| Title:
|
Phase field model for mode III crack growth in two dimensional elasticity (English) |
| Author:
|
Takaishi, Takeshi |
| Author:
|
Kimura, Masato |
| Language:
|
English |
| Journal:
|
Kybernetika |
| ISSN:
|
0023-5954 |
| Volume:
|
45 |
| Issue:
|
4 |
| Year:
|
2009 |
| Pages:
|
605-614 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
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. (English) |
| Keyword:
|
crack growth |
| Keyword:
|
phase field model |
| Keyword:
|
numerical simulation |
| MSC:
|
35K57 |
| MSC:
|
35Q74 |
| MSC:
|
74B20 |
| MSC:
|
74R10 |
| MSC:
|
81T80 |
| idZBL:
|
Zbl 1193.35007 |
| idMR:
|
MR2588626 |
| . |
| Date available:
|
2010-06-02T18:55:35Z |
| Last updated:
|
2013-09-21 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/140066 |
| . |
| Reference:
|
[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 |
| Reference:
|
[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. |
| Reference:
|
[3] B. Bourdin: Numerical implementation of the variational formulation of brittle fracture.Interfaces Free Bound. 9 (2007), 411–430. MR 2341850 |
| Reference:
|
[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 |
| Reference:
|
[5] M. Buliga: Energy minimizing brittle crack propagation.J. Elasticity 52 (1998/99), 3, 201–238. Zbl 0947.74055, MR 1700752 |
| Reference:
|
[6] C. M. Elliott and J. R. Ockendon: Weak and Variational Methods for Moving Boundary Problems.Pitman Publishing Inc. 1982. MR 0650455 |
| Reference:
|
[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 |
| Reference:
|
[8] A. A. Griffith: The phenomenon of rupture and flow in solids.Phil. Trans. Royal Soc. London A 221 (1920), 163–198. |
| Reference:
|
[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 |
| Reference:
|
[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 |
| Reference:
|
[11] R. Kobayashi: Modeling and numerical simulations of dendritic crystal growth.Physica D 63 (1993), 410–423. Zbl 0797.35175 |
| Reference:
|
[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 |
| Reference:
|
[13] A. Visintin: Models of Phase Transitions.Birkhäuser–Verlag, Basel 1996. Zbl 0903.35097, MR 1423808 |
| . |
