A visco-elasto-plastic evolution model with regularized fracture
Jakabčin, Lukáš1
1 Laboratoire Jean Kuntzmann, 51 rue des Mathématiques, Campus de Saint Martin d’Hères, BP 53, 38041 Grenoble cedex 09, France
ESAIM: Control, Optimisation and Calculus of Variations, Tome 22 (2016) no. 1, pp. 148-168.

We study a model for visco-elasto-plastic deformation with fracture, in which fracture is approximated via a diffuse interface model. We show that a discretized (in time) quasistatic evolution, converges to a solution of the continuous (in time) evolution, proving existence of a solution to our model.

Reçu le :
DOI : 10.1051/cocv/2015005
Classification : 49J40, 49J45, 74C10, 74R20
Mots-clés : Plasticity, regularized fracture, viscous dissipation
Jakabčin, Lukáš 1

1 Laboratoire Jean Kuntzmann, 51 rue des Mathématiques, Campus de Saint Martin d’Hères, BP 53, 38041 Grenoble cedex 09, France 
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Jakabčin, Lukáš. A visco-elasto-plastic evolution model with regularized fracture. ESAIM: Control, Optimisation and Calculus of Variations, Tome 22 (2016) no. 1, pp. 148-168. doi : 10.1051/cocv/2015005. http://www.numdam.org/articles/10.1051/cocv/2015005/

L. Ambrosio and V.M. Tortorelli, Approximation of functionals depending on jumps by elliptic functionals via Γ-convergence. Comm. Pure Appl. Math. XLIII (1990) 999–1036. | DOI | MR | Zbl

J.-F. Babadjian and M.G. Mora, Approximation of dynamic and quasi-static evolution problems in elasto-plasticity by cap models. Quart. Appl. Math. 73 (2015) 265–316. | DOI | MR | Zbl

J.-F. Babadjian, G.A. Francfort and M.G. Mora, Quasistatic evolution in non-associative plasticity - the cap model. SIAM J. Math. Anal. 44 (2012) 245–292. | DOI | MR | Zbl

E. Bonnetier, L. Jakabčin, S. Labbé and A. Replumaz, Numerical simulation of a class of models that combine several mechanisms of dissipation: fracture, plasticity, viscous dissipation. J. Comput. Phys. 271 (2014) 397–414. | DOI | MR | Zbl

B. Bourdin, Une formulation variationnelle en mécanique de la rupture, théorie et mise en oeuvre numérique. Th.D. thesis, Université Paris Nord, France (1998).

B. Bourdin, Numerical implementation of the variational formulation of brittle fracture. Interfaces Free Bound 9 (2007) 411–430. | DOI | MR | Zbl

B. Bourdin, G. Francfort and J.J. Marigo, Numerical experiments in revisited brittle fracture. J. Mech. Phys. Solids 48 (2000) 797–826. | DOI | MR | Zbl

B. Bourdin, G. Francfort and J.J. Marigo, The variational approach to fracture. J. Elasticity 91 (2008) 1–148. | DOI | MR | Zbl

G. Dal Maso and R. Toader, Quasistatic crack growth in elasto-plastic materials: the two-dimensional case. Arch. Ration. Mech. Anal. 196 (2010) 867–906. | DOI | MR | Zbl

N. Dunford and J.T. Schwartz, Linear operators. Part I. Wiley Classics Library. John Wiley and Sons Inc., New York (1988). | Zbl

L.C. Evans, Partial differential equations. Grad. Stud. Math. AMS, Rhode Island (1998).

G.A. Francfort and J.-J. Marigo, Revisiting brittle fracture as an energy minimization problem. J. Mech. Phys. Solids 46 (1998) 1319–1342. | DOI | Zbl

A. Griffith, The phenomena of rupture and flow in solids. Philos. Trans. R. Soc. Lond. Ser. A 221 (1920) 133–178.

F. Iurlano, A density result for GSBD and its application to the approximation of brittle fracture energies. Calc. Var. Partial Differ. Eqs. 51 (2014) 315–342. | DOI | Zbl

L. Jakabčin, Modélisation, analyse et simulation numérique de solides combinant plasticité, rupture et dissipation visqueuse. Th.D. thesis, Université de Grenoble, France (2014).

C.J. Larsen, C. Ortner and E. Suli, Existence of solutions to a regularized model of dynamic fracture. M3AS 20 (2010) 1021–1048. | Zbl

G. Peltzer and P. Tapponnier, Formation and evolution of strike-slip faults, rifts, and basins during the India-Asia collision: An experimental approach. J. Geophys. Res. 93 (1988) 15085–15117. | DOI

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