A gradient theory of grain boundary that accounts for surface gradient of the Burgers tensor

Abstract

This paper presents an extension of Gurtin’s theory of grain boundaries, which accounts for grain misorientation and grain-boundary orientation, by introducing the surface gradient of the grain boundary Burgers tensor. This new framework elucidates the behavior of grain boundaries within polycrystalline materials at the micron scale, enabling the characterization of jump discontinuities in the plastic distortion tensor and slip fields. These jumps, quantified by the grain boundary Burgers tensor, are expressed through orientation tensors tailored for each slip system within intersecting grains. By leveraging the principle of virtual power, we formulate both macroscopic and microscopic force balances, seamlessly augmenting them with pertinent constitutive relations compatible with the free-energy imbalance. It is shown that dislocation densities at grain boundaries become tractable through the surface gradient of the grain boundary Burgers tensor, particularly when slip plane normals are parallel to the grain boundary normal. Moreover, the incorporation of a surface gradient in our framework permits the presentation of the flow rule as a system of second-order partial differential equations in slips, navigating through appropriate boundary conditions.