Materials for engineering, 3rd Edition - (Malestrom)
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198<br />
<strong>Materials</strong> <strong>for</strong> <strong>engineering</strong><br />
agreement with equation [6.2], a material with a modest increase in stiffness<br />
but with no increase in fabrication costs.<br />
Continuous fibres<br />
If the composite consists of an array of continuous fibres parallel to the x-<br />
axis and the moduli of the two phases are E 1 and E 2 , one may again substitute<br />
in equation [6.1], but, in this case, the fibres will extend across the entire<br />
length of the unit cube, so A 2(x) = f f , the volume fraction of fibres in the<br />
material.<br />
Substitution in equation [6.1] yields:<br />
i.e.<br />
∫<br />
1<br />
1 dx<br />
=<br />
E E + ( E – E ) f<br />
c 0<br />
1 2 1 f<br />
E c = E 1 + (E 2 – E 1 )f f<br />
= E 2 f f + E 1 (1 – f f )<br />
But f f + f matrix = 1, so we can write<br />
E c = E 1 f matrix + E 2 f f [6.3]<br />
which predicts a linear ‘law of mixtures’ relationship <strong>for</strong> the modulus of the<br />
composite. This relationship has been verified experimentally with many<br />
fibre–resin systems.<br />
A direct comparison of equations [6.2] and [6.3] has been made in predicting<br />
the moduli of composites consisting of a matrix of copper containing either<br />
continuous wires of tungsten or equiaxed particles of tungsten. The measured<br />
data are compared with the theoretical predictions in Fig. 6.7.<br />
To give an example of greater industrial significance, Fig. 6.8 illustrates<br />
the improvements in specific stiffness achieved in aluminium-based MMCs<br />
containing either particulate SiC or aligned monofilament SiC. More than<br />
50% improvement is readily obtained <strong>for</strong> particulate composites and over<br />
100% <strong>for</strong> fibre-rein<strong>for</strong>ced systems.<br />
Continuous lamellae<br />
If the composite consists of an array of alternate lamellae of the two phases,<br />
then the analysis will be identical to the case <strong>for</strong> arrays of fibres and a law<br />
of mixtures will again be predicted.<br />
The above analyses are based on the assumption that the Poisson ratio is<br />
identical in the two phases. This is not strictly true, since different Poisson<br />
contractions will result in additional stresses which have not been considered<br />
here. The error in E c in the direction parallel to the fibres or laminae is likely
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