22 G. F. BECKER FINITE STRAIN IN ROCKS. 



In any case whatever one may express the axes A and C under the 

 forms A = A«, C = hp where « and p may be perfectly independent. 

 Then, since ABC - h 3 , B =- hjap. The values >*, \jap and p are the values 

 which A, B and C would have were there no dilation, and upon the 

 properties of « and p depend those of pure deformation, accompanied by 

 rotation: 



Shear. — A shear is the simplest possihle deformation. It may be de- 

 nned as an irrotational strain, unattended by dilation, in which one axis 

 of the strain ellipsoid retains its original length. The unit sphere is thus 

 converted into an ellipsoid, the axes of which are «, 1, l/«; and a is called 

 the ratio of shear. It is taken as greater than unity, excepting when it 

 is dealt with as an unknown quantity. 



In dealing with shears it is convenient to employ the following ab- 

 breviations : * 



2 S = « — u~ l ) 2 t = a -j- a -1 . 



These forms imply that * 2 — s 2 = 1. 



The displacement formulas for a shear, the contractile axis of which 

 makes an angle fl with ox are — 



%' = x (t — s cos 2 #) — ys sin 2 # ; i/ = y (a -f- s cos 2 #) — xs sin 2 # ; z* —z. 



To verify this statement consider that a = b, so that there is no rotation ; 

 (j = and (1 + c) (1 +/) — a0 =1) so that there is no dilation; tan 

 Q, -j- //) = tan 2 v = tan 2 #, showing that the axes of the strain ellipsoid 

 make angles »9 and > ( > 4- 90° with o x; finally b (1 +/) + a (1 + e) is nega- 

 tive, so that the minor axis of the strain ellipsoid makes an acute positive 

 angle with o x as required. 



When # = 90° these equations reduce to — 



x' = xa; y' = yja; z' = z, 



and when & = 45°, a case of importance, 



x' = xff — ys ; y 1 = ya — as ; 2' = 2. 



The quantity 2 s is called the amount of the shear. There are various 

 aspects of this quantity. One way of looking at it is as the sum of two 



distortions. The elongation of the major axis is «— 1 and the contrac- 



— • ■ 



*Let a = cot "co; then it is easy to see that o- = l/sin 2 td and s = cot 2 to. Here, as will be show n 

 later, 2 "to is the acute angle between the circular sections of the strain ellipsoid. The convenience 

 of s and a- depends upon this fact, and the significance of the formulas is increased by bearing it 

 in mind. The quantities s and a may lie regarded as hyperbolic sine and hyperbolic cusint "fan 

 area \J< = In a ; and then !mi° — ■ 2 "to is the corresponding transcendental angle. This view of the 

 functions, however, is not needful for the purposes of this discussion. 



