786 Dr. Chree on the Stresses in the Earth's Crust 



of stress. Further, if any calculation is to be made of the 

 internal stresses, there seems no alternative to the application 

 of elastic solid theory. The following results based on the 

 ordinary mathematical theory may thus be worth the con- 

 sideration of engineers and others interested in the practical 

 problem. 



I propose first to present the results applicable to a series 

 of mathematical problems, only one or two of which are abso- 

 lutely novel, and then to consider what their bearing is on 

 the problem now under discussion. Only the general 

 character of the mathematical steps will be indicated. 



§ 2. The two best known theories as to rupture, or more 

 strictly as to the limits of application of the elastic solid 

 theory, are : — 



(i.) The maximum stress difference theory, according to 

 which S the greatest value of S, where S is the difference 

 between the algebraically greatest and least of the three 

 principal stresses at a point, must not exceed a certain 

 experimental limit ; 



(ii.) The greatest strain theory, according to which the 

 limiting value attaches to the largest strain in any part of 

 the structure, assuming that to be an extension. 



Theory (i.) seems that least favourable to the permanence 

 of the structure in such problems as those to be considered 

 here, and attention is almost exclusively devoted to it. It 

 has the recommendation that there seems considerable experi- 

 mental evidence in favour of the view that it is the maximum 

 stress difference on wdiich depends the tendency to flow in 

 solids under severe non-uniform pressure *. 



flotation. 



§ 3. In isotropic material m and n represent the two elastic 

 constants in Thomson and Tait's notation. These are con- 

 nected with Young's modulus E, and Poisson's ratio v, by 

 the relations 



m/l=n/(l-2,)=E-i-{2(l + 1? )(l-2,)}. . . (1) 



In applications of spherical coordinates, r, 6, <£, the displace- 

 ment along the radius vector is denoted by u ; in most of the 



cases treated here the three principal stresses are the radial rr, 



and the two orthogonal stresses 66 and cjxj). Also <£(£ in the 



cases treated, with the exception of § 6, is equal to 66. The 

 three principal strains are then du/dr, u/r, and u/r. 

 * Todhunter and Pearson's ' History of Elasticity/ vol. ii. art. 247, &c 



