Biomedical Engineering Reference
In-Depth Information
(4.12H) to include the effect of the pore pressure
p
. The six-dimensional vector
(three-dimensional symmetric second order tensor)
A
is called the Biot effective
stress coefficient tensor. When
p
¼
0 the stress-strain-pore pressure relation (
8.1
)
coincides with the elastic strain-stress relation (5.12H). The porous elastic material is
treated as a composite of an elastic solid and a pore fluid. The Biot effective stress
coefficient vector
A
is related not only to the drained effective elastic constants of
porous matrix material
S
d
, but also to the elastic constants of the solid matrix material
S
m
. The solid matrix material elastic constants of the porous material are based on an
RVE that is so small that it contains none of the pores (Fig.
4.2
). The Biot effective
stress coefficient tensor
A
is related to the difference between effective drained elastic
constants
S
d
and the solid matrix material elastic compliance tensor
S
m
by the formula
A
¼ð1
C
d
S
m
ÞU
;
(8.3)
T
is the six-dimensional vector representation of the
three-dimensional unit tensor
1
. A derivation of (
8.3
) is given at the end of this
section. Note that the symbol
U
is distinct from the unit tensor in six dimensions that
is denoted by
1
;
U
is described in further detail in Sect. A.11, just before (A.165).
The components of
A
depend upon both the matrix and drained elastic constants.
The assumption concerning the symmetry of
S
d
is interesting and complex. If the
symmetry of
S
d
is less than transversely isotropic and/or its axis of symmetry is not
coincident with the transversely isotropic axis of symmetry of
C
d
, then the 6D
vector
C
d
in expression (
8.3
) has, in general, six nonzero components and the
solution to problems is more complicated. However if both
U
where
¼½
1
;
1
;
1
;
0
;
0
;
0
C
d
S
d
and
are
transversely isotropic with respect to a common axis, then
T
A
¼½A
1
; A
1
; A
3
;
;
;
;
0
0
0
(8.4)
where
A
1
¼
ðC
11
þ C
12
Þð S
11
þ S
12
þ S
13
Þ C
13
ð
2
S
13
þ S
33
Þ;
1
2
C
13
ð S
11
þ S
12
þ S
13
Þ C
33
ð
A
3
¼
2
S
13
þ S
33
Þ:
1
In the case when both
C
d
and
S
d
are isotropic, it follows that
A
is given by
aU
(
a
1
in 3D),
A
¼ aU
where
K
d
/K
m
a ¼½
1
ð
Þ;
(8.5)
where
a
is called the isotropic effective stress coefficient. In most situations there
appears to be little disadvantage in assuming that
S
d
is isotropic. The assumption does
not mean that the real matrix material is actually isotropic, it only means that there
is little error in assuming that it is isotropic because the principal determinant of
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