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A lower semicontinuity result for linearised elasto-plasticity coupled with damage in W1,γ, γ > 1

1 CMAP, École Polytechnique, UMR CNRS 7641, 91128 Palaiseau Cedex, France
2 TU München, Zentrum Mathematik-M7, Boltzmannstrasse 3, 85747 Garching, Germany

This contribution is part of the Special Issue: Variational Models in Elasticity
Guest Editors: Lucia De Luca; Marcello Ponsiglione
Link: https://www.aimspress.com/newsinfo/1369.html

Special Issues: Variational Models in Elasticity

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 $\gamma < n$.
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Keywords lower semicontinuity; elasto-plasticity; damage; functions of bounded deformation

Citation: Vito Crismale, Gianluca Orlando. A lower semicontinuity result for linearised elasto-plasticity coupled with damage in W1,γ, γ > 1. Mathematics in Engineering, 2020, 2(1): 101-118. doi: 10.3934/mine.2020006

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