Transverse shear effects for through-cracks in an elastic plate

Abstract

The state of stress in a thin elastic plate which contains through-cracks is studied with a view toward assessing the influence of transverse shear on the crack tip stress and deformation fields. Based on the assumption that the through-the-thickness extensional strain is uniform in the thickness direction, a crack tip boundary layer solution is obtained for the tensile opening mode of deformation (mode 1). As anticipated, the generalized plane stress solution prevails everywhere in the plane of the plate except near boundaries. Near points on any smooth boundary, including the crack faces away from the tips, the plane stress solution is only slightly perturbed. Near crack tips, however, a complicated boundary layer is developed. A standard separation of variables approach to the analysis of the boundary layer fails, but an exact solution is found in the form of a real integral. When evaluated, the solution shows a finite lateral contraction at the crack tip (in contrast to the generalized plane stress result), and the inner solution merges smoothly with the plane stress solution at distances from the tip of one-half to three-fourths of the plate thickness, depending on the value of Poisson's ratio. The results are consistent with experiments intended to establish the limitations of the plane stress interpretation of shadow spot data.