Page 144 - Applied Petroleum Geomechanics

Basic rock fracture mechanics 137

A formula differing from (Eq. 4.4) by a factor 0.8 has been derived by
Orowan (1934) from rather similar assumptions. In the case of plane strain,
it has the following form:
s ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2ET
P c ¼ (4.5)
2
pað1 n Þ
where n is Poisson’s ratio.
Sack (1946) extended Griffith’s theory to three dimensions and calcu-
lated the conditions of failure for a solid containing a plane crack bounded
by a circle: a “penny-shaped” crack, when one of the principal stresses is
acting normally to the plane of the crack. By treating the crack as an oblate
spheroid whose elliptic section has zero eccentricity, Sack proposed that
failure will occur when the tensile stress P normal to the crack exceeds the
critical value P c :
s ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
pET
P c ¼ (4.6)
2
2að1 n Þ
The three-dimensional model introduced by Sack thus gives a critical
tensile stress differing from the Griffith value (Eq. 4.4) by a factor
2 1/2
p/[2(1 n ) ].
4.2.2 Stress intensity factor and fracture toughness

The stress intensity factor, K, is used in fracture mechanics to describe the
stress state at a crack tip. It is related to the rate of crack growth and used to
establish failure criteria due to fracture. The stress intensity factor was
developed by George R. Irwin (Irwin, 1957), the father of fracture me-
chanics. It is one of the most fundamental and useful parameters in fracture
mechanics. For an elliptical crack, the stress intensity factor in Mode I
fracture (K I )isdefined as:
p ffiffiffiffiffi
K I ¼ s pa (4.7)

where s is the far-field stress; a is the half length of the crack.
A critical value of K can be obtained by experiments. Crack initiation
will take place if the stress intensity factor K reaches its critical value or
fracture toughness, K C . Fracture toughness is a property that describes the
ability of a material to resist fracture. The linear-elastic fracture toughness of
a material is determined from the stress intensity factor at which a thin crack