26 G. F. BECKER FINITE STRAIN IN ROCKS. 



shear coincides with oy, displacing / to .<•" and //' to ,'/". the ratio being <> Y . 

 then the displacement formulas :;: are — 



n -j ji V V 7 '' s 



X = x a. = xa x <j — ya s ; y = 2- = £_ — — 



a, a, a, 



This strain, although the resultant of two irrotational strains, is rota- 

 tional, since a — b is not zero. It is easy to see that this would probably 

 be the case, for the first shear alters the direction of every line excepting 

 those coinciding with its axes, and the direction of these is changed by 



the seeond shear. The rotation is given by — 



tan (y — ft) = ss 1 j ffff u 



where 2 <r, = « x + a^ 1 and 2 s l = a, — a~ x . 



It is an important fact that when the shears are of infinitesimal amount 

 this combination becomes irrotational. When a and a, differ infinitesi- 

 mally from unity, s = e, s t = e 1} a = 1, g x = 1 and tan {? — /-/.) = ec v an 

 infinitesimal of the second order.f 



The two finite shears are equivalent to the rotation stated above and a 

 simple shear of amount — 



2 \/ t"s{ + ^V. 



Plane undilational Strain. — The most general strain treated in this paper 

 may be considered as a perfectly general undilational strain in one plane, 

 combined with a shear at right angles to this plane and a dilation. The 

 more complex effects are confined to the principal plane in which rota- 

 tion occurs, and it is therefore desirable to reduce the plane undilational 

 strain to its simplest terms. 



One method of resolution consists in regarding the general strain as 

 compounded of elementary strains symmetrically oriented with reference 

 to the fixed axes, namely, an axial shear; a shear at 45° to ox; and a 

 scission, the unchanged direction of which coincides with one of the 

 axes. 



It is somewhat easier to test the results of analysis in this case than to 



* When the first shear makes an angle # with ox the formula.-; are — 



x" = xai (<r — s cos 2 d) — va-iS sin 2d; y" = " (a- + s cos 2 &) — — s sin 2 &. 



Here ab = -s 2 sin- 2 i>, anil is essentially positive. 



fWhen 1 1 - make an angle & and the ■-train i- infinii in {v — y.) = ee\ sin 2 i>, which 



is also an infinitesimal ol the second order, so that any two shears, and therefore any number of 

 shears of infinitesimal amount, combine to an irrotational strain. 



