On the Exactness of Truncated Crack-tip Stress Expansions

Abstract

Williams series appear to be the most favored analytical tool for the description of mechanical fields near crack-tips in planar domains. For practical use, these series are generally truncated. A common belief is to consider that the more terms are kept, the more accurate the representation will be. Based on closed-form series expressions, this belief is shown to be only partially true. Asymptotic expansions converge within series convergence disks as expected, but truncated series can also provide exact values for the stress field. This property can be easily observed with the map of relative error comparing truncated series solutions for stress with complex exact ones. The series remainder appears to be equal to zero on curves emanating from the crack-tip. Their number and their initial angles are shown to be related to the zeros of a Williams eigen-function that depends on the number of terms kept in the truncated series.