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A Young measures approach to quasistatic evolution for a class of material models with nonconvex elastic energies. (English) Zbl 1161.74010
Summary: Rate-independent evolution for material models with nonconvex elastic energies is studied without any spatial regularization of the inner variable; due to lack of convexity, the model is developed in the framework of Young measures. An existence result for the quasistatic evolution is obtained in terms of compatible systems of Young measures. We also show that this result can be equivalently reformulated in probabilistic language and leads to the description of the quasistatic evolution in terms of stochastic processes in a suitable probability space.

MSC:
74B20 Nonlinear elasticity
74G65 Energy minimization in equilibrium problems in solid mechanics
74H20 Existence of solutions of dynamical problems in solid mechanics
49Q20 Variational problems in a geometric measure-theoretic setting
49J55 Existence of optimal solutions to problems involving randomness
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References:
[1] E. Acerbi and N. Fusco, Semicontinuity problems in the calculus of variations. Arch. Rational Mech. Anal. 86 (1984) 125-145. Zbl0565.49010 MR751305 · Zbl 0565.49010 · doi:10.1007/BF00275731
[2] J.M. Ball, A version of the fundamental theorem for Young measures, in PDE’s and continuum models of phase transitions (Nice, 1988), Lecture Notes in Physics, Springer-Verlag, Berlin (1989) 207-215. Zbl0991.49500 MR1036070 · Zbl 0991.49500
[3] H. Brezis, Opérateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert. North-Holland, Amsterdam-London; American Elsevier, New York (1973). Zbl0252.47055 MR348562 · Zbl 0252.47055
[4] G. Dal Maso, G. Francfort and R. Toader, Quasistatic crack growth in nonlinear elasticity. Arch. Rational Mech. Anal. 176 (2005) 165-225. Zbl1064.74150 MR2186036 · Zbl 1064.74150 · doi:10.1007/s00205-004-0351-4
[5] G. Dal Maso, A. De Simone, M.G. Mora and M. Morini, Time-dependent systems of generalized Young measures. Netw. Heterog. Media 2 (2007) 1-36. Zbl1140.28001 MR2291810 · Zbl 1140.28001
[6] G. Dal Maso, A. De Simone, M.G. Mora and M. Morini, Globally stable quasistatic evolution in plasticity with softening. Netw. Heterog. Media (to appear). Zbl1156.74308 MR2425074 · Zbl 1156.74308 · doi:10.3934/nhm.2008.3.567 · aimsciences.org
[7] I. Fonseca, S. Müller and P. Pedregal, Analysis of concentration and oscillation effects generated by gradients. SIAM J. Math. Anal. 29 (1998) 736-756. Zbl0920.49009 MR1617712 · Zbl 0920.49009 · doi:10.1137/S0036141096306534
[8] G. Francfort and A. Mielke, Existence results for a class of rate-independent material models with nonconvex elastic energy. J. Reine Angew. Math. 595 (2006) 55-91. Zbl1101.74015 MR2244798 · Zbl 1101.74015 · doi:10.1515/CRELLE.2006.044
[9] M. Kočvara, A. Mielke and T. Roubíček, A rate-independent approach to the delamination problem. Math. Mech. Solids 11 (2006) 423-447. Zbl1133.74038 MR2245202 · Zbl 1133.74038 · doi:10.1177/1081286505046482
[10] A.N. Kolmogorov, Foundations of the Theory of Probability. Chelsea Publishing Company, 2nd edition, New York (1956). Zbl0074.12202 MR79843 · Zbl 0074.12202
[11] C. Miehe and M. Lambrecht, Analysis of microstructure development in shearbands by energy relaxation of incremental stress potentials: large-strain theory for standard dissipative solids. Internat. J. Numer. Methods Engrg. 58 (2003) 1-41. Zbl1032.74526 MR1999979 · Zbl 1032.74526 · doi:10.1002/nme.726
[12] C. Miehe, J. Schotte and M. Lambrecht, Computational homogenization of materials with microstructures based on incremental variational formulations, in IUTAM Symposium on Computational Mechanics of Solid Materials at Large Strains (Stuttgart, 2001), Solid Mech. Appl., Kluwer Acad. Publ., Dordrecht (2003) 87-100. Zbl1040.74039 MR1991327 · Zbl 1040.74039
[13] A. Mielke, Evolution of rate-independent systems, in Evolutionary equations, Vol. II, C.M. Dafermos and E. Feireisl Eds., Handbook of Differential Equations, Elsevier/North-Holland, Amsterdam (2005) 461-559. Zbl1120.47062 MR2182832 · Zbl 1120.47062
[14] A. Mielke and T. Roubíček, Rate-independent damage processes in nonlinear elasticity. Math. Models Methods Appl. Sci. 16 (2006) 177-209. Zbl1094.35068 MR2210087 · Zbl 1094.35068 · doi:10.1142/S021820250600111X
[15] A. Mielke, F. Theil and V.I. Levitas, A variational formulation of rate-independent phase transformations using an extremum principle. Arch. Rational Mech. Anal. 162 (2002) 137-177. Zbl1012.74054 MR1897379 · Zbl 1012.74054 · doi:10.1007/s002050200194
[16] M. Ortiz and E. Repetto, Nonconvex energy minimization and dislocation structures in ductile single crystals. J. Mech. Physics Solids 47 (1999) 397-462. Zbl0964.74012 MR1674064 · Zbl 0964.74012 · doi:10.1016/S0022-5096(97)00096-3
[17] P. Pedregal, Parametrized measures and variational principles. Progress in Nonlinear Differential Equations and their Applications 30. Birkhäuser Verlag, Basel (1997). Zbl0879.49017 MR1452107 · Zbl 0879.49017
[18] M. Valadier, Young measures, in Methods of nonconvex analysis (Varenna, 1989), Lecture Notes in Mathematics, Springer-Verlag, Berlin (1990) 152-188. Zbl0738.28004 MR1079763 · Zbl 0738.28004
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