Quasistatic damage evolution with spatial \(BV\)-regularization. (English) Zbl 1375.74009
Summary: An existence result for energetic solutions of rate-independent damage processes is established. We consider a body consisting of a physically linearly elastic material undergoing infinitesimally small deformations and partial damage. In [M. Thomas and A. Mielke, ZAMM, Z. Angew. Math. Mech. 90, No. 2, 88–112 (2010; Zbl 1191.35159)] an existence result in the small strain setting was obtained under the assumption that the damage variable \(z\) satisfies \(z \in W^{1,r}(\Omega)\) with \(r \in (1,\infty)\) for \(\Omega \subset \mathbb{R}^d\). We now cover the case \(r = 1\). The lack of compactness in \(W^{1,1}(\Omega)\) requires to do the analysis in \(BV(\Omega)\). This setting allows it to consider damage variables with values in {0,1}. We show that such a brittle damage model is obtained as the \(\Gamma\)-limit of functionals of Modica-Mortola type.
MSC:
74A45 | Theories of fracture and damage |
74R05 | Brittle damage |
74G65 | Energy minimization in equilibrium problems in solid mechanics |
35Q74 | PDEs in connection with mechanics of deformable solids |