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Quasi-static rate-independent evolutions: characterization, existence, approximation and application to fracture mechanics∗

Published online by Cambridge University Press:  04 August 2014

Matteo Negri
Matteo Negri*
Affiliation:
Department of Mathematics, University of Pavia, Via A. Ferrata 1, 27100 Pavia, Italy. matteo.negri@unipv.it
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Abstract

We characterize quasi-static rate-independent evolutions, by means of their graph parametrization, in terms of a couple of equations: the first gives stationarity while the second provides the energy balance. An abstract existence result is given for functionals ℱ of class C1 in reflexive separable Banach spaces. We provide a couple of constructive proofs of existence which share common features with the theory of minimizing movements for gradient flows. Moreover, considering a sequence of functionals ℱn and its Γ-limit ℱ we provide, under suitable assumptions, a convergence result for the associated quasi-static evolutions. Finally, we apply this approach to a phase field model in brittle fracture.

Keywords

Quasi-static evolutionsphase-field
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 20 , Issue 4 , October 2014 , pp. 983 - 1008
Copyright
© EDP Sciences, SMAI, 2014

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