A statistical analysis of strain hardening: The percolation limit and the Taylor equation

The two-parameter model for strain hardening has proven to be a powerful tool in explaining the main features of dislocation storage during plastic deformation. The well-established but empirical Taylor equation relates dislocation density to yield stress. The free path for dislocations relates strain to the increase in density and appears as an empirical constant. The goal of the present paper is to address the Taylor equation from a probabilistic framework; future work will explore the calculation of the free path length. The distribution of obstacles in the slip plane will be modelled as a plane Poisson process and the relationship between its Delaunay triangulation and dislocation storage will be established. The statistical properties of this triangulation allow determining the number of segments stored after dislocation bow-out as well as their length distribution. This naturally leads to the percolation limit for dislocation flow between impenetrable obstacles; together with a differential equation which describes the evolution of dislocation density as a function of stress, this provides a mathematical foundation for the Taylor equation in the case of forest hardening.