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.
Bensoussan, A., Lions, J. L., & Papanicolaou, G. (1978). Asymptotic analysis for periodic structures. Amsterdam: Nort Holland.
Benveniste, Y. (1987a). A new approach to the application of Mori-Tanaka theory in composite materials. Mechanics of Materials, 6, 147–157.
Benveniste, Y. (1987b). A differential effective medium theory with a composite sphere embedding. ASME Journal of Applied Mechanics, 54, 466–468.
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.
Benveniste, Y., Dvorak, G. J., & Chen, T. (1989). Stress fields in composites with coated inclusions. Mechanics of Materials, 7, 305–317.
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.
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.
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.
Berryman, J. G. (1980). Long wavelength propagation in composite elastic media II, Ellipsoidal inclusions. Journal of the Acoustical Society of America, 68, 1820–1831.
Boucher, S. (1974). On the effective moduli of isotropic two-phase elastic composites. Journal of Composite Materials, 8, 82–89.
Bruggeman, D. A. G. (1935). Berechnung verschiedener physikalisher Konstanten von heterogenen Substanzen I. Annalen der Physik, 24, 636–663.
Budiansky, B. (1965). On the elastic moduli of some heterogeneous materials. Journal of the Mechanics and Physics of Solids, 13, 223–227.
Budiansky, B., & O’Connell, R. J. (1976). Elastic moduli of a cracked solid. International Journal of Solids and Structures, 12, 81–97.
Chen, T., Dvorak, G. J., & Benveniste, Y. (1990). Stress fields in composites reinforced by coated cylindrically orthotropic fibers. Mechanics of Materials, 9, 17–32.
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.
Christensen, R. M. (1990). A critical evaluation for a class of micromechanics models. Journal of the Mechanics and Physics of Solids, 38, 379–404.
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).
Christensen, R. M., & Waals, F. M. (1972). Effective stiffness of randomly oriented fiber composites. Journal of Composite Materials, 6, 518–532.
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.
Cleary, M. P., Chen, I. W., & Lee, S. M. (1980). Self-consistent techniques for heterogeneous solids. ASCE Journal of Engineering Mechanics, 106, 861–867.
Daniel, I. M., & Ishai, O. (2006). Engineering mechanics of composite materials (2nd ed.). New York: Oxford University Press.
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.
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.
Einstein, A. (1905). Eine neue Berechnung der Moleküldimensionen. Annales de Physique, 19, 289–306.
Ferrari, M., & Johnson, G. C. (1989). Effective elasticities of short-fiber composites with arbitrary orientation distribution. Mechanics of Materials, 8, 67–73.
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.
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.
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.
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.
Hashin, Z. (1972). Theory of fiber reinforced materials. NASA CR-1974. Washington, DC: National Aeronautics and Space Administration, 690.
Hashin, Z. (1988). The differential scheme and its application to cracked materials. Journal of the Mechanics and Physics of Solids, 36, 719–734.
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.
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.
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.
Hatta, H., & Taya, M. (1986). Equivalent inclusion method for steady state heat conduction in composites. International Journal of Engineering Science, 24, 1159–1172.
Herakovich, C. T. (1998). Mechanics of fibrous composites. New York: Wiley.
Hershey, A. V. (1954). The elasticity of an isotropic aggregate of anisotropic cubic crystals. ASME Journal of Applied Mechanics, 21, 236–240.
Hill, R. (1963a). Elastic properties of reinforced solids: Some theoretical principles. Journal of the Mechanics and Physics of Solids, 11, 357–372 [1].
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.
Hill, R. (1965a). Continuum micromechanics of elastic-plastic polycrystals. Journal of the Mechanics and Physics of Solids, 13, 89–101.
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.
Hill, R. (1965c). A self-consistent mechanics of composite materials. Journal of the Mechanics and Physics of Solids, 13, 213–222.
Hirano, T., & Wakashima, K. (1995). Mathematical modeling and design. MRS Bulletin, 40-42.
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.
Hori, M., & Nemat-Nasser, S. (1993). Double-inclusion model and overall moduli of multi-phase composites. Mechanics of Materials, 14, 189–206.
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.
Kerner, E. H. (1956). The elastic and thermo-elastic properties of composite media. Proceedings of the Royal Society London, B69, 808–813.
Kröner, E. (1958). Berechnung der elastischen Konstanten der Vielkristalls aus den Konstanten der Einkristalls. Zeitschrift für Physik, 151, 504–518.
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.
Laws, N. (1973). On thermostatics of composite materials. Journal of the Mechanics and Physics of Solids, 21, 9–17.
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.
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.
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.
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.
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.
Laws, N., Dvorak, G. J., & Hejazi, M. (1983). Stiffness changes in composites caused by crack systems. Mechanics of Materials, 2, 123–137.
Lee, Y. -D., & Erdogan, F. (1994/1995). Residual thermal stresses in FGM and laminated thermal barrier coatings. International Journal of Fracture, 69, 145-165.
Markworth, A. J., & Saunders, J. H. (1995). A model of structure optimization for a functionally graded material. Materials Letters, 22, 103–107.
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.
McLaughlin, R. (1977). A study of the differential scheme for composite materials. International Journal of Engineering Science, 15, 237–244.
Milton, G. W. (2002). The theory of composites. Cambridge: Cambridge University Press.
Mori, T., & Tanaka, K. (1973). Average stress in matrix and average elastic energy of materials with misfitting inclusions. Acta Metallurgica, 21, 571–574.
Nemat-Nasser, S., & Hori, M. (1999). Micromechanics: Overall properties of hetero-geneous materials (2nd ed.). Amsterdam: Elsevier.
Norris, A. N. (1985). A differential scheme for effective moduli of composites. Mechanics of Materials, 4, 1–16.
Norris, A. N. (1989). An examination of the Mori-Tanaka effective medium approximation for multiphase composites. ASME Journal of Applied Mechanics, 56, 83–88.
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.
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.
Ozisik, M. N. (1968). Boundary value problems of heat conduction. Scranton: International Textbook Co.
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.
Postma, G. W. (1955). Wave propagation in a stratified medium. Geophysics, 20, 780–806.
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.
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.
Roscoe, R. (1952). The viscosity of suspensions of rigid spheres. British Journal of Applied Physics, 3, 267–269.
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.
Sanchez-Palencia, E. (1980). Homogenization techniques and vibration theory. Lecture Notes in Physics No. 127. Berlin: Springer.
Sasaki, M., & Hirai, T. (1991). Fabrication and properties of functionally gradient materials. Journal of the Ceramic Society of Japan, 99, 1002–1013.
Sayers, C. M. (1992). Elastic anisotropy of short-fibre reinforced composites. Journal of the Mechanics and Physics of Solids, 29, 2933–2944.
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.
Tanaka, K., & Mori, M. (1972). Note on volume integrals of the elastic field around an ellipsoidal inclusion. Journal of Elasticity, 2, 199–200.
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.
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.
Walpole, L. J. (1969). On the overall elastic moduli of composite materials. Journal of the Mechanics and Physics of Solids, 17, 235–251.
Walpole, L. J. (1981). Elastic behavior of composite materials: Theoretical foundations. In Advances in applied mechanics. New York: Academic, 21, 169–242.
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.
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.
Walsh, J. B. (1965). The effect of cracks on the compressibility of rock. Journal of Geophysical Research, 70, 381.
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.
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.
Weng, G. J. (1992). Explicit evaluation of Willis’ bounds with ellipsoidal inclusions. International Journal of Engineering Science, 30, 83–92.
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.
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.
Willis, J. R. (1981). Variational and related method for the overall properties of composites. In Advances in applied mechanics, 21, 1–78. Academic Press.
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.
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.
Zhao, Y. H., Tandon, G. P., & Weng, G. J. (1989). Elastic moduli for a class of porous materials. Acta Mechanica, 76, 105–130.
Zohdi, T. I., & Wriggers, P. (2005). An introduction to computational micromechanics. Berlin: Springer.
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.
Walker, K. P. (1993) Fourier integral representation of the Green function for anisotropic elastic half-space. Proc. Roy. Soc. London, A433, 367–389.
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.
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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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