THE PHENOMENA OF RUPTURE AND FLOW IN SOLIDS. 
169 
On substituting and integrating, it is found that, as « becomes large, the strain 
energy tends towards the value 
{2 (P — x ) e 2 “ -f (3 — p) cosh 2,4.(6) 
o/u 
Hence W, the increase of strain energy due to the cavity a 0 , is given by 
-nvH ? 2 
W = —— (3 — p) cosh 2a 0 .(7) 
8 yU 
or, on proceeding to the limit, a 0 = 0, 
\y — (3 — p) ttc 2 R 2 . ^ 
8 jU. 
for a very narrow crack of length 2c. 
II. —R = 0 (= P in the case of plane stress) a 0 — 0. 
Here the stresses are entirely unaltered by the crack, at every point of the plate 
except the two points x = dz c, ij = 0, where R«« — — Q. It follows that W = 0. 
III. —Q = R = 0, = 0. 
Here, again, the stresses are unaltered, and W = 0. 
The only positive quadratic function of P, Q and R which is compatible with these 
three particular cases is that given by equation (8) ; this is therefore the general form 
of W, and rupture is determined entirely by the stress R, perpendicular to the crack. 
A point of some interest, with regard to equation (8), may be noticed in passing. 
Since W cannot be negative it follows that, in real substances,' where is positive, 
3 — p must be positive. Hence «t cannot be negative in real isotropic solids. 
The potential energy of the surface of the crack, per unit thickness of the plate is 
U = 4cT.(9) 
where T is the surface tension of the material. 
Hence the total diminution of the potential energy of the system, due to the presence 
of the crack, is 
W - U = ( 3 ~ & 7rC " B ' 2 - 4cT.(10) 
8// 
The condition that the crack may extend is 
1 (W - U) = 0. 
dC 
or 
(3 — p) xcR 2 = 16/xT,.. . (11) 
