What Happened on September 30, 1822, and What Were its Implications for the Future of Continuum Mechanics?

Abstract

This contribution offers a discussion about the notion of stress in a general continuum as initially proposed in a magisterial paper by Cauchy in 1822 (but published only in 1828) without using arguments involving molecules. This is here presented in its historical context. Cauchy’s view is the currently accepted view among mechanicians and engineers although attempts (including by Navier and Cauchy himself) to start from a molecular description in the manner of Newton and Laplace were constantly offered in both nineteenth and twentieth centuries. The discussion introduces other secondary stress definitions such as those by Piola, Kirchhoff, and more recently Eshelby. The question naturally arises of what happens with the possibility to introduce other internal forces such as hyperstresses (in so-called gradient theories) and couple stresses (e.g., in Cosserat continua), and whether some introduced stresses have associated with them a meaningful boundary condition. Also pondered is the question whether one can identify a stress concept in physical approaches still considering interactions between point particles (lattice dynamics, kinetic theory, nonlocal theory, statistical-mechanics approach). The chapter is concluded by a more in depth discussion of the notion of stress-energy-momentum, culminating in that of pseudo-tensor of energy-momentum in gravitation theory.

Appendices

Appendix A

The “birth certificate of modern continuum mechanics”: Cauchy’s reading of his ideas at the September 30, 1822 session of the Académie des Sciences in Paris (Procès verbal de l’Académie des Sciences, Tome VII, Décembre 1822, Imprimerie d’Abbadia, Hendaye, 1916; kindly provided by Mrs Florence Greffe, Acad. Sc. Paris; May 2013) The remarkable roster of scientists among the above list of attending academicians is stupendous: e.g., Fourier, Magendie, Berthollet, Chaptal, Lamark, Laplace, Lacroix, Cauchy, Cuvier, Legendre, Prony, Poisson.

Appendix B

A. L. Cauchy—1823: Researches on the equilibrium and internal motion of solid bodies or fluids, whether elastic or non-elastic.

Bulletin of the Société Philomatique, pp. 9–13, 1823, Paris.

(Translation from the French by G.A. Maugin)

The present researches were undertaken on the occasion of the publication of a memoir by M. Navier on August 14, 1820. Its author, with a view to establishing the equilibrium equation of an elastic plane, had considered two kinds of forces, some produced by dilatation or contraction, and the other by the flexion of this plane. Moreover, he had supposed, in his computations, that both these forces are perpendicular to lines or faces on which they are exerted. It seemed to me that these two kinds of forces could be reduced to one kind only, which should be always called tension or pressure, and is of the same nature as the hydrostatic pressure exerted by a fluid at rest on the surface of a solid body. However, the new “pressure” will not always be perpendicular to the faces on which it act, and is not the same in all directions at a given point. Expanding this idea, I arrived soon at the following conclusions.

If in a solid body, whether elastic or not elastic, we succeed to render rigid—in thought. [GAM]—and invariable a small volume element bounded by any surfaces, this small element will be subjected on its different faces and in any point of each of these, to a determined pressure or tension. This pressure or tension will be of the same type as the pressure that a fluid exerts on an element of the boundary of a solid body, save for the difference that the pressure exerted by a fluid at rest on the surface of a solid body is directed normal to this surface, from the outside to the inside, and is independent at each point of the orientation of the surface with respect to the coordinate planes, while—in our case [GAM]—the pressure or tension exerted at a given point can be oriented perpendicularly or obliquely to this surface, sometimes from the outside to the inside if there is condensation [i.e., contraction, GAM] and sometimes from the inside to the outside if there is dilatation, and it can depend on the angle made by the surface with the relevant planes. Furthermore, the pressure or tension exerted on any plane can easily be deduced, in both amplitude and direction, from the pressures or tensions exerted on three given orthogonal planes. I had reached this point when M. Fresnel, who came to me to talk about his works devoted to the study of light and which he had presented only in part to the Institute, told me that, on his own, he had obtained laws in which elasticity varies according to the various directions issued from a unique point, a theorem similar to mine. However, the theorem in question was far from being sufficient for my projected object of study, at that period, that was to formulate the general equations of equilibrium and internal motion of a body; and it is only in recent times that I succeeded to establish the proper new principles that yielded this result, and that, now, I will make known.

From the above mentioned theorem, it follows that the pressure or tension at each point is equivalent to the inverse of the vector radius of an ellipsoid. Three pressures or tensions that we call principal correspond to the three axes of this ellipsoid, and we can show [This remark here is in agreement with the last researches of M. Fresnel (See the Bulletin of May 1822)] that each of these is perpendicular to the plane on which its acts. Among these principal pressures or tensions there are a maximum pressure or tension and a minimum one. The other pressures or tensions are distributed symmetrically about these three axes. Moreover, the pressure or tension normal to each plane, i.e., the component, perpendicular to a plane, of the pressure or tension exerted on this plane, is proportional to the inverse of the squared vector radius of a second ellipsoid. Sometimes, this second ellipsoid is replaced by two hyperboloids, one with one sheet, the other with two sheets, which have the same centre, the same axes, and are asymptotic at infinity with a common second- degree surface, of which the edges point in the direction for which pressure or normal tension reduces to zero.

This being said, if we consider a solid body of varying shape and subjected to arbitrary accelerating forces, in order to establish the equilibrium equations of this solid body it will be sufficient to write that there is equilibrium between the motive forces that act on an infinitesimal element along three axes of coordinates, and the orthogonal components of external pressure or tension that act on the faces of this element. We will thus obtain three equations of equilibrium that include, as a particular case, the corresponding equations for the equilibrium of fluids. But, in a general case, these equations contain six unknown functions of the coordinates x, y, z. It remains to determine the value of these six unknown quantities. But the solution of this last problem varies with the nature of the body and its more or less perfect elasticity. Now we shall explain how one can solve this problem for elastic bodies.

When an elastic body is in equilibrium by virtue of arbitrary accelerating forces, one must assume that each molecule has been displaced from the position it occupied when the body was in its natural state. As a consequence of these displacements, there are around each point different condensations or dilatations in different directions. But it is clear that each dilatation produces a tension, and each condensation produces a pressure. Furthermore, I prove that the various condensation or dilatation about this point, decreased by or augmented of the unit, become equal, up to the sign, to the vector radii of an ellipsoid. I call principal condensations or dilatations those that occur along the axes of this ellipsoid, about which the others are distributed symmetrically. This being set, it is clear that in an elastic body, tensions or pressures depending only on the condensations or dilatations, are directed in the same directions as the principal condensations or dilatations. In addition, it is natural to assume, at least when the displacements of molecules are small, that the principal tensions or pressures are proportional to the principal condensations and dilatations, respectively. Admitting this principle, we arrive immediately at the equilibrium equations of an elastic body. In the case of very small displacements, the component, perpendicular to a plane, of the pressure or tension exerted on that plane, always is in the same ratio with the condensation or dilatation that occurs in the same direction, and the formulas for equilibrium reduce to four partial differential equations of which each one determine separately the condensation or dilatation in volume, while each of the others serves to fix the displacement parallel to one of the coordinate axes.

The equations of equilibrium of an elastic body being set, it is now easy to deduce by ordinary means the equations of motion. The latter still are four in number, and each of them is a linear partial differential equation with an added variable term. These equations are integrated by use of methods that I exposed in a previous memoir. One of these equations contains only the unknown that represents the condensation or dilatation in volume. In the particular case where the acceleration force becomes constant and keeps everywhere the same direction, this equation reduces to the propagation of sound in air, with the only difference is that the constant it contains, instead of depending on the height of a supposedly homogeneous atmosphere, depends on the linear dilatation or condensation of a body in a given pressure. One must conclude from this that the speed of sound in an elastic body is constant, like in air, but it varies from one body to another one depending on the matter of which it is made. This constancy is all the more remarkable that the displacements of molecules considered successively in fluids and elastic solids obey different laws.

My memoir is concluded by the formation of the equations of the internal motion of solid bodies completely devoid of elasticity. To arrive at this it is sufficient to suppose that in these bodies the pressures or tensions about a point in motion do not depend any more on the total condensations or dilatations that correspond to the absolute displacement measured from the initial positions of the molecules, but only, after any lapse of time, on the very small condensations or dilatations that correspond to the respective displacement of the different points during a short interval of time. One therefore finds that the volume condensation is determined by an equation similar to that governing heat, what establishes a remarkable analogy between the propagation of the caloric [the supposed “fluid” carrying heat. GAM] and the vibrations of a body entirely devoid of elasticity.

In a forthcoming memoir, I shall give the application of the obtained formulas to the theory of elastic plates and strings.