Darrigol O. Worlds of flow. A history of hydrodynamics from the Bernoullis to Prandtl (794382), страница 37
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Cauchy's contemporariesthought differently. In the years following his publication, theorists of elasticity were notsatisfied with this purely macroscopic-continuum approach, even though they all adoptedCauchy's stress. In their eyes, the true foundation of elasticity had to remain molecular, as52it should be in Laplace's grand unification of physics.It would also be wrong to regard Cauchy's stress-strain approach as an indication that hesupported a continuist view of matter. For theological reasons, he was a finitistin mathematics and an atornist in physics.
That he first derived the equations of elasticitywithout reference to the molecular level only proves that he possessed the geometrical andalgebraic skills that make this route natural and easy. He in fact provided the most completeand rigorous molecular theory of elasticity, even before his first theory of elasticity waspublished. On this ground, he found himself again in competition with Poisson, undoubtedly the most aggressive supporter of the molecular approach. 535°Cauchy [1823], [1828a] par. 3.51 Cauchy [1828a].52Cf. Saint-Venant [1 864b], pp.
cliv-clv.53In his Torino lectures of 1833, Cauchy argued that extended molecules would be indefinitely divisible,against the principle that 'only God is infinite, everything is finite except him' (Cauchy [1833] pp. 36-7). However,he never used molecular considerations in print before his molecular theory of elasticity (I thank Bruno Belhostefor this information).WORLDS OF FLOW1223.4 Poisson: the rigors of discontinuity3.4.1Laplacian motivationsUnlike Navier and Cauchy, Poisson did not have engineering training and experience, forhe settled at the Polytechnique as a n!petiteur and then as a professor. His interest inelasticity came from his enthusiastic embrace of Laplace's molecular program.
His 1814theory of elastic plates was already molecular. Presumably stimulated by Navier's publications of 1 820/2 1 , he returned to this subject in the late 1820s. His memoir read on 14April 1 828 contains his famous plea for a mecanique physique: 54It would be desirable that geometers reconsider the main questions of mechanicsunder this physical point of view which better agrees with nature. In order to discoverthe general laws of equilibrium and motion, one had to treat these questions in a quiteabstract manner; in this kind of generality and abstraction, Lagrange went as far ascan be conceived when he replaced the physical connections of bodies with equationsbetween the coordinates of their various points: this is whatanalytical mechanics isabout; but next to this admirable conception, one could now erect a physical mechanics, whose unique principle would be to reduce everything to molecular actions thattransmit from one point to another the given action of forces and mediate theirequilibrium.Poisson's memoir of 1 828 can, to some extent, be seen as a reworking of Navier'smemoir of 1821 on the molecular derivation of the general equations of elasticity.
Bothauthors aimed at a derivation of the general equations and boundary conditions ofelasticity by the superposition of short-range molecular actions. However, there weresignificant differences in their assumptions and methods. Whereas the only molecularforces in Navier's calculations were those produced by the deformation of the solid,Poisson retained the total force j(R) between two molecules. Also, Poisson avoidedNavier's method of moments and instead directly summed the molecular forces actingon a given molecule.Cauchy worked on a similar molecnlar theory in the same period.
Competition wasso intense that Cauchy decided to deposit a draft of his calculation as a pli cachete atthe Academy, and Poisson decided to read his memoir in a still unripe form. Cauchy'sassumptions and methods were essentially the same as Poisson's; this should not surpriseus as they were both following Laplacian precepts without Navier's personal touch.
YetCauchy's execution surpassed Poisson's in rigor, elegance, and compactness.55By summation of the forces acting from one side of a given surface element to the otherside, the molecular theory leads to the stress system(3.23}where N is the original number of molecules per unit volume,S4Poisson [1829a] p. 361 . Cf. Amo1d [1983], Grattan-Guinness [1990] pp. 1015-25, Dahan [1992] chap. 10.55Cf.
Belhoste [1991] pp. 99-100.VISCOSITY123(3.24)and(3.25)(R is the length, and x; is the ith coordinate of the vector joining a molecule a situated atthe point at which the stress is computed and an arbitrary molecule{3). For an isotropicsolid, there are only two independent constants A and A', in terms of which the stresssystem reads(3.26)This result agrees with Cauchy's earlier macroscopic theory, except for the pressure A inthe original state. 563.4.2Sums versus integralsPoisson and Cauchy both investigated the limiting case of a continuous medium, in which�the sums (3.24) and (3.25) expressing the coefficients Aii and A k/ can be rigorously replacedby integrals. As Cauchy (but not Poisson) saw, isotropy follows without further assumption, and the coefficients A and A' are given byA =;N J002tR3dR(3.27)0andA' =27TN15Joo0Is dfK dR.RdR(3.28)Integrating the latter expression by parts yields the relationA + A' = lim R'i(R).R-0(3.29)Poisson and Cauchy both assumed the limit to be zero.
Then the medium loses its rigiditysince the transverse pressures disappear. As Cauchy further observed, the continuous limitof the stress has the form(3.30)56Cauchy [1828b], [1 829]; Poisson [1829a]. Cf. Saint-Venant [1 864b] pp. clv-clxi, [1 868a], Dahan [1992] chap.1 1 , Darrigol [2002a] pp. 121-4. Regarding the molecular definition of stress, see Saint-Venant's interventionmentioned later on p. 130.WORLDS OF FLOW124where p is the density in the deformed state. This means that the body is an elastic fluidwhose pressure varies as the square of the density.57In his own manner, Poisson also obtained the absence of transverse pressures in thecontinuum limit.
He used this conclusion to dismiss Navier's theory and to denounce thegeneral impossibility of substituting integrals for molecular sums in the new physicalmechanics. He also claimed to be the first to have offered a genuinely molecular theoryof elasticity. 583.4.3Navier's defenseThere followed a long, bitter polemic in theAnnales de chimie et de physique. Navier firstrecalled that Poisson and Laplace had had no quahns replacing sums with integrals in theirpast works.
The newer emphasis on a supposed rigor could only betray a desire to belittlehis own achievement. It was he, Navier, who in1 821'conceived the idea of a new question,one necessary to the computation of numerous phenomena that interest artists andphysicists.' It was he who 'recognized the principle on which this solution had to rest.'This principle, however, was not what Poisson thought it should be; it made the variationof intermolecular forces during a deformation of a solid body depend linearly on thevariation of molecular distances, but did not require that the molecules should interactthrough central forces only.
Consequently, Navier believed that his theory was immune toPoisson's arguments on sums versus integrals. 59Navier then counter-attacked Poisson for failing to provide a description of the forcefunctionj(R)that would account for the stability and elastic behavior of solids. Forexample, in order that the internal pressure vanishes in an unstrained solid Poissonrequired the vanishing of the sumI: Rf(R), without exhibiting a choice for f that metthis condition. If Poisson were willing to presuppose so much about the function/, NavierR reaches zero?+ A1 = 0, and allow the use of integrals instead of sums.60argued, why did he not consider a nonzero value of the limit of it'j whenThis would avoid the fatal AFrom this extract ofNavier's defense, one mayjudge that he was hesitating between twostrategies.
The first option was to deny the general applicability of the Laplacian doctrineof central forces, and to deal only with the forces that arise when an equilibriumunknown nature isofdisturbed. This option agreed with Navier's positivist sympathies andwith the style of applied physics that he embodied at the Ponts et Chaussees; and it couldaccommodate later, unforeseen changes in molecular theory. 6 1The second option was to admit the Laplacian reduction to central forces and to showthat appropriate results could nevertheless be obtained by substituting integrals for sums.Here Navier erred, because a Laplacian continuum, that is, a continuous set of material '57Cauchy [1 828b] p.266.58Poisson[1829a] pp. 397-8, 403-4."'Navier [1828a],[1828b], [1829a], [1829b]; Poisson [1828a], [1 828b].
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