Approximate mechanical impedance of a thin linear elastic slab

We consider the linear elasticity system for a body coated with a thin elastic layer. The effect of the thin layer on that body can be described through an impedance operator linking the displacement and the traction on the interface. This operator cannot be in general explicitly computed for an arbitrary shape. In this paper, we consider the case of a linear elastic slab with Hooke’s law. We compute by the Fourier transform the exact symbol of the impedance operator and prove that it is a pseudodifferential operator of order one. Then, we give a Padé-like approximation at order three (with respect to the thickness of the layer) for the impedance operator. The pattern of this approximation will be a helpful guide for approximating the impedance operator for a body whose boundary has nonzero curvature.