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macroscopic deformation of the solid framework
in the REV (Figure 3.3b-c). I call
σ
ij
S
p
L
+
δ
ij
σ
ij
S
f
ij
ε
the ''frame-
p
L
−
δ
ij
f
ij
to the applied
macroscopic stresses reflects the bulk mechan-
ical properties, in which the structure-sensitive
nature, typical of solid-liquid composites, comes
out. The constitutive equation is given, in a fully
3D anisotropic form, as
work strain.'' The response of
ε
=
+
p
L
p
L
1
3
k
S
p
L
(a)
(b)
(c)
f
ij
S
kl
p
L
ε
=
S
ijkl
(
σ
+
δ
kl
)
−
δ
ij
(3.3)
Fig. 3.4
Thought experiment to understand Equation
(3.3). (a) A macroscopic stress state described by a solid
stress
where S
ijkl
represents the mechanical compliance
tensor of the solid framework under a drained con-
dition,
k
S
represents the intrinsic bulk modulus
of the solid phase, and
σ
ij
S
(tension positive) and a liquid pressure
p
L
(compression positive) is considered to be a
superposition of (b) a pressure
p
L
uniformly applied to
both solid and liquid phases and (c) a differential stress
σ
δ
ij
is the Kronecker delta. A
summation convention for repeated subscripts is
employed. A ''drained condition'' refers to a liquid
kept at constant pressure, whereas an ''undrained
condition'' refers to a liquid where flux relative
to the solid framework is not allowed.
Equation (3.3) can be understood intuitively by
the following thought experiment. As schemat-
ically illustrated in Figure 3.4, the macroscopic
stress state specified by
S
ij
+
δ
ij
applied to the solid phase without changing
the liquid pressure (in other words, under a drained
condition). The solid phase in REV is surrounded by
two types of surface: the surface of the REV (shown by
the dotted line), and the phase boundary with the
liquid (continuous line). The solid stress
p
L
σ
ij
S
and liquid
pressure
p
L
are applied through the former and latter
surfaces, respectively. Modified from Takei (2009).
σ
ij
S
and
p
L
is considered
to be a superposition of the pressure
p
L
uniformly
applied to both the solid and liquid phases
(Figure 3.4b), and the differential stress
Therefore, I use here the term ''solid framework''
rather than ''skeleton.''
The idea of superposition, used in deriving
Equation (3.3), implicitly assumes that the intrin-
sic property of the solid phase is linear. However,
even when the solid phase is intrinsically linear,
various degrees of nonlinearity can be introduced
into Equation (3.3), because
S
ijkl
strongly depends
on the porosity and pore geometry. The structure-
sensitive character of
S
ijkl
is discussed in quanti-
tative terms in Section 3.4, where Equation (3.3)
is derived on the basis of a microstructural model.
Therefore, the validity of the thought experiment
considered here will be confirmed in Section 3.4.
S
ij
+
p
L
δ
ij
applied to the solid phase without changing the
liquid pressure, or in other words, in a drained
condition (Figure 3.4c). Equation (3.3) is obtained
by adding up the framework strains correspond-
ing to those two stress states (Figure 3.4b-c).
The second term on the right-hand side (RHS)
of Equation (3.3) represents
σ
f
ij
in response to the
uniform pressure (Figure 3.4b), which is equal
to the response of REV completely filled with
the solid phase (Nur & Byerlee, 1971), and hence
determined by
k
S
. The first term on the RHS
of Equation (3.3) represents
ε
f
ij
in response to
the differential stress, and hence
S
ijkl
describes
the response of the solid framework under a
drained condition (Figure 3.4c). In poroelasticity,
to emphasize the drained condition,
S
ijkl
is
sometimes called the ''skeleton property,'' where
skeleton means a solid framework with vacuum
pores. However, in a rheological context, a solid
framework with a drained melt is very different
from a solid framework with vacuum pores.
ε
3.3.2 Generalization to viscoelastic solids
Over a broad frequency range, from seismic wave
propagation (1
10
−
3
Hz) to mantle convection
and melt segregation (
−
10
−
9
Hz), the mechanical
response of mantle materials changes from elastic
to viscous, while the transitional property is
called anelasticity. Although there are still many
<
