A variational model for fracture mechanics: Numerical experiments

Journal of the Mechanics and Physics of Solids

Volumn 55, Issue 12, 2007, Pages 2513-2537

  Del Piero, Gianpietro ^a   Lancioni, Giovanni ^b   March, Riccardo ^c  

^a UNIVERSITY OF FERRARA   (Italy)

^b UNIVERSITÀ POLITECNICA DELLE MARCHE   (Italy)

^c CNR   (Italy)

Author keywords

Energy methods; Finite elements; Fracture mechanics; Free boundary problems; Variational calculus

Indexed keywords

BRITTLE FRACTURE; ELASTICITY; FINITE ELEMENT METHOD; MATHEMATICAL MODELS; OPTIMIZATION; PROBLEM SOLVING; VARIATIONAL TECHNIQUES;

DISCONTINUITY PROBLEM; ENERGY METHODS; FREE-BOUNDARY PROBLEMS;

FRACTURE MECHANICS;

EID: 35648997733     PISSN: 00225096     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.jmps.2007.04.011     Document Type: Article

References (31)

1. Ambrosio L. Existence theory for a new class of variational problems. Arch. Ration. Mech. Anal. 111 (1990) 291-322
2. k). Nonlinear Anal. 23 (1994) 405-425
3. Ambrosio, L., Braides, A., 1995. Energies in SBV and variational models in fracture mechanics. In: Cioranescu, D., et al. (Eds.), Homogenization and Applications to Material Sciences, GAKUTO International Series Mathematical Sciences and Applications, vol. 9, pp. 1-22.
4. Ambrosio L., and Tortorelli V.M. Approximation of functionals depending on jumps by elliptic functionals via Γ-convergence. Commun. Pure Appl. Math. 43 (1990) 999-1036
5. Ambrosio L., and Tortorelli V.M. On the approximation of free discontinuity problems. Boll. Un. Mat. Ital. 6-B (1992) 105-123
6. Ambrosio, L., Fusco, N., Pallara, D., 2000. Functions of Bounded Variation and Free Discontinuity Problems. In: Oxford Mathematical Monographs Oxford University Press, New York.
7. Ball J.M. Convexity conditions and existence theorems in nonlinear elasticity. Arch. Rational Mech. Anal. 63 (1977) 337-403
8. Bathe K.J. Finite Element Procedures in Engineering Analysis (1982), Prentice-Hall, Englewood Cliffs, NJ
9. Bellettini G., and Coscia A. Discrete approximation of a free discontinuity problem. Numer. Funct. Anal. Optim. 15 (1994) 105-123
10. Bourdin B., Francfort G.A., and Marigo J.J. Numerical experiments in revisited brittle fracture. J. Mech. Phys. Solids 48 (2000) 797-826
11. Buttazzo G. Semicontinuity, Relaxation and Integral Representation in the Calculus of Variations (1989), Longman, Harlow
12. Chambolle A. An approximation result for special functions with bounded deformation. J. Math. Pures Appl. IX Sér. 83 (2004) 929-954
13. Ciarlet, P.G., 1987. Mathematical Elasticity. Vol. I: Three-dimensional Elasticity. North-Holland, Amsterdam.
14. Crisfield, M.A., 1991. Non-linear Finite Element Analysis of Solids and Structures, Volume 1: Essentials. Wiley.
15. Dal Maso G. An introduction to convergence. Progress in Nonlinear Differential Equations and Their Applications vol. 8 (1993), Birkhauser, Boston MA
16. Dal Maso G., and Toader R. A model for the quasi-static growth of brittle fractures: existence and approximation results. Arch. Ration. Mech. Anal. 162 (2002) 101-135
17. Dal Maso G., Francfort G.A., and Toader R. Quasistatic crack growth in nonlinear elasticity. Arch. Ration. Mech. Anal. 176 (2005) 165-225
18. De Giorgi E., and Ambrosio L. Un nuovo funzionale del calcolo delle variazioni. Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. 82 (1988) 199-210
19. De Giorgi E., Carriero M., and Leaci A. Existence theorem for a minimum problem with free discontinuity set. Arch. Ration. Mech. Anal. 108 (1989) 195-218
20. Del Piero G. Some properties of the set of fourth-order tensors, with application to elasticity. J. Elasticity 9 3 (1979) 245-261
21. Ehlers W., and Eipper G. The simple tension problem at large volumetric strains computed from finite hyperelastic material lows. Acta Mech. 130 (1998) 17-27
22. Focardi M. On the variational approximation of free-discontinuity problems in the vectorial case. Math. Mod. Meth. Appl. Sci. 11 (2001) 663-684
23. Francfort G.A., and Larsen C. Existence and convergence for quasi-static evolution in brittle fracture. Comm. Pure Appl. Math. 56 (2003) 1465-1500
24. Francfort G.A., and Marigo J.J. Revisiting brittle fracture as an energy minimization problem. J. Mech. Phys. Solids 46 (1998) 1319-1342
25. Fusco, N., Leone, C., March, R., Verde, A., 2006. A lower semi-continuity result for polyconvex functionals in SBV. In: Proceedings of the Royal Society of Edinburg, Section A, Mathematics, vol. 136, pp. 321-336.
26. Griffith A.A. The phenomenon of rupture and flow in solids. Phil. Trans. R. Soc. London A 221 (1920) 163-198
27. Mumford D., and Shah J. Optimal approximations by piecewise smooth functions and associated variational problems. Comm. Pure Appl. Math. 42 (1989) 577-685
28. Oden J.T. Finite Elements of Nonlinear Continua (1972), McGraw-Hill, New York
29. Ogden, R.W., 1972. Large deformation isotropic elasticity, on the correlation of theory and experiment for compressible rubberlike solids. Proc. R. Soc. London A 326 (1972) 565-584, A 328 (1972) 567-583.
30. Ogden R.W. Non-linear Elastic Deformations (1984), Ellis Horwood, Chichester, UK
31. Simo J.C., and Pister K.S. Remarks on rate constitutive equations for finite deformation problems: computational implications. Comput. Methods Appl. Mech. Eng. 46 (1984) 201-215