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Numerical Study of Microstructures in Single-Slip Finite Elastoplasticity

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Abstract

A model problem in finite elastoplasticity with one active slip system in two dimensions is considered. It is based on the multiplicative decomposition of the deformation gradient and includes an elastic response, dissipation and linear hardening. The focus lies on deformation theory of plasticity, which corresponds to a single time step in the variational formulation of the incremental problem. The formation of microstructures in different regions of phase space is analyzed, and it is shown that first-order laminates play an important role in the regime, where both dissipation and hardening are relevant, with second- and third-order laminates reducing the energy even further. No numerical evidence for laminates of order four or higher is found. For large shear and bulk modulus, numerical convergence to the rigid-plastic regime is verified. The main tool is an algorithm for the efficient search for optimal microstructures, which are determined by minimization of the condensed energy. The presently used algorithm and code are extensions of those previously developed for the study of relaxation in sheets of nematic elastomers.

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Notes

  1. We use (quasi)convex envelope for functions and (quasi)convex hull for sets, following common practice in this context, see, for example, [8, 9].

References

  1. Ortiz, M., Repetto, E.A.: Nonconvex energy minimization and dislocation structures in ductile single crystals. J. Mech. Phys. Solids 47(2), 397–462 (1999)

    Article  MathSciNet  Google Scholar 

  2. Carstensen, C., Hackl, K., Mielke, A.: Non-convex potentials and microstructures in finite-strain plasticity. R. Soc. Lond. Proc. Ser. A 458(2018), 299–317 (2002)

    Article  MathSciNet  Google Scholar 

  3. Kröner, E.: Allgemeine Kontinuumstheorie der Versetzungen und Eigenspannungen. Arch. Ration. Mech. Anal. 4, 273–334 (1960)

    Article  MathSciNet  Google Scholar 

  4. Lee, E.H.: Elastic-plastic deformation at finite strains. J. Appl. Mech. 36, 1–5 (1969)

    Article  Google Scholar 

  5. Reina, C., Conti, S.: Kinematic description of crystal plasticity in the finite kinematic framework: a micromechanical understanding of \(F=F^e F^p\). J. Mech. Phys. Solids 67, 40–61 (2014)

    Article  MathSciNet  Google Scholar 

  6. Mariano, P.M.: Covariance in plasticity. R. Soc. Lond. Proc. Ser. A 469, 20130073 (2013)

    Article  MathSciNet  Google Scholar 

  7. Morrey Jr., C.B.: Quasi-convexity and the lower semicontinuity of multiple integrals. Pac. J. Math. 2, 25–53 (1952)

    Article  MathSciNet  Google Scholar 

  8. Dacorogna, B.: Direct Methods in the Calculus of Variations, vol. 78. Springer, Berlin (2007)

    MATH  Google Scholar 

  9. Müller, S.: Variational models for microstructure and phase transitions. In: Bethuel, F., et al. (eds.) Calculus of Variations and Geometric Evolution Problems, Springer Lecture Notes in Math. 1713, pp. 85–210. Springer, Berlin (1999)

    Google Scholar 

  10. Conti, S., Dolzmann, G.: An adaptive relaxation algorithm for multiscale problems and application to nematic elastomers. J. Mech. Phys. Solids 113, 126–143 (2018)

    Article  MathSciNet  Google Scholar 

  11. Parry, G.P.: On the planar rank-one convexity condition. Proc. R. Soc. Edinb. Sect. A 125(2), 247–264 (1995)

    Article  MathSciNet  Google Scholar 

  12. Kohn, R.V., Strang, G.: Optimal design and relaxation of variational problems. II. Commun. Pure Appl. Math. 39, 139–182 (1986)

    Article  MathSciNet  Google Scholar 

  13. Conti, S., Theil, F.: Single-slip elastoplastic microstructures. Arch. Ration. Mech. Anal. 178, 125–148 (2005)

    Article  MathSciNet  Google Scholar 

  14. Conti, S.: Relaxation of single-slip single-crystal plasticity with linear hardening. In: Gumbsch, P. (ed.) Multiscale materials modeling, pp. 30–35. Fraunhofer IRB, Freiburg (2006)

    Google Scholar 

  15. Conti, S., Dolzmann, G., Kreisbeck, C.: Relaxation of a model in finite plasticity with two slip systems. Math. Models Methods Appl. Sci. 23, 2111–2128 (2013)

    Article  MathSciNet  Google Scholar 

  16. Conti, S., Dolzmann, G., Kreisbeck, C.: Asymptotic behavior of crystal plasticity with one slip system in the limit of rigid elasticity. SIAM J. Math. Anal. 43, 2337–2353 (2011)

    Article  MathSciNet  Google Scholar 

  17. Braides, A.: \(\Gamma \)-Convergence for Beginners, Oxford Lecture Series in Mathematics and its Applications, vol. 22. Oxford University Press, Oxford (2002)

    Google Scholar 

  18. Dal Maso, G.: An Introduction to \(\Gamma \)-Convergence. Progress in Nonlinear Differential Equations and their Applications, 8. Birkhäuser Boston Inc., Boston (1993)

    Google Scholar 

  19. Press, W.H., Teukolsky, S.A., Vetterling, W.T., Flannery, B.P.: Numerical Recipes 3rd Edition: The Art of Scientific Computing, 3rd edn. Cambridge University Press, New York (2007)

    MATH  Google Scholar 

  20. Conti, S., Dolzmann, G., Klust, C.: Relaxation of a class of variational models in crystal plasticity. R. Soc. Lond. Proc. Ser. A 465, 1735–1742 (2009)

    Article  MathSciNet  Google Scholar 

  21. Pitteri, M., Zanzotto, G.: Continuum Models for Phase Transitions and Twinning in Crystals, Applied Mathematics (Boca Raton), vol. 19. Chapman & Hall/CRC, Boca Raton (2003)

    MATH  Google Scholar 

  22. Miehe, C., Lambrecht, M., Gürses, E.: Analysis of material instabilities in inelastic solids by incremental energy minimization and relaxation methods: evolving deformation microstructures in finite plasticity. J. Mech. Phys. Solids 52, 2725–2769 (2004)

    Article  MathSciNet  Google Scholar 

  23. Bartels, S., Carstensen, C., Hackl, K., Hoppe, U.: Effective relaxation for microstructure simulations: algorithms and applications. Comput. Methods Appl. Mech. Eng. 193, 5143–5175 (2004)

    Article  MathSciNet  Google Scholar 

  24. Carstensen, C., Conti, S., Orlando, A.: Mixed analytical-numerical relaxation in finite single-slip crystal plasticity. Contin. Mech. Thermod. 20, 275–301 (2008)

    Article  MathSciNet  Google Scholar 

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Acknowledgements

This work was partially supported by the Deutsche Forschungsgemeinschaft through the Sonderforschungsbereich 1060 “The mathematics of emergent effects,” project A5.

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Correspondence to Georg Dolzmann.

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Conti, S., Dolzmann, G. Numerical Study of Microstructures in Single-Slip Finite Elastoplasticity. J Optim Theory Appl 184, 43–60 (2020). https://doi.org/10.1007/s10957-018-01460-0

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