Understanding stress at the atomic scale through the principle of equivalence of virtual work

Although the virial stress, which was originally proposed by Clausius (1870) in the context of pressure in gases, is widely adopted in molecular dynamics simulations of fluids and solids, a rigorous and consistent derivation of stress based on atomic coordinates and momenta has proven to be elusive. As a result, inconsistencies have arisen in the calculation and interpretation of stress in molecular systems. In addition, systematic studies to verify the validity of the virial stress through numerical simulations are lacking, particularly for non-uniform fields and macroscopic dynamics. This dissertation addresses those issues. A rigorous derivation of stress in molecular dynamics (discrete systems) is presented based upon the principle of equivalence of virtual work, which equates the virtual work of applied tractions (and body forces) in the continuum setting and the virtual work of external forces in the atomistic setting, suitably averaged over time scales associated with thermal vibrations. Inertial effects are accounted for through the association of center-of-mass coordinates with continuum material points. The analysis applies both for the entire system and for sub-domains that include a sufficient number of atoms to approximate a continuum material point. The result is an expression for stress defined by a time average of terms that depend on atomic coordinates and accelerations as well as on the potential functions describing atomic interaction. Furthermore, if the interval for time averaging is sufficiently long, the stress can be expressed in terms of atomic coordinates and velocities, thereby eliminating explicit dependence on particle accelerations. Numerical simulations are presented for Lennard-Jones solids and various loading cases: (i) transients and thermal expansion associated with temperature variations and gradients for both free and constrained boundaries, (ii) non-uniform stress fields around inclusions, and (iii) cantilever beam vibrations which involve both temporal and spatial gradients. Consistent with the theoretical derivation, the atomic-scale expression involving moderate temporal and spatial averages is shown to agree precisely with continuum notions of stress in each simulation