# American Institute of Mathematical Sciences

February  2013, 6(1): 1-16. doi: 10.3934/dcdss.2013.6.1

## Relaxation and microstructure in a model for finite crystal plasticity with one slip system in three dimensions

 1 Institut für Angewandte Mathematik, Universität Bonn, Endenicher Allee 60，53115 Bonn, Germany 2 Fakultät für Mathematik, Universität Regensburg, 93040 Regensburg, Germany 3 Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, PA 15213, United States

Received  May 2011 Revised  July 2011 Published  October 2012

Modern theories in crystal plasticity are based on a multiplicative decomposition of the deformation gradient into an elastic and a plastic part. The free energy of the associated variational problems is given by the sum of an elastic and a plastic energy. For a model with one slip system in a three-dimensional setting it is shown that the relaxation of the model with rigid elasticity can be approximated in the sense of $\Gamma$-convergence by models with finite elastic energy and diverging elastic constants.
Citation: Sergio Conti, Georg Dolzmann, Carolin Kreisbeck. Relaxation and microstructure in a model for finite crystal plasticity with one slip system in three dimensions. Discrete & Continuous Dynamical Systems - S, 2013, 6 (1) : 1-16. doi: 10.3934/dcdss.2013.6.1
##### References:
 [1] J. M. Ball and F. Murat, $W^{1,p}$ quasiconvexity and variational prblems for multiple integrals,, J. Funct. Anal., 58 (1984), 225. doi: 10.1016/0022-1236(84)90041-7. Google Scholar [2] A. Braides, "$\Gamma$-Convergence for Beginners,", Oxford Lecture Series in Mathematics and its Applications 22. Oxford: Oxford University Press, (2002). Google Scholar [3] C. Carstensen, K. Hackl and A. Mielke, Non-convex potentials and microstructures in finite-strain plasticity,, R. Soc. Lond. Proc. Ser. A, 458 (2002), 299. doi: 10.1098/rspa.2001.0864. Google Scholar [4] S. Conti, Relaxation of single-slip single-crystal plasticity with linear hardening,, in, (2006), 30. Google Scholar [5] S. Conti, G. Dolzmann and C. Klust, Relaxation of a class of variational models in crystal plasticity,, Proc. R. Soc. Lond. Ser. A, 465 (2009), 1735. doi: 10.1098/rspa.2008.0390. Google Scholar [6] S. Conti, G. Dolzmann and C. Kreisbeck, Geometrically nonlinear models in crystal plasticity and the limit of rigid elasticity,, PAMM, 10 (2010), 3. doi: 10.1002/pamm.201010002. Google Scholar [7] S. Conti, G. Dolzmann and C. Kreisbeck, Asymptotic behavior of crystal plasticity with one slip system in the limit of rigid elasticity,, SIAM J. Math. Analysis, 43 (2011), 2337. doi: 10.1137/100810320. Google Scholar [8] S. Conti, G. Dolzmann and C. Kreisbeck, Relaxation of a model in finite plasticity with two slip systems,, Submitted (2011)., (2011). Google Scholar [9] S. Conti, G. Dolzmann and S. Müller, The div-curl lemma for sequences whose divergence and curl are compact in $W^{-1,1}$,, C. R. Acad. Sci. Paris, 349 (2011), 175. doi: 10.1016/j.crma.2010.11.013. Google Scholar [10] S. Conti and F. Theil, Single-slip elastoplastic microstructures,, Arch. Ration. Mech. Anal., 178 (2005), 125. doi: 10.1007/s00205-005-0371-8. Google Scholar [11] B. Dacorogna, "Direct Methods in the Calculus of Variations,", Applied Mathematical Sciences, (1989). Google Scholar [12] G. Dal Maso, "An Introduction to $\Gamma$-Convergence,", Birkhäuser, (1993). Google Scholar [13] E. De Giorgi, Sulla convergenza di alcune successioni d'integrali del tipo dell'area,, Rend. Mat., IV (1975), 277. Google Scholar [14] E. De Giorgi and T. Franzoni, Su un tipo di convergenza variazionale,, Atti Accad. Naz. Lincei Rend. Cl. Sci. Mat., 8 (1975), 842. Google Scholar [15] A. De Simone and G. Dolzmann, Macroscopic response of nematic elastomers via relaxation of a class of so(3)-invariant energies,, Arch. Ration. Mech. Anal., 161 (2002), 181. doi: 10.1007/s002050100174. Google Scholar [16] R. V. Kohn, The relaxation of a double-well energy,, Contin. Mech. Thermodyn, 3 (1991), 193. doi: 10.1007/BF01135336. Google Scholar [17] C. Kreisbeck, "Analytical Aspects of Relaxation for Models in Crystal Plasticity,", PhD thesis, (2010). Google Scholar [18] H. Le Dret and A. Raoult, The quasiconvex envelope of the Saint-Venant-Kirchhoff stored energy function,, Proc. R. Soc. Edinb., 125 (1995), 1179. Google Scholar [19] E. H. Lee, Elastic-plastic deformation at finite strains,, J. Appl. Mech., 36 (1969), 1. doi: 10.1115/1.3564580. Google Scholar [20] K. Lurie and A. Cherkaev, On a certain variational problem of phase equilibrium,, Material instabilities in continuum mechanics, (1988), 257. Google Scholar [21] S. Müller, Variational models for microstructure and phase transitions., in, (1999), 85. Google Scholar [22] F. Murat, Compacité par compensation,, Ann. Sc. Norm. Super. Pisa, 5 (1978), 489. Google Scholar [23] F. Murat, Compacité par compensation: condition necessaire et suffisante de continuite faible sous une hypothèse de rang constant,, Ann. Sc. Norm. Super. Pisa, 8 (1981), 69. Google Scholar [24] M. Ortiz and E. A. Repetto, Nonconvex energy minimization and dislocation structures in ductile single crystals,, J. Mech. Phys. Solids, 47 (1999), 397. doi: 10.1016/S0022-5096(97)00096-3. Google Scholar [25] A. C. Pipkin, Elastic materials with two preferred states,, Q. J. Mech. Appl. Math., 44 (1991), 1. doi: 10.1093/qjmam/44.1.1. Google Scholar [26] L. Tartar, Une nouvelle méthode de résolution d'équations aux dérivées partielles non linéaires,, Journ. d'Anal. non lin., 665 (1978), 228. doi: 10.1007/BFb0061808. Google Scholar [27] L. Tartar, Compensated compactness and applications to partial differential equations,, Nonlinear Analysis and Mechanics: Heriot-Watt Symp., 39 (1979), 136. Google Scholar [28] H. Triebel, "Interpolation Theory, Function Spaces, Differential Operators,", North-Holland Mathematical Library. Vol. 18. North-Holland Publishing Company, (1978). Google Scholar

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##### References:
 [1] J. M. Ball and F. Murat, $W^{1,p}$ quasiconvexity and variational prblems for multiple integrals,, J. Funct. Anal., 58 (1984), 225. doi: 10.1016/0022-1236(84)90041-7. Google Scholar [2] A. Braides, "$\Gamma$-Convergence for Beginners,", Oxford Lecture Series in Mathematics and its Applications 22. Oxford: Oxford University Press, (2002). Google Scholar [3] C. Carstensen, K. Hackl and A. Mielke, Non-convex potentials and microstructures in finite-strain plasticity,, R. Soc. Lond. Proc. Ser. A, 458 (2002), 299. doi: 10.1098/rspa.2001.0864. Google Scholar [4] S. Conti, Relaxation of single-slip single-crystal plasticity with linear hardening,, in, (2006), 30. Google Scholar [5] S. Conti, G. Dolzmann and C. Klust, Relaxation of a class of variational models in crystal plasticity,, Proc. R. Soc. Lond. Ser. A, 465 (2009), 1735. doi: 10.1098/rspa.2008.0390. Google Scholar [6] S. Conti, G. Dolzmann and C. Kreisbeck, Geometrically nonlinear models in crystal plasticity and the limit of rigid elasticity,, PAMM, 10 (2010), 3. doi: 10.1002/pamm.201010002. Google Scholar [7] S. Conti, G. Dolzmann and C. Kreisbeck, Asymptotic behavior of crystal plasticity with one slip system in the limit of rigid elasticity,, SIAM J. Math. Analysis, 43 (2011), 2337. doi: 10.1137/100810320. Google Scholar [8] S. Conti, G. Dolzmann and C. Kreisbeck, Relaxation of a model in finite plasticity with two slip systems,, Submitted (2011)., (2011). Google Scholar [9] S. Conti, G. Dolzmann and S. Müller, The div-curl lemma for sequences whose divergence and curl are compact in $W^{-1,1}$,, C. R. Acad. Sci. Paris, 349 (2011), 175. doi: 10.1016/j.crma.2010.11.013. Google Scholar [10] S. Conti and F. Theil, Single-slip elastoplastic microstructures,, Arch. Ration. Mech. Anal., 178 (2005), 125. doi: 10.1007/s00205-005-0371-8. Google Scholar [11] B. Dacorogna, "Direct Methods in the Calculus of Variations,", Applied Mathematical Sciences, (1989). Google Scholar [12] G. Dal Maso, "An Introduction to $\Gamma$-Convergence,", Birkhäuser, (1993). Google Scholar [13] E. De Giorgi, Sulla convergenza di alcune successioni d'integrali del tipo dell'area,, Rend. Mat., IV (1975), 277. Google Scholar [14] E. De Giorgi and T. Franzoni, Su un tipo di convergenza variazionale,, Atti Accad. Naz. Lincei Rend. Cl. Sci. Mat., 8 (1975), 842. Google Scholar [15] A. De Simone and G. Dolzmann, Macroscopic response of nematic elastomers via relaxation of a class of so(3)-invariant energies,, Arch. Ration. Mech. Anal., 161 (2002), 181. doi: 10.1007/s002050100174. Google Scholar [16] R. V. Kohn, The relaxation of a double-well energy,, Contin. Mech. Thermodyn, 3 (1991), 193. doi: 10.1007/BF01135336. Google Scholar [17] C. Kreisbeck, "Analytical Aspects of Relaxation for Models in Crystal Plasticity,", PhD thesis, (2010). Google Scholar [18] H. Le Dret and A. Raoult, The quasiconvex envelope of the Saint-Venant-Kirchhoff stored energy function,, Proc. R. Soc. Edinb., 125 (1995), 1179. Google Scholar [19] E. H. Lee, Elastic-plastic deformation at finite strains,, J. Appl. Mech., 36 (1969), 1. doi: 10.1115/1.3564580. Google Scholar [20] K. Lurie and A. Cherkaev, On a certain variational problem of phase equilibrium,, Material instabilities in continuum mechanics, (1988), 257. Google Scholar [21] S. Müller, Variational models for microstructure and phase transitions., in, (1999), 85. Google Scholar [22] F. Murat, Compacité par compensation,, Ann. Sc. Norm. Super. Pisa, 5 (1978), 489. Google Scholar [23] F. Murat, Compacité par compensation: condition necessaire et suffisante de continuite faible sous une hypothèse de rang constant,, Ann. Sc. Norm. Super. Pisa, 8 (1981), 69. Google Scholar [24] M. Ortiz and E. A. Repetto, Nonconvex energy minimization and dislocation structures in ductile single crystals,, J. Mech. Phys. Solids, 47 (1999), 397. doi: 10.1016/S0022-5096(97)00096-3. Google Scholar [25] A. C. Pipkin, Elastic materials with two preferred states,, Q. J. Mech. Appl. Math., 44 (1991), 1. doi: 10.1093/qjmam/44.1.1. Google Scholar [26] L. Tartar, Une nouvelle méthode de résolution d'équations aux dérivées partielles non linéaires,, Journ. d'Anal. non lin., 665 (1978), 228. doi: 10.1007/BFb0061808. Google Scholar [27] L. Tartar, Compensated compactness and applications to partial differential equations,, Nonlinear Analysis and Mechanics: Heriot-Watt Symp., 39 (1979), 136. Google Scholar [28] H. Triebel, "Interpolation Theory, Function Spaces, Differential Operators,", North-Holland Mathematical Library. Vol. 18. North-Holland Publishing Company, (1978). Google Scholar
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