Analytical Mechanics of Elastic Media

Abstract

Using the term “analytical mechanics” we usually mean the specific kind of presentation of mechanics of the finite systems of particles; the concept of constraints and that of the generalized coordinates plays an important role in this presentation. In this paper we are to deal with mechanics of uncountable systems of particles constituting material continua. However, our way of presentation is different from that used in the classical approach to continuum mechanics. We start from the dynamics of an uncountable system of homogenously deformable elastic particles and we impose certain restrictions on the motion or on the internal forces. Such system has much more general structure then the well known classical elastic continuum being characterized not only by its material properties but also by the restrictions imposed on the motion and the system of internal forces. It can be observed that all known “technical” or “approximate” theories of elasticity (theories of elastic plates, shells, rods as well as different finite element approaches, formulations based on the Ritz or Galerkin methods, etc.) can be directly derived form the analytical mechanics of elastic media presented in this contribution. At the same time, mechanics of such “structured” media is also of the more general nature than the mechanics of constrained continuum [1,2], where only restrictions for the deformations are taken into account.