A Σ-Algebra AND a CONCEPT OF LIMIT FOR BODIES

Abstract

Recent developments in mechanics of continua (the search for optimal shapes of bodies, homogenization theory, the study of the trabecular structure of bones, the dynamics of immiscible mixtures, etc.) render some of the introductory axioms of continuum mechanics inadequate. Not only does one need to give meaning to the join and meet of two bodies, but also to extend the consequent algebra so as to encompass the result of a countable sequence of operations of join or meet; and one should also be able to define the limit of a sequence of bodies. To achieve this goal we propose here to define a body ab initio through the assignment of a probability measure dπ. We realize that π leaves, generally, too much of the texture of the body unspecified; to make up for this deficiency, we suggest the use of appropriate texture measures, reminescent of Tartar's H-measures. 9 *

Differential Geometry and Continuum Mechanics

“…Alternatively, we may consider a sequence of bodies each member of which has a successively increasing number of regularly spaced voids whose limit, intended to represent the real body as in the search for an optimal shape, requires detailed investigation (c.p., [CP81,CM00]). Difficulties occur when endeavouring to model interaction between adjacent defects (possibly including extreme effects of collisions or coagulation of cavities).…”

“…The notion of flux seems to merge with that of volume density, and a boundary condition may become indistinguishable from equilibrated internal action. These important questions, considered in [CM00], are not exclusively theoretical, but, on the contrary, possess definite practical significance related, for example, to optimal shapes, granular materials, and suspensions.…”

Fluxes Across Parts of Fractal Boundaries

“…Vol. 74 (2006) Fluxes Across Parts of Fractal Boundaries 3 The reader is referred to [3,Section 6] and [2] for the discussion of nonstandard bodies in continuum mechanics.…”