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

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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

Sergio Conti ^1, , Georg Dolzmann ^2, and Carolin Kreisbeck ^3,

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

Abstract
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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.

Keywords: microstructure, relaxation, $\Gamma$-convergence., Single-crystal plasticity.

Mathematics Subject Classification: Primary: 49J45; Secondary: 74C1.

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 and 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), 225-253. doi: 10.1016/0022-1236(84)90041-7.
[2]	A. Braides, "$\Gamma$-Convergence for Beginners," Oxford Lecture Series in Mathematics and its Applications 22. Oxford: Oxford University Press, 2002.
[3]	C. Carstensen, K. Hackl and A. Mielke, Non-convex potentials and microstructures in finite-strain plasticity, R. Soc. Lond. Proc. Ser. A, Math. Phys. Eng. Sci., 458 (2002), 299-317. doi: 10.1098/rspa.2001.0864.
[4]	S. Conti, Relaxation of single-slip single-crystal plasticity with linear hardening, in "Multiscale Materials Modeling" (Ed. P. Gumbsch), Freiburg, Fraunhofer IRB, (2006), 30-35.
[5]	S. Conti, G. Dolzmann and C. Klust, Relaxation of a class of variational models in crystal plasticity, Proc. R. Soc. Lond. Ser. A, Math. Phys. Eng. Sci., 465 (2009), 1735-1742. doi: 10.1098/rspa.2008.0390.
[6]	S. Conti, G. Dolzmann and C. Kreisbeck, Geometrically nonlinear models in crystal plasticity and the limit of rigid elasticity, PAMM, 10 (2010), 3-6. doi: 10.1002/pamm.201010002.
[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), 2337-2353. doi: 10.1137/100810320.
[8]	S. Conti, G. Dolzmann and C. Kreisbeck, Relaxation of a model in finite plasticity with two slip systems, Submitted (2011).
[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, Ser. I, 349 (2011), 175-178. doi: 10.1016/j.crma.2010.11.013.
[10]	S. Conti and F. Theil, Single-slip elastoplastic microstructures, Arch. Ration. Mech. Anal., 178 (2005), 125-148. doi: 10.1007/s00205-005-0371-8.
[11]	B. Dacorogna, "Direct Methods in the Calculus of Variations," Applied Mathematical Sciences, 78, Springer-Verlag, Berlin, 1989.
[12]	G. Dal Maso, "An Introduction to $\Gamma$-Convergence," Birkhäuser, 1993.
[13]	E. De Giorgi, Sulla convergenza di alcune successioni d'integrali del tipo dell'area, Rend. Mat., IV (1975), 277-294.
[14]	E. De Giorgi and T. Franzoni, Su un tipo di convergenza variazionale, Atti Accad. Naz. Lincei Rend. Cl. Sci. Mat., 8 (1975), 842-850.
[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), 181-204. doi: 10.1007/s002050100174.
[16]	R. V. Kohn, The relaxation of a double-well energy, Contin. Mech. Thermodyn, 3 (1991), 193-236. doi: 10.1007/BF01135336.
[17]	C. Kreisbeck, "Analytical Aspects of Relaxation for Models in Crystal Plasticity," PhD thesis, University Regensburg, 2010.
[18]	H. Le Dret and A. Raoult, The quasiconvex envelope of the Saint-Venant-Kirchhoff stored energy function, Proc. R. Soc. Edinb., Sect. A, 125 (1995), 1179-1192.
[19]	E. H. Lee, Elastic-plastic deformation at finite strains, J. Appl. Mech., 36 (1969), 1-6. doi: 10.1115/1.3564580.
[20]	K. Lurie and A. Cherkaev, On a certain variational problem of phase equilibrium, Material instabilities in continuum mechanics, Proc. Symp. Edinburgh/Scotl, (1985/86), 257-268, Oxford Sci. Publ., Oxford Univ. Press, New York, 1988.
[21]	S. Müller, Variational models for microstructure and phase transitions. in "Calculus of Variations and Geometric Evolution Problems (1999)" (Eds. F. Bethuel et al.), Springer Lecture Notes in Math. Springer-Verlag, (1713), 85-210.
[22]	F. Murat, Compacité par compensation, Ann. Sc. Norm. Super. Pisa, Cl. Sci., Ser. (IV), 5 (1978), 489-507.
[23]	F. Murat, Compacité par compensation: condition necessaire et suffisante de continuite faible sous une hypothèse de rang constant, Ann. Sc. Norm. Super. Pisa, Cl. Sci., Ser. (IV), 8 (1981), 69-102.
[24]	M. Ortiz and E. A. Repetto, Nonconvex energy minimization and dislocation structures in ductile single crystals, J. Mech. Phys. Solids, 47 (1999), 397-462. doi: 10.1016/S0022-5096(97)00096-3.
[25]	A. C. Pipkin, Elastic materials with two preferred states, Q. J. Mech. Appl. Math., 44 (1991), 1-15. doi: 10.1093/qjmam/44.1.1.
[26]	L. Tartar, Une nouvelle méthode de résolution d'équations aux dérivées partielles non linéaires, Journ. d'Anal. non lin., Proc., Besancon 1977, Lect. Notes Math., 665 (1978), 228-241. doi: 10.1007/BFb0061808.
[27]	L. Tartar, Compensated compactness and applications to partial differential equations, Nonlinear Analysis and Mechanics: Heriot-Watt Symp., Vol. 4, Edinburgh 1979, Res. Notes Math., 39 (1979), 136-212.
[28]	H. Triebel, "Interpolation Theory, Function Spaces, Differential Operators," North-Holland Mathematical Library. Vol. 18. North-Holland Publishing Company, Amsterdam, 1978.

show all references

References:

[1]	J. M. Ball and F. Murat, $ W^{1,p}$ quasiconvexity and variational prblems for multiple integrals, J. Funct. Anal., 58 (1984), 225-253. doi: 10.1016/0022-1236(84)90041-7.
[2]	A. Braides, "$\Gamma$-Convergence for Beginners," Oxford Lecture Series in Mathematics and its Applications 22. Oxford: Oxford University Press, 2002.
[3]	C. Carstensen, K. Hackl and A. Mielke, Non-convex potentials and microstructures in finite-strain plasticity, R. Soc. Lond. Proc. Ser. A, Math. Phys. Eng. Sci., 458 (2002), 299-317. doi: 10.1098/rspa.2001.0864.
[4]	S. Conti, Relaxation of single-slip single-crystal plasticity with linear hardening, in "Multiscale Materials Modeling" (Ed. P. Gumbsch), Freiburg, Fraunhofer IRB, (2006), 30-35.
[5]	S. Conti, G. Dolzmann and C. Klust, Relaxation of a class of variational models in crystal plasticity, Proc. R. Soc. Lond. Ser. A, Math. Phys. Eng. Sci., 465 (2009), 1735-1742. doi: 10.1098/rspa.2008.0390.
[6]	S. Conti, G. Dolzmann and C. Kreisbeck, Geometrically nonlinear models in crystal plasticity and the limit of rigid elasticity, PAMM, 10 (2010), 3-6. doi: 10.1002/pamm.201010002.
[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), 2337-2353. doi: 10.1137/100810320.
[8]	S. Conti, G. Dolzmann and C. Kreisbeck, Relaxation of a model in finite plasticity with two slip systems, Submitted (2011).
[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, Ser. I, 349 (2011), 175-178. doi: 10.1016/j.crma.2010.11.013.
[10]	S. Conti and F. Theil, Single-slip elastoplastic microstructures, Arch. Ration. Mech. Anal., 178 (2005), 125-148. doi: 10.1007/s00205-005-0371-8.
[11]	B. Dacorogna, "Direct Methods in the Calculus of Variations," Applied Mathematical Sciences, 78, Springer-Verlag, Berlin, 1989.
[12]	G. Dal Maso, "An Introduction to $\Gamma$-Convergence," Birkhäuser, 1993.
[13]	E. De Giorgi, Sulla convergenza di alcune successioni d'integrali del tipo dell'area, Rend. Mat., IV (1975), 277-294.
[14]	E. De Giorgi and T. Franzoni, Su un tipo di convergenza variazionale, Atti Accad. Naz. Lincei Rend. Cl. Sci. Mat., 8 (1975), 842-850.
[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), 181-204. doi: 10.1007/s002050100174.
[16]	R. V. Kohn, The relaxation of a double-well energy, Contin. Mech. Thermodyn, 3 (1991), 193-236. doi: 10.1007/BF01135336.
[17]	C. Kreisbeck, "Analytical Aspects of Relaxation for Models in Crystal Plasticity," PhD thesis, University Regensburg, 2010.
[18]	H. Le Dret and A. Raoult, The quasiconvex envelope of the Saint-Venant-Kirchhoff stored energy function, Proc. R. Soc. Edinb., Sect. A, 125 (1995), 1179-1192.
[19]	E. H. Lee, Elastic-plastic deformation at finite strains, J. Appl. Mech., 36 (1969), 1-6. doi: 10.1115/1.3564580.
[20]	K. Lurie and A. Cherkaev, On a certain variational problem of phase equilibrium, Material instabilities in continuum mechanics, Proc. Symp. Edinburgh/Scotl, (1985/86), 257-268, Oxford Sci. Publ., Oxford Univ. Press, New York, 1988.
[21]	S. Müller, Variational models for microstructure and phase transitions. in "Calculus of Variations and Geometric Evolution Problems (1999)" (Eds. F. Bethuel et al.), Springer Lecture Notes in Math. Springer-Verlag, (1713), 85-210.
[22]	F. Murat, Compacité par compensation, Ann. Sc. Norm. Super. Pisa, Cl. Sci., Ser. (IV), 5 (1978), 489-507.
[23]	F. Murat, Compacité par compensation: condition necessaire et suffisante de continuite faible sous une hypothèse de rang constant, Ann. Sc. Norm. Super. Pisa, Cl. Sci., Ser. (IV), 8 (1981), 69-102.
[24]	M. Ortiz and E. A. Repetto, Nonconvex energy minimization and dislocation structures in ductile single crystals, J. Mech. Phys. Solids, 47 (1999), 397-462. doi: 10.1016/S0022-5096(97)00096-3.
[25]	A. C. Pipkin, Elastic materials with two preferred states, Q. J. Mech. Appl. Math., 44 (1991), 1-15. doi: 10.1093/qjmam/44.1.1.
[26]	L. Tartar, Une nouvelle méthode de résolution d'équations aux dérivées partielles non linéaires, Journ. d'Anal. non lin., Proc., Besancon 1977, Lect. Notes Math., 665 (1978), 228-241. doi: 10.1007/BFb0061808.
[27]	L. Tartar, Compensated compactness and applications to partial differential equations, Nonlinear Analysis and Mechanics: Heriot-Watt Symp., Vol. 4, Edinburgh 1979, Res. Notes Math., 39 (1979), 136-212.
[28]	H. Triebel, "Interpolation Theory, Function Spaces, Differential Operators," North-Holland Mathematical Library. Vol. 18. North-Holland Publishing Company, Amsterdam, 1978.

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