Biomedical Engineering Reference
In-Depth Information
K
3
E
) given in Table 6.2. The bulk modulus of steel (magnesium) is
166 GPa (80 GPa). The volume fraction of the spherical inclusions is the ratio of
the volume of one sphere, (4/3)
¼
EG
/(9
G
r
3
, to the volume of the unit cell, (5
r
)
3
, thus
p
f
s
¼
ð
4
=
3
Þp
r
3
¼
ð
4
Þp
125
¼
=
3
4
375
p ¼
0
:
0335
:
125
r
3
Substitution of the accumulated information into the formulas (
7.20
) one finds
that
K
eff
and
G
eff
,
:
:
80
16
77
K
eff
166
¼
166
1
335
G
eff
77
¼
15
ð
1
:
3
Þ
1
0335
1
þ
166
þð
4
=
3
Þ
77
;
1
77
;
80
166
16
1
þ
7
5
ð:
3
Þþ
2
ð
4
5
ð:
3
ÞÞ
are given by 161 GPa and 74 GPa, respectively. The effective Young's modulus
E
eff
is determined to be 103 GPa from the formula
E
¼
9
KG
/(3
K
þ
G
) given in
Table 6.2.
As a second example of the effective Hooke's law (
7.13
), consider a composite
made up of a linear elastic homogenous, isotropic solid matrix material containing
cylindrical cavities aligned in the
x
3
direction (Fig.
7.3
). Although the matrix
material is assumed to be isotropic, the cylindrical cavities aligned in the
x
3
direction require that the material symmetry of the composite be transverse isotropy
effective transversely isotropic engineering elastic constants is in the following
form:
2
4
3
5
eff
12
E
eff
1
eff
13
E
eff
3
1
E
eff
1
n
n
0
0
0
eff
12
E
eff
1
eff
13
E
eff
3
n
1
E
eff
1
n
0
0
0
eff
13
E
eff
3
eff
13
E
eff
3
n
n
1
E
eff
3
0
0
0
S
¼
:
1
2
G
eff
23
0
0
0
0
0
1
2
G
eff
23
0
0
0
0
0
þ n
eff
12
1
0
0
0
0
0
E
eff
1
The effective elastic constants are expressed in terms of the matrix elastic
constants and the volume fraction of cylindrical cavities, which is denoted by
f
c
.
f
c
is assumed to be small and the distribution of cavities dilute
and random. Terms proportional to the square and higher orders of
The volume fraction
f
c
are neglected.
When
f
c
is this small, several different averaging methods show, using a plane
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