Evolution of fractal structures in dislocation ensembles during plastic deformation

Based on the irreversible thermodynamics approach to dislocation plasticity of metals, a simple description of the dislocation density evolution and strain hardening was suggested. An analytical expression for the fractal dimension (FD) of a cellular (or tangled) dislocation structure evolving in the course of plastic deformation was obtained on the basis of the dislocation model proposed. This makes it possible to trace the variation of FD of the dislocation cell structure with strain by just measuring the macroscopic stress-strain curve. The FD behavior predicted in this way showed good agreement with the experimentally measured FD evolution at different stages of deformation of a Ni single crystal and a Cu polycrystal. One new result following from the present model is that the FD of the bulk dislocation structure in a deforming metal peaks at a certain strain close to the onset of necking. The significance of fractal analysis as an informative index to follow the spatial evolution of dislocation structures approaching the critical state is highlighted.