A Reshetnyak-type lower semicontinuity result for linearised elasto-plasticity coupled with damage in $W^{1,n}$

TL;DR

This paper establishes a lower semicontinuity result of Reshetnyak type for functionals in small-strain elasto-plasticity with damage, characterizing the limits of measures under weak convergence in relevant function spaces.

Contribution

It provides a novel lower semicontinuity theorem for coupled elasto-plasticity and damage models, characterizing measure limits in $W^{1,n}$ and $BD$ spaces.

Findings

Characterization of measure limits as $eta E u + ext{singular part}$

Use of concentration compactness to analyze measure convergence

Extension of Reshetnyak's theorem to coupled elasto-plasticity and damage

Abstract

In this paper we prove a lower semicontinuity result of Reshetnyak type for a class of functionals which appear in models for small-strain elasto-plasticity coupled with damage. To do so 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.