Page 149 - Applied Petroleum Geomechanics

142 Applied Petroleum Geomechanics

4.2.5 Stresses and displacements in an inclined fracture
For a mixed mode (a fracture is exposed to two or more different loading
systems), the resulting total stresses and displacements at the fracture tip can
be superimposed to obtain because the principle of superposition is appli-
cable in linear-elastic fracture mechanics. Using this feature the complex
fracture problems with complicated loadings can be solved. For a plane
cracked body having a line crack, arbitrary in size and shape (e.g., an in-
clined crack), and loaded arbitrarily along its outer boundaries (see Fig. 4.6),
the elastic stress components in the immediate vicinity of the crack front
can be obtained by superposing those resulting from Model I and Model II
loadings (Eqs. 4.9 and 4.16), i.e.,

2 3
q 3q
6 1 sin sin
2 3 2 7
6 2 7
s x
6 7
K I q 6 q 3q 7
7
4 s y 5 ¼ p ffiffiffiffiffiffiffi cos 6 1 þ sin sin 7
6
2pr 2 6 2 7
6 2 7
s xy 6 7
q
sin cos
4 3q 5
2 2
2 3

q q 3q
6 sin 2 þ cos cos 7
2 2 2
6 7
6 7
q q 3q
6 7
K II 6
þ p ffiffiffiffiffiffiffi 6 sin cos cos 7 (4.20)
7
2 2 2
2pr 6 7
6 7

q q 3q
6 7
cos 1 sin sin
4 5
2 2 2
(A) σ (B) σ h
y y
r x r x
θ θ
Jσ α Jσ σ H p α σ H
2a 2a
σ σ h
Figure 4.6 An inclined center-crack subjected to biaxial stresses. (A) far-field tensile
stresses; (B) far-field compressive stresses and internal pressure.