cluster of all types of fibre is now considered to be enclosed in a cylinder of radius b such that the volume fraction of fibres of type i within the cylinder of radius b is given by Vfi=niai2/b2. The volume fractions must satisfy the relation
It then follows that (4.140) may be written in the form
The coefficients of the 1/r terms in relations (4.141) and (4.143) must be identical so that
It then follows that the effective transverse shear modulus for the multiphase composite is given by
where
On using (4.1), the result (4.145) may also be written in the form
4.6 Other Effective Elastic Properties for Multiphase Fibre-reinforced Composites
Four independent effective elastic properties can now be estimated using relations (4.69), (4.67), (4.91) and (4.147), namely, νAeff,kTeff,μAeff,μteff. It is clear that Maxwell’s methodology has not provided an expression for the axial modulus EAeff of a multiphase unidirectionally fibre-reinforced composite. This problem has, however, been overcome [6] by considering a special case of aligned spheroidal inclusions (see Chapter 15 for details and (15.100)) where it has been shown that the effective axial Young’s modulus EAeff may be obtained from the following formula
where
and where values of νAeff and kTeff have already been determined. The transverse Young’s modulus ETeff and transverse Poisson’s ratio νteff can be estimated by making use of the following relations, corresponding to (4.18) and (4.47),
