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},
year = {2014},
publisher = {EDP Sciences},
volume = {20},
number = {4},
doi = {10.1051/cocv/2014004},
mrnumber = {3264231},
zbl = {1301.49017},
language = {en},
url = {https://www.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 - https://www.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 https://www.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, Tome 20 (2014) no. 4, pp. 983-1008. doi: 10.1051/cocv/2014004
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