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Preface: Rate-independent evolutions
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 |
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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