A Reshetnyak-type lower semicontinuity result for linearised elasto-plasticity coupled with damage in $W^{1,n}$

Abstract

In this paper we prove a lower semicontinuity result of Reshetnyak type for a class of functionals which appear in models for small-strain elasto-plasticity coupled with damage. To do so we characterise the limit of measures $\alpha_k\,\mathrm{E}u_k$ with respect to the weak convergence $\alpha_k\rightharpoonup \alpha$ in $W^{1,n}(\Omega)$ and the weak$^*$ convergence $u_k\stackrel{*}\rightharpoonup u$ in $BD(\Omega)$, $\mathrm{E}$ denoting the symmetrised gradient. A concentration compactness argument shows that the limit has the form $\alpha\,\mathrm{E}u+\eta$, with $\eta$ supported on an at most countable set.