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(Takei & McCarthy, 2010). The contiguity model
is, therefore, applicable to properties over a broad
range of frequency. As stated in Section 3.1, the
theoretical treatment of viscosity of a partially
molten rock is still limited to diffusion creep.
where
n
represents the exterior unit normal
vector on the grain surface, and
dS
represents a
small area on the grain surface. The scalar con-
tiguity (
ϕ
) introduced in Section 3.2, is related to
the contiguity tensor in the form
ϕ
=
(
ϕ
xx
+
ϕ
yy
+
ϕ
zz
)
3. The contiguity tensor can describe
contact anisotropy.
/
3.4.2 Description of microstructures
In the contiguity model, it is assumed that each
grain in a partially molten aggregate is a sphere
with radius
R
, and the contact state of that grain
with neighboring grains is described by a contact
function
X
C
defined on the grain surface as
3.4.3 Derivation of elastic and viscous
constitutive relationships
As illustrated in Figure 3.5, once the contact
function
X
C
is known, constitutive equations
relating framework strain
f
ε
ij
to the macroscopic
σ
ij
S
and
p
L
can be derived by solving the
behavior of each grain. Here, the scale smaller
than grain size is referred to as the ''micro-
scopic'' scale. Procedures A-C in Figure 3.5,
which connect the microscopic and macroscopic
mechanical fields, are specified in Takei (1998a)
for the case where each grain deforms elastically,
and in Takei & Holtzman (2009a) for the case
where each grain deforms viscously due to grain
boundary diffusion creep. The derivations of the
elastic and viscous constitutive relationships,
presented in those two papers, are summarized
here, in order to clarify the similarities and
differences between the two properties.
stresses
X
C
(
r
R
)
1 if the grain is in contact with solid at
r
R
0 if the grain is in contact with liquid at
r
R
(3.11)
(Figure 3.1c), where
r
R
represents a position on
the grain surface relative to the grain center.
The contact function
X
C
can describe various
contact states, such as the equilibrium geometry
and the stress-induced anisotropy introduced in
Section 3.2. The contiguity tensor
=
ϕ
ij
is defined by
R
2
r
1
X
C
n
i
n
j
dS
,
ϕ
ij
=
(3.12)
4
π
=
R
mechanical
constitutive
relationships
ij
n
j
σ
ij
ε
p
L
n
i
(3.8)
macroscopic stresses
macroscopic strain
of solid framework
X
C
X
C
=
0
=
1
Fig. 3.5
Map view of the
procedures to derive mechanical
constitutive relationships of
solid-liquid composites based
on the response of each solid
grain. The numbers refer to the
relevant equations in the text.
Modified from Takei (1998a).
Reproduced with permission of
the American Geophysical
Union.
(3.17)
C C'
(3.13)
A'
A
p
L
n
i
f
i
governing equations for
a grain deformation
u
i
(
r
)
B
elastic (3.14)
viscous (3.16)
microscopic stress
in a grain
microscopic deformation
of a grain
