Integral equations in direct and inverse problems of elastostatics for crack detection in layered structures

Abstract

This study deals with both direct and inverse problems for interfacial crack identification in laminates. The main focus is given to the case of an elastic substrate coated by a film made of a different elastic material. It is assumed that delamination can be developed on the interface between these materials. It is modelled as a combined open-sliding interface crack or by a pure sliding crack (slip). Its position may not be specified. The boundary conditions on the interface assume continuity of the stress vector across the whole interface and continuity of the displacements outside the crack. In the case of the slip the normal displacements are assumed to be continuous. The inverse boundary value considered is of the Cauchy type; it assumes that both stress and displacement vectors are known on the external boundary of the structure. In the case of the slip the problem is overdetermined on a part of the boundary, where three conditions are imposed, and undetermined on its remainder where just one condition is imposed. It is further referred to as a semi-inverse formulation. Therefore, these problems are ill-posed with the specified boundary conditions. Inverse, semi-inverse and direct problems are reduced to integral equations derived from basic properties of holomorphic functions followed by applications of Fourier transforms. Analytical solutions are found for the formulations considered and difficulties of numerical implementation are discussed in brief.