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

Abstract

a b s t r a c tFollowing 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 pointparticles.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.

Micromorphic Balances and Source-flux Duality

“…With no attempt to resolve that problem, the present note is focused on a source-flux construct introduced in [2] 1 . This construct establishes a fairly direct connection between Newton's and Cauchy's laws and provides an expression for Cauchy stress suggested by the statistical mechanics of point-particles [4]. Furthermore, it appears to offer a useful mathematical foundation for the "peridynamics" of Stilling and coworkers [5].…”

On the Thermoelectricity of W. Thomson: Towards a Theory of Thermoelastic Conductors

“…It is plausible that the simple-material model employed here will always allow a certain latitude in such definitions (cf. [14]). …”

“…Based on his view of thermo-electricity as irreversible, [29] represents the linear theory by means of his phenomenological relations for forces and fluxes, such as those in (14), by…”

“…For the present purposes, we consider the case of an isotropic conductor with forces and fluxes given by (14), so that − ∇φ e = y e − cĵ e + N eSĵS + R ee j e + R eS j S , and − ∇θ = y S − cĵ S − N eSĵe + R eS j e + R SS j S , where…”

On the Thermoelectricity of W. Thomson: Towards a Theory of Thermoelastic Conductors

“…For the present purposes, we consider the case of an isotropic conductor with forces and fluxes given by (14), so that − ∇φ e = y e − cĵ e + N eSĵS + R ee j e + R eS j S , and …”