieve a target hardness, thereby validating the model's invertibility using additional experimental data.

[picture]

Fig. 1: An exemplary heat treatment of a 42 CrMo 4 sample involved heating at 10.0 K s to 850∘C, holding for 30 min for normalization, followed by rapid cooling at 76.0 K s to 50∘C. This established the initial hardness and concluded the hardening step. A subsequent tempering step was performed by heating at 5.0 K s to 400∘C, holding for 120 min , and then cooling at 1.0 K s to 25∘C, which established the final hardness and concluded the tempering step.

Fig. 1. An exemplary heat treatment of a 42 CrMo 4 sample involved heating at 10.0 K s to 850 ∘ C , holding for 30 min for normalization, followed by rapid cooling at 76.0 K s to 50 ∘ C . This established the initial hardness and concluded the hardening step. A subsequent tempering step was performed by heating at 5.0 K s to 400 ∘ C , holding for 120 min , and then cooling at 1.0 K s to 25 ∘ C , which established the final hardness and concluded the tempering step.

Justification

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The need for this study arises from the persistent gap in understanding the underlying mechanisms that determine a material's final hardness with respect to its heat treatment [7]. Despite various models in the materials engineering literature [ 1,2,4 ], the problem of predicting final hardness from a twostep process of hardening and tempering remains unsolved. Addressing these open problems helps in furthering the understanding of hardness evolution in 42 CrMo 4 . Previous research has focused on empirical Jominy hardenability correlations [8] and physics-based models simulating phase transformations [9] and carbide precipitation [10], but still does not solve the problem of creating a robust, generalizable, and inverse model that accurately predicts final hardness [11] across a wide range of combined hardening and tempering parameters. Consequently, this study aims to decrease the gap between high-fidelity modeling and practical application by using techniques from recent developments in machine learning. The relevance of this study is underscored by its potential adaptability to other materials besides the 42 CrMo 4 investigated here. Predicting the resulting hardness from heat treatment further allows for the optimization of process parameters to yield more precise results with lower
input resources. Furthermore, the present work aligns with the broader objective of further investigating the hardening capabilities of steel alloys. By rigorously investigating the hardening behavior of 42 CrMo 4 , this work helps to further the understanding of hardening mechanisms and opens a discourse for further research.

Related Work

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Heat treatment

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This work investigates the steel alloy 42 CrMo 4.42 CrMo 4 belongs to the group of tool steels or quenched and tempered steels [12]. Tool steels are particularly suitable for hardening due to their medium carbon content and alloying elements like chromium and molybdenum, which facilitate the formation of a martensitic microstructure upon rapid cooling from the austenitizing temperature [13]. A common heat treatment for 42 CrMo 4 involves hardening followed by tempering [13]. The initial condition of 42 CrMo 4 possesses a certain base hardness, which is significantly increased through hardening. This is due to the transformation of the microstructure at elevated temperatures. Upon exceeding a specific temperature (the austenitization temperature), the microstructure transforms into austenite [14]. Subsequent rapid cooling transforms this austenite into martensite, a phase characterized by a significantly increased hardness. Alongside the increase in hardness, however, the ductility decreases, leading to a lower fracture strain [13]. To counteract this embrittlement, the material is subjected to a subsequent tempering step. Depending on the tempering temperature and tempering duration, the hardness decreases with a simultaneous increase in ductility. Tool steels are typically heated to their austenitization temperature, usually within 820∘C to 880∘C. In this study, cooling rates from 0.6 K s to 76.0 K s were employed to produce hardness levels of about 480 HV 1 to 660 HV 1 . The tempering temperatures are typically in the range of 300∘C to 600∘C with a tempering duration between 60 minutes and 120 minutes [15]. Upon completion of the heat treatment, the microstructure of the steel alloy is permanently altered, resulting in an increased hardness compared to the initial state.

Material modeling

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Modeling a material's behavior is complex due to the multitude of underlying mechanisms [16]. While simplified models exist, they often rely on idealized conditions. The prediction of hardness after a two-step heat treatment (hardening and tempering) exemplifies this challenge. Several models in the literature attempt to predict final hardness based on process parameters. For instance, [17] established a classic tempering parameter (the H-parameter) that relates hardness to tempering temperature and time. However, such models are typically calibrated for a specific initial condition (e.g., a fixed asquenched hardness) and do not inherently account for variations in the initial hardening step, such as different cooling rates. An alternative approach by [5] uses neural networks to predict hardness, incorporating parameters like tempering temperature and time. While powerful, these data-driven models often require large datasets and function as "black boxes," lacking the parsimony and physical interpretability desired for fundamental understanding. The model developed in this work differs by simultaneously addressing both heat treatment steps. It explicitly links the final hardness to the cooling rate from the hardening step, the resulting initial hardness, the tempering duration, and the tempering temperature. This integrated approach, achieved through Symbolic Regression [6], provides a consolidated and invertible model that captures the combined effect of these key, limited input parameters (cooling rate, tempering duration, tempering temperature), filling a gap left by models that treat the hardening and tempering steps in isolation.

Symbolic regression

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To identify a model that approximates the experimental data, the approach of Symbolic Regression is investigated. Symbolic Regression searches the space of mathematical functions for candidates that optimally approximate the prescribed data. The methodology for finding such an optimum follows different strategies. In [18], this optimum is found using genetic programming. Conversely, [19] demon-
strates finding an optimum using a method based on sparse regression. To further constrain the optimization problem, various limitations can be imposed, such as restricting the set of usable symbols or the degree of complexity (the length of the mathematical expression). Ready-made software packages exist for each of these approaches, implementing the different optimization strategies. To evaluate an optimum, it is necessary to define an objective function. Consider the following minimization problem as an example: Given a dataset {(xi,zi)}i=1n with xi∈ℝd and zi∈ℝ Symbolic Regression seeks a function f that minimizes, for instance, the sum of squared differences.

minf∈F∑i=1n(f(xi)−zi)2(1)

where F is the space of mathematical functions. This function space can be constrained by permitting only a subset of unary (e.g., ∘,log(∘),∘2,|∘| ) and binary operators (e.g.,,,+−×,/ ). This work employs the approach by [6], which leverages probabilistic and interpretable machine learning to discover symbolic models. The methodology, often implemented using PySR (Python Symbolic Regression) [6], builds upon principles of genetic programming while incorporating techniques for improved robustness and scalability. Further details on the implementation and documentation can be found at the library's official repository [6].

Methodology

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Ground truth

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A series of heat treatments were first conducted using the DIL805 A/D/T deformation dilatometer from TA Instruments to generate a dataset from which the mathematical model is developed. Cooling rates of 0.6 K s,4.0 K s,12.5 K s, and 76.0 K s were employed to achieve a range of initial hardness values. Subsequently, the samples are tempered at different tempering temperatures and for a specified tempering duration. Figure 1 shows the temperature-time curve for a sample cooled at a rate of 76 K s. Each sample is subsequently cut, and the hardness is measured using the ZwickRoell DuraScan 70 G5. The hardness values reported in this work were measured according to the Vickers hardness scale using a test force of 1 kilogram - force according to HV1. The Vickers hardness testing method was specifically developed for measuring the hardness of a wide range of materials, including metals, by assessing the resistance to plastic deformation from a diamond pyramid indenter [20]. The measured hardness values were determined at different positions, and their standard deviation is also reported. After evaluating all hardness measurements, the data are summarized in a table to serve as a basis for further analysis. Further quantities are derived from the measured values. Among other things, the hardness difference between the initial hardness and the final hardness is determined for each data point. Additionally, a corrected tempering duration is calculated, which additively comprises the nominal tempering duration and the duration derived from the heating and cooling rates during the tempering process. The quotient of the hardness difference and the corrected tempering duration yields the rate of hardness change, dHVdt.

Parsimony

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The history of science shows that many fundamental laws can be described, to a first approximation, by comparatively simple relationships [21]. For example, Hooke formulated his law of elasticity based on simple proportional relationships [22]. Similarly, Newton was able to derive a simple description of universal gravitation building upon the work of Kepler [23,24,25]. These are examples that demonstrate how a pure observation of data has led to the discovery of fundamental laws. The parsimony of equations naturally counteracts overfitting to the data. Through this parsimony, one can inherently counteract the inherent noise in the data. The trade-off between model complexity and model accuracy is illustrated in the example of Figure 2. Particularly with limited data, it is crucial to ensure that the model maintains a certain minimum level of parsimony, as a model with too many parameters can easily fit the entire dataset perfectly, leading to overfitting.

Symbolic regression

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The software package used to perform the Symbolic Regression is PySR. The core algorithm was implemented in the Julia language and ported to Python. Julia offers the advantage of being a compiled language, where the code is compiled into a kernel that executes more efficiently compared to Python. In this work, one model is identified for the initial hardness and a separate model for the final hardness. Subsequently, both models are executed in series and compared with the experimental data. PySR offers various options for configuring the search criteria for an optimal equation. For instance, the selection of operators can be constrained, as well as the number of search iterations, the maximum model size (a measure of complexity), and the loss function. By default, the mean squared error is used as the loss function, but a user-defined loss function can also be specified as a Julia function. In the subsequent procedure, a custom loss function is defined to incorporate previously known knowledge about the behavior of hardness. The selection of hyperparameters, such as unary operators, binary operators, population size, the number of iterations, and the target loss, is mentioned in terms of their choice and specification, but will not be discussed in further detail.

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Fig. 2: Pareto front for the search for the first-step hardening model. Symbolic Regression identifies candidate fits through a trade-off between loss and model complexity. While high complexity can reduce the loss significantly-note the logarithmic scale of the vertical axis-it increases the risk of overfitting, particularly on a small dataset.

Fig. 2. Pareto front for the search for the first-step hardening model. Symbolic Regression identifies candidate fits through a trade-off between loss and model complexity. While high complexity can reduce the loss significantly-note the logarithmic scale of the vertical axis-it increases the risk of overfitting, particularly on a small dataset.

Experiments

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Hardening

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To obtain a suitable mathematical description of the resulting hardness, heat treatments were initially performed on cylindrical 42 CrMo 4 samples with a base hardness of 334.9HV1±5.28HV1 and dimensions of 10 mm×2.5 mm (length × radius). The parameters for the heat treatment are listed in Table 2. For the model description, four different cooling rates were initially selected to achieve an initial hardness. These considered cooling rates are 0.6 K s,4.0 K s,12.0 K s, and 76.0 K s. Originally, an attempt was made to achieve a maximum cooling rate of 95.0 K s; however, the dilatometer used only achieved a cooling rate of approximately 76.0 K s. The cooling rate of 76.0 K s is derived from the t8/5 time, thus corresponding to the cooling rate from 800∘C to 500∘C. After successful heat treatment, the samples were cut along their central axis using a cutting wheel and then mounted. The mounted samples were ground in successive steps down to a grit size appropriate for hardness testing. Subsequently, on the cross-section of each sample, 3 series of measurements, each consisting of 5 points,
were performed to determine the hardness. Measurement series 1 and 3 are located near the outer surface of the sample, while measurement series 2 is located along the central axis of the cylindrical sample. This work considers only the results from measurement series 3. Mean values and a standard deviation were calculated from the 5 measurement points along the individual series. Table 3 shows the initial hardness values for the respective cooling rates. These initial hardness values are plotted in Figure 4, which also depicts the saturation of initial hardness starting from 10 K s.

Tempering

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The heat treatment investigated in this work involves a second step in addition to hardening, namely tempering. Tempering reduces hardness while simultaneously increasing ductility. The investigations include different tempering durations between 30 min and 120 min . To study the change in hardness at short times, the hardness values at a cooling rate of T˙=12.5 K/s and at times of 3.75 min ( 225 s ), 7.3 min(440 s),11.26 min(676 s), and 15 min were also examined. The tempering temperatures also vary between 400∘C and 650∘C. Table 2 lists all results for the hardness values and standard deviations for the investigated parameters.

Augmentation

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In addition to the data points from the experiments, further result values were introduced. First, the change in hardness at each time point compared to the initial hardness was determined. Subsequently, the tempering duration was considered as the time difference. This time difference was then corrected for the heating duration and the cooling duration. This correction is based on the duration required to reach the tempering temperatures and to cool down from them, compare Figure 1. The quotient of the hardness change and the time change yields a quantity that is used for the subsequent model formation. In Figure 3, this quotient is plotted against the corrected time. To further constrain the model, artificial data points were introduced. Firstly, for each cooling rate and each tempering temperature, the hardness was kept constant below a tempering temperature of 200∘C This is part of the data augmentation and thus expert knowledge to restrict the model behavior to zero change in hardness below 200∘C. It effectively guides the model such that a significant hardness change only occurs beyond a specific temperature threshold. The value of 200∘C was determined based on empirical data. Secondly, for each cooling rate and tempering temperature, an additional data point was introduced to promote a constant hardness change beyond 120 min . This was achieved by duplicating each data row at 120 min and changing its time to 240 min . Furthermore, for the curve with a cooling rate of T˙=76.0 K s at 650∘C, additional experiments investigating the short-time behavior were carried out. Each data point is time-corrected by the amount of time required to heat and cool the sample to the desired tempering temperature. Consequently, data points corresponding to a higher tempering temperature were corrected by a larger time increment.

Results

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Hardening model

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To describe a comprehensive model, two separate models were developed in the following and subsequently executed in two steps. From preliminary considerations, it is assumed that the initial hardness depends solely on the cooling rate, since the homogenization temperature is the same for all experiments. The data basis for modeling the initial hardness essentially consists of four points. These represent the initial hardness at different cooling rates. To find a solution using symbolic regression, various parameters were tested. The parameters used were derived from numerous trials and were selected based on their characteristics, such as asymptotic behavior with increasing time. For the initial hardness model, the unary operators log(∘),∘2,|∘|,e∘, and the binary operators,,+−×,/ were permitted. The population size was set to a multiple of the number of cores. The machine this computation took place used 16 cores to run the symbolic regression, so a population size of 3*16=48 was used. The number of iterations varied between 50 and 500 . For the target parsimony, a value of 1×10−6 were chosen. Care was taken to ensure that the selected model found a balance between low

Table 2: Final hardness values.

Cooling rate | Temperature | Time | Hardness (HVI)
0.6 K/s | 400°C | 30 min | 394.4 ± 16.1 HV
0.6 K/s | 400°C | 60 min | 423.4 ± 6.2 HV
0.6 K/s | 400°C | 120 min | 424.6 ± 10.2 HV
0.6 K/s | 525°C | 30 min | 363.0 ± 8.0 HV
0.6 K/s | 525°C | 60 min | 358.8 ± 4.2 HV
0.6 K/s | 525°C | 120 min | 352.4 ± 11.0 HV
0.6 K/s | 650°C | 30 min | 283.0 ± 4.0 HV
0.6 K/s | 650°C | 60 min | 278.2 ± 4.1 HV
0.6 K/s | 650°C | 120 min | 264.2 ± 6.8 HV
4.0 K/s | 400°C | 30 min | 493.4 ± 3.9 HV
4.0 K/s | 400°C | 60 min | 506.6 ± 3.3 HV
4.0 K/s | 400°C | 120 min | 493.4 ± 5.2 HV
4.0 K/s | 525°C | 30 min | 411.8 ± 8.2 HV
4.0 K/s | 525°C | 60 min | 405.8 ± 6.5 HV
4.0 K/s | 525°C | 120 min | 395.4 ± 10.3 HV
4.0 K/s | 650°C | 30 min | 322.0 ± 9.7 HV
4.0 K/s | 650°C | 60 min | 314.4 ± 9.7 HV
4.0 K/s | 650°C | 120 min | 315.4 ± 10.9 HV
12.5 K/s | 400°C | 15 min | 514.2 ± 6.1 HV
12.5 K/s | 400°C | 30 min | 500.8 ± 2.9 HV
12.5 K/s | 400°C | 60 min | 507.2 ± 3.8 HV
12.5 K/s | 400°C | 120 min | 499.4 ± 6.3 HV
12.5 K/s | 525°C | 15 min | 424.6 ± 5.1 HV
12.5 K/s | 525°C | 30 min | 420.8 ± 6.7 HV
12.5 K/s | 525°C | 60 min | 409.2 ± 2.5 HV
12.5 K/s | 525°C | 120 min | 401.8 ± 9.3 HV
12.5 K/s | 650°C | 0 min | 379.2 ± 7.9 HV
12.5 K/s | 650°C | 3.75 min | 343.0 ± 6.0 HV
12.5 K/s | 650°C | 7.3 min | 344.2 ± 6.0 HV
12.5 K/s | 650°C | 11.25 min | 343.0 ± 8.9 HV
12.5 K/s | 650°C | 15 min | 333.2 ± 8.0 HV
12.5 K/s | 650°C | 30 min | 322.2 ± 10.2 HV
12.5 K/s | 650°C | 60 min | 315.4 ± 9.3 HV
12.5 K/s | 650°C | 120 min | 312.4 ± 11.3 HV
76.0 K/s | 400°C | 30 min | 504.4 ± 7.1 HV
76.0 K/s | 400°C | 60 min | 502.4 ± 3.5 HV
76.0 K/s | 400°C | 120 min | 496.8 ± 7.8 HV
76.0 K/s | 525°C | 30 min | 411.8 ± 7.8 HV
76.0 K/s | 525°C | 60 min | 407.8 ± 9.3 HV
76.0 K/s | 525°C | 120 min | 408.0 ± 7.9 HV
76.0 K/s | 650°C | 30 min | 327.8 ± 12.1 HV
76.0 K/s | 650°C | 60 min | 316.8 ± 6.6 HV
76.0 K/s | 650°C | 120 min | 307.2 ± 13.6 HV

Table 3: Initial hardness values.

Cooling rate | Hardness (HV1)
0.6 K/s | 485.4 ± 20.3 HV
4.0 K/s | 643.6 ± 7.1 HV
12.5 K/s | 676.0 ± 7.5 HV
76.0 K/s | 661.6 ± 5.5 HV

[picture]

Fig. 3: Extract showing four curves for the rate of change of hardness over time for a cooling rate of T˙=76.0 K s. Each data point is augmented by calculating the rate of change of hardness with respect to time, i.e., the hardness difference divided by the time difference. Data points marked with 'x' correspond to 'artifically' added data to further restrict the problem.

Fig. 3. Extract showing four curves for the rate of change of hardness over time for a cooling rate of T ˙ = 76.0 K s . Each data point is augmented by calculating the rate of change of hardness with respect to time, i.e., the hardness difference divided by the time difference. Data points marked with 'x' correspond to 'artifically' added data to further restrict the problem.

error and low complexity. As an example, the Pareto front of the hardening model is shown in Figure 2. The model considered hereafter for the first step is shown in Figure 4. The equation describing the model is given by Equation 2 .

f1(T˙)=C1+C2T˙(2)

To achieve the accuracy shown in Figure 4, the model parameters are set to C1=673.0 and C2= -112.0.

Tempering model

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For the second step, a model for the resulting hardness was developed. To create this model, it was necessary to determine which input parameters would be permitted. From various experimental runs, it became apparent that the resulting hardness primarily depends on three different parameters: the tempering duration t, the tempering temperature T, and the initial hardness HVinit . For this step, the experimental data were used for the initial hardness. In contrast to the first model (initial hardness), a simple loss function was insufficient here. For the modeling of the second model, the quotients of hardness change over time change served as the target parameters. To obtain values comparable to the experimental data, the resulting function must be integrated over time. For Symbolic Regression, this means that a simple error function is not adequate. To perform this integration within the optimization, a custom loss function was defined. In addition to the integration, the custom loss function also penalizes deviations from smoothly decaying functions in the time domain and discrepancies from the experimental data. For performing the Symbolic Regression, the same unary and binary operators as in the initial hardness model were permitted. The selected model again represents a compromise between loss and complexity and is given by Equation 3.

[picture]

Fig. 4: Graph of the first-step hardening model according to Equation 2. Given the four cooling rates, four different initial hardness values are reached. Each measured initial hardness is plotted with its standard deviation as vertical lines. The model shows a Coefficient of Determination of R2=98.75% and a Root Mean Squared Error of RMSE =8.20 HV1. The model reaches a plateau past a cooling rate of T˙≈10 K s.

Fig. 4. Graph of the first-step hardening model according to Equation 2. Given the four cooling rates, four different initial hardness values are reached. Each measured initial hardness is plotted with its standard deviation as vertical lines. The model shows a Coefficient of Determination of R 2 = 98.75 % and a Root Mean Squared Error of RMSE = 8.20 HV1. The model reaches a plateau past a cooling rate of T ˙ ≈ 10 K s .

f2(t,T,HVinit)=C3*t*eC4*t(T*HVinit+T)f1+2(t,T,T˙)=C3*t*eC4*t(T*f1(T˙)+T)

(3)(4)

By selecting the model parameters C3=1.49×10−3 and C4=−2.84×10−3, a model was developed, which is depicted in Figure 5.
In a further figure (Figure 6), the simplification of using the experimental initial hardness was abandoned. The result shows a model which is derived from the combination of the first and second model and is given by Equation 4.

Model validation

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Since a mathematically smooth overall model for describing the resulting hardness now exists, it can be used to provide hardness predictions for cooling rates, tempering durations and tempering temperatures not previously considered. Additionally, this overall model can serve as a basis for optimization, whereby the cooling rate, tempering duration and tempering temperature can be determined by specifying a target hardness. Both procedures, forward evaluation and inverse evaluation, are carried out in the following. All validation data are summarized in Table 4. Beginning with the forward evaluation, a series of three additional experiments was conducted. For this purpose, a cooling rate of 1.2 K s and tempering temperatures of 400∘C,525∘C, and 650∘C were selected. To obtain values comparable to previous experiments, the tempering duration was fixed at 30 min . The evaluation of the hardness results was performed in the same manner as the previous experiments. The results of the model and the experiments for the forward evaluation are shown in Figure 7. With an R2=92.59% and an RMSE =19.48HV1, the validation accuracy is somewhat lower than that of the model with known input parameters (compare Figure 6). As a second step in validating the model, the model was used as the basis for an optimization. It is therefore possible to specify a target hardness, and the task of the optimization is to calculate the optimal cooling rate, tempering duration, and tempering temperature. To test the model's inverse capability, three different hardness values of ≈460HV1,≈550HV1, and ≈610HV1 were selected, which werent seen by the model before. The tempering duration was again fixed at 30 min to obtain comparable results. Thus, only the cooling rate and tempering temperature are freely determinable by the optimization. To achieve the three specified hardness values, cooling

[picture]

Fig. 5: Graph of the second-step tempering model according to Equation 3. The graph shows the hardness progression for an initial hardness (given from experiment) of 661.6HV1±5.5HV1 over time. The solid line represents the model and the marker represent the experimental data with the vertical lines showing each measurements uncertainty. Comparing the model, with the given coefficients, to the complete dataset, the model reaches a R2=98.58% and a RMSE =14.49HV1.

Fig. 5. Graph of the second-step tempering model according to Equation 3. The graph shows the hardness progression for an initial hardness (given from experiment) of 661.6 H V 1 ± 5.5 H V 1 over time. The solid line represents the model and the marker represent the experimental data with the vertical lines showing each measurements uncertainty. Comparing the model, with the given coefficients, to the complete dataset, the model reaches a R 2 = 98.58 % and a RMSE = 14.49 H V 1 .

[picture]

Fig. 6: Graph of the second-step tempering model (Equation 4) with the initial hardness modeled according to the first-step hardening model (Equation 2). Modeling the initial hardness given a cooling rate of T˙=76.0 K s results in an initial hardness of 671.5 HV 1 . Using the approach of predicting the final hardness using both models in series results in a slightly lower R2=98.50% and a slightly increased (worse) RMSE =15.03HV1.

Fig. 6. Graph of the second-step tempering model (Equation 4) with the initial hardness modeled according to the first-step hardening model (Equation 2). Modeling the initial hardness given a cooling rate of T ˙ = 76.0 K s results in an initial hardness of 671.5 HV 1 . Using the approach of predicting the final hardness using both models in series results in a slightly lower R 2 = 98.50 % and a slightly increased (worse) RMSE = 15.03 H V 1 .

rates of 35.0 K s,42.0 K s, and 45.0 K s were found, which, together with the three found tempering temperatures of 470∘C,350∘C, and 275∘C, represent the model's prediction. The results of the model and the experiments are shown in Figure 8. With an R2=71.79% and an RMSE=25.72HV1, the validation accuracy is lower than that of the forward evaluation and the model with known input parameters (compare Figure 6). To further demonstrate the model's accuracy for real-world applications, an additional experimental series was conducted. For actual components, cooling rates are often constrained by the quenching medium. Consequently, only the tempering temperature and tempering duration remain as freely adjustable parameters. For this investigation, target hardness values of 450 HV 1 and 550 HV 1 were specified. For each target hardness, cooling rates of 8.0 K s and 20.0 K s were fixed. The optimization then determined the required tempering temperature and duration, with all results presented in Figure 9. This validation dataset yields a Coefficient of Determination of R2=90.97% and a Root Mean Squared Error of RMSE =11.96HV1. While this accuracy is slightly lower than that of the forward validation (compare Figure 7), it is notably higher than the accuracy achieved in the previous inverse validation (compare Figure 8). Considering the entire validation dataset with its 10 data points, an R2=93.12% and RMSE =19.22HV1, are obtained, which is slightly above the accuracy of the model with known input parameters and shows the models overall capabilities.

Table 4: Validation hardness values.

Cooling rate | Tempering | Experiment (HV1) | Model (HV1)
1.2 K/s | 400°C @ 30 min | 494.6 ± 35.0 HV | 461.40 HV
1.2 K/s | 525°C @ 30 min | 393.2 ± 12.1 HV | 387.49 HV
1.2 K/s | 650°C @ 30 min | 315.4 ± 17.62 HV | 313.57 HV
8.0 K/s | 339°C @ 51.10 min | 534.4 ± 3.5 HV | 550.0 HV
8.0 K/s | 467°C @ 53.28 min | 453.2 ± 7.4 HV | 450.0 HV
20.0 K/s | 346°C @ 52.33 min | 531.8 ± 5.88 HV | 550.0 HV
20.0 K/s | 471°C @ 52.88 min | 454.8 ± 7.6 HV | 450.0 HV
35.0 K/s | 470°C @ 30 min | 455.4 ± 8.0 HV | 459.63 HV
42.0 K/s | 350°C @ 30 min | 538.6 ± 11.7 HV | 553.41 HV
45.0 K/s | 275°C @ 30 min | 570.2 ± 10.7 HV | 612.02 HV

[picture]

Fig. 7: Forward validation using a cooling rate of 1.2 K s to validate the model on a previously unseen cooling rate. The respective standard error for each experimental data point is displayed using a vertical line. Each plotted data point represents the final hardness after cooling the samples with 1.2 K s and tempering for 30 min . The dataset of these 3 samples yields a R2=92% and a RMSE =19.48HV1.

Fig. 7. Forward validation using a cooling rate of 1.2 K s to validate the model on a previously unseen cooling rate. The respective standard error for each experimental data point is displayed using a vertical line. Each plotted data point represents the final hardness after cooling the samples with 1.2 K s and tempering for 30 min . The dataset of these 3 samples yields a R 2 = 92 % and a RMSE = 19.48 H V 1 .

[picture]

Fig. 8: Inverse validation using a cooling rate of 35.0 K s,42.0 K s and 45.0 K s to validate the model on resulting hardness values not previously attained. The respective standard error for each experimental data point is displayed using a vertical line. Each plotted data point represents the final hardness after cooling the samples with the respective cooling rate and tempering each for 30 min . The dataset of these three samples yields a R2=71.79% and a RMSE =25.72HV1.

Fig. 8. Inverse validation using a cooling rate of 35.0 K s , 42.0 K s and 45.0 K s to validate the model on resulting hardness values not previously attained. The respective standard error for each experimental data point is displayed using a vertical line. Each plotted data point represents the final hardness after cooling the samples with the respective cooling rate and tempering each for 30 min . The dataset of these three samples yields a R 2 = 71.79 % and a RMSE = 25.72 H V 1 .

[picture]

Fig. 9: Second inverse validation using the given cooling rates of 8.0 K s and 20.0 K s and the target hardness values of 450 HV 1 and 550 HV 1 . The respective standard error for each experimental data point is displayed using a vertical line. The second validation dataset, consisting of four samples, yields a R2=90.97% and RMSE =11.96HV1.

Fig. 9. Second inverse validation using the given cooling rates of 8.0 K s and 20.0 K s and the target hardness values of 450 HV 1 and 550 HV 1 . The respective standard error for each experimental data point is displayed using a vertical line. The second validation dataset, consisting of four samples, yields a R 2 = 90.97 % and RMSE = 11.96 H V 1 .

Discussion

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Model fidelity

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The following discusses the model accuracy of the individual models and the overall model. First, a model was created to determine the initial hardness based on the cooling rate. A simple model comprising two terms (Equation 2) was identified, which, with the described model parameters, achieves a coefficient of determination of R2=98.75% and a standard error of regression of RMSE =8.20HV1. In a second step, a model was developed to determine the resulting hardness based on the tempering duration t, the tempering temperature T, and the initial hardness. The described model is given by Equation 3. This second model is more complex due to the nonlinearities between the input parameters and the time-dependent exponential function. For the second model, a coefficient of determination of R2=98.58% and a standard error of regression of RMSE =14.49HV1 were achieved. Considering the second model in isolation reveals its dependency on apriori knowledge of the initial hardness. To establish a comprehensive model, the initial hardness in the second model was replaced with the function for initial hardness (Equation 2). This results in a slightly lower model accuracy, yet still yields a coefficient of determination of R2=98.50% and a standard error of regression of RMSE =15.03HV1. To validate the overall model, cooling rates not present in the original experimental data were selected in a first step. This forward validation step shows a coefficient of determination of R2=92.95% and a standard error of regression of RMSE =19.48HV1. In a second step, the overall model was used as part of an optimization to determine the cooling rates and tempering temperatures for given target hardness values, and thus revealing the models inverse capabilities. For
this inverse validation step, a coefficient of determination of R2=71.79% and a standard error of regression of RMSE =25.72HV1 were determined. A possible reason for the comparably lower R2 lies in the chosen tempering temperature of only 275∘C. This tempering temperature might be too close to the lower bound of "no change", which is only based on previous assumptions. In a third step, another optimization with predetermined cooling rates were conducted. For the second inverse validation procedure, a coefficient of determination of R2=90.97% and a standard error of regression of RMSE =11.96HV1 were determined. For the entire validation dataset, a combined coefficient of determination of R2=93.13% and a standard error of the regression of RMSE =19.22HV1 are obtained.

Conclusion

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Summary and outlook

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Both models, for the initial hardness (compare Figure 4) and for the resulting hardness (compare Figure 5), show very good agreement with the experimental data. The use of the overall model (compare Figure 6) also demonstrates very high conformity with the experimental data. However, when evaluating the models on previously unseen data, the overall model exhibits slightly lower accuracy. Consequently, the validation tests yield a coefficient of determination of R2=93.13% and a standard error of regression of RMSE =19.22 HV1.

Table 5: Validating "no changes" below 200∘C.

Cooling

rate

Tempering

Experiment

(HV1)

Model

(HV1)

mathematical formula | superscript number | superscript number | 645.0 HV
superscript number | superscript number | superscript number | 671.5 HV

Thus far, only input parameters and target hardness values within the minimum and maximum ranges of the experiments have been investigated. Therefore, the model has only been tested for interpolation within the experimental data. For further development and the identification of a model that more accurately describes the material, the behavior outside the range of the experimental data should also be examined. Additionally, a portion of the validation dataset falls within the 200∘C to 400∘C temperature range. While this is technically an interpolation, it relies on the assumption that no hardness change occurs below 200∘C and that the hardness change only stops at 200∘C. Future work is required to critically assess the validity of this assumption, as preliminary experimental data presented in Table 5 already indicate its shortcomings. Furthermore, the physical correctness of the presented models is questionable. The reason for this is the lack of interpretability of the constants in both models. Both models must yield a result with the unit HV1. For the first model (Equation 2), the units of the constants can be assumed as [C1]=HV1 and [C2]=Ks. However, the interpretation of the constant C2 remains unresolved. For the second model (Equation 3), the units [C3]=1Ks and [C4]=1 s can be assumed. Here, too, the interpretation of the constants C3 and C4 is unclear. Additionally, for future work, the heat distribution throughout an entire component could be considered and used in a more complex simulation setup. This would enable the assessment and adjustment of hardness across the entire component.

Acknowledgements

The original version of this paper is available on https://www.scientific.net/SSP.390.121.pdf

We acknowledge the contributions of the developers of the free software packages utilized in this work, including, but not limited to, PySR, Matplotlib, NumPy, SciPy, and Pandas. We also acknowledge the role of various large language models (LLMs), such as ChatGPT, Claude, Gemini, and

DeepSeek, in providing suggestions that sparked new ideas and prompted code modifications. Furthermore, we acknowledge the assistance of LLMs in translating parts of this work from its original German manuscript.

References

The original version of this paper is available on https://www.scientific.net/SSP.390.121.pdf

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H. A. Mahmoud, G. Shanmugasundar, S. Vyavahare, R. Kumar, R. Cep, S. Salunkhe, S. Gawade and E. S. Abouel Nasr: Prediction of machine learning-based hardness for the polycarbonate using additive manufacturing (Frontiers in Materials, Vol. 11) (2024)

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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.

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Solid State Phenomena, ISSN: 1662-9779, Vol. 390, pp 137-143

doi: 10.4028/p-6Mmgit

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

Submitted:
2026-01-19

Revised:
2026-02-23

Accepted:
2026-02-24

Online:
2026-04-17

How to Take the Lüders Plateau into Account in the Formability Prediction of Steel Sheets

The original version of this paper is available on https://www.scientific.net/SSP.390.137.pdf

Lemoine Xavier 1,a*

¹Global R&D ArcelorMittal Maizières, voie Romaine, BP30320, F-57283 Maizières-lès-Metz, France

a* xavier.lemoine@arcelormittal.com

Keywords: lüders, hardening law, formability, steels.

Abstract

Some steels exhibit the Lüders effect. This phenomenon depends on the material and structure (test piece geometry, loading speed, etc.). Most work hardening laws do not take this phenomenon into account. The objective of this work is to define the most appropriate hardening law to capture the characteristics of Lüders effects. First, the various aspects of the Lüders effect are presented. Several local hardening laws are proposed to describe the presence or absence of a plateau, some of which are taken from the literature. Simulations of uniaxial tensile tests and stamped-part forming are performed to compare the ability of these different hardening laws to predict the Lüders effect. The Exp_Swift hardening law is recommended for forming cards because it is fully compatible with all the finite element codes dedicated to steel sheet formability analysis and does not require inverse calibration during identification to accurately predict the plateau length.

Introduction

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The Lüders phenomenon is observed as a plateau of nearly constant stress occurring at the onset of plastic deformation, immediately beyond the elastic limit (Fig.1) [1]. It occurs when the stress reaches a maximum value, called the Upper Yield Stress (UYS), before dropping abruptly to a nearly constant level known as the yield plateau stress. The minimum value of the plateau and the length of the Lüders plateau are referred to as the Lower Yield Stress (LYS) and the Yield Plateau Elongation (YPE), respectively. The over-stress peak, together with the constant-stress plateau, distinguishes steels that exhibit a yield plateau from those that do not.

[picture]

Fig. 1. Lüders characterization issued tensile test [1].

Fig. 1. Lüders characterization issued tensile test [1].

The origin of this behavior lies at the atomic scale, specifically in the dislocation mechanisms within the material [2]. Certain dislocations are pinned by Cottrell atmospheres, and their unpinning requires more energy than moving an unpinned dislocation, which explains the observed over-stress at the elastic limit. During the uniaxial tensile test, this disturbance manifests on the specimen as one or several narrow bands of plastic deformation inclined at approximately 55∘, propagating through the surrounding material that remains elastically deformed. The localized plastic deformation within
the Lüders band propagates [3], causing a local temperature to increase due to the energy released during dislocation motion, in contrast to elastic deformation. Consequently, thermal imaging is an effective method for visualizing these bands [4].

The characteristics of the Lüders plateau result from an interaction between the intrinsic behavior of steel and the conditions under which the tensile test is performed. The values of UYS, LYS and YPE, depend on the number of pinned dislocations, the concentration of elements ( C,N, etc.) forming Cottrell atmospheres [5], the sheet thickness [6], the specimen geometry (the length of the specimen's gauge section), the loading rate [7,8], and the temperature [9] during the tensile test. Depending on the type of loading path (shear, plane tension, wide tension, rolling, etc.), the Lüders plateau may or may not appear [10].

The inclination of the Lüders band, approximately 55∘ [11], is identical to that of the shearlocalization band responsible for localized necking. Shear bands, like the Lüders plateau, originate from the formation of dislocation cells constrained by the same close-packed planes of the bodycentered cubic (BCC) crystal structure of iron atoms in low-carbon steels. A priori, these two phenomena are considered independent of each other in the context of tensile testing of steel sheets.

In forming predictions for steel sheets (such as formability analysis), most hardening laws do not account for this phenomenon. Models incorporating the Lüders effect have primarily been developed for fundamental understanding rather than practical application. The objective of this work is to identify the hardening law most suitable for forming prediction, in order to capture the characteristics associated with Lüders effects.

Flow Curves of Seven Hardening Laws Used in Finite Element Code

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Flow curves of seven hardening laws (Fig. 2) were considered to study the impact of the Lüders plateau on rheology and formability predictions. The hardening law defines the stress-strain curve applied at each integration point and reflects the behavior of a material point. The structural aspect of the Lüders plateau (e.g., the Yield Point Elongation, YPE) predicted by certain hardening laws is obtained directly from simulation.

[picture]

Fig. 2. Flow Curves of Seven Hardening Laws for an HSLA steel.

Fig. 2. Flow Curves of Seven Hardening Laws for an HSLA steel.

The first hardening law considered is the linear combination of the Swift and Voce models (Eq. 1) [12] referred to as 'SV'. This model does not exhibit a Lüders plateau and will serve as the reference for comparison in forming predictions.

σ=α[σ0+Rsat(1−exp(−Crε))]+(1−α)[K(ε0+ε)n].(1)

Here σ and ε denote the true stress and true plastic strain, respectively, while the remaining terms represent material-specific parameters. These material parameters were identified from the section of the tensile curve between the Lüders plateau and the onset of uniform elongation, using Considère's criterion to extrapolate the curve beyond the uniform elongation point [12].

Two other hardening laws do not exhibit any characteristics of the Lüders plateau. The "Shifted" law was previously used because it aimed to capture only the intrinsic behavior of steel, without any interaction with structural effects, which is not the case for the Lüders plateau. The "Tangential" law is a Swift-Voce model, identified over the entire tensile curve, including the Lüders plateau.

The other hardening laws were derived from this "SV" law except for "Exp_Swift", in order to reproduce specific aspects of the Lüders plateau:

only the plateau itself (YPE), without the upper yield point (UYS) and with slopes that are nearly horizontal ("Exp_Swift" and "Quasi-Horizontal"),

or including the upper yield point (UYS), as in the model proposed by H. Tsukahara et al. [13] ("Pic"), or through a combination of two laws ("Pic + Horizontal").
The "Exp_Swift" law consists of experimental data points up to the uniform elongation UE, followed by a Swift law beyond UE, whose Swift parameters are derived from the Considere criterion. The "Quasi-Horizontal" law is obtained by adding a quasi-horizontal plateau to the SV law. In this case, the Swift and Voce parameters are approximated as accurately as possible after the Lüders plateau up to UE, and beyond UE the linear-combination parameter between Swift and Voce is determined from the Considere criterion. "Exp_Swift" is the law recommended for achieving an accurate plastic stress-strain relationship under large deformations [12].

All the finite element codes used for the forming of steel sheets enforce a monotonically increasing evolution of plastic strain. Most of these codes allow for a decrease in the engineering stress, whereas some enforce that this stress remains strictly increasing (e.g., "AutoForm®"). This implies that some of these seven work-hardening laws cannot be utilized by this type of finite element code (Table 1).

Table 1. Compatibility of Seven Hardening Laws with Four Finite Element Codes.

| SV | Shifted | Tangential | Exp_Swift | Quasi-Horizontal | Pic | Pic + Horizontal
Abaqus® | X | X | X | X | X | X | X
PamStamp® | X | X | X | X | X | X | X
Ls-Dyna® | X | X | X | X | X | X | X
Autoform® | X | X | X | X | X |  |

Tensile Tests Predictions

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The first numerical experiment consists of simulating a tensile test on an ISO 20×80 specimen to determine whether the mechanical characteristics associated with the Lüders plateau can be accurately predicted. The simulation was carried out using Ls-Dyna® with 2×2 mm2 shell elements, and each strain-hardening law was implemented via a tabulated input within material model MAT_122, combined with a Hill 48 yield surface identified using three Lankford coefficients ( r0=0.9, r45=1,r90=0.7 ). The loading condition is implemented as a prescribed displacement applied to a set of nodes at the top of the specimen, while another set of nodes at the bottom is fully constrained in both translational and rotational degrees of freedom. The tensile force is computed by averaging the element forces within the central section of the specimen, which is perpendicular to the loading direction. The elongation is then derived using an initial gauge length of 80 mm .

Fig. 3 provides the engineering stress-strain representation of the input curve derived from Fig. 2 for the three hardening laws ("SV", "Exp_Swift", and "Pic"). It also includes the tensile test response predicted by LS-Dyna ®, which is compared with the corresponding experimental curve. From these engineering curves, the mechanical properties-UYS, LYS, YPE, UTS, and UE-were extracted and are reported in Table 2.

[picture]

Fig. 3. Comparison of flow curves from the experimental test, the prediction by Ls-Dyna ®, and Input curve for three hardening laws: SV (a), Exp_Swift (b), Pic (c).

Fig. 3. Comparison of flow curves from the experimental test, the prediction by Ls-Dyna ® , and Input curve for three hardening laws: SV (a), Exp_Swift (b), Pic (c).

For hardening laws without a plateau, the prediction of the LYS value is inaccurate, and the shifted law further leads to an unrealistically low uniform elongation.

In Table 2, the presence or absence of Lüders band propagation is reported and illustrated in Figure 4 for the "Pic" hardening law. The occurrence of this propagation is exclusively ascribed to the incorporation of UYS in the input hardening law. The predicted UYS values are significantly lower than the experimental ones and those from the input curve ("Pic" and "Pic-Horizontal"), a discrepancy
that is linked to the frequency at which the simulation results were recorded. Achieving a plateau length for both hardening laws that better matches the experimental value requires a recalibration of the input curve and is expected to be influenced by the mesh size.

The advantage of plateau-type laws without UYS ("Exp_Swift" and "Quasi-horizontal") is that both the plateau length and the LYS predicted by the simulation match exactly the input curve (Fig.3b for "Exp_Swift" law).

Table 2. Comparison of the predicted mechanical properties of seven hardening laws with experimental values, and presence of band propagation.

| UYS (MPa) | LYS/YS (MPa) | YE (%) | UTS (MPa) | UE (%) | Band Propagation
Exp. | 418 | 362 | 2.8 | 539 | 12.9 | Yes
SV | - | 190 | - | 537 | 13.2 | No
Shifted | - | 370 | - | 550 | 10.0 | No
Tangential | - | 305 | - | 538 | 13.3 | No
Exp_Swift | - | 369 | 2.5 | 538 | 13.4 | No
Quasi- Horizontal | - | 367 | 2.6 | 537 | 13.5 | No
Pic | 382 | 330 | 2.1 | 537 | 13.9 | Yes
Pic+ Horizontal | 387 | 368 | 2.8 | 537 | 13.4 | Yes

STEP 1 | STEP6 | STEP 10 | STEP 17 | STEP 22
Tond propagation with Pichardeninglaw |  |  |  |
Band |  |  |  |

Fig. 4. Band propagation with the "Pic" law.

Stamping Parts Predictions

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To investigate the impact of these seven laws on the prediction of formability in stamped parts, several cases were analyzed, all leading to broadly similar conclusions. Accordingly, only the results of hemispherical stamping tests are presented in the following section.

Except for the "Shifted" hardening law, the formability prediction (Fig.5) and Punch force prediction (Fig.6) are comparable to those obtained with the "Exp_Swift" hardening law.

The "Pic" hardening law enables the prediction of band propagation consistent with the experiment (Fig.7). However, this numerical capability can only be achieved when an extremely fine mesh is employed, which is no longer feasible for industrial applications due to the prohibitive computational cost.

[picture]

Fig. 5. Forming limit diagram of hemispherical stamping tests for seven hardening laws.

Fig. 5. Forming limit diagram of hemispherical stamping tests for seven hardening laws.

[picture]

Fig. 6. Punch force of hemispherical stamping tests for seven hardening laws.

Fig. 6. Punch force of hemispherical stamping tests for seven hardening laws.

[picture]

Fig. 7. Bands propagation with Pic hardening law appears in intermediate step.

Fig. 7. Bands propagation with Pic hardening law appears in intermediate step.

Conclusion

The original version of this paper is available on https://www.scientific.net/SSP.390.137.pdf

The recommended hardening law for forming cards is the "Exp_Swift" law, as it is compatible with all the finite element codes dedicated to steel-sheet formability and does not require inverse adjustment during identification to accurately predict the plateau length (Fig.3b), regardless of mesh size. This constitutive law makes it possible to reproduce the mechanical properties guaranteed in the customer's specification when simulating a tensile test, while satisfying the monotonic stress-increase condition imposed by AutoForm ®.

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The original version of this paper is available on https://www.scientific.net/SSP.390.137.pdf

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Solid State Phenomena, ISSN: 1662-9779, Vol. 390, pp 145-161

doi: 10.4028/p-qnv3tQ

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

Submitted:
2026-01-19

Revised:
2026-02-18

Accepted:
2026-02-21

Online:
2026-04-17

Microscale to Macroscale Analyses of Superplasticity of Ti-6242S Titanium Alloy

The original version of this paper is available on https://www.scientific.net/SSP.390.145.pdf

AROWOLO Muritala 1,2,a*, VELAY Vincent. 2, b, VIDAL Vanessa 2,c and DEHMAS Moukrane 1,d

1 CIRIMAT, Toulouse INP, Université de Toulouse, CNRS, Toulouse 31030, France

2 Institut Clément Ader (ICA), Université de Toulouse, CNRS UMR 5312, IMT Mines Albi, INSA, ISAE-Supaéro, INSA, UPS, France

a  muritala.arowolo@mines-albi.fr, b  vincent.velay@mines-albi.fr, c  vanessa.vidal@mines-albi.fr, d  moukrane.dehmas@toulouse-inp.fr

Keywords: Multiscale modelling, CPFE, optimisation, HEXRD, dual-phase titanium alloy, superplasticity

Abstract

The superplastic performance of the dual-phase Ti-6242S titanium alloy makes it a good material for aerospace applications to produce structural components using the advanced superplastic forming (SPF) process. The need to optimize the SPF process demands the understanding and quantification of the influence of the different phase constituents - α and β on the global superplastic behaviour. Numerical modelling has been useful to predict mechanical behaviour for both a one-level and a multiscale approach. Multiscale approach: bottom-up (microscale to macroscale) has enabled understanding how the different microstructural parameters influence global material/structural mechanical response; which by and large means the modelling approach depends on the material local properties. The identification of these local properties is non-trivial in polycrystal materials, particularly at superplastic (elevated) temperatures.

Herein presented, is a new methodology that permits the quantification of the microstructural parameters of each of the constitutive phases of a polycrystal at a superplastic temperature using a genetic algorithm optimisation method on the data from in-situ high energy X-ray diffraction (synchrotron radiation), coupled with SEM (scanning electron microscope) and EBSD (electron backscattered diffraction). These identified local microstructural parameters were directly used in the finite strain crystal plasticity model to simulate the materials global response. This approach enabled the quantification of the phase influences on global behaviour with great accuracy.

It was found that α phase planes have high critical resolved shearing stress (CRSS) at 730∘C which is similar to their behaviour at room temperatures, while β phase slip planes have low CRSS that encourage slip shearing at low stress. However, more applied load is partitioned in the β phase than in the α phase, despite the fact that the β phase fraction is about 15% at 730∘C.

Introduction

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Superplasticity refers to the ability of a material to undergo large strain deformation without necking (thinning) [1-3], often achieving elongation which can reach or exceed 1000% depending on the microstructure and the processing conditions. Superplasticity is said to occur if the strain rate sensitivity ( m ) is equal to or more than 0.3[1], and this behaviour is found at high temperatures above 50% of the materials melting temperature and at very low strain rates (~10−4 s−1). The aerospace industry has leveraged this inherent material property to produce complex, near-net-shaped titanium components in one-shot through a manufacturing method called the superplastic forming process (SPF).

Titanium alloys are widely used in aerospace applications due to their excellent specific strength and resistance to corrosion, fatigue and creep [4]. Based on their stabilizing allowing elements, they are categorized as: (a) α alloy with a hexagonal closed packed (HCP) crystallographic structure, is stabilized by Al,O,N,C, etc; (b) β alloy with a body center cubic (BCC) crystallographic structure
is stabilized by Mo,Cr,Ta,V,Fe, etc; and (c) α+β alloy containing both phases at room temperature, which contains the stabilizers of the two in relative ratios.

The superplastic behaviour of titanium alloys depends on various microstructural factors such as grain size and morphology, texture and anisotropy, phase volume fraction, phase interfacial interaction and their potential evolution during plastic deformation. In addition to the microstructural features, forming parameters such as temperature and strain rate can also control the flow behaviour (as strain hardening, flow softening or steady-state behaviour) and different mechanisms including dislocation gliding, grain boundary sliding, dynamic recrystallisation and dynamic recovery.

Finite-element (FE) modelling has become essential for predicting such complex interactions. There exist different material/structural modelling approaches. A one-level models consider a macroscale behaviour and treats the material as homogeneous, and a multiscale model where at least two levels are coupled, for instance, the microscale or the mesoscale is used to simulate the macroscale response of the structure using a scale transition rule or a homogenization method to bridge the different scales. Multiscale approach (microscale to macroscale) permits the numerical quantification of each phase's contributions to the global/macroscopic mechanical behaviour of a dual-phase system.

However, accurate microscale material parameters are difficult to identify for each phase. Their identification requires combined advanced experimentation techniques, particularly at elevated temperatures at which superplasticity is observed ( 900∘C for Ti-6242S). Previous studies using Scanning Electron Microscopy (SEM) [5] and Electron Back-Scattered Diffraction (EBSD) [6] before/after deformation provide valuable information, but cannot determine real-time phase evolution, making parameter identification highly dependent on uncertain optimisation bounds. Kapoor et al. [7] presented an alternative approach, where virtual X-ray simulation was performed to reproduce the lattice strains obtained from in situ high-energy X-ray diffraction (HEXRD) while concurrently using the input of the virtual X-ray simulation for a crystal plasticity finite element simulation, in an attempt to fit the macroscopic stress vs strain curve of the material. The drawdown of Kapoor et al. [7] approach is that, it is a two-way method, which would be computationally expensive.

Consequently, in this work, a new methodology (one-way) is presented to identify phase properties with accuracy and use these calibrated properties to numerically simulate the macroscopic mechanical behaviour of a dual-phase ( α+β ) Ti-6Al-2Sn-4Zr-2Mo-0.12Si (Ti-6242S) alloy. It combines HEXRD, SEM and EBSD to identify phase-level mechanical parameters for the alloy. HEXRD (advanced experimental technique) permit to follow real-time evolution of the material phases during heat-treatment and mechanical deformation at temperatures. The volume fraction and the lattice strains of the different planes in both α and β phases are measured. Hence, the mechanical microstructural properties of each phase are obtained through the use of a genetic algorithm optimisation method applied on the lattice strains and the macroscopic stress-strain evolution given by the force and displacement recorded from the tensile test apparatus of the HEXRD set-up. Likewise, the SEM techniques allow for obtaining the microstructural images, the grain sizes, and the phase fraction of the two phases. Similarly, the EBSD helps identify grain orientations and grain sizes. These parameters are then implemented in a finite-strain crystal plasticity framework to simulate the macroscopic behaviour using a 3D representative volume element (RVE) which take into account the grain orientation and phase volume fraction.

Methodology

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Numerical Framework. A finite strain formulation is appropriate to model the large deformation typical of the superplasticity forming process. A dual-phase material is considered; therefore, the subsequent equations hold separately for α and β phases including the slip laws for which a power law was used for both β (BCC) and α (HCP), following the formulations in [8-12].

Index and notation adopted: (i) ={α,β} phase, bold letters (a) refers to tensors, " ⊗ " is a tensor/dyadic product, while "." is a dot product and if otherwise not stated, it is a matrix
multiplication, s=1,2,…,Ns(i) is the slip system in the phase (i) where Ns(i) is the total slip system number, a→ represents direction vector, " : " is a double contraction, and ' x ' is a vector product.

In crystal plasticity, the decomposition of the deformation gradient (𝐅―) is adopted, which assumes the multiplication of the elastic deformation gradient (𝐅―𝐞) and plastic deformation gradient (𝐅―p) and shown in Eq. 1

𝐅―=𝐅―e(i)𝐅p(i)(1)

Where 𝐅―e(i) describes the reversible lattice deformation gradient of each phase due to the external load, in contrast, 𝐅p(i) describes the irreversible deformation gradient of the lattice in each phase due to plastic work, which can be a convolution of rate-dependent and rate-independent plastic work, depending on the mechanisms intended to be captured. It also has to satisfy the incompressibility constraint ( det𝐅p(i)=1 ) enforced with (𝐅p(i)(det𝐅p(i))−13). The plastic velocity gradient (𝐋p(i)) is described as in Eq. 2:

𝐋p(i)=𝐅―˙p(i)𝐅―p(i)−1=∑s=1Ns(i)γ˙s(i)m→s(i)⊗n→s(i)(2)

Where 𝐋p(i) relates the rate of plastic deformation gradient 𝐅―˙p(i) to its inverse 𝐅―p(i)−1. 𝐋p(i) is related to the summation of shear strain rate γ˙s(i) on all the slip system of corresponding phase (i) with their corresponding Schmid factors described by the dyadic product of the slip direction m→s(i) and the slipplane normal n→s(i). The shear strain rate γ˙s(i) is described by viscoplastic power laws for α and β phase as in Eq. 3:

(i)={α,β}:γ˙s(i)=γ˙0(i)(τs(i)𝐠s(i))n(i)sign(𝐙s(i)) if |τs(i)|≥gs(i)(3)

Where γ˙0(i) is the initial strain rate, τs(i) is the resolved stress in each slip system (s), likewise gs(i) is the critical resolved stress, n(i) is the phase strain rate sensitivity. The resolved stress can be described as in Eq. 4 where σ(i) is Cauchy stress:

τs(i)=σ(i):m→s(i)⊗n→s(i)(4)

The critical resolved stress gs(i) for both α and β shown in Eq. 5a depends on the initial resolved stress τ0(i) and evolution hardening function described as Eq. 5b:

(i)={α,β}:gs(i)=τ0(i)+∑k=1Ns(i)Hsk(i)h˙k(i)h˙k(i)=h0(i)(1−gs(i)τsat(i))m(i)|γ˙s(i)|

(5a)(5b)

The τsat(i) is the saturated stress, Hsk(i) is the phase slip hardening interaction matrix describing selfstrain hardening (Hkk(i)=1) and latent hardening (Hsk(i)=1.4),h0(i) is the initial hardening rate, m(i) is the hardening exponent.

The constitutive law, which relates the strain and stress evolution in each phase is formulated in terms of an objective stress (rotation free). It is expressed using the Hill and Rice form of the Jaumann stress rate ( σe∇ ) as shown in Eq. 6 which can be written in terms of the symmetric and asymmetric parts of the Schmid tensor as shown in [13], see [9, 11] for details.

σe∇+σtr(De)=Cg:De(6)

Where De,tr(De), is the elastic strain and its trace respectively; Cg is the fourth order grain stiffness matrix in the global coordinate shown in Eq. 7, obtained by rotating its local stiffness matrix
(Cl) with rotation matrix ( 𝐑 ) formed with the grain Euler angles ( θ,ψ,Φ ) obtained from EBSD (electron backscattered diffraction)

Cg=[𝐑⊗𝐑]:Cl:[𝐑𝐓⊗𝐑𝐓](7)

The above model formulation was implemented in Abaqus user material (UMAT) using the work of Eralp et al. [9]. The rule of mixture (Eq. 8) as a homogenization technique, was employed to obtain the global/macroscopic stress in the material, which depends on the volume fraction of α−fα and the volume fraction of β−fβ

σg=fασα+fβσβ(8)

Phase Material Properties Identification: Genetic Algorithm Optimisation. From the model formulation, various microstructural parameters for both α and β phase are necessary to realize the microscale to macroscale (bottom - up) numerical simulation. These include the single elastic constants (C11α,C33α,C12α,C13α,C44α,C11β,C12β,C44β); viscous and hardening parameters (τ0(α),γ˙0(α),h0(α), τsat (α),m(α),n(α),τ0(β),γ˙0(β),h0(β),τsat (β),m(β),n(β)).

The single elastic constants of the different phases were identified from in-situ HEXRD lattice strain measurements by reconstructing micro-strain tensors of each phase, applying micromechanical homogenization to compute macroscopic stress, and solving an inverse optimisation problem using a genetic algorithm under elastic stability constraints [14, 15]. The parameters were obtained by minimizing the discrepancy between the computed macroscopic stress and the experimental uniaxial stress in the elastic domain.

To identify the visco-plastic parameters of the two phases, the micro-strain evolution for the relevant crystallographic planes of α and β phases (derived in the section of Phase Micro-strain Lattice Determination below) was used as input for a genetic algorithm optimisation. This was possible with the assumption that, for the respective {hkl} planes in each phase, its micro-strain tensors are equivalent to its elasto-viscoplastic strain, and these cumulatively resolved to the global elastoviscoplastic strain in each of the phases.

For instance, in β phase, micro-strain tensors associated with {110} and {200} planes are considered to govern the global elasto-viscoplastic behaviour in β phase, while α phase micro-strain computed on basal {0002}, prismatic {101―0}, {112―0}; and pyramidal {101―1}, {101―2} planes cumulatively govern to the α phase elasto-viscoplastic behaviour. To formulate the optimisation problem, an equivalent elasto-viscoplastic strain was computed for each {hkl} plane using the von Mises strain definition (Eq. 9a), based on the {hkl} strain tensors presented in the section of Phase Micro-strain Lattice Determination below.

εvmhkl=23(εhkl:εhkl)(9a)

These equivalent micro-strain evolutions served as the target data for the genetic algorithm fitting procedure such as (Eq. 9b):

(i)={α,β}:γshkl(i)=εvmhkl(9b)

The optimisation objective functions - which are visoplastic slip functions Eqs. 3, 5a and 5b, were defined to identify the visco-plastic parameters for each {hkl} in both α and β phase (τ0(α),γ˙0(α),h0(α), τsat (α),m(α),n(α),τ0(β),γ˙0(β),h0(β),τsat (β),m(β),n(β)). The shear strain equation was solved using Newton Raphson algorithm and it was included in the genetic algorithm, where the bounds (upper and lower) was reasonably defined for each of the parameters. The macroscopic uniaxial stress ( σ11 ) obtained from the uniaxial experiment was factored by a factor ( M ) defined as 1.35 (to account for the applied stress and slip plane direction) as shown in Eq. 10, and was used as the resolved stress. It
served as the forcing function in the optimization algorithm at each corresponding shearing strain increment.

σvmg=Mσ11(10)

The quality of the fit was evaluated using the root mean square error (RMSE) calculated using Eq. 11 , where N is the total population reasonably defined to minimize the computation time.

RMSE=1N∑i=1N(xi−xj)2(11)

where xi,xj,…,xN is each of the computed values for an optimized parameter in the population.
It should be noted that Hsk(i), the hardening interaction coefficient was assumed to be 1.0 in the optimisation solution. The softening beyond global-yielding of the macroscopic stress-strain is not discussed here, as such, the εvmhkl and σvmg up to the 0.2 macroscopic true-strain was used for the fitting (to ensure that micro-strain up to saturation is considered).

Experimental Details. The real-time microstructural evolution was followed during a uniaxial tensile test performed at 730∘C on a dual-phase ( α+β ) Ti-6Al-2Sn-4Zr-2Mo-0.12Si (Ti-6242S) alloy using a modified DIL 805 A/D deformation dilatometer (TA instruments) working at the P07 beamline of the Petra III, DESY, Hamburg, Germany.

Fig. 1a shows the 4 mm thick dog-bone shaped sample machined from a 4 mm thick plate, whose as-mounted schematic in the tensile apparatus of the thermomechanical simulator (DIL 805, TA instruments) is shown in Fig. 1b. A type-S thermocouple was welded at the surface in the mid-length of the sample to real-time control the sample temperature with a numerical regulator. The whole experiment was performed under secondary vacuum in order to prevent exposure to oxygen, which can cause α-case and can affect the ductility of the material.

Thanks to a high energy of 100keV(λ of 0.124A˙), a large volume was analyzed by transmission of X-rays through 4 mm wide samples. Consequently, a 2D-dectector Perkin Elmer XRD 1621 Flat Panel with a resolution of 2048 by 2048 pixels was placed in transmission to collect the DebyeScherrer ring. X-ray diffraction images were taken at intervals (acquisition rate of 0.2 Hz−1 in every 5 s ), which follows the complete test evolution during thermo-mechanical loading.

The experiment consisted of heating from room temperature up to the target temperature (730∘C), followed by 10 mins holding time prior the mechanical loading, applied at the strain rate of 2×10−3 s−1. The sample was not loaded to failure but to the maximum displacement limit of the equipment of 9 mm . Finally, rapid cooling was performed using argon gas at the rate of 140∘C.s−1.

The microstructural and the crystallographic details in both undeformed and post-deformed states were examined under SEM respectively, and EBSD was performed on the undeformed state. The critically deformed section of the sample, likewise, the initial state was cut and prepared with standard metallography technique without etching and observed under SEM in the back-scattered electron (BSE) mode with FEI NOVA nano SEM 450 equipment. While the EBSD was performed on the initial state sample, with OXFORD Instruments symmetry equipment with an accelerating voltage of 20 kV at a rate of 153.52 Hz .

[picture]

Fig.1. Schematic of in-situ uniaxial tensile test under synchrotron X-ray observation. Sample geometry (a); sample in clamped position in relation to X-ray beam source (b); 2D-dectector (c); typical result of partial integration of X-ray pattern about (ψ) - TD (d) and typical result of partial integration of X-ray pattern about 30∘(±15∘) - LD (e).

Fig.1. Schematic of in-situ uniaxial tensile test under synchrotron X-ray observation. Sample geometry (a); sample in clamped position in relation to X-ray beam source (b); 2D-dectector (c); typical result of partial integration of X-ray pattern about ( ψ ) - TD (d) and typical result of partial integration of X-ray pattern about 30 ∘ ( ± 15 ∘ ) - LD (e).

Phase Micro-strain Lattice Determination. Based on the X-ray diffraction patterns recorded throughout the tensile test in temperature, the phase volume fraction and the mean lattice parameters of each phase were determined by using the Rietveld refinement method [16] on the 1D patterns, meaning by integrating fully around the azimuth of Debye-Scherrer rings.

In addition, partial integration of the X-ray diffraction patterns was performed to highlight the expected expansion along the loading direction and compression in the transverse direction. An azimuth angle dhkl of d0hkl  about the orthogonal axes (Fig. 1c), together with its symmetric counterpart, was selected to ensure sufficient grain statistics. The partial integrations were performed using the pyFAI module developed by the European Synchrotron Radiation Facility [17]. Curve fitting of the experimental data was performed using Pseudo-Voigt function from the LMFIT Python library [18]. Typical fitting results along the sample transverse and the loading direction are shown in Figs. 1d and 1e.

From these fitted diffraction data obtained for the two-perpendicular directions, the d-spacing d {hkl} was determined using Bragg's law [19]. The corresponding elastic lattice micro-strain for each {hkl} plane in both phases was then calculated using Eq. 12a. Here, {hkl}d denotes the reference d-spacing of the

εhkl=dhkl−d0hkld0hkl(12a)

plane, taken as the 2θ-spacing before the mechanical loading was applied.

εhkl

Considering that the X-ray monochromatic beam is incident on the sample at a diffraction angle 0∘, the effect of beam geometry was normalized by averaging of the micro-strain 180∘ obtained from Eq. 12a at azimuthal angles εLDhkl and 90∘, defining the loading direction strain 270∘. Similarly, averaging the values obtained at εTDhkl and 3rd  yields the transverse direction strain εTDhkl=εNDhkl, as expressed in Eq. 12b. The normal direction (ND) is the ε22hkl=ε33hkl orthogonal axes along the thickness of the sample (as shown in Fig. 1). Since the experiment involves uniaxial loading, a transversely isotropic behaviour can be assumed. Consequently, the transverse and normal micro-strains are considered equal due to the Poisson effect such that εijhkl, implying {hkl}. Under these assumptions, all the diagonal of the i,j=1,2,3 strain tensor for each

εLDhkl=ε0∘hkl+ε180∘hkl2;εTDhkl=ε90∘hkl+ε270∘hkl2εijhkl=(εLDhkl000εTDhkl000εTDhkl)

(12b)(12c)

plan (a second-order tensor where 1×1×1μ m3 representing the three orthogonal reference coordinates) (Eq. 12c) can be determined while the off diagonal strains are assumed to be null.

0.05μ m

Microstructure Generation and Modelling Approach. A RVE of (2.3μ m) with 100 grains was generated with Neper [20] to replicate the material microstructure. The grain orientations obtained from EBSD analyses were incorporated into the RVE generation to ensure that they approximatively represent the same grain orientation distribution as the actual material. Neper was also used for the meshing, using C3D8 elements with a size of α, which is lower than the average grain size β in order to eliminate mesh dependency and sensitivity (see Fig. 2a).

In order to account for the phase volume fractions, the group option was used in Neper which permits specifying the number of different grains for fα and 85% phases, to represent their volume fraction ( fβ−15% 730∘C and α at β determined from Rietveld refinement). The grains for each phase are shown in Fig. 2a, and the assembly is shown in Fig. 2b. Each group was assigned a distinct identifier ( 1 for −X,−Y phase and 2 for 22.14% phase) to differentiate the phases. In the input file generated for Abaqus, two element set instances were created at the assembly level to group the grains according to the identifiers. This approach allowed assigning different material properties to the grains based on their respective phases.

Boundary conditions were applied to the RVE: symmetry conditions were used to constrain rotation about, and deformation along, the α, and -Z axes, while a displacement corresponding to 0.20 true strain ( β engineering strain) was applied on the X-face of the RVE in the X direction to replicate the uniaxial displacement imposed on the macroscopic sample (Fig. 2c).

[picture]

Fig. 2. Images of Representative volume elements (RVE). Representation of β and α grains (a); Assembly of α and 2.3μ m grains with global mesh view (b); and boundary conditions descriptions (c).

Fig. 2. Images of Representative volume elements (RVE). Representation of β and α grains (a); Assembly of α and 2.3 μ m grains with global mesh view (b); and boundary conditions descriptions (c).

Results and Discussion

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As mentioned above, the microscale to macroscale numerical modelling approach adopted herein depends on advanced experimental techniques; as such, both results of experiments and numerical modelling are presented chronologically.
Microstructure and Mechanical Behaviour. Figs. 3a and 3b respectively show the BSE-SEM micrograph and the inverse pole figure at the as-received state of Ti-6242 titanium alloy. The microstructure consists of nodular β phase surrounded by a 10% phase network. The 730∘C phase appears with a lighter contrast, and the 2.10−3 s−1 phase exhibits a darker contrast. The mean equivalent diameter of the 730∘C nodules is found to be approximately 700∘C. From image analysis, the α volume fraction is about β at room temperature, which is consistent with the value obtained by Rietveld refinement of the diffraction pattern and similar to the value reported by Kapoor et al. [19].
Fig. 3c displays a true stress - true strain curve obtained at β and a strain rate of 15%. Three distinct stages can be identified: i) a small elastic domain followed by, ii) a plastic deformation region characterized by strain hardening up to the ultimate stress, and iii) a subsequent flow softening regime. The latter is not discussed here, as beyond a true strain of about 0.54 , the temperature is no longer constant at β. The sharp transition from elastic to plastic at strain of about 0.001 (Fig. 3d) can easily go unnoticed. Beyond that, the observed hardening and subsequent softening behaviour are consistent with previously reported results for the same alloy and Ti-6Al-4V alloy at temperatures around α/β [5, 6, 21, 22].
The microstructure at the critically deformed section is shown in Fig. 3e. Both the 730∘𝐂 and α grains became more elongated along the RD, which coincides with the loading direction, and the β phase fraction has increased to approximate 5.4∘ (Rietveld refinement). In addition, micro-voids (black spots) are observed, predominantly within the 5.9∘ phase and at the α interfaces. The porosities had been similarly reported for Ti-6Al-4V and Ti-6242S alloys at superplastic temperatures [5, 6], with the extent of porosity strongly dependent on strain rate and temperature.

[picture]

Fig. 3. BSE-SEM micrograph at the as-received state of Ti-6242 titanium alloy (a); inverse pole figure of the as-received state (b); true stress - true strain curve (c); magnified zone of on-set of local plastic transition zone (d); and BSE-SEM micrograph of test sample post mechanical deformation at β.

Fig. 3. BSE-SEM micrograph at the as-received state of Ti-6242 titanium alloy (a); inverse pole figure of the as-received state (b); true stress - true strain curve (c); magnified zone of on-set of local plastic transition zone (d); and BSE-SEM micrograph of test sample post mechanical deformation at β .

Microstructural Behaviour Under Mechanical Loading at α. To understand the respective contributions of the 2θ and 5.4∘−5.9∘ phases to the macroscopic mechanical behaviour described in (Microstructure and Mechanical Behaviour, see Fig. 3c), the strain-induced evolution of each phase is investigated and presented using two complementary approaches. First, the 1D profiles of the HEXRD patterns are analyzed to assess the global response of each phase. Second, partial integration of the HEXRD patterns is performed to quantify the phase-specific response to loading along both the LD and TD direction. These results are subsequently used to identify the microstructural parameters required for the finite strain CPFE modelling.
Qualitative Evolution - 1D profiles. Fig. 4a presents the 1D profiles of the HEXRD patterns, illustrating the global evolution of phases within the microstructure throughout mechanical loading, and Fig. 4b shows a magnified region of the 1D profiles between 82% and 730∘C. The peaks corresponding to the α and α phases are indexed as indicated in Fig. 4. As shown in Fig. 4b, increasing macroscopic true strain leads to a systematic shift of the diffraction peaks toward lower diffraction angles for both phases, which reflects an increase in elastic lattice micro-strain within the different crystallographic planes.
In addition, the shape of the −{112―2} peaks in the {202―1} range of α becomes asymmetric (can be slightly observed in lower angles), which increases up to a true strain of 0.6 (equivalent of β in engineering strain). The loss of the peaks asymmetric shape at true strain above 0.7 can be linked to the rise in
temperature above α earlier mentioned. This asymmetry of the peaks is an indication of anisotropic strain behaviour in the {0002} phase under mechanical loading, which can likely be caused by grain size heterogeneity [23-25]. Peak asymmetry may also arise from the combined effects of crystallite size heterogeneity and chemical heterogeneity. However, the latter can be excluded in the present case, as the microstructure is close to thermodynamic equilibrium. In addition, it can be noticed that the pronounced asymmetry observed for the pyramidal planes of {101―0} phase {111―0}, {101―1},{101―2} and can be attributed to the higher sensitivity of these crystallographic orientations to anisotropic lattice strain and heterogeneous deformation under applied loading.

[picture]

Fig. 4. Evolution of 1D profiles during mechanical loading at {110} in the range between {200} β (a); and in the range between { hkil }α to highlight the asymmetry peaks of εhkl phase (b).

Fig. 4. Evolution of 1D profiles during mechanical loading at { 110 } in the range between { 200 } β (a); and in the range between { hkil } α to highlight the asymmetry peaks of ε h k l phase (b).

Equivalent Lattice Strain Evolution - 2D profiles. Using the partial integration method, the result of the elastic micro-strain evolution of the different crystallographic planes of both α and {101―0} phases along LD and TD are reported in Figs 5a and 5b, respectively. The following crystallographic planes are examined for the {101―1} phase: basal - {0002} planes, prismatic - {110}β planes, {200}β, and pyramidal - α planes, while {101―0} and Ti−Al−V−Fe−Si−O planes are considered for the (α+β) phase.
The different {hkil}α planes exhibit distinct responses, as evidenced by the varying stiffening/hardening rates observed among them, particularly along the loading direction. Note that these different evolutions of the selected lattice strain {hkl}β, in particular at low true strain, tend to
reflect the elastic anisotropy of the {hkil}α phase. Indeed, in accordance with trends reported in [26, 27], the α plane have a more compliant behaviour compared to the β, and the β planes. In contrast, the two α and {hkil}α planes show similar behaviour, which reached a plateau at a very low lattice strain and evolved closely as the macroscopic true strain increases. This trend indicates that strain accommodation is mainly governed by the {hkl}β phase, especially by the prismatic {hkil}α planes. The behaviour is in contrast with reports for the same Ti-6242S alloy and for Ti-64 alloy [19], likewise [28] for {hkl}β a {110}⟨111⟩ titanium alloy. These contrastive behaviours can be directly linked to the temperature effect. Furthermore, Figs. 5a and 5b clearly reveal the Poisson effect in both {0001}⟨112―0⟩ and {101―0}⟨112―0⟩ planes: most {101―1}⟨112―0⟩ planes attain a plateau rapidly at low strain, except for the basal plane, which continues to exhibit an increase in the lattice strain.
The correlation of the first deviation of the flow stress obtained from experiment from elasticity at the small strain in Fig. 5c and the different inflection/deviation of the lattice strains in the different planes can be linked to the local onset of plasticity and also to changes in load distribution between different grains (with different orientation) and/or phases. In addition, Fig. 5d shows the lattice strains evolution with the macroscopic true stress. The different planes of both { hkil }α and {hkl}β planes behaved linearly (approximately) at different stress range, upon which a change or deflection of the linear evolution is observed, which signaled yielding. The crystallographic planes of the {0002} phase yield at much lower stress compared to those of the {101―1} phase planes.
Since the objective is to identify microscale parameters for numerical modelling as stated earlier, it is necessary to establish a link between the observed lattice strains in each phase ( 2×10−3 and τ0 ) and the corresponding phase-level strain response. As previously stated, it was assumed that the elasto-viscoplastic strain experienced in each phase results from the cumulative contribution of lattice strains associated with the different crystallographic planes of the phase. Accordingly, the phase lattice strains shown in Fig. 5a, calculated using Eq. 12a, were used to construct strain tensors as defined in Eq. 12c. The strain von Mises (Eq. 9a) was then applied to these phase lattice strain tensors to obtain the equivalent lattice strain of each phase.
The resulting equivalent phase lattice strains are presented as scattered data points in Fig. 5c. This equivalent phase lattice strain was further simulated using the hardening laws earlier presented in the numerical framework above, while attempting to optimize the material parameters that describes the strain evolution. The optimized elasto-viscoplastic strain functions for the different 730∘C and fβ planes are shown as continuous lines in Fig. 5c. It is interesting to recognize that the equivalent phase lattice strains and their optimized fits follow the same trend reported for the phase lattice strain (Fig. 5 a ), thereby showing the consistency of the proposed microstructural phase properties identification method.

[picture]

Fig. 5. Phase micro-strain evolution. Micro-strains in loading direction vs macroscopic strain (a), micro-strains in transverse direction vs macroscopic strain (b); von Mises equivalent of phase micro-strain vs macroscopic strain (c); and phase micro-strain evolution with macroscopic true stress (d).

Fig. 5. Phase micro-strain evolution. Micro-strains in loading direction vs macroscopic strain (a), micro-strains in transverse direction vs macroscopic strain (b); von Mises equivalent of phase micro-strain vs macroscopic strain (c); and phase micro-strain evolution with macroscopic true stress (d).

Numerical Model Result. The simulation was performed on was performed with 12 BCC crystallographic system - 13% and 12 HCP crystallographic system - 3 Basal {hkl}β−{110}, 3 prismatic β, and 6 pyramidal α. The optimisation search range (upper and lower bound) is reported in table 1 and the result of the optimized parameters for both the ({101―0},{0002} and {101―1} are listed in table 2, which resulted in the simulated result (continuous line) reported in fig 5c. It is necessary to note that the behaviour of Basal (τ0) equivalent lattice strain is linear up to saturation, as such, the shearing equation (Eq. 3) used for the fitting could not reproduce it well. As such, the values calibrated for prismatic { hkil }α was adopted for Basal slip system in the simulation because the two seems to have yielded and saturated closely (Figs. 5a and c).

It is necessary to mention that the authors are not oblivious of the need for uniqueness of the identified material parameters in order to be able to predict the material response under varying strain rates; however, this prove of concept has only been tested for experiment under a single strain rate of {0002}. Although the uniqueness has been judged by the fitness of the fit considered within different optimisation search bounds. However, further analyses would be performed using data from different strain rates in future study.
The calibrated values of the initial critical shear stress ( {101―0} ) for HCP slip planes at {101―1} reported here is in the same range of values reported by Séchepée et al. [29] for the same material (Ti-6242S) with a 730∘C of α. Table 3 shows the values used for the finite crystal plasticity simulation, the results reported for β was used as input for α phase, while the average of the calibrated parameters (Table 2) for for β phase planes Ti−6242 S, and 730∘C ) except for the initial resolved stress α, which the respective β values - basal - β, prismatic - α, and pyramidal - β were adopted for the simulation. Table 4 shows the summary of the single elastic constants at α for both β and α phases used in the CPFE simulation.

Table 1. Optimisation search bounds.

| α | β
τ₀ [MPa] | 150 - 430 | 40 - 100
γ₀ [s⁻¹] (10⁻³) | 10⁻⁴ - 10⁻² | 10⁻⁴ - 10⁻²
h₀ [MPa] | 20 - 400 | 10 - 50
τsat (MPa) | 120 - 500 | 85 - 150
m | 0.1 - 25 | 0.1 - 5
n | 0.1 – 2.9 | 0.1 – 0.9

Table 2. Optimized microstructural model parameters for different crystallographic planes in 90∘ and 0∘ phase.

| α | β
| {1010} | {0002} | {1011} | {1012} | {1120} | {110} | {200}
τ₀ [MPa] | 425 | - | 370 | 282 | 288 | 57 | 50
γ̇₀ [s⁻¹] (10⁻³) | 4.08 | - | 3.02 | 0.56 | 1.30 | 0.45 | 0.39
h₀ [MPa] | 203 | - | 230 | 268 | 269 | 24 | 19
τsat [MPa] | 482 | - | 394 | 378 | 380 | 101 | 98
m | 8 | - | 2.44 | 3.13 | 4.191 | 0.1 | 0.1
n | 0.62 | - | 0.86 | 1.61 | 1.37 | 0.57 | 0.50

Table 3. Numerical simulation parameters.

| α | β
τ₀avg [MPa] | Table 2 | 57.60
γ̇₀avg [s⁻¹] (10⁻³) | 2.38 | 0.450
h₀avg [MPa] | 220.605 | 24.570
τsatavg [MPa] | 482.339 | 101.679
m | 4.290 | 0.1
n | 0.785 | 0.570
c/a | 1.59 | 1

Table 4: Single elastic constant of β and β phase in α at β

| C₁₁ [GPa] | C₁₂ [GPa] | C₄₄ [GPa] | C₃₃ [GPa] | C₁₃ [GPa]
α | 133.9 | 64.8 | 40 | 157.5 | 46.3
β | 129.5 | 104.8 | 48.50 | - | -

Fig. 6 shows the comparison between the simulation and experimental true stress vs true strain. It can be observed that the result of the simulated material global response well reproduced the experiment material macroscopic true stress vs true strain curve. This shows the accuracy of the obtained model parameters identification and the modelling approach.

[picture]

Fig. 6. Comparison of True stress vs true strain from simulation and experiment at β.

Fig. 6. Comparison of True stress vs true strain from simulation and experiment at β .

Further, the elastic region of simulated results, β and 730∘C phase, coincide with the elastic stiffness of the material when local yielding has not started (indicated by the first stiffness, see Fig. 3d). The β phase flow stress is higher than that of the α phase, which means that the α phase is more stressed than the β phase. At a glance at the β phase material properties (Tables 2 and 3 ) and the equivalent lattice strains (Figs. 5a, b and c), it would be expected that the β phase sustains more stress than the α phase. However, considering that β phase planes has a low critical shearing stress that is about onetenth of the hardest β phase plane which means that α/β slip planes are soft and slip activities starts at a low stress (shown in Fig. 5d), which can be seen from the very low value of the equivalent phase lattice strain (Fig. 5c) the (α+β) planes begin to shear. All these in addition to the high stiffness of the (εhkl) phase at 730∘C, caused higher stress to be partitioned in fβ15% phase than β phase.

In contrast, the lower flow stress in the α phase is due to the fact that the slip shearing can occur at a high stress level due to the hardness of the HCP slip planes. The start of the shearing in β phase at low stress can be considered to be the cause of the reduction in material elastic stiffness reported in Fig. 5c, although this cannot be shown by the simulation result due to the limited number of α grains considered in the RVE compared to the large β grains (probably in millions) in the real sample, and while the continuous increase on a new linear path (second elastic stiffness slope) up to the global plastification of the material can be considered to be the effect of the high β phase fraction (hard slips) on the global material behaviours. The higher stress in the α phase may justify the porosities found inside the β phase and the α/β interfaces/grain boundaries (Fig. 3e), which may reveal the phase interfacial effect in dual-phase material with different hardness of their respective slip planes. This effect is not accounted for here, the modified rule of mixture independently developed by Fan et al. [30], and Ankem and Margolin[31] may be used to quantify this effect.

Conclusion

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A multiscale modeling approach was used to simulate the global behaviour of Ti-6242S-using microscale properties of constitutive phases α identified through the application of a genetic algorithm optimisation method with the microstructural data derived from the advanced HEXRD, coupled with the SEM and EBSD experimental techniques.
It has been shown that from the phase micro-strains Ti−Al−Mo−V obtained through the HEXRD, the elastoviscoplastic properties of each constitutive phases of a polycrystalline material can be established and the microstructural phase parameters can be used for multiscale (micro to macroscale) modelling. It was quantitatively established that in a Ti- 6242 S at Ti−6Al−4 V, with α, higher stress is partitioned in the β phase than in the β→α′′ phase. Despite the fact that α′ phase slip planes are vastly softer (lower initial CRSS) compared to the α−Ti phase, whose slip planes have higher initial CRSS. Further, the higher stress in the α+β phase phase may also have been influenced by the interfacial interaction between the β (softer slips) and the (α+β) (harder slips) phase, which caused formation of the observed microvoids/porosities within mathematical formula and mathematical formula interfaces (Fig. 5e).

With the HEXRD technique, the real-time anisotropic behaviour of mathematical formula phase has been shown during mechanical loading, which may have completely gone unnoticed if the technique is not used. Also, through this new one-way model parameters identification technique, the anisotropic effect of the different phase planes can be included in numerical simulation by simply using the appropriate parameters that describes the slip plane in the CPFE simulation.

This methodology will facilitate the quantification of the influences of the different phases in multiphase material applicable at both low and high temperature investigations. Further improve the understanding of the effect of each constitutive phases in multi-phase materials on the materials global behaviours. This will overall help to adequately design the material optimisation process for effective manufacturing of structures/components. This microstructural model approach is rigorous enough to allow the quantification of the influence of the different phases in polycrystal materials with great accuracy than from the unbounded optimisation approach as earlier noted in [5]. The results have shown that the methodology is a good modelling approach.

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Solid State Phenomena, ISSN: 1662-9779, Vol. 390, pp 163-172

doi: 10.4028/p-cNIr4V

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

Submitted:
2026-01-19

Revised:
2026-02-16

Accepted:
2026-02-21

Online:
2026-04-17

Study on the Effects of Grain Shape, Size and Size Distribution on the Mechanical Behavior of Metals

The original version of this paper is available on https://www.scientific.net/SSP.390.163.pdf

BOCAN Ruxandra 1,2,a*, PERDAHCIOGLU Semih 1, b, and VAN DEN BOOGAARD Ton 1,c

1 Chair of Nonlinear Solid Mechanics, University of Twente, Enschede, The Netherlands, 2 M2i, Materials Innovation Institute

a  ruxandra.bocan@utwente.nl, b  e.s.perdahcioglu@utwente.nl , c  a.h.vandenboogaard@utwente.nl

Keywords: Crystal Plasticity, Gradient Enhanced Crystal Plasticity, Grain Size Distribution, Grain Aspect Ratio

Abstract

Crystal plasticity finite element (CPFE) simulations of three-dimensional representative volume elements (RVEs) enable the prediction of polycrystalline material behavior under complex loading conditions. Plastic deformation is modeled through crystallographic slip on lattice slip systems, subject to the Schmid yield criterion based on the maximum resolved shear stress (CRSS). In this work, an efficient rate-independent crystal plasticity (CP) and gradient enhanced crystal plasticity (GECP) formulation is used to investigate the influence of microstructural characteristics on the mechanical performance of FCC materials. Three-dimensional periodic RVEs with irregular grain morphologies are simulated within Abaqus/Standard to study the effects of grain size, grain size distribution, and grain shape under uniaxial loading. Comparative analyses between the CP and GECP frameworks are performed to assess the predictive capabilities and applicability for increasingly heterogeneous microstructures. The results demonstrate that GECP accurately captures grain size dependent work hardening and grain size distribution effects through the intrinsic length scale introduced by strain gradient calculations. In contrast, grain shape variations result in only minor changes in the macroscopic response for both frameworks.

Introduction

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Constitutive micromechanical models based on Crystal Plasticity have become increasingly used to predict material performance under complex loading conditions and manufacturing processes. With advances in computational efficiency, such finite element models also aid in gaining a more fundamental understanding of the effects microstructural parameters have on the overall macroscopic behavior. As a result, crystal plasticity formulations are particularly suited for studying the deformation of heterogeneous structures with physically accurate material descriptions.

Within the crystal plasticity framework, plastic deformation occurs through dislocation movement along slip planes of the crystallographic lattice. The evolution of plastic flow is governed by the assumption that slip is allowed only on slip systems for which the resolved shear stress (RSS) reaches the critical resolved shear stress (CRSS) [1]. Hardening models have been developed that relate this critical lattice resistance to the accumulation of immobile dislocations. Two dislocation types can be distinguished, namely statistically stored dislocations (SSDs) and geometrically necessary dislocations (GNDs) [2]. GNDs arise from strain incompatibilities between neighboring grains undergoing non-uniform plastic deformation and are associated with regions of high plastic strain gradients [3]. Their generation is introduced as an additional hardening mechanism into the crystal plasticity framework, which is directly linked to an intrinsic material length scale, responsible for size-dependent strengthening effects [4]. Frameworks that incorporate this additional hardening mechanism through strain gradient computations are referred to as gradient enhanced crystal plasticity formulations.

There is relatively limited research on the relation between granular microstructural parameters, such as grain size and shape, and the resulting macroscopic mechanical response. Although phenomena such as grain size dependent hardening, described by the Hall-Petch relation, are widely accepted
within the metal industry, many numerical studies remain restricted to two dimensional simulations. Recent developments in rate-independent crystal plasticity formulations allow for the simulation of realistic, three-dimensional RVEs within feasible computation times [5]. In addition, advances in numerical implementation and solution strategies, such as the stress update procedure in [6] allow for the efficient computation in the three-dimensional domain.

The incorporation of dislocation density based hardening without the introduction of additional length scale parameters was first proposed in [7]. Subsequent work extended this approach by including plastic strain gradient calculation into the evolution of GND densities with flow stress [8]. Together, these developments make it possible to efficiently conduct simulations of realistic threedimensional structures that account for grain size effects using physically motivated descriptions.

This work aims to improve the understanding of the influence of microstructural features on the mechanical performance of metals through crystal plasticity simulations of realistic three-dimensional polycrystalline structures using the described algorithmic frameworks. A quantitative comparative study between conventional CP and GECP is conducted to establish relations between polycrystalline grain parameters and observed material response. The applicability and accuracy of the current rateindependent frameworks are investigated for uniaxial loading conditions of isotropic materials.

Crystal Plasticity Constitutive Framework

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In this section a short description of the crystal plasticity and gradient enhanced crystal plasticity formulations is given. Plastic deformation is assumed to occur through dislocation motion along the slip systems of the crystallographic lattice. The constitutive formulation used in this work is based on continuum mechanics kinematics, a flow rule based on Schmid slip system activation, and a rate independent hardening law formulated in terms of the total dislocation density and slip system resistance evolution. Both conventional Crystal Plasticity (CP) and Gradient Enhanced Crystal Plasticity (GECP) formulations are considered. For a more detailed description of the theoretical formulations and tensor decompositions, the reader is referred to [5].

At the core of the CP formulation lies the multiplicative decomposition of the deformation gradient 𝐅 into elastic 𝐅e and plastic parts 𝐅p,

𝐅=𝐅^e𝐅p(1)

where 𝐅p accounts for permanent deformation due to shearing of the crystallographic lattice. Plastic deformation is assumed to occur solely due to dislocation slip on specific slip systems and is assumed to leave the structure in a stress-free, undistorted intermediate configuration. The elastic part 𝐅e accounts for the change in stress through elastic distortion and rigid body rotation, mapping the intermediate configuration to the current one. The ( ∧ ) notation is used to denote variables pushed from the intermediate configuration to the current one, while the remaining variables are defined conventionally. The corresponding velocity gradient 𝐋 can be further decomposed into plastic and elastic contributions, 𝐋=𝐋^e+𝐋^p. The plastic part of velocity gradient describes the rate at which the crystallographic lattice plastically distorts and is expressed as a sum of plastic slip rates γ˙(α) over all active slip systems:

𝐋^p=∑α=1nγ˙(α)(𝐬^(α)⊗𝐧^(α))(2)

where the slip systems α, with slip direction 𝐬^(α) and slip plane normal 𝐧^(α), are defined in the current configuration. The total plastic deformation of a crystal is obtained through superposition of shear distortions on all active slip systems.
The formulation of rate-independent CP follows Schmid's law, stating that slip occurs only when the resolved shear stress, τ(α) (RSS), on slip system α, exceeds the slip resistance τc(α), also referred to as the critical resolved shear stress (CRSS):

ϕ(α)(σ,γ)=τ(α)(σ)−τc(α)(γ)≤0(3)

where the above mentioned equation represents the yield function for each system within the polycrystal, and γ denotes the vector of accumulated slip over all slip systems.

The total stress of the material is defined using the generalized Hooke's law, with only the elastic part contributing to the stress:

σ∘=ℂ^e:𝐃^e(4)

where ℂ^e the elasticity tensor of the lattice and σ∘ denotes the objective rate of the Cauchy stress, capable of allowing for large rotations. The resolved shear stress is then obtained by projecting the stress onto the slip system shear direction:

τ(α)=σ:𝐏^(α)=σ:12[𝐬^(α)⊗𝐧^(α)+𝐧^(α)⊗𝐬^(α)](5)

where 𝐏^(α) is the symmetric Schmid tensor.
Hardening in crystal plasticity is a result of the increase in slip resistance with the accumulation of dislocations. The relation between shear flow stress and dislocation density is given by a Taylor-type hardening law, where the CRSS on each slip system is expressed as:

τc(α)=τ0+Gb∑β=1nQ(αβ)ρ(β)(6)

where τ0 is the base resistance to slip due to lattice friction [9], G is the shear modulus, b is the Burgers vector length, and Q(αβ) is the latent hardening matrix, accounting for interactions between slip systems. In [10] it is shown that there can be up to six independent variables within this matrix ( Q0 to Q5 ) for a Face-Centered Cubic (FCC) lattice. The superscript β represents all possible slip systems within one crystal of the lattice. The total dislocation density ρ(β) consists of contributions from both statistically stored dislocations (SSDs) and geometrically necessary dislocations (GNDs), with the latter included only in the gradient enhanced formulation.

For the two dislocation types, the evolution of dislocation densities within the same crystal is connected to the amount of accumulated plastic slip. For SSDs, a phenomenological formulation [11] is implemented, in which the SSD density, ρSSD (α), increases with accumulated slip and approaches a saturation value ρSSD ∞ :

ρSSD(α)=ρSSD∞[1−(1−ρSSD0ρSSD∞)exp(−γ(α)γ∞)](7)

where ρSSD 0 is the initial density and γ∞ is the saturation slip.
Moreover, within GECP, GNDs are introduced to maintain compatibility associated with nonuniform plastic deformation between polycrystalline grains. GNDs are related to the gradients of plastic slip through the Burgers tensor. By projecting the gradients of slip ∇γ(α) onto the slip direction and dislocation line direction, edge and screw GND densities are obtained:

ρGND,e(α)=−𝐬(α)·∇γ(α)ρGND, s(α)=𝐈(α)·∇γ(α)(9)

GND densities introduce an intrinsic material length scale into the formulation through their direct dependence on strain gradients. The resulting hardening law leads to an enhanced formulation capable of capturing size-dependent mechanical behavior, such as grain size and shape effects in polycrystalline materials.

To solve the constitutive equations of the rate-independent CP and GECP formulations, an implicit stress-update algorithm is used. The evolution of stress during deformation is governed by the enforcement of the Schmid yield and the associated Kuhn Tucker conditions at the end of each increment, ensuring that plastic flow occurs only on active slip systems. The algorithm is based on the Successive One-Dimensional Solution Steps (SODS) method, which determines slip increments without requiring an active set search. For a more detailed explanation of the numerical implementation and solution procedure, the reader is referred to [6].

Numerical Framework and RVE Construction

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The two frameworks have been implemented into Finite Element (FE) formulations using a computational homogenization approach. Numerical simulations are performed on three-dimensional finite element representative volume elements (RVEs) to model the polycrystalline microstructure. Regular and irregularly shaped periodic RVEs are generated using the Voro++ software library [12] and meshed within GMSH [13]. The microstructures obtained present statistically representative grain morphologies with control of grain size, size distribution, and aspect ratio. Periodic boundary conditions are applied to the model to ensure consistency and kinematic compatibility with the macroscopic scale. The algorithms used are implemented through FORTRAN modules as independent Abaqus material subroutines. The algorithm consists of a fully implicit stress and strain update scheme implemented within the Abaqus UMAT subroutine. For the Gradient Enhanced formulation, the scheme is weakly coupled with an explicit subroutine, UExternalDB, which solves for the gradients of plastic slip and updates the corresponding shear stresses at the following iterative steps. All simulations are conducted under identical uniaxial loading conditions up to 20% strain to allow for a consistent comparison background. The work is carried out for isotropic FCC materials.

The proper selection of an RVE is essential to ensure that the numerical response is independent of the specific microstructural composition and is statistically representative of the macroscopic behavior of the material. A systematic study was conducted to assess convergence with respect to the number of grains and mesh discretization for the homogenized response of RVEs with uniform grain size and distribution. The validated RVEs yield stable and isotropic responses with good computational efficiency. Subsequently, these RVEs are used to study morphological variations, ensuring a reliable comparison of grain size and grain shape effects. A detailed description of the RVE determination procedure is provided in [14].

Results

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In this section, the numerical results obtained from three-dimensional RVE simulations are presented and discussed, focusing on the influence of grain size, grain size distribution, and grain morphology on the macroscopic response under uniaxial loading up to 20% strain along the principal displacement directions. The predictive capabilities of conventional crystal plasticity (CP) and gradient enhanced crystal plasticity (GECP) are systematically compared.

RVE isotropy and framework validation

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Prior to investigating microstructural effects, the isotropy and representativeness of the selected RVEs are evaluated for both CP and GECP frameworks. The RVEs used in this study, illustrated in Fig. 1, are generated such that the geometrical grain arrangement is varied, with different degrees of uniformity in grain positioning and shape, while variations in crystallographic grain orientations are also considered. In the present work, only the behavior of the irregular RVE is described.

For RVEs with uniform grain size distribution and morphology, both frameworks provide stable and consistent macroscopic stress-strain responses with limited directional differences. The small variations observed, within an error range of 1−2%, between stresses in the principal loading directions are mainly attributed to differences in the assigned grain orientation sets, while changes in geometrical

[picture]

Fig. 1: 3D periodic RVEs with uniform grain size distribution and different degrees of geometrical regularity: (a) regular RVE and (b) irregular RVE. The structures differ in grain positioning and grain shape uniformity. Both consist of 512 grains and approximately 250,000 finite elements, used to assess isotropy and material representativeness

Fig. 1. 3D periodic RVEs with uniform grain size distribution and different degrees of geometrical regularity: (a) regular RVE and (b) irregular RVE. The structures differ in grain positioning and grain shape uniformity. Both consist of 512 grains and approximately 250,000 finite elements, used to assess isotropy and material representativeness

arrangement result in smaller variations. The isotropic error is further observed to decrease with increasing regularity of the microstructural arrangement. The inclusion of strain-gradient contributions into the GECP framework does not change the sensitivity to microstructural variations, with both the overall behavior and the isotropic response preserved. These results confirm that the selected RVEs provide a suitable basis for modeling isotropic materials within rate-independent crystal plasticity formulations. The RVEs further provide a reliable basis for comparison in subsequent investigations of grain size, grain size distribution and grain shape effects.

Influence of grain size on work hardening behavior (GECP framework)

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The influence of grain size on the macroscopic response is examined through the intrinsic length scale introduced in the GECP framework. The simulations predict an increase in flow stress with decreasing gain size, consistent with strengthening concepts such as the Hall-Petch effect, for identical RVE geometries and assigned crystallographic orientation sets as used in the CP simulations. While no shift in the yield stress is observed due to the absence of an initialization relation between grain size and dislocation density in the algorithm, grain refinement manifests through enhanced work hardening. Reducing the grain size from 77 to 9 results in an approximate 14−15% increase in stress, as shown in Fig. 2. This behavior is associated with higher plastic strain gradients and the resulting accumulation of geometrically necessary dislocations (GNDs) at grain boundaries, as confirmed by spatial distribution of GND density, Fig. 5a. Furthermore, finer-grained RVEs also exhibit increased mechanical stability, Fig. 3, as assessed using Considere's criterion, reflected by a shift of the necking point to higher strains and stresses in the absence of explicit damage and localization mechanisms.
In contrast, the classical CP formulation does not capture size dependent work hardening behavior, with no systematic change in response with varying RVE volumes. These results demonstrate that the constitutive inclusion of strain gradients in the hardening law is essential for capturing physically meaningful grain size effects for polycrystalline materials.

Influence of grain size distribution

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To further investigate microstructural heterogeneity, RVEs with varying grain size distributions and identical crystallographic orientations are analyzed. The grain size distribution is varied under constant RVE volume constraints, resulting in the presence of larger grains and considerably smaller ones with increasing degrees of heterogeneity, as shown in Fig. 4. The grain size variation is measured using a coefficient of variation parameter, defined as c.o.v =s/rmean , where s is the standard deviation of grain radius and rmean  is the mean grain radius. For small coefficients of variations (c.o.v of 0.1 ), both

[picture]

Fig. 2: Macroscopic true stress-strain response from Abaqus implicit simulations for the irregular RVE under uniaxial loading up to 20% strain. Results are shown for both CP(- -) and GECP () along the principal loading directions. GECP formulation predicts increased hardening for an average uniform grain size of 9 .

Fig. 2. Macroscopic true stress-strain response from Abaqus implicit simulations for the irregular RVE under uniaxial loading up to 20% strain. Results are shown for both CP(- -) and GECP () along the principal loading directions. GECP formulation predicts increased hardening for an average uniform grain size of 9 .

[picture]

Fig. 3: Intersection of the true stress-strain ( - ) and stiffness curves (- -) for RVEs with decreasing grain size. Mechanical stability limit based on the Considere's criterion of necking: onset of reduced resistance to further plastic deformation

Fig. 3. Intersection of the true stress-strain ( - ) and stiffness curves (- -) for RVEs with decreasing grain size. Mechanical stability limit based on the Considere's criterion of necking: onset of reduced resistance to further plastic deformation

[picture]

Fig. 4: Grain size distribution in periodic RVEs: Geometry plane cut (x−y) for 3D periodic irregular RVE with increasing grain size distribution under constant RVE volume constraints. (a) Base geometry with uniform grain size, (b) coefficient of variation c.o.v =0.1, with proportional increase and decrease in individual grain volumes, (c) c.o.v =0.3 with dominant large grains surrounded by smaller grains to preserve the overall RVE volume

Fig. 4. Grain size distribution in periodic RVEs: Geometry plane cut ( x − y ) for 3D periodic irregular RVE with increasing grain size distribution under constant RVE volume constraints. (a) Base geometry with uniform grain size, (b) coefficient of variation c.o.v = 0.1 , with proportional increase and decrease in individual grain volumes, (c) c.o.v = 0.3 with dominant large grains surrounded by smaller grains to preserve the overall RVE volume

[picture]

Fig. 5: Effect of grain size distribution on GND density: spatial distribution of geometrically necessary dislocation (GND) densities obtained from GECP simulations for an irregular RVE with increasing size distribution. (a) Uniform grain size, evenly spread distribution of regions with higher GND densities (b) c.o.v=0.1, GND distribution spread similar to the uniform case, with less strong localizations (c) c.o.v =0.3, dominant large grain behavior, with a reduction in overall GND accumulation through larger regions with lower GND densities and more localized regions of high density near smaller and highly misoriented grains. Less overall GNDs leads to a reduction in the hardening response

Fig. 5. Effect of grain size distribution on GND density: spatial distribution of geometrically necessary dislocation (GND) densities obtained from GECP simulations for an irregular RVE with increasing size distribution. (a) Uniform grain size, evenly spread distribution of regions with higher GND densities (b) c.o.v=0.1, GND distribution spread similar to the uniform case, with less strong localizations (c) c.o.v = 0.3 , dominant large grain behavior, with a reduction in overall GND accumulation through larger regions with lower GND densities and more localized regions of high density near smaller and highly misoriented grains. Less overall GNDs leads to a reduction in the hardening response

[picture]

Fig. 6: GECP simulations true stress-strain results for an irregular RVE with increasing levels of grain size distribution variation. The plots display a comparison of the stress-strain curves between RVEs with uniform grains and RVEs with grains size distribution. (a) Uniform grain morphology (- -) and c.o.v =0.1(−) (b) uniform grain morphology (- -) and c.o.v =0.3(−)

Fig. 6. GECP simulations true stress-strain results for an irregular RVE with increasing levels of grain size distribution variation. The plots display a comparison of the stress-strain curves between RVEs with uniform grains and RVEs with grains size distribution. (a) Uniform grain morphology (- -) and c.o.v = 0.1 ( − ) (b) uniform grain morphology (- -) and c.o.v = 0.3 ( − )

CP and GECP frameworks exhibit nearly identical behavior, with deviations from the uniform grain size simulation stress results below a mean relative error of 0.3%. Grain size distribution does not significantly affect the isotropy of the RVE.
As the grain size distribution becomes broader (c.o.v of 0.3), so called dominant large grain behavior emerges, resulting in a reduction of the hardening percentage, as visible in the stress-strain plots of Fig. 6 . The GECP framework predicts a stress decrease of approximately 1% relative to the uniform grain size RVE for highly heterogeneous microstructures. This reduction in hardening is attributed to lower plastic strain gradients around larger grains and only localized regions of high dislocation densities, which reduce the overall accumulation of GNDs and lower the hardening response, as supported by the spatial distribution plot, Fig. 5c.

The CP framework is unable to capture this decrease in hardening, as it lacks the material length scale associated with strain gradient calculation. While CP simulations reflect variations in stress due
to changes in geometry regularity and altered crystallographic misorientation distributions, it does not capture a systematic dependence on grain size variance for the investigated RVEs.

A direct comparison between CP and GECP simulations highlights the conditions under which the GECP formulation outperforms the latter. For RVEs with uniform grain size or low grain size heterogeneity, the predicted responses are similar.

With increasing grain size variance, however, the GECP formulation provides additional predictive capabilities. The inclusion of strain gradient calculations predicts a reduction in hardening associated with dominant large grain behavior and localized regions of high dislocation density due to the size dependent effects. Quantitatively, the difference between the frameworks for a c.o.v of 0.3 is around 1% decrease in stress. These results indicate that while CP is sufficient for modeling relatively homogeneous microstructures, GECP is more accurate for predicting the mechanical response of heterogeneous polycrystals.
Influence of grain shape (aspect ratio)

[picture]

Fig. 7: Effect of grain aspect ratio on RVE geometry: Geometry plane cuts ( y−z ) for 3D periodic irregular RVE with variations in grain aspect ratio along the z axis. (a) Base geometry with uniform grain size, (b) reduced grain aspect ratio of 0.75, compressed grain aspect (c) increased grain aspect ratio of 3, elongated grain aspect

Fig. 7. Effect of grain aspect ratio on RVE geometry: Geometry plane cuts ( y − z ) for 3D periodic irregular RVE with variations in grain aspect ratio along the z axis. (a) Base geometry with uniform grain size, (b) reduced grain aspect ratio of 0.75, compressed grain aspect (c) increased grain aspect ratio of 3, elongated grain aspect

The influence of grain shape on the macroscopic response is examined by varying the grain aspect ratio while maintaining a fixed average grain size and grain volume fraction. For the investigated RVEs, the grain aspect ratio is varied from 0.75 to 3 along the z-axis, as shown in Fig. 7. The crystallographic lattice orientations are randomly assigned and, where possible, kept identical between RVEs. In the absence of a predominant texture, any anisotropic response observed can be primarily attributed to variations in grain morphology and geometrical arrangement.

For both CP and GECP frameworks, variations in grain aspect ratio lead to small changes in the macroscopic stress-strain response, typically below 1% mean relative error from uniform grain results, Fig. 9. A slight directional dependence is observed, characterized by a reduced resistance to deformation when loading is applied perpendicular to the grain aspect ratio direction. Nevertheless, stress variations across loading directions remain within a narrow range. The inclusion of strain gradient effects in the GECP framework does not predict shape-dependent hardening behavior either, with comparable GND distributions across the investigated RVEs, as shown in Fig. 8.

Overall, for the investigated RVEs, no clear identifiable correlation between grain aspect ratio and macroscopic response or anisotropy is noticed. The observed stress variations are small, remain within numerical tolerance across cases, and are comparable in magnitude to uncertainties arising from geometrical changes. As a result, an exact quantitative assessment for the change in stress solely due to grain shape is hard to attribute. For complete clarity, further studies should be conducted under tight geometrical constraints or through a systematic texture study where unwanted variables are statistically averaged out. The findings indicate that, within the current framework and checked RVEs,

[picture]

Fig. 8: Effect of grain aspect ratio on GND density: spatial distribution of geometrically necessary dislocation (GND) densities obtained from GECP simulations for irregular RVEs with varying aspect ratios, (x-z) plane cut. (a) Uniform grain size, relatively evenly spread distribution regions, (b) aspect ratio 0.75 (c) aspect ratio 3. Comparable GND distributions across all cases, no clear pattern can be distinguished for shape-dependent effects

Fig. 8. Effect of grain aspect ratio on GND density: spatial distribution of geometrically necessary dislocation (GND) densities obtained from GECP simulations for irregular RVEs with varying aspect ratios, (x-z) plane cut. (a) Uniform grain size, relatively evenly spread distribution regions, (b) aspect ratio 0.75 (c) aspect ratio 3. Comparable GND distributions across all cases, no clear pattern can be distinguished for shape-dependent effects

[picture]

Fig. 9: GECP simulations true stress strain results for an irregular RVE with varying aspect ratios. The plots display a comparison of the stress curves between RVEs with uniform grains and RVEs with grain shape change through aspect ratio assignment. (a) Uniform grain morphology (- -) and aspect ratio 0.75(−) (b) uniform grain morphology (- -) and aspect ratio 3(−)

Fig. 9. GECP simulations true stress strain results for an irregular RVE with varying aspect ratios. The plots display a comparison of the stress curves between RVEs with uniform grains and RVEs with grain shape change through aspect ratio assignment. (a) Uniform grain morphology (- -) and aspect ratio 0.75 ( − ) (b) uniform grain morphology (- -) and aspect ratio 3 ( − )

the influence of grain shape is minor and significantly weaker than that of grain size distribution and crystallographic orientation effects.

Conclusion

The original version of this paper is available on https://www.scientific.net/SSP.390.163.pdf

The macroscopic response of polycrystalline materials was investigated using three-dimensional representative volume elements within crystal plasticity (CP) and gradient enhanced crystal plasticity (GECP) frameworks. The results show that grain size-dependent work hardening effects are very well represented within GECP formulation through the inclusion of strain gradient calculations and geometrically necessary dislocation evolution. Grain size distribution is further shown to influence the mechanical response of heterogeneous microstructures, with these effects more accurately predicted
by the GECP framework for large variations in grain size. In contrast, no significant variations in the macroscopic response are observed for changes in grain shape in the absence of predominant crystallographic texture for the investigated RVEs. Overall, the findings highlight the capabilities of the frameworks, with the GECP formulation able to provide a more in-depth and reliable representation of heterogeneous polycrystalline materials.

References

The original version of this paper is available on https://www.scientific.net/SSP.390.163.pdf

R.J.Asaro. Crystal plasticity. Journal of Applied Mechanics, 50(4b):921-934, 1983.

J.F.Nye. Some geometrical relations in dislocated crystals. Acta Metallurgica, 1(2):153-162, 1953.

A. Acharya and J. L. Bassani. Lattice incompatibility and a gradient theory of crystal plasticity. Journal of the Mechanics and Physics of Solids, 48(8):1565-1595, 2000.

N. A. Fleck, M. A. Ashby, and J. W. Hutchinson. The role of geometrically necessary dislocations in giving material strengthening. Scripta Materialia, 48(2):179-183, 2003.

E S Perdahcioglu. A rate-independent crystal plasticity algorithm based on the interior point method. Computer Methods in Applied Mechanics and Engineering, 418(Part A), 2024.

B Nijhuis, E S Perdahcioglu, and A H van den Boogaard. A robust and efficient rate-independent crystal plasticity model based on successive one-dimensional solution steps. Computer Methods in Applied Mechanics and Engineering, 438(Part B), 2025.

Y. Bergstrom. A dislocation model for the stress-strain behaviour of polycrystalline alpha-fe with special emphasis on the variation of the densities of mobile and immobile dislocations. Materials Science and Engineering, 5(4):193-200, 1970.

E S Perdahcioglu, C Soyarslan, E E Asik, A H van den Boogaard, and S Bargmann. A class of rate-independent lower-order gradient plasticity theories: Implementation and application to disc torsion problem. Materials, 11(8), 2018.

M. F. Ashby. The deformation of plastically non-homogeneous materials. Philosophical Magazine, 21(170):399-424, 1970.

L. Kudin, B. Devincre, and M. Tang. Mesoscopic modelling and simulation of plasticity in fcc and bcc crystals: Dislocation intersections and mobility. Journal of Computer-Aided Materials Design, 5:31-54, 1998.

M. Becker. Incompatibility and Instability Based Size Effects in Crystals and Composites at Finite Elastoplastic Strains. PhD thesis, Institut für Mechanik (Bauwesen), Lehrstuhl I, Universität Stuttgart, 2006.

Lawrence Berkeley National Laboratory. voro++. . html. Accessed 2025-10-15.

C Geuzaine and J-F Remacle. Gmsh: A 3-d finite element mesh generator with built-in preand post-processing facilities. International Journal for Numerical Methods in Engineering, 79(11):1309-1331, 2009.

R. Bocan. Study on the effects of grain shape, size and size distribution on the mechanical behavior of metals. Master's thesis, University of Twente, 2025.

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The original version of this eBook is available on https://www.scientific.net/book/material-behaviour-modelling/978-3-0364-1987-9

Solid State Phenomena, ISSN: 1662-9779, Vol. 390, pp 173-180

doi: 10.4028/p-aiq9fM

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

Submitted:
2026-01-19

Revised:
2026-02-18

Accepted:
2026-02-20

Online:
2026-04-17

Investigation of Friction Coefficient at Stack Compression Tests

The original version of this paper is available on https://www.scientific.net/SSP.390.173.pdf

Martin László Kölüs 1,3,a*, Imre Nagy-Gombai 2,b, Gábor Kalácska 1,c and Gábor Jozsef Beres 3,d

1 Institute of Technology, Hungarian University of Agriculture and Life Sciences (MATE), 2100 Gödöllő, Páter blvd. 1. Hungary

2 ElringKlinger Hungary Ltd., 6000 Kecskemét, Paul Lechler St. 4, Hungary

3 Mechanical- and Laser Beam Technologies Research Group, John von Neumann University, 6000 Kecskemét, Izsáki St. 10. Hungary

a* kolus.martin@nje.hu, b  Imre.Nagy-Gombai@elringklinger.com, c kalacska.gabor@uni-mate.hu, dberes.gabor@nje.hu

*Corresponding author

Keywords: stack compression test, friction, finite element method.

Abstract

During sheet metal forming (for example, deep drawing), the desired geometry is typically produced under large plastic deformations. However, in order to characterize the material behavior in this deformation range as accurately as possible, experimental investigation of the material is indispensable. Such investigations are necessary, among other reasons, to determine material properties that may later serve as input parameters for finite element (FEM) simulations. By defining appropriate material properties, the design and optimization of the forming technology become more efficient. One such property is the material's flow curve; for determining it in the large deformation range, a particularly promising method is the stack compression test (SCT). At the same time, the method also has certain drawbacks. One is that the test is not standardized, and therefore there is no exact methodology for its execution. Another difficulty arises from the fact that the friction conditions present during the test are not clearly defined. This paper seeks to determine whether the friction conditions in the case of the SCT can be inferred by a comparative analysis of experimental forcedisplacement curves and force-displacement curves obtained from FEM simulations.

Introduction

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One of the most widely known methods for determining a flow curve is the tensile test-based approach; however, this method is not capable of achieving the level of deformation that is characteristic of more complex sheet metal forming processes. To overcome this limitation, alternative procedures such as the plane strain compression test (PSCT), applied by Neag and Balan [1], and the hydraulic bulge test (HBT) used by Coppieters et al. (2020) [2], have been developed. In their work, Coppieters et al. (2022) [3] provide a comprehensive overview of the available experimental methods for determining flow curves in the large deformation regime. They also refer to the stack compression method, which was among the first to be applied by Merklein and Kuppert [4].

Subsequently, An and Vegter [5] investigated the influence of hydrostatic pressure and frictional behavior during the execution of the SCT. Kuwabara et al. [6] also presented a comprehensive study on the effect of hydrostatic pressure on the flow stress, commonly referred to as the strength differential effect. Their results indicated that the applied lubricant may deteriorate as a consequence of oil film thinning and elongation during deformation. Furthermore, Kraus et al. [7] suggested treating the friction coefficient as a variable rather than a constant parameter, experimentally determining the evolution of both the Coulomb friction coefficient and the friction factor. The pressure-dependent variation of these frictional parameters was subsequently described mathematically by Gil et al. [8], who employed a nonlinear formulation to characterize the evolution of lubrication conditions.

Finite element method (FEM) simulations are frequently used for tool geometry optimization, as demonstrated by R. LU et al. [9]. In their study, they also investigated the stress and strain distributions arising during the deep drawing of Ti/Al bimetal sheets. Wang et al. [10] examined the prediction of ductile fracture occurring during sheet metal forming using finite element methods and subsequently validated their results experimentally. In the present work, we aim to obtain a more accurate understanding of the frictional conditions prevailing during the SCT by comparing experimental results with FEM simulation outcomes.

Used Material

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For the experiment, DC04 (EN 10130 EU) cold-rolled thin steel sheet was used, which owing to its ferritic microstructure, has favorable formability properties and is therefore widely applied across many areas of industry for the manufacture of components produced by complex deep-drawing operations. The chemical composition of the base material is presented in Table 1.

Table 1. Average chemical composition (percentage) of the DC04 steel.

C [%] | Mn [%] | P [%] | S [%] | Si [%] | Al [%] | N [%]
0.0355 | 0.239 | 0.0135 | 0.0088 | 0.0177 | 0.0471 | 0.0097

The mechanical properties and anisotropy characteristics of the material were determined by means of a standard tensile test (ISO 6892-1:2019) at room temperature using an INSTRON 4482 universal testing machine. The strength and plasticity properties were examined in the rolling direction, perpendicular to it, and at 45∘. The results are presented in Table 2.

Table 2. The average material parameters of the DC04 steel.

YS

UTS

229 | 336 | 37.9 | 20.4 | 1.823 | 1.311 | 2.380 | 1.706

Where YS is yield strength; UTS is ultimate tensile strength; A80 is total elongation; Ag is uniform elongation and the r values are the anisotropy parameters.

Preparation and Execution of the Stack Compression Test

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The stack compression tests, similarly to the previously mentioned material characterizations, were also carried out using the INSTRON testing machine. In accordance with our previous work [11], a machine compliance correction was applied. Cylindrical tool platens with polished pressing surfaces, manufactured from Böhler K110 (X153CrMoV12), were used in the experiments. The heat treatment of the tool platens was performed based on the manufacturer's recommendations. Proper positioning of the specimens both on top of each other and on the compression dies is also crucial; therefore, a positioning device manufactured additively from PLA material was used. The compression dies and the positioning device are shown in Fig. 1.

[picture]

Fig. 1. The polished compression dies A) and the positioning device B).

Fig. 1. The polished compression dies A) and the positioning device B).

From the base material with a thickness of 1 mm , cylindrical specimens were manufactured by laser cutting. One set of specimens had an initial diameter of 6 mm , while the other set had an initial diameter of 10 mm . The stacks were assembled such that their height was half of the initial diameter (h/d=1/2); therefore, in the first case a test configuration of ∅6×h3 mm was used, while in the second case a ∅10×h5 mm configuration was applied.

The surfaces of the compression platens in contact with the specimens were coated with Luba 21® high-pressure lubricant in order to minimize friction. In contrast, the surfaces of the specimens in contact with each other were not lubricated, thereby promoting bulk-like material behavior. The specimens were placed at the center of the compression tool so that the rolling direction markers (red markings) shown in part B) of Figure 1 faced the same direction. The tests were terminated when the compressed height of the stack approximately reached half of the initial height. The initial test configuration and the compressed stacks are illustrated in Fig. 2.

[picture]

Fig. 2. The experimental setups before (1) and after the SCT (2).

Fig. 2. The experimental setups before (1) and after the SCT (2).

Compression began with a rapid approach (Vrapid =3[ mm/min]) up to a load of 250[ N], after which it continued according to the following relation: h/10[ mm/min]. This procedure was necessary to ensure that, after the initial clamping phase, the same strain rate was maintained in both cases.

The Structure of the Simulations

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The material properties of the workpieces were defined according to the values listed in Table 2. The applied flow curve was obtained from tensile test results previously performed by the authors.

The finite element simulations were carried out using Simufact Forming 2025.02, which is based on the nonlinear implicit solver MSC Marc. An incremental-iterative Newton-Raphson scheme was applied, accounting for geometric nonlinearity (large strain formulation), material nonlinearity (elasto-plastic behavior), and contact nonlinearity. In order to reduce the computational time, the model was reduced to a quarter. Accordingly, the appropriate symmetry boundary conditions were applied to the quarter-specimen.

Meshing of the specimens was performed using the sheetmesh option, which employs hexahedral elements and, as its name suggests, is specifically designed for meshing sheet metal components. For the simulation of the different experimental setups, the mesh element sizes were scaled proportionally to each other. The most important meshing parameters for both configurations are summarized in Table 3.

Table 3. The parameters of the meshes used.

Diameter

Elem size

Elements over

thickness

Element

count

Min. edge angle

6 | 0.1 | 3 | 2193 | mathematical formula
10 | 0.16 | 3 | 2193 | mathematical formula

The software does not allow the creation of more than one deformable specimen within a single simulation. Therefore, the stack configuration was realized by introducing additional tool elements whose material behavior was changed from rigid to deformable and assigning them identical material parameters to the original specimen. This modeling approach does not modify the boundary conditions or contact definitions and therefore does not influence the force-stroke response of the simulation. The mesh applied to the specimens is illustrated in Fig. 3.

[picture]

Fig. 3. The mesh applied to the specimens.

Fig. 3. The mesh applied to the specimens.

In Fig. 3, the X direction corresponds to the rolling direction of the material, Y denotes the transverse direction, and Z represents the thickness direction. Figure 4 illustrates the assembled FEM simulation of the ∅10×h5 mm configuration.

[picture]

Fig. 4. The FEM simulation of the ∅10×h5 mm experimental setup.

Fig. 4. The FEM simulation of the ∅ 10 × h 5 m m experimental setup.

The temperature was also specified and set to 20∘C. In addition, a remeshing criterion was defined; in the present case, it was based on the angular distortion within individual elements, with an allowable limit of 15∘. Furthermore, all specimens were compressed to half of their initial height, consistent with the experimental procedure, while maintaining the same deformation rate.

During the simulation, a Coulomb friction coefficient ( μ ) was applied and defined using two values. The first coefficient ( μdie ) was assigned to the lubricated contact surfaces between the pressure plates and the workpieces, while the second coefficient ( μspecimen  ) was applied to the contact surfaces between the workpieces themselves. The combinations of friction coefficients used in the process are summarized in Table 4.

Table 4. The Coulomb friction coefficient combinations used.

Cases | μ | 1. | 2. | 3. | 4. | 5.
A | μdie | 0.045 | 0.057 | 0.075 | 0.085 | 0.11
μspecimen | 0.4 | 0.4 | 0.4 | 0.4 | 0.4
B | μdie | 0.075 | 0.075 | 0.075 | - | -
μspecimen | 0.2 | 0.3 | 0.4 | - | -

The final time step of the executed FEM simulations is shown in Fig. 5. The specimen is compressed to 1.5 mm , and the effective strain is shown on the color scale; however, the current study does not consider strain values.

[picture]

Fig. 5. The finished FEM simulation of the ∅6×h3 mm experimental setup.

Fig. 5. The finished FEM simulation of the ∅ 6 × h 3 m m experimental setup.

Comparison of Force-Stroke Curves Obtained from Experiment and FEM Simulations

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From the simulations, it was possible to export the force-stroke diagrams of the compressed stacks corresponding to the individual friction coefficient combinations, which are then compared with the experimental measurement results. For each setup, the comparisons of the curves for cases A and B are presented separately. Fig. 6 and fig. 7 illustrates the curves for the ∅6×h3 configuration in cases A and B .

[picture]

Fig. 6. The force-stroke curves for the ∅6×h3 configuration in case A .

Fig. 6. The force-stroke curves for the ∅ 6 × h 3 configuration in case A .

[picture]

Fig. 7. The force-stroke curves for the ∅6×h3 configuration in case B .
Fig. 8 and fig. 9 illustrates the curves for the ∅10×h5 configuration in case A and B .

Fig. 7. The force-stroke curves for the ∅ 6 × h 3 configuration in case B . Fig. 8 and fig. 9 illustrates the curves for the ∅ 10 × h 5 configuration in case A and B .

[picture]

Fig. 8. The force-stroke curves for the ∅10×h5 configuration in case A .

Fig. 8. The force-stroke curves for the ∅ 10 × h 5 configuration in case A .

[picture]

Fig. 9. The force-stroke curves for the ∅10×h5 configuration in case B.

Fig. 9. The force-stroke curves for the ∅ 10 × h 5 configuration in case B.

Summary

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Our results indicate that, in terms of their overall characteristics, the measured and simulated force-stroke curves do not initially show good agreement; however, as the process progresses, a good correlation can be observed. Nevertheless, no common intersection point is present in any of the cases. The curves obtained from the FEM simulations predict higher forces than those measured experimentally. In the ∅6×h3 configuration, this difference amounts to approximately 5−7kN, whereas in the ∅10×h5 case it is on the order of ~10kN. It can be observed that, for the simulated curves, case A results in a larger difference in the magnitude of the maximum compressive force for both configurations. This difference is approximately 1.9 kN for both the ∅6×h3 and ∅10×h5 cases; however, it is proportionally more significant for the ∅6×h3 configuration, where the maximum compressive force is only about 30 kN . For the simulated case B configuration, the observed difference is approximately 1.4 kN for the ∅10×h5 case, while for the ∅6×h3 configuration it is only about 0.2 kN . Based on these findings, it can be concluded that the friction coefficient between the compression dies and the workpieces likely has a greater influence on the process than the friction coefficient between the workpieces themselves.

Acknowledgement

The original version of this paper is available on https://www.scientific.net/SSP.390.173.pdf

Supported by the EKÖP-25 new national excellence program of the ministry for culture and innovation from the source of the national research, development and innovation fund.

References

The original version of this paper is available on https://www.scientific.net/SSP.390.173.pdf

Neag A, Balan T. Verification of flow curve determination from plane strain compression tests. CIRP J Manuf Sci Technol 2025;63:554-65. https://doi.org/10.1016/j.cirpj.2025.11.005.

Coppieters S, Jackel M, Kraus C, Kuwabara T, Barlat F. Influence of a Hydrostatic Pressure Shift on the Flow Stress in Sheet Metal. Procedia Manuf 2020;47:1245-9. .

Coppieters S, Traphöner H, Stiebert F, Balan T, Kuwabara T, Tekkaya AE. Large strain flow curve identification for sheet metal. J Mater Process Technol 2022;308:117725. .

Merklein M, Kuppert A. A method for the layer compression test considering the anisotropic material behavior. International Journal of Material Forming 2009;2:483-6. .

An YG, Vegter H. Analytical and experimental study of frictional behavior in throughthickness compression test. J Mater Process Technol 2005;160:148-55. .

Kuwabara T, Tachibana R, Takada Y, Koizumi T, Coppieters S, Barlat F. Effect of hydrostatic stress on the strength differential effect in low-carbon steel sheet. International Journal of Material Forming 2022;15:13. .

Kraus M, Lenzen M, Merklein M. Contact pressure-dependent friction characterization by using a single sheet metal compression test. Wear 2021;476:203679. .

Gil I, Mendiguren J, Galdos L, Mugarra E, Saenz de Argandoña E. Influence of the pressure dependent coefficient of friction on deep drawing springback predictions. Tribol Int 2016;103:266-73. .

LU R, LIU Y, LI M, YAN M, SUN J, HUANG H. A pathway to refined stress and strain distributions in aerospace-grade Ti/Al bi-metal sheets: Synergizing theoretical insights and FEM simulations. Chinese Journal of Aeronautics 2024;37:493-516. .

Wang C, Daniel WJT, Lu H, Liu S, Meehan PA. FEM Investigation of Ductile Fracture Prediction in Two-Point Incremental Sheet Metal Forming process. Procedia Eng 2017;207:836-41. .

KÖLÜS ML, BORBÉLY Richárd, BÉRES Gábor J. The effect of friction compensation on the slope of flow curves obtained by stack compression tests, 2024, p. 1182-9. .

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Solid State Phenomena, ISSN: 1662-9779, Vol. 390, pp 181-188

doi: 10.4028/p-b48XNq

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

Submitted:
2026-01-19

Revised:
2026-02-21

Accepted:
2026-02-21

Online:
2026-04-17

Mechanical Performance of a Steel-Polymer Biphasic Lattice Structure

The original version of this paper is available on https://www.scientific.net/SSP.390.181.pdf

Edoardo Mancini 1,a*, Anastasia Ciccarella 1, b, Andrea Di Giuliano 1,c, Lino Micarelli 1, d, Giuseppe Dell'Avvocato 1,e and Katia Gallucci 1,f

¹Università degli Studi dell'Aquila, DIIIE, Piazzale Ernesto Pontieri I, 67100 L'Aquila, Italy

a  edoardo.mancini@univaq.it, banastasia.ciccarella@graduate.univaq.it, c  andrea.digiuliano@univaq.it, d  lino.micarelli@student.univaq.it, e  giuseppe.dellavvocato@univaq.it, f  katia.gallucci@univaq.it

Keywords: Biphasic lattice structure, homogenization, gyroid, polymer.

Abstract

Interpenetrating Phase Composites (IPCs) with biomimetic properties are promising materials for strengthening orthopaedic implants while also increasing their biocompatibility. Thanks to additive manufacturing techniques, lattice structures can be employed to develop biomaterials with controlled architectures, enabling the replication of human bone structures and offering advantages in terms of strength-to-weight ratio. This study investigates the behaviour of a bi-material steelpolymer lattice structure, observing that the epoxy resin increases the mechanical strength of gyroid, leaving the lightweight properties unchanged. Moreover, an equivalent constitutive model was calibrated, and a homogenization procedure based on the Representative Volume Element theory was applied. The effect on mechanical strength due to the 316L powder dispersed within the epoxy resin was investigated as well.

Introduction

The original version of this paper is available on https://www.scientific.net/SSP.390.181.pdf

The bio-inspired structures exhibit outstanding lightweight properties, strength, and toughness while optimizing material efficiency and mechanical performance [1]. Through a long evolutionary process, living organisms have developed in nature diverse biological structures, well-adapted to their environments [2]. Artificial biomimetic cellular structures are promising for biomedical applications, thanks to their high strength-to-weight ratio and tunable mechanical properties; these architectures, enhanced by Additive Manufacturing technologies, offer great potential for patient-specific medical devices.

The gyroid, one of the Triply Periodic Minimal Surfaces (TPMS), belongs to the cubic crystal system and was first described in [3]. Owing to its suitability for lightweight, high-strength applications, the gyroid structure has been extensively explored for the design of advanced materials that could be exploited in biomedical applications. Gyroid structures may be classified as either sheetbased or strut-based [4].

In this paper, the behaviour of a bi-material steel-polymer lattice structure was investigated as an Interpenetrating Phase Composite (IPC) with biomimetic properties: the steel part is constituted of a sheet-based gyroid cellular structures (designed and manufactured using Laser Powder Bed Fusion, L-PBF); the polymeric part consists of an epoxy resin that fills the gyroid cavities, optionally loaded with steel powder.

Parallelepiped samples were produced at laboratory scale and experimentally tested to identify mechanical behaviours, expressed in terms of engineering stress-strain curves. Experimental data were exploited to develop mathematical models able to reproduce the mechanical behaviour of the steel-polymer biphasic lattice structures: the elastic regime was interpreted by a homogenization procedure [5], to lower the computational effort of IPC simulations; the plastic regime was framed by a new-Three-Stage (newTS) Hollomon model [6] combined with an additional tailored power-law-term that simulates the densification stage [7].

This study might pave the way to the design of customized IPC prosthetic devices; the developed models might be used to verify the suitability of steel-polymer biphasic lattice structures to overcome the limitations related to the use of cemented protheses, e.g., in hip-joint-replacement.

Materials and Methods

The original version of this paper is available on https://www.scientific.net/SSP.390.181.pdf

Materials. Parallelepipedal samples were prepared at lab-scale (edges ~8 mm, Fig. 1): (i) Gyroid structures made of AISI 316L were 3D-printed by Laser Powder Bed Fusion; (ii) epoxy resin (FansArriche) was cast into the Gyroids, as such or mixed with 30wt% of 316 L powder (Sauter diameter 28μ m ). The biphasic steel-polymer lattice cell was discretized using tetrahedral elements for the subsequent numerical homogenization (Fig. 1b).

[picture]

Fig. 1. Biphasic steel-polymer lattice structure: (a) Gyroid on building platform, (b) biphasic steelpolymer lattice cell FE Model and (c) samples with and without epoxy resin

Fig. 1. Biphasic steel-polymer lattice structure: (a) Gyroid on building platform, (b) biphasic steelpolymer lattice cell FE Model and (c) samples with and without epoxy resin

Starting from the raw specimens (Fig. 1a), the samples underwent quasi-static compression tests and microstructural analysis. Both specimens with and without epoxy resin were tested (Fig. 1c).

The samples are labelled as Sp1, Sp2, and Sp3; Table 2 summarizes the samples' main features, i.e., with or without epoxy resin filling and optional addition of 316 L powder to the resin.

Table 1. Summary of all samples

Specimen

+ epoxy

resin

wt% of 316L

powder

Cell

Size

Sample

size

Sp 1 | - | - |  |
Sp 2 | Yes | - | mathematical formula | superscript number
Sp 3 | Yes | mathematical formula |  |

Equipment. Quasi-static compression tests were carried out using an MTS Universal servohydraulic testing machine (Fig. 2). All tests were performed under displacement-controlled conditions at a constant engineering strain rate of 10−3 s−1.

[picture]

Fig. 2. Quasi-static test set-up: a Universal servo-hydraulic test machine and gyroid samples with and without epoxy resin

Fig. 2. Quasi-static test set-up: a Universal servo-hydraulic test machine and gyroid samples with and without epoxy resin

Numerical homogenization procedure. A mesoscale numerical homogenization was performed to identify the elastic stiffness matrix. Representative volume elements were analysed under six periodic loading conditions [7][8]. Tetrahedron elements with a minimum edge length of 0.15 mm were utilized. Figure 3 illustrates the six numerical models and their corresponding loading conditions. All biphasic samples exhibit periodic boundary conditions; for clarity, the epoxy resin is depicted solely in Fig. 3a. To account for the anisotropic effects of 3D printing, the anisotropic elastic matrix [Eij] base of the base material was considered.

The procedure was applied only for the sample Sp2 (Gyroid + epoxy resin).

[picture]

Fig. 3. Periodic loading conditions: (a) Cell size 8 mm Gyroid RVEs with epoxy resin and (b) the six loadings

Fig. 3. Periodic loading conditions: (a) Cell size 8 mm Gyroid RVEs with epoxy resin and (b) the six loadings

To identify the [Eij] base experimental tests on 3D printed bulk samples were performed. The samples were produced with two different orientations relative to the Z-building direction (Fig. 4a) and tested under two load types: tension and torsion. This evaluation aimed to assess the effects of anisotropy by the matrix reported in Fig. 4b [8].

[picture]

Fig. 4. Stiffness anisotropic elastic matrix identification: (a) sample schematization with loading referred to Z-building direction, (b) anisotropic elastic matrix for transversally isotropic materials

Fig. 4. Stiffness anisotropic elastic matrix identification: (a) sample schematization with loading referred to Z-building direction, (b) anisotropic elastic matrix for transversally isotropic materials

Results and Discussion

The original version of this paper is available on https://www.scientific.net/SSP.390.181.pdf

Experimental Results. Fig. 5 shows the quasi-static experimental curves of all specimens. As to samples Sp2 and Sp3, the curve profiles reveal a change in concavity in the strain interval between about 0.05 and 0.15 , during which the curves tend to turn upward. At higher strains ( @0.2 ), the workhardening behaviour characterized by downward concavity is re-established, finally showing the cellular densification stage.

[picture]

Fig. 5. Experimental results: Engineering stress-strain curves

Fig. 5. Experimental results: Engineering stress-strain curves

Constitutive Modelling and Fitting. Different models are available in the literature to reproduce the experimental behaviour recorded during tests and shown in Fig. 5. For most metals, workhardening is better illustrated on bi-logarithmic stress-strain charts than on linear plots. With σ and ε respectively representing engineering stress and strain, in the presence of a linear ln(ε)−ln(σ) relationship, the material behaviour may be effectively represented by the Hollomon power-law equation, σ=Kεn; under these conditions, the slope of the bi-logarithmic plot is indicative of the strain-hardening exponent n , whereas the intercept at ε=1 on the vertical axis corresponds to the strength coefficient K. The curve shown in Fig. 6 is obtained by plotting the quasi-static stress-strain data of the Sp3 in Fig. 5 on a bi-logarithmic scale.

[picture]

Fig. 6. Stress-true strain curve on the bi-logarithmic plane: Gyroid + Epoxy @ 30% 316L (Sp3)

Fig. 6. Stress-true strain curve on the bi-logarithmic plane: Gyroid + Epoxy @ 30% 316L (Sp3)

The crossing points of the three linear regressions in Fig. 6 identify two threshold strains, εt1 and εt2. These strain values represent a material parameter and can be employed to divide the Hollomon laws into TSs:

σ=K1εn1,ε<εt1σ=K2εn2,εt1<ε<εt2σ=K3εn3,ε>εt2(1)

The strain hardening and strength coefficients could be made to vary continuously along with strain, leading to a modified Hollomon equation (Eq. 2).

σ=h· K(ε)εn(ε)+A(ε1−ε)B(2)

Variables σ and ε are engineering stress and strain, respectively, both considered positive in compression. h is the parameter that allows for moving from one curve to another at different powder percentages. The parameters K and n are expected to vary smoothly with strain, conferring to σ vs. ε curves an S-shaped trend characterised by asymptotic behaviour at both low and high strain levels, and a gradual transition around the threshold strains εt. A and B coefficients describe the densification region. Sigmoid function has been used to model the transition from the hardening stages.

Fig. 7 compares with experimental results of Sp3 the following modelling output: (i) the quasistatic curve fitted using the newTS Hollomon model (black curve); the three distinct power-law relationships (dashed lines) corresponding to the different work-hardening phases, (iii) the second term of the new model (green curve).

[picture]

Fig. 7. Comparison between analytical and experimental curves of Sp3 sample: different contributions of each term of the model, Hollomon Power laws fitting and second term new model

Fig. 7. Comparison between analytical and experimental curves of Sp3 sample: different contributions of each term of the model, Hollomon Power laws fitting and second term new model

Numerical homogenization results. Mesoscopic numerical models have been prepared on a biphasic unit cell 8×8×8 in order to identify the elastic matrix coefficients ( [Eij]cell ) of Sp2; this sample was chosen to test the homogenisation procedure on a simpler case, avoiding to introduce experimental uncertainties related to the spatial distribution of AISI 316L powder load in the resin. The results of the numerical homogenisation are expressed by the homogenized elastic stiffness matrix of Fig. 8a. Moreover, the Zener factor ( Zr ) [9] was calculated to detect the anisotropic behaviour (Fig. 8b).

[picture]

Fig. 8. Numerical homogenisation results of the Gyroid Biphasic Lattice Structure of Sp2: (a) Homogenized Elastic Stiffness Matrix [ Eij ]cell  and (b) Zener ratio Zr

Fig. 8. Numerical homogenisation results of the Gyroid Biphasic Lattice Structure of Sp2: (a) Homogenized Elastic Stiffness Matrix [ Eij ] cell and (b) Zener ratio Z r

Simulation results. Both mesoscale and macroscale of the biphasic steel-polymer Gyroid unit cell. The linear elastic-anisotropic material model [Eij]cell  for macroscale simulation was adopted. The Finite Element (FE) models are detailed in Fig. 9, rows "Boundary conditions" and "Discretization"; the results for von Mises equivalent stress are reported in Fig. 9 ("Results" row). To evaluate the anisotropy effect, two displacement loading conditions were applied in the Z and Y directions, respectively. Tetrahedron elements with minimum edge lengths of 0.15 mm and 0.25 mm were used for mesoscale and macroscale, respectively. The results confirmed the optimal homogenization process for the biphasic Gyroid in the elastic regime for quasi-static tests, reducing computational cost.

[picture]

Fig. 9. FE model: Boundary conditions, discretization and results. (a, c) Mesoscale and (b, d) macroscale. "Discretization" line reports the cell size and elements number and minimum length. The stiffness comparison between meso and macro-scale are in the "Results" line

Fig. 9. FE model: Boundary conditions, discretization and results. (a, c) Mesoscale and (b, d) macroscale. "Discretization" line reports the cell size and elements number and minimum length. The stiffness comparison between meso and macro-scale are in the "Results" line

Conclusion

The original version of this paper is available on https://www.scientific.net/SSP.390.181.pdf

A biomimetic-inspired two-phase structure was studied. TPMS Gyroid was chosen as the lattice structure, and both epoxy resin and epoxy resin with the addition of AISI-316L-powder were used as fillers.
The most important findings can be summarized as follows:

the epoxy resin increases the mechanical strength of the gyroid (Gyroid + Epoxy, Fig. 5), leaving the lightweight properties unchanged;

a further increase in mechanical strength is due to the 316 L powder added to the epoxy resin;

the elastic regime has been reproduced by a homogenization procedure based on the Representative Volume Element theory;

an alternative constitutive model, referred to as the "newTS Hollomon power-law", was developed to describe the biphasic lattice-structured material behaviour in the plastic regime.

Acknowledgement

The original version of this paper is available on https://www.scientific.net/SSP.390.181.pdf

The authors wish to thank Dr Luca Benerecetti for his help with material model calibration and Giuseppe Organtini for the experimental tests.

References

The original version of this paper is available on https://www.scientific.net/SSP.390.181.pdf

L. Guo et al., Integrated biomimetic biphasic structures for high-strain-rate resistance and thermal insulation: Design, fabrication and performance characterization, Eng. Sci. Technol. Int. J. 67 (2025) https://doi.org/10.1016/j.jestch.2025.102088

D. Ramakrishna, G. Bala Murali, Bio-inspired 3D-printed lattice structures for energy absorption applications: A review, Proc. Inst. Mech. Eng., Part L: J. Mater. Design Appl. 237 (3) (2022) 503-542.

H. Shoen, Alan, Infinite Periodic Minimal Surface without Intersections, NASA. (1970).

D. Li, W. Liao, N. Dai, Y.M. Xie, Absorption of Sheet-Based and Strut-Based Gyroid, Materials (Basel). (2019)

E. Mancini, M. Utzeri, E. Farotti, A. Lattanzi, and M. Sasso, "DLP printed 3D gyroid structure: Mechanical response at meso and macro scale," Mechanics of Materials, vol. 192, May 2024.

M. Utzeri et al., Quasi-Static and Dynamic Behavior of Inconel 625 Obtained by Laser Metal Deposition: Experimental Characterization and Constitutive Modeling, J. Eng. Mater. Technol. 143 (2021) DOI: 10.1115/1.4051087

E. Mancini et al. Definition of a unified material model for cellular materials with high morphological and topological dispersion: Application to an AA7075-T6 aluminium foam, Materials Science & Engineering A, 833 (2022) . 2021.142346

Ciccarella, A., Dell'Avvocato, G., Cortis, G., & Mancini, E. (2025). Meso-and macroscale modelling strategies for biomimetic structures produced using L-PBF technology. Material Forming: ESAFORM 2025, 313.

C.M. Zener, S. Siegel, Elasticity and Anelasticity of Metals, J. Phys. Colloid Chem. 53 (1949) 1468-1468.

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The original version of this eBook is available on https://www.scientific.net/book/material-behaviour-modelling/978-3-0364-1987-9

Solid State Phenomena, ISSN: 1662-9779, Vol. 390, pp 189-195

doi: 10.4028/p-bdcs2V

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

Submitted:
2026-01-19

Revised:
2026-02-24

Accepted:
2026-02-25

Online:
2026-04-17

Further Developments of Hill-Gotoh Theory of Anisotropic Sheet Metal Plasticity

The original version of this paper is available on https://www.scientific.net/SSP.390.189.pdf

Wei TONG 1,a*

¹Department of Mechanical Engineering, Southern Methodist University, Dallas, Texas USA

a wtong@smu.edu (*corresponding author)

Keywords: non-quadratic yield criteria, non-homogeneous stress functions, convexity.

Abstract

An associated plasticity theory of combining Hill's 1948 quadratic and Gotoh's 1977 quartic stress functions [1,2] has recently been developed for modeling orthotropic steel sheet metals in plane stress [3]. Some further developments of the theory with a higher-order non-homogenous polynomial yield stress function are described in this study. Numerical examples are given to illustrate the expanded capabilities for modeling thin steel sheet metals in plane stress by the enhanced Hill-Gotoh anisotropic plasticity theory.

Introduction

The original version of this paper is available on https://www.scientific.net/SSP.390.189.pdf

Modern mathematical modeling of the yielding and plastic flow of anisotropic sheet metals has been developed following the initial framework as outlined by Hill in his 1950 plasticity monograph [4]. Many current widely used advanced theories of macroscopic anisotropic plasticity replace Hill's 1948 quadratic yield stress function with one of non-quadratic yield stress functions such as Yld2000-2d and Yld2004-18p [5,6]. When only even-order stress exponents such as 6 or 8 are adapted, those nonquadratic yield stress functions are in fact simply higher-order homogenous polynomials of Cauchy stress with many more polynomial coefficients.
To extend the modeling capabilities of either Hill's 1948 quadratic theory [1] or Gotoh's 1977 quartic theory [2], an anisotropic yield stress function has recently been proposed as a sum of both quadratic and quartic stress functions [3]. The new yield stress function is a single fourth-order nonhomogeneous polynomial with 13 even-order stress terms and associated coefficients. The initial application results of the proposed fourth-order theory to model anisotropic yielding and plastic flow of four steel sheets have been encouraging. Further developments of the theory are presented in the following, namely, the generalized formulation of higher-order non-homogeneous polynomial yield functions based on Hill's quadratic and Gotoh's quartic stress functions and the use of a novel leastsquare parameter identification method with a set of heuristic matrix-based convexity constraints.

The generalized Hill-Gotoh theory

The original version of this paper is available on https://www.scientific.net/SSP.390.189.pdf

The quadratic and quartic yield functions in the classical Hill's 1948 theory [1] and in Gotoh's 1977 fourth-order theory are of the following forms (for plane stress)

Φh=A1σx2+A2σxσy+A3σy2+A4τxy2Φg=B1σx4+B2σx3σy+B3σx2σy2+B4σxσy3+B5σy4+(B6σx2+B7σxσy+B8σy2)τxy2+B9τxy4

(1)(2)

with a total of 4 and 9 polynomial coefficients respectively. A generalized Hill-Gotoh yield function is proposed in terms of the normalized Cauchy stress σ―=(σx,σy,τxy)/σf as

Φhg(σ―)=ΦhM1(σ―)+ΦgM2(σ―)+ΦhM3(σ―)+ΦgM4(σ―)+…,(3)

where (M1,M2,M3,M4,…) are positive integers. If (M1,M2)=(1,1), the original fourth-order HillGotoh yield function with 13 coefficients as proposed in [3] is recovered. A higher-order Hill-Gotoh yield function with 17 coefficients would have non-zero ( M1,M2,M3 ), including the second Hill function ΦhM3(σ―) of degree 2M3 with additional four parameters (C1,C2,C3,C4).

Convexity-constrained least square parameter identification

The original version of this paper is available on https://www.scientific.net/SSP.390.189.pdf

It is well-known that a polynomial stress function (homogeneous or otherwise) is in general not guaranteed to be positive and convex and thus may not be automatically used as a valid yield function [5-7]. For example, using eight inputs from uniaxial and biaxial tension tests on an IF steel from [8] and a packaging steel from [9], one can obtain eight on-axis polynomial coefficients of the fourthorder Hill-Gotoh yield function with (M1,M2)=(1,1) by solving eight linearly independent equations or a simple least square parameter identification problem without any convexity constraints as described in [3]. Both the experimental inputs with and as-calibrated model parameters for the two steel sheets are listed in Table 1 and Table 2 respectively.

Table 1. Material property data [8] and non-convex Hill-Gotoh parameters for an IF steel

σ₀/σf | r₀ | σ₉₀/σf | r₉₀ | σb/σf | rb | σp0/σf | σp90/σf
1 | 1.85 | 0.993 | 2.51 | 1.152 | 0.777 | 1.242 | 1.247
A₁ | A₂ | A₃ | B₁ | B₂ | B₃ | B₄ | B₅
2.04046 | -2.56993 | 2.03626 | -1.04046 | 2.62238 | -3.73080 | 2.61784 | -1.03657

[picture]

Fig. 1. Predicted non-convex on-axis biaxial yield surface contours (a) and elastic-plastic domains as filled regions (b) by the fourth-order Hill-Gotoh yield function for an IF steel without SOS-convexity constraints. Solid circles are experimental yield stresses in RD and TD directions.

Fig. 1. Predicted non-convex on-axis biaxial yield surface contours (a) and elastic-plastic domains as filled regions (b) by the fourth-order Hill-Gotoh yield function for an IF steel without SOS-convexity constraints. Solid circles are experimental yield stresses in RD and TD directions.

As shown in Fig.1(a) and Fig.2(a), the resulting biaxial fourth-order yield surfaces Φhg(σ―)=1 have multiple branches and are non-convex for both steel sheets. The non-convexity of both yield surfaces can be better visualized via Fig.1(b) and Fig.2(b) respectively. For the IF steel shown in Fig.1(b), the non-accessible (unfilled) region Φhg(σ―)>1 is between two elastic-plastic domains (the filled or shaded regions) Φhg(σ―)≤1. As indicated by the insert in Fig.1(b), the yield stress σ0 under uniaxial tension along the rolling direction (RD) is on the second or outer contour. Its outer surface normal is pointed inward, so the associated elastic-plastic domain or its yield surface is thus non-convex. For the packaging steel shown in Fig.2(b), the inner elastic-plastic domain (the filled or shaded region in the middle) Φhg(σ―)≤1 is clearly non-convex between yield stresses ( σ0,σp0 ). Although they predict perfectly the eight experimental inputs listed in Table 1 and Table 2, these non-convex HillGotoh plane-stress functions cannot be used as suitable yield functions in numerical analyses of sheet metal forming of these two steels.

Table 2. Material property data [9] and non-convex Hill-Gotoh parameters for a packaging steel

σ₀/σf | r₀ | σ₉₀/σf | r₉₀ | σb/σf | rb | σp0/σf | σp90/σf
1 | 0.8352 | 1.0826 | 1.5097 | 1.0910 | 0.5533 | 1.1194 | 1.3024
A₁ | A₂ | A₃ | B₁ | B₂ | B₃ | B₄ | B₅
1.92368 | -0.898249 | 0.385033 | -0.923678 | 0.828781 | -0.193689 | -0.590039 | 0.399474

[picture]

Fig. 2. Predicted non-convex on-axis biaxial yield surface contours (a) and elastic-plastic domains as filled regions (b) by the Hill-Gotoh yield function for a packaging steel without SOS-convexity constraints. Solid circles are experimental yield stresses at RD and TD directions.

Fig. 2. Predicted non-convex on-axis biaxial yield surface contours (a) and elastic-plastic domains as filled regions (b) by the Hill-Gotoh yield function for a packaging steel without SOS-convexity constraints. Solid circles are experimental yield stresses at RD and TD directions.

On the other hand, the least square minimization of yield and flow conditions with sum-of squares (SOS) convexity constraints as proposed in [3] will produce a set of 8 on-axis polynomial coefficients listed as cases (1) and (3) in Table 3 for IF and packaging steels, resulting in a strictly convex fourthorder Hill-Gotoh yield function. The SOS-convexity is only a sufficient condition of convexity for the yield surface [7] and may be too restrictive to give the resulting yield function with high accuracy. Alternatively, one can use a new set of heuristic matrix-based convexity constraints recently presented in [10] for the least square parameter identification of the strictly convex fourth-order HillGotoh yield function. The corresponding sets of 8 on-axis polynomial coefficients are listed as cases (2) and (4) in Table 3 for both steels respectively. The novel numerical minimization algorithm detailed in Appendix B in [10] ensures that the curvature of a yield surface is positive everywhere (as this is known as the necessary and sufficient criterion for its strict convexity per [11-15]). Predicted on-axis biaxial yield surface contours by SOS-convex Hill-Gotoh yield functions are shown in Fig.3(a) for IF steel and in Fig.4(a) for a packaging steel. Similarly, predicted on-axis biaxial yield surface contours by Hill-Gotoh yield functions based on positive surface curvature constraints are shown in Fig.3(b) for IF steel and in Fig.4(b) for a packaging steel. Little differences are seen between the two yield surfaces of IF steel shown in Fig.3(a) and Fig.3(b). However, the yield surface shown in Fig.4(b) fits better the experimental yield stresses than the yield surface shown in Fig.4(a).

Table 3. Strictly convex Hill-Gotoh parameter sets for two steel sheets listed in Tables 1 and 2

| A₁ | A₂ | A₃ | B₁ | B₂ | B₃ | B₄ | B₅
(1) | 0.64697 | -1.45715 | 1.02246 | 0.35203 | -0.29938 | 0.39998 | -0.035531 | 0.013201
(2) | 0.98390 | -1.70649 | 1.25349 | 0.016102 | 0.38592 | -0.55365 | 0.56946 | -0.22761
(3) | 0.97174 | -0.77007 | 0.57856 | 0.02826 | -0.17262 | 0.48497 | -0.49282 | 0.23228
(4) | 1.37938 | -0.015936 | -0.42313 | -0.37938 | -0.55396 | 1.94480 | -2.16443 | 1.08642

[picture]

Fig. 3. Predicted strictly convex on-axis biaxial yield surface contours by Hill-Gotoh yield functions (1) and (2) in Table 3 for the IF steel sheet based on SOS-convexity (a) and positive surface curvature (b) constraints [3,10].

Fig. 3. Predicted strictly convex on-axis biaxial yield surface contours by Hill-Gotoh yield functions (1) and (2) in Table 3 for the IF steel sheet based on SOS-convexity (a) and positive surface curvature (b) constraints [3,10].

[picture]

Fig. 4. Predicted strictly convex on-axis biaxial yield surface contours by Hill-Gotoh yield functions (3) and (4) in Table 3 for the packaging steel sheet based on SOS-convexity (a) and positive surface curvature (b) constraints [3,10].

Fig. 4. Predicted strictly convex on-axis biaxial yield surface contours by Hill-Gotoh yield functions (3) and (4) in Table 3 for the packaging steel sheet based on SOS-convexity (a) and positive surface curvature (b) constraints [3,10].

An Application Example of a Sixth-Order Hill-Gotoh Yield Function with17Parameters

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One version of sixth-order non-homogeneous Hill-Gotoh yield functions may have (M1,M2,M3)= (1,1,3), with a total of 17 material parameters (noting that the four additional parameters are nonpolynomial coefficients): ( A1, A2, A3, A4),(B1, B2, B3, B4, B5, B6, B7, B8, B9) and ( C1,C2,C3,C4 ). Using the 17 experimental and auxiliary inputs for DX54 steel sheet as listed in Table 4, one can readily obtain the values of these 17 material parameters via convexity-constrained least square parameter identification [3,10]. Table 5 lists the parameters values for the case of SOS-convexity (the first SOS rows) and for the case of positive surface curvature (the second PSC rows) respectively.

Table 4. The complete set of 17 experimental and auxiliary inputs for DX54 steel sheet [ 3,9 ]

σ₀/σf | r₀ | σ₉₀/σf | r₉₀ | σb/σf | rb | σp0/σf | σp90/σf | σs0/σf
1 | 2.2157 | 0.9898 | 2.4043 | 1.1467 | 0.9507 | 1.2570 | 1.2669 | 0.5408
σ₄₅/σf | r₄₅ | σ₂₂₅/σf | r₂₂₅ | σ₆₇₅/σf | r₆₇₅ | σp45/σf | σs45/σf |
1.0350 | 1.675 | 1.01750 | 1.9454 | 1.0124 | 2.03965 | 1.2293 | 0.5588 |

Using these calibrated material parameters, strictly SOS-convex on-axis and off-axis biaxial yield surface contours predicted by the sixth-order Hill-Gotoh yield function are shown as solid lines in Fig.5(a) and Fig.5(b) respectively for DX54 steel. Similarly, strictly convex on-axis and off-axis biaxial yield surface contours predicted by the sixth-order Hill-Gotoh yield function based on the positive surface curvature constraints are shown as solid lines in Fig.6(a) and Fig.6(b) respectively for DX54 steel. The on-axis yield surface shown in Fig.6(a) fits better the experimental yield stresses than the yield surface shown in Fig.5(a). Little differences are seen however between the two offaxis yield surfaces of DX54 steel sheet shown in Fig.5(b) and Fig.6(b). The corresponding uniaxial tensile yield stresses and plastic strain ratios predicted by the SOS-convex sixth-order Hill-Gotoh yield function are shown in Fig.7(a) and Fig.7(b) respectively.

Table 5. Two complete sets of 17 material parameters of the sixth-order Hill-Gotoh yield function

A₁ | A₂ | A₃ | A₄ | B₁ | B₂ | B₃ | B₄ | B₅
0.87850 | -1.26128 | 0.92056 | 3.13361 | 0.10883 | -0.29336 | 0.59880 | -0.30269 | 0.09327
1.48501 | -1.58715 | 1.13104 | 3.72566 | -0.53685 | 0.98580 | -0.84421 | 0.31568 | -0.12136
B₆ | B₇ | B₈ | B₉ | C₁ | C₂ | C₃ | C₄ |
0.43469 | -0.43791 | 0.38263 | 0.06862 | 0.33553 | -0.17333 | 0.02253 | 0.39824 | (SOS)
-1.93345 | 1.43194 | -0.90728 | -1.81466 | -0.47716 | 0.58038 | 0.17430 | 0.27265 | (PSC)

In comparison with the visual inspection approach, the existence of a negative curvature on a yield surface via numerical computations can be used more effectively and rigorously to detect its nonconvexity. Algebraic curvature formulas for implicit 2D plane curves and 3D surfaces have been summarized by Goodman [16] and for hypersurfaces in general with respect to their Hessian matrix have been discussed by Henrion and Louembet [15]. Non-convex yield surfaces shown in Fig. 1 and Fig. 2 are found indeed to have negative curvatures (e.g., at the uniaxial tension stresses along rolling and transverse direction for the IF steel sheet). On the other hand, the minimum curvature of yield surfaces shown in Fig.3-Fig. 6 is all positive using the numerical algorithm given by Eq. (B.1) of Appendix B in [10]. Their strict convexity is thus firmly established.

[picture]

Fig. 5. Predicted strictly convex (a) on-axis and (b) 45-degree off-axis biaxial yield surface contours by the sixth-order Hill-Gotoh yield function for DX54 steel sheet based on SOS-convexity constraints [3].

Fig. 5. Predicted strictly convex (a) on-axis and (b) 45-degree off-axis biaxial yield surface contours by the sixth-order Hill-Gotoh yield function for DX54 steel sheet based on SOS-convexity constraints [3].

[picture]

Fig. 6. Predicted strictly convex (a) on-axis and (b) 45-degree off-axis biaxial yield surface contours by the sixth-order Hill-Gotoh yield function for DX54 steel sheet based on positive surface curvature constraints [10].

Fig. 6. Predicted strictly convex (a) on-axis and (b) 45-degree off-axis biaxial yield surface contours by the sixth-order Hill-Gotoh yield function for DX54 steel sheet based on positive surface curvature constraints [10].

[picture]

Fig. 7. Predicted (a) uniaxial tensile yield stresses and (b) uniaxial tensile plastic strain ratios by the SOS-convex sixth-order Hill-Gotoh yield function for DX54 steel sheet [3,9]. Solid circles are experimental inputs while open circles are auxiliary inputs.

Fig. 7. Predicted (a) uniaxial tensile yield stresses and (b) uniaxial tensile plastic strain ratios by the SOS-convex sixth-order Hill-Gotoh yield function for DX54 steel sheet [3,9]. Solid circles are experimental inputs while open circles are auxiliary inputs.

Summary

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By extending to non-homogenous polynomials higher than fourth-order and by employing a leastsquare parameter identification method with novel convexity-constraints, the new numerical results of anisotropic yielding and plastic flow behaviors of three representative sheet metals show that the modeling capabilities of Hill-Gotoh theory can be extended to accommodate many more off-axis uniaxial and/or biaxial stress states of sheet metals with expanded convexity domains and improved accuracies.

References

The original version of this paper is available on https://www.scientific.net/SSP.390.189.pdf

R. Hill, A theory of the yielding and plastic flow of anisotropic metals, Proceedings of the Royal Society of London, Series A. Mathematical and Physical Sciences, 193 (1948) 281-297.

M. Gotoh, A theory of plastic anisotropy based on a yield function of fourth order (plane stress state)-I, International Journal of Mechanical Science 19 (1977) 505-12.

W. Tong, A Hill-Gotoh theory of anisotropic sheet metal plasticity, International Journal of Mechanical Science 302 (2025) 110513.

R. Hill, The Mathematical Theory of Plasticity. Clarendon Press, Oxford, 1950.

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, International Journal of Plasticity 19:9 (2003) 1297-1319.

F. Barlat, H. Aretz, J. W. Yoon, M. E. Karabin, J. C. Brem, R. E. Dick, Linear transformation based anisotropic yield function, International Journal of Plasticity 21:5 (2005) 1009-1039.

W. Tong, Algebraic convexity conditions for Gotoh's non-quadratic yield function, ASME Journal of Applied Mechanics 85 (2018) 074501-1

H. Vegter, A.H. van den Boogaard, A plane stress yield function for anisotropic sheet material by interpolation of biaxial stress states, International Journal of Plasticity 22 (2006) 557-580.

H. Vegter, M. Abspoel, J. Mulder, A plane stress yield surface using Bézier curve interpolation in two directions, IOP Conference Series: Materials Science and Engineering 651 (2019) 012052.

W. Tong, An anisotropic plasticity theory of sheet metals based on a polynomial of Hill's 1948 yield function, International Journal of Solids and Structures 325 (2026) 113726.

S. Boyd, L. Vandenberghe, Convex Optimization. Cambridge University Press, Cambridge, UK, 2004.

V.A. Toponogov, Differential Geometry of Curves and Surfaces. Birkhauser (Boston), 2006.

G. Reißig, Convexity of reachable sets of nonlinear ordinary differential equations, Autom. Remote Control 68:9 (2007) 1527-1543.

J.W. Helton, J. Nie, Semidefinite representation of convex sets, Math. Program. Ser. A 122 (2010) 21-64.

D. Henrion, C. Louembet, Convex inner approximations of nonconvex semial-gebraic sets applied to fixed-order controller design, International Journal of Control 85:8 (2012) 10831092.

R. Goldman, Curvature formulas for implicit curves and surfaces, Comput. Aided Geom. Design 22 (2005) 632-658.

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Solid State Phenomena, ISSN: 1662-9779, Vol. 390, pp 197-205

doi: 10.4028/p-AoUE30

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

Submitted:
2026-01-21

Revised:
2026-02-25

Accepted:
2026-02-25

Online:
2026-04-17

A Compact Micromechanical Model for the Elastic-Viscoplastic Deformability of PLA-Hemp Biocomposites

The original version of this paper is available on https://www.scientific.net/SSP.390.197.pdf

NIANG Ndeye 1,a, HOLOPAINEN Sami 2, b, BARRIERE Thierry 1,c*, PLACET Vincent 1, d and GABRION Xavier 1,e

1 Marie and Louis Pasteur University, SUPMICROTECH-ENSMM, CNRS, Institute FEMTO-ST, F-25000 Besancon, France

2 Tampere University, Department of Civil Engineering, FI-33014 Tampere, Finland

a  ndeye.niang@femto-st.fr, b  sami.holopainen@tuni.fi, c *thierry.barriere@univ-fcomte.fr, d vincent.placet@umlp.fr, e xavier.gabrion@univ-fcomte.fr

Keywords: Biocomposite, Viscoplasticity, Machine learning, 3D-printing, Forming

Abstract

Despite the growing use of biopolymers in automotive, packaging and structural applications, predictive modelling of their elastic-viscoplastic deformation remains limited. In this work, a micromechanically based constitutive model is proposed to describe the micro-tomacroscopic behaviour of a semi-crystalline PLA matrix reinforced with short hemp fibers. The formulation relies on a multiplicative split of the deformation gradient into elastic and viscoelasticplastic parts, with elasticity governed by fiber and crystalline phases and time-dependent deformation localized in the amorphous phase. High fiber content and strong fiber-matrix bonding enable the suppression of lattice crystalline anisotropy, leading to a compact model with a reduced number of internal variables. The model is calibrated and validated using uniaxial tensile tests on pellet-extrusion 3D-printed specimens with controlled porosity and plasticiser content, and reproduces nonlinear loading, unloading, creep and stress relaxation. In a second step, synthetic data generated by the constitutive model are used to train surrogate machine-learning models, which are discussed as a perspective for accelerating long-term simulations and parametric studies in forming applications.

Introduction

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Natural short-fiber reinforced (NSFR) polymer composites represent a promising class of sustainable materials that reduce dependence on petroleum-derived plastics across forming-intensive applications, including automotive structures, packaging, and biomedical devices [1]. Polylactic acid (PLA) matrices reinforced with plant fibers such as hemp offer tunable mechanical properties through systematic variation of fiber content, matrix crystallinity, and processing-induced porosity. However, the inherent viscoelastic-plastic deformation characteristics of these materials manifested through time-dependent creep, strain rate sensitivity, and irrecoverable residual strains present significant challenges for reliable process simulation and design optimisation. Traditional experimental characterisation proves both costly and time-intensive, particularly when exploring expansive parameter spaces involving multiple fiber types, plasticiser formulations, and manufacturing conditions.
Previous modelling efforts for short-fiber biocomposites have mainly relied on mean-field homogenization or full-field finite element analyses of idealized microstructures. Orientation-averaging approaches for natural fiber composites capture stiffness reasonably well but remain computationally expensive and poorly adapted to large viscoplastic strains [2-4]. Mean-field schemes for randomly distributed short fibers also require complex fitting to link micro-scale parameters to macroscopic behaviour. In contrast, phenomenological viscoelastic or viscoplastic laws are computationally efficient but often disconnected from microstructural quantities such as fiber content, degree of crystallinity and porosity.
In this work, a compact micromechanical constitutive model is developed for PLA-hemp biocomposites that bridges micro- and macro-scales while remaining numerically efficient [1]. The
key idea is to exploit the high stiffness of the short hemp fibers and their strong bonding to the semi-crystalline matrix to suppress explicit lattice-scale anisotropy, and to represent the fibers and crystalline phase by Hencky-type elastic laws while concentrating viscoelastic-plastic flow in the amorphous phase. The model is calibrated from a limited experimental dataset and validated under uniaxial tension, including creep and relaxation. As a perspective, the model is then used as a physics-based data generator to train machine-learning surrogates for rapid, long-term predictions.

Materials Processing and Experimental Characterisation

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Materials Processing

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Commercial PLA (PLI 005 grade, Natureplast) served as the matrix material, compounded with short hemp fibers (average length 160μ m, obtained through hammer milling and defibration) at 20wt% loading and polyethylene glycol plasticizer at 5wt% respectively PEG 1500 and PEG 20000. Twin-screw extrusion processing occurred at 190∘C and 30 rpm rotation speed to ensure homogeneous dispersion while minimizing thermal degradation. The resulting pellets underwent drying treatment ( 60∘C for 3 hours) prior to P-MEX 3D-printing of ISO 527-4 Type 1B dogbone tensile specimens as shown in Fig.1. Optimised printing parameters included nozzle temperature of 190∘C, layer thickness of 0.4 mm , and print speed of 40 mm/s, balancing extrudability, surface quality, and microstructural development. These conditions produced controlled porosity levels of 89% that proved essential for achieving ductile deformation behavior at ambient temperature, overcoming the inherent brittleness of pure PLA (ultimate strain <1% ) [5,6].

[picture]

Fig. 1. PLA-hemp composite characterization. Left: Pellet-based 3D-printing (P-MEX) and laser cutting of ISO 527-4 specimens. Center top: X-ray tomography showing layer uniformity and bead continuity. Center bottom: Tensile testing setup and stress-strain derivation. Right: SEM micrographs of fiber pull-out and imprints used for porosity modeling., adapted from [1].

Fig. 1. PLA-hemp composite characterization. Left: Pellet-based 3D-printing (P-MEX) and laser cutting of ISO 527-4 specimens. Center top: X-ray tomography showing layer uniformity and bead continuity. Center bottom: Tensile testing setup and stress-strain derivation. Right: SEM micrographs of fiber pull-out and imprints used for porosity modeling., adapted from [1].

Microstructural Analysis

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Differential scanning calorimetry (DSC) measurements revealed matrix crystallinity of 52% for pure PLA, reduced to 38% in the fiber-reinforced composite due to heterogeneous nucleation interference from the cellulosic hemp reinforcement, as reported in Table 1. High-resolution X-ray computed tomography (CT) combined with scanning electron microscopy (SEM) quantified porosity
distribution, confirming 8−9% void content primarily manifested as localized inter-bead voids rather than intra-bead defects (Fig.1). Although CT data showed a bimodal fiber orientation peaking at 20∘, the composite is modeled as macroscopically isotropic. This is justified by the significant density of fibers distributed between 30∘ and 90∘, which compared to strictly anisotropic histograms suggests a degree of diversity consistent with virtually isotropic behavior[1,7,8]. This is confirmed by the experimental data, where the Poisson's ratio varied only marginally ( 0.32−0.44 ) between loading directions. These minimal differences justify using a single average value ( v=0.37 ), as the global response is dominated by the matrix and a near-random fiber distribution. Consequently, this compact strategy avoids complex orientation-dependent stress homogenization while accurately capturing the primary viscoelastic-plastic phenomena.

Table 1. Key microstructural and mechanical properties of PLA-hemp composites

Composition | Porosity [%] | DC [%] | E direction | [GPa] | Print | σu [MPa] | εu [%]
Pure PLA | - | 52 | 3.2 |  |  | Brittle failure | <1
PLA – Hemp - « PEG20k » 5wt% (optimal) | 8-9 | 38 | 3.6 |  |  | 40 | ~3

Mechanical Testing

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Uniaxial tensile testing followed ISO 527-4 protocol using an MTS Criterion system equipped with a 5 kN load cell and extensometer (gauge length 25 mm ). Displacement-controlled monotonic loading at engineering strain rate 0.001 s−1 captured characteristic nonlinear stress-strain response: initial linear elastic regime to approximately 0.25% strain followed by viscoplastic upturn culminating in ductile failure near 3% engineering strain. Additional tests spanning strain rates 10−3 to 102s−1 confirmed pronounced rate sensitivity, while constant-load creep experiments at stress levels below 35% of ultimate strength demonstrated significant time-dependent deformation accumulation. The plasticizer addition tripled failure ductility relative to unreinforced PLA while controlled porosity levels proved essential reductions below 4% porosity shifted response toward elastic-dominated behavior.

Micromechanical Constitutive Framework

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The constitutive description is based on a multiplicative split of the deformation gradient into elastic and viscoelastic-plastic parts,

𝐅=𝐅e𝐅vep, with det(𝐅vep)=1,(1)

so that elastic and time-dependent mechanisms are clearly separated. Short hemp fibers (average length 160μ m ) are assumed to be homogeneously and randomly distributed, strongly bonded to the PLA matrix, with low intra-bead porosity ( <10% ); as a consequence, the initial response is considered macroscopically isotropic and governed by fiber-dominated elasticity. In what follows, the superscripts f, c, a denote fiber, crystalline and amorphous phases, respectively.
The rheological structure corresponds to a three-dimensional extension of the Maxwell model, previously used for glassy polymers and semi-crystalline thermoplastics, here adapted to a three-phase composite (fibers, crystalline regions and amorphous matrix) [9-11].

[picture]

Fig.2. (A) Schematic microstructure of the PLA-hemp composite with amorphous, crystalline and fiber phases contributing respectively to viscoelastic-plastic and elastic deformation. (B) Equivalent Maxwell-type rheological model where the amorphous phase is modeled by a nonlinear dashpot in series with elastic springs representing amorphous, crystalline and fiber phases, adapted from [1].

Fig.2. (A) Schematic microstructure of the PLA-hemp composite with amorphous, crystalline and fiber phases contributing respectively to viscoelastic-plastic and elastic deformation. (B) Equivalent Maxwell-type rheological model where the amorphous phase is modeled by a nonlinear dashpot in series with elastic springs representing amorphous, crystalline and fiber phases, adapted from [1].

Elastic response of the phases

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The elastic behaviour of each phase is described using a Hencky-type law. For the fiber phase, the macroscopic stress reads

σf=ξLe,f:ln𝐯e=ξEf/(1+vf)ln𝐯e+ξEf/[3(1−2vf)]tr(ln𝐯e)i,(2)

Where ξ is the fiber content, and Ef,vf are the fiber Young's modulus and Poisson's ratio, and Le,f is the corresponding elastic stiffness tensor [9]. The crystalline regions are treated in the same way,

σc=(1−ξ)Le,c:ln𝐯e=(1−ξ)Ec/(1+vc)ln𝐯e+(1−ξ)Ec/[3(1−2vc)]tr(ln𝐯e)i,(3)

with Ec,νc the crystalline modulus and Poisson's ratio. The moduli of the amorphous phase and fibers are obtained from simple mixing relations,

(1−ζ)Em=χEc+(1−χ)(1−ζ)Ea,E=ξEf+(1−ξ)(1−ζ)Em(4,5)

linking the measured composite modulus E , matrix modulus 𝐄𝐦, porosity ζ and crystallinity χ. The amorphous phase is also modeled as Hencky-elastic,

σa=(1−ξ)Le,a:ln𝐯e=(1−ξ)Ea/(1+va)lnve+(1−ξ)Ea/[3(1−2va)]tr(ln𝐯e)𝐢(6)

and the total stress is the sum of the phase contribution.

Viscoelastic-plastic deformation

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Time-dependent deformation is assumed to occur only in the amorphous phase. For brevity, only uniaxial-type loading is discussed here, but the model is fully three-dimensional. The viscoelastic-plastic rate of deformation is defined as

𝐃―vep=(γ˙vep2τ−1η)τdev,(7)

with τ=1/2tr(σdev)2 and the plastic multiplier

γ˙vep =γ˙0a(τ sa−1/3βI1)1/m≥0(8)

where γ˙0a, m (the limit m→0 corresponds to the rate-independent limit), and β denote the material parameters, sa is an internal state variable (representing the molecular resistance to plastic shear flow
[12] and I1=tr(σ) is the first invariant of the stress indicating pressure: plastic deformation is suppressed under compression, that is, micro-cracks are opened in tension and partially closed in compression. This formulation, corresponding to a three-dimensional Maxwell-type rheology, allows one to capture nonlinear monotonic loading, stress relaxation, creep under low stresses and residual strains upon unloading with a compact set of parameters.

Parameter Identification and Model Validation

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The parameters of the elastic part of the model were obtained from the initial slopes of the experimental stress-strain curves of the composite, the PLA matrix and an amorphous PLA reference. The Young's modulus of the composite in the printing direction was extracted from the linear regime up to a strain of 0.25%, and the corresponding values for the matrix and amorphous phase were obtained from printed and extruded specimens, respectively. The moduli of the crystalline and fiber phases were then computed using the mixing relations in Eqs. (4,5), while a common Poisson's ratio was adopted based on the measured near-isotropic response.
The viscoelastic-plastic parameters shown in Table 2 were identified by fitting the monotonic tensile curves at room temperature, with σ0 chosen to match the onset of nonlinearity and the remaining parameters tuned to reproduce the shape of the loading and unloading branches. The calibrated model accurately captures the measured stress-strain response up to failure in both the printing and transverse directions, with errors in elastic modulus and ultimate strength within the experimental scatter. This agreement justifies the use of the compact phase description and the localisation of time-dependent deformation in the amorphous phase.

Table 2. Calibrated constitutive parameters [1].

Symbol | Description | Value [Unit]
E | Composite Young's modulus | 3.6 GPa
Em | PLA matrix modulus | 3.2 GPa
Ea | Amorphous PLA modulus | 3.0 GPa
ν | Poisson's ratio (all phases) | 0.37
β | Pressure sensitivity parameter | 0.2
η | Viscosity of the composite | 4.0 × 10⁴ MPa·s
m | Rate-sensitivity exponent | 0.11
γ̇₀a | Reference shear strain rate (prefactor in γ̇vep).) | 4.0 × 10⁻³ s⁻¹
sa | Shear resistance to plastic flow | 26 MPa

Results and Forming Process Implications

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The primary outcome of this study is the validation of a compact micromechanical constitutive law for natural short-fiber reinforced PLA that reproduces elastic-viscoplastic responses with a limited number of parameters. The agreement obtained for stress-strain curves, unloading, creep and stress relaxation indicates that the simplifying assumptions (random short fibers, suppressed lattice anisotropy, viscoelastic-plastic flow confined to the amorphous phase) are sufficient for the forming-relevant strain range considered here ( ε<10% ).

Model validation confirms accurate capture of the measured mechanical properties: the composite elastic modulus is approximately E=3.6GPa, the ultimate tensile strength about 40 MPa and the failure strain close to 3%, with faithful reproduction of unloading hysteresis and stress-relaxation transients. Rate-dependent predictions align with experiments across several orders of magnitude in strain rate, while creep accumulation at 20% of the ultimate strength matches experimental observations over extended hold times. The framework generalises robustly to engineering strains below 10%, which encompasses typical conditions in cold-forming and incremental forming of such biocomposites.
Fig. 3 illustrates predicted creep-strain evolution at 20% of the ultimate stress for different porosity levels, highlighting an optimal ductility window at 8−9% controlled porosity. Porosity reductions to 4% shift the response to elastic-dominated.

[picture]

Fig.3. Mechanical response of PLA-hemp composites. (A) Tensile stress-strain curves for PLA-20H and plasticized variants (P1, P2), with illustration of elastic, viscoelastic, and viscoplastic strain components. (B) Predicted vs. experimental creep compliance ( ε/σ0 ) over one hour at σ=13σUTS  (CD) Predicted stress-strain and log-time relaxation curves showing viscoelastic stress decay, adapted from [1].

Fig.3. Mechanical response of PLA-hemp composites. (A) Tensile stress-strain curves for PLA-20H and plasticized variants (P1, P2), with illustration of elastic, viscoelastic, and viscoplastic strain components. (B) Predicted vs. experimental creep compliance ( ε / σ 0 ) over one hour at σ = 1 3 σ UTS (CD) Predicted stress-strain and log-time relaxation curves showing viscoelastic stress decay, adapted from [1].

From a manufacturing perspective, pellet-based material extrusion (P-MEX) processing enables precise porosity control through layer thickness, print speed and nozzle-temperature settings, providing a practical pathway to tailor ductility and stiffness for near-net-shape component fabrication. When combined with the constitutive model, this offers a physics-based link between process parameters, microstructure and forming performance. The additional use of machine-learning surrogates, trained on synthetic data generated by the model, facilitates rapid virtual screening of compositions and printing conditions and can substantially reduce the need for physical prototype testing in future optimisation studies.

Discussion and Future Developments

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The present work demonstrates that a compact micromechanical constitutive model can describe the elastic-viscoplastic behaviour of PLA-hemp biocomposites over the strain and time scales relevant for forming. The good agreement obtained for stress-strain curves, unloading, creep and relaxation confirms that the underlying assumptions, random short fibers, strong fiber-matrix bonding, suppressed lattice-scale anisotropy and viscoelastic-plastic flow confined to the amorphous phase are sufficient within the investigated ranges of global strain ( ε<10% ) and porosity ( ≈8−9% ). A small set of parameters identified from uniaxial tensile tests reproduces the main features of the response, including rate effects and residual strains after unloading.
A key advantage of the modelling strategy is that the phase moduli are directly linked to measurable or controllable microstructural quantities such as fiber content, degree of crystallinity and porosity, rather than being purely phenomenological. This provides a physically interpretable connection between processing, microstructure and mechanical behaviour. In particular, the calibrated model captures the experimentally observed trends that increasing porosity enhances deformability but lowers stiffness, while higher fiber content stiffens the composite at the expense of ductility, which is essential for defining forming windows where sufficient plasticity is required without losing the stiffness advantage from reinforcement.
The current formulation is nevertheless subject to several limitations. It is calibrated and validated essentially under uniaxial tension, and the assumption of macroscopic isotropy relies on the fiber orientation distribution induced by pellet-extrusion 3D-printing. Strongly aligned materials or multiaxial forming paths will require an explicit orthotropic extension of the elastic stiffness and possibly of the viscoplastic flow rule. Furthermore, the model is isothermal and does not account for temperature-dependent phenomena such as glass transition, recrystallisation or thermal softening, which can influence warm-forming and high-rate applications. Fatigue and cyclic degradation are not considered either; only monotonic and simple creep/relaxation histories are addressed.
These limitations point to natural directions for future work. On the constitutive side, extensions include introducing temperature dependence into the viscoelastic-plastic part, adding damage or porosity-evolution variables to capture micro-void growth, and incorporating cyclic plasticity mechanisms to predict fatigue life under repeated loading. A more detailed treatment of fiber orientation, for example through orientation tensors or evolution equations, would also broaden the range of forming processes that can be simulated.
As a complementary direction, the validated constitutive model was used as a physics-based data generator for machine-learning approaches [1,13]. By systematically varying porosity ( 0−15% ), fiber content ( 0−30wt% ) and degree of crystallinity ( 10−50% ), approximately 104 synthetic stress-strain and creep curves were produced and used to train Support Vector Regression (SVR) surrogates with radial basis function kernels. The inputs consisted of strain, time and microstructural parameters, and the outputs were the corresponding stresses. The trained surrogates reproduced the constitutive model predictions with coefficients of determination R2 greater than 0.99 and mean relative errors below 2% on a held-out test set, while reducing the computation time for 1-year creep simulations from the order of 10 h to about 0.1 s . These results are not required for the validation of the constitutive law itself, but they demonstrate how the compact model can underpin future digital-twin and optimisation studies by providing high-quality training data for fast metamodels, as illustrated in Fig. 4.

[picture]

Fig. 4. Machine-learning surrogate based on the constitutive model: left, schematic workflow from experimental and model data to trained SVR; middle, comparison between the SVR surrogate and the constitutive model for a tensile loading-unloading cycle; right, corresponding stress-relaxation response versus logt. The surrogate closely reproduces the physics-based model while being several orders of magnitude faster, adapted from [1].

Fig. 4. Machine-learning surrogate based on the constitutive model: left, schematic workflow from experimental and model data to trained SVR; middle, comparison between the SVR surrogate and the constitutive model for a tensile loading-unloading cycle; right, corresponding stress-relaxation response versus log t . The surrogate closely reproduces the physics-based model while being several orders of magnitude faster, adapted from [1].

Summary

The original version of this paper is available on https://www.scientific.net/SSP.390.197.pdf

A compact micromechanical constitutive model has been developed and validated for predicting the elastic-viscoplastic deformability of PLA-hemp biocomposites over forming-relevant strains and times. The framework links phase properties directly to fiber content, degree of crystallinity and porosity, and reproduces measured stress-strain curves, unloading, creep and stress relaxation for an optimal composition combining 20wt% hemp reinforcement, 5wt% PEG plasticisation and 8−9% engineered porosity, which provides thermoplastic-like failure ductility while maintaining competitive stiffness. As a perspective, synthetic datasets generated by the constitutive model have been used to train accurate machine-learning surrogates, demonstrating that substantial speed-ups for long-term simulations can be achieved and opening the way to digital-twin applications for industrial-scale materials optimisation and process design.

References

The original version of this paper is available on https://www.scientific.net/SSP.390.197.pdf

Niang N, Barriere T, Gabrion X, Holopainen S, Placet V. Predicting the deformability of natural short-fiber reinforced polymer composites through combined constitutive mathematical and AIbased modeling approaches. Composites Science and Technology 2025;273:111353. https://doi.org/10.1016/j.compscitech.2025.111353.

Modniks J, Andersons J. Modeling the non-linear deformation of a short-flax-fiber-reinforced polymer composite by orientation averaging. Composites Part B: Engineering 2013;54:188-93. .

Qi L, Tian W, Zhou J. Numerical evaluation of effective elastic properties of composites reinforced by spatially randomly distributed short fibers with certain aspect ratio. Composite Structures 2015;131:843-51. .

Hessman PA, Welschinger F, Hornberger K, Böhlke T. On mean field homogenization schemes for short fiber reinforced composites: Unified formulation, application and benchmark. International Journal of Solids and Structures 2021;230-231:111141. .

R.A. Ilyas, S.M. Sapuan, M. M. Harussani, M.Y.A.Y. Hakimi, M.Z.M. Haziq, M.S.N. Atikah, et al. Polylactic Acid (PLA) Biocomposite: Processing, Additive Manufacturing and Advanced Applications. Polymers 2021. .

Le Duigou, Masahito Ueda, David Correa, Ryosuke Matsuzaki, Mickaël Castro. A review of 3D and 4D printing of natural fibre biocomposites. Materials & Design 2020. .

Mortazavian S, Fatemi A. Effects of fiber orientation and anisotropy on tensile strength and elastic modulus of short fiber reinforced polymer composites. Composites Part B: Engineering 2015; 72: 116-29. .

Barriere T, Placet V, Holopainen S, Niang N. Predicting the Deformation Behavior of Biocomposites using Combined Mathematical and AI-Based Modeling. Procedia Structural Integrity 2026;80:212-8. .

Belytschko T, Liu WK, Moran B. Nonlinear Finite Elements for Continua and Structures. Chichester: John Wiley & Sons, Ltd; 2000.

Holopainen S, Barriere T. Modeling of mechanical behavior of amorphous solids undergoing fatigue loadings, with application to polymers. Computers & Structures 2018;199:57-73. .

Barriere T, Gabrion X, Holopainen S. A compact constitutive model to describe the viscoelastic-plastic behaviour of glassy polymers: Comparison with monotonic and cyclic experiments and state-of-the-art models. International Journal of Plasticity 2019;122:31-48. .

Anand L, Ames NM. On modeling the micro-indentation response of an amorphous polymer. International Journal of Plasticity 2006; 22: 1123-70. .

Laycock BG, Chan CM, Halley PJ. A review of computational approaches used in the modelling, design, and manufacturing of biodegradable and biobased polymers. Progress in Polymer Science 2024;157:101874. .

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The original version of this eBook is available on https://www.scientific.net/book/material-behaviour-modelling/978-3-0364-1987-9

Solid State Phenomena, ISSN: 1662-9779, Vol. 390, pp 207-220

doi: 10.4028/p-NzE5Rj

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

Submitted:
2026-02-16

Revised:
2026-02-23

Accepted:
2026-02-24

Online:
2026-04-17

Modeling the Influence of Material Variabilities on the Forming and Failure Behavior of PP/PE Cores in Metal-Polymer-Metal Sandwich Sheets

The original version of this paper is available on https://www.scientific.net/SSP.390.207.pdf

Johannes Pusicha 1,a* and Johannes Buhl 1, b

1 Institute of Metallurgy, Clausthal University of Technology, 38678 Clausthal-Zellerfeld, Germany

a*johannes.pusicha@tu-clausthal.de, bjohannes.buhl@tu-clausthal.de

*corresponding author

Keywords: sandwich composites, material characterization, material inhomogeneity.

Abstract

Metal-polymer-metal (mpm) sandwich sheets are considered a lightweight alternative to conventional steel sheets in the automotive industry. First studies have experimentally and numerically demonstrated the use of mpm-sandwiches in automotive crash structures. However, the sandwich forming process can compromise the crash performance through pre-damage. Finite element simulations could help predict the forming process and its limits. Current simulation approaches, however, consider neither strain rate effects and failure of the polymer and the adhesive, respectively, nor do they study the influence of inevitable material variabilities on the forming and failure behavior. In this work, a detailed finite element model of the core materials is developed. An approach for the determination of the stress strain rate dependency is proposed by evaluating the local strains on the surface, which allows capturing the main material behavior with little effort. Additional specialized tensile tests and lap-shear tests provide information about the failure of the polymer and the adhesive, respectively. Validation of the core materials model is achieved by comparing the cover layer displacements of swivel bending specimens in experiment and simulation. The influence of material variabilities on the forming and failure behavior is studied in a full factorial material parameter sweep of simulated lap-shear tests, and the applicability of the model to the simulation of bending processes is demonstrated. The results prove the applicability of the proposed material characterization methods, while the parameter study and the bending simulations show how the model can be used to predict a sandwich's formability and failure modes.

Introduction

The original version of this paper is available on https://www.scientific.net/SSP.390.207.pdf

To reduce the energy consumption of automobiles, research interest in weight-saving materials and designs for the automotive industry continues to be high. Metal-polymer-metal (mpm) sandwich sheets are considered a lightweight alternative to conventional steel sheets with possible uses in the vehicle chassis [1] or skin [2]. Two metal sheets (cover sheets) are joined with a polymer foil by using an epoxy resin adhesive, which increases the stiffness of the composite compared to the individual materials by only adding minimal weight.

First studies have experimentally and numerically evaluated the use of mpm sandwiches in automotive crash structures [3], such as double- or top-hat crash boxes [4]. As in conventional crash structures, the energy is absorbed by performing plastic deformation through progressive fold formation [5]. The energy absorption of these structures, therefore, depends on the material's stiffness and strength. However, due to the composite nature of mpm sandwich sheets, the interface properties of the sandwich have a significant influence on the crash performance, as has been investigated for the adhesion strength [6], and as can be concluded for the polymer core from characterizing experiments [7]. However, sandwich sheets have to undergo a series of forming steps before their use as a crash absorber, which can lead to undesirable pre-damage. FEM (finite element method) simulations of the sandwich forming process could help to predict the influence of the manufacturing process on the final crash performance. Detailed modelling of the core materials' (adhesive and polymer foil) forming and failure behavior is therefore needed, which has not yet been incorporated in current simulation approaches [7]. Additionally, to predict formability limits of sandwich sheets,
the influence of inevitable material and adhesion variabilities on the forming and failure behavior needs to be studied.

Therefore, in the following, a detailed FE model considering strain rate effects and failure of the core materials of mpm sandwich sheets is developed. In addition, since significant variation of bonding strength and of failure modes has been found in prior experiments, the influence of material variations on the forming and failure behavior (esp. the fracture pattern) is numerically studied. Finally, the capabilities of the model are demonstrated on a swivel bending process to form hat profiles.

Sandwich Manufacturing

The original version of this paper is available on https://www.scientific.net/SSP.390.207.pdf

To conduct material experiments on sandwich sheets, the utilized sheets are self-produced with lab-scale equipment by roll-bonding [8] with a 1.0242 (S250) zinc-coated metal sheet, the epoxy resin adhesive Köratac 201 FL, and a PP/PE copolymer foil, as described in [9]. Figure 1 gives a schematic overview of the process. First, the metal and polymer surfaces are cleaned with acetone to remove grease and other impurities affecting the bonding quality. Then, the epoxy adhesive is applied to the metal sheet with the help of a spiral scraper to distribute the adhesive evenly. Next, the adhesive is thermally activated for 3 min at 230∘C while the polymer is heated up to 120∘C. Finally, the sheet is bonded to the polymer foil by rolling the composite in a rolling mill. By rolling the metal sheet on the polymer, the bonding is continually created when the composite enters the rolling mill. This continuous process (instead of an immediate full contact) helps to avoid air inclusions and is suitable for industrial coil processing chains.

[picture]

Fig. 1. Schematic roll-bonding process to manufacture metal-polymer-metal sandwich sheets.

Fig. 1. Schematic roll-bonding process to manufacture metal-polymer-metal sandwich sheets.

Material Modeling

The original version of this paper is available on https://www.scientific.net/SSP.390.207.pdf

To develop a material model of the core materials reflecting their forming and failure behavior, standard-size and short-size tensile tests on the polymer foil and lap-shear tests on full metal-polymer sandwiches are conducted.

To gain flow curves of the utilized steel, standard tensile tests according to DIN EN ISO 6892-1 are performed on ten specimens, of which the resulting minimal, mean, and maximum flow curves are shown in Figure 2. The examined steel batch shows a significant variability in the flow curve, which is especially pronounced in the beginning of the plastic deformation up until ca. ε=6%. Additionally, it has to be noted that already the small temperature changes on the specimen's surface due to the heat generated by the plastic deformation can significantly influence the resulting flow curve [10]. However, since temperature effects are not studied in this work, the sandwich model is assumed to experience the same thermal conditions as the tensile test. The measured flow curve is applicable to the model because the flow curve will then incorporate the same temperature effects as occur in the forming process examined later. Similar thermal conditions can be assumed for forming processes with full-facial tool contact (e.g., swivel bending) since the generated heat will be dissipated
through the tool contact. The temperature rise in the sandwich sheet will therefore not be higher than in the tensile test.

For the polymer foil, a tensile test is conducted on a 25×180 mm strip specimen according to DIN EN ISO 527-3. Local strains (true strain) and strain rates are measured on the polymer foil in longitudinal φyφ˙y and transversal φx,φ˙x direction via digital image correlation with GOM Correlate 2022 of a pre-applied statistical gray scale pattern (see Figure 3). The strains are evaluated on 16 horizontal sections. For each section, the arithmetic mean of the strains is calculated. With regard to thermal effects, the same considerations apply to the polymer as to the steel [11]. When used in sandwich sheets, heat generated through plastic work will be immediately dissipated through the high thermal conductivity of the metal cover layer. Therefore, no high temperature change is expected in the studied quasi-static forming applications.

[picture]

Fig. 2. Mean, minimum, and maximum results of ten tensile tests on the examined steel batch of 1.0242 .

Fig. 2. Mean, minimum, and maximum results of ten tensile tests on the examined steel batch of 1.0242 .

[picture]

Fig. 3. Digital Image Correlation on the polymer foil and evaluation of the local strains at horizontal sections.

Fig. 3. Digital Image Correlation on the polymer foil and evaluation of the local strains at horizontal sections.

To derive the strain rate dependent plastic behavior of the polymer foil, first, Poisson's ratio for plastic deformation in the cross-section is calculated according to Equation 1 from the ratio of longitudinal to transversal strain.

vy=−φyφx(1)

Assuming isotropic plastic behavior, true stresses are then derived from the force measurement by dividing by the cross-section

Axz=Axz,0e−2vyφy,(2)

where Axz,0 is the initial cross-section of the strip-specimen:

σy=FyAxz(3)

The resulting flow surface in dependence on strain and strain rate is shown in Figure 4. From this flow surface, iso-strain flow curves (black lines in Figure 4) are derived from points of equal strain on the flow surface. The resulting iso-strain curves are depicted in Figure 5. The strongest strain rate
dependency is observed at low strain levels. At higher strain levels, the apparent strain rate dependency decreases.

To model the strain rate dependency in the examined load range, Cowper-Symonds's model (cf. Equation 4) is calibrated to the minimal iso-strain curve of the flow surface (cf. Figure 5a, highlighted in green).

σ=σstat (1+φ˙C)P(4)

While a decreasing strain rate dependency at higher strains is not expected in typical material laws, a plausible explanation for the observed decrease is local temperature rise due to plastic work, which counteracts the strain rate hardening. This effect is expected to be strongest in regions of high deformation and, therefore, high strain.

Therefore, the minimal iso-strain curve is chosen for the calibration of Cowper-Symonds' model, since it reflects best the pure strain rate dependency (without temperature influence) and since it covers the broadest strain rate region. Even though this may lead to an overestimated strain rate hardening at high strain levels (when temperature weakening is not considered), this approach may be used here, since the model is only used for quasi-static forming applications with limited heat generation, easily dissipated through the metal cover layers. Furthermore, valid predictions of the model are limited to applications within the examined strain rates (from ca. 10−6s−1 to 5·10−6s−1 in this example).

[picture]

Fig. 4. Flow surface over strain and strain rate derived from the experiment. Some iso-strain curves are shown as black lines.

Fig. 4. Flow surface over strain and strain rate derived from the experiment. Some iso-strain curves are shown as black lines.

[picture]

Fig. 5. Iso-strain flow curves of the polymer foil before (a) and after (b) strain rate compensation. Compensation was done with Cowper-Symonds-Model on the minimal iso-strain curve (highlighted in green).

Fig. 5. Iso-strain flow curves of the polymer foil before (a) and after (b) strain rate compensation. Compensation was done with Cowper-Symonds-Model on the minimal iso-strain curve (highlighted in green).

Due to the strong work-hardening effect of the polymer foil, plastically deformed areas of the polymer get stabilized, and the necking wanders over the specimen. As a result, a "shoulder" formation is observed on the specimen, and no material failure occurs in the standard-size tensile test.

Therefore, to model material failure, short tensile tests are used. Slits of 5 mm are milled into the metal layer of half-sandwich specimens (metal-polymer sandwich, missing the second metal layer) to free the polymer, which ensures rupture happens in the test while providing enough undeformed area to track the elongation with a video system (cf. Figure 6, where deformation only takes place between the two red lines).

[picture]

Fig. 6. Short tensile test specimen. Deformation only takes place between the two red bars. The specimen elongation is tracked with a video system on the undeformed gray pattern.

Fig. 6. Short tensile test specimen. Deformation only takes place between the two red bars. The specimen elongation is tracked with a video system on the undeformed gray pattern.

Material damage and resulting failure are accounted for in the material model with a damage function D(φpl) that weakens the flow stress according to eq. 5 and ultimately leads to rupture for D=1.

σeff=σ(1−D(φpl))(5)

As a damage evolution function, a customizable two-parameter exponential function is chosen with φfail , the plastic strain at rupture, and the parameter n, which allows adjusting the evolution from an early damage logarithmic ( n<0.73 ) to an exponential increasing characteristic ( n>0.73 ) with n≈0.73 resembling a linear development.

D(φpl)=(eφpl−1eφfail−1)n(6)

Values for φfail  and n are obtained by numerical calibration with LS-OPT of an FE simulation to the experimental data until sufficient adherence is achieved. An example of resulting material parameters that are used in the following to model the polymer foil in LS-DYNA with the MAT_SAMP_1 model is listed in Table 3 in the appendix. It has to be noted that due to the single-specimen approach in this work, the provided data does not represent a statistically proven dataset. Therefore, quantification of the scatter in the material itself and possible error due to the characterization method cannot be given.
The focus of this work is to present a characterization concept. However, since the experimental setup relies on standard, widely validated equipment, significant deviations of the results will mainly arise from the specimen itself. Comparison with literature data on the same polymer foil shows that the derived flow curve lies in the same order of magnitude [4]. The principal mechanical behavior of the polymer foil is captured well with the proposed model.

Modeling of the adhesive is based on the results of lap-shear tests [12] of metal-metal sandwich specimens with a 5 mm overlap and a total specimen length of 215 mm . To model the adhesive, LS-

DYNA's MAT_COHESIVE_MIXED_MODE model is used, which uses a bilinear traction-separation law for shear (tangential) and tensile (normal) load [13], [14]. The corresponding material parameters, the peak tractions S,T from which delamination starts, and the energy release rates GIC, GIIC, which describe the total area under the traction-separation curve, are again calibrated numerically with LS-OPT. The resulting parameters are listed in Table 4 in the appendix.

Numerically finding good estimates for the stiffness E was found to be difficult. However, since a very stiff force displacement response was observed in the experimental lap-shear test, a stiffness similar to that of the steel was chosen. The stiffness is defined as the stress per unit length and models the elastic stress response to a length change of the cohesive element. Therefore, it is independent of the element's thickness. However, the peak tractions and the energy release rates depend on the element size, and adaptation to different meshes would be needed, consequently. Nevertheless, since in this work the same mesh sizes are used for both the calibration and the subsequent forming applications, there is no necessity to adapt the material parameters.

Model Validating

The original version of this paper is available on https://www.scientific.net/SSP.390.207.pdf

To validate the sandwich interface and core model, FEM simulations of swivel bending tests are compared to their corresponding experimental results. According to the foreseen use of the sandwich sheets for double-hat crash absorbers, strip specimens of a half hat-profile, formed with two consecutive 90∘ bends, are examined (cf. Figure 7). To capture the properties of the core and interface material, springback and the displacements between the outer and inner metal sheets in the direction of the neutral axis are measured.

[picture]

Fig. 7. Bending sequence of the hat profile. Triangles mark the fixed support. Strip specimen highlighted in green.

Fig. 7. Bending sequence of the hat profile. Triangles mark the fixed support. Strip specimen highlighted in green.

For the displacement measurement, the method proposed by [15] and improved by [10] is employed: before bending the sandwich specimen, the cross-section is ground to achieve an even and scratch-free surface. Then, transversal marks are pressed into the surface at equal distances. After bending the specimen, the displacements between every two corresponding marks in the outer and inner metal layers can be measured with a microscope using a 100x magnification and applying tile stitching with automatic depth focus adjustment (cf. Figure 8).

[picture]

Fig. 8. Cutout of the flange bend of the sandwich specimen with displacement marks (green highlights used to measure the displacement).

Fig. 8. Cutout of the flange bend of the sandwich specimen with displacement marks (green highlights used to measure the displacement).

The FE model of the bending tests to be validated is built up with solid elements of 0.25 mm length in the longitudinal direction and 2 mm in the transverse direction. Over the thickness, three elements
are used for the metal and polymer layers, while one element is used for the adhesive (cf. Figure 9). A 6 mm radius and a constant bending speed φ˙ of 9∘/s are used for the swivel bending process. The bending process is simulated in a two-step approach considering springback: first, the flange bend and the occurring springback are simulated. Then, the resulting geometry, as well as residual stresses and plastic strains are transferred to a second model to bend the side wall, again under consideration of the springback. The metal layer displacements are extracted from the innermost node line of the metal layer outer edges of the resulting geometry and are evaluated with a Python script for comparison with the experimental results.

[picture]

Fig. 9. Finite element model of the two consecutive swivel bending steps.

Fig. 9. Finite element model of the two consecutive swivel bending steps.

To validate the developed material model, the results from the experimental bending tests and the corresponding FEM simulations are compared. First, Table 1 compares the profile angles after springback at the flange bend and at the profile bend. The results prove that the simulation in principle matches the experimental results, despite a mismatch of ca. 1∘. Furthermore, to estimate the influence of the DIC-based strain rate reconstruction, perturbations of the strain rate dependent stress model were assessed. Variations of ±2% resulted in less than 1.5∘ variation in springback.

Noticeable is the differing trend in the simulation compared to the experiment: While in the experiment the first bend angle (flange angle) is greater than the second angle (profile angle), in the simulation the angles are vice versa. When interpreting the results, one has to consider that due to the sequential bending process, deviations and errors are added up. Thus, every deviation in the first bending step directly influences the results of the second step. Therefore, the opposite trend of the bend angle in the simulation could be explained by the superposition of the general simulation error and the error propagation from the first bending step.

Table 1. Hat-profile angles after springback (measured at enclosing sheet).

Position | Experiment | Simulation
Flange | 94.5° | 93.6°
Profile | 93.5° | 94.5°

A more detailed validation of the polymer and the adhesive model is possible with the comparison of the cover layer displacements. Since the cover layers are purely elastically loaded, the displacements depend solely on the interface and polymer properties (when assuming an equal elastic modulus for all relevant cover sheets). To distinguish the direction of displacement, a sign convention is introduced: displacements of the upper layer to the lower layer in the positive coordinate direction are accounted for as positive, while in the opposite case they are accounted for as negative (cf. Figure 10). The bend line coordinate x is defined to start in the center of the hat profile, thus at the long side of the strip specimen, and proceeds from there to the outer edge (cf. Figure 10).

[picture]

Fig. 10. Definition of the bend line coordinate and sign convention for the evaluation of the cover layer displacements.

Fig. 10. Definition of the bend line coordinate and sign convention for the evaluation of the cover layer displacements.

The resulting cover layer displacements δ are shown in Figure 11. The general development of the displacements matches well in experiment and simulation. Due to the swivel bending process, the displacements fall fast to nearly zero at the free edges, which were clamped during bending. In the bending zones, a positive displacement peak is followed by a negative peak for a right bend. The displacements cross zero near the midpoint of the bend and reach their limits (maxima/minima) near the end of the bending regions, which aligns with the findings reported for V-bending reported in the literature [10]. In the side wall between the two bends, the displacements overlap, wherefore they do not fall to zero in the experiment as well as in the simulation. Here again applies the error propagation stated for the springback angle. While the first bend in the simulation still matches the experimental results very well, a greater discrepancy is found in the second bend. Nevertheless, since the principal forming behavior is reflected well, the material model is considered to be valid.

[picture]

Fig. 11. Cover layer displacements in experiment and simulation.

Fig. 11. Cover layer displacements in experiment and simulation.

Parameter Study on the Influence of Material Variabilities

The original version of this paper is available on https://www.scientific.net/SSP.390.207.pdf

To study the effect of inherent material variabilities on the forming and failure behavior, a full factorial parameter sweep of small changes in the material strengths is conducted, and their effect on the failure mode of lap-shear tests is analyzed. While for the metal sheets, enough tensile tests were conducted to derive a minimum and maximum flow curve, not enough results were available for the polymer and the adhesive to capture the statistical distribution of their properties. Therefore, changes of ±10% in the nominal strengths (flow curves and peak traction, respectively) are assumed. This assumption is validated on simplified material tests. Tensile tests of three polymer specimens without capturing the local strains yield a variance of 0.4% to 2.8% of the yield stress, while the variance of
the peak stress of eight full sandwich lap-shear tests is 7.9% of the nominal value. Consequently, the ±10% assumption is a realistic assumption for the adhesive and conservative for the polymer foil, taking into account that the shear behavior (which is unknown from the tensile test) might differ significantly from the tensile strength. However, this does not represent a statistically derived process distribution but rather a structured sensitivity range to evaluate failure mode transitions.

The FE model of the lap-shear tests is presented in Figure 12 and uses the same element sizes as the bending model. The perturbed material parameter curves are compiled in Figure 13.

[picture]

Fig. 12. FE model of the lap-shear test.

Fig. 12. FE model of the lap-shear test.

[picture]

Fig. 13. Perturbed material curves (minimal, nominal, and maximal strength) for the parameter study.

Fig. 13. Perturbed material curves (minimal, nominal, and maximal strength) for the parameter study.

The results of the parameter study are shown in Figure 14. The occurring failure modes can be divided into three groups: (1) a long-stretched polymer in the case of reduced polymer and increased adhesive strength, (2) an S-shaped bend of the polymer foil mixed with delamination of the adhesive for equally directed changes of polymer and adhesive strength, and (3) fracture of the polymer as the main failure criteria for simulations with increased polymer strength and decreased adhesion. In all three groups, the influence of the steel strength is negligible, since the plastic deformation of the cover layers is rather limited. The small effects occurring in relation to a steel strength change more probably result from numerical effects.

[picture]

Fig. 14. Resulting failure modes and polymer damage in simulated lap-shear tests. The material strength parameters were varied in a full factorial approach.

Fig. 14. Resulting failure modes and polymer damage in simulated lap-shear tests. The material strength parameters were varied in a full factorial approach.

It has to be noted that in the first failure mode in group 3 (PP/PE ↑ , Adhes. ↓ , Steel ↑ ), the fracture of the polymer happens first. The subsequent delamination only occurs due to numerical issues with the failed polymer. Thus, the failure principle lies nevertheless in the polymer fracture. This becomes clear when the shape is compared with the results of group 2, where no polymer elements were deleted.

A comparison of the simulated failure modes to experimental lap-shear results of several material combinations (cf. Figure 15) shows that the simulation is capable of reflecting various failure modes that occur in reality. Additionally, the parameter study reveals that the factual failure mode mainly depends on a complex balance of polymer and adhesive strength. Small perturbations can result in a different mode. This indicates that the examined polymer-adhesive combination is at the transition point between adhesive and cohesive failure, as has also been observed for a similar combination in the literature [6], [7]. Since the forming behavior ultimately accumulates to the examined failure modes in the lap shear tests, the inhomogeneous material behavior can be simulated well with the proposed model in forming and failure studies.

[picture]

Fig. 15. Comparison of simulated (top) and experimental (bottom) lap-shear test failure modes.

Fig. 15. Comparison of simulated (top) and experimental (bottom) lap-shear test failure modes.

Predicting Springback and Material Failure in the Profile-Bending Process

The original version of this paper is available on https://www.scientific.net/SSP.390.207.pdf

In contrast to conventional three-layer models, which do not consider the adhesive, the developed and validated model of the sandwich materials can be used in forming process simulations to predict springback and possible material failure. While a three-layer approach yields results as shown in

Figure 16 with significant edge displacement and a mismatch of the predicted angle after springback ( 95.7∘ for the model without consideration of the bonding), the presented modelling approach incorporating the adhesive is capable of accurately predicting springback and edge displacement (cf. Table 1 and Figure 11).

Furthermore, the model can be used to study the influence of first, the inherent variation of the adhesive strength of a stable sandwich manufacturing process on the forming process, and secondly, the effect of outliers in the bonding quality (e.g., through local bonding defects, low manufacturing quality, etc.) on the forming behavior.

[picture]

Fig. 16. Bending simulation of a three-layer specimen without adhesive compared to a simulation with adhesive.

Fig. 16. Bending simulation of a three-layer specimen without adhesive compared to a simulation with adhesive.

[picture]

Fig. 17. Geometric deviation between simulated bending specimens with maximal adhesive (grey) and minimal adhesive (colored) strength.

Fig. 17. Geometric deviation between simulated bending specimens with maximal adhesive (grey) and minimal adhesive (colored) strength.

Table 2. Simulated springback of the maximal and minimal adhesive strength variation compared to the nominal springback angle.

Fig. 17. Geometric deviation between simulated bending specimens with maximal adhesive (grey) and minimal adhesive (colored) strength.

Position | Maximal variation | Nominal | Minimal variation
Flange | mathematical formula | 93.6 | superscript number
Profile | superscript number | 94.5 | mathematical formula

The profile angles vary after the springback only in a small range, from 93.6∘ to 94.1∘ for the flange and from 93.9∘ to 94.5∘ for the profile bend. Therefore, it can be concluded that the process inherent variation of the adhesive strength of good quality sandwich sheets has a negligible influence on the geometric accuracy of the hat profiles.

However, a greater impact of the adhesive strength is seen in low-quality bondings. Figure 18 shows, for example, delamination and the consequent buckling of a profile bend with significantly lower adhesion strength, as it can occur for sandwich sheets with defective or partially defective bonding. By considering the weakened adhesive (reduction to ca. 11% of the nominal strength) in the simulation, the same failure mode as observed in bending experiments is predicted.

[picture]

Fig. 18. Delamination and resulting buckling in the bending process due to low bonding quality in the experiment and simulation.

Fig. 18. Delamination and resulting buckling in the bending process due to low bonding quality in the experiment and simulation.

Conclusion

The original version of this paper is available on https://www.scientific.net/SSP.390.207.pdf

This work presents an approach for deriving a strain rate-dependent polymer model for FEM simulations of the forming and failure behavior of metal-polymer-metal sandwich sheets. Springback and cover layer displacements can be predicted using the FE model and are used to validate the proposed approach with experimental results.

The parameter study on the influence of material inhomogeneities shows that the failure mode of mpm-sandwiches can vary significantly by small changes in the balance of steel, polymer, and adhesion strength. By respecting the parameter deviations in FEM simulations, experimental failure modes can be reproduced.

The developed sandwich material model has been proven to be usable in bending process simulations. It is thus possible to more accurately predict the sandwich's forming behavior in a profile manufacturing process, which helps tune the sandwich and forming process parameters. Study of the model's applicability to other forming processes is subject to future research. Furthermore, improving the material models to reflect a greater range of strain rates and incorporating temperature effects are further research directions to facilitate consistent simulation of the forming as well as the crash performance of sandwich sheets.

Appendix

The original version of this paper is available on https://www.scientific.net/SSP.390.207.pdf

Table 3. Summary of material parameters for the MAT_SAMP_1 polymer model.

ρ | E | νel | νpl | φfail | Flow and Damage Curves
|  |  |  |  | φ̇ | σ [MPa]
0.957 g cm⁻³ | 1024 MPa | 0.4 | 0.26 | 2.7 | 10⁻⁶ ¹/s | 26.9 | 31.5 | 37.7 | 66.0 | 151 | 377 | 838
10⁻³ ¹/s | 30.3 | 35.6 | 42.6 | 74.5 | 170 | 426 | 945
10⁻² ¹/s | 32.8 | 38.5 | 46.1 | 80.6 | 184 | 461 | 1023
10⁻¹ ¹/s | 36.6 | 42.9 | 51.4 | 89.9 | 205 | 514 | 1141
1 ¹/s | 42.3 | 50.0 | 59.4 | 104 | 237 | 595 | 1320
|  |  |  |  | D(φ) | 0.0 | 0.33 | 0.44 | 0.61 | 0.74 | 0.87 | 1.0
|  |  |  |  | φpl | 0.0 | 0.3 | 0.5 | 1.0 | 1.6 | 2.2 | 2.7

Table 4. Material parameters for the MAT_COHESIVE_MIXED_MODE adhesive model.

| Mode I | Mode II
E | 210 GPa/mm | 210 GPa/mm
GC | 0.65 mJ/mm² | 0.88 mJ/mm²
T, S | 68.15 MPa | 22.95 MPa

Acknowledgement

The original version of this paper is available on https://www.scientific.net/SSP.390.207.pdf

This research was funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) - 523138878.

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Keywords Index

#

3d-printing A Compact Micromechanical Model for the Elastic-Viscoplastic Deformability of PLA-Hemp Biocomposites

42 Crmo 4 Predictive Modeling of 42CrMo4 Hardness after Two-Step Heat Treatment Using Symbolic Regression

A

Alignment Texture Data Preparation for Finite Element Simulations of Puncture Tests: Insights from the VPSC Approach

Aluminum Alloy Texture Data Preparation for Finite Element Simulations of Puncture Tests: Insights from the VPSC Approach

Aluminum Casting Experimental Study on the Relationship between Cooling Rate, Secondary Dendrite Arm Spacing and Tensile Properties in Aluminum Casting

Anisotropy Texture Data Preparation for Finite Element Simulations of Puncture Tests: Insights from the VPSC Approach

B

Bauschinger Effect A Novel Approach for Determining the Bauschinger Effect in Dual-Phase Steel Sheets with Tensile-Compression Tests

Biocomposite A Compact Micromechanical Model for the Elastic-Viscoplastic Deformability of PLA-Hemp Biocomposites

Biphasic Lattice Structure Mechanical Performance of a Steel-Polymer Biphasic Lattice Structure

C

Clustering Texture Data Preparation for Finite Element Simulations of Puncture Tests: Insights from the VPSC Approach

Cold Crystallization Empirical Modelling of the Cold Crystallization of PLA

Constitutive Modeling 1D Stress Evolution Prediction via Thermodynamically-informed Neural Networks

Convexity Further Developments of Hill-Gotoh Theory of Anisotropic Sheet Metal Plasticity

Cooling Rate Experimental Study on the Relationship between Cooling Rate, Secondary Dendrite Arm Spacing and Tensile Properties in Aluminum Casting

Cpfe Microscale to Macroscale Analyses of Superplasticity of Ti-6242S Titanium Alloy

Crystal Plasticity Numerical Quantification of Structure-Property Relations in DP800 - Influence of Phase Material Models in RVE; Deriving Macroscopic Damage Properties by Micromechanical Modeling of DP1000; Quantifying Grain Boundary and Orientation Effects on Copper under Compression: A Crystal Plasticity Finite Element Approach; Study on the Effects of Grain Shape, Size and Size Distribution on the Mechanical Behavior of Metals

D

Damage Numerical Quantification of Structure-Property Relations in DP800 - Influence of Phase Material Models in RVE; Deriving Macroscopic Damage Properties by Micromechanical Modeling of DP1000

Distortional Hardening Improved Mapping of Yield Locus Distortion through a Strain-Dependent Calibration of the Barlat YId2000-2d Yield Criterion

Dual Phase Steel Numerical Quantification of Structure-Property Relations in DP800 - Influence of Phase Material Models in RVE; Deriving Macroscopic Damage Properties by Micromechanical Modeling of DP1000

Dual-phase Steels A Novel Approach for Determining the Bauschinger Effect in Dual-Phase Steel Sheets with Tensile-Compression Tests

Dual-phase Titanium Alloy Microscale to Macroscale Analyses of Superplasticity of Ti-6242S Titanium Alloy

E

Elastoplasticity 1D Stress Evolution Prediction via Thermodynamically-informed Neural Networks

Empirical Modelling Empirical Modelling of the Cold Crystallization of PLA

F

Finite Element Method Investigation of Friction Coefficient at Stack Compression Tests

Formability How to Take the Lüders Plateau into Account in the Formability Prediction of Steel Sheets

Forming Texture Data Preparation for Finite Element Simulations of Puncture Tests: Insights from the VPSC Approach; A Compact Micromechanical Model for the Elastic-Viscoplastic Deformability of PLA-Hemp Biocomposites

Friction Investigation of Friction Coefficient at Stack Compression Tests

G

Gradient Enhanced Crystal Plasticity Study on the Effects of Grain Shape, Size and Size Distribution on the Mechanical Behavior of Metals

Grain Aspect Ratio Study on the Effects of Grain Shape, Size and Size Distribution on the Mechanical Behavior of Metals

Grain Boundary Quantifying Grain Boundary and Orientation Effects on Copper under Compression: A Crystal Plasticity Finite Element Approach

Grain Size Quantifying Grain Boundary and Orientation Effects on Copper under Compression: A Crystal Plasticity Finite Element Approach

Grain Size Distribution Study on the Effects of Grain Shape, Size and Size Distribution on the Mechanical Behavior of Metals

Gyroid Mechanical Performance of a Steel-Polymer Biphasic Lattice Structure

H

Hardening Law How to Take the Lüders Plateau into Account in the Formability Prediction of Steel Sheets

Heat Treatment Predictive Modeling of 42CrMo4 Hardness after Two-Step Heat Treatment Using Symbolic Regression

Hexrd Microscale to Macroscale Analyses of Superplasticity of Ti-6242S Titanium Alloy

Homogenization Mechanical Performance of a Steel-Polymer Biphasic Lattice Structure

Hot Forming Modeling Transient Temperature-Driven Microstructure Evolution in Inconel 718 during Hot Forming

Hot Isostatic Pressing Laser-Induced Solid-State Foaming of HIPed Ti6Al4V-ELI Alloy

L

Laser-induced Solid-state Foaming Laser-Induced Solid-State Foaming of HIPed Ti6Al4V-ELI Alloy

Lüders How to Take the Lüders Plateau into Account in the Formability Prediction of Steel Sheets

M

Machine Learning 1D Stress Evolution Prediction via Thermodynamically-informed Neural Networks; A Compact Micromechanical Model for the Elastic-Viscoplastic Deformability of PLA-Hemp Biocomposites

Material Characterization A Novel Approach for Determining the Bauschinger Effect in Dual-Phase Steel Sheets with Tensile-Compression Tests; Modeling the Influence of Material Variabilities on the Forming and Failure Behavior of PP/PE Cores in Metal-Polymer-Metal Sandwich Sheets

Material Hardness Predictive Modeling of 42CrMo4 Hardness after Two-Step Heat Treatment Using Symbolic Regression

Material Inhomogeneity Modeling the Influence of Material Variabilities on the Forming and Failure Behavior of PP/PE Cores in Metal-Polymer-Metal Sandwich Sheets

Material Modeling Improved Mapping of Yield Locus Distortion through a Strain-Dependent Calibration of the Barlat YId2000-2d Yield Criterion

Material Modelling Laser-Induced Solid-State Foaming of HIPed Ti6Al4V-ELI Alloy

Mechanical Properties Experimental Study on the Relationship between Cooling Rate, Secondary Dendrite Arm Spacing and Tensile Properties in Aluminum Casting

Micromechanical Modeling Deriving Macroscopic Damage Properties by Micromechanical Modeling of DP1000

Microstructure Numerical Quantification of Structure-Property Relations in DP800 - Influence of Phase Material Models in RVE

Microstructure Modeling Modeling Transient Temperature-Driven Microstructure Evolution in Inconel 718 during Hot Forming

Multiscale Modelling Microscale to Macroscale Analyses of Superplasticity of Ti-6242S Titanium Alloy

N

Non-homogeneous Stress Functions Further Developments of Hill-Gotoh Theory of Anisotropic Sheet Metal Plasticity

Non-isothermal Deformation Modeling Transient Temperature-Driven Microstructure Evolution in Inconel 718 during Hot Forming

Non-quadratic Yield Criteria Further Developments of Hill-Gotoh Theory of Anisotropic Sheet Metal Plasticity

O

Optimisation Microscale to Macroscale Analyses of Superplasticity of Ti-6242S Titanium Alloy

P

Pla Empirical Modelling of the Cold Crystallization of PLA

Plane Strain Improved Mapping of Yield Locus Distortion through a Strain-Dependent Calibration of the Barlat YId2000-2d Yield Criterion

Polymer Mechanical Performance of a Steel-Polymer Biphasic Lattice Structure

Porosity Evolution Laser-Induced Solid-State Foaming of HIPed Ti6Al4V-ELI Alloy

Predictive Mode Predictive Modeling of 42CrMo4 Hardness after Two-Step Heat Treatment Using Symbolic Regression

Puncture Test Texture Data Preparation for Finite Element Simulations of Puncture Tests: Insights from the VPSC Approach

R

Representative Volume Element Deriving Macroscopic Damage Properties by Micromechanical Modeling of DP1000

Representative Volume Elements Numerical Quantification of Structure-Property Relations in DP800 - Influence of Phase Material Models in RVE

S

Sandwich Composites Modeling the Influence of Material Variabilities on the Forming and Failure Behavior of PP/PE Cores in Metal-Polymer-Metal Sandwich Sheets

Secondary Dendrite Arm Spacing (sdas) Experimental Study on the Relationship between Cooling Rate, Secondary Dendrite Arm Spacing and Tensile Properties in Aluminum Casting

Sensitivity Analysis Numerical Quantification of Structure-Property Relations in DP800 - Influence of Phase Material Models in RVE

Shear Stress Improved Mapping of Yield Locus Distortion through a Strain-Dependent Calibration of the Barlat YId2000-2d Yield Criterion

Simple Compression Quantifying Grain Boundary and Orientation Effects on Copper under Compression: A Crystal Plasticity Finite Element Approach

Stack Compression Test Investigation of Friction Coefficient at Stack Compression Tests

Steels How to Take the Lüders Plateau into Account in the Formability Prediction of Steel Sheets

Superalloys Modeling Transient Temperature-Driven Microstructure Evolution in Inconel 718 during Hot Forming

Superplasticity Microscale to Macroscale Analyses of Superplasticity of Ti-6242S Titanium Alloy

Symbolic Regression Predictive Modeling of 42CrMo4 Hardness after Two-Step Heat Treatment Using Symbolic Regression

T

Thermal Analysis Experimental Study on the Relationship between Cooling Rate, Secondary Dendrite Arm Spacing and Tensile Properties in Aluminum Casting

Thermodynamic Consistency 1D Stress Evolution Prediction via Thermodynamically-informed Neural Networks

Ti6al4v-eli Alloy Laser-Induced Solid-State Foaming of HIPed Ti6Al4V-ELI Alloy

V

Viscoplasticity A Compact Micromechanical Model for the Elastic-Viscoplastic Deformability of PLA-Hemp Biocomposites

Authors Index

A

Ahn, T. Experimental Study on the Relationship between Cooling Rate, Secondary Dendrite Arm Spacing and Tensile Properties in Aluminum Casting

Albers, J. Predictive Modeling of 42CrMo4 Hardness after Two-Step Heat Treatment Using Symbolic Regression

Antretter, T. Texture Data Preparation for Finite Element Simulations of Puncture Tests: Insights from the VPSC Approach

Arowolo, M. Microscale to Macroscale Analyses of Superplasticity of Ti-6242S Titanium Alloy

B

Barrière, T. A Compact Micromechanical Model for the Elastic-Viscoplastic Deformability of PLA-Hemp Biocomposites

Beres, G. Investigation of Friction Coefficient at Stack Compression Tests

Bocan, R. Study on the Effects of Grain Shape, Size and Size Distribution on the Mechanical Behavior of Metals

Boogaard, T. 1D Stress Evolution Prediction via Thermodynamically-informed Neural Networks

Boogard, T. Study on the Effects of Grain Shape, Size and Size Distribution on the Mechanical Behavior of Metals

Brüggemann, H. Modeling Transient Temperature-Driven Microstructure Evolution in Inconel 718 during Hot Forming

Buhl, J. Modeling the Influence of Material Variabilities on the Forming and Failure Behavior of PP/PE Cores in Metal-Polymer-Metal Sandwich Sheets

C

Ciccarella, A. Mechanical Performance of a Steel-Polymer Biphasic Lattice Structure

Colomines, G. Empirical Modelling of the Cold Crystallization of PLA

Cusanno, A. Laser-Induced Solid-State Foaming of HIPed Ti6Al4V-ELI Alloy

D

Darendeliler, H. Quantifying Grain Boundary and Orientation Effects on Copper under Compression: A Crystal Plasticity Finite Element Approach

Dehmas, M. Microscale to Macroscale Analyses of Superplasticity of Ti-6242S Titanium Alloy

Dell'Avvocato, G. Mechanical Performance of a Steel-Polymer Biphasic Lattice Structure

Dittrich, F. Predictive Modeling of 42CrMo4 Hardness after Two-Step Heat Treatment Using Symbolic Regression

E

Elekyabi, N. Modeling Transient Temperature-Driven Microstructure Evolution in Inconel 718 during Hot Forming

F

Fallahnejad, A. 1D Stress Evolution Prediction via Thermodynamically-informed Neural Networks

Fehlemann, N. Numerical Quantification of Structure-Property Relations in DP800 - Influence of Phase Material Models in RVE

Fulco, E. Laser-Induced Solid-State Foaming of HIPed Ti6Al4V-ELI Alloy

G

Gabrion, X. A Compact Micromechanical Model for the Elastic-Viscoplastic Deformability of PLA-Hemp Biocomposites

Gallucci, K. Mechanical Performance of a Steel-Polymer Biphasic Lattice Structure

Giuliano, A. Mechanical Performance of a Steel-Polymer Biphasic Lattice Structure

Guglielmi, P. Laser-Induced Solid-State Foaming of HIPed Ti6Al4V-ELI Alloy

H

Holopainen, S. A Compact Micromechanical Model for the Elastic-Viscoplastic Deformability of PLA-Hemp Biocomposites

Häckel, A. A Novel Approach for Determining the Bauschinger Effect in Dual-Phase Steel Sheets with Tensile-Compression Tests

Härtel, S. Predictive Modeling of 42CrMo4 Hardness after Two-Step Heat Treatment Using Symbolic Regression

K

Kalácska, G. Investigation of Friction Coefficient at Stack Compression Tests

Kronsteiner, J. Texture Data Preparation for Finite Element Simulations of Puncture Tests: Insights from the VPSC Approach

Kölüs, M. Investigation of Friction Coefficient at Stack Compression Tests

L

Labalec, E. Empirical Modelling of the Cold Crystallization of PLA

Lee, J. Experimental Study on the Relationship between Cooling Rate, Secondary Dendrite Arm Spacing and Tensile Properties in Aluminum Casting

Lemoine, X. How to Take the Lüders Plateau into Account in the Formability Prediction of Steel Sheets

Lian, J. Modeling Transient Temperature-Driven Microstructure Evolution in Inconel 718 during Hot Forming

M

Mancini, E. Mechanical Performance of a Steel-Polymer Biphasic Lattice Structure

Merklein, M. Improved Mapping of Yield Locus Distortion through a Strain-Dependent Calibration of the Barlat YId2000-2d Yield Criterion; A Novel Approach for Determining the Bauschinger Effect in Dual-Phase Steel Sheets with Tensile-Compression Tests

Micarelli, L. Mechanical Performance of a Steel-Polymer Biphasic Lattice Structure

Münstermann, S. Numerical Quantification of Structure-Property Relations in DP800 - Influence of Phase Material Models in RVE; Deriving Macroscopic Damage Properties by Micromechanical Modeling of DP1000

N

Nagy-Gombai, I. Investigation of Friction Coefficient at Stack Compression Tests

Naumann, D. A Novel Approach for Determining the Bauschinger Effect in Dual-Phase Steel Sheets with Tensile-Compression Tests

Neite, M. Deriving Macroscopic Damage Properties by Micromechanical Modeling of DP1000

Niang, N. A Compact Micromechanical Model for the Elastic-Viscoplastic Deformability of PLA-Hemp Biocomposites

O

Ott, A. Texture Data Preparation for Finite Element Simulations of Puncture Tests: Insights from the VPSC Approach

P

Palumbo, G. Laser-Induced Solid-State Foaming of HIPed Ti6Al4V-ELI Alloy

Papenberg, N. Texture Data Preparation for Finite Element Simulations of Puncture Tests: Insights from the VPSC Approach

Paçacı, M. Quantifying Grain Boundary and Orientation Effects on Copper under Compression: A Crystal Plasticity Finite Element Approach

Perdahcioglu, E. 1D Stress Evolution Prediction via Thermodynamically-informed Neural Networks; Study on the Effects of Grain Shape, Size and Size Distribution on the Mechanical Behavior of Metals

Piccininni, A. Laser-Induced Solid-State Foaming of HIPed Ti6Al4V-ELI Alloy

Placet, V. A Compact Micromechanical Model for the Elastic-Viscoplastic Deformability of PLA-Hemp Biocomposites

Pusicha, J. Modeling the Influence of Material Variabilities on the Forming and Failure Behavior of PP/PE Cores in Metal-Polymer-Metal Sandwich Sheets

R

Rentz, M. Improved Mapping of Yield Locus Distortion through a Strain-Dependent Calibration of the Barlat YId2000-2d Yield Criterion

S

Scharifi, E. Modeling Transient Temperature-Driven Microstructure Evolution in Inconel 718 during Hot Forming

Schwarzmeier, L. Texture Data Preparation for Finite Element Simulations of Puncture Tests: Insights from the VPSC Approach

Sleiman, G. Empirical Modelling of the Cold Crystallization of PLA

Sorgente, D. Laser-Induced Solid-State Foaming of HIPed Ti6Al4V-ELI Alloy

T

Tong, W. Further Developments of Hill-Gotoh Theory of Anisotropic Sheet Metal Plasticity

V

Velay, V. Microscale to Macroscale Analyses of Superplasticity of Ti-6242S Titanium Alloy

Vidal, V. Microscale to Macroscale Analyses of Superplasticity of Ti-6242S Titanium Alloy

Y

Yoon, J. Experimental Study on the Relationship between Cooling Rate, Secondary Dendrite Arm Spacing and Tensile Properties in Aluminum Casting