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

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 $W^{1,n}(\mathrm{\Omega })$ and the weak${}^{\ast }$ convergence $u_{k}u$ in $BD(\mathrm{\Omega })$, $\mathrm{E}$ 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.