The Rayleigh-Ritz Technique and the Lagrange Equations in Continuum Mechanics: Formulations for Material and Non-Material Volumes

Abstract

In the present Lecture, we use the Rayleigh-Ritz technique in connection with the Lagrange equations in order to approximate the partial differential equations of continuum mechanics by means of a system of ordinary differential equations in time. We start from the local form of the equation of balance of momentum, from which we proceed to an extended Hamilton principle, eventually ending up with the Lagrange equations. We lay special emphasis upon the functional dependencies to be considered in the Rayleigh-Ritz technique with respect to the spatial and the material descriptions of continuum mechanics. In analogy to the notion of a material-time derivative, we define material variations and material partial derivatives of the quantities and entities that enter the Lagrange equations, and we relate these derivatives to local partial derivatives, the latter being particularly feasible with respect to the spatial formulation of continuum mechanics. Having formulated the Lagrange equations for the case of a material volume, we present a re-formulation for problems that are posed with respect to non-material volumes. As an example for such a problem, we shortly treat the coiling of a strip. The presented re-formulation of the Lagrange equations for non-material volumes represents an alternative derivation of recent results by Irschik and Holl (2002).