ON THE MATHEMATICAL THEORY OF THE INTERNAL 

 FRICTION AND LIMITING STRENGTH OF ROCKS 

 UNDER CONDITIONS OF STRESS EXISTING IN THE 

 INTERIOR OF THE EARTH 



LOUIS VESSOT KING 

 McGill University, Montreal 



INTRODUCTION 



That solid bodies could be permanently deformed and made to 

 flow without rupture under sufficiently great stress has long been 

 known. The extensive experiments of Tresca on the flow of metals 1 

 (1864-72) directed the attention of several mathematicians of the 

 time to the subject. Tresca announced as a result of his experi- 

 ments the simple law that a stressed solid would commence to flow 

 as soon as the maximum shearing stress exceeded a limiting value K 

 characteristic of the solid. This hypothesis was incorporated into 

 the elastic solid theory by Saint- Venant 2 and others. The hope 

 was expressed by these writers that by effecting the solution of 

 simple problems in "plasticodynamics," corresponding to the 

 experimental arrangements employed, it might be possible, not 

 only to verify the theoretical results, but also to determine a specific 

 constant K characteristic of the various metals and related in an 

 intimate manner to other physical constants. It was found pos- 

 sible, however, to solve only a very limited number of extremely 

 simple problems: (1) circular cylinder under uniform pressure over 

 the plane ends or subject to uniform lateral pressure ; (2) cylindrical 

 shell constrained to remain of constant length and subject to uni- 

 form internal and external pressure; (3) circular cylinder twisted 

 beyond the elastic limit; (4) bar of rectangular section bent by a 

 suitable distribution of forces to take the form of a circular arc. 



1 H. Tresca, Par. Mem. Sav. Etr., XX (1872), 75 ff. and 281 ff. A summary of 

 Tresca's experiments is given by L. S. Ware, Journal of the Franklin Institute, LXXIII 

 (1877), 4i8 f. 



2 Saint-Venant, Comptes Renins, LXVII (1868), 131 ff., 203 ff., 278 ff.; LXVIII 

 (1869), 221 ff., 290 ff. * 



638 



