Atomic-level stress in an inhomogeneous system

The distribution of local stress in an atomic system can be calculated from its mechanical definition. While this leads, for a homogeneous system, to an expression equivalent to that derived from the virial theorem (thermodynamic definition), when local inhomogeneities are present the equivalence can be demonstrated only in the sense of a volume average which effectively eliminates the inhomogeneities. Atomistic simulation results are presented to show that for a relaxed crystal with a planar free surface, the normal stress on each atomic plane parallel to the surface, when calculated according to the mechanical definition, vanishes uniformly as expected, whereas if the virial expression were applied the resulting stress distribution shows unphysical oscillations in the surface region.