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Anisotropic Elastic Solids

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Micromechanics of Composite Materials

Part of the book series: Solid Mechanics and Its Applications ((SMIA,volume 186))

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Abstract

Properties of composite materials and their constituents often depend on both position and direction in a fixed system of coordinates. In the terminology of solid mechanics, such materials are heterogeneous and anisotropic. This chapter is concerned with the directional dependence, defined by certain material symmetry elements, and reflected in eight distinct forms of the stiffness and compliance matrices of elastic solids. Such materials include, for example, reinforcing fibers, particles and their coatings, or fibrous composites and laminates represented on the macroscale by homogenized solids with equivalent or effective elastic moduli. Identification of the positions of zero-valued coefficients, and of any connections between nonzero coefficients in those matrices is of particular interest. Different classes of crystals exhibit a much larger range of symmetries, derived from spatial arrangement of their lattices. Broader expositions of these topics can be found in several books, such as Love (1944), Lekhnitskii (1950), Green and Atkins (1960), Nye (1957, 1985), Hearmon (1961), Ting (1996), and Cowin and Doty (2007).

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References

  • Ahmad, F. (2002). Invariants and structural invariants of the anisotropic elasticity tensor. Quarterly Journal of Mechanics and Applied Mathematics, 55, 597–606.

    Article  MathSciNet  MATH  Google Scholar 

  • Almgren, R. F. (1985). An isotropic three-dimensional structure with Poisson’s ratio = −1. Journal of Elasticity, 15, 427–430.

    Article  Google Scholar 

  • Backus, G. A. (1970). Geometrical picture of anisotropic elastic tensors. Reviews of Geophysics and Spacephysics, 8, 633–671.

    Article  Google Scholar 

  • Baerheim, R. (1993). Harmonic decomposition of the anisotropic elasticity tensor. Quarterly Journal of Mechanics and Applied Mathematics, 46, 511–523.

    Article  MathSciNet  Google Scholar 

  • Baughman, R. H., Shacklette, J. M., Zakhidov, A. A., & Stafstrom, S. (1998). Negative Poisson’s ratio as a common feature of cubic materials. Nature, 426, 667.

    Google Scholar 

  • Boulanger, P., & Hayes, M. (1998). Poisson’s ratio for orthotropic materials. Journal of Elasticity, 50, 87–89.

    Article  MATH  Google Scholar 

  • Budiansky, B., & Kimmel, E. (1987). Elastic moduli of lungs. ASME Journal of Applied Mechanics, 54, 351–358.

    Article  MATH  Google Scholar 

  • Chadwick, P., Vianello, M., & Cowin, S. C. (2001). A new proof that the number of linear elastic symmetries is eight. Journal of the Mechanics and Physics of Solids, 49, 2471–2492.

    Article  MathSciNet  MATH  Google Scholar 

  • Christensen, R. M. (1987). Sufficient symmetry conditions for isotropy of the elastic moduli tensor. Journal of Applied Mechanics, 54, 772–777.

    Article  Google Scholar 

  • Cowin, S., & Doty, S. (2007). Tissue mechanics. New York: Springer.

    Book  MATH  Google Scholar 

  • Cowin, S. C., & Mehrabadi, M. M. (1987). On the identification of material symmetry for anisotropic materials. Quarterly Journal of Mechanics and Applied Mathematics, 40, 451–476.

    Article  MathSciNet  MATH  Google Scholar 

  • Cowin, S. C., & Mehrabadi, M. M. (1989). Identification of the elastic symmetry of bone and other materials. Journal of Biomechanics, 22, 503–515.

    Article  Google Scholar 

  • Cowin, S. C., & Mehrabadi, M. M. (1995). Anisotropic symmetries of linear elasticity. Applied Mechanics Reviews, 48, 247–285.

    Article  Google Scholar 

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

    Google Scholar 

  • Evans, K. E. (1991). Auxetic polymers: A new range of materials. Endeavour, New Series, 15, 170–174.

    Article  Google Scholar 

  • Evans, K. E., Nkansah, M. A., Hutchinson, I. J., & Rogers, S. C. (1991). Molecular network design. Nature, 353, 124.

    Article  Google Scholar 

  • Green, A. E., & Atkins, J. E. (1960). Large elastic deformations and non-linear continuum mechanics. Oxford: Clarendon Press, pp. 11, 14, 15.

    Google Scholar 

  • Guo, C. Y., & Wheeler, L. (2006). Extreme Poisson’s ratios and related elastic crystal properties. Journal of the Mechanics and Physics of Solids, 54, 690–707.

    Article  MathSciNet  MATH  Google Scholar 

  • Hayes, M., & Shuvalov, A. (1998). On the extreme values of Young’s modulus, the shear modulus, and Poisson’s ratio for cubic materials. Journal of Applied Mechanics, 65, 786–787.

    Article  MathSciNet  Google Scholar 

  • Hearmon, R. F. S. (1961). Introduction to applied anisotropic elasticity. Oxford: Clarendon Press.

    Google Scholar 

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

    Google Scholar 

  • Hill, R. (1948). A theory of the yielding and plastic flow of anisotropic metals. Proceedings of the Royal Society London, A193, 281–297.

    Article  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 

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

    Article  Google Scholar 

  • Lakes, R. (1987). Foam structures with a negative Poisson’s ratio. Science, 235, 1038–1040.

    Article  Google Scholar 

  • Lakes, R. (1993). Advances in negative Poisson’s ratio materials. Advanced Materials, 5, 293–296.

    Article  Google Scholar 

  • Lakes, R. (2000). Deformations in extreme matter. Science, 288, 1976–1977.

    Article  Google Scholar 

  • Lekhnitskii, S. G. (1950). Theory of elasticity of an anisotropic body, Gostekhizdat Moscow (English translation published by Holden-Day, San Francisco, 1963).

    Google Scholar 

  • Li, Y. (1976). The anisotropic behavior of Poisson’s ratio, Young’s modulus, and shear modulus in hexagonal materials. Physica Status Solidi, 38, 171–175.

    Article  Google Scholar 

  • Love, A. E. H. (1944). A treatise on the mathematical theory of elasticity (4th ed.). New York: Dover.

    MATH  Google Scholar 

  • Mulhern, J. F., Rogers, T. G., & Spencer, A. J. M. (1967). A continuum model for fibre-reinforced plastic materials. Proceedings of the Royal Society London, A301, 473–492.

    Article  Google Scholar 

  • Mulhern, J. F., Rogers, T. G., & Spencer, A. J. M. (1969). A continuum theory of a plastic-elastic fibre-reinforced material. International Journal of Engineering Science, 7, 129–152.

    Article  MATH  Google Scholar 

  • Nye, J. F. (1957, 1985). Physical properties of crystals. Their representation by tensors and matrices. Oxford: Oxford University Press.

    Google Scholar 

  • Sirotin, Y. I., & Shakol’skaya, M. P. (1982). Fundamentals of crystal physics. Moscow: MIR Publishers.

    Google Scholar 

  • Smith, G. F., & Rivlin, R. (1958). The strain-energy function for anisotropic elastic materials. Transactions of the American Mathematical Society, 88, 175–193.

    Article  MathSciNet  MATH  Google Scholar 

  • Spencer, A. J. M. (1972). Deformation of fibre-reinforced materials. London: Oxford University Press.

    Google Scholar 

  • Srinivasan, T. P., & Nigam, S. D. (1969). Invariant elastic constants for crystals. Journal of Mathematics and Mechanics, 19, 411–420.

    Google Scholar 

  • Ting, T. C. T. (1987). Invariants of anisotropic elastic constants. Quarterly Journal of Mechanics and Applied Mathematics, 40, 431–438.

    Article  MathSciNet  MATH  Google Scholar 

  • Ting, T. C. T. (1996). Anisotropic elasticity: Theory and applications. Oxford: Oxford University Press.

    MATH  Google Scholar 

  • Ting, T. C. T. (2003). Generalization of Cowin-Mehrabadi theorems and direct proof that the number of linear elastic symmetries is eight. International Journal of Solids and Structures, 40, 7129–7142.

    Article  MathSciNet  MATH  Google Scholar 

  • Ting, T. C. T. (2004). Very large Poisson’s ratio with a bounded transverse strain in anisotropic elastic materials. Journal of Elasticity, 77, 163–176.

    Article  MathSciNet  MATH  Google Scholar 

  • Ting, T. C. T. (2005a). The stationary values of Young’s modulus for monoclinic and triclinic materials. Journal of Mechanics, 21, 249–253.

    Article  Google Scholar 

  • Ting, T. C. T. (2005b). Explicit expression of the stationary values of Young’s modulus and the shear modulus for anistropic elastic materials. Journal of Mechanics, 21, 255–266.

    Article  Google Scholar 

  • Ting, T. C. T., & Barnett, D. M. (2005). Negative Poisson’s ratios in anisotropic linear elastic media. ASME Journal of Applied Mechanics, 72, 929–931.

    Article  MathSciNet  MATH  Google Scholar 

  • Ting, T. C. T., & Chen, T. (2005). Poisson’s ratio for anisotropic elastic materials can have no bounds. Quarterly Journal of Mechanics and Applied Mathematics, 58, 73–82.

    Article  MathSciNet  MATH  Google Scholar 

  • Ting, T. C. T., & He, Q. C. (2006). Decomposition of elasticity tensors and tensors that are structurally invariant in three dimensions. Quarterly Journal of Mechanics and Applied Mathematics, 59, 323–341.

    Article  MathSciNet  MATH  Google Scholar 

  • Tsai, S. W., & Wu, E. M. (1971). A general theory of strength for anisotropic materials. Journal of Composite Materials, 5, 58–80.

    Article  Google Scholar 

  • Turley, J., & Sines, G. (1971). The anisotropy of Young’s modulus, shear modulus and Poisson’s ratio in cubic materials. Journal of Physics, 4, 264–271.

    Google Scholar 

  • Voigt, W. (1910). Lehrbuch der Kristallphysik. Leipzig: B. G. Teubner.

    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. (1985a). The stress-strain law of a textured aggregate of cubic crystals. Journal of the Mechanics and Physics of Solids, 33, 363–370.

    Article  MATH  Google Scholar 

  • Zener, C. (1948). Elasticity and anelasticity of metals. Chicago: University of Chicago Press.

    Google Scholar 

  • Zheng, Q. S., & Chen, T. (2001). New perspective on Poisson’s ratios of elastic solids. Acta Mechanica, 150, 191–195.

    Article  MATH  Google Scholar 

Download references

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Authors and Affiliations

  1. Mechanical, Aeronautical and Nuclear Engineering, Rensselaer Polytechnic Institute, Troy, NY, USA

    George J. Dvorak

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  1. George J. Dvorak
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Dvorak, G.J. (2013). Anisotropic Elastic Solids. In: Micromechanics of Composite Materials. Solid Mechanics and Its Applications, vol 186. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-4101-0_2

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  • DOI: https://doi.org/10.1007/978-94-007-4101-0_2

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