MAXIMUM STRAIN. 33 



When strain'begins the major axis of the shear ellipse makes an angle, 

 v , with ox, and the undistorted planes then make an angle of 45° with 

 the major axis or angles v ± 45° with ox. When the strain is complete 

 the major axis makes an angle, v, with ox, and the undistorted planes 

 make angles co with this axis. But before strain began this last axial 

 fiber "made an angle, //., with ox, and the particles constituting the last 

 undistorted plane then made an angle, 90° — to, with >).. Thus in the 

 undistorted mass the angles bounding the wedge through which the cir- 

 cular sections will sweep are v ± 45° and <>. ± (90° — ro). 



On the side of the minor axis toward which rotation takes place this 

 range is therefore — 



v 4- 45° - | n + 90° - ro 1 = ro - 45° + i^A 



and on the opposite side of the minor axis the range is — 



| ,, _ (90° - en) } - (v — 45°) = ro - 45° - V JZ^. 



The difference of range is thus the angle of rotation, and is actual when- 

 ever the strain is a rotational one. 



In a simple shear, then, there is no difference in range, and the range 

 on each side is en — 45°. In the case of scission or shearing motion it is 

 easy to see that 2 (w — 45°) = v — //., so that the range is zero on the side 

 from which rotation takes place, and one and the same set of fibers are 

 exposed to maximum tangential strain throughout the process of strain, 

 while the other circular section sweeps through the maximum possible 

 angle. In any case of plane strain the difference in range is at once 

 assigned by the angle of rotation, so that for two shears in the same 

 plane at an angle of 45° the difference is measured by tan (v — //) = 

 ss 1 \o<j v 



For plane strains the value of en may be simply expressed in terms of 

 the displacement coefficients. It is easy to see that — 



-j = la a 2 en = B\A. 



Hence also — 



4 AB (1 + e) (1 H - -«/>. (l)] 



tan 2 en = {A _ B y - 4 ^ _j y + (ft + by2 (■>) 



Case of Strain in three Dimensions. — It has been pointed out already 

 that the relative motions of the particles in the xy plane due to a shear 

 a are unaffected by an axial shear p in the B (.'plane. The sole effect of 

 the second shear, so far as the x // plane is concerned, is to change the 

 length of all lines parallel to the common axis of the shears uniformly in 



