Analytical determination of coefficients in crack-tip stress expansions for a finite crack in an infinite plane medium

Abstract

Williams series expansion provides a general framework for the description of stress state near a crack-tip. To each cracked configuration corresponds a sequence of coefficients depending on the geometry and the load. Analytical definitions are widely available for the first two terms leading to finite energy in crack-tip area (stress intensity factor, T-stress), but rarely for higher order terms (either regular or singular ones). In the case of cracks of finite length, the radius of convergence of Williams series limits the area of validity for the asymptotic expansion. This paper presents closed-form expressions for the whole sequences of coefficients related to the problem of a finite crack in an infinite plane medium with mode I and mode II remote load. Identification of coefficients is based on expansions of Westergaard’s exact complex solutions. Closed-form crack-tip expansions are given using power and Laurent series for points respectively inside and outside a specific disk related to the geometry. Validity of the expressions derived is assessed with the conclusive comparison of analytical series to complex solutions for mode I and mixed mode problems. The existence of distinct domains of convergence for power and Laurent series is emphasized and their radial and angular bases are shown to be different.