leads to the relation
As
it then follows from (2.47) that the local form (2.44) for the total energy balance equation may also be expressed
The energy balance equation may, therefore, be written in the more compact form
where the total energy per unit volume χ is given by
which is the sum of the internal energy, potential energy and kinetic energy per unit volume, and where the total energy flux Jχ is given by
On using (2.51), relation (2.52) may also be written in the form
which identifies the mechanisms whereby energy can enter or leave the system, namely advection (the term χv), as heat (the term h) and as external work (the term −σ.v).
The energy balance equation (2.50) implies energy conservation as the energy within any region V always remains fixed during any nonequilibrium process provided that there is no energy flow across the surface S bounding the region V. This result is easily established by integrating (2.50) over the region V, and then making use of the divergence theorem.
2.9 Equations of State for Hydrostatic Stress States
The balance equations of mass, momentum and energy introduce a variety of physical variables, but they do not involve in any way the properties of materials. An equation of state must now be introduced, implying the existence of a variety of material properties, depending upon the complexity of the material. If elastic effects are to be included in models, then it is through an equation of state that they must be introduced. Before using strain as a thermodynamic state variable, it is useful first to introduce some key thermodynamic relationships that can be applied when the stress field is hydrostatic, i.e. shear stresses are absent, characterised by a pressure p, such that the stress tensor has the form σ=−pI where I is the second-order unit tensor.
2.9.1 Global Thermodynamic Relations
It is essential to understand fully the nature of thermodynamic principles that underpin modelling procedures. It is assumed now that the system studied is multi-component and uniform, and that the temperature and pressure are uniform as would be the case for many equilibrium states of the system. At equilibrium, the classical thermodynamic relations for a uniform multi-component mixture, where Mk,k=1,…,n denote the total masses of the various species, are given by
where U is the internal energy of the system and F and G are the corresponding Helmholtz and Gibbs energies defined by
The brackets {} in (2.54) denote a set of state variables associated with the various species defined for k=1,…,n. The state variables S, T, V and p are the entropy, thermodynamic (i.e. absolute) temperature, volume and pressure, respectively, and the parameters μk,k=1,…,n, are the mass-based chemical potentials. Each of the relations (2.54) defines, for media having uniform state variables and hydrostatic stress states (i.e. in the absence of shear stresses), an equation of state for three different but equivalent selections of the independent state variables. It follows from the three differential relations in (2.54) that
