FRICTION AND LIMITING STRENGTH OF ROCKS 



647 



sion of this hypothesis due to Navier (the so-called internal-friction 

 theory) replaces (12) by the condition 



S<K+ixN, 



H being a new constant somewhat analogous to the coefficient of 

 friction of mechanics. In order to 

 discover the relation between the 

 principal stresses at the elastic limit, 

 it is necessary to find the direction 

 (I, m, n) which makes (S— /xN) a 

 maximum and equate the result to 

 K. Suppose the principal stresses to 

 be all of the same sign, two of 

 them equal, yy = xx, and zz>xx 

 (corresponding to the state of 

 affairs in the cylindrical rock speci- 

 mens under test). We then have, 



Fig. 3 



writing /=sin 6 cos <$>, m= sin 6 sin <f>, n = cos 6, 



S 3 +N 2 = xx 2 sin 2 6+zz 2 cos 2 0, N = xx sin 2 0+zz cos 2 \ , . 

 S = (zz—xx) sin 6 cos J 



S — pN=(zz—xx) sin cos 6— fi(xx sin 6 -\-zz cos 2 6) (14) 



This expression reaches a maximum when 



cot 20= —/a, 

 in which circumstances 



(S — pN) ma x. = \{zz cot 6— xx tan 6), 



(i5) 



(16) 



and the relation between the principal stresses at breakdown is 

 given by 



zz = 2K tan 6-\- xx tan 2 6 (17) 



where 6 is given in terms of jx (the coefficient of friction) by (15). 

 This result indicates that the material in question will break down 

 along a family of cones of semivertical angle a = ^7r— 6. 



2. Discussion of observations. — In the experiments of Adams and 

 Bancroft the cylindrical rock specimens were subjected to end loads 

 transmitted by the steel pistons. As a result of the intense pressure 



