Biomedical Engineering Reference
In-Depth Information
Figure 3.2
Elastic modulus bounds for a
material with
E
2
=
100 GPa and
E
1
=
10, 1,
or 0.1 GPa for modulus mismatch (
E
2
/
E
1
)
factors of 10, 100, or 1000.
Voigt - Reuss
Hashin - Shtrikman
100
E
2
/E
1
=
10
80
60
40
20
0
0.0
0.2
0.4
0.6
0.8
1.0
100
E
2
/E
1
=
100
80
60
40
20
0
0.0
0.2
0.4
0.6
0.8
1.0
100
E
2
/E
1
=
1000
80
60
40
20
0
0.0 0.2 0.4 0.6
Volume fraction second phase, V
2
0.8
1.0
where
AR
d
is the particle aspect ratio,
V
F
is the volume fraction reinforcing
phase, and the coefficients
η
L
and
η
T
are related to the filler and matrix phase
elastic moduli (
E
F
,
E
M
)as
=
l
/
(
E
F
/
E
M
)
−
1
(
E
F
/
E
M
)
+
2
AR
η
=
(3.13)
L
(
E
F
/
E
M
)
−
1
η
=
(3.14)
T
(
E
F
/
E
M
)
+
2
The Halpin-Tsai expressions are empirical and this reasonable but imperfect
agreement is to be expected at intermediate volume fractions of reinforcement
phase [71].
A more recent model is for a composite with a staggered high aspect-ratio
reinforcing phase [72, 73]. The longitudinal modulus is expressed as
V
F
V
F
G
M
AR
2
−
1
4(1
−
V
F
)
E
L
=
E
F
+
+
(1
−
V
F
)
E
M
(3.15)
where
G
M
is the shear modulus of the protein phase and other terms are as defined
above. This expression gives low values (i.e., beneath the other lower bounds) for
