ROTATIONAL STRAINS. 21 



is negative. The two coincide when this quantity is zero, or when 

 4 ab -f (e — f) 3 = 0. The value of tan /- then reduces to ± y —alb, 

 showing that a and b must have opposite signs. This particular case 

 occurs in the strain often known as shearing motion, as, for example, 

 when a rivet is shorn by tension of the plates which it connects. It will 

 be discussed later. 



The condition of no rotation can be derived from tan /.. The equation 

 represents two lines, and if v. x and ■/.,, are the two angles, tan x 1 tan y-. 2 = 

 — a I b. If there is no rotation, the axial lines are lines of unchanged 

 direction and tan x t tan *.,= — 1, or a = 6.* 



SIMPLE STRAINS. 



Pare Rotation. — If the mass undergoes rotation without strain, each of 

 the axes is equal to unity, and h has the same value. Then by (3), e ==f 

 and a + b = 0, and by (5), (1 -f e)' 2 = 1 — a 2 . Hence tan (v — p.) = 

 al\/ 1 — ft' 2 , or sin (y ■-- ,"■) = a. This result can also be derived imme- 

 diately from the displacement formulas. 



Dilation. — When the only strain is dilation, A--B = C —h, whether or 

 not the displacements cause rotation. • Then by (3) e =/and a + b == 0. 

 By (2) also (1 + ''■)"' + a 2 — (1 +(/) 2 - The rotation is then given by — 



a a 



J 1 + « -,/ h* — a 2 



When there is no rotation, so that the displacements cause pure dilation, 

 ft = b = and e =f= g = h — 1. 



In dealing with dilations it is usually convenient to consider h, the 

 ratio of dilation, as greater than unity, excepting when its value is 

 unknown. The volume of a compressed mass is then l//t 3 , which does 

 not vanish unless the ratio of dilation is infinite. 



♦The length of the lines of unchanged length exhibits a somewhat remarkable relation. Let k 

 be the length of such a line. Then— 



x l = y- = k, 



x y 

 and by the displacement formulas— 



?/ _ k — (1 +e) _ a 



x b k-(l+f) 

 This gives— 



2£ = l + e + l +/± tfiao + (e—f)*. 



If A.'j and k« are the two values of k, then — 



fcj fc 2 =(l +/) (1 + e) — ab, 



which by (2) is the product of the semi-axes or A B. Thus the product of these lines remains in- 

 variable, whether or not they coincide with the axes. 



