Mixed mode fracture in power law hardening materials near Mode I

Abstract

Mixed mode fracture in power law hardening materials near mode I loading conditions is investigated for the case of plane strain. It is demonstrated by application of full-field finite element analysis and two-parameter asymptotic analysis that the asymptotic mixed mode solution of Shih (1974), where the effects of arbitrary loading combine to enter the leading term of the asymptotic series, does not apply close to mode I. Instead, a `mode I dominant' higher order asymptotic form applies, where the leading term is the pure mode I symmetric HRR solution and the second term is antisymmetric. The significant difference between the two asymptotic solutions is the singular nature of the antisymmetric part of the stresses and strains. Therefore, near mode I the antisymmetric part of the local fields does not contribute to the J-integral. The transition from the mixed mode asymptotic solution to this two-term solution occurs when the ratio of the radial stress on the upper crack surface to that on the lower crack surface, i.e., σ<sub>rr</sub>(r,+π)/σ<sub>rr</sub>(r,−π), for the limit as r approaches zero, switches from −1 to +1. Budiansky and Rice (1973) predicted this change in sign somewhere between mode I and mode II. This sign change corresponds to the singular nature of the antisymmetric fields switching from the leading HRR eigenvalue to a weaker second term eigenvalue. Based on full-field finite element results, the far-field loading for which this `jump' occurs is very difficult to identify, as it originates deep within the plastic zone at unrealistically small distances from the crack tip. At the physical length scale on the order of the crack tip opening displacement, the transition from the mode II dominant asymptotic solution to the mode I dominant solution appears to be continuous, not abrupt. Identification of this transition is estimated from the full-field results by making use of the two asymptotic solutions. These results identify a breakdown in the HRR theory for the representation of the antisymmetric part of the stresses for mixed mode fracture near mode I. This breakdown is explained by the switching from a mixed mode solution to a higher order solution. Unfortunately, the practical application of this higher-order solution is limited to hardening powers of approximately 2<n<3.