A lower semicontinuity result for linearised elasto-plasticity coupled with damage in W1,γ, γ > 1
Vito Crismale, 1 CMAP, École Polytechnique, UMR CNRS 7641, 91128 Palaiseau Cedex, France, Gianluca Orlando, 2 TU München, Zentrum Mathematik-M7, Boltzmannstrasse 3, 85747 Garching, Germany, <sup>†</sup><b>This contribution is part of the Special Issue:</b> Variational Models in Elasticity, Guest Editors: Lucia De Luca; Marcello Ponsiglione, Link: <a href="" target="_blank">https://www.aimspress.com/newsinfo/1369.html</a> · Mathematics in Engineering · 2019
We prove the lower semicontinuity of functionals of the form $ \begin{equation*} \int \limits_\Omega \! V(\alpha) {\rm d} |{\rm E} u| \, , \end{equation*} $ with respect to the weak converge of $\alpha$ in $W^{1, \gamma}(\Omega)$, $\gamma \gt 1$, 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 $\gamma \lt n$.