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The energy balance equation is derived from the first law of thermodynamics which requires that, for a fixed region V bounded by a closed surface S, the rate of change of the sum of the internal energy and kinetic energy is balanced by the sum of the heat flowing across the external surface S, and the rate of working of the external tractions acting on S and of the body force acting in V. There are two types of stored energy that must be considered. The first is the internal energy that accounts for the strain energy stored owing to elastic deformation and the energy of the thermal agitations of atoms in the solid. The second is the kinetic energy arising from the local average motion of the medium. In addition, the heat flow across the external boundary and the mechanical work done by the applied tractions must be considered. Body forces such as that caused by gravity also need to be taken into account. A final type of energy is the local heating that can arise, for example, from the flow of electric currents when electrodynamic effects are otherwise neglected. However, this local heating term is a very useful theoretical device for imposing precise isothermal conditions in a simulation. Owing to various energy dissipation processes, heat will be generated locally in the medium, leading to temperature variations and heat flow. Imagine that it is possible to add or remove heat at every point of the medium, and that this may be controlled to maintain a constant uniform temperature at all points. This approach to achieving isothermal conditions requires the use of a distribution of local heat sources or sinks. An alternative method is to impose isothermal conditions only on the external boundary, and to assume that the thermal conductivity is effectively infinite so that the heat generated within the medium by dissipation processes can flow immediately out of the system, thus maintaining a uniform temperature.

In the absence of electrodynamic effects, the global form of the energy balance equation is written as

(2.38)

where υ is the specific internal energy, h is the heat flux vector and r is a local rate of heat supply per unit mass. In (2.38), the symbol v² is used to denote the value of the scalar product v . v. On using (2.27) the energy balance equation (2.38) may be written as

(2.39)

On using the divergence theorem, it then follows that

(2.40)

As relation (2.40) must be satisfied for all regions V, this leads to the following local form for the energy balance equation

(2.41)

which must be satisfied at every point in the medium for all times t > 0. On using the continuity equation (2.30), the local energy balance equation reduces to the form

(2.42)

From the equation of motion (2.37) it follows, on taking the scalar product with the velocity vector v, that

(2.43)

where a superscript T denotes that the transpose of the tensor must be used. The symbol: is defined here such that for any second-order tensors a and b the double scalar product a:b≡aijbji. An alternative definition a:b≡aijbij is sometimes used in the literature. On using (2.43), together with the symmetry of the stress tensor, (2.42) reduces to the form

(2.44)

where d is the symmetric rate of deformation tensor defined by

(2.45)

Relation (2.44) is the well-known local form of the internal energy balance equation for a continuous medium.