436 



IX. DYNAMICS OF DEFORMABLE BODIES. 



The determinant of the substitution is skew symmetrical. 

 The ratio of expansion is 



37) 



V 

 ~V 



Ado , 



Ado , 1 



which is equal to 1 plus terms involving do 2 which are negligible, 

 agreeing with the result that rotation as a rigid body is unaccompanied 

 by change of volume. 



167. General Small Strain. We shall now consider small 

 strains in general, that is, strains in which the displacements of all 

 points are small quantities whose squares and products may be 

 neglected. 



Let the components of the displacements or shifts be 



38) 



x x = u, y y = v, s z = w, 



so that if we now write the coefficients of the strain as 



39) y' = \x 4- (1 - 



we have the shifts given by the substitution 



a. 2 y 



40) 



w= 



and if the a's, &'s and c's are small, u, v, w will be small quantities 

 of the same order. The ratio of dilatation is by 30) 



41) = 



The quantity 



42) 



terms of higher order, 



1 4- &i 4- &2 + c s + terms of higher order. 



IT! -\r 



is the increment of volume per unit volume, and will be referred to 

 simply as the dilatation and denoted by (5. 



