Disk-shaped shear crack in an elastic body with initial stresses

Abstract

This article examines a three-dimensional static problem for a circular disk-shaped crack in an elastic body with initial stresses. The problem is examined within the framework of a three-dimensional linearized formulation [2, 4], where additional loads are represented as constituting an arbitrary shear field. We will solve the problem by the approach used in [4]. In accordance with the latter, for an elastic body with initial stresses and weakened by a crack of a certain form, the problem is reduced to a mixed problem of harmonic potential theory. The solution is obtained in general form for compressible and incompressible bodies with an elastic potential of arbitrary form for the theory of finite strains and two variants of the theory of small initial strains. We examined two examples for different cases of appli cation of external loads to compressible and incompressible materials. Relations are presente~ to describe the effect of initial stresses on the stress intensity factors. It is shownthat with both an arbitrary shear load and with different special types of shear load, the initial stresses have an effect on the stress distribution near the edge of the crack (on the stress intensity factors). This is in contrast to axisymmetric problems [3, 5] and the problem of a normal-rupture crack with a nonaxisymmetric load [5]. This distinguishes the result obtaine~ for three-dimensional static problems from the result for the two-dimensional problem [4], where initial stresses have no effect on the stress intensity factors for any loading variant (normal rupture, longitudinal and transverse shear).