Page 147 - Applied Petroleum Geomechanics

140 Applied Petroleum Geomechanics

Using Hooke’s law, the displacement components at the fracture tip can
be obtained from Eq. (4.9):
q 3q
2 3
u K I r 6 2 2 7
r ffiffiffiffiffiffi ð2k 1Þcos cos
¼ 6 7 (4.12)
v 4G 2p 4 q 3q 5
ð2k þ 1Þsin sin
2 2
where u and v are the displacements along x- and y-axes, respectively; G is
the shear modulus.
For plane strain: k ¼ 3 4n and w ¼ 0.
n R
For plane stress: k ¼ (3 n)/(1 þ n) and w ¼ ðs x þ s y Þdz
E
where w is the displacement along z-axis.
The principal stresses (s 1 , s 2 , s 3 ) at the fracture tip can be obtained from
Eqs. (4.9)e(4.11):

2
q 3
1 þ sin

s 1 K I q 6 2 7
¼ p ffiffiffiffiffiffiffi cos 6 7 (4.13)
s 2 2pr 2 4 q 5
1 sin

2
K I q
For plane strain: s 3 ¼ nðs 1 þ s 2 Þ¼ 2n ffiffiffiffiffiffiffip cos (4.14)
2pr 2
For plane stress: s 3 ¼ 0 (4.15)
4.2.4.2 Model II fracture
For the mode II fracture under in-plane shear stresses in the far field, the
stresses in the vicinity of the crack tip (point A in Fig. 4.5) can be expressed
as follows (Whittaker et al., 1992):
2 3
q q 3q
6 sin 2 þ cos cos 7
2 2 2
2 3 6 7
s x 6 7
q q 3q
6 7
6
7
4 s y 5 ¼ p K II 6 sin cos cos 7 (4.16)
ffiffiffiffiffiffiffi 6
7
2 2 2
2pr 6 7
s xy 6 7
6 q q 3q 7
cos 1 sin sin
4 5
2 2 2
For plane strain: s z ¼ n(s x þ s y )
For plane stress: s z ¼ s xz ¼ s yz ¼ 0.
where K II is the mode II fracture tip stress intensity factor, and
p ffiffiffiffiffi
K II ¼ s i pa; s i is the in-plane shear stresses in the far field.