Cutting-out method in the problem of longitudinal shear of anisotropic half-space with a crack

Abstract

The previously developed direct cutting-out method in application to isotropic materials, in particular to bodies with thin inhomogeneities in the form of cracks and thin deformable inclusions is extended to the case of taking into account the possible anisotropy of the material. The basis of the method is to modulate the original problem of determining the stress state of a limited body with thin inclusions by means of a technically simpler to solve problem of elastic equilibrium of an infinite space with a slightly increased number of thin inhomogeneities, which in turn form the boundaries of the investigated body. By loaded cracks we model the boundary conditions of the first kind, and by absolutely rigid inclusions embedded into a matrix with a certain tension – the boundary conditions of the second kind. Using the method of the jump functions and the interaction conditions of a matrix with inclusion, the problem is reduced to a system of singular integral equations, the solution of which is carried out using the method of collocations. Approbation of the developed approach is carried out on the problem of elastic equilibrium of anisotropic (orthotropic in direction of shear) half-space with a symmetrically loaded very flexible inclusion (a crack) at jammed half-space boundary. The influence of inhomogeneity orientation and the half-space material on the generalized stress intensity factors were studied.