Classical Infinitesimal Theory Of Elasticity

Abstract

The classical linearized theory of elasticity provides a model that is useful in the study of infinitesimal deformations of an elastic material. In this chapter we remark briefly on the relationship between this linearized theory for infinitesimal deformations and the exact theory of the elastic simple material.

Helmholtz seems to have been the first to state that a theory can be no more than a mathematical model for nature.

Truesdell (1973)

To compare conclusions, we must first have them. The only way to get a conclusion from a mathematical theory is by logic, by mathematical steps. Any conclusion gotten otherwise, as for example by “physical intuition”, blind teamwork on huge machines, or other gilded guessing is really not a conclusion from the theory. At best it is itself some other theory, not a consequence of the one we study. At worst, it is wrong. Once we recognize that a theory is a mathematical model, we recognize also that only rigorous mathematical conclusions from a theory can be accepted in tests of the justice of that theory...

Therefore, in physical theory mathematical rigor is of the essence. Being human beings and hence fallible, we may not always achieve this rigor, but we must attempt it. A result partly proved and honestly presented as such, like a tunnel drilled partly through a mountain, may be useful in giving the next man a better place from which to start, or, if fortune frowns, in showing him that to drive further in this direction is futile.

Truesdell (1980)