Stable crack growth under mixed-mode conditions

Abstract

T he problem of a plane strain crack growing quasi-statically in an elastoplastic solid under mixed-mode conditions is studied. Two possibilities are considered: interfacial crack growth, corresponding to the case where the phases on either side of the crack plane arc dissimilar, but tougher than the interface; and asymmetric crack growth in a homogeneous solid, corresponding to the case where the phases are identical. but the direction of crack propagation is determined by a weak surface in the material. The constitutive behavior of the two phases is characterized by the J 2-flow theory of plasticity with linear hardening, but with generally distinct elastic and plastic properties. The goal is to characterize the asymptotic near-tip stress and deformation fields of the problem. To this end, a variable-separable form is assumed for the most singular term in an asymptotic expansion of the solution, and the associated eigenvalue problem is solved. The strength of the singularity in the radial distance from the crack tip serves as the eigenvalue, and the angular variations of the fields pose as the corresponding eigenfunctions. It is found that within this class of solutions only two distinct solutions are possible in general. These two solutions possess slightly different singularity strengths, and it is found that the angular variations of the corresponding stress and deformation fields are uniquely determined up to arbitrary amplitude factors. Thus, the “mixity” parameter (measuring the relative proportions of the shear and tensile stresses straight ahead of the crack) associated with these two asymptotic near-tip solutions can only take in general one of two discrete values. These mixity values normally correspond to either a tensile-like mode with dominating tensile stresses in the line ahead of the crack, or a shear-like mode with the opposite behavior. A direct implication of this result is that the effect of the applied remote boundary conditions can only be sensed at the tip of the crack through the amplitude of the (otherwise determinate) asymptotic fields. This result in turn suggests a ductile mechanism for explaining the dependence of the toughness of a nominally brittle interface on the phase angle of the applied loading during crack growth. The asymptotic predictions are compared with numerical full-field results for the problem of crack growth along a brittle-ductile interface, under small-scale yielding and mixed-mode conditions. Novel physical and mathematical arguments arc provided to help elucidate the surprising results.