64 G. P. BECKER — FINITE STRAIN IN ROCKS. 



the masb Avas rigidly supported and that it was subject to no lateral con- 

 straint. Thus great caution must be exercised in applying this or any 

 similar method to geological occurrences.* 



Distortion, on Planes of maximuip Strain. — It has already been pointed 

 out that the planes of maximum strain are not in general undistorted 

 planes. Consider one of the planes making an angles -j- v with the fixed 

 plane and inquire what ellipse on this plane would answer to a circle of 

 unit radius in the unstrained solid. Prior to strain this material plane 

 made an angle 03 with that radius of the sphere which ultimately forms 

 the minor axis of the ellipsoid. This radius also originally made an 

 angle — p- with o x. Hence it is easily seen that the original angle of the 

 plane to o x is 90° ■ — ■ (ro — (i). 



If y is the original position of the extremity of the unit radius drawn 

 on the intersection of this plane with that of x y — 



y = cos(m — //.). 

 If y' is the corresponding ordinate after strain — 



y' = (1 +/) cos ( en— //.). 

 If the altered length of this radius is denoted by D, its value is given by — 



n „// -; t _i_ ^ ( l+,f) cog (ro — ,a) 



JJ = V Sill (OJ + v) == - — -. : r 



J ' v J sin {cu + v) 



It is easy to see that D is in general less than unity. Were the strain 

 plane and undilational, D would he unity, and this case is realized in 

 simple scission (or shearing motion), Avhieh may be tolerably frequent 

 among rocks. For a simple shear D would also he unity, but this is a 

 strain probably seldom realized. Whenever a compressive strain is 

 accompanied by two shears the radius in question undergoes contraction 

 and is less than unity. 



On the other hand, the unit radius parallel to o z is elongated to 

 1 +p/ = l + e=Cby an inclined force. 



Thus the ellipse on a plane making an angle o> -f- v with ox, whose 

 major axis is C'and whose minor axis is D, corresponds to a circle in the 

 original mass. The strain involves a compression in the direction of 

 M -f- v and an elongation in the direction o z. 



* La'eral Constraint.— [fa mass were no! only supported on a rigid foundation but confined by rigid 

 walls perpendicular to th,e fixed foundation and parallel to the horizontal component of the force, 

 the strain is also easilj calculated on Poisson's hypothesis. Evidently g = 0, and it is easj w>see 

 that /= — 3e. Of course, 6 retains the same value as if there were no lateral constraint. 1 am no! 

 aware that any particularly interesting results arise in this case, which differs from plain undila- 

 tional strain only in the fact that there is cubical compression. It applies t" Mr Sharpe's theory of 

 slaty cleavage. 



