THE PHENOMENA OF RUPTURE AND FLOW IN SOLIDS. 171 
Equations (17) show that the effect of the small stress dR on the elliptic cavity is to 
deform it into another ellipse. If a and b are the major and minor semi-axes of the 
ellipse, when the plate is subjected to a stress R, then, by (17), 
da _ 2b i 
dR ~ E I 
db_ _ 2a \ 
dR ~ Ej 
on making use of the relation b = a tanh a„. 
The solution of these simultaneous differential equations is 
, 2R . , . , 2R 1 
a = a 0 cosh — + o 0 smh^=- i 
Jli ili 
& = % sinh —-- + b 0 cosh— 
(18) 
(19) 
where a 0 and b 0 are the values of a and b in the unstrained state. 
With the help of equations (19) it is possible to find the maximum stress, F, due to 
an applied stress, R, taking account of the change in the shape of the cavity. From (2) 
whence 
dF _ 2 a 
dR~^b 
F = 2 
,K a 0 cosh sinh ^jpdR 
2R 
io a, sinh ^ -j- 6 n cosh ~ 
E E 
( 20 ) 
= E log 
(cosh 
2R 
E 
■^2 sinh 
2R\ 
E /’ 
( 21 ) 
and in the case of a narrow crack which is elliptic only near its ends, ~ may, as in (14), 
On 
be replaced by ,\J fi. 
In the general case, where Q is not equal to R, the quantity R — Q must be added 
to the value of F given by (21). 
Formulae (19) and (21) are not, of course, exactly true. The application of integration 
to equations (18) and (20) involves the assumption that the strains are so small that 
they can be superposed. If the strains are finite, this involves an error in the stresses 
depending on the square of the strains. In the case of ordinary solids, it is improbable 
that this assumption can alter the calculated stress by as much as 1 per cent. 
vol. ccxxi.— a. 2 B 
