T Baseline plus kappa Subscript m Baseline EndFraction comma upper P 2 equals StartFraction 1 Over k Subscript upper T Baseline plus mu Subscript m Baseline EndFraction comma upper P 3 equals StartStartFraction nu Subscript upper A Baseline OverOver StartFraction 1 Over k Subscript upper T Baseline EndFraction plus StartFraction 1 Over mu Subscript m Baseline EndFraction EndEndFraction comma upper P 4 equals StartStartFraction alpha Subscript upper T Baseline plus nu Subscript upper A Baseline alpha Subscript upper A Baseline OverOver StartFraction 1 Over k Subscript upper T Baseline EndFraction plus StartFraction 1 Over mu Subscript m Baseline EndFraction EndEndFraction comma 2nd Row upper P 5 equals StartFraction 1 Over mu Subscript upper A Baseline plus mu Subscript m Baseline EndFraction comma upper P 6 equals StartFraction 1 Over mu Subscript t Baseline plus mu Subscript m Superscript asterisk Baseline EndFraction comma upper P 7 equals upper E Subscript upper A Baseline plus StartFraction 4 k Subscript upper T Baseline left-parenthesis nu Subscript upper A Baseline right-parenthesis squared mu Subscript m Baseline Over k Subscript upper T Baseline plus mu Subscript m Baseline EndFraction comma where mu Subscript m Superscript asterisk Baseline equals StartFraction k Subscript upper T Superscript m Baseline mu Subscript m Baseline Over k Subscript upper T Superscript m Baseline plus 2 mu Subscript m Baseline EndFraction period EndLayout"/>(4.154)
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)–(4.69), (4.91), (4.147) and (4.152) are generated. Thus, most multiphase properties can be derived from corresponding results for two-phase properties.
It should be noted that it has not been possible to derive the axial thermal expansion coefficient for a multiphase unidirectional composite, although the concentric cylinder model of a composite generates this property for a two-phase composite (see Appendix B). The challenge now is to determine the property combination P8 (≡αAeff) that enables the two-phase result (B.52) (see Appendix B) to be extended to multiphase composites using a relation of the form (4.158). Following inspection of the multiphase results so far obtained, one might expect that the required mixtures relation for a two-phase composite might have the following form:
It can be shown that the conjectured relation (4.159) for just two phases is an equivalent form of the result (B.52) derived using the concentric cylinder model of a two-phase unidirectional composite. Relation (4.158) can then be used to generate the following corresponding expression which is conjectured to be valid for multiphase composites: