Application of 3-D weight functions—II. The stress field and energy of a shear dislocation loop at a crack tip

The General weight function expressions given in Gao (J. Mech. Phys. Solids 37, 133, 1989), referred to here as part I, for combined-mode crack-dislocation interaction problems in the three-dimensional regime are applied to solve for the stress field and energy of a shear dislocation loop emerging from the tip of a half-plane crack. The results are compared to the previously proposed approximate estimates for shear loops by Anderson and Rice (J. Mech. Phys. Solids 35, 743, 1987), who solved exactly for prismatic opening dislocation loops that are co-planar with the crack and also for the analogous 2-D cases of general crack tip-parallel line dislocations. The energy results are presented in terms of a correction factor m, following Anderson and Rice, to the usual estimate of energy for an emergent crack tip loop as half the energy of a full loop (identified as the emergent loop and its image relative to the crack front) in an uncracked solid. For a full circular shear loop the energy is U = [(2 − ν)μb 2 r/4(1-ν)] In (8r/e2 r 0), where r 0 denotes the core cut-off parameter and μ, ν are the shear modulus and Poisson ratio. Thus for a semicircular loop emerging from the crack tip, the energy is expressed as U = [(2 − ν)μb 2 r/8(1-ν] In (8mr/e2r0 ), where the constant m depends on the orientation angle Ψ of the Burgers vector relative to a line normal to the crack tip and the inclination angle φ of the dislocated plane relative to the crack plane. The m factors are calculated at selected angles φ for rectangular and semicircular loops. This involves multiple numerical integrations based on the weight functions of part I, first to obtain the stress field and then to integrate it over the dislocated area to get the energy, and requires a large amount of computing CPU time. An approximate formula for m is proposed for general inclined dislocation loops, based on known 2-D results for m factors for arbitrary angles φ calculated by Anderson and Rice (1987) and the 3-D m(φ = 0) results given here for shear dislocation loops in the crack plane. It compares well to the exact results.