# Three-point correlation bounds on the effective bulk modulus of isotropic multicomponent materials

## Three-point correlation bounds based on minimum energy principles are constructed to give estimates on the effective elastic bulk modulus of disordered multicomponent materials. The constructed trial fields are extensions of Hashin-Shtrikman polarization ones used in our previous approach and lead to tighter bounds. Some examples of applications are presented. | Vietnam Journal of Mechanics, VAST, Vol. 34, No. 2 (2012), pp. 67 – 77 THREE-POINT CORRELATION BOUNDS ON THE EFFECTIVE BULK MODULUS OF ISOTROPIC MULTICOMPONENT MATERIALS Pham Duc Chinh, Vu Lam Dong Institute of Mechanics, VAST Abstract. Three-point correlation bounds based on minimum energy principles are constructed to give estimates on the effective elastic bulk modulus of disordered multicomponent materials. The constructed trial fields are extensions of Hashin-Shtrikman polarization ones used in our previous approach and lead to tighter bounds. Some examples of applications are presented. Key words: Isotropic multicomponent material, effective elastic moduli, three-point correlation parameters. 1. INTRODUCTION Determination of effective elastic moduli of composite materials, especially those with disordered irregular micro-geometries, is a difficult task, and in many cases one has to construct upper and lower bounds on the moduli, using limited available statistical information about the micro-structure of the composites. That can be done with the help of variational approach [1 - 6]. The most simple bounds are Voigt-Reuss-Hill arithmetic average and harmonic average ones based on minimum energy and complementary energy principles, with constant trial stress and strain fields. Hashin-Shtrikman [1] constructed tight bounds that also involve only the information on the volume proportions and components properties of the isotropic composites, developing their special variational principles and ingenious polarization trial fields. Starting from the classical minimum energy principles and using Hashin-Shtrikman polarization trail fields, Pham [2] obtained the bounds tighter than Hashin-Shtrikman ones that involve additional three-point correlation parameters describing the micro-structure of a composite. In this work, we use new trial field more general than Hashin-Shtrikman polarization ones to derive three-point bounds on the effective elastic bulk modulus .

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