Baseline equals 2 mu Subscript t Superscript eff Baseline left-parenthesis 1 plus nu Subscript t Superscript eff Baseline right-parenthesis comma StartFraction 1 Over k Subscript upper T Superscript eff Baseline EndFraction equals StartFraction 2 left-parenthesis 1 minus nu Subscript t Superscript eff Baseline right-parenthesis Over upper E Subscript upper T Superscript eff Baseline EndFraction minus StartFraction 4 left-parenthesis nu Subscript upper A Superscript eff Baseline right-parenthesis squared Over upper E Subscript upper A Superscript eff Baseline EndFraction period"/>(4.150)
It follows that
so that
enabling values of ETeff and νTeff to be determined.
4.7 Relationship between Two-phase and Multiphase Formulae
An interesting question is whether the formulae for the effective properties of multiphase composites can be derived from results that are valid only for two-phase composites. The results (4.10), (4.66)–(4.69), (4.91), (4.147) and (4.152) are mixtures relations of the type
for the following combinations of properties:
The properties PJ are effective properties of the unidirectional composite whereas PJf and PJm are the corresponding properties for the fibre and matrix, respectively. As Vf+Vm=1, relation (4.153) may now be written in the form
implying that the effective property PJ of a two-phase composite is the sum of the matrix value PJm and the product of the property difference PJf−PJm and the fibre volume fraction Vf. This approach can be extended to multiphase composites having N types of fibre reinforcement by generalising (4.155) to the following form:
where PJf(i) is the value of the property PJ for the fibre of type i and Vfi is the corresponding fibre volume fraction. As, for the multiphase composite,
relation (4.156) may be expressed in the mixtures form
When the result (4.158) is applied to the properties defined in (4.154), relations (4.10), (4.66)–(
