of Flow in a Strained Elastic Solid. 505 



be subject to a shearing-strain of amount — ; i. e. the angle 



bad will be diminished by an amount^- , where p is the inten- 



J n 



sity of the shearing-strain on the element, and n is the 

 coefficient of rigidity of the metal. The effect of this shearing- 

 strain on the element may be described as a rotation of a d 

 through an angle s 1} and of a b through an angle s 2 , where 



Sl + s 2 = — ; s 1 and s 2 may be equal, when the displacement of 



c will be wholly parallel to ox; or they may be unequal, 

 when c will have a displacement parallel to o x and a resolved 

 displacement parallel to oy, which will be + ve or — ve accord- 

 ing as si is less than or greater than s 2 , and which will be a 

 maximum if either s x or s 2 = 0. 



The value of s x + s 2 , and the ratio of ^ to s 2 for any element, 

 appears to be determined by the following considerations. 

 If a side a b of the element be produced in both directions so 

 as to cut the sides of the bar, there is seen to be a tendency 

 of that part of the bar below this line to slide over that part 

 which is above. Assuming the resulting shearing-strain to 

 be uniform over the length l x of this oblique section, the 

 shearing-strain at any element in this direction, i. e. s l3 will 

 clearly be inversely proportional to \ ; s 2 will also be inversely 

 proportional to l 2j where l 2 is the length of an oblique section 



parallel to a d, and s x -f s 2 will be proportional to —yj say 



to -• 



P 



Suppose, now, we take a series of elements, a 1 6 1 c 1 ^ 1 — 

 a l b 2 c 2 d 2 touching one another and lying across the bar. If 

 we imagine the points a v a 2 , a 3 , &c. fixed on a line parallel to 

 oy, it is clear that the displacements of c 1? c 2 , c 3 , &c. will be pro- 

 portional to the sum of the shearing-strains in each element. 

 If s L + s 2 is the same for each element, and s 1 = s 2 , the points 

 c 1? c 2 , c 5 , &c. will remain on a line parallel to oy. If, however, 

 s ± + s 2 becomes greater as we approach the middle, and less 

 again towards the further side, the points c 1? c 2 , c 3 , &c. will 

 be found on a line concave upwards ; if s Y + s 2 diminishes 

 towards the centre these points will lie on a line concave 

 downwards. If the strains should exceed the limit of elasticity 

 the line through c 1? c 2 , c s , &c. will be permanently curved.^ 



If the points %, a 2 , a s , . . . , instead of being in one straight 

 line have unequal downward displacements, the form of the 

 curves obtained as above may be modified. 



If, now, we consider a series of consecutive elements, 

 °\ ^1 ^1 di — a x b 2 c 2 d 2 , lying in a line parallel to o x y if Si=Sj for 



