Calculus of Variations and Geometric Measure Theory

M. Negri

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

created by negri on 08 May 2013
modified on 11 Sep 2014


Published Paper

Inserted: 8 may 2013
Last Updated: 11 sep 2014

Journal: ESAIM Control Optim. Calc. Var.
Volume: 20
Number: 4
Pages: 983-1008
Year: 2014
Doi: 10.1051/cocv/2014004


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 $\mathcal{F}$ of class $C^1$ in reflexive separable Banach spaces. We provide a couple of constructive proofs of existence which share commune features with the theory of minimizing movements for gradient flows. Moreover, considering a sequence of functionals $\mathcal{F}_n$ a its $\Gamma$-limit $\mathcal{F}$ 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.