400 



RICE 



ART. K 



{du/dy')r}' and the angle QPM has for its trigonometrical 

 tangent the value du/dy'. The figure shows that this strain is 

 what is called a "shear." A bar shaped element of volume 

 which is extended parallel to the axis OZ' (perpendicular to the 

 plane of the paper) and whose section by the plane OX'Y' is 

 P'Q'R'S' (Fig. 2), is displaced to a position whose section is PQRS. 

 This is equivalent to a simple displacement of the bar as a 

 whole from P'Q'R'S' to PMNS and a real strain or change of 

 shape from PMNS to PQRS. This latter is the "shear" and 

 its magnitude is measured by the tangent of the angle QPM 

 (or simply by the angle itself when the strain is so small that the 

 tangent of the angle and its radian measure are practically 

 identical), i.e., by du/dy'. If w is a linear function of y', the 



O'MO R'NR 



Fig. 2 



shear is homogeneous throughout the body; otherwise it is 

 heterogeneous and the amount of shearing varies from point to 

 point of the body. 



When we undertake a general analysis of strain these special 

 cases give us a hint how to proceed. The point P' whose co- 

 ordinates are x', y', z' experiences a displacement whose com- 

 ponents we represent by u{x' ^ y', z'), v{x', y', z'), w{x', y', z'), 

 for the displacement must have some functional relationship 

 with the position of P' if analysis is to be possible at all.* 



* Will the reader please note that we are, for the time being, referring 

 the body before and after the strain to the same axes OX', OY', OZ'. 

 Formally Gibbs' procedure is a little wider since he refers the body after 



