Homogenisation of high-contrast brittle materials (Q2099375)

The author analyzes the large-scale behavior of high-contrast composite materials which can undergo fracture. For a given open and bounded set \( \Omega \subset \mathbb{R}^{n}\) with Lipschitz boundary, she introduces the collection \(\mathcal{A}(\Omega )\) of all open subsets of \(\Omega \) and the localized energy functionals \(\mathcal{F}_{k}(u,U)=\int_{U_{k}}f_{k}(x, \nabla u)dx+\alpha _{k}\int_{U\setminus U_{k}}f_{k}(x,\nabla u)dx+\int_{S_{u}\cap U_{k}}g_{k}(x,\nu _{u})d\mathcal{H}^{n-1}+\beta _{k}\int_{S_{u}\cap (U\setminus U_{k})}g_{k}(x,\nu _{u})d\mathcal{H}^{n-1}\) if \(u\in SBV^{p}(U)=\{u\in SBV(U):\nabla u\in L^{p}(U)\) and \(\mathcal{H} ^{n-1}(S_{u})<+\infty \}\), \(\mathcal{F}_{k}(u,U)=+\infty \) otherwise in \( L^{1}(\Omega )\), \(S_{u}\) being the approximate discontinuity set of \(u\in SBV^{p}(U)\), where \(U_{k}=U\cap \varepsilon _{k}E\), \(\nu _{u}\) is the generalized normal to \(S_{u}\), \(f_{k}:\mathbb{R}^{n}\times \mathbb{R} ^{n}\rightarrow \lbrack 0,+\infty )\) are Caratheodory functions such that there exist \(p>1\) and \(0<c_{1}\leq c_{2}<+\infty \) such that for every \( (x,\xi )\in \mathbb{R}^{n}\times \mathbb{R}^{n}\) and every \(k\in \mathbb{N} :c_{1}\left\vert \xi \right\vert ^{p}\leq f_{k}(x,\xi )\leq c_{2}(1+\left\vert \xi \right\vert ^{p})\) and \(f_{k}(x,0)=0\) for every \(x\in \mathbb{R}^{n}\) and every \(k\in \mathbb{N}\), \(g_{k}:\mathbb{R}^{n}\times \mathbb{S}^{n-1}\rightarrow \lbrack 0,+\infty )\) are Borel functions such that there exist \(0<c_{3}\leq c_{4}<+\infty \) such that for every \((x,\nu )\in \mathbb{R}^{n}\times \mathbb{S}^{n-1}\) and every \(k\in \mathbb{N} :c_{3}\leq g_{k}(x,\nu )\leq c_{4}\) and \(g_{k}(x,\nu )=g_{k}(x,-\nu )\), for every \((x,\nu )\in \mathbb{R}^{n}\times \mathbb{S}^{n-1}\) and every \(k\in \mathbb{N}\), and \(\alpha _{k},\beta _{k}\in \lbrack 0,1]\). The author defines the notion of convergence for a sequence \((u_{k})\subset L^{1}(\Omega )\) to a function \(u\in L^{1}(\Omega )\) if there exists a sequence \((\widetilde{u}_{k})\subset L^{1}(\Omega )\) such that \(\widetilde{u} _{k}=u_{k}\) a.e. in \(\Omega _{k}\) and \(\widetilde{u}_{k}\) converges to \(u\) in \(L^{1}(\Omega )\), then the notion of sequential \(\Gamma \)-convergence for a sequence \(\mathcal{F}_{k}:L^{1}(\Omega )\rightarrow \lbrack 0,+\infty ] \) to \(\mathcal{F}\), with respect to the previously defined convergence, if for every \(u\in L^{1}(\Omega )\) the two following conditions are satisfied: for every \((u_{k})\subset L^{1}(\Omega )\) converging to \(u\) \(\mathcal{F} (u)\leq \lim \inf_{k\rightarrow +\infty }\mathcal{F}_{k}(u_{k})\), and if there exists \((\overline{u}_{k})\subset L^{1}(\Omega )\) converging to \(u\) such that \(\mathcal{F}(u)\geq \lim \sup_{k\rightarrow +\infty }\mathcal{F} _{k}(\overline{u}_{k})\). She recalls the compactness result for the above-defined localized energy functionals \(\mathcal{F}_{k}\): there exists a subsequence \((\mathcal{F}_{k_{j}})\subset (\mathcal{F}_{k})\) such that the corresponding functionals \(\mathcal{F}^{\prime }(\cdot ,U)=\Gamma -\lim \inf_{k\rightarrow +\infty }\mathcal{F}_{k}(u,U)\) and \(\mathcal{F}^{\prime \prime }(\cdot ,U)=\Gamma -\lim \sup_{k\rightarrow +\infty }\mathcal{F} _{k}(u,U)\) satisfy \(\mathcal{F}_{-}^{\prime }=\mathcal{F}_{-}^{\prime \prime }\), with \(\mathcal{F}_{-}^{\prime }(u,U)=\sup \{\mathcal{F}^{\prime }(u,V):V\subset \subset U\), \(V\in \mathcal{A}(\Omega )\}\), \(\mathcal{F} _{-}^{\prime \prime }(u,U)=\sup \{\mathcal{F}^{\prime \prime }(u,V):V\subset \subset U\), \(V\in \mathcal{A}(\Omega )\}\).\ She proves properties, an integral representation of a \(\Gamma \)-limit of sequences \((\mathcal{F}_{k})\) and a \(\Gamma \)-convergence result for sequences \((\mathcal{F}_{k})\). In the last part of her paper, the author describes two examples of such situations concerning periodic porous brittle materials, or periodic brittle high-contrast materials with soft or weak inclusions.