Efficient homogenization techniques for elastic composites: Maxwell scheme vs. effective field method

The work is devoted to the comparative analysis of four homogenization techniques (the original and generalized Maxwell schemes and the one-particle and multi-particle effective field methods) in application to the problem of calculation of the effective elastic stiffness tensor of matrix composites. The generalized Maxwell scheme and multi-particle effective field method reduce the problem to calculation of elastic fields in a finite volume of the composite embedded in the infinite homogeneous matrix medium and subjected to a constant external stress or strain field. For the solution of this problem, an efficient numerical method based on Gaussian approximating functions and fast Fourier transform technique is used. The results of the methods are compared for composites with ellipsoidal inclusions which elastic moduli are much smaller or much larger than the moduli of the matrix. The case of hybrid composites with two different families of ellipsoidal inclusion is also considered. It is shown that for various shapes and elastic properties of inclusions, the generalized Maxwell scheme and multi-particle effective field method give close results. But in the case of hard inclusions of large volume fractions the results of these methods deviate substantially from the original Maxwell scheme that does not take into account interactions between inclusions.