Particular Solutions, Stress Singularities, and Stress Decay

Abstract

Not all boundary value problems are amenable to a simple analytical solution. This is particularly the case when the geometry of the boundary on which the boundary conditions are specified contains a corner. A crack tip is a special corner. When an analytical solution for the entire region is not available, asymptotic solutions near the corner can be obtained which provide useful information on the nature of stress singularities at the corner. They also provide more accurate numerical solutions by a finite element scheme in which the asymptotic solution at the corner is employed in a special element at the corner with regular elements elsewhere (see, for example, Stolarski and Chiang, 1989). Other information that can be obtained by an asymptotic analysis is the decay factor of stress at a large distance from a point at which a self-equilibrated load is applied (Crafter, et al., 1993). For singularities that arise in non-linear elastic materials the reader is referred to the book by Antman (1995). As in Chapter 8 most solutions can be expressed in a real form with the aid of identities presented in Chapters 6 and 7.