Continuum Formulations of Conservation Laws - Continuum Mechanics of Anisotropic Materials - page 69

3.5 The Conservation of Energy

The idea of energy and its balance or conservation is central in science yet energy is
not considered to be a precisely defined term. A precise definition would imply that
all types of energy are known, and we do not think that they are. In this regard,
review the quote fromHerbert Callen at the beginning of the chapter. Some energies,
such as kinetic energy, are well known and readily identified in any given situation.
It is possible to define energy as any member of the set consisting of all energies
which are recognized by science, energies such as heat energy, kinetic energy,
atomic energy, chemical energy, electromagnetic energy, etc. As science identifies
each new energy, it would become a member of this set of energies.
The conservation of energy is therefore viewed here more as a basic method of
science rather than as a basic fact in the sense that the charge of an electron is a
scientific fact. The conservation of energy is viewed here as a method of checking
energetic interactions and discovering new energies. Whenever one approaches a new
scientific problem, one tries to select or invent energies such that, by setting their sum
equal to a constant, some aspect of the physical phenomenon is correctly described.
In the continuum theories, the known energies will include kinetic energy, heat
energy, chemical energy, electromagnetic energy, and so forth. The total energy E
of a system consists of the sum of all the energies we choose to recognize or define
and the remainder of the total energy of a system is said to be the internal energy U
of the system. That is to say, all the energies that are not singled out and explicitly
defined are placed in the category of internal energy. The total energy E of an object
consists of a kinetic energy,

ð
O rð

1
2

K ¼ v v Þ d v ; (3.41)

and an internal energy U ,

E ¼ K þ U ; (3.42)

where U consists of all energies except kinetic.
The principle of conservation of energy is the statement that the total energy of
an object is constant. It is more convenient to reformulate the conservation of
energy as a balance of rates: the rate of increase of the total energy of an object is
equal to the rate of energy flux into the object. The flux of energy into a object
occurs in two ways, first through the mechanical power P of the surface tractions
and action-at-a-distance forces and, second, through a direct flow of heat Q into the
object. With these definitions and conventions established, the conservation of
energy may be written in the form

¼ K þ U ¼ E

P þ Q ; (3.43)