168 
MR. A. A. GRIFFITH ON 
h is the modulus of transformation 
V 
c 2 (cosh 2a — cos 2(3) 
fx\ is the modulus of rigidity of the material. 
E is Young’s modulus. 
a- is Poisson’s ratio. 
p = 3 — 4 <j in the case of plane strain, and 
3 — or 
i + 
- in the case of plane stress. 
<T 
The state of uniform stress existing at points far from the crack (i.e. where a is 
large) will be specified by the three principal tensions P, Q and R. P is normal to the 
plate, and in the case of plane stress it is the same everywhere. Q and It are parallel 
respectively to the axes of x and y, and It is positive. 
The strain energy of the plate is a quadratic function of P, Q and It, and hence, in 
accordance with the theorem proved above, the increase of strain energy due to the 
crack must be a positive quadratic function of P, Q and It. The general form of this 
function may be found by evaluating a sufficient number of particular cases. 
The following particular cases are sufficient:— 
I.—Q = It (and P = 0 in the case of plane stress). 
Boundary of crack given by a = a 0 . 
The stresses are 
R„„ = 
R 
w 
p sinh 2a (cosh 2a — cosh 2a 0 ) 
(cosh 2a — cos 2 (3f 
p sinh 2a (cosh 2a + cosh 2a,-, — 2 cos 2/3) 
(cosh 2a — cos 2(3f 
(1) 
a _ -n sin 2(3 (cosh 2a — cosh 2a 0 ) 
(cosh 2a — cos 2/3) 2 
. . (2) 
(3) 
while the displacements are given by 
u* 
h 
cTl 
8/x 
{(P 
1) cosh 2a — (p + 1) cos 2/3 + 2 cosh 2a 0 } 
Up 
h 
= 0 
• (4) 
The strain energy of the material within the ellijise a, per unit thickness of plate is 
