Plane strain elastic-ideally plastic crack fields for mode I quasistatic growth at large-scale yielding—I. A new family of analytical solutions

Abstract

I n an effort to extend the theoretical foundations for understanding and quantitatively describing stable plane strain tensile crack growth in ductile materials to large-scale and general yielding situations, we derive a new family of solutions for the near-tip stress and deformation fields of such growing cracks, within the most analytically tractable model of isotropic, incompressible, elastic-ideally plastic PrandtlReuss-Mises material. One limiting member of this new family is the previously-derived “modified Prandtl field” solution that is believed to apply under small-scale yielding conditions. We provide evidence that the present solution family, which involves a parameter that is unspecified by the near-tip analysis, may be able to describe growing crack fields for the entire range of yielding extent from small-scale through general yielding conditions in potentially arbitrary Mode I plane strain geometries. Except for the “modified Prandtl field” limiting case, all members of the new solution family exhibit: (i) singular straining in the “constant stress” plastic sector ahead of the growing crack tip as well as in the “centered fan” plastic sectors that lie above and below the growing tip; and (ii) completely continuous near-tip stress and velocity fields. While the solution family derived is the result of a leading-order (in distance, r, from the moving crack tip) asymptotic analysis of the continuum mechanical field equations, we extend this solution explicitly to large r within the region of principal plastic deformation, thereby elucidating the radial dependence of the stress field (not specified by a leading-order analysis in nonhardening materials), and providing a framework for assessing the influence of far-field boundary conditions (specimen geometry and loading) on near-tip fields. Also, the new solution family is shown to provide a natural, fundamentalsbased generalization, possibly to arbitrary yielding extent and plastic constraint level, of an existing successful well-contained-yielding ductile crack growth criterion.