#### Vol. 12, No. 4, 2017

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Polarization approximations for elastic moduli of isotropic multicomponent materials

### Duc Chinh Pham, Nguyen Quyet Tran and Anh Binh Tran

Vol. 12 (2017), No. 4, 391–406
##### Abstract

Polarization approximations (PA) are proposed for the macroscopic elastic moduli of $d$-dimensional ($d=2$ or 3) isotropic multicomponent materials. Specifically we use Hashin–Shtrikman-type polarization trial fields, which were constructed earlier from the minimum energy principles to bound the effective moduli of the composites, as the approximate solutions to the field equations. The approximations contain free reference parameters, which can be found analytically, numerically, or experimentally, from the reference effective moduli at dilute and/or finite volume proportions of the component materials. In the basic one-point reference parameter version, the approximations should obey Hashin–Shtrikman (HS) bounds for all the ranges of the volume proportions of the component materials. In the refined versions involving variable reference parameters to improve the accuracy of the scheme, the approximations satisfy HS bounds over the ranges of components’ volume proportions between the extreme reference points. We provide numerous numerically and experimentally based examples to illustrate the applications of the proposed approach.

##### Keywords
isotropic composite material, macroscopic elastic moduli, polarization approximation, reference parameter