On material velocities and non-locality in the thermo-mechanics of continua

Following is a discussion of several issues surrounding the continuum modeling of discrete particulate systems, particularly the uncertainty in the definition of continuum-level material particles and velocities. The work is motivated in part by various proposals for the introduction of supplemental velocity fields into the thermo-mechanics of single-component fluids. A review and modification are given of the relevant continuum field equations, based on the Eulerian rather than the conventional Lagrangian description, and a connection is made to the well-known statistical mechanics of Kirkwood and coworkers for point-particles.The Noetherian technique of Green and Naghdi is employed to derive the momentum balance directly from the particle-level energy balance, and it is shown that non-barycentric effects in particle motion engender stress asymmetry. It is conjectured that this and related effects are generally bound up with higher-gradient effects and the breakdown of the simple-material model for continuum thermo-mechanics. To illustrate this, a theory of non-local linear viscoelasticity is presented and compared with Brenner’s “bi-velocity hydrodynamics” and Müller’s “extended thermodynamics” for heat flux and stress. These models are shown to represent expansions in spatial wave vector, representing a Knudsen number, and temporal frequency, representing a Deborah number. Suggestions are made for further work to explore the associated effects of couple-stress and the consequences of departures of entropy flux from heat flux.