STRAINED ELASTIC SOLIDS 



423 



To obtain them we visualize a very small rectangular parallel- 

 opiped (Fig. 6) of the medium in the state of strain which has the 

 point P at its center. It is bounded by six rectangular faces 

 parallel in pairs to the planes of reference OYZ, OZX, OXY. 

 The local axis of x through P cuts one face parallel to OYZ in a 

 point Q and the other in a point U, such that PQ = PU = ^, 

 the coordinates of Q being x -\- ^,y,z and oiU,x — ^, y, z. The 

 local axes of y and z each cut two faces, in the points R, V and S, 

 W, respectively, RV being equal to 2??, and SW to 2^. Thus the 

 volume of the parallelopiped is 8^??^ , its sides being 2^, 2?/, 2f and 

 its faces having the areas 477^, 4f^, 4^??. Let Xx, ... ^z be 

 the values of the "stress-constituents" at P. At Q they are 



At U they are 

 ^^ ~ bx ^' 



Fig. 6 



dXy 



dXy 



dZz 

 dx 



dZz 



and similar formulae give the values at R, V, S, W. If we 

 assume the values at Q to be the average values over the face 

 containing Q, then the medium outside the parallelopiped exerts 

 a pull on it across this face in the direction of OX of amount 





477f, 



