structural geology, propagation mechanics and - Stanford School of ...

Comparison of the two approaches establishes the formal equivalence of the very
eccentric ellipsoidal inclusion with the embedded layer.
With the x1-axis normal to the layer—i.e. corresponding to the short axis of the
inclusion (Figure 2.11a)—Cocco and Rice (2002) note that εij b = εij ∞ if neither i nor j is 1
and σij b = σij ∞ if either i or j is 1, where the superscripts “b” and “∞” denote values of
stress (σ) and strain (ε) inside the model band (layer/inclusion) and in the far field,
respectively. The relations of particular relevance here are
σ b
ε b
∞
11 = σ 11
∞
22 = ε 22
(1a)
∞
, ε ε
(1b)
b
33 = 33
Hooke’s law for the linear elastic relation between strain and stress is
1 ⎧ ν ⎫
ε ij = ⎨σ
ij − σ kkδ
ij ⎬
2G
⎩ 1+
ν ⎭
where the repeated index denotes summation, and δij =1 if i = j, and 0 if i ≠ j. The remote
stresses and strains are related by (2). In the band, (2) relates the stresses to the elastic
strains. The total strain inside the band is given by
b total b elastic p
[ ε ] [ ε ] + ε
ij
= (3)
ij
ij
where εij p denotes the plastic strain representing compaction within the band, which
equals zero unless i = j = 1 as stipulated in the field-based conceptual model. Equations
(1), (2) and (3) can be combined to eliminate the elastic strains and yield expressions for
the stresses inside the model band in terms of the far-field stresses and elastic constants:
σ b
σ
∞
11 = σ 11
b
22
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⎢
⎣
62
2ν
b
G
−
( 1+
ν ) G ( 1+
ν )
b
b
s
2ν
s
s
⎤⎫
⎥⎬
+
⎦⎭
(2)
(4a)
(4b)