upper A Superscript eff Baseline equals upper V Subscript f Baseline upper E Subscript upper A Superscript f Baseline alpha Subscript upper A Superscript f Baseline plus upper V Subscript m Baseline upper E Subscript m Baseline alpha Subscript m Baseline plus 2 lamda left-parenthesis nu Subscript m Baseline minus nu Subscript upper A Superscript f Baseline right-parenthesis left-parenthesis alpha Subscript m Baseline plus nu Subscript m Baseline alpha Subscript m Baseline minus alpha Subscript upper T Superscript f Baseline minus nu Subscript upper A Superscript f Baseline alpha Subscript upper A Superscript f Baseline right-parenthesis upper V Subscript f Baseline upper V Subscript m Baseline comma"/>(4.190)
where the parameter λ is defined by
4.10 Bounds for Two-phase Fibre-reinforced Composites
Bounds for the effective properties of unidirectional fibre-reinforced composites have been derived using variational techniques by Hashin and Shtrikman [7], and in the review by Hashin [3]. In this section, these bounds are now associated with the results in this chapter derived using Maxwell’s methodology. The results of Section 4.8, derived using Maxwell’s methodology or using the concentric cylinder model of a composite, correspond exactly to one of the bounds. Furthermore, if one bound is known, then the other is obtained simply by interchanging the properties and volume fractions. It is of interest to determine the condition that specifies whether the extreme value is an upper or lower bound.
4.10.1 Thermal Conductivity
The thermal conductivity bounds are very easily determined from the result (4.183), which is such that the denominator of the third term on the right-hand side is the only term to change value when the properties and volume fractions are interchanged. It then follows that the bounds may be written as
whenever κm≤κTf. Whenever κm≥κTf, the bounds defined by (4.193) should be reversed.
4.10.2 Axial Young’s Modulus
The bounds for the axial Young’s modulus are given by
