FRICTION AND LIMITING STRENGTH OF ROCKS 641 



calibration-curve, just as the copper cylinders of the crusher-gauge 

 are calibrated under known end pressures in a testing machine. 



MATHEMATICAL DISCUSSION OF THE OBSERVATIONS OF ADAMS AND 

 BANCROFT DURING THE ELASTIC STAGE 



Although the experiments which form the subject of the present 

 discussion were all carried out when both rock and nickel-steel had 

 been deformed beyond the elastic limit, it is not without interest, 

 especially in view of further experiments on the subject, to follow 

 out the distribution of stresses in the rock specimen and in the 

 nickel-steel throughout the elastic stage. The necessary theory 

 from which the formulas given below are derived has been given by 

 the writer in a previous paper. 1 As in that discussion, it is sufficient 

 for the present purpose to consider the ideal problem of plane stress, 

 that is, one in which the end pressures and lateral pressures are such 

 that the displacements at the outer surfaces, both of the rock 

 specimen and of the nickel-steel jackets, are everywhere symmetrical 

 with respect to the axis and everywhere constant for a given load. 

 In reality the nickel-steel jacket shows a bulge over the center of 

 the specimen. As long as this is not too great the analysis will give 

 an approximate representation of the state of stress in the central 

 portion of the specimen and nickel-steel jacket at which the measure- 

 ments of displacement were taken by means of a sensitive exten- 

 someter. The justification for this mode of treatment has already 

 been noticed in the writer's paper previously referred to in its 

 application to a similar problem. 



We denote by rr the stress component along the radius r; by 68, 

 the component at right angles to r; and by zz, that along the axis. 

 According to Lame's notation, [x is the modulus of rigidity of the 

 rock specimen and X one of the moduli of elasticity such that 

 k= (X+§ju) is the modulus of compression. Poisson's ratio is 

 denoted by <r=|X/(X+/i). We denote by accented symbols the 

 corresponding elastic constants for nickel-steel. In the problem 

 under discussion we denote by b the radius of the rock specimen and 



1 L. V. King, "On the Limiting Strength of Rocks under Conditions of Stress 

 Existing in the Earth's Interior," Journal of Geology, XX (February-March, 1912), 

 121-26. 



