Page 196 - Plastics Engineering

Mechanical Behaviour of Composites 179
Using equation (3.9) and the fact that the thickness ratios will be equal to
the corresponding volume fractions

(3.12)

(3.13)

Fig. 3.7 shows how the longitudinal and transverse moduli vary with volume
fraction for a unidirectional fibre composite.

0 0.2 0.4 0.6 0.8 1
Volume fractiin of fibres (V,)

Fig. 3.7 Variation of longitudinal and transverse modulus in unidirectional composites

In practical terms the above analysis is too simplistic, particularly in regard
to the assumption that the stresses in the fibre and matrix are equal. Generally
the fibres are dispersed at random on any cross-section of the composite (see
Fig. 3.8) and so the applied force will be shared by the fibres and matrix but
not necessarily equally. Other inaccuracies also arise due to the mismatch of
the Poisson’s ratios for the fibres and matrix. Several other empirical equations
have been suggested to take these factors into account. One of these is the
Halpin-Tsai equation which has the following form

(3.14)