34 G. P. BECKER — FINITE* STRAIN IX ROCKS. 



the ratio 3. Hence if before the imposition of the ft shear a Line made an 

 angle w with A, this shear will alter the angle w to, say, a», and— 



tan <o = ft- 1 tan us = 1 / a/3 = B j h. (10) 



Lines making the angle to with .1 will not be undistorted when .-; differs 

 from unity, but they will be lines of maximum tangential strain what- 

 ever may be the value of ft. 



The value of w cannot easily be determined immediately from the dis- 

 placement coefficients. It can be expressed in terms of the axes for 

 /,,,/ ; W = B 1 1 AC, but the value of .8/0 is a complicated one, on account 

 of the inclination of the plane B C. 



Rotation is supposed to be confined to the axis 02, and is therefore 

 unaffected by the shear ft. Hence for strain in three dimensions, as well 

 as in plane strain, the difference of range of the planes of maximum 

 strain measured in the unstrained solid is the angle of rotation, v — //.. 



Numerical Example of Strain. — The application of the formulas devel- 

 oped may he illustrated by an example. Let — 



a =0.1; 6 = 0.3; 1 + «==1.2; l-f/=0.7; 1+g =1.1. 



This is a rotational strain, since b > a. Equations (6) also show that 

 v + (x = 38° 40' and v — /x == — 6° T. If the displacements constituted 

 a pure rotation, sin (y — <>■) would equal a. As tins is not the ease, there 

 is strain. Formula (5) gives h = 0.962, so that the strain is a compres- 

 sive one. If deformation were confined to the xy plane, 1 H- g would 

 equal h. Hence there are two shears. To find them it is most con- 

 venient to determine the axes of the ellipsoid from (3), which gives 

 A = 1.275. B = 0.635, C= 1.1. Then also a = I / h = 1.325, j= C h = 

 1.143. Equation (1) shows that the major axis makes a positive acute 

 angle with ox. The rotation, dilation and the ratios of the two shears 

 are now known. 



To resolve the rotation and the a shear into component, plane, undila- 

 tional strains, let %, b v e 1 and^ be the displacements which would pro- 

 duce only the a shear and the rotation. Then formula (8) leads to these 

 values — ■ 



„, =0.0695 : 6, = 0.2872 ; 1 + e, = 1.2572 : 1 -\-f 1 = 0.8113, 



which give for the elementary plane strains — 



a 2 = 0.9168 ; o... = 1.2524 ; s, = 0.0708. 



The a shear with the rotation is therefore equivalent to a shear with its 

 contractile axis coinciding with oy of ratio 1.2-324, together with a shear 

 the tensile axis of which makes a positive angle of 4-3° with ox, its ratio 



