tailieunhanh - Construction of bounds on the effective shear modulus of isotropic multicomponent materials

In our previous paper, we constructed bounds on the effective bulk modulus of isotropic multicomponent composites using minimum energy principles and modified Hashin-Shtrikman polarization trial fields. In this paper, following the variational approach, we construct more sophisticated bounds on the effective shear modulus. Applications to particular models are presented. | Volume 35 Number 4 4 Vietnam Journal of Mechanics, VAST, Vol. 35, No. 4 (2013), pp. 275 – 283 CONSTRUCTION OF BOUNDS ON THE EFFECTIVE SHEAR MODULUS OF ISOTROPIC MULTICOMPONENT MATERIALS Vu Lam Dong∗ , Pham Duc Chinh Institute of Mechanics, VAST, 18 Hoang Quoc Viet, Hanoi, Vietnam ∗ E-mail: vldong@ Abstract. In our previous paper, we constructed bounds on the effective bulk modulus of isotropic multicomponent composites using minimum energy principles and modified Hashin-Shtrikman polarization trial fields. In this paper, following the variational approach, we construct more sophisticated bounds on the effective shear modulus. Applications to particular models are presented. Keywords: Isotropic multicomponent material, effective shear elastic modulus, threepoint correlation parameters. 1. INTRODUCTION Macroscopic (effective) elastic moduli keff and µeff of isotropic multicomponent materials are important mechanical properties of the materials. It is difficult to find exactly these moduli because of complicated micro-geometries of composites. The most wellknown estimates are the volume-weighted arithmetic or harmonic average formulae of Voigt and Reuss (Hill first order) bounds and Hashin-Shtrikman (second order) bounds [1–5]. Pham [3] extended Hashin-Shtrikmans inequalities to incorporate a number of coefficients depending on the fluctuation fields to improve the bounds. In [1] we had constructed new bounds for effective bulk elastic modulus of isotropic multicomponent materials which involve three-point correlation parameters. Continuing the research in this direction we will use more general multi-free parameter trial fields to construct new tight bounds on effective shear elastic properties of isotropic multicomponent materials. Applications of the bounds are performed for some representative material models. 2. CONSTRUCTION OF NEW BOUNDS The α-component of the multicomponent composite has elastic moduli kα, µα, α = 1, ., N . The local elastic