Bounds for the Elastic Shear Moduli of Isotropic and Quasi-Symmetric Composites,and Exact Modulus of the Coated Sphere Assemblage

Abstract

A relation between the geometric parameters appearing in the bounds for isotropic composites is found, which leads to some new results. Improved upper and lower bounds on the effective shear modulus of isotropic multicomponent materials are deduced, which are expressed in the moduli and volume fractions of the components and reduce to Hashin-Shtrikman bounds in the case of well-ordered composites (i.e., when the extremal bulk and shear moduli belong to the same phases). The respective improved bounds for the subclass of quasi-symmetric composites are given. The shear modulus of Hashin's celebrated coated sphere assemblage is determined. The model also verifies the optimality of the bounds for isotropic two-component materials over a certain range of parameters, including those of the well-ordered case.

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References

[1] Hashin, Z. and Shtrikman, S. : A variational approach to the elastic moduli of multiphase materials. J. Mech. Phys. Solids, 11, 127-140 (1963).

[2] Hill, R. : Elastic properties of reinforced solids: Some theoretical principles. J. Mech. Phys. Solids, 11, 357-372 (1963).

[3] Walpole, L. J. : On bounds for the overall elastic moduli of inhomogeneous systems. J. Mech. Phys. Solids, 14, 152-162 (1966).

[4] Hashin, Z. : The elastic moduli of heterogeneous materials. J. Appl. Mech., 29, 143-150 (1962).

[5] Roscoe, R. : Isotropic composites with elastic or viscoelastic phases: General bounds for the moduli and solutions for special geometries. Rheol. Acta, 12, 404-411 (1973).

[6] Lurie, K. A. and Cherkaev, A. K. : The problem of formation of an optimal isotropic multicomponent composite. Preprint 895, loffe Physical-Technical Institute, Leningrad, 1984.

[7] Norris, A. N. : A differential scheme for the effective moduli of composites. Mech. Mat., 4, 1-16 (1985).

[8] Milton, G. W. : Modeling the properties of composites by laminates, in Homogenization and Effective Moduli of Materials and Media, pp. 150-174, ed., J. L. Ericksen et al., Springer-*rlag, New York, 1986.

[9] Francfort, G. A. and Murat, F. : Homogenization and optimal bounds in linear elasticity. Arch. Rational Mech. Anal., 94,307-334 (1986).

10.

[10] Milton, G. W. and Phan-Thien, N. : New bounds on effective elastic moduli of two-component materials. Proc. Roy. Soc. Lon. A, 380, 305-331 (1982).

11.

[11] Pham, D. C. : Bounds on the effective shear modulus of multiphase materials. Int. J. Eng. Sci., 31, 11-17 (1993).

12.

[12] Pham, D. C. : Bounds for the effective conductivity and elastic moduli of fully-disordered multicomponent materials. Arch. Rational Mech. Anal., 127, 191-198 (1994).

13.

[13] Pham, D. C. : Overall properties of planar quasisymmetric randomly inhomogeneous media: Estimates and cell models. Phys. Rev. E, 56, 652-660 (1997).

14.

[14] Pham, D. C. and Phan-Thien, N. : Bounds and extremal elastic moduli of isotropic quasi-symmetric multicomponent materials. Int. J. Eng. Sci., 36, 273-281 (1998).

15.

[15] Miller, M. N. : Bounds for effective bulk modulus of heterogeneous materials. J. Math. Phys., 10, 2005-2013 (1969).

16.

[16] Silnutzer, N.: Ph.D. thesis, University of Pennsylvania, Philadelphia, 1972.

17.

[17] Pham, D. C. : Improved bounds for the elastic moduli of perfectly-random composites. J. Elasticity, 41, 1-12 (1995).

18.

[18] Pham, D. C. : On macroscopic conductivity and elastic properties of perfectly-random cell composites. Int. J. Solids Structures, 33, 1745-1755 (1996).

19.

[19] Christensen, R. M. : Mechanics of Composite Materials, John Wiley, New York, 1979.

20.

[20] Pham, D. C. : Bounds on the effective properties of some multiphase matrix mixtures of coated sphere geometry. Mech. Materials, 27, 249-260 (1998).