Crack tip singular fields in ductile crystals with taylor power-law hardening. I: Anti-plane shear

Asymptotic singular solutions of the HRR type are presented for anti-plane shear cracks in ductile crystals. These are assumed to undergo Taylor hardening with a power-law relation between stress and strain at sufficiently large strain. Results are given for several crack orientations in fcc and bcc crystals. The neartip region divides into angular sectors which are the maps of successive flat segments and vertices on the yield locus. Analysis is simplified by use of new general integrals of crack tip singular fields of the HRR type. It is conjectured that the single crystal HRR fields are dominant only over part of the plastic region immediately adjacent to the crack tip, even at small scale yielding, and that their domain of validity vanishes as the perfectly plastic limit is approached. This follows from the fact that while in the perfectly plastic limit the HRR stress states approach the correct discontinuous distributions of the complete elasticideally plastic solutions for crystals (Rice and Nikolic, J. Mech. Phys. Solids 33, 595 (1985)), the HRR displacement fields in that limit remain continuous. Instead, the complete elastic-ideally plastic solutions have discontinuous displacements along planar plastic regions emanating from the tip in otherwise elastically stressed material. The approach of the HRR stress fields to their discontinuous limiting distributions is illustrated in graphical plots of results. A case examined here of a fcc crystal with a crack along a slip plane is shown to lead to a discontinuous near-tip stress state even in the hardening regime. Through another limiting process, the asymptotic solution for the near-tip field for an isotropic material is also derived from the present single crystal framework.