A pair-consistent differential scheme for the elastic moduli of concentrated rigid-sphere composites

Abstract

Abstract The differential scheme provides a convenient continuation of dilute composite mechanics to finite particle concentration, but its conventional form is exact only at first order in volume fraction. For rigid, perfectly bonded spheres, we construct a minimal modification that also reproduces the complete two-sphere coefficients of Chen and Acrivos. The correction accounts for both direct pair interaction and the change of Poisson ratio during incremental addition. In an incompressible matrix, the conventional scheme predicts a second-order shear coefficient of 35/8, whereas the pair calculation gives 5.01; the proposed scheme recovers the latter exactly. A separate packing factor is then introduced whose expansion begins at third order, so that neither exact low-concentration coefficient is disturbed. Numerical results distinguish the effect of pair consistency at moderate concentration from the common divergence of the moduli near a prescribed jamming fraction. The construction gives a tractable route from exact low-concentration mechanics to concentrated-composite models without an intractable term-by-term many-particle expansion.