Constrained ephemeral continua

Abstract

The theory of ephemeral continua was proposed to model bodies for which the basic tenet of permanence of material elements fails. The goal of the proposal was, principally, to lessen the impact of critical arguments against the imposition of the principle of material frame-indifference in continuum mechanics. Those arguments were based on the remark that, in any case, any mathematical model of reality is necessarily observer dependent; however, as it happens, they were urged on by noticing that some corollaries in the kinetic theory of gases appear to contradict requirements of frame-indifference. The proposed theory cures that wound via a definition of peculiar velocities which assures their exact observer independence. Besides, that theory o¤ers a wider base, allowing one to breed models for bodies where the processes are irremediably influenced by events at a lower scale, models based so far on a bottom up approach with partly uncertain steps up from the standard case. Here, we proceed via a top down approach, by introducing progressively stronger constraints. Largely, earlier results are confirmed, securer ground is given to some interpretations or alternatives offered. Briefly, the general requirements for any linear constraint to have physical content are stated and their immediate implications are derived. These requirements are applied to study certain special cases of the general constraint. Connections are provided between the consequent simplifications and results available by bottom-up approaches, so to speak—including, in particular, hypocontinua and Navier–Stokes-$\alpha ,\beta$ continua. To show that the value of our general approach transcends the building of an all-encompassing framework, we conclude by deriving the reduced balance laws consistent with the most general version of the constraint. These laws open an easy approach to obtaining potentially useful special cases.