Existence and Convergence for Quasi-Static Evolution in Brittle Fracture

Communications on Pure and Applied Mathematics

Volumn 56, Issue 10, 2003, Pages 1465-1500

  Francfort, Gilles A ^a,c   Larsen, Christopher J ^b,d  

^a UNIVERSITÉ PARIS 13   (France)

^b WORCESTER POLYTECHNIC INSTITUTE   (United States)

^c UNIVERSITÉ PARIS 13   (France)

^d Worcester Polytechnic Institute   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0042859940     PISSN: 00103640     EISSN: None     Source Type: Journal    
DOI: 10.1002/cpa.3039     Document Type: Article

References (24)

1. Ambrosio, L. A compactness theorem for a new class of functions of bounded variation. Boll. Un. Mat. Ital. B (7) 3 (1989), no. 4, 857-881.
2. Ambrosio, L.; Fusco, E.; Pallara, D. Functions of bounded variation and free discontinuity problems. Oxford Mathematical Monographs. Clarendon, Oxford University Press, New York, 2000.
3. Bourdin, B.; Chambolle, A. Implementation of an adaptive finite-element approximation of the Mumford-Shah functional. Numer. Math. 85 (2000), no. 4, 609-646.
4. Bourdin, B.; Francfort, G.; Marigo, J.-J. Numerical experiments in revisited brittle fracture. J. Mech. Phys. Solids 48 (2000), no. 4, 797-826.
5. Bower, A. F.; Ortiz, M. Solution of three-dimensional crack problems by a finite perturbation method. J. Mech. Phys. Solids 38 (1990), no. 4, 443-480.
6. Braides, A. Approximation of free-discontinuity problems. Lecture Notes in Mathematics, 1694. Springer, Berlin, 1998.
7. Bucur, D.; Varchon, N. Boundary variation for a Neumann problem. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 29 (2000), no. 4, 807-821.
8. Carriero, M.; Leaci, A. Existence theorem for a Dirichlet problem with free discontinuity set. Nonlinear Anal. 15 (1990), no. 7, 661-677.
9. Chambolle, A. A density result in two-dimensional linearized elasticity and applications. Arch. Rat. Mech. Anal., in press.
10. Chambolle, A.; Doveri, F. Continuity of Neumann linear elliptic problems on varying two-dimensional bounded open sets. Comm. Partial Differential Equations 22 (1997), no. 5-6, 811-840.
11. Dal Maso, G.; Morel, J.-M.; Solimini, S. A variational method in image segmentation: existence and approximation results. Acta Math. 168 (1992), no. 1-2, 89-151.
12. Dal Maso, G.; Toader, R. A model for the quasi-static growth of brittle fractures: existence and approximation results. Arch. Ration. Mech. Anal. 162 (2002), no. 2, 101-135.
13. De Giorgi, E.; Carriero, M.; Leaci, A. Existence theorem for a minimum problem with free discontinuity set. Arch. Rational Mech. Anal. 108 (1989), no. 3, 195-218.
14. Evans, L. C.; Gariepy, R. F. Measure theory and fine properties of functions. Studies in Advanced Mathematics. CRC, Boca Raton, Fla., 1992.
15. Francfort, G. A.; Marigo, J.-J. Revisiting brittle fracture as an energy minimization problem. J. Mech. Phys. Solids 46 (1998), no. 8, 1319-1342.
16. Giacomini, A.; Ponsiglione, M. Preprint. Scuola Internazionale Superiore di Studi Avanzati (SISSA), Trieste, Italy, 2003.
17. Griffith, A. The phenomena of rupture and flow in solids. Phil. Trans. Roy. Soc. London 221-A (1920), 163-198.
18. 1,1 and optimal jump microstructure in BV relaxation. SIAM J. Math. Anal. 29 (1998), no. 4, 823-848.
19. Larsen, C. J. Stability of relative minimality under SBV compactness. In preparation.
20. Maddalena, F.; Solimini, S. Lower semicontinuity properties of functionals with free discontinuities. Arch. Ration. Mech. Anal. 159 (2001), no. 4, 273-294.
21. Marigo, J.-J.; Truskinovsky, L. The dependence of the fiber pull-out on the surface energy. In preparation.
22. Mumford, D.; Shah, J. Optimal approximations by piecewise smooth functions and associated variational problems. Comm. Pure Appl. Math. 42 (1989), no. 5, 577-685.
23. Murat, F. The Neumann sieve. Nonlinear variational problems (Isola d'Elba, 1983), 24-32. Research Notes in Mathematics, 127. Pitman, Boston, 1985.
24. Ziemer, W. P. Weakly differentiable functions. Sobolev spaces and functions of bounded variation. Graduate Texts in Mathematics, 120. Springer, New York, 1989.