28 G. F. BECKER FINITE STRAIN IN ROCKS. 



By substituting x = x — 2 s^y and // = y these become the equations of 

 a simple shear at an angle of 45° to o x. Finally — 



x' = x — 2 s^, ]( = y, 



are the equations of a simple scission. 



Thus the most general plane undilational strain is resoluble into an 

 axial shear, a shear at 45° to o x, and a scission in the direction of one of 

 the axes. 



When a= b this general strain reduces to a single shear. Jib I a = 

 (1 4- e) I (1 +/) = ( >-i the strain reduces to two shears or, in other words, 

 the scission vanishes. If a = and (1 + e) / (1 4- /) = « 3 2 the strain is an 

 axial shear combined with a scission.* 



*A second Resolution.— The above method of resolution is the most convenient for computation, 

 but it fails to disclose a relation of much geological significance. It is a fact that any plane undila- 

 tional strain is resoluble either into two shears at an angle # or into a shear and a scission at an 

 angle <f> The significant difference between these two combinations is that the two shears cause a 

 relatively small rotation which is an infinitesimal of the second order when the strain is infinitesi- 

 mal, while the shear and scission produce a large rotation which is of the same order as the strain 

 when this is infinitesimal. The criterion discriminating the two classes of strains is exceedingly 

 simple. When a and b have the same sign the strain is invariably equivalent to two shears. When 

 a and 6 have opposite signs the strain is invariably equivalent to a shear and a scission. As in the 

 case of the other resolution, it is easiest to discriminate changes of notation from equations of 

 condition synthetically. 



Let a and b have the same sign. Then to show that the strain is compounded of two shears one 

 may proceed as follows : Adopt the notation — 



, (!+/) +«(! + «) . Sin2& = -Yba . 



a^ 



2 j/ ab s 2 a 



Each of these expressions is possible whenever a and b have the same signs, and then only. In 

 addition, the condition of piano undilational strain is (1 + e) (1 +/) — ab = 1. Here, then, is a 

 number of equations just sufficient to determine a, 6, e and/. Remembering that 1 + e and 1 +/ 

 are necessarily positive, they give — 



a- **"* ; b=-a s s 2 sin2&; 1 + e = a 8 (<r 2 - S 2 cos 8*) ; 1 +/= *' + Sa - cos 2 *. 



Is is easily seen that these values answer to an axial shear of ratio a a and a second shear of ratio 02 

 at an angle # with x. 

 Let a and b have opposite signs. This is implied in the expression— 



1 _|_ y! —at, 



and the condition of plane undilational strain is (1 f e, (I +/) — ab = 1. Purely notative are the 

 following: 



a 3 ■ a 3 Si 



These four equations give— 



ffi = Si (lTcos2<fr). & = _ a3 . Sl ( 1±C0S 9^) ; i + c = a;j (i +Sl sin24>); 1 +/= 1 ~ s l 8iw2<fr . 

 0-3 "3 



These values answer to an axial shear of ratio a 3 and a scission of ratio a^ The direction of the 

 scission makes an angle <f> with ox if the given values of a and b arc satisfied by choosing the upper 

 sign in these expressions. In tin' opposite case the direction of the scission makes an angle 

 with o y. 



