Stress singularity due to traction discontinuity on an anisotropic elastic half-plane

Abstract

In a half-plane problem with x 1 paralleling with the straight boundary and x 2 pointing into the medium, the stress components on the boundary whose acting plane is perpendicular to x 1 direction may be denoted by t 1 = [ σ 11, σ 12, σ 13] T. Stress components σ 11 and σ 13 are of more interests since σ 12 is completely determined by the boundary conditions. For isotropic materials, it is known that under uniform normal loading σ 11 is constant in the loaded region and vanishes in the unloaded part. Under uniform shear loading, σ 11 will have a logarithmic singularity at the end points of shear loading. In this paper, the behavior of the stress components σ 11 and σ 13 induced by traction-discontinuity on general anisotropic elastic surfaces is studied. By analyzing the problem of uniform tractions applied on the half-plane boundary over a finite loaded region, exact expressions of the stress components σ 11 and σ 13 are obtained which reveal that these components consist of in general a constant term and a logarithmic term in the loaded region, while only a logarithmic term exists in unloaded region. Whether the constant term or the logarithmic term will appear or not completely depends on what values of the elements of matrices Ω and Γ will take for a material under consideration. Elements for both matrices are expressed explicitly in terms of elastic stiffness. Results for monoclinic and orthotropic materials are all deduced. The isotropic material is a special case of the present results.