Continuum Formulations of Conservation Laws - Continuum Mechanics of Anisotropic Materials - page 52

Chapter 3
Continuum Formulations of Conservation Laws
The theme for this chapter is captured by a quote fromHerbert Callen's topic (Callen
1960 ) on thermodynamics. Callen introduces the conservation of energy and the
concept of internal energy in the following paragraph: “The development of the
principle of conservation of energy has been one of the most significant
achievements in the evolution of physics. The present form of the principle was
not discovered in one magnificent stroke of insight but has been slowly and labori-
ously developed over two and a half centuries. The first recognition of a conserva-
tion principle, by Leibnitz in 1693, referred only to the sum of the kinetic energy
((1/2)mv 2 ) and the potential energy (mgh) of a simple mechanical mass point in the
terrestrial gravitational field. As additional types of systems were considered, the
established form of the conservation principle repeatedly failed, but in each case it
was found possible to revive it by the addition of a new mathematical term - a “new
kind of energy.” Thus consideration of charged systems necessitated the addition of
the Coulomb interaction energy (Q 1 Q 2 /r) and eventually of the energy of the
electromagnetic field. In 1905 Einstein extended the principle to the relativistic
region, adding such terms as the relativistic rest-mass energy. In the 1930's Enrico
Fermi postulated the existence of a new particle, called the neutrino, solely for the
purpose of retaining the energy conservation principle in nuclear reactions. Con-
temporary research in nuclear physics seeks the form of interaction between
nucleons within a nucleus in order that the conservation principle may be formulated
explicitly at the subnuclear level. Despite the fact that unsolved problems of this
type remain, the energy conservation principle is now accepted as one of the most
fundamental, general, and significant principles of physical theory.”
3.1 The Conservation Principles
Conservation principles will be cast here in the form of a balance or accounting
statement for the time rate of change of a quantity in a system. On the plus side of
the accountant's ledger are the amount of the quantity coming into the system and