Derivation of In-plane Mixed-Mode Asymptotic Solutions for an Unsteadily Fast-Propagating Crack Tip Using a Symbolic Manipulation System.

Abstract

Explicit expressions for in-plane mixed-mode asymptotic solutions of an unsteadily fast-propagating crack tip had not been derived so far, due to seemingly insurmountable mathematical and algebraic difficulties To overcome these difficulties, a symbolic manipulation system was used in the present study. The explicit expressions for the mixed-mode transient stress and displacement fields up to the first five terms (n=0∼4) in asymptotic series were successfully derived. It was found that the transient effects do not appear in the first three terms (rigid body (n=O), singular (n=1) and constant-stress (n=2) terms), but appear only in the higher-order terms greater than or equal to the third order (ng3). The transient effects in the near-tip stress distributions were also visualized using the symbolic manipulation system. It was found that both effects of the crack acceleration and of the changes in the asymptotic coefficients increase the stress distribution ahead of the propagating crack tip. These stress-increases in the unsteady part may clarify the problem of the crack-acceleration effects in the dynamic fracture toughness versus crack velocity curves, which is one of important unsolved problems in dynamic fracture mechanics.