Biomedical Engineering Reference
In-Depth Information
1
, where
S
is now the anisotropic
elastic compliance tensor, is the Reuss lower bound on the effective (isotropic)
bulk modulus of the anisotropic elastic material
S
and that the Voigt effective bulk
modulus of an anisotropic elastic material,
K
Veff
¼ðU S
1
Hill (
1952
) showed that
K
Reff
¼ðU
S
U
Þ
UÞ=
9 , is the upper
bound,
1
S
S
1
ðU
U
K
Veff
¼ðU
U
Þ
¼
K
Reff
K
eff
Þ=
9
:
(8.20)
In the case of isotropy the two bounds coincide with the isotropic bulk modulus,
K
, thus:
S
1
U
U
1
K
Reff
¼
1
K
R
¼
1
K
eff
¼
1
K
Veff
¼
1
1
K
V
¼ðU S
UÞ
¼
9
3
ð
1
2
nÞ
1
K
;
¼
(8.21)
E
where
E
is the isotropic Young's modulus and
is the Poisson's ratio and where the
subscript eff is no longer applicable because these
K
's are the actual bulk moduli
rather than the effective bulk moduli. The Reuss effective bulk modulus of an
anisotropic elastic material
K
Reff
occurs naturally in anisotropic poroelastic theory
as shown, but not noted, by Thompson and Willis (
1991
). In analogy with this result
(
8.19
),
U
n
U
is defined as the inverse of the effective bulk modulus (1/
K
) when
the material is not isotropic. Thus, for the orthotropic drained elastic compliance
tensor
S
S
d
, and the matrix of the orthotropic elastic compliance tensor
S
d
, the
following definitions, which were employed in (
8.17
), are introduced:
n
23
n
31
n
12
1
1
E
1
þ
1
E
2
þ
1
E
3
2
2
2
K
eff
¼ U S
d
U ¼
E
2
E
3
E
1
;
(8.22)
m
23
m
31
m
12
1
1
E
1
þ
1
E
2
þ
1
E
3
2
n
2
n
2
n
K
eff
¼ U
S
d
U
¼
E
2
E
3
E
1
:
(8.23)
,(
8.18
), may be recast in terms of the
effective bulk moduli. Using (
8.3
) note that,
Using these notations the formula for
L
1
K
Reff
2
S
d
A
A
K
Reff
þ U
S
d
C
d
S
d
U
¼
;
(8.24)
where (
8.19
) and (
8.22
) have been employed. Substituting this result into (
8.18
) and
employing (
8.17
) to remove
C
eff
, it follows that
1
K
Reff
þ f
1
K
f
1
K
Reff
U
S
d
C
d
S
d
U
L ¼
:
(8.25)
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