THE PHENOMENA OF RUPTURE AND FLOW IN SOLIDS. 
167 
3. Application of the Theory to a Cracked Plate. 
The necessary analysis may be performed in the case of a flat homogeneous isotropic 
plate of uniform thickness, containing a straight crack which passes normally through 
it, the plate being subjected to stresses applied in its plane at its outer edge. 
If the plate is thin, its state is one of “ plane stress,” and in this case it may, without 
additional complexity, be subjected to any uniform stress normal to its surface, in 
addition to the edge tractions. If it is not thin, it may still be dealt with provided it is 
subjected to normal surface stresses so adjusted as to make the normal displacement 
zero. Here the plate is in a state of “plane strain.” The equations to the two states 
are of the same form,* differing only in the value of the constants ; they will therefore 
be taken together. 
The strain energy may be found, with sufficient accuracy, in the general case where 
the edge-tractions are arbitrary ; it is necessary in the present application, however, 
for the resulting stress-system to be symmetrical about the crack, as otherwise it is 
not obvious that the latter will remain straight as it spreads. The only stress 
distribution which will be considered, therefore, is that in which the principal stresses 
in the plane of the plate, at points far from the crack, are respectively parallel and 
perpendicular to the crack, and are the same at all such points. This is equivalent 
to saying that, in the absence of the crack, the plate would have been subjected to 
uniform principal stresses in and perpendicular to its plane. It is also necessary, on 
physical grounds, for the stress perpendicular to the crack and in the plane of the plate 
to be a tension, otherwise the surfaces of the crack are forced together instead of being 
separated, and they cannot remain free from traction. 
In calculating the strain energy of the plate use will be made of the solution obtained 
by Prof. Inglis for the stresses in a cracked plate, to which reference has already been 
made. The notation of Prof. Inglis’s paper will be employed. In that notation 
a, (3, are elliptic co-ordinates defined by the family of confocal ellipses ; a — const, 
and the orthogonal family of hyperbolae /3 = const. The crack is represented by the 
limiting ellipse or focal line a — 0. The axis of x coincides with the major axes, and 
the axis of y with the minor axes of the ellipses. The cartesian co-ordinates x, y, are 
connected with the elliptic co-ordinates a, /3, by the relation 
x -f iy — c cosh (a + i/3). 
R aa , u a , are the tensile stress and displacement respectively along the normal to 
a — const. 
R^, are the corresponding quantities in the case of the normal to (3 — const. 
S a3 is the shear stress in the directions of these normals, 
c is the half-length of the focal line. 
* A. E. H. Love, ‘ Mathematical Theory of Elasticity,’ 2nd ed., p. 205. 
