Augustin-Louis Cauchy, 2007. "On the Equations Expressing the Conditions of Equilibrium, or the Laws of Interior Motion, of an Elastic or Nonelastic Solid", Classics of Elastic Wave Theory, Michael A. Pelissier, Henning Hoeber, Norbert van de Coevering, Ian F. Jones
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In researching equations expressing the conditions of equilibrium or the laws of interior motion of solid or fluid bodies, we can consider these bodies either as continuous masses whose density varies from one point to another by imperceptible degrees or as systems of distinct material points, separated from one another by extremely small distances. It is in the former sense that fluids have been considered in a previous article and in various papers on mechanics published to date. We intend to consider solid bodies in the same sense here as well.
Therefore, let M be the mass of a solid body in equilibrium, m an infinitely small particle or portion taken at random within this mass, x, y, z the coordinates of the particle m on three rectangular axes, and ρ the density of the solid body at the point (x, y, z). If we call p′, p″, p‴ the pressures or tensions exerted at point (x, y, z) and on the side of the positive coordinates, against three planes parallel to the y, z, the z, x, and the x, y planes, the algebraic projections of these forces p′, p″, p‴ on the coordinate axes will be symmetrical [see page 47 of the second volume], and consequently could be represented by the quantities
Furthermore, if, after having passed a plane through point (x, y, z), we draw from this point and onto each of the half-axes perpendicular to the plane two segments —the first being inversely proportional to the
Figures & Tables
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: