# Theory of Elasticity

• Thomas H. Dawson

## Abstract

An inspection of the mechanical laws described in the last chapter reveals one scalar equation expressing the conservation of mass, three scalar equations expressing the balance of linear momentum, and three scalar equations expressing the balance of angular momentum, thus giving a total of seven equations in all. If we count the number of unknowns introduced into these equations, we find, however, one scalar density function, three scalar components of displacement or velocity, and nine scalar components of stress, so that, assuming the body forces are given, we thus have 13 unknowns and only seven equations connecting them. To determine the unknowns, it is therefore necessary to employ six additional equations. These equations provide information on the response of the particular material under consideration and are known as constitutive relations.

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1. Malvern, L. E., Introduction to the Mechanics of a Continuous Medium. Prentice-Hall, Englewood Cliffs, New Jersey, 1969. Chapter 6 gives a general discussion of constitutive relations. Section 6.7 discusses the principle of material indifference.Google Scholar
2. Love, A. E. H., Mathematical Theory of Elasticity. Cambridge University Press, Cambridge, England, 1927. This is a classical treatise on linear elasticity containing all important developments up to 1927. An excellent historical introduction is included at the beginning.
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4. Long, R. R., Mechanics of Solids and Fluids. Prentice-Hall, Englewood Cliffs, New Jersey, 1961. The equations of linear elasticity are developed in Chapter 5 and selected problems are considered in Chapter 6.
5. Leigh, D. C., Nonlinear Continuum Mechanics. McGraw-Hill Book Co., New York, 1968. Chapter 8 discusses general principles for constitutive equations, including the principle of material indifference.Google Scholar
6. Truesdell, C., and W. Noll, “The Non-Linear Field Theories of Mechanics,” in Encyclopedia of Physics, Vol. III/3. Springer-Verlag, Berlin, 1965. A very advanced treatise on continuum mechanics including detailed development of various constitutive relations. Section 19 provides an interesting discussion of the principle of material indifference.Google Scholar