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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. 

Abstract The influence of grain size distribution on ductile intergranular crack growth resistance is investigated using full-field microstructure-based finite element calculations and a simpler model based on discrete unit events and graph search. The finite element calculations are carried out for a plane strain slice with planar grains subjected to mode I small-scale yielding conditions. The finite element formulation accounts for finite deformations, and the constitutive relation models the loss of stress carrying capacity due to progressive void nucleation, growth, and coalescence. The discrete unit events are characterized by a set of finite element calculations for crack growth at a single-grain boundary junction. A directed graph of the connectivity of grain boundary junctions and the distances between them is used to create a directed graph in J-resistance space. For a specified grain boundary distribution, this enables crack growth resistance curves to be calculated for all possible crack paths. Crack growth resistance curves are calculated based on various path choice criteria and compared with the results of full-field finite element calculations of the initial boundary value problem. The effect of unimodal and bimodal grain size distributions on intergranular crack growth is considered. It is found that a significant increase in crack growth resistance is obtained if the difference in grain sizes in the bimodal grain size distribution is sufficiently large.