Comparison of the Helmholtz, Gibbs, and Collective-modes methods to obtain nonaffine elastic constants

TL;DR

This paper compares three theoretical methods for calculating nonaffine elastic constants in lattices, verifying their equivalence through examples and simulations to ensure consistency in linear elastic regimes.

Contribution

It provides a detailed comparison of Helmholtz, Gibbs, and collective-modes methods for elastic constants, highlighting their equivalence in linear elastic and equilibrium conditions.

Findings

All three methods yield identical elastic constants in the linear elastic limit.

Explicit verification on linear chains confirms theoretical equivalence.

Numerical simulation of a non-centrosymmetric crystal supports the analytical results.

Abstract

We review and compare the Born-Huang and the Lemaitre-Maloney's theories that lead to analytical expressions for elastic constants, accounting for affine and nonaffine deformations in a lattice. The Born-Huang method is based on Helmholtz energy while the Lemaitre-Maloney's formalism focus on Gibbs force. Although starting from different perspectives, in the linear elastic limit, and in equilibrium, elastic material constants must be the same in all these methods. This is explicitly verified on examples of linear chains, and numerical simulation of a non-centrosymmetric crystal.