Classes of ephemeral continua

Abstract

The qualifier “ephemeral” was proposed for continuous models of bodies, such as gases, for which the generally tacit axiom of permanence of material elements fails to apply. Consequently, to their scrutiny, a Eulerian (local) approach is mandatory, such as one adopted, eg, in molecular dynamics. Within the scheme of ephemeral continua, we discuss here 3 essential subclasses of bodies: (1) those undergoing energy‐preserving processes (in this sense hyperelastic), (2) hypoelastic bodies inspired by a type proposed by Truesdell, and (3) a number of minor ones. We re‐examine the essential issues of the general format focusing on the proposal of appropriate concepts of strains and strainings.

Continua with partially constrained microstructure

“…Furthermore, the peculiarities of the suspensions of rigid rotating granules in a fluid matrix and of pseudo-Cosserat continua are here explicitly obtained, while we can already recover other different instances for solids with nano-pores and soil mechanics (see, e.g., [34,41,42]), as well as in the theory of ephemeral continua [12].…”

“…Different specifications of those measures imply different models, as standard continua or media with edge interactions (see [14,60]). Moreover, if we consider more general tensor-valued measures into the expression (12) of working (depending on the smoothness of velocity fields (ẋ, ν)), then we can introduce "stresses" which may not derive from boundary tractions (as when a physically significant connection cannot be defined on the manifold M [10]) or that may be non-local [11]: a general introduction to these very subtle questions can be found in [18,63].…”

A discrete-to-continuum model of protein complexes

“…We choose to be such that best fits the overall protein kinetic energy; in other words, with N the number of amino acids in the protein, is an argument minimizing the difference where, as usual, the vertical bars indicate modulus. This last choice for recalls one in the discrete-to-continuum description of sparse phases (Capriz 2008 ), when we homogenize at continuum scale the fluctuating motion of a cluster of disconnected grains (Capriz and Mariano 2014 ; Capriz and Giovine 2017 ; Capriz and Mariano 2018 ).…”

Homogenization of Complex Lattices for Metamaterials: Open Problems and Conjectures

“…The associated affine deformation gradient G, itself a function of x and t, is a solution to the equation Ġ = BG , with initial datum that can be selected equal to the second-rank identity tensor; the superposed dot indicates total time derivative. The resulting equations are such that the scheme is, in general, multi-field (compare [48] with [49,50] for different settings where these ideas apply). In the presence of internal constraints, the picture simplifies because the resulting balances collapse, reducing the equations (also depending on whether we consider fluctuations c := w * − v − By).…”

“…In this case, although the prototypical discrete structure is taken as a simple, statistically periodical (i.e., the distribution of masses is independent of the specific space window) discrete one, the resulting representation becomes always multi-field: the gross deformation describes the motion of the window center of mass, and a tensor field represents the around-mass-center motion of mass points into the window. Pertinent interactions emerge, as shown in the example of Figure 2, and they have to be appropriately balanced [47,49,50].…”