Page 152 - Applied Petroleum Geomechanics

Basic rock fracture mechanics 145

intensity factors can be obtained as follows (Rice, 1968; Abou-Sayed et al.,
1978):
p ffiffiffiffiffi
2
2
K I ¼ paðp 0 s H cos a s h sin aÞ (4.28)

p ffiffiffiffiffi 1
K II ¼ pa ðs H s h Þsin 2 a (4.29)
2
4.2.6 Plastic zone and fracture process zone at the
fracture tip
4.2.6.1 Plastic process zone at the fracture tip
Fracture tip plastic zone or yield zone can be obtained by applying the
failure criteria to the stresses at the fracture tip. For metallic materials, the
Von Mises failure criterion can be used and is given by the following
expression:
2 2 2 2
ðs 1 s 2 Þ þðs 2 s 3 Þ þðs 1 s 3 Þ ¼ 2s (4.30)
0
where s 1 , s 2 , and s 3 are the principal stresses at the fracture tip region; s 0 is
the yield strength.
Substituting the principal stresses at the fracture tip (Eqs. 4.13e4.15)
into Eq. (4.30), the plastic zone at the fracture tip for Mode I fracture can be
solved as follows:
2

1 K I 3

2
For plane stress: rðqÞ¼ 1 þ sin q þ cos q (4.31)
4p s 0 2
2

1 K I 3 2

2
For plane strain: rðqÞ ¼ sin q þ ð1 þ cos qÞð1 2nÞ
4p s 0 2
(4.32)
It can be seen that as the fracture intensity factor increases, the size of the
plastic zone that develop around the tip will be greater. The plastic zone
length (r p ), the distance to the boundary ahead of the crack tip, can be
solved from Eqs. (4.31) and (4.32) when q ¼ 0 as follows:
2

1 K I
For plane stress: r p ¼ (4.33)
2p s 0
2

1 K I 2

For plane strain: r p ¼ ð1 2nÞ (4.34)
2p s 0