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Nonlinear Composites and Microstructure Evolution

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Mechanics for a New Mellennium

Abstract

Recently developed methods for estimating the effective behavior of nonlinear composites are reviewed. The methods follow from variational principles expressing the effective behavior of the given nonlinear composites in terms of the behavior of suitably chosen “linear comparison” composites. These methods allow the use of classical bounds and estimates (e.g. Hashin-Shtrikman, effective medium approximations) for linear materials to generate corresponding information for nonlinear ones. Comparisons are made with numerical simulations for metalmatrix composites, showing that the new methods are significantly more accurate than earlier ones, especially at high nonlinearity and heterogeneity contrast. The methods can be extended to incorporate evolution of the microstructure and its influence on the effective response under finite-strain conditions. An application to a forming process involving a porous metal is considered for illustrative purposes.

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References

  • Beran, M. 1965. Use of the variational approach to determine bounds for the effective permittivity of random media. Nuovo Cimento 38, 771–782.

    Article  Google Scholar 

  • Bornert, M., and P. Ponte Castañeda. 1998. Second-order estimates of the self-consistent type for viscoplastic polycrystals. Proceedings of the Royal Society of London A 356, 3035–3045.

    Article  ADS  Google Scholar 

  • Buryachenko, V., and A. M. Lipanov. 1989. Prediction of nonlinear flow parameters for multicomponent mixtures. Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki 4, 63–68.

    Google Scholar 

  • Chu, T., and Hashin, Z. 1971. Plastic behavior of composites and porous media under isotropic stress. International Journal of Engineering Science 9, 971–994.

    Article  MATH  Google Scholar 

  • deBotton, G., and P. Ponte Castañeda. 1995. Variational estimates for the creep behavior of polycrystals. Proceedings of the Royal Society of London A 448, 421–442.

    Article  Google Scholar 

  • Dvorak, G., and Y. Bahei-El-Din. 1987. A bimodal plasticity theory of fibrous composite materials. Acta Mechanica 69, 219–241.

    Article  MATH  Google Scholar 

  • Gărăjeu M., J. C. Michel, and P. Suquet. 2000. A micromechanical approach of damage in viscoplastic materials by evolution in size, shape and distribution of voids. Computer Methods in Applied Mechanics and Engineering 183, 223–246.

    Article  MATH  Google Scholar 

  • Gilormini, P., M. Nebozhyn, and P. Ponte Castañeda. 2000. Accurate estimates for the creep behavior of hexagonal polycrystals. Acta Materialia 49, 329–337.

    Article  Google Scholar 

  • Gurson, A. 1977. Continuum theory of ductile rupture by void nucleation and growth: Part I—Yield criteria and flow rules. Journal of Engineering Materials and Technology 99, 1–15.

    Article  Google Scholar 

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

    Article  MathSciNet  ADS  Google Scholar 

  • Hashin, Z., and S. Shtrikman. 1963. A variational approach to the theory of the elastic behavior of multiphase materials. Journal of the Mechanics and Physics of Solids 11, 127–140.

    Article  MathSciNet  MATH  ADS  Google Scholar 

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

    Article  MATH  ADS  Google Scholar 

  • Hill, R. 1965. Continuum micro-mechanics of elastoplastic polycrystals. Journal of the Mechanics and Physics of Solids 13, 89–101.

    Article  MATH  ADS  Google Scholar 

  • Hu, G. 1996. A method of plasticity for general aligned spheroidal voids or fiberreinforced composites. International Journal of Plasticity 12, 439–449.

    Article  MATH  Google Scholar 

  • Hutchinson, J. W. 1976. Bounds and self-consistent estimates for creep of polycrystalline materials. Proceedings of the Royal Society of London A 348, 101–127.

    Article  MATH  ADS  Google Scholar 

  • Kailasam, M., N. Aravas, and P. Ponte Castañeda. 2000. Constitutive models for porous metals with developing anisotropy and applications to deformation processing. Computer Modeling in Engineering and Sciences 1, 105–118.

    Google Scholar 

  • Masson, R., M. Bornert, P. Suquet, and A. Zaoui. 2000. An affine formulation for nonlinear composites and polycrystals. Journal of the Mechanics and Physics of Solids 48, 1203–1227.

    Article  MathSciNet  MATH  ADS  Google Scholar 

  • Michel, J. C., and P. Suquet. 1992. The constitutive law of nonlinearviscous and porous materials. Journal of the Mechanics and Physics of Solids 40, 783–812.

    Article  MathSciNet  MATH  ADS  Google Scholar 

  • Michel, J. C., H. Moulinec, and P. Suquet 1999. Effective properties of composite materials with periodic microstructure: A computational approach. Computer Methods in Applied Mechanics and Engineering 172, 109–143.

    Article  MathSciNet  MATH  Google Scholar 

  • Michel, J. C., H. Moulinec, and P. Suquet. 2000. A computational method based on augmented Lagrangians and fast Fourier transforms for composites with high contrast. Computer Modeling in Engineering and Sciences 1, 79–88.

    MathSciNet  Google Scholar 

  • Milton, G., and S. K. Serkov. 2000. Bounding the current in nonlinear conducting composites. Journal of the Mechanics and Physics of Solids 48, 1295–1324.

    Article  MathSciNet  MATH  ADS  Google Scholar 

  • Molinari, A., G. R. Canova, and S. Ahzi. 1987. A self-consistent approach of the large deformation polycrystal viscoplasticity. Acta Metallurgica Materialia 35, 2983–2994.

    Article  Google Scholar 

  • Moulinec, H., and P. Suquet. 1998. A numerical method for computing the overall response of nonlinear composites with complex microstructure. Computer Methods in Applied Mechanics and Engineering 157, 69–94.

    Article  MathSciNet  MATH  Google Scholar 

  • Nebozhyn, M. V., P. Gilormini, and P. Ponte Castañeda. 1999. Variational self-consistent estimates for viscoplastic polycrystals with highly anisotropic grains. Comptes Rendus de l’Académie des Sciences, Paris IIB 328, 11–17.

    Google Scholar 

  • Parteder, E., H. Riedel, and R. Kopp. 1999. Densification of sintered molybdenum during hot upsetting experiments and modeling. Materials Science and Engineering A 264, 17–25.

    Article  Google Scholar 

  • Ponte Castañeda, P. 1991. The effective mechanical properties of nonlinear isotropic composites. Journal of the Mechanics and Physics of Solids 39, 45–71.

    Article  MathSciNet  MATH  Google Scholar 

  • Ponte Castañeda, P. 1992. New variational principles in plasticity and their application to composite materials. Journal of the Mechanics and Physics of Solids 40, 1757–1788.

    Article  MathSciNet  MATH  Google Scholar 

  • Ponte Castañeda, P. 1996a. Exact second-order estimates for the effective mechanical properties of nonlinear composites. Journal of the Mechanics and Physics of Solids 44, 827–862.

    Article  MathSciNet  MATH  Google Scholar 

  • Ponte Castañeda, P. 1996b. Variational methods for estimating the effective behavior of nonlinear composite materials. In Continuum Models and Discrete Systems (CMDS 8) (K. Z. Markov, ed.). Singapore: World Scientific, 268–279.

    Google Scholar 

  • Ponte Castañeda, P., and P. Suquet. 1998. Nonlinear composites. Advances in Applied Mechanics 34, 171–302.

    Article  Google Scholar 

  • Ponte Castañeda, P., and E. Tiberio. 2000. A second-order homogenization method in finite elasticity and applications to black-filled elastomers. Journal of the Mechanics and Physics of Solids 48, 1389–1411.

    Article  MathSciNet  MATH  ADS  Google Scholar 

  • Ponte Castañeda, P., and J. R. Willis. 1999. Variational second-order estimates for nonlinear composites. Proceedings of the Royal Society of London A 455, 1799–1811.

    Article  ADS  MATH  Google Scholar 

  • Ponte Castañeda, P., and M. Zaidman. 1994. Constitutive models for porous materials with evolving microstructure. Journal of the Mechanics and Physics of Solids 42, 1459–1497.

    Article  MathSciNet  MATH  Google Scholar 

  • Suquet, P. 1993. Overall potentials and extremal surfaces of power-law or ideally plastic materials. Journal of the Mechanics and Physics of Solids 41, 981–1002.

    Article  MathSciNet  MATH  ADS  Google Scholar 

  • Suquet, P. 1995. Overall properties of nonlinear composites: A modified secant moduli theory and its link with Ponte Castañeda’s nonlinear variational procedure. Comptes Rendus de l’Académie des Sciences, Paris IIB 320, 563–571.

    MATH  Google Scholar 

  • Suquet, P. 1997. Effective properties of nonlinear composites. In Continuum Micromechanics (P. Suquet, ed.), CISM Lecture Notes 377. New York: Springer, 197–264.

    Google Scholar 

  • Suquet, P., and P. Ponte Castañeda. 1993. Small-contrast perturbation expansions for the effective properties of nonlinear composites. Comptes Rendus de l’Académie des Sciences, Paris II 317, 1515–1522.

    MATH  Google Scholar 

  • Talbot, D. R. S., and J. R. Willis. 1985. Variational principles for inhomogeneous nonlinear media. IMA Journal of Applied Mathematics 35, 39–54.

    Article  MathSciNet  MATH  Google Scholar 

  • Talbot, D. R. S., and J. R. Willis. 1992. Some simple explicit bounds for the overall behavior of nonlinear composites. International Journal of Solids and Structures 29, 1981–1987.

    Article  MathSciNet  MATH  Google Scholar 

  • Talbot, D. R. S., and J. R. Willis, 1997. Bounds of third order for the overall response of nonlinear composites. Journal of the Mechanics and Physics of Solids 45, 87–111.

    Article  MathSciNet  MATH  ADS  Google Scholar 

  • Willis, J. R. 1981. Variational and related methods for the overall properties of composites. Advances in Applied Mechanics 21, 1–78.

    Article  MathSciNet  MATH  Google Scholar 

  • Willis, J. R. 1983. The overall response of composite materials. Journal of Applied Mechanics 50, 1202–1209.

    Article  MATH  Google Scholar 

  • Willis, J. R. 1992. On methods for bounding the overall properties of nonlinear composites: correction and addition. Journal of the Mechanics and Physics of Solids 40, 441–445.

    Article  MathSciNet  ADS  Google Scholar 

  • Willis, J. R. 2000. The overall response of nonlinear composite media. European Journal of Mechanics A/Solids 19, S165–S184.

    Google Scholar 

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Author information

Authors and Affiliations

  1. Department of Mechanical Engineering and Applied Mechanics, University of Pennsylvania, Philadelphia, Penn., USA

    Pedro Ponte Castañeda

  2. Laboratoire de Mécanique et d’Acoustique, CNRS, Marseille, France

    Pierre Suquet

Authors
  1. Pedro Ponte Castañeda
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  2. Pierre Suquet
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Editor information

Editors and Affiliations

  1. Department of Theoretical and Applied Mechanics, University of Illinois at Urbana-Champaign, USA

    Hassan Aref  & James W. Phillips  & 

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© 2001 Kluwer Academic Publishers

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Ponte Castañeda, P., Suquet, P. (2001). Nonlinear Composites and Microstructure Evolution. In: Aref, H., Phillips, J.W. (eds) Mechanics for a New Mellennium. Springer, Dordrecht. https://doi.org/10.1007/0-306-46956-1_17

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  • DOI: https://doi.org/10.1007/0-306-46956-1_17

  • Publisher Name: Springer, Dordrecht

  • Print ISBN: 978-0-7923-7156-4

  • Online ISBN: 978-0-306-46956-5

  • eBook Packages: Springer Book Archive

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