Hostname: page-component-78c5997874-g7gxr Total loading time: 0 Render date: 2024-11-10T13:25:28.632Z Has data issue: false hasContentIssue false
  • English
  • Français

A Young measures approach to quasistatic evolutionfor a class ofmaterial models with nonconvexelastic energies

Published online by Cambridge University Press:  26 April 2008

Alice Fiaschi
Alice Fiaschi*
Affiliation:
SISSA, via Beirut 2-4, 34014 Trieste, Italy; fiaschi@sissa.it
Get access

Abstract

Rate-independent evolution for material models with nonconvexelastic energies is studied without any spatial regularization ofthe inner variable; due to lack of convexity, the model is developedin the framework of Young measures. An existence result for thequasistatic evolution is obtained in terms of compatible systems ofYoung measures. We also show as this result can be equivalentlyreformulated with probabilistic language and leads to thedescription of the quasistatic evolution in terms of stochasticprocesses on a suitable probability space.

Keywords

Quasistatic evolutionrate-independent processeselastic materialsincremental problemsYoung measures
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 15 , Issue 2 , April 2009 , pp. 245 - 278
Copyright
© EDP Sciences, SMAI, 2008

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Acerbi, E. and Fusco, N., Semicontinuity problems in the calculus of variations. Arch. Rational Mech. Anal. 86 (1984) 125–145. CrossRef
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.
H. Brezis, Opérateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert. North-Holland, Amsterdam-London; American Elsevier, New York (1973).
Dal Maso, G., Francfort, G. and Toader, R., Quasistatic crack growth in nonlinear elasticity. Arch. Rational Mech. Anal. 176 (2005) 165–225. CrossRef
Dal Maso, G., De Simone, A., Mora, M.G. and Morini, M., Time-dependent systems of generalized Young measures. Netw. Heterog. Media 2 (2007) 1–36.
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).
Fonseca, I., Müller, S. and Pedregal, P., Analysis of concentration and oscillation effects generated by gradients. SIAM J. Math. Anal. 29 (1998) 736–756. CrossRef
Francfort, G. and Mielke, A., Existence results for a class of rate-independent material models with nonconvex elastic energy. J. Reine Angew. Math. 595 (2006) 55–91.
Kočvara, M., Mielke, A. and Roubíček, T., A rate-independent approach to the delamination problem. Math. Mech. Solids 11 (2006) 423–447. CrossRef
A.N. Kolmogorov, Foundations of the Theory of Probability. Chelsea Publishing Company, 2nd edition, New York (1956).
Miehe, C. and Lambrecht, M., 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. CrossRef
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.
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.
Mielke, A. and Roubíček, T., Rate-independent damage processes in nonlinear elasticity. Math. Models Methods Appl. Sci. 16 (2006) 177–209. CrossRef
Mielke, A., Theil, F. and Levitas, V.I., A variational formulation of rate-independent phase transformations using an extremum principle. Arch. Rational Mech. Anal. 162 (2002) 137–177. CrossRef
Ortiz, M. and Repetto, E., Nonconvex energy minimization and dislocation structures in ductile single crystals. J. Mech. Physics Solids 47 (1999) 397–462. CrossRef
P. Pedregal, Parametrized measures and variational principles. Progress in Nonlinear Differential Equations and their Applications 30. Birkhäuser Verlag, Basel (1997).
M. Valadier, Young measures, in Methods of nonconvex analysis (Varenna, 1989), Lecture Notes in Mathematics, Springer-Verlag, Berlin (1990) 152–188.