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An accurate description of the evolution of dislocation networks is an essential part of discrete and continuum dislocation dynamics models. These networks evolve by motion of the dislocation lines and by forming junctions between these lines via cross slip, annihilation and junction reactions. In this work, we introduce these dislocation reactions into continuum dislocation models using the theory of de Rham currents. We introduce dislocations on each slip system as potentially open lines whose boundaries are associated with junction points and, therefore, still create a network of collectively closed lines that satisfy the classical relations and for the dislocation density tensor and the plastic distortion . To ensure this, we leverage Frank’s second rule at the junction nodes and the concept of virtual dislocation segments. We introduce the junction point density as a new state variable that represents the distribution of junction points within the crystal containing the dislocation network. Adding this information requires knowledge of the global structure of the dislocation network, which we obtain from its representation as a graph. We derive transport relations for the dislocation line density on each slip system in the crystal, which now includes a term that corresponds to the motion of junction points. We also derive the transport relations for junction points, which include source terms that reflect the topology changes of the dislocation network due to junction formation. 

For the past century, dislocations have been understood to be the carriers of plastic deformation in crystalline solids. However, their collective behavior is still poorly understood. Progress in understanding the collective behavior of dislocations has primarily come in one of two modes: the simulation of systems of interacting discrete dislocations and the treatment of density measures of varying complexity that are considered as continuum fields. A summary of contemporary models of continuum dislocation dynamics is presented. These include, in order of complexity, the two-dimensional statistical theory of dislocations, the field dislocation mechanics treating the total Kröner–Nye tensor, vector density approaches that treat geometrically necessary dislocations on each slip system of a crystal, and high-order theories that examine the effect of dislocation curvature and distribution over orientation. Each of theories contain common themes, including statistical closure of the kinetic dislocation transport equations and treatment of dislocation reactions such as junction formation. An emphasis is placed on how these common themes rely on closure relations obtained by analysis of discrete dislocation dynamics experiments. The outlook of these various continuum theories of dislocation motion is then discussed. 

The equations of dislocation transport at finite crystal deformation were developed, with a special emphasis on a vector density representation of dislocations. A companion thermodynamic analysis yielded a generalized expression for the driving force of dislocations that depend on Mandel (Cauchy) stress in the reference (spatial) configurations and the contribution of the dislocation core energy to the free energy of the crystal. Our formulation relied on several dislocation density tensor measures linked to the incompatibility of the plastic distortion in the crystal. While previous works develop such tensors starting from the multiplicative decomposition of the deformation gradient, we developed the tensor measures of the dislocation density and the dislocation flux from the additive decomposition of the displacement gradient and the crystal velocity fields. The two-point dislocation density measures defined by the referential curl of the plastic distortion and the spatial curl of the inverse elastic distortion and the associate dislocation currents were found to be more useful in deriving the referential and spatial forms of the transport equations for the vector density of dislocations. A few test problems showing the effect of finite deformation on the static dislocation fields are presented, with a particular attention to lattice rotation. The framework developed provides the theoretical basis for investigating crystal plasticity and dislocation patterning at the mesoscale, and it bears the potential for realistic comparison with experiments upon numerical solution.