Preamble

Analternative strategy for relating continuum field values to their microscopic origins is outlined. Spatial averaging of additive molecular quantities is effected in terms of ‘cells’. In contrast to weighting function methodology, linear momentum balance for macroscopic regions is established before deducing the local form. The existence of a traction field on the boundary of such regions is derived via the assumption of short-range molecular interactions. The corresponding interaction stress tensor is obtained in the standard manner of continuum mechanics. Unlike balances obtained in terms of weighting functions, which hold for any given pair (∈, Δ) of length–time scales, fields obtained as cellular averages exist only if their values are somewhat insensitive to changes in ∈, Δ, and cell shape.

Cellular Averaging

Recall Subsection 4.4.1 in which selection of an appropriate weighting function [namely a mollified version of relations (4.4.4)] delivered spatial cellular averages. The analyses of Sections 4.2, 5.2 through 5.6, 6.2 and 6.3, 7.2 through 7.5, 8.4 through 8.6, 8.9 and 9.2 through 9.7 apply to such a choice. Thus, in particular, the standard relations [which express mass conservation (4.2.16), momentum balance (2.7.30) with T = Tw, where Tw is given by (5.5.20), and energy balance (6.2.75)] hold with field values defined in terms of cellular molecular averages. The associated (i.e., cell-based) notion of material point, and motion thereof, derives from the corresponding velocity field vw defined in Section 5.2.