Plastic Stress Intensity Factors in Steady Crack Growth

The asymptotic stress and deformation fields of a crack propagating steadily and quasi-statically into an elastic-plastic material, characterized by J₂-flow theory with linear strain-hardening, were first determined by Amazigo and Hutchinson (1977) for the cases of mode III and mode I (plane strain and plane stress). Their solutions were approximate in that they neglected the possibility of plastic reloading on the crack faces. This effect was taken into account by Ponte Castan˜eda (1987b), who also introduced a new formulation for the (eigenvalue) problem in terms of a system of first order O.D.E.’s in the angular variations of the stress and velocity components. The strength of the power-type singularity, serving as the eigenvalue, and the angular variations of the field were determined as functions of the hardening parameter. The above analysis, however, does not determine the amplitude factor of these near-tip asymptotic fields, or plastic stress intensity factor. In this work, a simple, approximate technique based on direct application of a variational statement of compatibility is developed under the assumption of small scale yielding. A trial function for the stress function of the problem, that makes use of the asymptotic information in the near-tip and far-field limits, is postulated. Such a trial function depends on arbitrary parameters that measure the intensity of the near-tip fields and other global properties of the solution. Application of the variational statement then yields optimal values for these parameters, and in particular determines the plastic stress intensity factor, thus completing the knowledge of the near-tip asymptotic fields. The results obtained by this novel method are compared to available finite element results.