Chapter 9 - Towards a Variational Mechanics of Dissipative Continua?

Abstract

This chapter focuses on balancing the laws derived by use of Noether's theorem to study mechanics of continua. The main element of variational mechanics is given in the mechanics of continua of solid type. Inhomogeneous hyper-elasticity provides the standard of the formulation. Possible means to include relevant dissipative mechanisms, like heat conduction and thermodynamically irreversible inelasticity has been outlined. This exploits the notion of thermodynamics and shows how the somewhat strange thermo-elasticity ‘without dissipation’ follows thereafter. The variational formulation is shown to yield the classical thermo-elasticity of anelastic conductors in the appropriate reduction. A Euclidean four-dimensional space-time formulation of the canonical balance laws of energy and material momentum has also been discussed. Finally, an analogy between the obtained canonical balance laws and the Hamiltonian equations that gives the kinetic-wave theory of nonlinear dispersive waves in inhomogeneous materials is established. Furthermore, generalizations to the material behavior involving coupled fields, non-locality and evolving microstructure are mentioned.