Strain gradient visco-plasticity with dislocation densities contributing to the energy

Abstract

We consider the energetic description of a visco-plastic evolution and derive an existence result. The energies are convex, but not necessarily quadratic. Our model is a strain gradient model in which the curl of the plastic strain contributes to the energy. Our existence results are based on a time-discretization, the limit procedure relies on Helmholtz decompositions and compensated compactness.

A Global div-curl-Lemma for Mixed Boundary Conditions in Weak Lipschitz Domains

“…Let us also mention that the div-curl-lemma is particularly useful to treat homogenization of problems arising in plasticity, see, e.g., a recent preprint on this topic [20], for which the preprint [21] provides the important key div-curl-lemma. Unfortunately, in [21,20] a H 1 (Ω)-detour is used as the core argument for the proofs. The same detour is utilized in the recent contribution [11] where div-curl-type lemmas are presented which also allow for inhomogeneous boundary conditions.…”

“…The same detour is utilized in the recent contribution [11] where div-curl-type lemmas are presented which also allow for inhomogeneous boundary conditions. This unnecessarily high regularity assumption of H 1 (Ω)-fields excludes results like [11,21,20] to be applied to important applications which are stated, e.g., in Lipschitz domains.…”

Well-posedness of a one-dimensional nonlinear kinematic hardening model

“…At the macroscopic level, the reader is referred to Mielke & Müller [29] and [36] for existence results for incremental problems under boundedness assumptions on the curl of the plastic strain. In the viscoplastic setting, Röger & Schweizer [34] prove that such bounds on the curl allow to find a suitably weak evolution notion.…”

Quasistatic evolution for dislocation-free finite plasticity

“…This notion of solution features stability and energy balance on the time-discrete level as well as semistability relation with respect to elastic deformations and an energy inequality in the time-continuous limit. A similar notion has been considered in [12] in the quasistatic setting and in [54] for viscoplasticity and is weaker than the concept of energetic solutions [47]. Still, it implies the validity of the quasistatic equilibrium system as well as the dissipative character of the evolution.…”