Biomedical Engineering Reference
In-Depth Information
This approach is reasonable because, for many continuum theories, the averaging
arguments are intuitively justifiable. This is generally not the case for biological tissues
and nanomechanics in general. In almost all continuum theories the notation for RVE
averaging is not employed, but it is implied. Thus in continuummodels the notation of
the typical field variable
f
really means
, the RVE average of
f
. In particular, in any
continuum theory involving the use of the stress tensor
T
, it is really the RVE averaged
T
h
f
i
hi
that is being represented, even though an RVE has not been specified.
Problem
7.3.1 Find the average stress
hTi
in a heterogeneous cube of volume
b
3
if each face
of the cube is subjected to a different pressure. The pressure on the face
normal to
e
i
and -
e
i
is
p
i
for
i
in a
heterogeneous cube when all surfaces are subjected to the same pressure.
¼
1, 2, 3. Also, find the average stress
h
T
i
7.4 Effective Elastic Constants
As a first example of the effective Hooke's law (
7.13
), consider a composite
material in which the matrix material is isotropic and the inclusions are spherical
in shape, sparse in number (dilute), and of a material with different isotropic elastic
constants. In this case the effective elastic material constants are also isotropic and
the bulk and shear moduli,
K
eff
and
G
eff
, are related to the matrix material bulk and
shear moduli,
K
m
and
G
m
, and Poisson's ratio
n
m
and to the inclusion bulk and shear
moduli,
K
i
and
G
i
, by (Christensen, 1971; pp. 46-47)
f
s
K
i
G
i
G
m
K
m
1
f
s
15
ð
1
n
m
Þ
1
K
eff
K
m
¼
G
eff
G
m
¼
1
þ
G
m
;
1
G
m
;
(7.20)
K
i
K
m
K
m
þð
4
=
3
Þ
G
i
1
þ
7
5
n
m
þ
2
ð
4
5
n
m
Þ
where
f
s
is the porosity associated with the spherical pores. Thus if the porosity
f
s
and the matrix and inclusion constants
K
m
,
G
m
,
K
i
and
G
i
are known, the
formulas (
7.20
) may be used to determine the effective bulk and shear moduli,
K
eff
and
G
eff
, recalling that for an isotropic material the Poisson's ratio
n
m
is related
to
K
m
and
G
m
by
2
G
m
), see Table 6.2.
As a simple example of these formulas consider the case when
n
m
¼
(3
K
m
-2
G
m
)/(6
K
m
þ
1/3 and,
since there are only two independent isotropic elastic constants,
G
m
and
K
m
are
related. A formula from Table 6.2 may be used to show that
G
m
¼
n
m
¼
(3/8)
K
m
.
n
m
¼
1/3 and
G
m
¼
(3/8)
K
m
in (
7.20
), they simplify as follows:
Substituting
f
s
K
i
G
i
G
m
K
m
1
f
s
15 1
G
eff
G
m
¼
K
eff
K
m
¼
þ
;
:
1
1
(7.21)
7
G
i
G
m
8
þ
K
i
2
3
1
þ
K
m
1
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