Geoscience Reference
In-Depth Information
z
y
x
(a)
10
0
10
0
1/2
ϕ
shear modulus
shear modulus
N
/
µ
sk
/
µ
S
µ
S
10
−
1
k
sk
/k
S
K
b
/
k
S
bulk modulus
bulk modulus
10
−
2
2
10
−
1
ϕ
bulk viscosity
ξ
sk
/
η
cc
bulk viscosity
ξ
/
η
cc
10
−
3
η
sk
/
η
cc
η
cc
shear viscosity
η
/
shear viscosity
1/2
ϕ =
1-2.3
φ
10
−
4
10
−
2
10
−
2
10
−
1
10
0
0
0.1
0.15
0.05
Contiguity,
ϕ
Melt fraction,
φ
(b)
(c)
Fig. 3.6
The elastic and viscous properties of a solid framework, calculated for a tetrakaidekahedral grain model
with 14 circular contact faces (Figure 3.6a), are shown as functions of contiguity
φ
(Figure 3.6c). The results in (c) are obtained from those in (b) by assuming an equilibrium melt geometry and hence
assuming the
ϕ
(Figure 3.6b) and melt fraction
1
/
2
. The definitions of
k
sk
,
ϕ
-
φ
relationship given by
ϕ
=
1
−
2
.
3
φ
µ
sk
,
ξ
sk
,and
η
sk
and those of
K
b
,
N
,
ξ
,and
η
are given in Equations (3.9) and (3.10), respectively.
η
sk
approaches the intrinsic viscosity of a solid,
which is
infinitely high dissolution/precipitation reaction
rate. If these factors are finite, and if the melt frac-
tion is smaller than a critical value
∞
(incompressible) for the bulk viscos-
η
cc
for the shear viscosity. This singular
behavior of viscosity at
φ
c
, the rate-
limiting process changes from diffusion through
the grain boundaries to dissolution/precipitation
reaction for
ity and
1 can be understood
intuitively (Figure 3.7); even at a very small melt
fraction, the connected network of melt works
as a fast diffusion pathway and significantly de-
creases viscosity as a result of the short-circuit
effect. Equations (3.16) implicitly assume a liquid
diffusivity that is infinitely high as well as an
ϕ
→
ξ
sk
and to diffusion though the liquid
for
η
sk
deviate from the val-
ues shown in Figure 3.6b and rapidly approach
η
sk
.At
φ<φ
c
,
ξ
sk
and
∞
and
η
cc
, respectively (Takei & Holtzman, 2009b).
Therefore, the ''mathematical singularity'' can
