Biomedical Engineering Reference
In-Depth Information
an orientation efficiency factor (
η
0
) and length efficiency parameter (
η
L
)is
often used:
E
c
¼ Z
o
Z
L
E
f
V
f
þ
E
m
ð
1
V
f
Þ
½
7
:
1
in which E
c
, E
f
and E
m
are the elastic moduli of the composite, fibers and
matrix, respectively, and V
f
is the fiber volume fraction. The value of
η
0
for
three-dimensional random orientation of fibers is 0.2 (Matthews and
Rawlings, 1999) and the value of
η
L
depends upon the fiber length (l),
diameter (D), interfiber spacing (2R) and shear modulus of the matrix (G
m
):
tanh
½ð
1
=
2
Þb
l
Z
L
¼
1
½
7
:
2
ð
1
=
2
Þb
l
1
=
2
8G
m
b ¼
½
:
7
3
E
f
D
2
log
e
ð
2R
=
D
Þ
Applying the modified rule of mixtures on CNT-glass composites by
taking three different values of CNTs (1000, 500 and 100GPa), and the
elastic modulus of glass (70GPa) provides the composite elastic moduli
shown in Fig. 7.12 (Subhani, 2012b). This wide range of values reflect the
expected differences between arc-grown and CVD-grown CNTs (Yu et al.,
2000a, Zhao and Zhu, 2011); note that the hollow core is generally excluded
when reporting the elastic modulus of CNTs but its effect should be
considered in the context of composite reinforcement. Higher composite
values should be obtained (dotted line 1 in Fig. 7.12) if CNTs possess the
theoretically estimated elastic modulus of 1000GPa; at 500GPa there is a
modest improvement (dotted line 2) while a value of 100GPa decreases the
elastic modulus of the composites (dotted line 3) to a value less than the
glass matrix; many CNT-glass/glass-ceramic composites in Fig. 7.12 indeed
demonstrate an elastic modulus less than that of the matrix glass. This
simple analysis assumes that the transverse properties of CNTs are neglible,
which may be reasonable given relatively weak interfacial bonding and the
hollow/collapsible nature of the structure. In addition, the length of the
CNTs should be considered in more detail; it is worth noting that CNTs
near the critical length may be favored for toughening but are sub-optimal
for stiffening. For an increased composite modulus in three-dimensional
randomly oriented CNT-glass/glass-ceramic matrix composites, a high
intrinsic elastic modulus (500-1000GPa) is required; the value is higher than
most CVD-grown CNTs, but the graphitizing effects of high-temperature
sintering on the carbon lattice may be beneficial. The porosity generated by
insufficient densification also contributes to reducing the elastic modulus.
