A lower semicontinuity result for linearised elasto-plasticity coupled with damage in \(W^{1,\gamma}\), \(\gamma > 1\). (English) Zbl 1502.35164
Summary: 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 \(\gamma < n\).
MSC:
| 35Q74 | PDEs in connection with mechanics of deformable solids |
| 74A45 | Theories of fracture and damage |
| 74C05 | Small-strain, rate-independent theories of plasticity (including rigid-plastic and elasto-plastic materials) |
| 74R20 | Anelastic fracture and damage |
| 35R09 | Integro-partial differential equations |
| 35A01 | Existence problems for PDEs: global existence, local existence, non-existence |
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