Maxwell's far-field methodology predicting elastic properties of multiphase composites reinforced with aligned transversely isotropic spheroids

Abstract

Maxwell's methodology, developed to estimate the effective electrical conductivity of isotropic particulate composites, was used with a far-field elasticity result of Eshelby to derive closed-form formulae for effective transversely isotropic elastic properties of multiphase composites comprising aligned transversely isotropic spheroidal inclusions embedded in an isotropic matrix. Very simple expressions were derived for the effective shear moduli. Closed-form analytical results for all elastic constants are shown, using exact numerical methods, to be identical to more complex results derived by Qiu and Weng on applying Mori–Tanaka theory to spheroidal reinforcements. This is a contradictory result as Maxwell's approach neglects inclusion interactions, whereas Mori–Tanaka theory is designed, to some extent, to take such interactions into account. The rational conclusion is that inclusion interaction effects for volume fractions of practical relevance do not affect the far-field to any significant degree so that Maxwell's methodology, when combined with Eshelby's analysis, has much wider applicability than expected. Results for isotropic composites having distributions of spherical particles, and transversely isotropic composites having distributions of aligned fibres, correspond with known expressions, and can coincide with, or lie between, variational bounds for all volume fractions. A new simple expression having a ‘mixtures’ structure was obtained for the axial modulus of multiphase fibre-reinforced composites that reduces to concentric cylinders estimates when there are just two phases. To demonstrate accuracy, property results for a variety of composites are compared with accurate numerical results in the literature for two-phase composites having reinforcement volume fractions in the range of 0 to 0.7.

Keywords:

• Maxwell methodology
• Eshelby analysis
• elastic properties
• particulate composite
• spheroidal inclusion
• multiphase composite

Acknowledgements

The author would like to acknowledge many fruitful technical interactions with Professor Anthony Kelly, CBE, FRS, over many years of collaboration in a wide variety of theoretical fields relating to the properties of, and damage formation in, composite materials. In particular, Professor Kelly is thanked for encouragement to investigate the literature for property predictions that enabled the author to become aware of the ingenious Maxwell methodology for predicting effective conductivities of particulate composites, which has led to results in a recent joint publication and then to the ideas and results presented in this paper. The author would also like to acknowledge suggestions by referees that have enabled improvements to this paper.