1) If one considers an elastic body, and supposing that forces are applied to points of this body and that these are in mutual equilibrium, one may ask what changes the shape of the body has undergone as a result of the action of these forces. One may also ask, supposing that after the change in shape the body is released, what are the laws of the oscillatory motion which occur as a result of the forces which constitute elasticity.
The solution to these questions is composed of two parts: (1) the search for the differential equations which express the laws of the equilibrium or of the motion and (2) the integration of these equations. The search for the differential equations is the objective of this memoir. This must be founded on an exact understanding of the nature of those forces because of which a body is elastic, and of the internal structure of this body.
One considers an elastic solid body as a collection of material molecules positioned at very small distances. These molecules exert on one another two opposing actions, to wit, an intrinsic force of attraction and a force of repulsion resulting from the principle of heat. Between a molecule M and any neighboring molecule M′ there exists an action P, which is the difference between those two forces. In the initial state of the body, all the actions P are null or reciprocally cancel each other out, because the molecule M is at rest. When the
Figures & Tables
Classics of Elastic Wave Theory
In this chapter, we give a brief synopsis of each of the classic papers referred to in this collection. Where relevant, we reproduce the basic equations, recast in modern notation. Supporting works also are referred to. They are listed in the “General References” section.
Table 1 is a quick outline of the key contributions of each paper reprinted in this book.
Robert Hooke, “Potentia Restitutiva, or Spring” (Oxford, 1678)
The article by Robert Hooke, “Potentia Restitutiva, or Spring,” contains the statement of the proportional relation between stress and strain universally referred to as Hooke’s law. Although the English language has evolved somewhat since 1678, the article does not require translation. Hooke describes a variety of experiments, accompanied by illustrations, confirming the stress/strain relation over a wide range of applied loads. He emphasizes the great generality of his results.
Based on his experimental work from 1660 onward, Hooke first published his law in 1676 in the form of an anagram in Latin,
which he later revealed to be “ut tensio sic vis.” Roughly translated, this means “as the force, so is the displacement” (Love, 1911; Boyce and DiPrima, 1976).
In his treatise, Hooke examined the behavior of springs, so his first casting of the equations dealt with the restoring force on a spring, for a given displacement: