Rate-independent elastoplasticity at finite strains and its numerical approximation
- Mielke, Alexander
- Roubíček, Tomáš
2010 Mathematics Subject Classification
- 35K85 49S05 74S05 65M60 74A30 74C15 74M15
- Plasticity, quasistatic evolution, energetic solutions, dissipation distance, hardening, polyconvexity, Ciarlet-Nečas condition, Signorini contact, finite-element approximation, Gamma-convergence, Lavrentiev phenomenon, 2nd-grade nonsimple materials
Gradient plasticity at large strains with kinematic hardening is analyzed as quasistatic rate-independent evolution. The energy functional with a frame-indifferent polyconvex energy density and the dissipation are approximated numerically by finite elements and implicit time discretization, such that a computationally implementable scheme is obtained. The non-selfpenetration as well as a possible frictionless unilateral contact is considered and approximated numerically by a suitable penalization method which keeps polyconvexity and simultaneously by-passes the Lavrentiev phenomenon. The main result concerns the convergence of the numerical scheme towards energetic solutions. In the case of incompressible plasticity and of nonsimple materials, where the energy depends on the second derivative of the deformation, we derive an explicit stability criterion for convergence relating the spatial discretization and the penalizations.
- Math. Models Methods Appl. Sci., 26 (2016), pp. 2203--2236.