STRAINED ELASTIC SOLIDS 



399 



and the strain is "heterogeneous." Nevertheless, on account 

 of the proviso mentioned above, we can regard the strain as 

 being homogeneous throughout any assigned physically small 

 element of volume. If the length actually contracts, the 

 extension du/dx' is negative. 



As another simple example consider again the case in which 

 all particles are displaced parallel to OX', but now taking the 

 displacement to be a function of y', the distance of the particle 

 from a plane parallel to which the displacement takes place. 

 Now choose Q', the neighbor of P', to be a point such that 

 P'Q' is perpendicular to the direction of the displacements. 



M Q 







Fig. 1 



Thus if x', y', z' are the coordinates of P' and x, y, z are the 

 coordinates of its displaced position P, 



X = x' + u(y'), y = y', z = z'. 



Also if x', y' + t]', z' are the coordinates of the undisplaced 

 position Q' of the "neighbor," its displaced coordinates are 



x' + u{y' + 7,0, y' + -n', z'- 



The displacement P'P is u{y') and the displacement Q'Q is 

 u{y' + r]') or u{y') + (du/dy'W. Hence MQ in Fig. 1 is 



