Dislocation dynamics and work hardening of fractal dislocation cell structures

The dislocation dynamics during multiple slip deformation is formulated in terms of a simple stochastic model for the evolution of the densities of mobile and immobile dislocations. Randomness results in modified effective dislocation multiplication and reaction rates which account for the topology of the evolving microstructure. Depending on the intensity of local strain-rate fluctuations, two types of solution can be distinguished: (1) At low noise levels homogeneous dislocation structures develop which are described by a single characteristic length scale, i.e. the mean dislocation spacing. This is the case of b.c.c. metals deformed at low temperature. (2) Above a critical noise level self-similar dislocation cell patterns are found which are characterized by a lower cut-off length, i.e. the minimum dislocation spacing in the cell walls, and scale invariance beyond that cut-off. This case refers to rate-insensitive f.c.c. metals, where fractal dislocation structures have been identified recently [P. Hنhner, K. Bay, M. Zaiser, Phys. Rev. Lett. 81 (1998) 2470]. The model yields critical deformation conditions for fractal dislocation patterning and enables one to establish relations between the evolution of the fractal dimension of the cell structure, the strain-hardening behaviour, and the underlying dislocation dynamics. This is achieved without postulating a priori that the dislocation microstructure be heterogeneous.