The formation of dislocation loops at crack tips in three dimensions

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Abstract
The formation of finite dislocation loops in three dimensions (3D) at a crack tip in the crack plane under mode II (or mode III) loading is considered. As suggested by Schoeck (1990) the formation of a dislocation in 2D is treated in the Peierls model where the incipient dislocation is described by a continuous distribution of parallel infinitesimal dislocations. The resulting atomic displacement ahead of the crack tip is determined by the interatomic potential and is of the general type arctg(x/w) where we treat the dislocation width w as adjustable parameter. The finite incipient loop in 3D is modelled by a distribution of infinitesimal dislocation loops (Kroupa 1962) but with varying Burgers vectors. The saddle-point configuration for the formation of rectangular incipient loops is obtained numerically as function of the stress intensity K by studying the energy contours in phase-space. The free enthalpy of activation to emission increases continuously when the stress intensity K e drops below K 0 e the value for emission in 2D, but for energetic reasons cannot drop essentially below 2K 0 e/3. In all metals considered, including the b.c.c. transition metals, it is found that in mode II (or mode III) dislocation emission could occur before ‘shear-cleavage’. The numerical values obtained for Si are in excellent agreement with the experimental observations.