LIMITING STRENGTH OF ROCKS UNDER STRESS 133 
external pressure that the maximum stress-difference in the neigh- 
borhood of the cavity is 3p. This result is independent of the 
elastic constants and of the radius of the cavity. If then the 
pressure in the medium exceeds } the limiting stress-difference of 
the rock, the cavity will break down. The value of S given in 
(24) for the limiting stress-differences may be employed provided 
the radius of the cavity be small. 
(iit) On the Stability of a Spherical Cavity in a Medium under 
Pressure 
That the result quoted in (ii) should be independent of the radius 
of the cavity seems contrary to experience. The explanation of 
this anomaly lies in the fact that the question turns on an exami- 
nation of stability rather than on a purely static consideration 
of stress. A problem of a similar kind is that of the collapse of a 
thin-walled cylinder (e.g. a boiler flue) under external pressure. 
The result shows that for a cylindrical shell of infinite length collapse 
will not occur as long as the pressure does not exceed 
4h (t\3 
p= tt () (25) 
where # and ¢ denote the rigidity modulus and Poisson’s ratio for 
the material and t/a is the ratio of the thickness to the diameter. 
The vibrations in the neighborhood of a spherical cavity in an 
infinite medium under hydrostatic pressure do not appear to have 
been explicitly worked out; the problem would correspond to the 
external solution of the problem of the vibrations of a solid elastic 
sphere. In so far as the stability depends on the radial vibrations 
of the cavity, a solution could easily be adapted from Poisson’s 
solution for the case of a solid elastic sphere. The result of an 
investigation on stability would give a result analogous to (2 5) for 
a cylindrical shell. When the pressure p is specified the condition 
of stability will lead to a limiting radius which will be determined 
by the elastic constants of the medium. A cavity having a greater 
ULove, op. ctt., 530. 
2 Cf. Love, “‘ Gravitational Stability of the Earth,” Phil. Trans., 207 A, 1908. 
3 Poisson, ‘“‘ Mémoire sur équilibre et le mouvement des corps élastiques,”’ Mém. 
Paris Acad., t. 8, 1829. 
