Extreme cases in boundary homogenization for the linear elasticity system (Q6645645)

The authors consider the linear elasticity problem: \(-\frac{\partial \sigma_{ij}^{\varepsilon}}{\partial x_{j}}=f_{i}\), posed in a 3D bounded domain \(\Omega \subset \mathbb{R}^{3+}=\{x\in \mathbb{R}^{3}:x_{3}>0\}\), with the boundary conditions: \(u_{\varepsilon}=0\), on \(\Gamma_{\Omega}\), \(\sigma_{ij}^{\varepsilon}n_{j}=0\), on \(\Sigma \setminus \bigcup T^{\varepsilon}\), \(\sigma_{ij}^{\varepsilon}n_{j}+\beta (\varepsilon)M_{ij}u_{j}^{\varepsilon}=0\), on \(\bigcup T^{\varepsilon}\), where \(\partial \Omega\) is the Lipschitz boundary of \(\Omega\), \(\Sigma\) the non-empty part of the boundary in contact with the plane \(\{x_{3}=0\}\), \(\Gamma_{\Omega}\) the rest of the boundary of \(\Omega\), \(\bigcup T^{\varepsilon}\) a collection of domains \(r_{\varepsilon}\)-homothetic of fixed domains \(T^{p}\), \(p=1,\ldots,\mathfrak{M}\), and \(\varepsilon\)-periodically distributed on \(\{x_{3}=0\}\), and \(\lim_{\varepsilon \rightarrow 0}\frac{r_{\varepsilon}}{\varepsilon^{2}}=r_{0}\), \(\lim_{\varepsilon \rightarrow 0}r_{\varepsilon}\beta (\varepsilon)=\beta^{0}\), and \(\lim_{\varepsilon \rightarrow 0}\frac{\beta (\varepsilon)r_{\varepsilon}^{2}}{\varepsilon^{2}}=\beta^{\ast}\). The authors introduce the variational formulation associated with this problem. \N\NThe purpose of the paper is to describe the asymptotic behavior of the solution to the above problem, in some extreme cases associated with values of the parameters \(r_{0},\beta^{0},\beta^{\ast}\). In the case \(\beta^{\ast}=0\), the authors prove that the solution \(u^{\varepsilon}\) to the above problem converges in the strong topology of \((H^{1}(\Omega))^{3}\) to the solution \(u^{0}\) to the problem \(\int_{\Omega}\sigma_{ij,x}(u^{0})e_{ij,x}(v)dx=\int_{\Omega}f_{i}v_{i}dx\), \(\forall v\in V\), the completion of \(\{v\in (C^{1}(\Omega))^{3}:v=0\) on \(\Gamma_{\Omega}\}\) with respect to the norm associated to the linear elastic energy. For the proof, the authors prove uniform estimates on the solution \(u^{\varepsilon}\) and they pass to the limit in the terms of the variational formulation it satisfies. In the case where \(r_{0}=0\) and \(\beta^{\ast}\neq 0\), the authors prove a similar result assuming a further hypothesis on the domains \(\bigcup T^{\varepsilon}\). The authors then consider the case where \(r_{0}=+\infty\) and \(\beta^{\ast}=+\infty\), the solution \(u^{\varepsilon}\) converging in \((H^{1}(\Omega))^{3}\) to the solution \(u^{0}\) to the above limit problem but now posed in \((H_{0}^{1}(\Omega))^{3}\). The paper ends with some results concerning the spectral problem associated with this problem.\N\NFor the entire collection see [Zbl 1544.65013].