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The Quarterly Journal of Mechanics and Applied Mathematics 1976 29(2):163-177; doi:10.1093/qjmam/29.2.163
© 1976 by Oxford University Press
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THE OVERALL ELASTIC MODULI OF A DILUTE SUSPENSION OF SPHERES

J. R. WILLIS and J. R. ACTON

( School of Mathematics, University of Bath
School of Physics, University of Bath )

A composite material consisting of an elastic matrix with particles of different elastic moduli embedded in it is considered. The overall moduli, which relate the mean stress in the composite to the mean strain, are expressed in terms of the solution of an integral equation involving only integrals which converge. The formulation is then applied to problems for which the particles are of known shape but whose spatial distribution is known only in statistical terms. Such problems can be solved approximately, for moderately dilute suspensions, by admitting only interactions between small groups of particles. In particular, estimates of the overall moduli correct to second order in the concentration can be obtained by considering the average, over the ensemble of all possible configurations, of the interaction between pairs of particles. In contrast to some previous work, this ensemble averaging has no associated convergence difficulties, since it models convergent spatial integrals in the integral equation. Approximate analytic results are given for a suspension of isotropic spheres in an isotropic matrix, which compare tolerably well with an exact result that is available for a special case. Results are also given for a well-separated suspension of aligned ellipsoids in an anisotropic matrix demonstrating, even for this geometry, their sensitivity to the statistical assumptions employed.


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