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Mathematics in Engineering

2020, Volume 2, Issue 1: 101-118. doi: 10.3934/mine.2020006
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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
  • 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 $\gamma \lt n$.


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