Rate-independent elastoplasticity at finite strains and its numerical approximation

Abstract

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 is approximated numerically by finite elements and implicit time discretization, such that a computationally implementable scheme is obtained. The nonself-penetration as well as a possible frictionless unilateral contact is considered and approximated numerically by a suitable penalization method which keeps polyconvexity and simultaneously bypasses the Lavrentiev phenomenon. The main result concerns the convergence of the numerical scheme toward 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.

Two-well rigidity and multidimensional sharp-interface limits for solid–solid phase transitions

“…The additional higher order penalization situates our analysis within the framework of nonsimple materials, whose characteristic feature is an elastic stored energy density dependent on second order derivatives of the deformations. Starting from the seminal works by R.A. Toupin [59,60], these materials have been the subject of an intense research activity in nonlinear elasticity due to their enhanced compactness properties [5,48,54]. On the one hand, the penalization factor η will be chosen "large enough" to exploit the second order regularization also in the present two-wells setting.…”

Damage model for plastic materials at finite strains

“…The proof can be found in [ To show convergence (42) we argue as in [21,Lemma 3.5]. We know that ⇀ in 1 (0, ) and ( ) = lim sup →∞ ( ) for a.e.…”

A note on locking materials and gradient polyconvexity

“…One possibility to incorporate this restriction is to let the energy depend (on parts) of the second gradient of the deformation. The canonical way to do so is to let the energy density be a convex function of the second gradient (cf., e.g., [6,9,41,48,54]). Yet, here we propose a different approach inspired by the notion of polyconvexity due to Ball [4].…”

“…The contribution of the higher gradient is usually associated to interfacial energies, as in e.g. [6,9,41,48,54] which work with an energy functional of the type J(y) = Ω (w(∇y(x)) + γ|∇ 2 y(x)| d )dx, (1.9) for some γ > 0 and d > 1. Now, if d > n, any deformation of finite energy will satisfy L(F ) ≤ 0 with L from (1.6) and depending only on the energy bound by Sobolev embedding.…”