Theory of the Elasticity of the Professor Enrico Betti1
Published:January 01, 2007
To determine the relations that must exist between forces that act on a homogeneous elastic solid body and the deformations of its elements so that equilibrium is maintained, we will apply the following principle of Lagrange: In order that a system, whose virtual motions are interchangeable, is in equilibrium, it is necessary and sufficient that the mechanical work done from forces in any virtual motion is equal to zero.
Let X, Y, Z be the components of accelerating forces that act on every point of the body; L, M, N the components of the force acting on every point of its surface; and r constant density. We associate a virtual motion u, v, w with every point of the deformed body, and we denote by δu,δv, δw the variations this motion will take. Clearly, the work done by this motion by the given forces will be
where S is the volume occupied by the body and σ is its surface. The work done by elastic forces will be equal to the increase in potential of the whole body, that is,
Whence from the principle of Lagrange
Now, denoting by α, β, γ the cosines of the angles that the normal directed toward the interior of the body forms with the positive axes,
Substituting in it equation 11 and setting the coefficients δu, δv, δw of the triple and double integrals equal to zero separately, the equations of equilibrium are obtained:
To these equations, we will add the others which express
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: