A formulation of continuum mechanics as a dimensional reduction of a finite-dimensional dynamical system

Abstract

In the paper a generalized formulation of the continuum mechanics is suggested. The generalization consists in the assumption that the energy balance equation is not satisfied for all subbodies of a body but only for their chosen family. This formulation leads to fields in the continuum which create a finite-dimensional space. With the help of the chosen family of subbodies, a volume of averaging related to the continuum model is defined. This volume is connected with a more elementary dynamical system which takes part in determination of the form of constitutive equations. In general, the mechanical model of the continuum is seen as a dimensional reduction of the more elementary dynamical system related to another continuum or to a discrete set of material points.