Variational Principles for Non-Material Systems Within an Arbitrary Lagrangian Eulerian Description of Motion

Abstract

Dynamics of axially moving continua, such as beams, cables and strings, can be modeled by use of an Arbitrary La-grangian Eulerian (ALE) approach. Within a Finite Element framework, an ALE element is indeed a non-material system, i.e. a mass flow occurs at its boundaries. This article presents the dynamic description of such systems and highlights the peculiarities that arise when applying standard mechanical principles to non-material systems. Starting from D’Alembert’s principle, Hamilton’s principle and Lagrange’s equations for a non-material system are derived and the significance of the additional transport terms discussed. Subsequently, the numerical example of a length-changing beam is illustrated. Energetic considerations show the complex dynamic behavior non-material systems might exhibit.