Skip to main content Accessibility help
×
Home
Hostname: page-component-55597f9d44-n4bck Total loading time: 0.283 Render date: 2022-08-13T22:23:09.428Z Has data issue: true Feature Flags: { "shouldUseShareProductTool": true, "shouldUseHypothesis": true, "isUnsiloEnabled": true, "useRatesEcommerce": false, "useNewApi": true } hasContentIssue true

A Young measures approach to quasistatic evolution for a class ofmaterial models with nonconvex elastic energies

Published online by Cambridge University Press:  26 April 2008

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

Abstract

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.

Type
Research Article
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) 125145. 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) 165225. 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) 136.
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) 736756. 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) 5591.
Kočvara, M., Mielke, A. and Roubíček, T., A rate-independent approach to the delamination problem. Math. Mech. Solids 11 (2006) 423447. 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) 141. 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) 177209. 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) 137177. CrossRef
Ortiz, M. and Repetto, E., Nonconvex energy minimization and dislocation structures in ductile single crystals. J. Mech. Physics Solids 47 (1999) 397462. 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.

Save article to Kindle

To save this article to your Kindle, first ensure coreplatform@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about saving to your Kindle.

Note you can select to save to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

A Young measures approach to quasistatic evolution for a class of material models with nonconvex elastic energies
Available formats
×

Save article to Dropbox

To save this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you used this feature, you will be asked to authorise Cambridge Core to connect with your Dropbox account. Find out more about saving content to Dropbox.

A Young measures approach to quasistatic evolution for a class of material models with nonconvex elastic energies
Available formats
×

Save article to Google Drive

To save this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you used this feature, you will be asked to authorise Cambridge Core to connect with your Google Drive account. Find out more about saving content to Google Drive.

A Young measures approach to quasistatic evolution for a class of material models with nonconvex elastic energies
Available formats
×
×

Reply to: Submit a response

Please enter your response.

Your details

Please enter a valid email address.

Conflicting interests

Do you have any conflicting interests? *