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A method for solving three dimensional discrete dislocation plasticity boundary-value problems using a monopole representation of the dislocations is presented. At each time step, the displacement, strain and stress fields in a finite body are obtained by superposition of infinite body dislocation fields and an image field that enforces the boundary conditions. The three dimensional infinite body fields are obtained by representing dislocations as being comprised of points, termed monopoles, that carry dislocation line and Burgers vector information. The image fields are obtained from a three dimensional linear elastic finite element calculation. The implementation of the coupling of the monopole representation with the finite element method, including the interaction of curved dislocations with free surfaces, is presented in some detail because it differs significantly from an implementation with a line based dislocation representation. Numerical convergence and the modeling of dislocation loop nucleation for large scale computations are investigated. The monopole discrete dislocation plasticity framework is used to investigate the effect of size and initial dislocation density on the torsion of wires with diameters varying over three orders of magnitude. Depending on the initial dislocation source density and the wire diameter, three regimes of torsion–twist response are obtained: (i) for wires with a sufficiently small diameter, plastic deformation is nucleation controlled and is strongly size dependent; (ii) for wires with larger diameters dislocation plasticity is dislocation interaction controlled, with the emergence of geometrically necessary dislocations and dislocation pile-ups playing a key role, and is strongly size dependent; and (iii) for wires with sufficiently large diameters plastic deformation becomes less heterogeneous and the dependence on size is greatly diminished. 

A method for solving general boundary-value problems involving discrete dislocations is introduced. Plastic flow emerges from the motion of dislocations in an incremental fashion. At each increment, the displacement, strain and stress fields in the body are obtained by superposition of the infinite medium fields associated with individual dislocations and an image field that enforces boundary conditions. Dislocations are represented as monopoles and dislocation events are treated as a transportation map problem. Long-range interactions are accounted for through linear elasticity with a core regularization procedure. At the current state of development of the method, no ad hoc short-range interactions are included. An approximate loop nucleation model is used for large-scale computations. The image problem is solved using a finite element formulation with the following features: (i) a single Cholesky decomposition of the global stiffness matrix, (ii) a consistent enforcement of traction and displacement boundary conditions, and (iii) image force interpolation using an efficient BB-tree algorithm. To ensure accuracy, we explore stable time steps and employ monopole splitting techniques. Special attention is given to the interaction of curved dislocations with arbitrary domain boundaries and free surfaces. The capabilities of the framework are illustrated through a wire torsion problem.