A lower semicontinuity result for linearised elasto-plasticity coupled with damage in $W^{1,\gamma}$, $\gamma>1$

TL;DR

This paper establishes a lower semicontinuity result for functionals modeling elasto-plasticity with damage, crucial for proving existence of solutions, especially for subcritical exponents where previous results were lacking.

Contribution

It provides the first lower semicontinuity result for these functionals in the subcritical case where b3 < n, advancing the mathematical understanding of elasto-plasticity models.

Findings

Proves lower semicontinuity for b3 > 1

Addresses the subcritical case b3 < n for the first time

Supports existence proofs for quasi-static evolutions in damage models

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 $\gamma < n$.