Thomas, Marita Quasistatic damage evolution with spatial \(BV\)-regularization. (English) Zbl 1375.74009 Discrete Contin. Dyn. Syst., Ser. S 6, No. 1, 235-255 (2013). 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. Cited in 24 Documents 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 Keywords:partial damage; energetic formulation; gamma-convergence; Modica-Mortola functionals; existence; rate-independent damage Citations:Zbl 1191.35159 PDFBibTeX XMLCite \textit{M. Thomas}, Discrete Contin. Dyn. Syst., Ser. S 6, No. 1, 235--255 (2013; Zbl 1375.74009) Full Text: DOI