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Complete-damage evolution based on energies and stresses. (English) Zbl 1209.74010
The purpose of the paper is to study a rate-independent damage model. The material fills in a bounded and Lipschitz domain \(\Omega \) of \(\mathbb{R}^{d} \). The state of the system is described by its displacement \(\widetilde{u}\) and by the scalar damage variable \(z\), which lies between 0 (totally damaged) and 1 (no damage). The author considers the stored energy functional \(\mathcal{E}(t,u,z)=\int_{\Omega }W(x,\mathbf{e}_{D}(t,x)+\mathbf{ e}(u)(x),z(x))dx+\mathcal{G}(z)\), with \(\mathcal{G}(z)=\int_{\Omega }b(x,z(x)+G(x,\nabla z(x))dx\). Here \(\widetilde{u}(t)=u_{D}(t)+u(t)\) where \(u \) vanishes on a fixed part \(\Gamma _{D}\) of the boundary of \(\Omega \), \( \mathbf{e}(u)\) is the linearized strain tensor, \(\mathbf{e}_{D}(u)=\mathbf{e} (u_{D})\), \(W\), \(b\) and \(G\) are Caratheodory functions which satisfy smoothness and growth conditions, among other hypotheses. In order to get a coercivity property, the function \(W\) is replaced by \(W_{\delta }\) with \( W_{\delta }(x,e,z)=W(x,e,z)+\delta |e|^{p}\), for a positive \(\delta \).
The author first defines an appropriate notion of energetic solution for this regularized problem. The first main result of the paper proves an existence and uniqueness result, the energetic solution \((u_{\delta },z_{\delta })\) lying in the space \(L^{\infty }([0,T],W^{1,p}(\Omega ,\mathbb{R}^{d})\times W^{1,r}(\Omega ))\), with \(z\in BV([0,T],L^{1}(\Omega ))\). The main goal of the paper is the describe the asymptotic behaviour of \((u_{\delta },z_{\delta })\) when \(\delta \) goes to 0. The author here uses a parametrized \(\Gamma \)-convergence tool. The paper ends with the description of some examples and of some possible extensions.

MSC:
74C05 Small-strain, rate-independent theories of plasticity (including rigid-plastic and elasto-plastic materials)
35K65 Degenerate parabolic equations
35K85 Unilateral problems for linear parabolic equations and variational inequalities with linear parabolic operators
49S05 Variational principles of physics
74R05 Brittle damage
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