Approximation of Dirichlet-to-Neumann operator for a planar thin layer and stabilization in the framework of couple stress elasticity with voids

Abstract

In this paper, we start from a two dimensional transmission model problem in the framework of couple stress elasticity with voids which is defined in a fixed domain ${\mathrm{\Omega }}_{-}$ juxtaposed with a planar thin layer ${\mathrm{\Omega }}_{+}^{\mathit{\delta }}$ . We first derive a first approximation of Dirichlet-to-Neumann operator for the thin layer ${\mathrm{\Omega }}_{+}^{\mathit{\delta }}$ by using the techniques of asymptotic expansion with scaling, which allows us to approximate the transmission problem by a boundary value problem doesn’t take into account any more the thin layer ${\mathrm{\Omega }}_{+}^{\mathit{\delta }}$ , called approximate impedance problem; after that, we prove an error estimate between the solution of the transmission problem and the solution of the approximate impedance problem.