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${}^{\ast }$ convergence $u_k\stackrel{\ast }{\rightharpoonup }u$ 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.

12 pages, 2 figures