Previous |  Up |  Next


crack growth; phase field model; numerical simulation
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 $\epsilon >0$ 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.
[1] L. Ambrosio and V. M. Tortorelli: On the approximation of free discontinuity problems. Boll. Un. Mat. Ital. 7 (1992), 6-B, 105–123. MR 1164940
[2] B. Bourdin: The variational formulation of brittle fracture: numerical implementation and extensions. Preprint 2006, to appear in IUTAM Symposium on Discretization Methods for Evolving Discontinuities (T. Belytschko, A. Combescure, and R. de Borst eds.), Springer.
[3] B. Bourdin: Numerical implementation of the variational formulation of brittle fracture. Interfaces Free Bound. 9 (2007), 411–430. MR 2341850
[4] B. Bourdin, G. A. Francfort, and J.-J. Marigo: Numerical experiments in revisited brittle fracture. J. Mech. Phys. Solids 48 (2000), 4, 797–826. MR 1745759
[5] M. Buliga: Energy minimizing brittle crack propagation. J. Elasticity 52 (1998/99), 3, 201–238. MR 1700752 | Zbl 0947.74055
[6] C. M. Elliott and J. R. Ockendon: Weak and Variational Methods for Moving Boundary Problems. Pitman Publishing Inc. 1982. MR 0650455
[7] G. A. Francfort and J.-J. Marigo: Revisiting brittle fracture as an energy minimization problem. J. Mech. Phys. Solids 46 (1998), 1319–1342. MR 1633984
[8] A. A. Griffith: The phenomenon of rupture and flow in solids. Phil. Trans. Royal Soc. London A 221 (1920), 163–198.
[9] M. Kimura, H. Komura, M. Mimura, H. Miyoshi, T. Takaishi, and D. Ueyama: Adaptive mesh finite element method for pattern dynamics in reaction-diffusion systems. In: Proc. Czech–Japanese Seminar in Applied Mathematics 2005 (M. Beneš, M. Kimura, and T. Nakaki, eds.), COE Lecture Note Vol. 3, Faculty of Mathematics, Kyushu University 2006, pp. 56–68. MR 2277123
[10] M. Kimura, H. Komura, M. Mimura, H. Miyoshi, T. Takaishi, and D. Ueyama: Quantitative study of adaptive mesh FEM with localization index of pattern. In: Proc. of the Czech–Japanese Seminar in Applied Mathematics 2006 (M. Beneš, M. Kimura, and T. Nakaki, eds.), COE Lecture Note Vol. 6, Faculty of Mathematics, Kyushu University 2007, pp. 114–136. MR 2277123
[11] R. Kobayashi: Modeling and numerical simulations of dendritic crystal growth. Physica D 63 (1993), 410–423. Zbl 0797.35175
[12] A. Schmidt and K. G. Siebert: Design of Adaptive Finite Element Software. The Finite Element Toolbox ALBERTA (Lecture Notes in Comput. Sci. Engrg. 42.) Springer–Verlag, Berlin 2005. MR 2127659
[13] A. Visintin: Models of Phase Transitions. Birkhäuser–Verlag, Basel 1996. MR 1423808 | Zbl 0903.35097
Partner of
EuDML logo