Asymptotic analysis of steady dynamic crack growth in an elastic-plastic material

Abstract

A ttention is focussed on the asymptotic stress and deformation fields near the edge of a mode I crack propagating steadily in an elastic-perfectly plastic incompressible material under plane strain conditions. Features of previously reported results on this problem are reviewed, with a view toward establishing discriminating characteristics. An asymptotic solution valid for all crack speeds is constructed. This solution has the properties that the angular variation of stress and particle velocity around the crack tip are continuous and the plastic strains are bounded at the crack tip. The construction depends on the boundedness of hydrostatic stress at the crack tip, and the conditions under which this stress measure is bounded are discussed in an Appendix. The possibility of an asymptotic field with logarithmically singular plastic strain at the crack edge is examined. While fields with this feature can be constructed by admitting discontinuities in stress and particle velocity, it is shown that the sequence of mechanical states experienced in the discontinuities is inconsistent with the principle of maximum plastic work. The dynamic asymptotic field does not reduce to the generally accepted quasi-static asymptotic field in the limit as the crack speed goes to zero. It has been established for the equivalent case of mode III elastic-plastic crack growth that the domain of validity of the dynamic asymptotic field vanishes as the crack speed vanishes, and the implication is that the mode I result has the same property.