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 as this result can be equivalently reformulated with probabilistic language and leads to the description of the quasistatic evolution in terms of stochastic processes on a suitable probability space.

Keywords: quasistatic evolution, rate-independent processes, elastic materials, incremental problems, Young measures

@article{COCV_2009__15_2_245_0, author = {Fiaschi, Alice}, title = {A {Young} measures approach to quasistatic evolution for a class of material models with nonconvex elastic energies}, journal = {ESAIM: Control, Optimisation and Calculus of Variations}, pages = {245--278}, publisher = {EDP-Sciences}, volume = {15}, number = {2}, year = {2009}, doi = {10.1051/cocv:2008030}, mrnumber = {2513086}, zbl = {1161.74010}, language = {en}, url = {http://archive.numdam.org/articles/10.1051/cocv:2008030/} }

TY - JOUR AU - Fiaschi, Alice TI - A Young measures approach to quasistatic evolution for a class of material models with nonconvex elastic energies JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2009 SP - 245 EP - 278 VL - 15 IS - 2 PB - EDP-Sciences UR - http://archive.numdam.org/articles/10.1051/cocv:2008030/ DO - 10.1051/cocv:2008030 LA - en ID - COCV_2009__15_2_245_0 ER -

%0 Journal Article %A Fiaschi, Alice %T A Young measures approach to quasistatic evolution for a class of material models with nonconvex elastic energies %J ESAIM: Control, Optimisation and Calculus of Variations %D 2009 %P 245-278 %V 15 %N 2 %I EDP-Sciences %U http://archive.numdam.org/articles/10.1051/cocv:2008030/ %R 10.1051/cocv:2008030 %G en %F COCV_2009__15_2_245_0

Fiaschi, Alice. A Young measures approach to quasistatic evolution for a class of material models with nonconvex elastic energies. ESAIM: Control, Optimisation and Calculus of Variations, Volume 15 (2009) no. 2, pp. 245-278. doi : 10.1051/cocv:2008030. http://archive.numdam.org/articles/10.1051/cocv:2008030/

[1] Semicontinuity problems in the calculus of variations. Arch. Rational Mech. Anal. 86 (1984) 125-145. | MR | Zbl

and ,[2] 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. | MR | Zbl

,[3] Opérateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert. North-Holland, Amsterdam-London; American Elsevier, New York (1973). | MR | Zbl

,[4] Quasistatic crack growth in nonlinear elasticity. Arch. Rational Mech. Anal. 176 (2005) 165-225. | MR | Zbl

, and ,[5] Time-dependent systems of generalized Young measures. Netw. Heterog. Media 2 (2007) 1-36. | MR | Zbl

, , and ,[6] Globally stable quasistatic evolution in plasticity with softening. Netw. Heterog. Media (to appear). | MR | Zbl

, , and ,[7] Analysis of concentration and oscillation effects generated by gradients. SIAM J. Math. Anal. 29 (1998) 736-756. | MR | Zbl

, and ,[8] Existence results for a class of rate-independent material models with nonconvex elastic energy. J. Reine Angew. Math. 595 (2006) 55-91. | MR | Zbl

and ,[9] A rate-independent approach to the delamination problem. Math. Mech. Solids 11 (2006) 423-447. | MR | Zbl

, and ,[10] Foundations of the Theory of Probability. Chelsea Publishing Company, 2nd edition, New York (1956). | MR | Zbl

,[11] 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. | MR | Zbl

and ,[12] 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. | MR | Zbl

, and ,[13] 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. | MR | Zbl

,[14] Rate-independent damage processes in nonlinear elasticity. Math. Models Methods Appl. Sci. 16 (2006) 177-209. | MR | Zbl

and ,[15] A variational formulation of rate-independent phase transformations using an extremum principle. Arch. Rational Mech. Anal. 162 (2002) 137-177. | MR | Zbl

, and ,[16] Nonconvex energy minimization and dislocation structures in ductile single crystals. J. Mech. Physics Solids 47 (1999) 397-462. | MR | Zbl

and ,[17] Parametrized measures and variational principles. Progress in Nonlinear Differential Equations and their Applications 30. Birkhäuser Verlag, Basel (1997). | MR | Zbl

,[18] Young measures, in Methods of nonconvex analysis (Varenna, 1989), Lecture Notes in Mathematics, Springer-Verlag, Berlin (1990) 152-188. | MR | Zbl

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