E. B. Christoffel, 2007. "On the Propagation of Shock Waves through Elastic Solids", Classics of Elastic Wave Theory, Michael A. Pelissier, Henning Hoeber, Norbert van de Coevering, Ian F. Jones
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1. In the present treatise, I will apply the principles developed in my previous work (page 81 of this volume) to the propagation of shock waves in elastic solids. I begin with some preliminary remarks on how the phenomena to be developed here are to be understood, because the following analysis cannot be allowed to suffer from any ambiguity on this point.
I consider a homogeneous elastic body free of external forces in equilibrium and denote for this case by xyz the orthogonal coordinates of one of its constituent particles m, by ρ the constant density, by ℜ its volume, and by its surface area. If the equilibrium of the body is disturbed, then as usual, I will represent the coordinates of the particular particle m at time t by x + u, y + v, and z + w. The variables uvw, the displacement components of m, will be interpreted as functions of txyz; that is, they will be associated with points xyz of the volume ℜ at rest, so that at each point in time t, these determine the actual position of the particle m whose equilibrium coordinates are xyz.
We accordingly restrict all the phenomena to be investigated here to this volume at rest ℜ and its surface . In particular, for the propagation of shock waves, the surface of discontinuity at time t will not be defined as the actual surface which describes the dis-continuity occurring at this time.
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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: