| Xx = | ∂²φ3 | + | ∂²φ2 | , ..., Yz = − | ∂²φ1 | , ... ; |
| ∂y² | ∂z | ∂y∂z |
and it is clear that the functions φ1, φ2, φ3 can be adjusted in an infinite number of ways so that the bounding surface of the body may be free from traction.
93. Initial stress due to body forces becomes most important in the case of a gravitating planet. Within the earth the stress that arises from the mutual gravitation of the parts is very great. If we assumed the earth to be an elastic solid body with moduluses of elasticity no greater than those of steel, the strain (measured from the unstressed state) which would correspond to the stress would be much too great to be calculated by the ordinary methods of the theory of elasticity (§ 75). We require therefore some other method of taking account of the initial stress. In many investigations, for example those of Lord Kelvin and Sir G.H. Darwin referred to in § 83, the difficulty is turned by assuming that the material may be treated as practically incompressible; but such investigations are to some extent incomplete, so long as the corrections due to a finite, even though high, resistance to compression remain unknown. In other investigations, such as those relating to the propagation of earthquake shocks and to gravitational instability, the possibility of compression is an essential element of the problem. By gravitational instability is meant the tendency of gravitating matter to condense into nuclei when slightly disturbed from a state of uniform diffusion; this tendency has been shown by J.H. Jeans (Phil. Trans. A. 201, 1903) to have exerted an important influence upon the course of evolution of the solar system. For the treatment of such questions Lord Rayleigh (Proc. R. Soc. London, A. 77, 1906) has advocated a method which amounts to assuming that the initial stress is hydrostatic pressure, and that the actual state of stress is to be obtained by superposing upon this initial stress a stress related to the state of strain (measured from the initial state) by the same formulae as hold for an elastic solid body free from initial stress. The development of this method is likely to lead to results of great interest.
Authorities.—In regard to the analysis requisite to prove the results set forth above, reference may be made to A.E.H. Love, Treatise on the Mathematical Theory of Elasticity (2nd ed., Cambridge, 1906), where citations of the original authorities will also be found. The following treatises may be mentioned: Navier, Résumé des leçons sur l’application de la mécanique (3rd ed., with notes by Saint-Venant, Paris, 1864); G. Lamé, Leçons sur la théorie mathématique de l’élasticité des corps solides (Paris, 1852); A. Clebsch, Theorie der Elasticität fester Körper (Leipzig, 1862; French translation with notes by Saint-Venant, Paris, 1883); F. Neumann, Vorlesungen über die Theorie der Elasticität (Leipzig, 1885); Thomson and Tait, Natural Philosophy (Cambridge, 1879, 1883); Todhunter and Pearson, History of the Elasticity and Strength of Materials (Cambridge, 1886-1893). The article “Elasticity” by Sir W. Thomson (Lord Kelvin) in 9th ed. of Encyc. Brit. (reprinted in his Mathematical and Physical Papers, iii., Cambridge, 1890) is especially valuable, not only for the exposition of the theory and its practical applications, but also for the tables of physical constants which are there given.
(A. E. H. L.)
[1] The sign of M is shown by the arrow-heads in fig. 19, for which, with y downwards,
| EI | d²y | + M = 0. |
| dx² |
[2] The figure is drawn for a case where the bending moment has the same sign throughout.
[3] M0 is taken to have, as it obviously has, the opposite sense to that shown in fig. 19.