A generalized formulation of the theory of anti-plane deformation of a linear elastic solid was developed in which the Lamé constants depended upon a single spatial variable. The theory was applied to the study of a generalized screw dislocation, a semi-infinite crack and a finite crack which was located in a half-space region where the elastic inhomogeneity was depth-dependent. These situations were relevant to the study of the surface defects that were associated with functionally graded materials.

Some Generalized Anti-Plane Strain Problems for an Inhomogeneous Elastic Half-Space. A.J.M.Spencer, A.P.S.Selvadurai: Journal of Engineering Mathematics, 1998, 34, 403-16