Stress in ordered systems: Ginzburg-Landau type density field theory

TL;DR

This paper develops a Ginzburg-Landau based theoretical framework to derive stress tensors and elastic responses in ordered systems, applicable to defected crystals, and verifies predictions with numerical simulations.

Contribution

It introduces a new method for calculating stress and elastic properties in ordered systems within a Ginzburg-Landau density field theory, including defected crystals.

Findings

Derived explicit stress tensor expressions for various crystal symmetries.

Showed elastic constants relate to reciprocal lattice symmetries.

Verified stress-strain predictions through numerical deformation simulations.

Abstract

We present a theoretical method for deriving the stress tensor and elastic response of ordered systems within a Ginzburg-Landau type density field theory in the linear regime. This is based on spatially coarse graining the microscopic stress which is determined by the variation of a free energy with respect to mass displacements. We find simple expressions for the stress tensor for phase field crystal (PFC) models for different crystal symmetries in two and three dimensions. Using tetradic product sums of reciprocal lattice vectors, we calculate elastic constants and show that they are directly related to the symmetries of the reciprocal lattices. We also show that except for bcc lattices, there are regions of model parameters for which the elastic response is isotropic. The predicted elastic stress-strain curves are verified by numerical strain-controlled bulk and shear deformations. Since the method is independent of a reference state, it extends also to defected crystals. We exemplify this by considering an edge and screw dislocation in the simple cubic lattice.