On the circumferential shear stress around circular and elliptical holes

Abstract

The circumferential shear stress along the boundary of a traction-free circular hole in an infinite isotropic solid under antiplane shear is twice the circumferential shear stress along the corresponding circle in an infinite solid without a hole, under the same loading conditions. This result, originally derived by Lin et al. (Yielding, damage, and failure of anisotropic solids. EGF Publication, Mechanical Engineering Publications, London, 1990) by invoking the extended circle theorem of Milne-Thomson, is rederived in this paper using the Fourier series analysis, in the spirit of the original Kienzler and Zhuping (J Appl Mech 54:110–114, 1987) derivation of the plane strain result for the hoop stress around the hole. The formula is also deduced by the complex potential method, without explicitly invoking the extended circle theorem. The ratio of the circumferential stresses along the boundary of a noncircular hole and the congruent curve in a solid without a hole depends on the loading. For an elliptical hole in an infinite solid under remote loading ${{\sigma_{xz}^{0}}}$, this ratio is constant and equal to 1 + b/a; under ${{\sigma_{yz}^{0}}}$ it is equal to 1 + a/b, where a and b are the semi-axes of the ellipse. These constant values coincide with the values of the stress concentration factor in the two cases. If ${{\sigma_{xz}^{0}}}$ and ${{\sigma_{yz}^{0}}}$ are both applied, the stress ratio varies along the boundary of the ellipse; a closed form expression for this variation is determined. The portions of the elliptical boundary along which the strain energy density is increased or decreased by the creation of the hole are determined for different types of loading and different aspect ratios of the ellipse. An analytical expression for the stress concentration factor is then obtained for any orientation of the ellipse relative to the applied load.