Homogenization of the Neumann problem for the Lamé equations of linear elasticity in domains with a periodic system of channels of small length (Q877519)

The author presents an homogenization result for a linear elastic 2D problem.\ The domain \(\Omega _{\varepsilon }\) under consideration consists of two pieces \(\Omega _{\varepsilon }^{+}\) and \(\Omega ^{-}\) separated by a thin layer of width \(\varepsilon ^{q}\) and perforated by \(\varepsilon \) -periodically distributed channels of thinness \(a_{\varepsilon }\). The elastic problem under consideration is \(-\partial (A^{mk}\partial u_{\varepsilon }/\partial x_{k})/\partial x_{m}+\rho u_{\varepsilon }=f\) in \( \Omega _{\varepsilon }\), with homogeneous Neumann type boundary conditions on \(\partial \Omega _{\varepsilon }\). The author first proves uniform estimates on the displacement field \(u_{\varepsilon }\) in \(L^{2}(\Omega _{\varepsilon })\), in \(H^{1}(\Omega ^{-})\) and in \(H^{1}(\Omega _{\varepsilon }^{+})\). He also proves uniform estimates on \(v_{\varepsilon }(x_{1},x_{2})=u_{\varepsilon }(x_{1}+\varepsilon ^{q},x_{2})\) in \( H^{1}(\Omega ^{+})\). This is done using Korn's inequality in the whole domain and the variational formulation of the problem. The asymptotic behaviour of the solution depends on the limit \(\beta =\lim_{\varepsilon \rightarrow 0}a_{\varepsilon }\varepsilon ^{-1-q}\). The main results of the paper indeed distinguish between the cases \(\beta =0\) or \(\beta >0\). The convergence results are deduced from the variational formulation of the problem and some Poincaré-Wirtinger inequality in the channels, through the construction of appropriate test-functions.