24 G. F. BECKER FINITE STRAIN IN ROCKS. 



mentary strains. This will be demonstrated by a proof that any relation 

 whatever between the axes A, B and C of the ellipsoid whose volume 

 is proportional to h 3 can be brought about by two such shears. Let 

 A = ha, B = hy, and let C = hp, where A and B are entirely arbitrary. 

 Then since ABC = h 3 = BJr<i..\, it is evident that 1 / ap = y, or B = hj ap. 

 Now, if a shear of ratio a is applied axially in the x y plane to the sphere, 

 ^ + V~ + z? ' = h 2 , if will reduce this mass to the ellipsoid .r / <r + i/V + 

 2 2 = K 1 . If a second shear of ratio ft is applied axially in the y z plane 

 it will further reduce the second axis in the ratio p and elongate the 

 third axis in the same ratio. Thus the two shears yield an ellipsoid 

 %* I a 2 -p T/VyS 2 -f z 1 j p 1 = /r, and the axes of this ellipsoid are ha } hj aft and 

 hft, or A, B and C. 



A converse proposition is also important. Any number of shears 

 applied axially to a sphere can 011I3 7 modify the relations of the axes to 

 values A, B and C, the volume of the mass remaining proportional to 

 ABC= h 6 . Hence any number of axial shears are reducible to two and 

 not to three, as one might be inclined to surmise. This resolution may 

 take place mathematically with any one of the axes as the common axis 

 of the two shears. In most cases, however, considerations of symmetry 

 point to one of the axes as that common to the two shears. 



A simple shear produces relative motion of particles or fibers only in 

 its own plane. Its only effect on fibers in planes at right angles to its 

 own is to elongate them uniformly in one direction without any tendency 

 to the causation of relative motion. Hence .the effects of each shear must 

 be considered in its own plane, and the relative motion produced by each 

 of two shears in orthogonal planes is independent. 



Shea ring Motion or Scission. — A " shearing motion" is the rather ill-chosen 

 designation of a strain nearly corresponding to that which occurs when 

 a bar or plate is shorn by a pair of shears, or when a rivet yields perpen- 

 dicularly to its axis, say, in a bursting boiler. The term is not happy, 

 because it seems to indicate that there are shears not accompanied by 

 motion. It is, of course, from this strain that the term shear was de- 



