Properties for Design of Composite Structures. Neil McCartney

plus StartFraction 1 Over mu Subscript m Superscript asterisk Baseline EndFraction EndEndFraction plus StartStartFraction upper V Subscript m Baseline OverOver StartFraction 1 Over mu Subscript m Baseline EndFraction plus StartFraction 1 Over mu Subscript m Superscript asterisk Baseline EndFraction EndEndFraction comma"/>(4.145)

      where

      (4.146)

      On using (4.1), the result (4.145) may also be written in the form

(4.147)

      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

(4.148)

      where

      (4.149)

      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),

      (4.150)

      It follows that

      (4.151)

      so that

(4.152)

      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

(4.153)

      for the following combinations of properties:

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