How to compute the atomic stress objectively?

TL;DR

This paper introduces a Lagrangian atomic stress definition that objectively computes Cauchy stress in atomistic simulations, avoiding velocity-related controversies inherent in traditional virial stress methods.

Contribution

It proposes a fundamental Lagrangian atomic stress consistent with classical continuum mechanics, and clarifies the limitations of the classical virial stress.

Findings

Lagrangian atomic stress aligns with classical Cauchy stress.

Classical virial stress is actually Eulerian virial stress, valid only under certain conditions.

Lagrangian virial stress excludes velocities to avoid nonobjectivity.

Abstract

Atomistic simulation has been a powerful study tool in mechanics research, but how to objectively compute the atomic stress equivalent to Cauchy stress is still controversial, especially on the velocity-related part in the virial stress definition. In this paper, by strictly following the classical definition of the Cauchy stress for continuum medium, the fundamental Lagrangian atomic stress is proposed and can be used to obtain the correct Cauchy stress under any circumstances. Furthermore, the Lagrangian virial stress is proposed, which is still in virial form but does not include velocities to avoid controversial velocity treatments. It is also found that the widely used classical virial stress is actually the Eulerian virial stress, which includes the velocities of atoms, and is valid only when the impulse-momentum theorem is applicable to estimate the internal forces. However this requirement for the Eulerian atomic stress can not always be met in practical cases, such as the material volume element in rotation and the examples presented in this paper, but the proposed Lagrangian atomic stress can avoid these velocity-related nonobjectivities.