The Virial Stress Is Not a Measure of Mechanical Stress
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The Virial Stress Is Not a Measure of Mechanical Stress
Min Zhou The George W. Woodruff School of Mechanical Engineering Georgia Institute of Technology, Atlanta, GA 30332-040, U.S.A. ABSTRACT The virial stress is the most commonly used definition of stress in discrete molecular systems. This stress includes two parts. The first part depends on the mass and velocity of atomic particles, reflecting an assertion that mass transfer causes mechanical stress to be applied on stationary spatial surfaces external to an atomic particle system. The second part depends on interatomic forces and atomic positions, providing a continuum measure for the internal mechanical interactions between particles. For the simple conditions of rigid translation and uniform tension, the virial stress yields apparently erroneous interpretations. It is shown that the virial stress violates balance of momentum and does not possess physical significance as a measure for mechanical interaction between material points. The lack of physical significance is both at the individual atom level in a time-resolved sense and at the system level in a statistical sense. As a stress-like quantity, the virial stress has the geometric interpretation of being a measure for momentum change in a fixed spatial region. It is demonstrated that the interatomic force term alone is a valid stress measure and can be identified with the Cauchy stress. The discussions here focus on an error in the theoretical derivation of the virial stress that led to the inclusion of the kinetic energy term and on the conceptual flaws in the argument commonly used for including it. The result here demonstrates that, for gas systems in macroscopic steady state, the isotropic statistical average of the internal stress tensor over time and space is not equal to the negative of the macroscopic pressure tensor for the system. The statistical average of the stress is equal to the negative of the pressure only at absolute zero when all molecular motions cease. INTRODUCTION The continuum stress interpretation of atomic force fields is important since it allows the intensity and nature of internal interactions in materials to be measured. One of the commonly used definitions of stress in molecular dynamical (MD) systems is the virial stress. This stress is based on the virial theorem of Clausius (1870) and includes two parts (cf. Tsai 1979 and Rowlinson & Widom 1982). Specifically, the average virial stress over a volume Ω around a particle i at position ri is
Π=
1æ 1 ö − mi u i ⊗ u i + å rij ⊗ fij ÷ , ç Ωè 2 j ( ≠i ) ø
W2.6.1 Downloaded from https://www.cambridge.org/core. Access paid by the UCSB Libraries, on 17 Dec 2017 at 07:00:41, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1557/PROC-731-W2.6
(1)
where, mi is the mass of i, u i is the displacement of i relative to a reference position, u i = dui / dt represents material time derivative of u i , rij = rj − ri , and ⊗ denotes the tensor product of two vectors. Her
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