Asymptotic Analysis of the Crack Tip Stress Field (Consideration of Higher Order Terms)

This paper presents multi-parameter asymptotic description of the stress field near the tip of a central crack in a linear-elastic plate under: (1) normal tensile stress; (2) transverse shear; (3) mixed mode deformation in the full range of mixed modes of loading, from the opening mode fracture to antiplane shear. A multi-parameter expansion of the stress tensor components including higher order terms has been constructed. All the scale (amplitude) factors—coefficients of the complete Williams asymptotic expansion—have been determined as functions of the crack length and parameters of loading. The expansion constructed and formulas obtained for the expansion coefficients can be used for keeping any preassigned number of terms in asymptotic representations of mechanical fields at a crack tip in a plate. The number of components to keep at different distances from the tip of defect was subjected to analysis. The angles of crack propagation under conditions of mixed-mode loading were calculated using a multi-parameter expansion of stress field by means of (1) the maximum tangential stress criterion and (2) the criterion of minimum elastic strain energy density.