On inverse crack problems in elastostatics

Abstract

In solid mechanics, nondestructive testing has been an important technique in gathering information about unknown cracks, or defects in material. From a mathematical point of view, this is described as an inverse problem of partial differential equations, that is, the problem is to extract information about the location and shape of an unknown crack from the surface displacement field and traction on the boundary of the elastic material. By using the enclosure method introduced by Prof. Ikehata we can derive the extraction formula of an unknown linear crack from a single set of measured boundary data. Then, we need to have precise properties of a solution of the corresponding boundary value problem; for instance, an expansion formula around the crack tip. In this paper we consider the inverse problem concentrating on this point.Let Ω be a bounded convex domain in R 2 , representing a homogeneous linearized elastic plate. We denote by γ, which is the straight line segment P Q for any P ∈ ∂Ω and Q ∈ Ω the crack in Ω. Let Q be a point of intersection of an extension of the crack P Q and ∂Ω, γ denote P Q . Then, Ω is divided into two parts Ω + and Ω − by γ (See, Figure 1). By u = (u i ) i=1,2 and σ = (σ ij ) i,j=1,2 we denote the displacement vector and the stress tensor, respectively. Then, it is well known that σ has a singularity at the crack tip. In particular, coefficients of singular terms in an expansion of σ around the crack tip are important in fracture mechanics as stress intensity factors which are characterizing parameters of fracture.