Mixed-mode near-tip fields for elastic perfectly plastic solids.

Abstract

The small-scale yielding mixed-mode near-tip fields of a stationary crack in an elastic perfectly plastic solid are examined. Full field finite element analyses, employing a small-strain version of the $J\sb2$ incremental plasticity theory, are carried out to elucidate the details of the limiting stress states near the crack tip. Then, the asymptotic structures of the near-tip fields are generalized, and the corresponding asymptotic solutions are constructed. The current near-tip asymptotic solutions are in excellent agreement with the finite element results in every respect for the whole range of mixed-mode loading. Under both plane-strain and plane-stress conditions, elastic sectors are introduced in constructing the asymptotic near-tip fields. In plane strain there exists one elastic sector bordering the upper crack face for near mode I mixed-mode loading, while in plane stress there exist two elastic sectors bordering both crack faces for pure mode I and near mode I mixed-mode loading, and one elastic sector bordering the lower crack face for near mode II mixed-mode loading. Within the elastic sectors the stresses are nonsingular. The details of the mixed-mode plastic zone sizes and shapes are given. Implications on mixed-mode fractures are discussed. In addition, some issues pertaining to the finite element procedure such as the implementation of the small-scale yielding assumption are addressed in the light of the computational results. A perfect-plasticity solution to the mixed-mode near-tip fields in plane stress is also presented. The mode I crack-tip field, derived from this analysis, differs from Hutchinson's solution by a constant stress sector ahead of the crack tip. The relevance of the near mode II solutions to some important features of the mixed-mode crack-tip fields from early dominant singularity analyses is discussed.