MAXIMUM STRAIN. 31 



The circular sections of the shear ellipsoid for which the ratio is a make 

 an angle with the major axis whose cotangent is a* If this angle is 

 called m, the amount of shear is — 



2 s = a — «-' = cot w — - tan us = 2 cot 2 us = 2 tan 1 90° — 2 us). 



Here s, or half the so-called amount of shear, appears as measured by 

 the divergence from 90° of the angle 2 us between the circular sections of 

 the shear ellipsoid. A right angle is the value which 2 m assumes when 

 the strain is infinitesimal. 



The original position of the particles constituting the planes of no dis- 

 tortion, relatively to the fibers which coincide with the axes of the ellipse, 

 bears a simple relation to en. Suppose the shear to be axial and that the 

 sphere .>','' + V\ + z i = ^' i s converted into the ellipsoid x 2 /a 2 -f y 2 <r -f 

 z 2 = /r, so that yj ->\ = "'///•'7 then the original position of the material 

 plane forming the circular section of the shear ellipsoid was d l tan us = 

 „.= fan (90° — US). 



Thus these material planes made before shear the same angle with the 

 minor axis of the ellipsoid which they make after strain with the major 

 axis. 



Planes of maximum Strain. — It is instructive to regard the planes of no 

 distortion from another point of view. Consider any two very thin plane 

 layers in the unstrained mass which include between them the axis o z, 

 and let the angle which they make with, ox be <p. After strain these 

 planes will still be planes; they will make an angle cc' with ox and 

 hni <p = o?- tan <p' or 



, / ,n (a? — 1) tan a 



Ian ((f — w ) = V— ' — . — H. 



a? + tan 2 if 



The greater the angle cr — y' becomes, the greater must be the tangential 

 strain. Now this angle and its tangent are greatest when tan <p — a or 

 when tan <p r = 1/ a = tan us. Thus the undistorted planes arc those for 

 winch tangential strain is a maximum. For the axes, on the other hand. 

 <p — <p' = 0, and there is no tangential strain. 



Angular Range of undistorted Plane*. — Though at the end of a shear or 

 other plane strain there are planes which have the same dimensions as 

 before strain, it is not true that these planes have under-one no distor- 

 tion. On the contrary, there is but one strain in which any lines escape 



* The intersections of the shear ellipse with the circle of equal area are points in these sections, 

 since the radii of the ellipse retain their original length, say unity. These intersecl ions are given 

 by- 



a- 



whence a = ± x I y. 



VI— Bull. Oeol. Soc. Am., Vol. 4, 1892. 



