Modelling crack tip plastic zones and brittle-ductile transitions

Plasticity at crack tips may be modelled as self-organising arrays of dislocations emitted from a source near the crack tip. For materials where dislocation motion is slow and any friction stress is low, the modelled arrays are far from equilibrium. If dislocation motion is fast, and there is a substantial dislocation pinning stress, τf, the arrays are quasi-static and the model predicts behaviour very similar to those of earlier static models. A power-law relation connects the shielded crack tip stress intensity, k, to the pinning stress, τf, the length of the dislocation array, d and the size of the ‘dislocation free zone’ near the crack tip. Using either of two fracture criteria, (a) k = KIc, or (b) the stress at some point ahead of the crack tip exceeds a local fracture stress, the variation of stress intensity at fracture KF with temperature is predicted to be controlled by the temperature variation of the yield stress: KF ∝ σy−0.57.