Darrigol O. Worlds of flow. A history of hydrodynamics from the Bernoullis to Prandtl (794382), страница 34
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This work, as well as his edition of Gauthey's works,brought to his attention the empirical inadequacies of the existing theoretical treatments ofthe elasticity of solid bodies. Previous calculations of the compression, extension, andflexion of beams had assumed the existence of mutually-independent longitudinal fibersthat resisted extension or compression by a proportional tension or pressure; otherwise,they relied on an even cruder idealization in which the beam was replaced by a line or bladewith a curvature-driven elastic response. Navier worked to improve the fiber-basedreasoning in order to address the practically essential question of rupture.
He still taughtthis point of view in the course he began to teach at the Ponts et Chaussees in1 8 19, although he also told his students that the true foundation of elasticity should bemolecular.2725Cf. Heilbron [1993] pp. 1-16, Fox [1971], [1974], Crosland [1967], Grattan-Guinness [1990] chap. 7.26Navier, in Belidor [1819] pp.
x-xi, 208n. Ibid. on p. 215n Navier rejected Daniel Bernoulli's and Belidor'skinetic interpretation of pressure.27Cf. Prony [1864] pp. xliii-xliv, Saint-Venant [1864b] pp. civ-dx. On the history of elasticity, see alsoTruesdell [1960], Todhunter and Pearson [1886-1893], Timoshenko [1953], Benvenuto [1991]. On Navier's course,cf. Picon [1992] pp.
482-495.112WORLDS O F FLOWIn August1 820,Navier submitted to the Academy of Sciences a memoir on vibratingplates in which he still reasoned in terms of continuous deformations. The problem ofvibrating plates had occupied several excellent minds since the German acoustician ErnstChladni, with fme sand and a violin bow, had revealed their nodal lines to FrenchAcademicians in1 808.Whereas Sophie Germain and Lagrange still reasoned on thebasis of presumptive relations between curvature and restoring force, in1 8 14Laplace'sclose disciple Simeon Denis Poisson offered a first molecular theory. He considered a twodimensional array of molecules, and computed the restoring force acting on a givenmolecule by summing the forces exerted by the surrounding displaced molecules. Toperform this sum, Poisson assumed, as Laplace had done in his theory of capillarity,that the sphere of action of a molecule was very small compared to a macroscopicdeformation and nevertheless contained a very large number of molecules.
Consequently,he replaced the molecular sums with integrals and retained only low-order terms in theTaylor expansion of the deformation?8Poisson's analysis confirmed the differential equation used by Lagrange and Germain.Navier, nonetheless, sought another derivation, because he believed that the boundaryconditions were still in doubt. In his memoir of1820,he applied Lagrange's method ofmoments, which has the advantage of simultaneously yielding the equation of motion andthe boundary conditions. As we saw in Chapter1,in his seminal memoir on fluid motion,Lagrange had used this method to derive Euler's equations and the appropriate boundaryconditions.
In hiswork)Mechanique analitique,he had also introduced the moment (virtualJJ F8dS of the elastic tension F that arises in response to the stretching of an elasticmembrane.29For simplicity, Navier assumed that the local deformation of a plate could be decomposed into flexion and isotropic stretching.
To Lagrange's expression for the moment ofthe tension caused by the stretching, he added the moment of the elastic tensions andpressures that arise in response to the flexion of a plate of finite thickness (the fibers on oneside of the plate are compressed while those on the other side are extended). Lastly, heobtained the equation of equilibrium and the boundary conditions balancing the totalmoment of a virtual deformation with the moment of the external forces.303.2.4The general equations of elasticityOn the one hand, Navier admired Lagrange's method for its power to yield the boundaryconditions.
On the other, he approved of Laplace's and Poisson's molecular program. Afew months after submitting his memoir on elastic plates, he managed to combine thesetwo approaches. Presumably, he first rederived the moments for the elastic plate bysumming molecular moments. Having done so, he realized that this procedure could easilybe extended to an arbitrary, small deforniation of a three-dimensional body. He therebyobtained the general equations of elasticity for an isotropic body (with one elastic constantonly).
In the memoir he read on14May1 82 1 , he gave twodifferent derivations of these28Poisson [1814]. Cf. Saint-Venant [1864b] pp. ccliii-<:clviii, Dahan [1992] chap. 4, Grattan-Guinness [1990]vol. 1, pp. 462-5.29Navier [1 820], [1 823a]; Lagrange [1788] pp. 139-45, 158--62 (membrane), 438-41.
Cf. Dahan [1992] pp. 50-1.3°Cf. Saint-Venant [1864b] pp. cclix-<:elx, Grattan-Guinness, [1990] vol. 2, pp. 977-83.1 13VISCOSITYFig. 3.3.Diagram for displacements in an elastic body.equations. The first derivation was by a direct summation of the forces acting on the givenmolecules, and the second was by the balance of virtual moments. This second route,Navier's favorite, goes as followsYFor a solid in its natural state of equilibrium, the moment of molecular forces vanishes.After a macroscopic deformation such that a particle (i.e., a small portion of the solid)originally located at r goes to the pointaandr + u(r), the vector R",e joining the two moleculesf3 alters by 8R",e = u(r,e) - u(r") (see Fig. 3.3).
To first order in u, the correspondingchange of distance8Ra,eis given by the projectionuRof the vectoru(r,e) - u(r")onto theline joining the two molecules. Navier assumed that, for small deformations, the forcebetween two molecules varied by an amount proportional to the change in their distance,the proportionality coefficient being a rapidly-decreasing function,P(Ra,e)of their distance. This restoring force must be attractive for an increase of distance, and repulsive fora decrease of distance.Now consider a virtual displacement w(r) of the particles of the solid. To first order in u,the deformationmoleculesaandu{3,implies a change of momentwhereWR-<P8RwRfor the forces between theis the projection of the differenceline joining these two molecules (the indicesaandw(r,e) - w(r")onto thef3 affecting R are dropped to simplifythe notation; an attraction is understood to be positive).
Consequently, the total momentof molecular forces after the deformation is(3.2)Exploiting the rapid decrease of the function ,P(R), Navier replaced uR with its first-orderTaylor approximationate ofR,R-1 x;xjaiuj(r"). In this tensor notation, x; denotes the ith coordin8; is the partial derivation with respect to the ith coordinate ofr, and summationover repeated indices is understood. With a similar substitution forWR,we have(3.3)31 Navier [1 827] (read on 14 May 1821), [1823b]. Cf. Saint-Venant [1 864b] pp.
cxlvii-cxlix, Dahan [1992] chap.8, Grattan-Guinness [1990] pp. 983-5. In the extract of his memoir on elastic plates (Navier [1823a]), Navierassumed a molecular foundation for the flexion moment.1 14WORLDS OF FLOWNavier then replaced the sum over {3 in eqn (3.2) by a volume integral weighted by thenumber N of molecules per unit volume (since his calculation of the moment M waslimited to first order in u, he could neglect the variation of N caused by the deformation).Separating the integration over R and that over angular variables yieldsL cfJuRWR = 2NK(8;Uj0iWj + O;U;OjWj + a;ujajw;),f3withK=(3.4)�� I cp(R)KidR.(3.5)I Ty8iWjdT,(3.6)In order to obtain the total molecular moment M, Navier then performed the sum over a,which he also replaced by an integraL The result can be put in the formM=with(3.7)where By is the unit tensor.By analogy with Lagrange's hydrodynamic reasoning, Navier then integrated by partsto obtain(3.8)The deformed solid is in equilibrium ifand only if this moment is balanced by the moment ofthe applied forces, which may include an internal force density f (such as gravity) and anoblique pressure P on the surface of the solid.
For virtual displacements that occur entirelywithin the body, the balance requires thatjj 8;Tif = 0 or, in vector notation,-f + KN2[ilu + 2\7(17 u)] = 0.·(3.9)The second term represents the restoring force that acts on a volume element of thedeformed solid. According to d'Alembert's principle, the equations of motion of the elasticsolid are simply obtained by equating this force to the acceleration times the mass of theelement.
For virtual displacements at the surface of the body, the balance of the surfaceterm of eqn (3.8) with the moment J -P · w dS of the oblique external pressure gives theboundary condition(3.10)Navier, of course, used Cartesian notation, which gives a forbidding appearance to hiscalculation. However, the basic structure of his reasoning was as simple as the aboverendering suggests.
The only step in the tensor calculation that may imply more thanNavier had in mind is the introduction of the tensor Tif to prepare for the partialVISCOSITY1 15integration of eqn (3.6). Navier treated each term of this equation separately. He nonetheless wrote the following Cartesian version of eqn (3.10):[ ( � :�> �) (:, �:) (� �) l[ (dx' y ) (dx' y d.z') ( y d.z')][ (� �) (: �) (� :: �) ] ,X' = e cos l 3Y1= e cos lZ' = e cos z+ cos m++ cos n+d '+ cos m da' + 3 db' + de'd''db' + da+ cos m++ cos n++d+ cos n de'1 + db'+,(3. 1 1)+3which gives the local response of the solid to an oblique external pressure. 323 .2.5A new hydrodynamic equationSoon after presenting this memoir on elasticity, Navier thought of adapting his newmolecular technique to fluid mechanics. First considering a fluid in equilibrium, heassumed a force j(R) that acted between every pair of molecules and which decreasedrapidly with the distance R (an attraction being understood as positive).
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