Elastic-plastic analysis of combined mode I, II and III crack-tip fields under small-scale yielding conditions

Abstract

Within the context of the small-strain approach, combined mode I, II and III near-tip fields of stationary cracks in power-law hardening materials are investigated. We use a finite element technique to obtain asymptotic angular stress solutions for combined mode I and II perturbed from mode III. These perturbation solutions with the same stress singularities as those of pure mode III are presented for different hardening materials. The perturbation results further suggest that the order of crack-tip stress singularities varies smoothly with changing mode mixity. We also employ full-field finite element compulations to study the small-scale yielding near-tip fields for several combinations of prescribed remote mode I. II and III elastic K fields. These solutions verify an interesting pattern which agrees with the previous solutions for combined mode I and III loading as well as those for combined mode II and III loading: well within the plastic zone, under near mode I mixed-mode loadings, the in-plane stresses are slightly more singular than r-t/(n + 1) while the out-of-plane shear stresses are slightly less singular than r-t/(n + 1), where r is the radial distance to the lip and n is the strain hardening exponent of the material. To explain the complex behavior of the near-tip stresses, we introduce an effective in-plane shear stress and an effective out-of-plane shear stress to quantify the in-plane and out-of-plane plastic shear at different orientations in a consistent manner. The full-field solutions also corroborate the observation that the singularities of the in-plane slresses and the out-of-plane shear stresses vary smoothly with mode mixity.