A lower semicontinuity result for linearised elasto-plasticity coupled with damage in <em>W</em>^{1,<em>γ</em>}, <em>γ</em> &gt; 1

Abstract

We prove the lower semicontinuity of functionals of the form \begin{equation*} \int \limits_\Omega \! V(\alpha) d |E u| \, , \end{equation*} with respect to the weak converge of $\alpha$ in $W^{1,\gamma}(\Omega)$, $\gamma > 1$, and the weak* convergence of $u$ in $BD(\Omega)$, where $\Omega \subset \mathbb{R}^n$. These functional arise in the variational modelling of linearised elasto-plasticity coupled with damage and their lower semicontinuity is crucial in the proof of existence of quasi-static evolutions. This is the first result achieved for subcritical exponents $\gamma < n$.

Keywords

• lower semicontinuity
• elasto-plasticity
• damage
• functions of bounded deformation