Darrigol O. Worlds of flow. A history of hydrodynamics from the Bernoullis to Prandtl (794382), страница 39
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P. Wormeley, quoted in Cannone and Frielander [2003] p. 7).( a)(b)Fig. 3.4.(a) Navier's projected Pant des Invalides on the River Seine and (b), the anchoring system for the chains. From Navier [1830] plates.WORLDS OF FLOW128one henceforth exalted practice in its most material aspects, and pretended thathigher mathematics could not help, as if, when it comes to results, it made sense todistinguish between the more or less elementary or transcendent procedures that ledto them in an equally logical manner.
Somesavants supported or echoed theseunfounded criticisms.Some engineers were indeed fighting the theoretical approach that Navier embodied. InI 833, the Ingenieur en chefdes Ponts et Chaussees, Louis Vicat,already acclaimed for hisimprovement of hydraulic limes, cements, and mortars, performed a number of experiments on the rupture of solids. His declared aim was 'to determine the causes of theimperfection of known theories, and to point out the dangers of these theories to theconstructors who, having had no opportunity to verify them, would be inclined to lendthem some confidence.' He measured the deformations and the critical charge for variouskinds of loading, and observed the shapes of the broken parts. He believed to have refutedCoulomb's and Navier's formulas for the collapse of pillars, as well as Navier's formulasfor the flexion and the torsion of prisms.
Moreover, he charged Coulomb and Navier witherroneous conceptions of the mode of rupture.683.5.2Vicat's rupturesVicat distinguished three ways in which the aggregation of a solid could be destroyed: pull(tirage), pressure (pression), and sliding (glissement). He called the corresponding forces(force tirante), sustaining force (force portante), and transverse force (forcetransverse). This last force (our shearing stress) he defined as 'the effort which tends topulling forcedivide a body by making one of its parts slide on the other (so to say), without exerting anypressure nor pull outside the face of rupture.' The usual theories of sustaining beams, Vicatdeplored, ignored the slides and transverse forces, even though they controlled the ruptureof short beams under transvers load.
A important exception was Charles AugustinCoulomb, of whose theory Vicat however disapproved. 69Vicat published his memoir in theAnnales des Ponts et Chaussees, but also ventured tosend a copy for review to the Academy of Sciences. The reviewers, Prony and Girard,defended their friends Coulomb and Navier, arguing that Vicat had used granular,inflexible materials and short beams for which the incriminated formulas were not intended. They judged that Vicat's measurements otherwise confirmed existing theories.They also emphasized that only Coulomb's theory could justify the use of reduced-scalemodels, on which Vicat's conclusions partly depended. 70In his response, Vicat compared the two Academicians to geometers who' would declarethe law 'surface equals half-product of two side lengths' to apply to any triangle becausethey had found it to hold for rectangular triangles.
In a less ironic tone, he showed thatsome of his measurements did contradict the existing theories in their alleged domain ofvalidity. Navier himself did not respond to Vicat's aggression. However, some modifications in his course at the Ponts et Chaussees suggest that he took Vicat's conclusions on the68Saint-Venant [1 864a] p. 1xviii; Vicat [1833] p. 202.
On Vicat, his work on limes, cements, and mortars, and hisimplicit criticism ofNavier's conception of suspended bridges, cf. Picon [1992] pp. 364-71, 384-5.69Vicat [1833] p. 201. Cf. Benvenuto [1998] pp. 18-19.70Prony and Girard [1 834].VISCOSITY129importance of slides and transverse forces seriously. His former student Saint-Venantcertainly did.713.5.3Molecules, slides, and approximationsAdh6mar BarnS de Saint-Venant had an 'X+Ponts' training, and an exceptional determination to reconcile engineering with academic science. His mathematical fluency and hisreligious dedication to the improvement of his fellow citizens' material life determined thisattitude.
He rejected both the narrow empiricism ofVicat and the arbitrary idealizations ofFrench rational mechanics. His own sophisticated strategy may be summarized in thefollowing five steps. 72,c(i) Start with the general mechanics of bodies as they are in nature, which is to be basedon the molecular conceptions of Laplace, Poisson, and Navier.(ii) Determine the macroscopic kinematics of the system, and seek molecular definitionsfor the corresponding macroscopic dynamics.(iii) Find macroscopic equations of motion, if possible, by summing over molecul�s, orelse by macroscopic symmetry arguments. The molecular level is thus black-boxed inadjustable parameters.(iv) Develop analytical techniques and methods of approximation to solve these equations in concrete situations.(v) Test consequences and specify adjustable parameters by experimental means.Saint-Venant developed this methodology while working on elasticity and tryingto improve on Navier's methods.
He regarded the first, molecular step as essential fora clear definition of the basic concepts of mechanics and for an understanding ofthe concrete properties of matter. In his mind, the most elementary interaction was thedirect attraction or repulsion of two mass points. Consequently, there could be nocontinuous solid (as Poisson and Cauchy had proved in 1 828). Matter had to be discontinuous, and all physics had to be reduced to central forces acting between non-contiguouspoint-atoms. 73In the second, kinematic step Saint-Venant characterized the macroscopic deformationsof a quasi-continuum in harmony with Vicat's analysis of rupture. Cauchy had introducedthe quantities eif = 8;uj + OjU;, but only to determine the dilation or contraction(1 /2)eijdx;dxj of a segment dr of the body.
While studying a carpentry bridge on theRiver Creuze in 1 823, and later in his lectures at the Pouts et Chaussees, Saint-Venant gavea precise geometrical definition of Vicat's slides and took them into account in a computation of the flexion of beams. According to this definition, the jth component of slide(glissement) in a plane perpendicular to the ith axis is, at a given point of the body, thecosine of the angle that two concrete lines of the body intersecting at this point andoriginally parallel to the ith and jth axes make after the deformation (see Fig.
3.5). Tofirst order in u, this is the same as Cauchy's eif. Saint-Venant used the slides not only to72Cf. Boussinesq and Flamant (1886], Melucci [1996], Darrigol (2001].71 Vicat [1834].73Saint-Venant [1834], [1844].WORLDS OF FLOW130Fig. 3.5.LLe;;Geometrical meaning of Saint-Venant's slide ey with respects to the orthogonal axes i and} (in theplane of the figure).investigate the limits of rupture, but also to develop a better intuition of the internaldeformations in a bent or twisted prism.74Saint-Venant defined the pressure on a surface element dS as the resultant of the forcesbetween any two molecules such that the line joining them crosses the surface element.This pressure has the form rii dSj, which defines the stress system rii.
Saint-Venant thendetermined the relation between these stresses and the strainseii, in one of Cauchy's twomanners. Although the first manner, based on symmetry only, was simpler, he believedthat only the second, molecular manner could give the correct number of independentelastic constants (one constant instead of two in the isotropic case). 75On this very theoretical basis, Saint-Venant struggled to solve concrete problems ofengineering. He was aware of the great variety of available strategies of approximationthat could help in this task:Between mere groping and pure analysis, there are many intermediaries: the methodsof false position, the variation of arbitrary constants, the solutions by series orcontinuous fractions, the methods of successive approximations, integration by thecomputation of areas or by the formulas of Legendre and Thomas Simpson, thereduction of the equations to more easily soluble ones by the choice of an unknown ofwhich one may neglect a few powers or some fuoction in a first approximation,graphical' procedures, figurative curves drawn on squared paper, the use of curvilinear coordinates, etc.
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