The origin of the distinction between microscopic formulas for stress and Cauchy stress

TL;DR

This paper clarifies the fundamental difference between microscopic formulas for stress and the true Cauchy stress, providing a new, consistent atomistic formula that aligns with classical mechanics and removes previous ambiguities.

Contribution

It demonstrates that Cauchy stress can be unambiguously derived from internal force density, resolving longstanding confusion and proposing a new physically meaningful atomistic stress formula.

Findings

Cauchy stress results from internal force density

New atomistic stress formula is physically meaningful and well-defined

The proposed formula satisfies conservation laws and aligns with classical stress concepts

Abstract

Stress is calculated routinely in atomistic simulations. The widely used microscopic stress formulas derived from classical or quantum mechanics, however, are distinct from the concept of Cauchy stress, i.e., the true mechanical tress. This work examines various atomistic stress formulations and their inconsistencies. Using standard mathematic theorems and the law of mechanics, we show that Cauchy stress results unambiguously from the definition of internal force density, thereby removing the long-standing confusion about the atomistic basis of the fundamental property of Cauchy stress, and leading to a new atomistic formula for stress that has clear physical meaning and well-defined values, satisfies conservation law, and is fully consistent with the concept of Cauchy stress.