Continuum Formulations of Conservation Laws - Continuum Mechanics of Anisotropic Materials - page 70

where P Q is the rate of energy supply. Equation ( 3.43 ) is a global statement of

energy conservation and we will need a point form of the principle in continuum
mechanics applications. In the point form representation all the variables will be
intensive in the conventional thermodynamic use of that word. In thermodynamics
an extensive variable is a variable that is additive over the system, e.g., volume or
mass, and an intensive variable is a variable that is not additive over the system,
e.g., pressure or temperature. To understand these definitions consider adding
together two identical masses occupying the same volume at the same temperature
and pressure. When two masses have been added together the resulting system has
double the volume and double the mass, but it still has the same temperature and
pressure. Extensive variables can be made into intensive variables by dividing them
by the density of the particle. Thus density or specific volume (the reciprocal of
density) is the intensive variable associated with the normally extensive variable
mass. The internal energy U , an extensive variable, is represented in terms of the
specific internal energy

þ

e , an intensive variable, by the following volume integral

ð
O re

U ¼ d v : (3.44)

Integral representations of the mechanical power P and the non-mechanical or
heat power Q supplied to a object O are necessary in order to convert the global
form of the energy conservation principle ( 3.43 ) to a point form. Heat is transferred
into the object at a rate— q per unit area; the vector q is called the heat flux vector.
The negative sign is associated with q because of the long-standing tradition in
thermodynamics that heat coming out of a system is positive while heat going into a
system is negative. The internal sources of heat such as chemical reactions and
radiation are represented by the scalar field r per unit mass. Using these
representations the total non-mechanical power supplied to an object may be
written as the sum of a surface integral and a volume integral,

ð ð

O r

Q ¼ q n d a þ r d v : (3.45)

@ O

This integral representation for the heat supplied to the object distinguishes
between the two possible sources of heat, the internal and the external. Applying
the divergence theorem (A184) to the surface integral in ( 3.45 ) it is easy to see that
Q may also be represented by the volume integral

ð
O ðr

Q ¼ r r q Þ d v : (3.46)

The mechanical power P delivered to the object is represented in integral
form by