FRICTION AND LIMITING STRENGTH OF ROCKS 645 



When a specimen of Carrara marble is inserted we have 1 a — o . 2744, 

 /x=3.i54Xio 6 pounds per sq. in., whence /?= 1.889 and 



U'/c = o.i76X(P/n'), or 2V (inches) = 3. 3iXio~ s XW (pounds) (8) 



In another set of experiments (referred to as 0.33-centimeter 

 wall), 6=1.00 cm., 6=1.33 cm., giving 



i 8=2. 9 6X jr - -,. 



1—CTJX' 



In the case of tallow filling we find as before, 



Z77c = o.Q8oX(P/fO or 2V (inches) = 1.96X10- 7 XPT (pounds) (9) 

 and in the case of the Carrara marble specimen 

 C/'A = o.i47X(P>') or 2V (inches) = 2.94Xio- 8 XTF (pounds) (10) 



In Fig. 2 are compared the observed and theoretical stress-strain 

 diagrams corresponding to the cases calculated out in equations 

 (8) to (10). In the case of tallow filling, the initial slope of the 

 observed curves agrees approximately with the calculated slope. 

 In the case of the marble filling, the agreement is within the limits 

 of error involved in measuring these extremely small strains. 



MATHEMATICAL DISCUSSION OF THE OBSERVATIONS OF ADAMS AND 

 BANCROFT DURING THE PLASTIC STAGE 



i. Navier's theory of internal friction. — Let xx, yy, and zz be the 

 principal stresses in the solid at a point P measured toward the 

 origin (Fig. 3). Let S be the shearing ^stress in a plane whose 

 direction cosines with respect to the direction of the three principal 

 stresses are (/, m, n). Let N be the stress normal to this plane. 

 We then have 



S 2 -\-N 2 = I 2 'xx 2 + m 2 yy 2 + n 2 zz 2 I 

 and J (11) 



TV = l 2 xx-\- m 2 yy-\-n 2 zz 1 



Generalizing somewhat on Navier's hypothesis of elastic break- 

 down, we may state that the material will not break down as long as 



S<K (12) 



1 Adams and Coker, "Elastic Constants of Rocks," Publication No. 46 of the 

 Carnegie Institution of Washington, 1906, p. 69. 



