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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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