An evaluation of discrepancies between CPFE simulations and mean-field approximations for dual phase materials

Shahrzad Mirhosseini¹ *, E. H. Atzema^1,2, A. H. van den Boogaard¹

¹Chair of Nonlinear Solid Mechanics, University of Twente, 7522 NB Enschede, The Netherlands.

²Tata Steel Research and Development, 1970 CA, IJmuiden, The Netherlands.

*corresponding author

DOI:

https://doi.org/10.7494/cmms.2025.3.1024

Abstract:

This paper explores the discrepancies observed between 2D and 3D crystal plasticity finite element (CPFE) simulations and mean-field approximations in terms of macroscopic flow curves. Two hypotheses are proposed to address the discrepancies: (1) the type of yield function in the mean-field approach (2) differences in stress states between the two methodologies. Based on the first hypothesis, the type of yield function may influence the stress-strain partitioning in the mean-field approach. Consequently, the von Mises criterion is replaced with the Hershey yield function. To test the second hypothesis, CPFE simulations are extended to 3D to achieve comparable stress states in both methods. This analysis reveals that the exact shape of the yield function has a marginal impact on the discrepancies, whereas the proper 3D stress distribution significantly reduces them. This comprehensive study also uncovers a limitation of the mean-field approach in terms of accuracy in the prediction of macroscopic material response and stress partitioning for a two-phase polycrystalline material.

Cite as:

Mirhosseini, S., Atzema, E., & van den Boogaard, A. (2025). An evaluation of discrepancies between CPFE simulations and mean-field approximations for dual phase materials. Computer Methods in Materials Science, 25(3), 19-33. https://doi.org/10.7494/cmms.2025.3.1024

Article (PDF):

Keywords:

Crystal plasticity, Finite element simulations, Mean-field model, Hershey yield function, Arbitrarily-shaped RVEs, Periodic boundary conditions

Publication dates:

Received: 08.06.2025, Accepted: 15.09.2025, Published: 26.11.2025

Publication type:

Original scientific paper

References:

Asaro, R. J. (1983). Crystal plasticity. Journal of Applied Mechanics, 50(4b), 921–934.

Asik, E. E. (2019). Damage in Dual Phase Steels [PhD thesis, University of Twente]. University of Twente Research Information. https://doi.org/10.3990/1.9789036548816

Becker, R. (1991). Analysis of texture evolution in channel die compression – I. Effects of grain interaction. Acta Metallurgica et Materialia, 39(6), 1211–1230. https://doi.org/10.1016/0956-7151(91)90209-J

Bronkhorst, C. A., Kalidindi, S. R., & Anand, L. (1992). Polycrystalline plasticity and the evolution of crystallographic texture in FCC metals. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, 341(1662), 443–477. https://doi.org/10.1098/rsta.1992.0111

Eshelby, J. D. (1957). The determination of the elastic field of an ellipsoidal inclusion, and related problems. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 241(1226), 376–396. https://doi.org/10.1098/rspa.1957.0133

Eshelby, J. D. (1959). The elastic field outside an ellipsoidal inclusion. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 252(1271), 561–569. https://doi.org/10.1098/rspa.1959.0173

Hershey, A. V. (1954). The plasticity of an isotropic aggregate of anisotropic face-centered cubic crystals. Journal of Applied Mechanics, 21(3), 241–249. https://doi.org/10.1115/1.4010900

Hill, R. (1965). A self-consistent mechanics of composite materials. Journal of the Mechanics and Physics of Solids, 13(4), 213–222. https://doi.org/10.1016/0022-5096(65)90010-4

Hirth, J. P., & Lothe, J. (1982). Theory of Dislocations (2nd ed.). Wiley.

Hosford, W. F. (1996). On the crystallographic basis of yield criteria. Textures and Microstructures, 26–27, 479–493. https://doi.org/10.1155/TSM.26-27.479

Hutchinson, J. W. (1970). Elastic-plastic behaviour of polycrystalline metals and composites. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 319(1537), 247–272. https://doi.org/10.1098/rspa.1970.0177

Lebensohn, R. A., & Tomé, C. N. (1993). A self-consistent anisotropic approach for the simulation of plastic deformation and texture development of polycrystals: Application to zirconium alloys. Acta Metallurgica et Materialia, 41(9), 2611–2624. https://doi.org/10.1016/0956-7151(93)90130-K

Lebensohn, R. A., Liu, Y., & Castañeda, P. P. (2004). On the accuracy of the self-consistent approximation for polycrystals: Comparison with full-field numerical simulations. Acta Materialia, 52(18), 5347–5361. https://doi.org/10.1016/j.actamat.2004.07.040

Lebensohn, R. A., Castañeda, P. P., Brenner, R., & Castelnau, O. (2011). Full-field vs. homogenization methods to predict microstructure–property relations for polycrystalline materials. In S. Ghosh, D. Dimiduk (Eds.), Computational Methods for Microstructure- Property Relationships (pp. 393–441). Springer New York, NY. https://doi.org/10.1007/978-1-4419-0643-4_11

Lloyd, S. P. (1982). Least squares quantization in PCM. IEEE Transactions on Information Theory, 28(2), 129–137. https://doi.org/10.1109/TIT.1982.1056489

Logé, R. E., Bernard, P., Huang, K., Bag, S., & Bernacki, M. (2012). Mean field and finite element modeling of static and dynamic recrystallization. Materials Science Forum, 715–716, 737–737. https://doi.org/10.4028/www.scientific.net/MSF.715-716.737

Mirhosseini, S., Perdahcıoğlu, E., Atzema, E., & Boogaard, T., van den (2021). On the multiscale analysis of a two phase material: crystal plasticity versus mean field. In A.-M. Habraken (Ed.), ESAFORM 2021: Proceedings of the 24th International Conference on Material Forming. 14–16 April 2021.

Mirhosseini, S., Perdahcıoğlu, E. S., Atzema, E. H., & Boogaard, A. H., van den (2023). Response of 2D and 3D crystal plasticity models subjected to plane strain condition. Mechanics Research Communications, 128, 104047. https://doi.org/10.1016/j.mechrescom.2023.104047

Niskanen, I. (2023). Mean-field crystal plasticity compared to full-field crystal plasticity in fatigue modelling [Master’s thesis, University of Oulu]. OuluREPO. https://urn.fi/URN:NBN:fi:oulu-202306212698

Peirce, D., Asaro, R. J., & Needleman, A. (1982). An analysis of nonuniform and localized deformation in ductile single crystals. Acta Metallurgica, 30(6), 1087–1119. https://doi.org/10.1016/0001-6160(82)90005-0

Perdahcıoğlu, E. S. (2008). Consitutive Modeling of Metastable Austenitic Stainless Steel [PhD thesis, University of Twente]. https://doi.org/10.3990/1.9789036527699

Perdahcıoğlu, E. S. (2024). A rate-independent crystal plasticity algorithm based on the interior point method. Computer Methods in Applied Mechanics and Engineering, 418(A), 116533. https://doi.org/10.1016/j.cma.2023.116533

Perdahcıoğlu, E. S., Mirhosseini, S., & Boogaard, A. H., van den. (2021). Self-consistent, polycrystal rate-independent crystal plasticity modeling for yield surface determination. In A.-M. Habraken (Ed.), ESAFORM 2021: Proceedings of the 24th International Conference on Material Forming. 14–16 April 2021.

Roters, F., Eisenlohr, P., Bieler, T. R., & Raabe, D. (2011). Crystal Plasticity Finite Element Methods in Materials Science and Engineering. Wiley.

Rycroft, Ch. H. (2009). Voro++: A Three-Dimensional Voronoi Cell Library in C++. https://doi.org/10.2172/946741

Sadd, M. H. (2009). Elasticity: Theory, Applications, and Numerics. Academic Press.

Segurado, J., Lebensohn, R. A., & LLorca, J. (2018). Computational homogenization of polycrystals. Advances in Applied Mechanics, 51, 1–114. https://doi.org/10.1016/bs.aams.2018.07.001

Taylor, G. I. (1938). Plastic strain in metals. Journal of the Institute of Metals, 62, 307–324.

Tehrani, J. (2016). Derivative of eigenvectors of a matrix with respect to its components [Online forum post]. MathOverflow.
https://mathoverflow.net/questions/229425/derivative-of-eigenvectors-of-a-matrix-with-respect-to-its-components