Geoscience Reference
In-Depth Information
viscosity, caused by a microstructural anisotropy,
explains well the low angle of the bands without
invoking power law viscosity. This ''anisotropic
viscosity'' model provides a better explanation
of various experimental observations than the
''power law viscosity'' model. The results of
Takei and Holtzman (2009c) demonstrate that
grain scale melt redistribution significantly
affects the macro or meso-scale melt redistri-
bution, and hence the melt distribution under
stress has to be investigated on the multiscale.
So far, theoretical works that predict the grain
scale melt distribution under stress are limited in
number (e.g., Hier-Majumder
et al
., 2004; Takei
& Holtzman, 2009b), and the underlying physics
is poorly understood.
The experiments described here were per-
formed under a confining pressure large enough
to avoid brittle fracture. Brittle fracture is not
plausible in the upper mantle, so although dis-
equilibrium geometries may result from cracks
and dikes, they are not usually taken into account
under upper mantle conditions.
x
LAB
solidus
seismic
wave
solid flow
melt flow
(a)
(b)
(c)
Fig. 3.3
(a) A small representative elementary volume
(REV) at each point
x
in the solid-liquid two-phase
system. Schematic images for (b) shear and (c) volum-
etric deformations of the solid framework in REV.
3.3 Phenomenological Representation
taking an average for each phase within the REV
(phasic average) (Drew, 1983; McKenzie, 1984).
Solid displacement
u
S
, melt displacement
u
L
,
solid stress
In this section, I will take a phenomenological
approach to the microstructure-sensitive, vis-
coelastic, and multi-phase behavior of partially
molten rocks in terms of their constitutive
relationships. Then, in Section 3.4, I will pro-
vide a detailed derivation of the constitutive
relationships based on a microstructural model.
σ
ij
S
(tension positive), melt pressure
p
L
(compression positive), and melt volume fraction
φ
, are all defined rigorously by such a phasic aver-
age. These macroscopic variables are governed by
the mass and momentum conservation equations
and the constitutive relationships that describe
the mutual interaction between the solid and
liquid phases through the phase boundary. The
equations are explained in detail in an earlier
review of two-phase theory (Takei, 2009).
Of all these governing equations, the most rele-
vant here is the constitutive equation that relates
the macroscopic strain of the solid:
3.3.1 Mechanical constitutive relationships
in a two-phase system
The dynamics of partially molten mantle, includ-
ing melt segregation and seismic wave propaga-
tion, can be treated by the theory of a two-phase
system. I will discuss themechanical properties of
partially molten rocks based on such a theoretical
framework. In the two-phase theory, as shown in
Figure 3.3, a small representative elementary vol-
ume (REV), that contains a number of solid grains,
is considered at each point
x
in the dynamic
system. The macroscopic quantities at
x
are de-
fined for solid and liquid phases separately by
∂
,
u
j
∂
u
i
1
2
x
i
+
∂
f
ij
ε
=
(3.2)
∂
x
j
σ
ij
S
and
p
L
.As
to the macroscopic stresses
f
ij
is defined for the
macroscopic variable
u
S
(
x
), and represents the
shown by Equation (3.2),
ε
