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 evolutions, phase-field
@article{COCV_2014__20_4_983_0, author = {Negri, Matteo}, title = {Quasi-static rate-independent evolutions: characterization, existence, approximation and application to fracture mechanics}, journal = {ESAIM: Control, Optimisation and Calculus of Variations}, pages = {983--1008}, publisher = {EDP-Sciences}, volume = {20}, number = {4}, year = {2014}, doi = {10.1051/cocv/2014004}, mrnumber = {3264231}, zbl = {1301.49017}, language = {en}, url = {http://archive.numdam.org/articles/10.1051/cocv/2014004/} }
TY - JOUR AU - Negri, Matteo TI - Quasi-static rate-independent evolutions: characterization, existence, approximation and application to fracture mechanics JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2014 SP - 983 EP - 1008 VL - 20 IS - 4 PB - EDP-Sciences UR - http://archive.numdam.org/articles/10.1051/cocv/2014004/ DO - 10.1051/cocv/2014004 LA - en ID - COCV_2014__20_4_983_0 ER -
%0 Journal Article %A Negri, Matteo %T Quasi-static rate-independent evolutions: characterization, existence, approximation and application to fracture mechanics %J ESAIM: Control, Optimisation and Calculus of Variations %D 2014 %P 983-1008 %V 20 %N 4 %I EDP-Sciences %U http://archive.numdam.org/articles/10.1051/cocv/2014004/ %R 10.1051/cocv/2014004 %G en %F COCV_2014__20_4_983_0
Negri, Matteo. Quasi-static rate-independent evolutions: characterization, existence, approximation and application to fracture mechanics. ESAIM: Control, Optimisation and Calculus of Variations, Volume 20 (2014) no. 4, pp. 983-1008. doi : 10.1051/cocv/2014004. http://archive.numdam.org/articles/10.1051/cocv/2014004/
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