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Numerical Study of Microstructures in Single-Slip Finite Elastoplasticity

  • Sergio Conti
  • Georg DolzmannEmail author
Article
  • 19 Downloads

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.

Keywords

Elastoplasticity Single slip Quasiconvexity Relaxation 

Mathematics Subject Classification

49J45 74B20 74C15 

Notes

Acknowledgements

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

References

  1. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Kröner, E.: Allgemeine Kontinuumstheorie der Versetzungen und Eigenspannungen. Arch. Ration. Mech. Anal. 4, 273–334 (1960)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Lee, E.H.: Elastic-plastic deformation at finite strains. J. Appl. Mech. 36, 1–5 (1969)CrossRefzbMATHGoogle Scholar
  5. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Mariano, P.M.: Covariance in plasticity. R. Soc. Lond. Proc. Ser. A 469, 20130073 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Morrey Jr., C.B.: Quasi-convexity and the lower semicontinuity of multiple integrals. Pac. J. Math. 2, 25–53 (1952)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Dacorogna, B.: Direct Methods in the Calculus of Variations, vol. 78. Springer, Berlin (2007)zbMATHGoogle Scholar
  9. 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. 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)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Parry, G.P.: On the planar rank-one convexity condition. Proc. R. Soc. Edinb. Sect. A 125(2), 247–264 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Kohn, R.V., Strang, G.: Optimal design and relaxation of variational problems. II. Commun. Pure Appl. Math. 39, 139–182 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Conti, S., Theil, F.: Single-slip elastoplastic microstructures. Arch. Ration. Mech. Anal. 178, 125–148 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 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. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 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. 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. 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)zbMATHGoogle Scholar
  20. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 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)zbMATHGoogle Scholar
  22. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Carstensen, C., Conti, S., Orlando, A.: Mixed analytical-numerical relaxation in finite single-slip crystal plasticity. Contin. Mech. Thermod. 20, 275–301 (2008)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Universität BonnBonnGermany
  2. 2.Universität RegensburgRegensburgGermany

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