A novel continuum dislocation density field-based crystal plasticity theory

In this work, a novel dislocation density field-based crystal plasticity formulation, that incorporates up-scaled continuum dislocation density fields to represent all possible characters of the dislocation density, is presented. The continuum dislocation field theory, formulated assuming large strain kinematics, is based on an all-dislocation concept, whereby individual dislocation density types are described as vector fields. The evolutionary behaviour of the dislocation density fields is defined in terms of a glide component derived from the classical general conservation law for dislocation density fields proposed originally by Kroner and Acharya, a rotation one given by the curl of the dislocation velocity vector as proposed originally by Ngan and co-workers, and a statistically stored dislocation source/sink component. The proposed formulation has been numerically implemented within the finite element method using a modified up-wind stabilisation approach, which results on eight extra independent nodal degrees of freedom per slip system, associated with the upscaled dislocation densities of pure edge and screw character. Furthermore, the required boundary conditions are simple: initial total dislocation density and dislocation flux or velocity, without the need for higher order boundary conditions. Full details of the numerical implementation of the crystal plasticity theory are given. The theory is then used to investigate classical boundary value problems involving 1D and 2D dislocation pile-ups, and the development of boundary layers in a deforming single crystal strip under constrained conditions. The main general understanding that emerges from the case studies is that, as either the dislocation pile-ups or the boundary layers develop with deformation, geometrically necessary dislocations are generated and lead to an overall behaviour that is size -dependent. Furthermore, it is shown that, in the absence of sources or sinks, which are statistically stored in nature, that total number of dislocations in the pileup problems is preserved. The numerical predictions of the 1D pile -up and strip simple shear problems are in good agreement with published solutions.