Geoscience Reference
In-Depth Information
The anelastic relaxation due to grain boundary
sliding is assumed to be in an ''unrelaxed'' state
for elastic constitutive relationships, and ''fully
relaxed'' for viscous constitutive relationships.
Therefore, at the grain-to-grain contact faces, the
tangential stress is nonzero and the tangential
displacement continuous in the elastic formula-
tion, but zero and discontinuous in the viscous
formulation. Procedure A relates the macroscopic
strain to the microscopic deformation such that
the deformation of each grain
u
i
is constrained
by the framework strain
In Equations (3.14), the boundary condition at
X
C
1 represents the displacement compatibil-
ity given by procedure A, and the condition at
X
C
=
0 represents the stress continuity between
the solid and melt phases. Viscous deformation
of a grain during grain boundary diffusion creep is
governed by the following diffusion equation on
the grain-to-grain contact faces
=
2
f
r
30
R
−
3
η
cc
˙
u
r
at
X
C
(
r
R
)
∇
=−
=
1
u
r
(
r
R
)
=
ε
ij
r
j
n
i
at
X
C
(
r
R
)
=
1
(3.16)
f
ε
ij
at the grain-to-grain
contact faces, as follows
f
r
(
r
R
)
at
X
C
(
r
R
)
=
0
=
0,
f
ij
r
j
u
i
(
r
R
)
at
X
C
(
r
R
)
=
ε
=
1
(elastic)
where
f
r
(
f
i
n
i
) represents the radial component
of the differential traction and
=
η
cc
represents the
shear viscosity of the melt-free granular aggregate
due to grain boundary diffusion creep (Coble creep
viscosity). The tangential component of traction
is zero due to the occurrence of grain boundary
sliding. Equations (3.14) and (3.16) both describe
the linear responses of each grain.
Procedure C relates the microscopic to the
macroscopic stresses. The macroscopic solid
stress
f
ij
r
j
n
i
at
X
C
(
r
R
)
u
r
(
r
R
)
=
ε
=
1
(viscous),
(3.13)
where,
u
r
(
u
i
n
i
) represents the radial component
of
u
. In the viscous formulation, only the radial
component of
u
is constrained, because the tan-
gential component is discontinuous due to the
occurrence of grain boundary sliding.
Procedure B relates the microscopic deforma-
tion to the microscopic stresses on the basis
of the physical rules governing the grain defor-
mation. The procedure is quite different for the
elastic and viscous cases. Elastic deformation of a
grain is governed by the mechanical equilibrium,
the constitutive relationships, and the boundary
conditions, as follows:
=
σ
ij
S
is calculated from the average micro-
scopic stress within the grain. By expressing the
volume integral of stress in terms of the surface
integral of traction, the following relationships
are obtained
f
i
n
j
+
f
j
n
i
dS
(elastic)
S
ij
p
L
3
σ
+
δ
ij
=
R
2
8
π
τ
ij
,
j
=
0
at
r
<
R
r
=
R
τ
ij
=
λ
S
u
k
,
k
δ
ij
+
µ
S
(
u
i
,
j
+
u
j
,
i
) t
r
≤
R
f
r
n
i
n
j
dS
S
ij
p
L
3
σ
+
δ
ij
=
(viscous).
R
2
4
π
f
ij
r
j
u
i
(
r
R
)
at
X
C
(
r
R
)
=
ε
=
1
r
=
R
(3.17)
By solving Equations (3.13)-(3.17), the elastic
and viscous constitutive relationships can be
derived in exactly the same form as Equations
(3.8), where
S
e
ijkl
and
S
v
ijkl
are derived as func-
tions of the contact function
X
C
. Takei (1998a)
and Takei and Holtzman (2009a) solved these
problems analytically using approximations that
are considered physically and mathematically
reasonable. In the elastic case, Equations (3.14)
cannot be solved analytically, because the bound-
ary conditions are defined partly by displacement
f
i
(
r
R
)
at
X
C
(
r
R
)
=
=
0
0,
(3.14)
where
τ
ij
represents the microscopic stress in the
grain (tension positive),
f
i
(
=
τ
ij
n
j
) the traction on
the grain surface (positive outwards), and
f
i
the
differential traction defined by
f
i
p
L
n
i
.
=
f
i
+
(3.15)
The parameters
µ
S
repre-
sent the intrinsic Lame constants of the grain.
λ
S
(
=
k
S
−
2
µ
S
/
3) and
