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Micromechanics and effective moduli of elastic composites containing randomly dispersed ellipsoidal inhomogeneities

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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.

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References

  1. Hashin, Z.: Analysis of composite materials — a survey. J. Appl. Mech.50, 481–505 (1983).

    Google Scholar 

  2. Hashin, Z., Shtrikman, S.: On some variational principles in anisotropic and nonhomogeneous elasticity. J. Mech. Phys. Solids10, 335–342 (1962).

    Google Scholar 

  3. Hashin, Z., Shtrikman, S.: A variational approach to the theory of the elastic behaviour of polycrystals. J. Mech. Phys. Solids10, 343–352 (1962).

    Google Scholar 

  4. Hashin, Z., Shtrikman, S.: A variational approach to the theory of the elastic behaviour of multiphase materials. J. Mech. Phys. Solids11, 127–140 (1963).

    Google Scholar 

  5. Walpole, L. J.: On bounds for overall elastic moduli of inhomogeneous systems — I. J. Mech. Phys. Solids14, 151–162 (1966).

    Google Scholar 

  6. Walpole, L. J.: On bounds for overall elastic moduli of inhomogeneous systems — II. J. Mech. Phys. Solids14, 289–301 (1966).

    Google Scholar 

  7. Walpole, L. J.: On the overall elastic moduli of composite materials. J. Mech. Phys. Solids17, 235–251 (1969).

    Google Scholar 

  8. Walpole, L. J.: The elastic behaviour of a suspension of spherical particles. Q. J. Mech. Appl. Math.25, 153–160 (1972).

    Google Scholar 

  9. 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).

    Google Scholar 

  10. Willis, J. R.: Bounds and self-consistent estimates for the overall properties of anisotropic composites. J. Mech. Phys. Solids25, 185–202 (1977).

    Google Scholar 

  11. 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.

    Google Scholar 

  12. Silnutzer, N.: Effective constants of statistically homogeneous materials. Ph.D. Thesis, Univ. of Pennsylvania, 1972.

  13. Milton, G. W.: Bounds on the transport and optical properties of a two-component composite material. J. Appl. Phys.52, 5294–5304 (1981).

    Google Scholar 

  14. Milton, G. W.: Bounds on the electromagnetic, elastic, and other properties of two-component composites. Phys. Rev. Lett.46, 542–545 (1981).

    Google Scholar 

  15. Milton, G. W.: Bounds on the elastic and transport properties of two-component composites. J. Mech. Phys. Solids30, 177–191 (1982).

    Google Scholar 

  16. 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).

    Google Scholar 

  17. 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).

    Google Scholar 

  18. 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).

    Google Scholar 

  19. Torquato, S.: Random heterogeneous media: microstructure and improved bounds on effective properties. Appl. Mech. Rev.44, 37–76 (1991).

    Google Scholar 

  20. Ponte Castaneda, P., Willis, J. R.: On the overall properties of nonlinearly viscous composites. Proc. R. Soc. London Ser.A 416, 217–244 (1988).

    Google Scholar 

  21. Ponte Castaneda, P.: The effective mechanical properties of nonlinear isotropic composites. J. Mech. Phys. Solids39, 45–71 (1991).

    Google Scholar 

  22. Willis, J. R.: On methods for bounding the overall properties of nonlinear composites. J. Mech. Phys. Solids39, 73–86 (1991).

    Google Scholar 

  23. Hill, R.: A self-consistent mechanics of composite materials. J. Mech. Phys. Solids13, 213–222 (1965).

    Google Scholar 

  24. Budiansky, B.: On the elastic moduli of some heterogeneous materials. J. Mech. Phys. Solids13, 223–227 (1965).

    Google Scholar 

  25. Budiansky, B., O'Connell, R. J.: Elastic moduli of a cracked solid. Int. J. Solids Struct.12, 81–97 (1976).

    Google Scholar 

  26. Roscoe, R.: The viscosity of suspensions of rigid spheres. Brit. J. Appl. Phys.3, 267–269 (1952).

    Google Scholar 

  27. Roscoe, R.: Isotropic composites with elastic or viscoelastic phases: general bounds for the moduli and solutions for special geometries. Rheol. Acta12, 404–411 (1973).

    Google Scholar 

  28. McLaughlin, R.: A study of the differential scheme for composite materials. Int. J. Eng. Sci.15, 237–244 (1977).

    Google Scholar 

  29. Hashin, Z.: The differential scheme and its application to cracked materials. J. Mech. Phys. Solids36, 719–734 (1988).

    Google Scholar 

  30. Mori, T., Tanaka, K.: Average stress in matrix and average elastic energy of materials with misfitting inclusions. Acta Metall.21, 571–574 (1973).

    Google Scholar 

  31. 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).

    Google Scholar 

  32. 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).

    Google Scholar 

  33. 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).

    Google Scholar 

  34. 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).

    Google Scholar 

  35. Benveniste, Y.: On the Mori-Tanaka's method in cracked bodies. Mech. Res. Com.13, 193–201 (1986).

    Google Scholar 

  36. Zhao, Y. H., Tandon, G. P., Weng, G. J.: Elastic moduli for a class of porous materials. Acta Mech.76, 105–131 (1989).

    Google Scholar 

  37. 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).

    Google Scholar 

  38. 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).

    Google Scholar 

  39. 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).

    Google Scholar 

  40. 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).

    Google Scholar 

  41. 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.

    Google Scholar 

  42. Christensen, R. M.: A critical evaluation for a class of micromechanics models. J. Mech. Phys. Solids38, 379–404 (1990).

    Google Scholar 

  43. Dewey, J. M.: The elastic constants of materials loaded with non-rigid fillers. J. Appl. Mech.18, 578–581 (1947).

    Google Scholar 

  44. Kerner, E. H.: The elastic and thermo-elastic properties of composite media. Proc. R. Soc. London Ser.B 69, 807–808 (1956).

    Google Scholar 

  45. 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).

    Google Scholar 

  46. 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.

  47. 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).

    Google Scholar 

  48. 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).

    Google Scholar 

  49. 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).

    Google Scholar 

  50. 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).

    Google Scholar 

  51. 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).

    Google Scholar 

  52. Nemat-Nasser, S., Taya, M.: On effective moduli of an elastic body containing periodically distributed voids. Q. Appl. Math.39, 43–59 (1981).

    Google Scholar 

  53. 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).

    Google Scholar 

  54. Iwakuma, T., Nemat-Nasser, S.: Composites with periodic microstructure. Comp. Struct.126, 13–19 (1983).

    Google Scholar 

  55. Nunan, K. C., Keller, J. B.: Effective elasticity tensor of a periodic composite. J. Mech. Phys. Solids32, 259–280 (1984).

    Google Scholar 

  56. Sangani, A. S., Lu, W.: Elastic coefficients of composites containing spherical inclusions in a periodic array. J. Mech. Phys. Solids35, 1–21 (1987).

    Google Scholar 

  57. Hashin, Z.: The elastic moduli of heterogeneous materials. J. Appl. Mech.29, 143–150 (1962).

    Google Scholar 

  58. Sen, A. K., Torquato, S.: Effective conductivity of anisotropic two-phase composite media. Phys. Rev.B 39, 4504–4515 (1989).

    Google Scholar 

  59. Ju, J. W., Chen, T. M.: Effective elastic moduli of two-phase composites containing randomly dispersed spherical inhomogeneities. Acta Mech.103, 123–144 (1994).

    Google Scholar 

  60. Eshelby, J. D.: Elastic inclusions and inhomogeneities. In: Progress in solid mechanics (Sneddon, I. N., Hill, R., eds.). Amsterdam: North-Holland 1961.

    Google Scholar 

  61. Mura, T.: Micromechanics of defects in solids. The Hague: Martinus Nijhoff 1982.

    Google Scholar 

  62. 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).

    Google Scholar 

  63. 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).

    Google Scholar 

  64. Hashin, Z.: On elastic behavior of fibre reinforced materials of arbitrary transverse phase geometry. J. Mech. Phys. Solids13, 119–134 (1965).

    Google Scholar 

  65. Benveniste, Y.: A new approach to the application of Mori-Tanaka's theory in composite materials. Mech. Mater.6, 147–157 (1987).

    Google Scholar 

  66. Norris, A. N.: An examination of the Mori-Tanaka effective medium approximation for multiphase composites. J. Appl. Mech.56, 83–88 (1989).

    Google Scholar 

  67. Krajcinovic, D., Fanella, D.: A micromechanical damage model for concrete. Eng. Fract. Mech.25, 585–596 (1986).

    Google Scholar 

  68. Ju, J. W.: A micromechanical damage model for uniaxially reinforced composites weakened by interfacial arc microcracks. J. Appl. Mech.58, 923–930 (1991).

    Google Scholar 

  69. Hori, H., Nemat-Nasser, S.: Overall moduli of solids with microcracks: load-induced anisotropy. J. Mech. Phys. Solids331, 155–171 (1983).

    Google Scholar 

  70. Laws, N., Dvorak, G. J., Hejazi, M.: Stiffness changes in unidirectional composites caused by crack systems. Mech. Mater2, 123–137 (1983).

    Google Scholar 

  71. 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).

    Google Scholar 

  72. Sumarac, D., Krajcinovic, D.: A self-consistent model for microcrack-weakened solids. Mech. Mater.6, 39–52 (1987).

    Google Scholar 

  73. Sumarac, D., Krajcinovic, D.: A mesomechanical model for brittle deformation processes. Part II. J. Appl. Mech.56, 57–62 (1989).

    Google Scholar 

  74. Krajcinovic, D., Sumarac, D.: A mesomechanical model for brittle deformation processes. Part I. J. Appl. Mech.56, 51–62 (1989).

    Google Scholar 

  75. Ju, J. W.: On two-dimensional self-consistent micromechanical damage models for brittle solids. Int. J. Solids Struct.27, 227–258 (1991).

    Google Scholar 

  76. 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).

    Google Scholar 

  77. 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).

    Google Scholar 

  78. Kachanov, M.: Elastic solids with many cracks: a simple method of analysis. Int. J. Solids Struct.23, 23–43 (1987).

    Google Scholar 

  79. 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).

  80. 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).

  81. 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).

    Google Scholar 

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  1. J. W. Ju & T. M. Chen

    Present address: Department of Civil Engineering and Operations Research, Princeton University, 08 544, Priceton, NJ, USA

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    1. J. W. Ju
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    2. T. M. Chen
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    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

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