Three-Dimensional Stress Distribution Around an Elliptical Crack Under Arbitrary Loadings

TLDR

The study extends classical elliptical crack analysis by deriving solutions for an infinite solid with an elliptical crack under arbitrary shear loads through superposition of general loading cases. Stress‑intensity factors \(K_j\) (j = 1, 2, 3) are calculated from the derived stress equations to assess fracture strength in elastic solids with cracks. The resulting three‑dimensional stress field near the crack edge is expressed in polar coordinates (r, θ), exhibiting the same angular distribution and inverse‑square‑root singularity as in two‑dimensional problems, a property that holds for any smooth discontinuity plane and clarifies the extension of fracture‑mechanics theories to three dimensions.

Abstract

As a companion problem to that of a flat elliptical crack subject to a uniform tension perpendicular to the crack plane, this paper deals with the case of arbitrary shear loads. Upon superposition, solutions to problems of an infinite solid containing an elliptical crack subjected to loads of a general nature may be obtained. It is shown that the three-dimensional stresses near the crack border can be expressed explicitly in terms of a convenient set of coordinates r and θ defined in a plane normal to the edge of the crack. In such a plane, the local stresses in a solid are found to have the same angular distribution and inverse square-root stress singularity as those in a two-dimensional body under the action of in-plane stretching and out-of-plane shear. This result will, in general, hold for any plane of discontinuity bounded by a smooth curve. Such information provides a clear interpretation of current fracture-mechanics theories to three dimensions. In particular, stress-intensity factors kj(j = 1, 2, 3), used in the Griffith-Irwin theory of fracture, are evaluated from the stress equations for determining the fracture strength of elastic solids with cracks or flaws.