Carbon Nanotube Reinforced Composites: Metal and Ceramic ...
Carbon Nanotube Reinforced Composites: Metal and Ceramic ...
Carbon Nanotube Reinforced Composites: Metal and Ceramic ...
Create successful ePaper yourself
Turn your PDF publications into a flip-book with our unique Google optimized e-Paper software.
92j 3 Physical Properties of <strong>Carbon</strong> <strong>Nanotube</strong>–<strong>Metal</strong> Nanocomposites<br />
material due to a rise or drop in temperature. Mathematically, it can be written as:<br />
a ¼ Dl<br />
lDT<br />
ð3:1Þ<br />
where a is the CTE, Dl the thermal expansion displacement, l the original length <strong>and</strong><br />
DT the temperature change.<br />
The CTE of the microcomposites can be predicted from several thermo-elastic<br />
models such as Kerner, Turner <strong>and</strong> Schapery provided that the corresponding<br />
effective elastic moduli of the composite are known. The Kerner model assumes<br />
the reinforcement is spherical <strong>and</strong> wetted by a uniform layer of matrix. The composite<br />
is considered as macroscopically isotropic <strong>and</strong> homogeneous [15]. The CTE of<br />
the composite can be expressed as:<br />
ac ¼ ap þ amVm þðap amÞVpVmL ð3:2Þ<br />
L ¼<br />
Kp Km<br />
VpKp þ VmKm þ 3<br />
4 KpKmGm<br />
ð3:3Þ<br />
where V is the volume fraction, a the CTE of the component. The subscripts c, p <strong>and</strong><br />
m represent the composite, particulate reinforcement <strong>and</strong> matrix, respectively.<br />
K <strong>and</strong> G are the bulk <strong>and</strong> shear modulus of the components of the composite,<br />
respectively <strong>and</strong> relate to the Young s modulus <strong>and</strong> the Poisson s ratio u of isotropic<br />
materials by:<br />
K ¼<br />
G ¼<br />
E<br />
3ð1 2uÞ<br />
ð3:4Þ<br />
E<br />
: ð3:5Þ<br />
2ð1 þ uÞ<br />
The Turner model assumes homogeneous strain throughout the composite <strong>and</strong><br />
that only uniform hydrostatic stresses exist in the phases [16]. The CTE of the<br />
composite is given by:<br />
ac ¼ amVmKm þ apVpKp<br />
: ð3:6Þ<br />
VmKm þ VpKp<br />
Shapery s model gives upper ( þ ) <strong>and</strong> lower ( ) limits on the CTE [17]. The upper<br />
limit of CTE can be expressed as:<br />
ð Þ<br />
where KC a ðþÞ<br />
C ¼ ap þðam apÞ KmðKp<br />
ð Þ<br />
KC Þ<br />
ð Þ<br />
KC ðKp KmÞ<br />
ð3:7Þ<br />
is Hashin <strong>and</strong> Shtrikman lower limit to the bulk modulus of the<br />
composite given by [18, 19]:<br />
ð Þ<br />
KC ¼ Km<br />
Vp<br />
þ 1 Vm<br />
þ Kp Km Km þ 4 3Gm : ð3:8Þ