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BELL SYSTEM TECHNICAL JOURNAL 



approaches zero, the right hand side of (10) is one order smaller than the 

 left hand side and hence 



T = T 



(11) 



The same argument applies to the other terms. Hence the stress com- 

 ponents of (7) can be written in the symmetrical form 



r. 



T. 



T. 



(12) 



The last form is a short hand method for reducing the number of indices 

 in the stress tensor. The reduced indices 1 to 6, correspond to the tensor 

 indices if we replace 



llbyl; 22 by 2; 33 by 3; 23 by 4; 13 by 5; 12 by 6. 



This last methcd is the mcst common way for writing the stresses. 



1.2 Strain Component, 



The types of strain present in a body can be specified by considering two 

 points P. and ^ of a medium, and calculating their separation in the strained 

 condition. Let us consider the point P at the origin of coordinates and the 

 point Q having the coordinates x, y and z as shown by Fig. 3. Upon strain- 



Fig. 3. — Change in length and position of a hne due to strain in a solid body. 



ing the body, the points change to the positions P', Q'. In order to specify 

 the strains, we have to calculate the difTerence in length after straining, or 

 have to evaluate the distance P'Q'-P Q. After the material has stretched 

 the point P' will have the coordinates ^i , 7?i , f 1 , while Q' will have the 

 coordinates -v + I2 ; v + 772 ; 2 + ^> . But the displacement is a continuous 

 function of the coordinates .r, y and z so that we have 



^2 = ^1 + ^ X + / >' + ^ 3- 

 dx dy dz 



