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

In a half-plane problem with x1 paralleling with the straight boundary and x2 pointing into the medium, the stress components on the boundary whose acting plane is perpendicular to x1 direction may be denoted by t1 = [σ11, σ12, σ13]T. Stress components σ11 and σ13 are of more interests since σ12 is completely determined by the boundary conditions. For isotropic materials, it is known that under uniform normal loading σ11 is constant in the loaded region and vanishes in the unloaded part. Under uniform shear loading, σ11 will have a logarithmic singularity at the end points of shear loading. In this paper, the behavior of the stress components σ11 and σ13 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 σ11 and σ13 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.