Continuous distribution of moving dislocations

Abstract

A deformation field due to a moving single dislocation is expressed by line integrals along the dislocation line. Proper definitions for a continuous distribution of moving dislocations and its velocities enable us to extend the expression for the deformation field due to a single dislocation, to the deformation field due to a continuous distribution of dislocations. The fundamental relationship between plastic strains and dislocation density tensors, and the relationship between plastic strain rates and velocities of a continuous distribution of dislocations are determined from a formulation similar to the law of conservation of matter. Also it is found that a continuous distribution of dislocations must satisfy a certain boundary condition on the free boundary of the material. The relationship between the total strain, and the elastic and plastic strains, is discussed by referring to their definitions in the mathematical theory of plasticity.