An Improved Heat Equation to Model Ductile-to-Brittle Failure Mode Transition at High Strain Rates Using Fully Coupled Thermal-Structural Finite Element Analysis

Abstract:

In this paper a universal heat equation for fully coupled thermal structural finite element analysis of deformable solids capable of predicting ductile-to-brittle failure mode transition at high strain rates is presented. In the problem mathematical formulation appropriate strain measures describing the onset and the growth of ductile and total damage and heat generation rate per unit volume to model dissipation-induced heating have been employed, which were extended with the heat equation. The model was implemented into a finite element code utilizing an improved weak form for updated Lagrangian formulation, an extended NoIHKH material model for cyclic plasticity of metals applicable in wide range of strain rates and the Jaumann rate in the form of the Green-Naghdi rate in the co-rotational Cauchy’s stress objective integration. The model verification showed excellent agreement with the modelled experiment at low strain rates. Plastic bending of a cantilever has been studied at higher strain rates. A few selected analysis results are presented and briefly discussed.

• Periodical: Defect and Diffusion Forum (Volume 354)
• Pages: 1-23
• DOI: https://doi.org/10.4028/www.scientific.net/DDF.354.1
• Online since: June 2014
• Authors: Ladislav Écsi*, P. Élesztős
• Keywords: Ductile-to-Brittle Failure Mode Transition, Finite Strain Elastoplasticity, Fully Coupled Thermal Structural Finite Element Analysis, Improved Weak, Internal Damping, Strong Forms, Wide Range of Strain Rates

References

[1] J. J. Skrzypek, A. W. Ganczarski, F. Rusttichelli, H. Egner, Advanced materials and structures for extreme operating conditions, Springer Verlag, Berlin, 2008.

[2] A. Needleman and V. Tvergaard, An analysis of dynamic, ductile crack growth in a double edge cracked specimen. Int. J. Fracture, 49 (1991), 41-47.

DOI: 10.1007/bf00013502

[3] A. Needleman and V. Tvergaard, A numerical study of void distribution effect on dynamic, ductile crack growth. Engng. Fracture Mech., 38 (1991), 157-173.

DOI: 10.1016/0013-7944(91)90079-g

[4] A. Needleman and V. Tvergaard, Mesh effects in the analysis of dynamic ductile crack growth. Engng. Fracture Mech., 47 (1994), 75-91.

DOI: 10.1016/0013-7944(94)90239-9

[5] A. Needleman and V. Tvergaard., Analysis of a brittle-ductile transition under dynamic shear loading, Int. J. Solids Structures, 32(17-18), (1995), 2571-2590.

DOI: 10.1016/0020-7683(94)00283-3

[6] G.A., Maugin, The thermomechanics of plasticity and fracture, Cambridge University Press, Cambridge, 1992.

[7] J.E. Field, S.M. Walley*, W.G. Proud, H.T. Goldrein, C.R. Siviour, Review of experimental techniques for high rate deformation and shock studies, International Journal of Impact Engineering, 30 (2004) 725–775.

DOI: 10.1016/j.ijimpeng.2004.03.005

[8] H.N. Krishna Teja Palleti a, S. Gurusamy a, Santosh Kumar a, R. Soni b, B. John b, R. Vaidya b, A. Bhoge b, N.K. Naik, Ballistic impact performance of metallic targets, Materials and Design, 39 (2012) 253-263.

DOI: 10.1016/j.matdes.2012.02.033

[9] Bidyapati Mishra a,*, P.K. Jena a, B. Ramakrishna a, V. Madhu a, T.B. Bhat a, N.K. Gupta, Effect of tempering temperature, plate thickness and presence of holes on ballistic impact behavior and ASB formation of a high strength steel, International Journal of Impact Engineering, 44 (2012) 17-28.

DOI: 10.1016/j.ijimpeng.2011.12.004

[10] T. Børvik, M. Langseth, O.S. Hopperstad, K.A. Malo, Ballistic penetration of steel plates, International Journal of Impact Engineering, 22. (1999) 855-886.

DOI: 10.1016/s0734-743x(99)00011-1

[11] V.N. Wijayathunga ∗, D.C. Webb, Experimental evaluation and finite element simulation of explosive forming of a square cup from a brass plate assisted by a lead plug, Journal of Materials Processing Technology, 172 (2006) 139–145

DOI: 10.1016/j.jmatprotec.2005.09.014

[12] D.J. Mynors*, B. Zhang, Applications and capabilities of explosive forming, Journal of Materials Processing Technology, 125–126 (2002) 1–25.

DOI: 10.1016/s0924-0136(02)00413-2

[13] F. Abdi, K. Bowcutt, C. Godines, J. Bayandor, Collision provoked failure sequencing in space reentry vehicles, Computers and Structures, 89 (2011) 930–939.

DOI: 10.1016/j.compstruc.2011.01.021

[14] M.J. Buehler, F.F. Abraham, H. Gao, "Hyperelasticity governs dynamic fracture at a critical length scale, Nature, 426, (2003) 141-146.

DOI: 10.1038/nature02096

[15] J.T. Foster, W. Chen, V.K. Luk, Dynamic crack initiation toughness of 4340 steel at constant loading rates, Engineering Fracture Mechanics, 78 (2011) 1264–1276.

DOI: 10.1016/j.engfracmech.2011.02.019

[16] B. N. Cox, H. Gao, D. Gross, D. Rittel, Modern topics and challenges in dynamic fracture, Journal of the Mechanics and Physics of solids, 53 (2005) 565–596.

DOI: 10.1016/j.jmps.2004.09.002

[17] A. Conde, B. J. Fernandez, J. J. De Damborenea, Characterization of the SCC behaviour of 8090 Al-L1 alloy by means of the slow-strain-rate technique, Corrosion Science, Vol. 40, No. 1, (1998) 91-102.

DOI: 10.1016/s0010-938x(97)00117-0

[18] F. Trebuňa, F. Šimčák, J. Bocko, P. Trebuňa, M. Pástor, Patrik Šarga, Analysis of crack initiation in the press frame and innovation of the frame to ensure its further operation, Engineering Failure Analysis, 18 (2011) 244–255.

DOI: 10.1016/j.engfailanal.2010.09.004

[19] B. Li, A. Kidane, G. Ravichandran, M. Ortiz, Verification and validation of the Optimal Transportation Meshfree (OTM) simulation of terminal ballistics, International Journal of Impact Engineering, 42 (2012) 25-36.

DOI: 10.1016/j.ijimpeng.2011.11.003

[20] B. Li, F. Habbal and M. Ortiz, Optimal transportation meshfree approximation schemes for fluid and plastic flows, Int. J. Numer. Meth. Engng; 83, (2010) 1541–1579.

DOI: 10.1002/nme.2869

[21] S. Li, W.K. Liu, A. J. Rosakis, T. Belytschko, W. Hao, Mesh-free galerkin simulations of dynamic shear band propagation and failure mode transition, International Journal of Solids and Structures, No. 39, p.1213–1240.

DOI: 10.1016/s0020-7683(01)00188-3

[22] J.C. Simo, T.J.R. Hughes, Computational inelasticity, Springer, NY, 1998.

[23] P. Perzyna, Fundamental problems in viscoplasticity, in Advances in Applied Mechanics, Vol. 9, Academic, Press, New York, p.243–377.

DOI: 10.1016/s0065-2156(08)70009-7

[24] P. Perzyna, Thermodynamic theory of viscopplasticity, in Advances in Applied Mechanics, Vol. 11, Academic, Press, New York, p.313–354.

[25] J.D. Clayton, Continuum multiscale modeling of finite deformation plasticity and anisotropic damage in polycrystals, Theoretical and Applied Fracture Mechanics, No. 45, p.163–185.

DOI: 10.1016/j.tafmec.2006.03.001

[26] A.G. Varghese, R.C. Batra, Constitutive equations for thermomechanical deformations of glassy polymers, International Journal of Solids and Structures, No. 46, p.4079–4094.

DOI: 10.1016/j.ijsolstr.2009.08.006

[27] A.D. Mulliken, M.C. Boyce, Mechanics of the rate-dependent elastic–plastic deformation of glassy polymers from low to high strain rates, International Journal of Solids and Structures, No. 43, p.1331–1356.

DOI: 10.1016/j.ijsolstr.2005.04.016

[28] J. Richeton, S. Ahzi, K.S. Vecchio, F.C. Jiang, A. Makradi, Modeling and validation of the large deformation inelastic response of amorphous polymers over a wide range of temperatures and strain rates, International Journal of Solids and Structures, No. 44, p.7938–7954.

DOI: 10.1016/j.ijsolstr.2007.05.018

[29] R.C. Batra, B.M. Love, Mesoscale analysis of shear bands in high strain rate deformations of tungsten/nickel-iron composites, Journal of Thermal Stresses, Vol. 28, No. 6-7, pp.747-782.

DOI: 10.1080/01495730590932724

[30] A. Rusinek, R. Zaera, J.R. Klepaczko, Constitutive relations in 3-D for a wide range of strain rates and temperatures – Application to mild steels, International Journal of Solids and Structures, No. 44, p.5611–5634.

DOI: 10.1016/j.ijsolstr.2007.01.015

[31] R. Zaera, J.A. Rodriguez-Martinez, A. Casado, J. Fernández-Sáez, A. Rusinek, R. Pesci, A constitutive model for analyzing martensite formation in austenitic steels deforming at high strain rates, International Journal of Plasticity, No. 29, p.77–101.

DOI: 10.1016/j.ijplas.2011.08.003

[32] M. Kuroda, A. Uenishi, H. Yoshida, A. Igarashi, Ductility of interstitial-free steel under high strain rate tension: Experiments and macroscopic modeling with a physically-based consideration, International Journal of Solids and Structures, No. 43, p.4465–4483.

DOI: 10.1016/j.ijsolstr.2005.06.076

[33] G.Z. Voyiadjis, F.H. Abed, A coupled temperature and strain rate dependent yield function for dynamic deformations of bcc metals, International Journal of Plasticity, No. 22, pp.1398-1431.

DOI: 10.1016/j.ijplas.2005.10.005

[34] S. Nemat-Nasser, Plasticity, A treatise on finite deformation of heterogenous inelastic materials, Cambridge University Press, Cambridge, 2004.

[35] J. Lemaitre, J.L. Chaboche, Mechanics of solid materials, Cambridge university press, Cambridge, 1994.

[36] P. Rosakis, A.J. Rosakis, G. Ravichandrian, J. Hoodowany, A thermodynamic internal variable model for the partition of plastic work into heat and stored energy in metals, J. Mech. Phys. ​Solids, 48, (2000) 581-607.

DOI: 10.1016/s0022-5096(99)00048-4

[37] E. H. Dill, The finite element method for mechanics of solid with ANSYS applications, CRC Press, London, 2012.

[38] X. Lu, S.V. Hangund, A nonequilibrium irreversible thermodynamics model for material damping, Int. J. Solids. Struct, 44, (2006) 3278-3303.

DOI: 10.1016/j.ijsolstr.2006.09.021

[39] L. Écsi, P. Élesztős, Moving toward a more realistic material model of a ductile material with failure mode transition, Mat.-wiss. u. Werkstofftech., 43, No. 5, (2012) 379-387.

DOI: 10.1002/mawe.201200969

[40] L. Écsi, P. Élesztős, Constitutive equation with internal damping for materials under cyclic and dynamic loadings using a fully coupled thermal-structural finite element analysis, Int. J. Multiphysics, 3 2 (2009) 155-165.

DOI: 10.1260/175095409788837829

[41] R. D. Cook, D. S. Malkus, M. E. Plesha and R. J. Witt, Concepts and applications of finite element analysis, 4. Ed., John Wiley & Sons Ltd., N.Y., 2001.

[42] L.C. Evans, Partial differential equations, American Mathematical Society, Providence, Rhode Island, 1998.

[43] G.R. Johnson, W.H. Cook, A Constitutive model and data for metals subjected to large strains, strain rates, and high pressures, Proceedings Of The 7th International Symposium On Ballistics, The Hague, The Netherlands, 1983.

[44] W.K. Rule, S.E. Jones, A revised form of the Johnson-Cook strength model, International Journal Of Impact Engineering, Vol.21, No. 8, pp.609-624.

DOI: 10.1016/s0734-743x(97)00081-x

[45] F.J. Zerilli, R.W. Armstrong, Dislocation mechanics based constitutive relations for material dynamics calculations, Journal Of Applied Physics, Vol.61, pp.1816-1825.

DOI: 10.1063/1.338024

[46] A. Rusinek , R. Zaera, J.R. Klepaczko, Constitutive relations in 3-D for a wide range of strain rates and temperatures – Application to mild steels, International Journal of Solids and Structures, No. 44, p.5611–5634.

DOI: 10.1016/j.ijsolstr.2007.01.015

[47] R. Zaera, J.A. Rodriguez-Martinez, A. Casado, J. Fernández-Sáez, A. Rusinek, R. Pesci, A constitutive model for analyzing martensite formation in austenitic steels deforming at high strain rates, International Journal of Plasticity, No. 29, p.77–101.

DOI: 10.1016/j.ijplas.2011.08.003

[48] A. Rusinek, J.A. Rodriguez-Martinez, Thermo-viscoplastic constitutive relation for aluminium alloys, modeling of negative strain rate sensitivity and viscous drag effects, Materials and Design, No. 30, p.4377–4390.

DOI: 10.1016/j.matdes.2009.04.011

[49] J.R. Klepaczko, A practical stress–strain–strain rate-temperature constitutive relation of the power form, J. Mech. Working Technol.. No. 15, p.143–165.

DOI: 10.1016/0378-3804(87)90031-3

[50] A. Rusinek, J.R. Klepaczko, Shear testing of sheet steel at wide range of strain rates and a constitutive relation with strain-rate and temperature dependence of the flow stress. Int. J. Plasticity, No. 17, pp.87-115.

DOI: 10.1016/s0749-6419(00)00020-6

[51] L. Écsi, Extended NOIHKH model usage for cyclic plasticity of metals, Engineering mechanics, Roč. 13, č. 2, 83-92.

[52] J. Lemaitre, Handbook of material behavior models, Vol. 1, Deformations of materials, Academic press, London, 2001.

[53] E.A. De Souza Neto, D. Perić, D.R.J. Owen, Computational methods for plasticity, Theory and applications, John Wiley & Sons Ltd., Singapore, 2008.

[54] M. Šilhavý, The mechanics and thermodynamics of continuous media, Springer-Verlag, Berlin Heidelberg, 1997.

[55] I. Müller, Thermodynamics, Pitman publishing LTD., London, 1985.

[56] P. Haupt, Continuum mechanics and theory of materials, second ed., Springer-Verlag, Berlin Heidelberg, 2002.

[57] J. F. Nye. Physical properties of crystals: Their representation by tensors and matrices. Clarendon Press. Oxford. 1957.

[58] J. Bonet, R.D. Wood, Nonlinear continuum mechanics for finite element analysis, 2nd. Ed., Cambridge University Press, Cambridge, 2008.

[59] J.R. Klepazko, A. Rusinek, Experiments on heat generation during plastic deformation and stored energy for TRIP steels, Materials and Design, 30 (2009) 35-48.

DOI: 10.1016/j.matdes.2008.04.048

[60] F. Trebuňa, F. Šimčák, Príručka experimentálnej mechaniky, Edícia vedeckej a odbornej literatúry, TypoPress, Košice, 2007.

[61] A. Služalec, An evaluation of the internal dissipation factor in coupled thermoplasticity, Int. J. Non-Lin. Mech., 25 4 (1990) 395-403.

[62] T. Kuwabara, S. Ikeda, K. Kuroda , Measurement and analysis of differential work hardening in cold-rolled steel sheet under biaxial tension, J. Mater. Proc. Technol., No. 80-81, pp.517-523.

DOI: 10.1016/s0924-0136(98)00155-1

[63] W. Müller, Beitrag zur Charakterisierung von Blechwerkstoffen unter merachsiger Beamspruchung, Springer Verlag, Berlin-Heidelberg, 1996.

[64] H. Franke, Lexikon der Physik, Franckh'sche Verlagshandlung, W. Keller & Co, Stuttgart, 1959.

[65] M.L. Pastor, X. Balandraud, M. Grédiac, J.L. Robert, Applying infrared thermography to study the heating of 2024-T3 aluminium specimens under fatigue loading, Infrared Phys. Techn. 51 (2008) 505-515.

DOI: 10.1016/j.infrared.2008.01.001