440 



IX. DYNAMICS OF DEFORMABLE BODIES. 



strain known as a uniform expansion for which the strain -ellipsoid 

 is a sphere and the dilatation <? = 3s. 



The part (b) of the pure strain represents a strain which, like 

 a rotation, is unaccompanied by a dilatation, but which differs from 

 a rotation in that it involves a change of form. We shall consider 

 it, as we have the part (a) in three parts. 

 . In the first, 



u = 



v = g z x, w = 



G 



every point is shifted in the X direction through a distance pro- 

 portional to its distance from the XZ-plane, while it is shifted 

 in the Y direction through a distance proportional to its distance 



from the YZ- plane. Points at 

 unit distance from the two 

 named planes are shifted both 

 ways by the same amount g x , 

 so that the new positions of 

 the planes XZ, YZ make 

 with the old angles whose 

 tangents or sines are equal 

 to g z . 



The square OACS (Fig. 145) 

 becomes the rhombus OA'C'B', 

 which is symmetrical about 

 the diagonal OC bisecting the 

 angle XOY. The diagonals 

 AS and OC maintain their 

 directions unchanged, and are 

 accordingly two of the axes of the strain, the axis OZ being the 

 third. The stretch -ratio along OC is 



Fig. 145. 



OC'-OC CC' 



OC 



OC 



C"C 

 BC 



SB' 

 OB 



as may be seen by inspection of the figure. The stretch along the 

 perpendicular axis OE is negative, 



OE'-OE EE' EE" 



OE 



OE 



AE 



The stretch along the ^-axis is zero. Accordingly the sum of the 

 three stretches along the axes of the strain is zero. Such a strain, 

 involving a distortion but no expansion and depending upon a constant #, 

 is called a simple shear. The plane of the shear is the plane parallel 

 to which all points are displaced, in this case the XY- plane. 



A shear may be defined as a stretch along one axis combined 

 with a squeeze of equal magnitude along a perpendicular axis, and 



