Influence of the Higher Order Terms in Williams’ Series Expansion of the Stress Field on the Stress-Strain State in the Vicinity of the Crack Tip. Part Ii

Abstract

The description of mechanical fields at the vicinity of a bi-dimensional crack-tip can be performed using the classic Williams asymptotic series expansion. While the general structure is well known, complete expressions are rarely available for specific crack problems. The paper is devoted to the multi-parameter description of the stress field in the vicinity of two collinear crack of different length in an infinite isotropic elastic medium subjected to 1) Mode I loading; 2) Mode II loading; 3) mixed (Mode I + Mode II) mode loading. The multi-parameter asymptotic expansions of the stress field are obtained. The procedure used in the paper relates Williams series coefficients and the complex potentials of the plane elasticity. The amplitude coefficients of the multi-parameter series expansion are found in the closed form. Having obtained the coefficients of the Williams series expansion one can keep any preassigned number of terms in the asymptotic series. Asymptotic analysis of number of the terms in the Williams asymptotic series which is necessary to keep in the asymptotic series at different distances from the crack tip. It is shown that the more distance from the crack tip the more terms in the Williams asymptotic expansion need to be kept. Complete closed-form expressions can be used to derive, test and improve numerical and experimental approaches involving higher order terms in crack-tip expansions.