*Published Paper*

**Inserted:** 23 sep 2019

**Last Updated:** 13 dec 2019

**Journal:** Mathematics in Engineering

**Volume:** 2

**Number:** 1

**Pages:** 101-118

**Year:** 2020

**Doi:** 10.3934/mine.2020006

**Abstract:**

We prove the lower semicontinuity of functionals of the form \[
\int \limits_\Omega \! V(\alpha) \, \mathrm{d}

\mathrm{E} u

\, , \] 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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