Page 143 - Applied Petroleum Geomechanics

136 Applied Petroleum Geomechanics

size. Griffith (1921) found that this seemed contrary to the well-known fact
that larger cracks are propagated more easily than smaller ones. This
anomaly led Griffith to a theoretical analysis of fractures based on the point
of view of minimum potential energy (Fischer-Cripps, 2007). Griffith
proposed that the reduction in strain energy due to the formation of a crack
must be equal to or greater than the increase in surface energy required by
the new crack faces. According to Griffith, there are two conditions
necessary for crack growth:
(1) The bonds at the crack tip must be stressed to the point of failure. The
stress at the crack tip is a function of the stress concentration factor,
which is dependent on the ratio of its radius of curvature to its length.
(2) For an increment of crack extension, the amount of strain energy
released must be greater than or equal to that required for the surface
energy of the two new crack faces.
Based on the fact that tensile strength of an actual material is much
lower than that theoretically predicted, Griffith (1921) postulated that
typical brittle materials inevitably contain numerous submicroscopic flaws,
microcracks, or other discontinuities of heterogeneity, which are distributed
with random orientation throughout the volume of the material (Whittaker
et al., 1992). These cracks serve as stress concentrators, and fracture initi-
ation is caused by the stress concentrations at the tips of these minute
internal cracks. These cracks have since been referred to as the Griffith flaws
or the Griffith cracks.
The fundamental concept of the Griffith theory is that the bounding
surfaces of a solid possess a surface tension, just as those of a liquid do, and
when a crack spreads the decrease in the strain energy is balanced by an
increase in the potential energy due to this surface tension (Sneddon, 1946).
The calculation of the effect of the presence of a crack on the energy of
an elastic body is based on Inglis’s solution (Inglis, 1913) of the two-
dimensional equations of elastic equilibrium in the space bounded by
two concentric ellipses, the crack being then taken to be an ellipse of zero
eccentricity. Denoting the surface tension of the material of the solid body
by T, the length of the crack by 2a, and Young’s modulus of the material of
the body by E, Griffith showed that, in the case of plane stress, the crack
will spread when the tensile stress P, applied normally to the direction of
the crack, exceeds the critical value P c :
2ET
r ffiffiffiffiffiffiffiffiffi
P c ¼ (4.4)
pa