Reconstruction of Elliptic Voids in the Elastic Wedge: The Antiplane Problem

Abstract

In this article we study the reconstruction of the geometry of elliptic voids located in the elastic wedge, in frames of an antiplane two-dimensional problem. The wedge domain is bounded by a pair of straight boundaries. The first boundary is horizontal, and the second one is inclined at some angle. We assume that a known point force is applied to the horizontal boundary surface of the wedge, whose other face is fixed or is free from load. It is also assumed that the defect is unknown and that we can measure the shape of the surface over a certain finite-length interval of the same side where the outer force is applied. Such measurements can in principle be performed experimentally, but for our numerical experiments we carried out them by solving the respective direct problem, with the use of boundary element method (BEM). Then it is shown that an explicit form of Green's function can be constructed when the interior angle of the wedge is an integer part of 180°. For these cases, we construct an algorithm to restore the position, the size, and the orientation of the elliptic void. Some numerical examples demonstrate the good stability of the proposed algorithm.