Stress field of plane dislocation pile-ups in anisotropic theory of elasticity: PMM vol. 39, n≗5, 1975, pp. 942–950

Abstract

The stresses at each point of an infinite anisotropic elastic medium with defects described effectively by dislocation pile-ups (the pile-ups themselves, cracks, deformation twins, fine crystals of martensitic type) are determined within the framework of the plane problem of the theory of elasticity. The stresses are expressed in terms of the dislocation density in the pile-ups for an arbitrary realization of the defects, taking into account their total Burgers vector. The interaction between a source of the linear force-dislocation type and the mentioned defects (taking account of the image force) is investigated. An expression is presented for the stress around pile-ups described discretely. A number of papers ([1 – 7], for example) is devoted to a theoretical study of the stress fields of defects of pile-up type. Attention is turned in some [4 – 7] to the similarity between the type of stress and the dislocation distribution in the pile-up. Nevertheless, no general expression has been obtained for the stress field in an anisotropic elastic and even isotropic medium with pile-ups although there is a solution for a slit-like crack [1] (valid, however, only in particular cases). By using a simpler method than in [1], more exact results are obtained in this paper which are applicable to all defects of the plane dislocation pile-up type for an arbitrary realization of the defects. In contrast to the expressions in [1] which are valid only near the blocked ends of cracks with zero Burgers vector and do not accurately take account of the interaction with other defects, the expressions obtained yield the stress in a whole medium with defects of the pile-up type in external fields which can be described by linear combinations of rational-fraction functions. Let us note that the influence of anisotropy on the dislocation distribution and on the stress, not essential in a number of cases [1, 2], can play a major part in describing such processes as the excitation of defects by definite development systems in a field of others, the rotation and branching of cracks, the distribution of impurities and the separation of phases around defects, the formation of relaxation zones around them, etc.