A method for solving small-strain plasticity problems was presented in which plastic flow was represented by the collective motion of a large number of discrete dislocations. The dislocations were modelled as line defects in a linear elastic medium. At each instant, superposition was used to represent the solution in terms of the infinite-medium solution for the discrete dislocations, and a complementary solution that enforced the boundary conditions on the finite body. The complementary solution was non-singular, and was obtained from a finite-element solution of a linear elastic boundary value problem. The lattice resistance to dislocation motion, dislocation nucleation and annihilation were incorporated into the formulation via a set of constitutive rules. Obstacles which could lead to dislocation pile-ups were also accounted for. The deformation history was calculated in a linear pile-up incremental manner. Plane-strain boundary value problems were solved for a solid having edge dislocations on parallel slip planes. Monophase and composite materials which were subjected to simple shear parallel to the slip plane were analyzed. A peak was typically found in the shear stress versus shear strain curve, after which the stress fell to a plateau at which the material deformed steadily. The plateau was associated with the localization of dislocation activity into more or less isolated systems. The results for composite materials were compared with solutions for a phenomenological continuum slip characterization of plastic flow.

Discrete Dislocation Plasticity: a Simple Planar Model. E.Van der Giessen, A.Needleman: Modelling and Simulation in Materials Science and Engineering, 1995, 3[5], 689-735