Darrigol O. Worlds of flow. A history of hydrodynamics from the Bernoullis to Prandtl (794382), страница 36
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At anyinstant, the molecules of a fluid are in positions that slightly deviate from an equilibriumconfiguration that continuaily changes over time. Thus, the molecular force function 1/JVRin Navier's moment formula(3. I 6) does not refer to the actual distance of the molecules,but to the distance that they have in the nearest equilibrium configuration; and the4°Coumot [1 828] pp. 1 1-14.41Ibid. pp. 1 1-12; Navier [1823c] p. 392: 'La force repulsive qui s'etablit entre les deux molecules depend de lasituation du point M [lieu de la premiere molecule], puisqu'elle doit balancer la pression, qui peut varier danslesdiverses parties du fluide.'42Coumot [1 828] p.
12.VISCOSITY119difference between these two distances obviously depends o n the macroscopically-impressed motion. This is how the fluid velocity enters the expression of Navier's molecularforces, even though the true forces depend only on the distances between the molecules.43Unfortunately, Navier never explained as much-so that none of his successors (exceptSaint-Venant) could make sense of his calculation. In Cournot's eyes, the premises ofNavier's equation seemed as arbitrary as its applicability to concrete problems wasdifficult to judge:44M.
Navier himself only gives his starting principle as a hypothesis that can be solelyverified by experiment. However, if the ordinary formulas of hydrodynamics resistanalysis so strongly, what should we expect from new, far more complicated formulas? The author can only arrive at numerical applications after a large number ofsimplifications and particular suppositions. The applications no doubt show greatanalytical skill; but can we judge a physical theory and the truth of a principle afteraccumulating so many approximations? In one word, will the new theory of M.Navier make the science of the distribution and expense of waters less empirical? Ido not feel able to answer such a question. I can only recommend the reading of thismemoir to all who are interested in this kind of application.3.33.3.1Cauchy: stress and strainThe stress systemLike Navier, Augustin Cauchy was an 'X+ Pouts' with superior mathematical training andwith engineering experience.
However, his poor health and mathematical genius soonconfined him to purely academic activities. In1 822,his study of Navier's memoir onelastic plates led him to new considerations that still constitute the basis of elasticitytheory. If we are to trust Cauchy's own account, then what triggered his main inspirationwas Navier's appeal to two kinds of restoring forces produced by extension and flexion.45The second kind of force, Cauchy surmised, could be avoided if forces of the first kindwere no longer assumed to be perpendicular to the sections on which they acted. With thisinsight, he then imitated Euler's hydrodynamics and reduced all elastic actions to pressuresacting on the surface of portions of the body: The only difference was the non-normality ofthe pressure.
Previous students of elasticity, in particular Coulomb and Young, hadalready considered tangential pressures (our shearing stresses) in specific problems suchas the rupture of beams. In his memoir of May1821on a molecular derivation of thegeneral equations of elasticity, Navier had introduced oblique external pressures andboundary conditions that entailed the Cauchy stress system.
Whether or not Cauchy reliedon such anticipations, he was the first to base the theory of elasticity on a generaldefinition of internal stresses.4643Navier [1 823c] p. 390.44Cournot [1828] pp. 13-14.45Cauchy [1823]. Cf. Belhoste [1991] pp. 93-102; Grattan-Guinness [1990] pp. 1005-13.46Cauchy [1823], [1827b]. Cf. Truesdell [1968], Dahan [1992] chap. 9.
In his memoir on elastic plates [1820],Navier noted that in general the pressures would not be parallel to the faces of the element. Fresnel's theory oflightwas perhaps another source ofCauchy's inspiration, cf. Belhoste [1991] pp. 94-5. The stress-strain ternrinology isWilliam Rankine's.
Cauchy and contemporary French writers used the words pressionltension and condensation/dilatation.WORLDS OF FLOW120As in hydrodynamics, Cauchy introduced the pressures (or tensions) that act on avolume element by recourse to the forces that would act on its surface after an imaginarysolidification of the element. For a unit surface element normal to thejth axis, denote by Tijthe ith component of the force acting on the negative side (xj < 0) of the element.
Notethat, with this convention, adopted by most ofCauchy's followers, a tension is understoodto be positive. Cauchy proved three basic theorems in a manner that is still used in modemtexts on elasticity.The first theorem stipulates that the pressure on an arbitrary surface element dS is givenby the sum TifdSi. In modem words, the stress system Tij is a tensor of second rank.This results from the fact that the resultant of the pressures acting on the pyramidalvolume element {x, y, z, > 0; ax + {3y + yz < e} would be of second order in the smallquantity e, and therefore could not be balanced by the resultant of a volume force (whichis of third order) if the theorem were not true.
Cauchy's second theorem states thesymmetry of the stress system, namely Tij = Tj;, without which the resultant of the pressuretorques on a cubic element of the solid would be of third order and therefore could not bebalanced by the torque of any volume force, which is of fourth order. Thirdly, and mostobviously, the resultant of the pressures acting on a (cubic) volume element is OjTij per unitvolume.47Strain and motion3.3.2As Cauchy knew from the theory of quadratic forms (which he had recently applied toinertial moments), the symmetry of the pressure system implies the existence of threeprincipal axes for which the pressures are normal (in modem terins, the stress tensor isthen diagonal).
Cauchy used this property to relate the pressure system to the localdeformations of the system. If u(r) is the displacement of a solid particle at the point ofspace r, he showed, then the first-order variation in the distance between two points whosecoordinate differences have the very small values dx; is given by dx; dxj O;Uj. In modemterms, this quadratic form is associated with the symmetric tensor eif = 8;uj + OjU;. Thistensor has three principal axes, which means that the local deformation is reducible tothree dilations or contractions along three orthogonal axes.48Cauchy then argued that, for an isotropic body, the principal axes of the tensors Tij andeij were necessarily identical.
He further assumed that the pressure ratios between two suchaxes were equal to the dilation ratios. This implies that the two tensors are proportional.Lastly, Cauchy assumed that the proportionality coefficient was a constant independent ofthe deformation, which is a generalization of Hooke's Jaw. He thus obtained an equationof equilibrium similar to Navier's equation (3.9), though without the factor 2 in the"V("V u) term. The boundary conditions immediately result from the balance of internaland external pressures.49·48Cauchy [1823], [1827c].47Cauchy [1827b], [1827dj.49Cauchy [1823], [1828a]. Cauchy introduced the word 'isotrope' in 1839/40, for example in Cauchy [1840].VISCOSITY3.3.3121The perfectly inelastic bodyIn a last section, Cauchy considered the case of a 'non-elastic body', defined as a bodyfor which the stresses at a given instant only depend on the change of form experienced by thebody in a very small time interval preceding this instant.
He found it natural to assume thatthe stress tensor was proportional to the tensor representing the velocity of deformation(again reasoning with respect to principal axes). For an incompressible body, the resultingequation of motion is the one Navier had given for viscous fluids, save for the pressure term.Cauchy, however, mentioned neither Navier's result nor any similarity between a real fluidand his 'non-elastic body'.
Instead, he noted that, for very slow motion, the linearizedequation of motion was identical to Fourier's equation for the motion of heat, and claimed'a remarkable analogy between the propagation of caloric and the propagation ofvibrations50in a body entirely deprived of elasticity.'3.3.4Finalfoundations?Cauchy announced his theory of elasticity on 30 September 1 822, and published it insummary form the following year. He waited six more years before complete publicationin his own, self-serving journal, the Exercices de mathematiques.
The reason for this delaymay have been the courtesy of waiting for Navier's memoir of 1821 to be published. In thefinal version of his theory, Cauchy proposed the more general, two-constant relation(3.22)·between stress and deformation. This allowed him to retrieve Navier's equation of equilibrium as the particular case for which K' = K". The two-constant theory is the one nowaccepted for isotropic elasticity. 51Cauchy's memoirs on elasticity were written with incomparable elegance and rigor. Forthis reason, and also because of their strikingly modem appearance, they have often beenregarded as the first and final foundation of this part of physics.
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