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Basic rock fracture mechanics 151

the crack differ from those of the two-dimensional case by a numerical
factor only.”
4.3.2 General solution for fracture width of the
Griffith fracture
The possibility of solving crack (fracture) problems by reducing them to a
mixed boundary values for a half-plane or a half-space was first pointed out
by Sneddon (1946) and Sneddon and Elliott (1946). Sneddon (1966)
considered a case of stress field in the xy-plane owing to the application of a
symmetrical pressure p(x) to the faces of the Griffith fracture jxj L, y ¼ 0,
as shown in Fig. 4.9. The symmetrical pressure means that p(x) is an even
function of x and that the pressure on the face y ¼ 0 of the crack is
identical to that applied on the face y ¼ 0þ. It is sufficient to calculate the
components of the stress tensor at the point (x, y) in the half-plane y 0
when the line y ¼ 0 is subjected to the boundary conditions:
s yy ðx; 0Þ¼ pðxÞ; 0 x L
u y ðx; 0Þ¼ 0; x > L
u xy ðx; 0Þ¼ 0; x L

These are readily established by considering the symmetry of the
problem. Sneddon’s integral equations can be applied to obtain the
analytical solution of the fracture width (displacements of two sides of
the fracture) caused by the internal pressure p(x), i.e.,
2 Z L
8ð1 n Þ tgðtÞdt
wðxÞ¼ p ffiffiffiffiffiffiffiffiffiffiffiffiffiffi; jxj < L (4.46)
pE x t x 2
2
where g(t) is a function of the internal pressure (p(x)or s y ) in the fracture,
Z t pðxÞdx Z t s y dx
and gðtÞ¼ p ffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ p ffiffiffiffiffiffiffiffiffiffiffiffiffiffi (Sneddon, 1966). The internal
2
2
0 t x 2 0 t x 2
pressure may not be a constant or uniformly distributed. If p(x) is a constant
(e.g., p(x) ¼ p 0 ), then g(t) ¼ pp 0 /2. Substituting it into Eq. (4.46), the solu-
tion of Eq. (4.45) can be obtained.
4.3.3 3-D solution for a penny-shaped fracture

A penny-shaped crack is a typical 3-D fracture in the interior of an infinite
2
2
2
2
elastic medium, occupying the circle r ¼ x þ y ¼ L in the plane z ¼ 0,
under the action of an internal pressure as a function of the radius r
(Fig. 4.10). For this 3-D circular fracture with a constant internal pressure