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A phase field model for fracture. (English) Zbl 1394.74176
Summary: The variational formulation of brittle fracture as formulated for example by G. A. Francfort and J. J. Marigo [J. Mech. Phys. Solids 46, No. 8, 1319–1342 (1998; Zbl 0966.74060)], where the total energy is minimized with respect to any admissible crack set and displacement field, allows the identification of crack paths, branching of preexisting cracks and even crack initiation without additional criteria. For its numerical treatment a continuous approximation of the model in the sense of \(\Gamma\)-convergence has been presented by B. Bourdin [Interfaces Free Bound. 9, No. 3, 411–430 (2007; Zbl 1130.74040)]. In the regularized Francfort-Marigo model cracks are represented by an additional field variable (secondary variable) \(s\in [0,1]\) which is \(0\) if the material is cracked and 1 if it is undamaged.
In this work, we reinterpret the crack variable as a phase field order parameter and address cracking as a phase transition problem. The crack growth is governed by the evolution equation of the order parameter which resembles the Ginzburg-Landau equation. The numerical treatment is done by finite elements combined with an implicit Euler scheme for the time integration.

74R99 Fracture and damage
74S05 Finite element methods applied to problems in solid mechanics
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