Geodesic
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

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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.

DOI : 10.1051/cocv/2014004
Classification : 49J27, 74R10, 58E30
Keywords: quasi-static evolutions, phase-field 
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     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://geodesic.mathdoc.fr/articles/10.1051/cocv/2014004/}
}
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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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