Geoscience Reference
In-Depth Information
unknown variables in anelasticity, its potential
importance at seismic frequencies has been rec-
ognized, and anelasticity is now a subject of
intensive study (e.g., Karato, 1993; Jackson
et al
.,
2002; McCarthy
et al
., 2011). Therefore, the gen-
eral viscoelastic representation of
S
ijkl
to cover all
elastic, anelastic, and viscous regimes, is useful in
understanding the recent findings on anelasticity.
Linear viscoelastic behavior is usually repre-
sented by a creep function that shows a time-
dependent evolution of strain in response to a
unit-step of stress. The time derivative of the
creep function provides the impulse response.
Strain corresponding to an arbitrary stress his-
tory can be calculated by using the convolution
of the impulse response and the stress history.
Because the convolution becomes a product in
the Fourier domain, the general viscoelastic form
of Equation (3.3) can be written as
(Nowick & Berry, 1972), where
represents the
timescale and
S
e
ijkl
and
S
v
ijkl
represent the elastic
and viscous compliance tensors, respectively.
Equations (3.6) show that a line spectrum in
X
ijkl
(
τ
) causes a Debye peak in
S
2
ijkl
.When
τ
X
ijkl
(
) is not a line spectrum, but varies slowly
with ln
τ
τ
(often the case for rock anelasticity),
)
2
)and
)
2
) can be
factors 1
/
(1
+
(
ωτ
ωτ/
(1
+
(
ωτ
approximated by a step function H(
−
ln
τ
−
ln
ω
)
and a delta function (
), respec-
tively, and Equations (3.6) can be approximated as
π/
2)
δ
(ln
τ
+
ln
ω
τ
=
ω
−
1
S
ijkl
(
S
ijkl
+
ω
)
=
X
ijkl
(τ )
d
ln
τ
(3.7)
τ
=
0
S
ijkl
ω
2
X
ijkl
(
S
ijkl
(
τ
=
ω
−
1
)
ω
=
+
)
(Nowick & Berry, 1972). Equations (3.7) are
clear and useful: the contributions of elasticity,
anelasticity, and viscosity to S
∗
ijkl
are clearly
described by
S
e
ijkl
,
X
ijkl
(
1
3
k
S
p
L
(
f
ij
(
S
ijkl
(
S
kl
(
+
p
L
(
), and
S
v
ijkl
, respectively.
The modulus dispersion and attenuation caused
by anelasticity
X
ijkl
(
ε
ω
)
=
ω
)(
σ
ω
)
ω
)
δ
kl
)
−
ω
)
δ
ij
(3.4)
τ
) are clearly shown by the
first and second equations, respectively.
τ
where
ω
(
=
2
π
f
) represents the angular frequency,
f
ij
(
S
ij
(
ε
ω
σ
ω
), and
p
L
(
ω
and
) are complex numbers
representing the Fourier transforms of
),
f
ij
(
t
),
S
ij
(
t
),
and
p
L
(
t
), respectively. The complex compliance
ε
σ
3.3.3 Elastic and viscous constitutive
relationships
So far, much more of the theoretical and experi-
mental work on partially molten rocks has been
spent in trying to understand elasticity
S
e
ijkl
and
viscosity
S
v
ijkl
rather than anelasticity
X
ijkl
(
S
ijkl
(
S
ijkl
(
iS
ijkl
(
ω
)
=
ω
)
+
ω
)
(3.5)
represents the Fourier transform of the impulse
response, which is, generally, a 4-rank tensor
describing a behavior of each
τ
).
The purely elastic and viscous constitutive
relationships are obtained as the high- and
low-frequency limits, respectively, of the general
viscoelastic formulation of Equations (3.6)-(3.7).
When
f
ij
component
ε
S
kl
p
L
in response to each
δ
kl
component. In
Equation (3.4), the anelastic relaxation of the
intrinsic bulk modulus
k
S
is neglected. The fact
that the impulse response is nonzero only at t
σ
+
,
S
ijkl
=
S
ijkl
ω
→∞
and when
ω
→
0,
0
(i.e. causality) results in the Kramers-Kronig
relationship between
S
1
ijkl
and
S
2
ijkl
.There-
lationship is implicitly expressed by using the
relaxation spectrum
X
ijkl
(
≥
S
ijkl
=
ω
−
1
S
ijkl
. Hence, Equation (3.4) is written
in the time domain as
i
f
ij
1
3
k
S
p
L
S
ijkl
(
S
kl
p
L
ε
=
σ
+
δ
kl
)
−
δ
ij
(elastic)
τ
) as follows
f
ij
S
ijkl
(
S
kl
p
L
ε
=
σ
+
δ
kl
)
(viscous),
(3.8)
τ
=∞
1
S
ijkl
(
=
S
ijkl
+
ω
)
X
ijkl
(τ )
)
2
d
ln
τ
f
ij
represents the framework strain-rate.
In the viscous equation, the amplitude of strain
is generally much larger than the amplitude of
elastic strain, and hence the second term on the
RHS of Equation (3.4) is neglected. For an isotropic
1
+
(
ωτ
where
ε
τ
=
0
τ
=∞
S
ijkl
ω
ωτ
S
ijkl
(
ω
)
=
X
ijkl
(
τ
)
)
2
d
ln
τ
+
1
+
(
ωτ
τ
=
0
(3.6)
