or, per unit volume of the little prism a'bc', we require work equa to Κσ. But this quantity must equal the previous K'o or K' = K, the result experimentally ascertained by Tresca. Saint-Venant concludes the note as follows: Ce raisonnement me paraît, aussi, justifier l'hypothèse, hardie au premier aperçu, mais, en y réfléchissant, très-rationnelle, de l'égalité des résistances à l'extension et à la compression permanente, par unité superficielle des bases des prismes qu'on y soumet; bien entendu, sous la condition générale, que tout ceci suppose remplie, de mouvements excessivement lents, ou tels que leur vitesse n'entre pour rien dans les résistances aux déformations qu'ils produisent. In a footnote he refers to a method by which the flow-lines of a plastic material might be obtained experimentally. It must be noted that the proof assumes the coefficients K1, K of resistance to squeeze- and stretch-set to be equal, otherwise we should have 2 K1+ K12 = 2K', The reader may compare Coulomb's results on shearing and tractive strength referred to on p. 877 of our first volume. [237.] Formules des augmentations que de petites déformations d'un solide apportent aux pressions ou forces élastiques, supposées considérables, qui déjà étaient en jeu dans son intérieur.-Complément et modification du préambule du mémoire: Distribution des élasticités autour de chaque point, etc. qui a été inséré en 1863 au Journal de Mathématiques, (see our Arts. 127–152). This memoir is published in the Journal de Mathématiques, Tome xvi. 1871, pp. 275-307, and is divided into two parts; the Première Partie (pp. 275-291) is occupied with correcting an error which Brill and Boussinesq had pointed out in the memoir of 1863 (see our Art. 130); the Deuxième Partie deals with the relations between the elastic constants xxxx, etc. and the six components of initial strain. It occupies pp. 291–307 and forms the subject of a note on pp. 355 and 391 of the Comptes rendus, T. LXXII. 1871, [238.] The error in question was really indicated in our first volume (see Art. 1619*), namely that the true relations between the strains, s, o'y and the shift-fluxions are in their most general form of the types1: ww} ...(i), o'yz (1 + Sy) (1 + Sz) = Vz + Wy +UyUz + VyVz+WyWz but that these are not the values taken by Saint-Venant in his memoirs of 1847 and 1863: see our Arts. 1622* and 130. Accordingly Saint-Venant's attempt to deduce Cauchy's equations from a multi-constant hypothesis is erroneous. The full value of the potential energy is • = Po + xxo (8x + §8x2) + + ...... ...... + ở g (1 + 8y) (1+8)+ + $1 as assumed in the memoir of 1863 (see our Art. 130). But the expression (ii) has been deduced only from molecular considerations on the rariconstant hypothesis. The fact is that we can on the multi-constant hypothesis expand in linear and quadratic terms of the strain-components x, y, Ez, Nyz, Nzx, Nxy of our Art. 1619*, as Green in fact did (Collected Papers, pp. 298-9), but we cannot determine to what extent the resulting coefficients are functions of the initial stress-components. This apparently requires us also to make some molecular assumption. [239.] Starting with expression (ii) for the potential energy, we should arrive at the equations of Cauchy (as Saint-Venant had done in his memoir of 1863 by a double self-correcting error), but we must renounce the hope of arriving at (ii) on the simple assumption of a generalised Hooke's Law. We may note one or two further points in the first part of the memoir : (a) To the second order of small quantities, This was first noticed by Brill: see p. 279 of Saint-Venant's memoir. 1 o'z differs from the σ of our Art. 1621*, it being the cosine and not the cotangent of the slide-angle. See Saint-Venant's definition of slide in Art. 1564*. (b) If we assume that the work-function may be expanded in powers of 8, 8, 8z, σzy, σxz, σy and write then we are throwing a portion of ..(iv), involving initial stresses into ☀,, which thus differs from the 1 of (ii). We thus obtain for the stresses But xx and yz, while being of the same form as Cauchy's x1, *1 [see our Art. 129, (ii)], will in reality have constants increased by the corresponding initial stresses, as is shewn by the rari-constant investigation. Thus : It is the impossibility of determining on the multi-constant theory how these initial stresses occur in the changed values of the constants, which throws us back on rari-constancy for a proof of (ii). Results (vi) combined with (v) convert the latter into Cauchy's formulae: see our Art. 129, (i). [240.] The second part of the memoir deals with the following problem: If xxxx\, \xxxy\, \xxyz\, etc. are the elastic constants when there is an initial state of stress xx, x, etc. it is required to determine these constants in terms of \xxxx\, \xxx\。, \xxx\。, etc. the elastic constants before this initial state of stress. Saint-Venant deals with the problem on rari-constant lines. We have, with abbreviated symbols (see our Art. 143): Further we have, if xo, yo, o be the position of the molecule m relative to a second before the initial strain, u。, vo, wo its shift due to that strain, and x, y, z the relative position after the strain, Here Иход... denote du/dx...., and since the stresses axo, zo are given functions of ux vy...uz...etc., we can express the new coefficients [x1..... in terms of the old ... and the initial stresses. These results are obviously only a more general case of the formulae of our Art. 616*. The following pages 297-304 are concerned with other modes of looking at these results or expressing the stresses in terms of them. [241.] Let us take as a special case that of a bar of primitively isotropic material subjected to a traction xx, there being an initial traction x. We have Thus, 24=3λ(158), 1341=13416=3λ, \x2y2| = \x2x2\ = d (1 + 8∞), \y22 =λ(1-38%), \y3z|=|x2yz|= etc. = 0. Substituting in the traction-type as given by Cauchy's formula, Eqn. (i) Art. 129, we have or or xxo {1 + 8x − 28y} + 3λ (1 − 5 8x) 8x + 2λ (1 + 8x) Sy, yy = 0 = 3λs, + λ (1 + Sx) 8x + λ (1 − 3 8x) Sy. Whence we find from the second equation: 8(4-8)=(1+8x) 8x Sy Substituting in the first we easily deduce Thus if E be the new stretch-modulus, we have E = E - 13 xx。• 0 This shows that a large initial traction can alter to some extent the value of the stretch-modulus. It slightly decreases it. Saint-Venant obtains in our notation but I do not think this result is correct. It would denote an increase of the stretch-modulus. Saint-Venant in fact puts the stretch-squeeze ratio after the initial stress = 1, (thus on p. 305 he writes S Sy -Sa), but it seems to me that this ratio = == (1 + Sxo)/(4 — § Sx。) = − ‡ (1 + ¥ Sxo), and is only =-1/4 when So=xx/E=0, or, when there is no initial stress. The matter is one of theoretical rather than practical interest, for supposing E were 30,000,000 lbs. per sq. inch, it is unlikely that could be at most more than 40,000 to 60,000 lbs. per sq. inch; hence the change in E would not amount to more than 140,000 to 200,000 lbs., or at most to 1/150 of E, which with the want of uniformity in any material is in practice almost within the limits of experimental error. |