| Abstract: |
| In this talk I will present a lower semicontinuity result of Reshetnyak type for a class of functionals which appear in models for small-strain elasto-plasticity coupled with damage. The plastic potentials that we consider depend on Sobolev functions $\alpha$, the damage variables, and on bounded Radon measures $p$, the plastic strains. The main difficulty in the proof of the lower semicountinuity of the plastic potential lies in the lack of continuity of the damage variable.
To prove the result, obtained in collaboration with Vito Crismale, 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. This is the key for the proof of the result. |
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