Existence of solutions to an elasto-viscoplastic model with kinematic hardening and r-Laplacian fracture approximation
Jakabčin, Lukáš1
1 Laboratoire Jean Kuntzmann, 51 rue des Mathématiques, Campus de Saint-Martin d’Hères BP 53, Grenoble-Alpes, 38041 Grenoble, cedex 09, France.
ESAIM: Mathematical Modelling and Numerical Analysis , Tome 50 (2016) no. 2, pp. 455-473.

This paper deals with an existence theorem for a model describing an elasto-viscoplastic evolution of a 2D material with linear kinematic hardening and fracture where the Griffith fracture energy is regularized using a r-Laplacian.

Reçu le :
DOI : 10.1051/m2an/2015053
Classification : 74R20, 49J40, 74C10
Mots-clés : Fracture, plasticity, kinematic hardening
Jakabčin, Lukáš 1

1 Laboratoire Jean Kuntzmann, 51 rue des Mathématiques, Campus de Saint-Martin d’Hères BP 53, Grenoble-Alpes, 38041 Grenoble, cedex 09, France. 
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     title = {Existence of solutions to an elasto-viscoplastic model with kinematic hardening and $r${-Laplacian} fracture approximation},
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Jakabčin, Lukáš. Existence of solutions to an elasto-viscoplastic model with kinematic hardening and $r$-Laplacian fracture approximation. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 50 (2016) no. 2, pp. 455-473. doi : 10.1051/m2an/2015053. http://archive.numdam.org/articles/10.1051/m2an/2015053/

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

J.F. Babadjian and V. Millot, Unilateral gradient flow of the Ambosio-Tortorelli functional by minimizing movements. Ann. Inst. Henri Poincaré (C) Anal. Non Lin. 31 (2014) 779–822. | DOI | Numdam | 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. Ph.D. thesis, Université Paris Nord (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

V. Chrismale, Globally stable quasistatic evolution for a coupled elasto-plastic damage model. Preprint CVGMT (2015). | MR

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

G. Dal Maso and R. Scala, Quasistatic evolution in perfect plasticity as limit of dynamic processes. J. Dyn. Differ. Equations 26 (2014) 915–954. | DOI | MR | Zbl

G. Dal Maso, A. De Simone, M.G. Mora and M. Morini, A vanishing viscosity approach to quasistatic evolution in plasticity with softening. Arch. Ration. Mech. Anal. 189 (2008) 469–544. | DOI | MR | Zbl

L.C. Evans, Partial Differential Equations. Grad. Stud. Math. AMS, Rhode Island (1998).

M. Focardi, On the variational approximation of free discontinuity problems in the vectorial case. Math. Models Methods Appl. Sci. 11 (2001) 663–684. | DOI | MR | Zbl

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

A. Giacomini, Ambrosio−Tortorelli approximation of quasi-static evolution of brittle fracture. Calc. Var. Partial Differ. Equations 22 129–172. | DOI | MR | Zbl

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

L. Jakabčin, A visco-elasto-plastic model with regularized fracture. ESAIM: COCV 22 (2016) 148–168. | Numdam | MR | Zbl

C. J. Larsen, C. Ortner and E. Suli, Existence of solutions to a regularized model of dynamic fracture. Math. Models Methods Appl. Sci. 20 (2010) 1021–1048. | DOI | MR | Zbl

A. Mainik, A. Mielke, Existence results for energetic models for rate-independent systems. Calc. Var. Partial Differ. Equations 22 (2005) 73–99. | DOI | MR | 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) 15 085–15, 117.

P. Suquet, Sur les équations de la plasticité: existence et régularité des solutions. J. Mécanique 20 (1981) 3–39. | MR | Zbl

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