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

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. I. INTRDUCTIONEnergy, force, stress are basic concepts in the characterization of the state of condensed matter. Stress, in particular, is a key concept that links theory, simulation, and experiment. It has also been a subject of theoretical interest, as it can be used to establish correspondence between classical and quantum mechanics and between particle and continuum mechanics. Both the classical and quantum mechanical virial theorems show that the systemwide average stress in many-body systems is determined by the kinetic energy and the virial of the potential, [1][2][3][4][5][6][7][8] generally referred to as the kinetic and potential part of stress, respectively. In contrast, the atomistic formula for local stress remains a subject of debate. The critical issue is that no correspondence has been established between atomistic formulas for local stress and the fundamental concept of Cauchy stress. [8][9][10][11] Cauchy stress, also known as the true mechanical stress, is defined as the force that the material on one side of a surface element exerts on the material on the other side, divided by the area of the surface.12 It is a measure of the intensity of internal forces, has a clear physical origin, is the actual physical quantity measured in experiments, and is applicable on all scales. Atomistic stress formulas, on the other hand, were derived from classical or quantum mechanics as a function of the forces and positions of atoms, and is what have been used in ab initio calculations 13 as an intrinsic property of the quantummechanical ground state of matter, or in classical molecular dynamics or coarse-grained atomistic simulations to predict strength 14 , fracture toughness 15, 16 , hardness 17 , or to quantify the effect of local stress on ferroelectricity 18 , thermal conductivity 19,20 , phase transition 21, 22 , etc. There are numerous computational efforts that have attempted to understand the difference between various atomistic formulas for local stress and that between atomistic stress and Cauchy stress 8-11, 19, 23-25 . For example, FIG. 1 compares the stress at a dislocation core predicted by atomistic simulations using two most popular atomistic formulas with that by classical elasticity. While both atomistic s...

On the equivalence of the two foundational formulations for atomistic flux in inhomogeneous transport processes

“…that is also consistent with the intuition that the forces of interaction through the surface of a body are responsible for changing the body's linear momentum as portrayed in other works [16,52]. It is also worth noting that the result of equation (2.9) is consistent with the method of planes (MOP) when the surface in question is an infinite plane, when the surface is finite the local aspect of the relationship between flux and the line-plane intersection problem, that is distinct from MOP, becomes evident as seen in figure 1.…”

“…Closed-form expressions appeared in later developments, for example, in the paper by Miller [49], and in the statistical mechanics book by Kreuzer [50], with the difference between two δ functions being expressed as a line integral of the distribution's action, as: (2.4) where r kl = r k − r l . Recently, using the scaling property of Dirac delta listed by Paul Dirac in his Principles of Quantum Mechanics [48], or simply using the fundamental theorem of line integrals, equation (2.4) has been rewritten as a path integral with the path L kl being a line segment from r k to r l [15,16], as…”

Physical foundation and consistent formulation of atomic-level fluxes in transport processes

“…Comparisons of MOP and VA using MD simulations show a good agreement for local pressure, after sufficient time averaging, in a spatially homogeneous Lennard-Jones fluid 34,38 . By contrast, significant differences in the calculated results of local stress and heat flux between popular VA formulas and surface formulas are demonstrated for solids with defects or interfaces 36,37,39 .…”

“…The control volume formulation 35 is another effort that describes fluxes as a planar average by linking the time derivatives of the conserved quantity in a volume element to the fluxes across the bounding surface of the volume; this work demonstrates that at the limit of zero volume the formulation returns to the IK point-function formulation. Using the integral form of conservation laws, formulas for atomic-level local stress and heat flux have recently been derived as a surface average for three dimensional inhomogeneous systems in transient transport processes 36,37 .…”

Nanoscale self-healing mechanisms in shape memory ceramics

“…Stress calculation at atomic scale has been a subject of theoretical debate, basically because there is a lack of linkage between the atomistic stress formulation and the fundamental concept of Cauchy stress, which is the actual physical quantity measured in experiments 41 . In quantum and classical molecular mechanics, the local stresses are usually calculated based on virial theorem, which is the most commonly used definition of stress in discrete particle systems 42 .…”