Saint-Venant's principle and the anti-plane elastic wedge

Abstract

The stress field due to self-equilibrating loading on the inner or outer radius of a wedge sector, consistent with anti-plane deformation, will be affected by two agencies: a geometric effect of increasing or decreasing area and decay as anticipated by Saint-Venant's principle. When the load is applied to the inner radius the two effects are acting in concert; however, when the load is applied to the outer radius the two effects act in opposition. For a wedge angle in excess of the half-space the geometric effect is dominant over Saint-Venant decay and the stress increases the greater the distance from the loaded outer radius, indicating a breakdown in Saint-Venant's principle. For the wedge angle 2α = 360°, the unique inverse square root stress singularity at the crack tip, which is at the heart of linear elastic fracture mechanics, can be attributed to this breakdown of Saint-Venant's principle for just one eigenmode.