In this chapter we state various axioms which form the basis for a thermo-mechanical theory of continuum bodies. These axioms provide a set of balance laws which describe how the mass, momentum, energy and entropy of a body change in time under prescribed external influences. We first state these laws in global or integral form, then derive various corresponding local statements, primarily in the form of partial differential equations. The balance laws stated here apply to all bodies regardless of their constitution. In Chapters 6–9 these laws are specialized to various classes of bodies with specific material properties, via constitutive models.

The important ideas in this chapter are: (i) the balance laws of mass, momentum, energy and entropy for continuum bodies; (ii) the difference between the integral form of a law and its local Eulerian and Lagrangian forms; (iii) the axiom of material frame-indifference and its role in constitutive modeling; (iv) the idea of a material constraint and its implications for the stress field in a body; (v) the balance laws relevant to the isothermal modeling of continuum bodies.

Motivation

In order to motivate the contents of this chapter it is useful to recall some basic ideas from the mechanics of particle systems. To this end, we consider a system of N particles with masses mi and positions xi as illustrated in Figure 5.1. It will be helpful to think of these particles as the atoms making up a continuum body.