Theoretical Interpretation of the Virial Stress

Abstract

The virial stress for atomic systems includes two parts. The first part depends on the mass and velocity (or, in some versions, the fluctuation part of the velocity) of atomic particles. The second part depends on interatomic forces and atomic positions. Historic derivations of the virial stress include generalization from the virial theorem of Clausius (1870) for gas pressure and solution of the spatial equation of balance of momentum. The virial stress is a stress-like measure for momentum change in space. This paper shows that, contrary to the generally accepted view, the virial stress is not a measure for mechanical force between material points and cannot be regarded as a measure for mechanical stress in any sense. The lack of physical significance is both at the individual atom level in a timeresolved sense and at the system level in a statistical sense. It is demonstrated that the interatomic force term alone is a valid stress measure and can be identified with the Cauchy stress. The proof in this paper consists of two parts. First, for the simple conditions of rigid translation, uniform tension, and tension with thermal oscillations, the virial stress yields clearly erroneous interpretations of stress. Second, the conceptual flaw in the generalization from the virial theorem for gas pressure to stress and the confusion over spatial and material equations of balance of momentum in theoretical derivations of the virial stress that led to its erroneous acceptance as the Cauchy stress are pointed out. Interpretation of the virial stress as a measure for mechanical force violates balance of momentum and is inconsistent with the basic definition of stress. The versions of the virial stress formula that involve total particle velocity and the thermal fluctuation part of the velocity are demonstrated to be measures of spatial momentum flow relative to, respectively, a fixed reference frame and a moving frame with a velocity equal to the part of particle velocity not included in the virial formula.