STRAINED ELASTIC SOLIDS 427 



in a right-handed sense; that of the second is 



( 



aZr \ 



also in a right-handed sense. That of the third is 



/ dYz \ 



in a left-handed sense and that of the fourth is 



also in a left-handed sense. Thus the four shearing tractions 

 yield a couple around the local axis of x in the right-handed 

 sense of amount 



S^vUZy - Yz). 



Turning now to the body forces we see that even if their action 

 on the element is not symmetrical about P (as would be the 

 case for example with gravity forces) they can yield in com- 

 parison with the moments arising from the shearing forces only 

 a vanishingly small couple, since about the local axis of x, for 

 instance, this couple must have an order of magnitude which 

 cannot be greater than the product of Fy, 8^r/f and i;, or Fz, 

 8^7?^ and 77. Since ^ry^^ or ^tj^^ is small compared to ^i)^ when 

 ^, rj and ^ are small, these contributions are evanescent in 

 comparison with that written above, when the volume con- 

 sidered is small. Thus the total couple on the parallelopiped 

 has components around the three axes given by 



{Zy - F^)8^r,r, {Xz - ZxM-n^, {Yx - Xy)S^-n^. 



But in equilibrium these components must be zero, and so 



Yz = Zy, Zx = Xz, Xy = Yx' (30) 



This demonstrates that there are only six independent strain- 

 constituents, as already stated. 



