Numerical implementation of the variational formulation for quasi-static brittle fracture

Interfaces and Free Boundaries

Volumn 9, Issue 3, 2007, Pages 411-430

  Bourdin, Blaise ^a  

^a LOUISIANA STATE UNIVERSITY   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 35648980407     PISSN: 14639963     EISSN: None     Source Type: Journal    
DOI: 10.4171/IFB/171     Document Type: Article

References (28)

1. ALBERTI, G. Variational models for phase transitions, an approach via Γ-convergence. Calculus of Variations and Partial Differential Equations, G. Buttazzo et al, (ed.), Springer (2000), 95-114. Zbl 0957.35017 MR 1757697
2. AMBROSIO, L., & TORTORELLI, V. M. Approximation of functionals depending on jumps by elliptic functionals via Γ-convergence. Comm. Pure Appl. Math. 43 (1990), 999-1036. Zbl 0722.49020 MR 1075076
3. AMBROSIO, L., & TORTORELLI, V. M. On the approximation of free discontinuity problems. Boll. Un. Mat. Ital. B (7) 6 (1992), 105-123. Zbl 0776.49029 MR 1164940
4. BELLETTINI, G., & COSCIA, A. Discrete approximation of a free discontinuity problem. Numer. Funct. Anal. Optim. 15 (1994), 201-224. Zbl 0806.49002 MR 1272202
5. BOURDIN, B., & CHAMBOLLE, A. Implementation of an adaptive finite-element approximation of the Mumford-Shah functional. Numer. Math. 85 (2000), 609-646. Zbl 0961.65062 MR 1771782
6. BOURDIN, B., FRANCFORT, G. A., & MARIGO, J.-J. Numerical experiments in revisited brittle fracture. J. Mech. Phys. Solids 48 (2000), 797-826. Zbl 0995.74057 MR 1745759
7. BOURDIN, B., FRANCFORT, G. A., & MARIGO, J.-J. The Variational Approach to Fracture (reprinted from J. Elasticity). Springer, to appear.
8. BOURDIN, B. Une méthode variationnelle en mécanique de la rupture, théorie et applications numériques. Ph.D. thesis, Université Paris Nord, Institut Galilée (1998); http://www.math.lsu.edu/~bourdin.
9. BOURDIN, B. Image segmentation with a finite element method. M2AN Math. Model. Numer. Anal. 33 (1999), 229-244. Zbl 0947.65075 MR 1700033
10. BRAIDES, A., Γ-convergence for Beginners. Oxford Lecture Ser. Math. Appl. 22, Oxford Univ. Press, Oxford (2002). Zbl pre01865939 MR 1968440
11. CAMACHO, G. T., & ORTIZ, M. Computational modelling of impact damage in brittle materials. Int. J. Numer. Meth. Engrg. 33 (1996), 2899-2938. Zbl 0929.74101
12. CHAMBOLLE, A. An approximation result for special functions with bounded variations. J. Math. Pures Appl. 83 (2004), 929-954. Zbl pre02118455 MR 2074682
13. CHAMBOLLE, A. Addendum to "An approximation result for special functions with bounded deformation": the n-dimensional case. J. Math. Pures Appl. 84 (2005), 137-145. Zbl pre02118455 MR 2112874
14. CHAMBOLLE, A., GIACOMINI, A., & PONSIGLIONE, M. Crack initiation in elastic bodies. Arch. Ration. Mech. Anal., 2005, to appear.
15. DAL MASO, G. An Introduction to Γ- convergence. Birkhäuser, Boston (1993). Zbl 0816.49001 MR 1201152
16. DAL MASO, G., FRANCFORT, G. A., & TOADER, M. Quasi-static evolution in brittle fracture: the case of bounded solutions. Calculus of Variations: Topics from the Mathematical Heritage of E. De Giorgi, Quad. Mat. 14, Dept. Mat., Seconda Univ. Napoli, Caserta (2004), 245-266. Zbl pre02159267 MR 2118420
17. DEL PIERO, G., LANCIONI, G., & MARCH, R. A variational model for fracture mechanics: numerical experiments. J. Mech. Phys. Solids, 2007, to appear.
18. FRANCFORT, G. A., & LARSEN, C. Existence and convergence for quasi-static evolution in brittle fracture. Comm. Pure Appl. Math. 56 (2003), 1465-1500. Zbl 1068.74056 MR 1988896
19. FRANCFORT, G. A., & MARIGO, J.-J. Revisiting brittle fracture as an energy minimization problem. J. Mech. Phys. Solids 46 (1998), 1319-1342. Zbl 0966.74060 MR 1633984
20. GIACOMINI, A. Ambrosio-Tortorelli approximation of quasi-static evolution of brittle fractures. Calc. Var. Partial Differential Equations 22 (2005), 129-172. Zbl 1068.35189 MR 2106765
21. GIACOMINI, A., & PONSIGLIONE, M. A discontinuous finite element approximation of quasi-static growth of brittle fractures. Numer. Funct. Anal. Optim. 24 (2003), 813-850. Zbl 1046.74048 MR 2011593
22. GRIFFITH, A. A. The phenomena of rupture and flow in solids. Philos. Trans. Roy. Soc. London Ser. A 221 (1921), 163-198.
23. MODICA, L., & MORTOLA, S. Il limite nella Γ-convergenza di una famiglia di funzionali ellittici. Boll. Un. Mat. Ital. A (5) 14 (1977), 526-529. Zbl 0364.49006 MR 0473971
24. --convergenza. Boll. Un. Mat. Ital. B (5) 14 (1977), 285-299. Zbl 0356.49008 MR 0445362
25. MOES, N., DOLBOW, J., & BELYTSCHKO, T. A finite element method for crack growth without remeshing. Int. J. Numer. Meth. Engrg. 46 (1999), 131-150. Zbl 0955.74066
26. NEGRI, M. A finite element approximation of the Griffith model in fracture mechanics. Numer. Math. 95 (2003), 653-687. Zbl 1068.74080 MR 2013123
27. TONEGAWA, Y. On stable critical points for a singular perturbation problem. Comm. Anal. Geom. 13 (2005), 439-459. Zbl pre05000499 MR 2154826
28. XU, X. P., & NEEDLEMAN, A. Numerical simulations of fast crack-growth in brittle solids. Internat. J. Solids Structures 42 (1994), 1397-1434, Zbl 0825.73579