Modelling crack tip plastic zones and brittle-ductile transitions

Plasticity at crack tips has been modelled as self-organising arrays of dislocations emitted from a source near the crack tip. The model takes into account dislocation shielding, but not blunting. For materials where dislocation motion is slow, and the friction stress is high, the modelled arrays are far from equilibrium. The brittle-ductile transition is then controlled by dislocation velocity. If dislocation motion is fast, and there is a substantial dislocation pinning stress, tau(f), 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, tau(f), the length of the dislocation array, d, and the size of the "dislocation free zone" near the crack tip. Close to the crack tip, the stress sigma(yy) proportional to k.(x(-1/3)). Using either of two fracture criteria, (a) k = K-1c, or (b) the stress at some point ahead of the crack tip exceeds a local fracture stress, the variation of stress intensity at fracture K-F with temperature is predicted to be controlled by the temperature variation of the yield stress: K-F proportional to sigma(y)(-0.57)