Asymptotic analysis of the crack tip stress field (consideration of higher order terms)

This paper deals with the multi-parameter asymptotic description of the stress field near the crack tip of a finite crack in an infinite isotropic elastic plane medium subject to 1) tensile stress; 2) in-plane shear; 3) mixed mode loading for a wide range of mode-mixing situations (Mode I and Mode II). The multi-parameter series expansion of the stress tensor components containing higher order terms has been constructed. All the coefficients of the multi-parameter series expansion of the stress field are given. The main focus is on the discussion of the influence of considering the higher-order terms of the Williams expansion. Analysis of the higher order terms in the stress field is made. It is shown that the larger distance from the crack tip, the more terms are necessary to be kept in the asymptotic series expansion.
Therefore, it can be concluded that several more higher-order terms of the Williams expansion must be used for the stress field description when the distance from the crack tip is not small enough. The crack propagation direction angle has been calculated. Two fracture criteria: maximum tangential stress criterion and the strain energy density criterion, are used. The multi-parameter form of two commonly used fracture criteria is introduced and tested. Thirty and more terms of the Williams expansion enable the angle to be calculated more precisely.