Biomedical Engineering Reference
In-Depth Information
Fig. 8.3
Cartoon of the
loading (
8.11
) for a cube of
material (only a cross-section
is visible). Observe that the
loadings (
8.10
) and (
8.11
)
may be superposed to produce
the total loading (
8.9
),
illustrated in Fig.
8.1
. This
loading consists of a surface
traction that is the sum of
the actual surface traction
t
¼
T
n
and the superposed pore
pressure,
t
pn
on
O
o
, the external surface. The
surface traction on the pores
is zero
¼
T
n
þ
and the total strain due to the loadings (
8.10
) plus (
8.11
) is the total strain due to the
loading (
8.9
)
E
ð
8
:
9
Þ
¼ E
ð
8
:
10
Þ
þ E
ð
8
:
11
Þ
¼ S
d
T
ðS
d
S
m
ÞU
þ
p
:
(8.14)
Figures
8.1
,
8.2a
, and
8.3
correspond to the loadings (
8.9
), (
8.10
), and (
8.11
)
respectively, and the strains
E
ð
8
:
9
Þ
,
E
ð
8
:
10
Þ
, and
E
ð
8
:
11
Þ
have the same respective
correspondence. It is easy to see that (
8.14
) may be rewritten as
E
¼ S
d
ðT
þð1
C
d
S
m
ÞUp
Þ;
(8.15)
¼ E
ð
8
:
9
Þ
is the total strain. Comparing this result with (
8.1
) it may be seen
that the
Biot effective stress coefficient tensor A
is given by (
8.3
). Note that this
proof is based on superposition that is a characteristic of linear systems and thus
applies to all the considerations of linear compressible poroelasticity, isotropic or
anisotropic, but fails when the deformations are no longer infinitesimal.
where
E
Problems
8.2.1. Show that the representation (
8.4
) for
A
in the case when both
C
d
and
S
d
are
transversely isotropic with respect to a common axis reduces to (
8.5
) when
both
C
d
and
S
d
are isotropic.
8.2.2. Derive the expressions (
8.8
) for
K
d
and
G
d
in terms of the volume fraction
f
,
matrix bulk and shear moduli,
K
m
and
G
m
, and Poisson's ratio
m
from the
n
formulas (7.20).
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