444 IX. DYNAMICS OF DEFORMABLE BODIES. 



A combination of two simple shears in planes at right angles 

 obtained from 52 b) by putting g z = 0, has the elongation quadric 



71) g x yg -f g y xg = 0, 

 which breaks up into the two planes 



72) # = and g x y 



at right angles to each other. It is to be noticed that the cone of 

 the resultant of three simple shears in mutually perpendicular planes 

 does not so break up. 



We have seen that we require nine constants to specify the 

 general homogeneous strain, of which three belong to the rotation, 

 six to the pure strain. Let us consider the number of data required 

 to specify a simple pure strain. To specify a uniform dilatation we 

 require only the constant of dilatation tf; for a simple stretch, the 

 direction of the axis, involving two data, and the magnitude of the 

 stretch, making three in all; for a simple shear, four data, the 

 magnitude of the shear, two to fix the plane of the shear and one 

 additional for an axis. Consequently we may always represent a 

 general strain as the resultant of three simple expansions, or of two 

 simple shears and a uniform dilatation. 



169. Heterogeneous Strain. If the displacements are not 

 given by linear functions of the coordinates, the strain is said to be 

 heterogeneous. In this case we may examine the relative displacements 

 of two neighboring points. Let the coordinates of the first point P 

 be before the strain x, y, z, and after it x -f u, y + v, z + w, and 

 those of the second, , be before x + f, y + g, + h, and after 

 % + f -\- u', y 4- g -4- v\ z + h + w 1 . If the point Q be referred to P 

 as an origin both before and after the strain, it has as relative co- 

 ordinates before f, g, h, and after f -f u' u, g + v f v, h -f w' w, 

 so that the relative displacements are u' u, v' v, w f w. Now 

 u, v, w may be any functions of the coordinates x, y, 3 of P, but 

 they must be continuous, otherwise the body would be split at 

 surfaces of discontinuity. Accordingly u\ v',w* being the values of 

 u, v, w for x -\- f f y -\- g, % -\- h may be developed by Taylor's theorem, 

 so that, neglecting terms of order higher than the first in /*, #, li 



i ~c-u . du . ^ du 



u u = f-x \- g- h -o-> 

 1 dx ' y oy ' dz 



rroN f r ^V , fiv . * dv 



73) v ' v = fjr- + g^r -f h -5-9 



1 dx ' y dy ' dz 



3w , 7 dw 



