Stress and strain fields near cracks in solids with stress state-dependent elastic properties under conditions of anti-plane shear

Deformation fields near the tip of a crack in a material with properties depending on the type of stress state are presented for conditions of longitudinal shear. These properties are revealed in rocks, concretes, refractory ceramics, cast iron, structural graphite, some composites, and many other microheterogeneous materials. The behavior of heterogeneity depends on the loading conditions, and as a result, the deformation properties of these materials are stress state-dependent. Moreover, shear and volume deformation processes are interrelated in these materials. Therefore, it is impossible to distinguish between the fields corresponding to shear strain and bulk strain in solids. For the study of stress, strain, and displacement fields at a crack tip, corresponding constitutive equations are used to simulate the dependence of equivalent elastic properties on the type of stress state and to describe the relationship between shear strains and volumetric deformations. The traditional approach based on anti-plane strain hypothesis cannot be used for solving problems under longitudinal shear conditions because shear strains are not independent of bulk strains. Therefore, the corresponding representations for displacements have been proposed. Nonlinear constitutive relations are used, and conditions for a unique solution to the boundary value problem have been shown. The obtained solutions are compared with the known solutions for a linear elastic solid. The opening of crack faces is observed under conditions of out-of-plane shear, and the dependence of the crack opening on the sensitivity of material properties to the stress state is studied.