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Ambrosio-Tortorelli approximation of quasi-static evolution of brittle fractures. (English) Zbl 1068.35189
The author defines a notion of quasi-static evolution for the elliptic approximation of the Mumford-Shah functional proposed by L. Ambrosio and V. M. Tortorelli [Commun. Pure Appl. Math. 43, No. 8, 999–1036 (1990; Zbl 0722.49020), Boll. Unione Mat. Ital., VII. Ser., B 6, No. 1, 105–123 (1992; Zbl 0776.49029)]. The quasi-static evolution for the Ambrosio-Tortorelli functional is obtained through a discretization in time procedure: each step is performed using a variational argument which gives the minimal properties. The author proves that this regular evolution converges to a quasi-static growth of brittle fractures in linearly elastic bodies in the sense of Francfort-Larsen.

##### MSC:
 35B27 Homogenization in context of PDEs; PDEs in media with periodic structure 35R35 Free boundary problems for PDEs 74R10 Brittle fracture 49J45 Methods involving semicontinuity and convergence; relaxation 74G65 Energy minimization in equilibrium problems in solid mechanics
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