Geoscience Reference
In-Depth Information
change the results in these numerical studies,
selection of an appropriate constitutive model be-
comes important. So far, the contiguitymodel can
explain the observed behavior of
3.6 Anelasticity
3.6.1 Several relaxation mechanisms
better than the
mixture models. However, in order tomake a firm
decision on which model is better, more data for
the bulk viscosity are needed. In particular, a set of
bulk and shear viscosities, both measured for the
same sample, would be useful to test the models.
Also, further refinement of the contiguity model
is needed to provide better estimates of the critical
fraction of melt below which
η
Various defects in rocks cause anelastic relax-
ation, and these include point defects, disloca-
tions, grain boundaries, and the presence of a
melt phase. The relaxation caused by point de-
fects is considered to be small (Karato & Spetzler,
1990), and hence the relaxation spectrum
X
ijkl
(
τ
)
introduced in Section 3.3.2 is expressed as
ξ
increases rapidly.
GB
ijkl
X
GB
(
disl
ijkl
X
disl
(
X
ijkl
(
τ
)
=
τ
)
+
τ
)
melt
ijkl
X
melt
(
+
τ
),
(3.24)
3.5.3 Summary of the properties of a texturally
equilibrated system
β
ijkl
and
X
β
(
where
β
=
GB
,
disl
,
melt
) repre-
sent the total relaxation strength and distribu-
tion of relaxation timescales, respectively, for
each mechanism
τ
)(
Multiple models, based on the same underlying
physics, have been developed to study the
elastic properties of partially molten rocks. Any
differences in the results of these studies come,
therefore, simply from differences in the pore
geometries assumed when applying the models.
The use of an ''equivalent aspect ratio'' (Section
3.5.1) helps in treating the various pore geome-
tries systematically. So far, for the elasticity of
texturally equilibrated partially molten rocks, the
following quantitative results have been obtained
consistently from microstructural analysis, the-
oretical modeling, and ultrasonic measurements:
.
X
β
(
β
τ
) is normalized to unity:
τ
=∞
X
β
(
τ
)
d
ln
τ
=
1.
τ
=
0
At grain boundaries, the tangential compo-
nent of stress is relaxed by grain boundary
sliding, resulting in a large reduction in the
shear modulus of polycrystalline aggregates.
Using a simple spherical grain model,
the
total
relaxation strength is estimated to be
ν
S
)
-
1
(4
µ
S
)
-
1
(
GB
ijkl
ν
S
)2
−
1
(7
=
3(7
−
10
+
5
δ
ik
δ
jl
+
δ
il
δ
jk
−
3) (Zener, 1941). This result
can also be obtained by solving the elastic
contiguity model under the relaxed state of
grain boundary sliding. The fully relaxed shear
modulus is 0.55
2
δ
ij
δ
kl
/
ln
V
S
/φ
∼−
2.2 and
R
SP
=
ln
V
S
/
ln
V
P
=
1
.
3
−
5.
In contrast, the multiple models for studying
viscosity in partially molten rocks have been
developed with different underlying physics. The
deformation mechanisms that have been mod-
eled are fundamentally different, and this results
in large differences in the calculated viscosities.
Although the viscous contiguity model explains
the existing data better than the mixture models,
more experimental data are needed, especially
for bulk viscosities. Although experimental data
for shear viscosities have been measured in both
the diffusion and dislocation creep regimes, the
theoretical treatment is limited to diffusion
creep, and the lack of a theoretical formulation
for the effect of melt on dislocation creep is a gap
that needs to be filled.
1.5, corresponding to
α
∼
0.1 and
k
S
/
k
f
∼
µ
S
for
ν
S
=
0.25. Although the
GB
total reduction
is large, a reduction at
ijkl
a certain frequency
f
(
f
1Hz, for example) is
determined by the product of
=
GB
ijkl
and the
f
)
-
1
(Equation 3.7). Therefore, the detailed shape of
X
GB
(
integration of
X
GB
(
τ
)from
τ
=
0to
τ
=
(2
π
) is important in estimating the reduction of
seismic wave velocity. Grain boundary sliding is
disturbed by various obstacles such as grain edges
and grain boundary steps, where grain boundary
sliding causes overlaps and/or gaps. Therefore,
the sliding occurs in two stages called elastically
accommodated grain boundary sliding and dif-
fusionally accommodated grain boundary sliding
τ
