Approximate mechanical impedance of a thin linear elastic slab

An elastic deformable body covered with a thin elastic layer is considered. In the literature, it is well known that the effect of thin layers can be described by impedance boundary conditions. In this paper, we consider a planar geometry, and compute the mechanical impedance defined on the exterior surface of the elastic slab. It is given through a relation between the traction tensor and the displacement. In Fourier space it is a pseudodifferential operator of order one. After computing the exact mechanical impedance, we give an approximate impedance problem at order three with respect to the thickness of the thin elastic layer.