The J-integral and geometrically necessary dislocations in nonuniform plastic deformation

A two-dimensional discrete dislocation simulation method is developed to simulate the configuration of geometrically necessary dislocations in a beam subject to combined tension and bending. We confirm that the density of geometrically necessary dislocations is very small under pure tension, and under bending it is 80% higher than that estimated by the Nye–Ashby relation. Our calculations also show that the statistical average of the J-integral is zero under pure tension and its magnitude increases monotonically with the plastic curvature of deformation in combined tension and bending. This may have interesting implications on the strain gradient plasticity theories aiming to model geometrically necessary dislocations.