Geodesic
Phase field model for mode III crack growth in two dimensional elasticity
Kybernetika, Tome 45 (2009) no. 4, pp. 605-614
Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

Voir la notice de l'article

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.
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.
Classification : 35K57, 35Q74, 74B20, 74R10, 81T80
Keywords: crack growth; phase field model; numerical simulation 
@article{KYB_2009_45_4_a4,
     author = {Takaishi, Takeshi and Kimura, Masato},
     title = {Phase field model for mode {III} crack growth in two dimensional elasticity},
     journal = {Kybernetika},
     pages = {605--614},
     year = {2009},
     volume = {45},
     number = {4},
     mrnumber = {2588626},
     zbl = {1193.35007},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/KYB_2009_45_4_a4/}
}
TY  - JOUR
AU  - Takaishi, Takeshi
AU  - Kimura, Masato
TI  - Phase field model for mode III crack growth in two dimensional elasticity
JO  - Kybernetika
PY  - 2009
SP  - 605
EP  - 614
VL  - 45
IS  - 4
UR  - http://geodesic.mathdoc.fr/item/KYB_2009_45_4_a4/
LA  - en
ID  - KYB_2009_45_4_a4
ER  - 
%0 Journal Article
%A Takaishi, Takeshi
%A Kimura, Masato
%T Phase field model for mode III crack growth in two dimensional elasticity
%J Kybernetika
%D 2009
%P 605-614
%V 45
%N 4
%U http://geodesic.mathdoc.fr/item/KYB_2009_45_4_a4/
%G en
%F KYB_2009_45_4_a4
Takaishi, Takeshi; Kimura, Masato. Phase field model for mode III crack growth in two dimensional elasticity. Kybernetika, Tome 45 (2009) no. 4, pp. 605-614. http://geodesic.mathdoc.fr/item/KYB_2009_45_4_a4/

[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

[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

[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

[5] M. Buliga: Energy minimizing brittle crack propagation. J. Elasticity 52 (1998/99), 3, 201–238. | MR | Zbl

[6] C. M. Elliott and J. R. Ockendon: Weak and Variational Methods for Moving Boundary Problems. Pitman Publishing Inc. 1982. | MR

[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

[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

[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

[11] R. Kobayashi: Modeling and numerical simulations of dendritic crystal growth. Physica D 63 (1993), 410–423. | Zbl

[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

[13] A. Visintin: Models of Phase Transitions. Birkhäuser–Verlag, Basel 1996. | MR | Zbl