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and partly by traction. Therefore, procedures A-C
in Figure 3.5 are followed in counterclockwise or-
der: Equations (3.13) and (3.17) are inverted so that
f
i
3.4.4 Microstructure-sensitive behavior of
elasticity and viscosity
That the elasticity and viscosity tensors,
Equations (3.18) and (3.19), are derived as
functions of the contact function means that
the contact function
X
C
, or more practically,
contiguity
at
X
C
σ
ij
S
p
L
=
1 is constrained by
+
δ
ij
(proce-
f
ij
is calculated from
u
i
at
X
C
dure C'), and
1
(procedure A'). Then, procedure B becomes the
elastic deformation of a sphere under a given trac-
tion, and this can be solved analytically using
generalized spherical harmonics. In the viscous
case, unlike the elastic case, procedures A-C are
followed in clockwise order: by assuming each
contact face to be a circular plane surrounded by
the wetted region, the Poisson equation in (3.16)
can be solved analytically.
The elastic compliance tensor is given in an
analytical form as follows:
ε
=
, is the essential geometrical factor
in determining the elastic and viscous properties
of the solid framework. Figure 3.6b shows these
properties as functions of contiguity
ϕ
,where
X
C
is assumed to have 14 circular contact faces
(Figure 3.6a). In Figure 3.6b, the results are shown
for
k
sk
and
ϕ
µ
sk
, corresponding to a Poisson's ratio
of
0.25 in the solid. More general results
are presented in Takei (1998b); the variations
of
k
sk
and
ν
S
=
µ
sk
, with
ϕ
(0.03
≤
ϕ
≤
1) and
ν
S
(0.05
0.45) are given as fitting formulae.
The results for
≤
ν
S
≤
∞
l
1
1
1)
m
2
l
+
3(
−
ξ
sk
and
η
sk
are closely fitted by
S
ijpq
=
1
α
=−
1
β
=−
1
l
=
0
m
=−
l
2
ξ
sk
/η
cc
=
0.37
ϕ
l
(
i
j
)
D
αβ
χ
β
−
m
.
(3.20)
T
−
1
iji
j
χ
−
α
m
l
(
p
q
)
T
−
1
·
(3.18)
p
q
pq
2
η
sk
/η
cc
=
0.2
ϕ
where subscripts
i
,
j
,
p
,and
q
,aswellasthosewith
a prime, are used in a Cartesian coordinate system
(
x
,
y
,
z
), and a summation convention for repeated
subscripts is employed for those subscripts. The
RHS of Equation (3.18) consists of the geometrical
factors (
T
ijpq
-
1
and
Figure 3.6b demonstrates that elasticity and vis-
cosity differ significantly in their dependence on
ϕ
, even though both decrease with a decrease in
the contiguity (
2
)ismuchmore
ϕ
). Viscosity (
∝
ϕ
2
),
and the reason for this is clear from the deriva-
tions. Because a macroscopic stress applied to the
solid framework is supported by the contact faces,
a decrease in the contact area increases the con-
tact stress (the stress concentration effect). The
mild decrease in the elastic moduli
k
sk
and
1
/
structure sensitive to
ϕ
than is elasticity (
∝
ϕ
χ
α
m
l
(
pq
)
), determined by the con-
tact function
X
C
, and the intrinsic factor (
D
αβ
),
determined by the intrinsic elasticities,
k
S
and
µ
S
. Explicit forms of these factors are given in
Takei (1998a). The viscosity tensor is given in an
analytical form as follows:
µ
sk
,
with decreasing
is caused by this stress concen-
tration effect. In addition, for viscosity, a decrease
in the contact area decreases the length of the
diffusion path through the grain boundaries (the
short-circuit effect). The significant decrease in
viscosities
ϕ
a
f
R
4
nf
45
16
η
cc
n
i
n
j
n
f
k
n
l
C
ijkl
=
(
S
ijkl
)
−
1
=
f
=
1
(3.19)
where
nf
represents the number of the circular
contact faces,
a
f
represents the radius of the
f
th
(
f
is caused
by the combination of the stress concentration
effect and the short-circuit effect.
Another remarkable difference between elas-
ticity and viscosity is the singularity. Without
the presence of a melt (
ξ
sk
and
η
sk
with decreasing
ϕ
,
nf
) contact face, and
n
f
represents
the exterior unit normal vector on the grain sur-
face at the center of the
f
th
contact face. Similar
to the elastic case, the RHS of Equation (3.19)
consists of the geometrical factors (
a
f
/
=
1, 2,
...
µ
sk
ap-
proach the intrinsic elastic moduli of a solid,
k
S
and
ϕ
=
1),
k
sk
and
R
and
n
f
)
and the intrinsic factor (
η
cc
).
µ
S
(Figure 3.6b). However, neither
ξ
sk
nor
