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Numerical implementation of the variational formulation for quasi-static brittle fracture. (English) Zbl 1130.74040
Summary: This paper presents the analysis and implementation of variational formulation of quasi-static brittle fracture mechanics proposed by G. A. Francfort and J.-J. Marigo [J. Mech. Phys. Solids 46, No. 8, 1319–1342 (1998; Zbl 0966.74060)]. We briefly present the model itself, and its variational approximation in the sense of \(\Gamma\)-convergence. We propose a numerical algorithm based on alternate minimizations and prove its convergence under restrictive assumptions. We establish a new necessary condition for optimality for the entire time evolution from which we derive the backtracking algorithm. We give an analysis of the backtracking algorithm on a simple problem. We present realistic numerical simulation of a traction experiment on a fiber-reinforced matrix, and of the propagation of cracks in a perforated sample under mode-I loading.

74R10 Brittle fracture
74G65 Energy minimization in equilibrium problems in solid mechanics
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