On some variational principles in mechanics of continuous media: PMM vol. 37, n≗6, 1973, pp. 963–973

Abstract

Some problems of the analytical mechanics of continuous media are considered. The kinematc constraints, restricting the motion of elements of a continuous medium are separated into internal and external constraints; the internal surface stresses in a continuous medium are treated as reactive forces of the internal constraints. Since the work of these latter on possible displacements of elements of the continuous medium is not zero in the general case, then the internal constraints in a continuous medium should be referred to the category of nonideal constraints. The general equation of the dynamics of a continuous medium expressing the d'Alembert-Lagrange variational principle and including all the dynamic laws is examined. The work of the internal surface stresses in this equation can be given by using the first and second laws of thermodynamics, whereupon the general equation of the dynamics of continuous media can be represented in two other forms. Furthermore, an extension of the Gauss and Chetaev principles to continuous media is given.