### Mathematics in Engineering

2020, Issue 1: 101-118. doi: 10.3934/mine.2020006
Research article Special Issues

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

• Received: 20 June 2019 Accepted: 23 September 2019 Published: 27 November 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 $\gamma \lt n$.

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

### Related Papers:

• 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 $\gamma \lt n$.

###### 通讯作者: 陈斌, bchen63@163.com
• 1.

沈阳化工大学材料科学与工程学院 沈阳 110142

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