Finite segment method for dislocation mechanics

Abstract

Equilibrating dislocation arrays in a static stress field is a fundamental operation in analyzing the plastic deformation of metal crystals and polycrystals. The problem is compounded when the individual dislocation is allowed to adopt an equilibrated shape, rather than relying on two-dimensional analyses based on straight dislocation geometries. In the three-dimensional problem, the Peach-Koehler formula may be used to compute the force at any point on a dislocation of arbitrary shape, but the result depends on the shape via the dislocation self-stress. Equilibrating within this framework is necessarily an iterative process of approximating the dislocation shape, assessing the unbalanced forces, and reducing these forces by reshaping the dislocation line. The idea of approximating a curved dislocation by a series of finite straight segments is fundamental, and has been explored by researchers. The dislocation problem described above may be formulated analogously to standard finite element formulations for solid mechanics problems. In this approach, which the authors call the Finite Segment Method (FSM), an arbitrary dislocation is discretized into a series of straight line segments joined at nodes. For simplicity, they only consider a linear, isotropic infinite continuum here. The system potential energy is derived, and forces are obtained by differentiation. Equilibriummore » is found identifying a stationary energy state via numerical iteration. This approach is applied to example problems including the Orowan Bypass mechanism and dislocation generation and equilibrium at crack tips and misfitting particles.« less