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σ
b
33
1+
ν b =
2
G
G
b
s
( 1−ν
)
b
⎧
⎨
⎩
∞ ∞ ( σ −σ )
22
2
( 1−ν
s )
( 1+
ν )
33
s
G
G
b
s
∞ ∞ ( σ + σ )
22
33
∞ ⎡
+ σ11⎢
⎣
2ν
b
G
−
( 1+
ν ) G ( 1+
ν )
b
b
s
2ν
s
s
⎤⎫
⎥⎬
−
⎦⎭
where the subscripts “b” and “s” denote the elastic parameters of the band and
surrounding sandstone, respectively.
The state of stress at the point adjacent to and immediately outside the tip of the
model band along the x2-axis can now be obtained from the conditions of displacement
and traction continuity at the interface:
σ = σ
(5a)
t
22
t
11
b
11
b
22
t b
ε = ε , ε = ε
(5b)
33
33
where the superscript “t” refers to the values just outside the tip. In the second equation
of (5b), ε33 t can be replaced by ε33 ∞ from the second equation of (1b). Again, the elasticity
relations can be used to eliminate the elastic strains and yield expressions for the stresses
at the tip of the band in terms of the uniform stresses inside, the elastic constants and the
nonzero plastic strain component ε11 p :
σ
G
ν
⎛ G
⎜ −
⎝
⎞
⎟
⎠
G
ν −ν
t s b s b s s b s b
s p
11 = σ 11 + σ 22 1 σ kk
ε11
Gb
( 1 ν s ) ⎜ G ⎟
+
+
(6a)
−
b Gb
( 1−ν
s )( 1+
ν b ) ( 1−ν
s )
σ = σ
(6b)
σ
t
22
G
b
22
ν
⎛ G
⎜ −
⎝
⎞
⎟
⎠
G
ν −ν
t s b s b s s b s b
s p
33 = σ 33 + σ 22 1 σ kk
ν sε11
Gb
( 1 ν s ) ⎜ G ⎟
+
+
(6c)
−
b Gb
( 1−ν
s )( 1+
ν b ) ( 1−ν
s )
By substituting (4) into (6), expressions for the stresses at the tip in terms of the
remote values also can be derived. It can be seen from (6) that, for given values of the
elastic parameters and remote stresses, σ11 t and σ33 t are linear functions of ε11 p , with
slopes of 2Gs/(1-νs) and [2Gs/(1-νs)] νs, respectively, while σ22 t remains constant. In fact,
for ε11 p > 10 -3 and any realistic range of elastic parameters and remote stresses, the final
63
2G
2G
(4c)