Crack tip singular fields in ductile crystals with taylor power-law hardening

an asymptotic singular solution of the HRR type is presented for mode I tensile cracks in ductile single 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 a crack on the (010) plane with its tip along the [10 1 ] direction, and for a crack on the (101) plane with its tip along the same [10 1 ] direction in a fee crystal. The yield surfaces for both of these orientations are identical and thus, for the “small strain” formulation, the same macroscopic solution applies to both. The near-tip region is divided into angular sectors which are maps of successive flat segments and vertices of the yield surface. While the solution here involves 14 different sectors referring to stress states corresponding to flat and vertex segments of the yield locus, pRice 's (Mech. Mater. 6,714, 1987) asymptotic solution for the elastic-ideally plastic crystals involved only 7 sectors which mapped into the vertex points of the yield surface. The perfectly plastic limit of the HRR fields here reduce to 7 stress states of pRice (1987). In this limit, the HRR displacement fields remain continuous resulting in a discontinuous yet bounded and nonzero strain field. In contrast, the elastic-ideally plastic solutions have discontinuous shear displacements across sector boundaries. Furthermore the contours of constant effective strain here have various peaks and troughs at sector boundaries and lean backward relative to the direction of crack growth. Conversely, in the recent finite element solutions for elastic-ideally plastic single crystals by Hawk (preliminary summary of results is included in Rice et al., Int. J. Fracture, in press. 1989), the plastic zones lean forward and the strain field is consistent with a Dirac singular form similar to Rice's (1987). Thus it is conjectured that, similar to the anti-plane shear case of pRice and pSaeedvafa (J. Mech. Phys. Solids 36, 189, 1988), the single crystal HRR fields are dominant only over part of the plastic region immediately adjacent to the crack tip, and that their domain of validity vanishes as the perfectly plastic limit is approached.