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Estimates of Mechanical Properties of Composite Materials

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Micromechanics of Composite Materials

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Abstract

Together with the methods described in the previous chapter, overall moduli and local field averages in the phases can be estimated by one of several approximate methods, which use different models of the microstructure. Among those described here are variants of the average field approximation, or AFA, which rely on strain or stress field averages in solitary ellipsoidal inhomogeneities, embedded in large volumes of different comparison media L 0. Among the most widely used procedures are the self-consistent and Mori-Tanaka methods, and the differential scheme, described in Sects. 7.1, 7.2 and 7.3. Those are followed by several double inclusion or double inhomogeneity models in Sect. 7.4, and by illustrative comparison with finite element evaluations for functionally graded materials in Sect. 7.5.

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References

  • Babuska, I. (1975). Homogenization and application: Mathematical and computational problems. In B. Hubbard (Ed.), Numerical solution of partial differential equations –III. New York: Academic.

    Google Scholar 

  • Bensoussan, A., Lions, J. L., & Papanicolaou, G. (1978). Asymptotic analysis for periodic structures. Amsterdam: Nort Holland.

    MATH  Google Scholar 

  • Benveniste, Y. (1987a). A new approach to the application of Mori-Tanaka theory in composite materials. Mechanics of Materials, 6, 147–157.

    Article  Google Scholar 

  • Benveniste, Y. (1987b). A differential effective medium theory with a composite sphere embedding. ASME Journal of Applied Mechanics, 54, 466–468.

    Article  Google Scholar 

  • Benveniste, Y., & Dvorak, G. J. (1989). On a correspondence between mechanical and thermal effects in two-phase composites. In Micromechanics and inhomogeneity (The Toshio Mura 65th anniversary volume, pp. 65–81). New York: Springer.

    Google Scholar 

  • Benveniste, Y., Dvorak, G. J., & Chen, T. (1989). Stress fields in composites with coated inclusions. Mechanics of Materials, 7, 305–317.

    Article  Google Scholar 

  • Benveniste, Y., Chen, T., & Dvorak, G. J. (1990). The effective thermal conductivity of composites reinforced by coated cylindrically orthotropic fibers. Journal of Applied Physics, 67, 2878–2884.

    Article  Google Scholar 

  • Benveniste, Y., Dvorak, G. J., & Chen, T. (1991a). On the effective properties of composites with coated cylindrically orthotropic fibers. Mechanics of Materials, 12, 289–297.

    Article  Google Scholar 

  • Benveniste, Y., Dvorak, G. J., & Chen, T. (1991b). On diagonal and elastic symmetry of the approximate effective stiffness tensor of heterogeneous media. Journal of the Mechanics and Physics of Solids, 39, 927–946.

    Article  MathSciNet  MATH  Google Scholar 

  • Berryman, J. G. (1980). Long wavelength propagation in composite elastic media II, Ellipsoidal inclusions. Journal of the Acoustical Society of America, 68, 1820–1831.

    Article  MATH  Google Scholar 

  • Boucher, S. (1974). On the effective moduli of isotropic two-phase elastic composites. Journal of Composite Materials, 8, 82–89.

    Article  Google Scholar 

  • Bruggeman, D. A. G. (1935). Berechnung verschiedener physikalisher Konstanten von heterogenen Substanzen I. Annalen der Physik, 24, 636–663.

    Article  Google Scholar 

  • Budiansky, B. (1965). On the elastic moduli of some heterogeneous materials. Journal of the Mechanics and Physics of Solids, 13, 223–227.

    Article  Google Scholar 

  • Budiansky, B., & O’Connell, R. J. (1976). Elastic moduli of a cracked solid. International Journal of Solids and Structures, 12, 81–97.

    Article  MATH  Google Scholar 

  • Chen, T., Dvorak, G. J., & Benveniste, Y. (1990). Stress fields in composites reinforced by coated cylindrically orthotropic fibers. Mechanics of Materials, 9, 17–32.

    Article  Google Scholar 

  • Chen, T., Dvorak, G. J., & Benveniste, Y. (1992). Mori-Tanaka estimates of the overall elastic moduli of certain composite materials. ASME Journal of Applied Mechanics, 59, 539–546.

    Article  MATH  Google Scholar 

  • Christensen, R. M. (1990). A critical evaluation for a class of micromechanics models. Journal of the Mechanics and Physics of Solids, 38, 379–404.

    Article  Google Scholar 

  • Christensen, R. M., & Lo, K. H. (1979). Solutions for effective shear properties in three phase sphere and cylinder models. Journal of the Mechanics and Physics of Solids, 27, 315–330. Erratum ibid. 34, 639 (1986).

    Google Scholar 

  • Christensen, R. M., & Waals, F. M. (1972). Effective stiffness of randomly oriented fiber composites. Journal of Composite Materials, 6, 518–532.

    Google Scholar 

  • Christensen, R. M., Schantz, H., & Schapiro, J. (1992). On the range of validity of the Mori-Tanaka method. Journal of the Mechanics and Physics of Solids, 40, 69–73.

    Article  Google Scholar 

  • Cleary, M. P., Chen, I. W., & Lee, S. M. (1980). Self-consistent techniques for heterogeneous solids. ASCE Journal of Engineering Mechanics, 106, 861–867.

    Google Scholar 

  • Daniel, I. M., & Ishai, O. (2006). Engineering mechanics of composite materials (2nd ed.). New York: Oxford University Press.

    Google Scholar 

  • Drugan, W. J., & Willis, J. R. (1996). A micromechanics-based nonlocal constituive equation and estimates of representative volume element size for elastic composites. Journal of the Mechanics and Physics of Solids, 44, 497–524.

    Article  MathSciNet  MATH  Google Scholar 

  • Dunn, M., & Ledbetter, H. (2000). Micromechanically based acoustic characterization of the fiber orientation distribution of morphologically textured short fiber composites: Prediction of thermomechanical and physical properties. Materials Science and Engineering A, 285, 56–61.

    Article  Google Scholar 

  • Einstein, A. (1905). Eine neue Berechnung der Moleküldimensionen. Annales de Physique, 19, 289–306.

    Google Scholar 

  • Ferrari, M., & Johnson, G. C. (1989). Effective elasticities of short-fiber composites with arbitrary orientation distribution. Mechanics of Materials, 8, 67–73.

    Article  Google Scholar 

  • Finot, M., & Suresh, S. (1996). Small and large deformation of thick and thin-film multi-layers: Effects of layer geometry, plasticity and compositional gradients. Journal of the Mechanics and Physics of Solids, 44, 683–722.

    Article  Google Scholar 

  • Fukui, Y., Takashima, K., & Ponton, C. B. (1994). Measurement of Young’s modulus and internal friction of an in situ Al-Al/Ni functionally gradient material. Journal of Materials Science, 29, 2281–2288.

    Article  Google Scholar 

  • Giannakopoulos, A. E., Suresh, S., Finot, M., & Olsson, M. (1995). Elastoplastic analysis of thermal cycling: Layered materials with compositional gradients. Acta Metallurgica et Materialia, 43, 1335–1354.

    Article  Google Scholar 

  • Gusev, A. A. (1997). Representative volume element size for elastic composites: A numerical study. Journal of the Mechanics and Physics of Solids, 45, 1449–1459.

    Article  MATH  Google Scholar 

  • Hashin, Z. (1972). Theory of fiber reinforced materials. NASA CR-1974. Washington, DC: National Aeronautics and Space Administration, 690.

    Google Scholar 

  • Hashin, Z. (1988). The differential scheme and its application to cracked materials. Journal of the Mechanics and Physics of Solids, 36, 719–734.

    Article  MathSciNet  MATH  Google Scholar 

  • Hashin, Z., & Rosen, B. W. (1964). The elastic moduli of fiber reinforced materials. ASME Journal of Applied Mechanics 31E, 223–232. Errata, 1965, ibid., 32E, 219.

    Google Scholar 

  • Hashin, Z., & Shtrikman, S. (1962a). On some variational principles in anisotropic and nonhomogeneous elasticity. Journal of the Mechanics and Physics of Solids, 10, 335–342.

    Article  MathSciNet  Google Scholar 

  • Hashin, Z., & Shtrikman, S. (1962b). A variational approach to the theory of the elastic behaviour of polycrystals. Journal of the Mechanics and Physics of Solids, 10, 343–352.

    Article  MathSciNet  Google Scholar 

  • Hatta, H., & Taya, M. (1986). Equivalent inclusion method for steady state heat conduction in composites. International Journal of Engineering Science, 24, 1159–1172.

    Article  MATH  Google Scholar 

  • Herakovich, C. T. (1998). Mechanics of fibrous composites. New York: Wiley.

    Google Scholar 

  • Hershey, A. V. (1954). The elasticity of an isotropic aggregate of anisotropic cubic crystals. ASME Journal of Applied Mechanics, 21, 236–240.

    MATH  Google Scholar 

  • Hill, R. (1963a). Elastic properties of reinforced solids: Some theoretical principles. Journal of the Mechanics and Physics of Solids, 11, 357–372 [1].

    Google Scholar 

  • Hill, R. (1964). Theory of mechanical properties of fibre-strengthened materials: I. Elastic behavior. Journal of the Mechanics and Physics of Solids, 12, 199–212.

    Article  MathSciNet  Google Scholar 

  • Hill, R. (1965a). Continuum micromechanics of elastic-plastic polycrystals. Journal of the Mechanics and Physics of Solids, 13, 89–101.

    Article  MATH  Google Scholar 

  • Hill, R. (1965b). Theory of mechanical properties of fibre-strengthened materials – III. Self-consistent model. Journal of the Mechanics and Physics of Solids, 13, 189–198.

    Article  Google Scholar 

  • Hill, R. (1965c). A self-consistent mechanics of composite materials. Journal of the Mechanics and Physics of Solids, 13, 213–222.

    Article  Google Scholar 

  • Hirano, T., & Wakashima, K. (1995). Mathematical modeling and design. MRS Bulletin, 40-42.

    Google Scholar 

  • Hirano, T., Teraki, J., & Yamada, T. (1990). On the design of functionally gradient materials. In: M. Yamanouchi, M. Koizumi, T. Hirai, & I. Shiota (Eds.), Proceedings of the 1st International Symposium on Functionally Gradient Materials, pp. 5-10.

    Google Scholar 

  • Hori, M., & Nemat-Nasser, S. (1993). Double-inclusion model and overall moduli of multi-phase composites. Mechanics of Materials, 14, 189–206.

    Article  Google Scholar 

  • Hu, G. K., & Weng, G. J. (2000). The connections between the double inclusion model and the Ponte Castaneda-Willis, Mori Tanaka, and Kuster-Toksoz models. Mechanics of Materials, 32, 495–503.

    Article  Google Scholar 

  • Kerner, E. H. (1956). The elastic and thermo-elastic properties of composite media. Proceedings of the Royal Society London, B69, 808–813.

    Article  Google Scholar 

  • Kröner, E. (1958). Berechnung der elastischen Konstanten der Vielkristalls aus den Konstanten der Einkristalls. Zeitschrift für Physik, 151, 504–518.

    Article  Google Scholar 

  • Kröner, E., Datta, B. K., & Kessel, D. (1966). On the bounds of the shear modulus of macroscopically isotropic aggregates of cubic crystals. Journal of the Mechanics and Physics of Solids, 14, 21–24.

    Article  Google Scholar 

  • Laws, N. (1973). On thermostatics of composite materials. Journal of the Mechanics and Physics of Solids, 21, 9–17.

    Article  Google Scholar 

  • Laws, N. (1974). The overall thermoelastic moduli of transversely isotropic composites according to the self-consistent method. International Journal of Engineering Science, 12, 79–87.

    Article  MATH  Google Scholar 

  • Laws, N. (1980). The elastic response of composite materials. Physics of Modern Materials, I. International Atomic Energy Agency, Vienna, IAEA-SMR 46/107, pp. 465–520.

    Google Scholar 

  • Laws, N., & Dvorak, G. J. (1987). The effect of fiber breaks and penny shaped cracks on the stiffness and energy release in unidirectional composites. International Journal of Solids and Structures, 23, 1269–1283.

    Article  Google Scholar 

  • Laws, N., & McLaughlin, R. (1978). Self-consistent estimates for viscoelastic creep compliances of composite materials. Proceedings of the Royal Society of London, A359, 251–273.

    MathSciNet  Google Scholar 

  • Laws, N., & McLaughlin, R. (1979). The effect of fiber length on the overall moduli of composite materials. Journal of the Mechanics and Physics of Solids, 27, 1–13.

    Article  MATH  Google Scholar 

  • Laws, N., Dvorak, G. J., & Hejazi, M. (1983). Stiffness changes in composites caused by crack systems. Mechanics of Materials, 2, 123–137.

    Article  Google Scholar 

  • Lee, Y. -D., & Erdogan, F. (1994/1995). Residual thermal stresses in FGM and laminated thermal barrier coatings. International Journal of Fracture, 69, 145-165.

    Google Scholar 

  • Markworth, A. J., & Saunders, J. H. (1995). A model of structure optimization for a functionally graded material. Materials Letters, 22, 103–107.

    Article  Google Scholar 

  • Markworth, A. J., Parks, W. P., & Ramesh, K. S. (1995). Review: Modelling studies applied to functionally graded materials. Journal of Materials Science, 30, 2183–2193.

    Article  Google Scholar 

  • McLaughlin, R. (1977). A study of the differential scheme for composite materials. International Journal of Engineering Science, 15, 237–244.

    Article  MATH  Google Scholar 

  • Milton, G. W. (2002). The theory of composites. Cambridge: Cambridge University Press.

    Book  MATH  Google Scholar 

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

    Article  Google Scholar 

  • Nemat-Nasser, S., & Hori, M. (1999). Micromechanics: Overall properties of hetero-geneous materials (2nd ed.). Amsterdam: Elsevier.

    Google Scholar 

  • Norris, A. N. (1985). A differential scheme for effective moduli of composites. Mechanics of Materials, 4, 1–16.

    Article  Google Scholar 

  • Norris, A. N. (1989). An examination of the Mori-Tanaka effective medium approximation for multiphase composites. ASME Journal of Applied Mechanics, 56, 83–88.

    Article  MATH  Google Scholar 

  • Norris, A. N., Callegari, A. J., & Sheng, P. (1985). A generalized differential effective medium theory. Journal of the Mechanics and Physics of Solids, 33(6), 525–543.

    Article  MATH  Google Scholar 

  • Oskay, C., & Fish, J. (2007). Eigendeformation-based reduced order homogenization for failure analysis of heterogeneous materials. Computer Methods in Applied Mechanics and Engineering, 196, 1216–1243.

    Article  MathSciNet  MATH  Google Scholar 

  • Ozisik, M. N. (1968). Boundary value problems of heat conduction. Scranton: International Textbook Co.

    Google Scholar 

  • Ponte Castaneda, P., & Willis, J. R. (1995). The effect of spatial distribution on the effective behavior of composite materials and cracked media. Journal of the Mechanics and Physics of Solids, 43, 1919–1951.

    Article  MathSciNet  MATH  Google Scholar 

  • Postma, G. W. (1955). Wave propagation in a stratified medium. Geophysics, 20, 780–806.

    Article  Google Scholar 

  • Reiter, T., & Dvorak, G. J. (1998). Micromechanical models for graded composite materials: II Thermomechanical loading. Journal of the Mechanics of Physics of Solids, 46, 1655–1673.

    Article  MATH  Google Scholar 

  • Reiter, T., Dvorak, G. J., & Tvergaard, V. (1997). Micromechanical models for graded composite materials. Journal of the Mechanics and Physics of Solids, 45, 1281–1302.

    Article  Google Scholar 

  • Roscoe, R. (1952). The viscosity of suspensions of rigid spheres. British Journal of Applied Physics, 3, 267–269.

    Article  Google Scholar 

  • Russel, W. B. (1973). On the effective moduli of composite materials: Effect of fiber length and geometry at dilute concentrations. Zeitschrift für Angewandte Mathematik und Physik, 24, 581.

    Article  Google Scholar 

  • Sanchez-Palencia, E. (1980). Homogenization techniques and vibration theory. Lecture Notes in Physics No. 127. Berlin: Springer.

    Google Scholar 

  • Sasaki, M., & Hirai, T. (1991). Fabrication and properties of functionally gradient materials. Journal of the Ceramic Society of Japan, 99, 1002–1013.

    Article  Google Scholar 

  • Sayers, C. M. (1992). Elastic anisotropy of short-fibre reinforced composites. Journal of the Mechanics and Physics of Solids, 29, 2933–2944.

    MATH  Google Scholar 

  • Suquet, P. (1987). Elements of homogenization for inelastic solid mechanics. In E. Sanchez-Palencia & A. Zaoui (Eds.), Homogenization techniques for composite media. New York: Springer.

    Google Scholar 

  • Tanaka, K., & Mori, M. (1972). Note on volume integrals of the elastic field around an ellipsoidal inclusion. Journal of Elasticity, 2, 199–200.

    Article  Google Scholar 

  • Tanaka, K., Tanaka, Y., Enomoto, K., Poterasu, V. F., & Sugano, Y. (1993a). Design of thermoelastic materials using direct sensitivity and optimization methods: Reduction of thermal stresses in functionally gradient materials. Computer Methods in Applied Mechanics and Engineering, 106, 271–284.

    Article  MATH  Google Scholar 

  • Tanaka, K., Tanaka, Y., Watanabe, H., Poterasu, V. F., & Sugano, Y. (1993b). An improved solution to thermoelastic material design infunctionally gradient materials: Scheme to reduce thermal stresses. Computer Methods in Applied Mechanics and Engineering, 109, 377–389.

    Article  MATH  Google Scholar 

  • Walpole, L. J. (1969). On the overall elastic moduli of composite materials. Journal of the Mechanics and Physics of Solids, 17, 235–251.

    Article  MATH  Google Scholar 

  • Walpole, L. J. (1981). Elastic behavior of composite materials: Theoretical foundations. In Advances in applied mechanics. New York: Academic, 21, 169–242.

    Google Scholar 

  • Walpole, L. J. (1984). Fourth-rank tensors of the thirty-two crystal classes; multiplication tables. Proceedings of the Royal Society London A, 391, 149–179.

    Article  MathSciNet  MATH  Google Scholar 

  • Walpole, L. J. (1985c). The analysis of the overall elastic properties of composite materials. In B. A. Bilby, K. J. Miller, & J. R. Willis (Eds.), Fundamentals of deformation and fracture: Eshelby memorial symposium (pp. 91–107). Cambridge: Cambridge University Press.

    Google Scholar 

  • Walsh, J. B. (1965). The effect of cracks on the compressibility of rock. Journal of Geophysical Research, 70, 381.

    Article  MATH  Google Scholar 

  • Weng, G. J. (1984). Some elastic properties of reinforced solids, with special reference to isotropic ones containing spherical inclusions. International Journal of Engineering Science, 22, 845–856.

    Article  MATH  Google Scholar 

  • Weng, G. J. (1990). The theoretical connection between Mori-Tanaka’s theory and the Hashin-Shtrikman-Walpole bounds. International Journal of Engineering Science, 28, 1111–1120.

    Article  MathSciNet  MATH  Google Scholar 

  • Weng, G. J. (1992). Explicit evaluation of Willis’ bounds with ellipsoidal inclusions. International Journal of Engineering Science, 30, 83–92.

    Article  MathSciNet  MATH  Google Scholar 

  • Williamson, R. L., Rabin, B. H., & Drake, J. T. (1993). Finite element analysis of thermal residual stresses at graded ceramic-metal interfaces. Part 1. Model description and geometrical effects. Journal of Applied Physics, 74, 1311–1320.

    Article  Google Scholar 

  • Willis, J. R. (1980). A polarization approach to the scattering of elastic waves – I. Scattering by a single inclusion. II. Multiple scattering from inclusions. Journal of the Mechanics and Physics of Solids, 28, 287–327.

    Article  MathSciNet  MATH  Google Scholar 

  • Willis, J. R. (1981). Variational and related method for the overall properties of composites. In Advances in applied mechanics, 21, 1–78. Academic Press.

    Google Scholar 

  • Withers, P. J. (1989). The determination of the elastic field of an ellipsoidal inclusion in a transversely isotropic medium, and its relevance to composite materials. Philosophical Magazine, 59, 759–781.

    Article  Google Scholar 

  • Wu, T. T. (1966). The effect of inclusion shape on the elastic moduli of a two-phase material. International Journal of Solids and Structures, 2, 1–8.

    Article  Google Scholar 

  • Zhao, Y. H., Tandon, G. P., & Weng, G. J. (1989). Elastic moduli for a class of porous materials. Acta Mechanica, 76, 105–130.

    Article  MATH  Google Scholar 

  • Zohdi, T. I., & Wriggers, P. (2005). An introduction to computational micromechanics. Berlin: Springer.

    Book  Google Scholar 

  • Zohdi, T. I., Oden, J. T., & Rodin, G. J. (1996). Hierarchical modeling of heterogeneous bodies. Computer Merthods in Applied Mechanics and Engineering, 138, 273–298.

    Article  MathSciNet  MATH  Google Scholar 

  • Walker, K. P. (1993) Fourier integral representation of the Green function for anisotropic elastic half-space. Proc. Roy. Soc. London, A433, 367–389.

    Article  Google Scholar 

  • Ghosh, S., Lee, K., Raghavan, P. (2001). A multi-level computational model for multi-scale damage analysis in composite and porous materials Intl. J. Solids. Struct., 38, 2335–2385.

    Article  MATH  Google Scholar 

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Dvorak, G.J. (2013). Estimates of Mechanical Properties of Composite Materials. In: Micromechanics of Composite Materials. Solid Mechanics and Its Applications, vol 186. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-4101-0_7

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