Neil McCartney

Properties for Design of Composite Structures

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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

      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(4.146)

      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

      k Subscript upper T Superscript m Baseline equals k Subscript m Baseline plus one-third mu Subscript m Baseline comma(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),

      upper E Subscript upper T Superscript eff 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

      StartFraction 1 plus nu Subscript upper T Superscript eff Baseline Over upper E Subscript upper T Superscript eff Baseline EndFraction equals StartFraction 1 Over 2 mu Subscript upper T Superscript eff Baseline EndFraction comma StartFraction 1 minus nu Subscript upper T Superscript eff Baseline Over upper E Subscript upper T Superscript eff Baseline EndFraction equals StartFraction 1 Over 2 k Subscript upper T Superscript eff Baseline EndFraction plus StartFraction 2 left-parenthesis nu Subscript upper A Superscript eff Baseline right-parenthesis squared Over upper E Subscript upper A Superscript eff Baseline EndFraction comma(4.151)

      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: