The mathematical theory of elasticity based on the idealization presented in this chapter is a remarkable classical branch of the mechanics of continua, which has advanced far in the more than 200 years it has been studied. In this chapter a concise presentation of the fundamentals of this theory will be given, bearing in mind readers who have not met it before, and it will also serve as a preparation for the next chapter, where we discuss the mathematical modeling of fracture phenomena, which nowadays is the principal area of attention.

The fundamental idealization

A crucially important property of a deformable solid continuum is that it is possible for it to possess non-trivial stress distributions even when the body is at rest, i.e. when the velocity is everywhere equal to zero.

The theory of elasticity as a science is older than fluid mechanics. Its basic law, which was developed to a fundamental model, was formulated by Robert Hooke more than 300 years ago, in the article Hooke (1678).

Readers already know the formulation of Hooke's law for an elastic rod. A rod is an elastic body whose length l is substantially larger than its cross-sectional size s and which has a constant cross-section area S (see Figure 5.1, taken from the book of Galileo Galilei (1638)). Let us take the longitudinal direction of the rod as the x1 axis of a system of orthonormal Cartesian coordinates.