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Phase field model for mode III crack growth in two dimensional elasticity

Takeshi Takaishi;  Masato Kimura — 2009

Kybernetika

A phase field model for anti-plane shear crack growth in two dimensional isotropic elastic material is proposed. We introduce a phase field to represent the shape of the crack with a regularization parameter ϵ > 0 and we approximate the Francfort–Marigo type energy using the idea of Ambrosio and Tortorelli. The phase field model is derived as a gradient flow of this regularized energy. We show several numerical examples of the crack growth computed with an adaptive mesh finite element method.

Computational studies of non-local anisotropic Allen-Cahn equation

Michal Beneš;  Shigetoshi Yazaki;  Masato Kimura — 2011

Mathematica Bohemica

The paper presents the results of numerical solution of the Allen-Cahn equation with a non-local term. This equation originally mentioned by Rubinstein and Sternberg in 1992 is related to the mean-curvature flow with the constraint of constant volume enclosed by the evolving curve. We study this motion approximately by the mentioned PDE, generalize the problem by including anisotropy and discuss the computational results obtained.

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