Volume 20, Number 4, October-December 2014
|Page(s)||983 - 1008|
|Published online||04 August 2014|
Quasi-static rate-independent evolutions: characterization, existence, approximation and application to fracture mechanics∗
Department of Mathematics, University of Pavia,
Via A. Ferrata 1, 27100
Received: 8 May 2013
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.
Mathematics Subject Classification: 49J27 / 74R10 / 58E30
Key words: Quasi-static evolutions / phase-field
This material is based on work supported by ERC under Grant No. 290888 “Quasistatic and Dynamic Evolution Problems in Plasticity and Fracture”, by GNAMPA under project “Modelli variazionali per la propagazione di fratture, la delaminazione e il danneggiamento” and by MIUR under project “Calculus of Variations”.
© EDP Sciences, SMAI, 2014
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