We discuss methods to describe the evolution of dislocation systems in terms of a limited number of continuous field variables while correctly representing the kinematics of systems of flexible and connected lines. We show that a satisfactory continuum representation may be obtained in terms of only four variables. We discuss the consequences of different approximations needed to formulate a closed set of equations for these variables and propose a benchmark problem to assess the performance of the resulting models. We demonstrate that best results are obtained by using the maximum entropy formalism to arrive at an optimal estimate for the dislocation orientation distribution based on its lowest-order angular moments.

Continuum dislocation dynamics (CDD) aims at representing the evolution of systems of curved and connected dislocation lines in terms of density-like field variables. Here we discuss how the processes of dislocation multiplication and annihilation can be described within such a framework. We show that both processes are associated with changes in the volume density of dislocation loops: dislocation annihilation needs to be envisaged in terms of the merging of dislocation loops, while conversely dislocation multiplication is associated with the generation of new loops. Both findings point towards the importance of including the volume density of loops (or 'curvature density') as an additional field variable into continuum models of dislocation density evolution. We explicitly show how this density is affected by loop mergers and loop generation. The equations which result for the lowest order CDD theory allow us, after spatial averaging and under the assumption of unidirectional deformation, to recover the classical theory of Kocks and Mecking for the early stages of work hardening.

Metal plasticity is governed by the motion of dislocations, and predicting the interactions and resulting collective motion of dislocations is a major task in understanding and modeling plastically deforming materials. This task has, despite all the efforts and advances of the last few decades, not yet been fully accomplished. The reason for this is that discrete models which describe the dislocation system with high accuracy are only computationally feasible for small systems, small strains, and high strain rates. Classical continuum models do not suffer from these restrictions but lack sufficiently detailed information about dislocation microstructure. In this paper we present the steps that are needed for averaging systems of discrete dislocations toward a continuous and hence more efficient representation. Our main emphasis lies on investigating the effects of averaging on the description of stress fields and dislocation interactions. We show how the evolution of continuous dislocation fields can then be appropriately described by a dislocation density-based model and validate our results by comparison with discrete dislocation dynamic simulations.

Plasticity is governed by the evolution of, in general anisotropic, systems of dislocations. We seek to faithfully represent this evolution in terms of density-like variables which average over the discrete dislocation microstructure. Starting from T. Hochrainer's continuum theory of dislocations (CDD) [Hochrainer 2015], we introduce a methodology based on the 'Maximum Information Entropy Principle' (MIEP) for deriving closed-form evolution equations for dislocation density measures of different order. These equations provide an optimum representation of the kinematic properties of systems of curved and connected dislocation lines with the information contained in a given set of density measures. The performance of the derived equations is benchmarked against other models proposed in the literature, using discrete dislocation dynamics simulations as a reference. As a benchmark problem we study dislocations moving in a highly heterogeneous, persistent slip-band-like geometry. We demonstrate that excellent agreement with discrete simulations can be obtained in terms of a very small number of averaged dislocation fields containing information about the edge and screw components of the total and excess (geometrically necessary) dislocation densities. From these the full dislocation orientation distribution which emerges as dislocations move through a channel-wall structure can be faithfully reconstructed.developed by Kröner (1958) and Nye (1953). This theory is formulated in a geometrically rigorous manner and provides generic relationships between the dislocation microstructure, the plastic distortion and the associated internal stress fields. The fundamental object of the theory is the dislocation density tensor α which is defined as the curl of the plastic distortion, α = −curlβ pl . This theory was extended by Mura (1963) who formulated a kinematic equation of evolution for the dislocation density tensor,where v is the dislocation velocity vector. In this form the theory provides a full description of dislocation microstructure evolution and of plastic deformation for situations where all dislocations are geometrically necessary dislocations (GND), i.e., where they can be envisaged as contour lines of the plastic shear strain on the respective slip systems (e.g. Sedláček et al., 2003;Xiang, 2009;Zhu and Xiang, 2015). In the general case where dislocations of multiple orientations and slip systems are present, such a field theory, in order to fully capture the evolution of the dislocation microstructure, requires a spatial resolution that is well below the spacing of the individual dislocation lines in order to make the dislocation velocity field v uniquely defined. (See e.g. Xia and El-Azab (2015) and Zhang et al. (2015) for such implementations). A similar spatial resolution is also required for phase field approaches to dislocation microstructure evolution who directly simulate the evolution of the shear strain fields (Wang et al., 2001;Rodney et al., 2003). Such simulations, which evidently incur a high comput...

In the current paper we modify the evolution equations of the simplified continuum dislocation dynamics theory presented in [T. Hochrainer, S. Sandfeld, M. Zaiser, P. Gumbsch, Continuum dislocation dynamics: Towards a physical theory of crystal plasticity. J. Mech. Phys. Solids. (in print)] to account for the nature of the so-called curvature density as a conserved quantity. The derived evolution equations define a dislocation flux based crystal plasticity law, which we present in a fully three-dimensional form. Because the total curvature is a conserved quantity in the theory the time integration of the equations benefit from using conservative numerical schemes. We present a discontinuous Galerkin implementation for integrating the time evolution of the dislocation state and show that this allows simulating the evolution of a single dislocation loop as well as of a distributed loop density on different slip systems.