Asymptotic analysis and optimization of an elastic body surrounded by thin layers

Abstract

We consider an elastic body surrounded by thin elastic layers along a part of its boundary. We study the asymptotic behavior of the structure as the maximum thickness of the layers tends to zero. We derive an effective boundary integral energy involving a matrix of Borel measures not charging polar sets and having the same support contained in the boundary. We characterize this matrix for three special cases: periodic layers, layers which are determined by a given nonnegative function $h$, and layers with abrupt changes along self similar fractals. We then consider an optimal control problem, which consists in determining the shape of the best material distribution around the elastic body, under the maximal work of external loads, and characterize the optimal zones on its boundary where possible elastic layers could take place.