Summary
A micromechanical framework is proposed to investigate effective mechanical properties of elastic multiphase composites containing many randomly dispersed ellipsoidal inhomogeneities. Within the context of the representative volume element (RVE), four governing micromechanical ensemble-volume averaged field equations are presented to relate ensemble-volume averaged stresses, strains, volume fractions, eigenstrains, particle shapes and orientations, and elastic properties of constituent phases of a linear elastic particulate composite. A renormalization procedure is employed to render absolutely convergent integrals. Therefore, the micromechanical equations and effective elastic properties of a statistically homogeneous composite are independent of the shape of the RVE. Various micromechanical models can be developed based on the proposed ensemble-volume averaged constitutive equations. As a special class of models, inter-particle interactions are completely ignored. It is shown that the classical Hashin-Shtrikman bounds, Walpole's bounds, and Willi's bounds for isotropic or anisotropic elastic multiphase composites are related to the “noninteracting” solutions. Further, it is demonstrated that the Mori-Tanaka methodcoincides with the Hashin-Shtrikman bounds and the “noninteracting” micromechanical model in some cases. Specialization to unidirectionally aligned penny-shaped microcracks is also presented. An accurate, higher order (in particle concentration), probabilistic pairwise particle interaction formulation coupled with the proposed ensemble-volume averaged equations will be presented in a companion paper.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Hashin, Z.: Analysis of composite materials — a survey. J. Appl. Mech.50, 481–505 (1983).
Hashin, Z., Shtrikman, S.: On some variational principles in anisotropic and nonhomogeneous elasticity. J. Mech. Phys. Solids10, 335–342 (1962).
Hashin, Z., Shtrikman, S.: A variational approach to the theory of the elastic behaviour of polycrystals. J. Mech. Phys. Solids10, 343–352 (1962).
Hashin, Z., Shtrikman, S.: A variational approach to the theory of the elastic behaviour of multiphase materials. J. Mech. Phys. Solids11, 127–140 (1963).
Walpole, L. J.: On bounds for overall elastic moduli of inhomogeneous systems — I. J. Mech. Phys. Solids14, 151–162 (1966).
Walpole, L. J.: On bounds for overall elastic moduli of inhomogeneous systems — II. J. Mech. Phys. Solids14, 289–301 (1966).
Walpole, L. J.: On the overall elastic moduli of composite materials. J. Mech. Phys. Solids17, 235–251 (1969).
Walpole, L. J.: The elastic behaviour of a suspension of spherical particles. Q. J. Mech. Appl. Math.25, 153–160 (1972).
Beran, M. J., Molyneux, J.: Use of classical variational principles to determine bounds for the effective bulk modulus in heterogeneous media. Q. Appl. Math.24, 107–118 (1966).
Willis, J. R.: Bounds and self-consistent estimates for the overall properties of anisotropic composites. J. Mech. Phys. Solids25, 185–202 (1977).
McCoy, J. J.: On the displacement field in an elastic medium with random variations of material properties. Recent Advances in Engineering Sciences, vol. 5, New York: Gordon and Breach 1970.
Silnutzer, N.: Effective constants of statistically homogeneous materials. Ph.D. Thesis, Univ. of Pennsylvania, 1972.
Milton, G. W.: Bounds on the transport and optical properties of a two-component composite material. J. Appl. Phys.52, 5294–5304 (1981).
Milton, G. W.: Bounds on the electromagnetic, elastic, and other properties of two-component composites. Phys. Rev. Lett.46, 542–545 (1981).
Milton, G. W.: Bounds on the elastic and transport properties of two-component composites. J. Mech. Phys. Solids30, 177–191 (1982).
Milton, G. W., Phan-Thien, N.: New bounds on effective elastic moduli of two-components materials. Proc. R. Soc. London Ser.A 380, 305–331 (1982).
Torquato, S., Lado, F.: Effective properties of two-phase disordered composite media. II. Evaluation of bounds on the conductivity and bulk modulus of dispersions of impenetrable spheres. Phys. Rev. B33, 6428–6434 (1986).
Sen, A. K., Lado, F., Torquato, S.: Bulk properties of composite media. II. Evaluation of bounds on the shear moduli of suspensions of impenetrable spheres. J. Appl. Phys.62, 4135–4141 (1987).
Torquato, S.: Random heterogeneous media: microstructure and improved bounds on effective properties. Appl. Mech. Rev.44, 37–76 (1991).
Ponte Castaneda, P., Willis, J. R.: On the overall properties of nonlinearly viscous composites. Proc. R. Soc. London Ser.A 416, 217–244 (1988).
Ponte Castaneda, P.: The effective mechanical properties of nonlinear isotropic composites. J. Mech. Phys. Solids39, 45–71 (1991).
Willis, J. R.: On methods for bounding the overall properties of nonlinear composites. J. Mech. Phys. Solids39, 73–86 (1991).
Hill, R.: A self-consistent mechanics of composite materials. J. Mech. Phys. Solids13, 213–222 (1965).
Budiansky, B.: On the elastic moduli of some heterogeneous materials. J. Mech. Phys. Solids13, 223–227 (1965).
Budiansky, B., O'Connell, R. J.: Elastic moduli of a cracked solid. Int. J. Solids Struct.12, 81–97 (1976).
Roscoe, R.: The viscosity of suspensions of rigid spheres. Brit. J. Appl. Phys.3, 267–269 (1952).
Roscoe, R.: Isotropic composites with elastic or viscoelastic phases: general bounds for the moduli and solutions for special geometries. Rheol. Acta12, 404–411 (1973).
McLaughlin, R.: A study of the differential scheme for composite materials. Int. J. Eng. Sci.15, 237–244 (1977).
Hashin, Z.: The differential scheme and its application to cracked materials. J. Mech. Phys. Solids36, 719–734 (1988).
Mori, T., Tanaka, K.: Average stress in matrix and average elastic energy of materials with misfitting inclusions. Acta Metall.21, 571–574 (1973).
Taya, M., Chou, T.-W.: On two kinds of ellipsoidal inhomogeneities in an infinite elastic body: an application to a hybrid composite. Int. J. Solids Struct.17, 553–563 (1981).
Taya, M., Mura, T.: On stiffness and strength of an aligned short-fiber reinforced composite containing fiber-end cracks under uniaxial applied stress. J. Appl. Mech.48, 361–367 (1981).
Taya, M.: On stiffness and strength of an aligned short-fiber reinforced composite containing penny-shaped cracks in the matrix. J. Comp. Mat.15, 198–210 (1981).
Weng, G. J.: Some elastic properties of reinforced solids, with special reference to isotropic ones containing spherical inclusions. Int. J. Eng. Sci.22, 845–856 (1984).
Benveniste, Y.: On the Mori-Tanaka's method in cracked bodies. Mech. Res. Com.13, 193–201 (1986).
Zhao, Y. H., Tandon, G. P., Weng, G. J.: Elastic moduli for a class of porous materials. Acta Mech.76, 105–131 (1989).
Weng, G. J.: The theoretical connection between Mori-Tanaka's theory and the Hashin-Shtrikman-Walpole bounds. Int. J. Eng. Sci.28, 1111–1120 (1990).
Qiu, Y. P., Weng, G. J.: On the application of Mori-Tanaka's theory involving transversely isotropic spheroidal inclusions. Int. J. Eng. Sci.289, 1121–1137 (1990).
Christensen, R. M., Lo, K. H.: Solutions for effective shear properties in three phase sphere and cylinder models, J. Mech. Phys. Solids27, 315–330 (1979).
Laws, N., Dvorak, G. J.: The effect of fiber breaks and aligned penny-shaped cracks on the stiffness and energy release rates in unidirectional composites. Int. J. Solids Struct.23, 1269–1283 (1987).
Nemat-Nasser, S., Hori, M.: Elastic solids with microdefects. In: Micromechanics and inhomogeneity (Weng, G. J., Taya, M., Abe, H., eds.), pp. 297–320. New York: Springer 1990.
Christensen, R. M.: A critical evaluation for a class of micromechanics models. J. Mech. Phys. Solids38, 379–404 (1990).
Dewey, J. M.: The elastic constants of materials loaded with non-rigid fillers. J. Appl. Mech.18, 578–581 (1947).
Kerner, E. H.: The elastic and thermo-elastic properties of composite media. Proc. R. Soc. London Ser.B 69, 807–808 (1956).
Eshelby, J. D.: The determination of the elastic field of an ellipsoidal inclusion, and related problems. Proc. R. Soc. London Ser.A 241, 376–396 (1957).
Hashin, Z.: The moduli of an elastic solid, containing spherical particles of another elastic material. In: IUTAM Non-homogeneity in elasticity and plasticity symposium (Olszak, W., ed.), pp. 463–478, Warsaw 1959.
Batchelor, G. K., Green, J. T.: The determination of the bulk stress in a suspension of spherical particles to orderc 2. J. Fluid Mech.56, 401–427 (1972).
Willis, J. R., Acton, J. R.: The overall elastic moduli of a dilute suspension of spheres. Q. J. Mech. Appl. Math.29, 163–177, (1976).
Chen, H.-S., Acrivos, A.: The solution of the equations of linear elasticity for an infinite region containing two spherical inclusions. Int. J. Solids Struct.14, 331–348 (1978).
Chen, H.-S., Acrivos, A.: The effective elastic moduli of composite materials containing spherical inclusions at non-dilute concentrations. Int. J. Solids Struct.14, 349–364 (1978).
Rodin, G. J., Hwang, Y.-L.: On the problem of linear elasticity for an infinite region containing a finite number of non-intersecting spherical inhomogeneities. Int. J. Solids Struct.27, 145–159 (1991).
Nemat-Nasser, S., Taya, M.: On effective moduli of an elastic body containing periodically distributed voids. Q. Appl. Math.39, 43–59 (1981).
Nemat-Nasser, S., Taya, M.: On effective moduli of an elastic body containing periodically distributed voids: comments and corrections. Q. Appl. Math.43, 187–188 (1985).
Iwakuma, T., Nemat-Nasser, S.: Composites with periodic microstructure. Comp. Struct.126, 13–19 (1983).
Nunan, K. C., Keller, J. B.: Effective elasticity tensor of a periodic composite. J. Mech. Phys. Solids32, 259–280 (1984).
Sangani, A. S., Lu, W.: Elastic coefficients of composites containing spherical inclusions in a periodic array. J. Mech. Phys. Solids35, 1–21 (1987).
Hashin, Z.: The elastic moduli of heterogeneous materials. J. Appl. Mech.29, 143–150 (1962).
Sen, A. K., Torquato, S.: Effective conductivity of anisotropic two-phase composite media. Phys. Rev.B 39, 4504–4515 (1989).
Ju, J. W., Chen, T. M.: Effective elastic moduli of two-phase composites containing randomly dispersed spherical inhomogeneities. Acta Mech.103, 123–144 (1994).
Eshelby, J. D.: Elastic inclusions and inhomogeneities. In: Progress in solid mechanics (Sneddon, I. N., Hill, R., eds.). Amsterdam: North-Holland 1961.
Mura, T.: Micromechanics of defects in solids. The Hague: Martinus Nijhoff 1982.
Zhu, Z. G., Weng, G. J.: Creep anisotropy of a metal-matrix composite containing dilute concentration of aligned spheroidal inclusions. Mech. Mater.9, 93–105 (1990).
Tandon, G. P., Weng, G. J.: Average stress in the matrix and effective moduli of randomly oriented composites. Composite Sci. Tech.27, 111–132 (1986).
Hashin, Z.: On elastic behavior of fibre reinforced materials of arbitrary transverse phase geometry. J. Mech. Phys. Solids13, 119–134 (1965).
Benveniste, Y.: A new approach to the application of Mori-Tanaka's theory in composite materials. Mech. Mater.6, 147–157 (1987).
Norris, A. N.: An examination of the Mori-Tanaka effective medium approximation for multiphase composites. J. Appl. Mech.56, 83–88 (1989).
Krajcinovic, D., Fanella, D.: A micromechanical damage model for concrete. Eng. Fract. Mech.25, 585–596 (1986).
Ju, J. W.: A micromechanical damage model for uniaxially reinforced composites weakened by interfacial arc microcracks. J. Appl. Mech.58, 923–930 (1991).
Hori, H., Nemat-Nasser, S.: Overall moduli of solids with microcracks: load-induced anisotropy. J. Mech. Phys. Solids331, 155–171 (1983).
Laws, N., Dvorak, G. J., Hejazi, M.: Stiffness changes in unidirectional composites caused by crack systems. Mech. Mater2, 123–137 (1983).
Laws, N., Brockenbrough, J. R.: The effect of micro-crack systems on the loss of stiffness of brittle solids. Int. J. Solids Struct.23, 1247–1268 (1987).
Sumarac, D., Krajcinovic, D.: A self-consistent model for microcrack-weakened solids. Mech. Mater.6, 39–52 (1987).
Sumarac, D., Krajcinovic, D.: A mesomechanical model for brittle deformation processes. Part II. J. Appl. Mech.56, 57–62 (1989).
Krajcinovic, D., Sumarac, D.: A mesomechanical model for brittle deformation processes. Part I. J. Appl. Mech.56, 51–62 (1989).
Ju, J. W.: On two-dimensional self-consistent micromechanical damage models for brittle solids. Int. J. Solids Struct.27, 227–258 (1991).
Ju, J. W., Lee, X.: On three-dimensional self-consistent micromechanical damage models for brittle solids. Part I: Tensile loadings. J. Eng. Mech. ASCE117, 1495–1515 (1991).
Lee, X., Ju, J. W.: On three-dimensional self-consistent micromechanical damage models for brittle solids. Part II: Compressive loadings. J. Eng. Mech. ASCE117, 1516–1537 (1991).
Kachanov, M.: Elastic solids with many cracks: a simple method of analysis. Int. J. Solids Struct.23, 23–43 (1987).
Ju, J. W., Chen, T. M.: On effective elastic moduli of two-dimensional brittle solids with interacting microcracks. Part I: Basic formulations. J. Appl. Mech. (in press).
Ju, J. W., Chen, T. M.: On effective elastic moduli of two-dimensional brittle solids with interacting microcracks. Part II: Evolutionary damage models. J. Appl. Mech. (in press).
Ju, J. W., Tseng, K. H.: A three-dimensional statistical micromechanical theory for brittle solids with interacting microcracks. Int. J. Damage Mech.1, 102–131 (1992).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Ju, J.W., Chen, T.M. Micromechanics and effective moduli of elastic composites containing randomly dispersed ellipsoidal inhomogeneities. Acta Mechanica 103, 103–121 (1994). https://doi.org/10.1007/BF01180221
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01180221