Plane strain elastic-ideally plastic crack fields for mode I quasistatic growth at large-scale yielding—II. Global analytical solutions for finite geometries

Abstract

T his paper continues the effort begun in drugan and chen [ J. Mech. Phys. Solids 37, 1 (1989)] (Part I) to extend the theoretical foundations for understanding and quantitatively describing stable plane strain tensile crack growth to large-scale yielding and general yielding situations in ductile materials. We first confirm the veracity and physical appropriateness of Part I's infinite series (in distance, r, from the crack tip) global representation of the growing crack stress field in extensions of “centered fan” plastic regions in isotropic, incompressible elastic-ideally plastic Prandtl-Reuss-Mises material. This is accomplished both by showing that this series almost always possesses a physically significant radius of convergence, and by calculating higher-order (in r) material velocity and plastic dissipation corrections. We then propound a method for employing this stress field representation to construct approximate global analytical solutions for crack growth in finite geometries at general yielding. The method is illustrated for the specific, practically important geometry of a crack approaching a straight, perpendicular surface; the applied loads considered are all combinations of bending and tension on the uncracked ligament, and all levels of uniform normal stress on the perpendicular surface, that together produce symmetric near-tip crack opening and general yielding of the ligament. An important product of these solutions is approximate analytical relations between Part I's near-tip characterizing parameter m and the applied loading. These display interesting features: in situations where the applied bending dominates, m is shown (at least approximately) to be independent of the applied moment/tension ratio ; whereas when the applied tension dominates, m does depend on this ratio. (The manner in which m varies with applied loading and geometry has a significant effect on crack growth criteria, as will be shown in Part III.) Our global solutions also provide specific analytical estimates of the radius of validity of the leading-order (in r) asymptotic solutions for differing far-field conditions ; one particular, striking result is that this radius is exceedingly small for a purely tensile uncracked ligament at general yielding, but is enormously increased by application of a small bending moment. Comparisons between our approximate global analytical solutions and full-field numerical finite element solutions verify the analytical results and demonstrate their remarkable accuracy.