Application of 3-D weight functions—I. Formulations of crack interactions with transformation strains and dislocations

Abstract

I n this paper we formulate the three-dimensional elastic problem of a half plane crack interacting with zones of transformation strain and dislocations. The stress intensity factors induced at the crack tip are discussed. The analysis is based on R ice's ( Int. J. Solids Struct. 21, 781, 1985b) development, using three-dimensional “weight function” theory, of elastic crack tip interactions with internal stress sources and on B ueckner's ( Int. J. Solids Struct. 23, 57, 1987) solution for the complete set of weight functions for a half plane crack. The formulae for the shear mode weight functions by Bueckner are simplified significantly. Explicit formulae are given for the intensity factors induced by arbitrarily distributed 3-D transformation strains, which reduce to the solutions known in the literature for 2-D transformation strains. Results are also specified to calculate the intensity factor distribution due to rectangular and semicircular crack-tip dislocation loops, and compared to those previously estimated by A nderson and R ice ( J. Mech. Phys. Solids 35, 743, 1987). The three-dimensional results are novel and in part II of this series (G ao and R ice, J. Mech. Phys. Solids 37, 155, 1989) we use the formulations presented here to calculate the self energy of a 3-D shear dislocation loop emerging from the crack tip.