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In this work, some popular analytic formulas such as Maxwell (MA), Mori-Tanaka approximation (MTA), and a recent method, named the Polarization approximation (PA) will be applied to estimate the elastic moduli for some porous media. These approximations are simple and robust but can be lack reliability in many cases. The Hashin-Shtrikman (H-S) bounds do not supply an exact value but a range that has been admitted by researchers in material science. Meanwhile, the effective properties by unit cell method using the finite element method (FEM) are considered accurate. Different shapes of void inclusions in two or three dimensions are employed to investigate. Results generated by H-S bounds and FEM will be utilized as references. The comparison suggests that the method constructed from the minimum energy principle PA can give a better estimation in some cases. The discussion gives out some remarks which are helpful for the evaluation of effective elastic moduli.

On the respect to the Hashin-Shtrikman bounds of some analytical methods applying to porous media for estimating elastic moduli