On the Theory of Direct and Inverse Problems in the Elastic Rectangle: Anti-Plane Case

In this article, we study the antiplane deformation of the boundary surface of a rectangular domain in the presence of a void and a shear force on the outer boundary surface. For a formulated inverse problem, we develop some analytical results and use them to solve the problem numerically for various elliptic geometrical configurations. The analytical method allows us to give an efficient representation for Green's function in the rectangular domain. Then we derive the same Green's function by an alternative method based on Fourier series expansions. Finally, for a number of configurations, we demonstrate the comparison between real and reconstructed defects.