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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

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 & 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.  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, Math. Phys. Eng. Sci., 458 (2002), 299-317. doi: 10.1098/rspa.2001.0864.  Google Scholar

[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. 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, Math. Phys. Eng. Sci., 465 (2009), 1735-1742. 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), 3-6. 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), 2337-2353. 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). 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, Ser. I, 349 (2011), 175-178. 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), 125-148. doi: 10.1007/s00205-005-0371-8.  Google Scholar

[11]

B. Dacorogna, "Direct Methods in the Calculus of Variations," Applied Mathematical Sciences, 78, Springer-Verlag, Berlin, 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), 277-294. 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), 842-850.  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), 181-204. doi: 10.1007/s002050100174.  Google Scholar

[16]

R. V. Kohn, The relaxation of a double-well energy, Contin. Mech. Thermodyn, 3 (1991), 193-236. doi: 10.1007/BF01135336.  Google Scholar

[17]

C. Kreisbeck, "Analytical Aspects of Relaxation for Models in Crystal Plasticity," PhD thesis, University Regensburg, 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., Sect. A, 125 (1995), 1179-1192.  Google Scholar

[19]

E. H. Lee, Elastic-plastic deformation at finite strains, J. Appl. Mech., 36 (1969), 1-6. 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, Proc. Symp. Edinburgh/Scotl, (1985/86), 257-268, Oxford Sci. Publ., Oxford Univ. Press, New York, 1988.  Google Scholar

[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.  Google Scholar

[22]

F. Murat, Compacité par compensation, Ann. Sc. Norm. Super. Pisa, Cl. Sci., Ser. (IV), 5 (1978), 489-507.  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, Cl. Sci., Ser. (IV), 8 (1981), 69-102.  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), 397-462. 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), 1-15. 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., Proc., Besancon 1977, Lect. Notes Math., 665 (1978), 228-241. doi: 10.1007/BFb0061808.  Google Scholar

[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.  Google Scholar

[28]

H. Triebel, "Interpolation Theory, Function Spaces, Differential Operators," North-Holland Mathematical Library. Vol. 18. North-Holland Publishing Company, Amsterdam, 1978.  Google Scholar

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.  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, Math. Phys. Eng. Sci., 458 (2002), 299-317. doi: 10.1098/rspa.2001.0864.  Google Scholar

[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. 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, Math. Phys. Eng. Sci., 465 (2009), 1735-1742. 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), 3-6. 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), 2337-2353. 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). 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, Ser. I, 349 (2011), 175-178. 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), 125-148. doi: 10.1007/s00205-005-0371-8.  Google Scholar

[11]

B. Dacorogna, "Direct Methods in the Calculus of Variations," Applied Mathematical Sciences, 78, Springer-Verlag, Berlin, 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), 277-294. 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), 842-850.  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), 181-204. doi: 10.1007/s002050100174.  Google Scholar

[16]

R. V. Kohn, The relaxation of a double-well energy, Contin. Mech. Thermodyn, 3 (1991), 193-236. doi: 10.1007/BF01135336.  Google Scholar

[17]

C. Kreisbeck, "Analytical Aspects of Relaxation for Models in Crystal Plasticity," PhD thesis, University Regensburg, 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., Sect. A, 125 (1995), 1179-1192.  Google Scholar

[19]

E. H. Lee, Elastic-plastic deformation at finite strains, J. Appl. Mech., 36 (1969), 1-6. 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, Proc. Symp. Edinburgh/Scotl, (1985/86), 257-268, Oxford Sci. Publ., Oxford Univ. Press, New York, 1988.  Google Scholar

[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.  Google Scholar

[22]

F. Murat, Compacité par compensation, Ann. Sc. Norm. Super. Pisa, Cl. Sci., Ser. (IV), 5 (1978), 489-507.  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, Cl. Sci., Ser. (IV), 8 (1981), 69-102.  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), 397-462. 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), 1-15. 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., Proc., Besancon 1977, Lect. Notes Math., 665 (1978), 228-241. doi: 10.1007/BFb0061808.  Google Scholar

[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.  Google Scholar

[28]

H. Triebel, "Interpolation Theory, Function Spaces, Differential Operators," North-Holland Mathematical Library. Vol. 18. North-Holland Publishing Company, Amsterdam, 1978.  Google Scholar

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