minus mu Subscript m Baseline right-parenthesis squared upper V Subscript f Baseline upper V Subscript m Baseline Over upper V Subscript m Baseline mu Subscript upper A Superscript f Baseline plus upper V Subscript f Baseline mu Subscript m Baseline plus zero width space zero width space mu Subscript m Baseline EndFraction less-than-or-equal-to mu Subscript upper A Superscript eff Baseline less-than-or-equal-to upper V Subscript f Baseline mu Subscript upper A Superscript f Baseline plus upper V Subscript m Baseline mu Subscript m Baseline minus StartFraction left-parenthesis mu Subscript upper A Superscript f Baseline minus mu Subscript m Baseline right-parenthesis squared upper V Subscript f Baseline upper V Subscript m Baseline Over upper V Subscript m Baseline mu Subscript upper A Superscript f Baseline plus upper V Subscript f Baseline mu Subscript m Baseline plus zero width space zero width space mu Subscript upper A Superscript f Baseline EndFraction comma"/>(4.198)
which are valid only if μm≤μAf, and the bounds are reversed if μm≥μAf.
4.10.7 Axial Thermal Expansion
The bounds for the axial thermal expansion are given by
where
These bounds are valid only if (νAf−νm)(α^Tf−α^Tm)(μtf−μm)≥0, and the bounds are reversed if (νAf−νm)(α^Tf−α^Tm)(μtf−μm)≤0.
4.10.8 Transverse Thermal Expansion
The bounds for the transverse thermal expansion are given by
which are valid only if (kTf−kTm)(α^Tf−α^Tm)(μtf−μm)≤0, and the bounds are reversed if (kTf−kTm)(α^Tf−α^Tm)(μtf−μm)≥0.
4.11 Comparison of Predictions with Known Results
One of the more remarkable aspects of the results derived in this chapter is that Maxwell’s methodology, when applied to two-phase fibre-reinforced composites where the reinforcing cylindrical fibres all have the same size, is that for the case of transverse thermal conductivity, bulk and shear moduli, transverse thermal expansion and transverse thermal conductivity, the effective properties correspond exactly to one of the bounds derived using variational techniques. This bound is also identical to estimates for the properties obtained using the composite cylinders assemblage method. These results are remarkable as one might expect Maxwell’s methodology to be accurate only for sufficiently low volume fractions of reinforcement. The reason for this is that the methodology involves the examination of the stress and displacement fields in the matrix at large distances from the assembly of parallel fibres, and assuming that the perturbing effect of each fibre can be approximated by assuming all fibres are located at the same point. The nature of this approximation is such that one can expect that all interactions between fibres in the cluster are neglected, and that the resulting effective properties will be accurate only for low volume fractions. This appears to contradict the finding that the estimates correspond exactly with one of the variational bounds and with the results arising from the use of the composite cylinders assemblage model. The latter techniques are expected to lead to reasonable estimates of effective composite properties for all volume fractions. It should also be noted from (4.183) that κeff→κf when Vf→1. These facts lead to the conclusion that results derived using Maxwell’s methodology are not restricted to low fibre volume fractions.
The effective properties of a two-phase composite, derived using Maxwell’s methodology, may be expressed in the form of a mixtures estimate plus a correction term, as seen from the results in Section 4.8. It should be noted that the correction term is always proportional to the product VfVm of volume fractions, and a term that involves the square of property differences for the case of conductivity, bulk and shear moduli, and the product of the bulk compressibility difference and expansion coefficient difference for the case of thermal expansion. The form of these
