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Homogenization and optimal bounds in linear elasticity

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References

  • G. E. Backus [1] “Long-Wave Elastic Anisotropy Produced by Horizontal Layering”, J. Geophys. Res., 1962, V. 67, p. 4427–4440.

    Google Scholar 

  • A. Bensoussan, J. L. Lions, & G. Papanicolaou [1] Asymptotic Analysis for Periodic Structures, North-Holland, Amsterdam, 1978.

    Google Scholar 

  • R. M. Christensen [1] Mechanics of Composite Materials, Wiley Interscience, New York, 1979.

    Google Scholar 

  • P. Germain [1] Mécanique des Milieux Continus, Masson, Paris, 1973.

    Google Scholar 

  • K. Golden & G. Papanicolaou [1 ] “Bounds for Effective Parameters of Heterogeneous Media by Analytic Continuation”, Commun. Math. Phys., 1983, V. 90, p. 473–491.

    Google Scholar 

  • Z. Hashin [1] “Analysis of Composite Materials, A Survey”, J. Appl. Mech., 1983, V. 50, p. 481–505.

    Google Scholar 

  • Z. Hashin & S. Shtrikman [1] “A Variational Approach to the Theory of the Elastic Behaviour of Multiphase Materials”, J. Mech. Phys. Solids, 1963, V. 11, p. 127–140.

    Google Scholar 

  • R. Hill [1] “Elastic Properties of Reinforced Solids: Some Theoretical Principles”, J. Mech. Phys. Solids, 1963, V. 11, p. 357–372.

    Google Scholar 

  • Y. Kantor & D. Bergman [1] “Improved Rigorous Bounds on the Effective Elastic Moduli of a Composite Material”, J. Mech. Phys. Solids, 1984, V. 32, p. 41–62.

    Google Scholar 

  • R. J. Knops & L. E. Payne [1] Uniqueness Theorems in Linear Elasticity, Springer-Verlag Tracts in Natural Philosophy, V. 19, Berlin, Heidelberg, New York, 1971.

  • K. A. Lurié & A. V. Cherkaev [1] “Exact Estimates of Conductivity of Composites Formed by Two Materials Taken in Prescribed Proportion”, Proc. Royal Soc. Edinburgh A, 1984, V. 99, p. 71–87.

    Google Scholar 

  • K. A. Lurié & A. V. Cherkaev [2] “The Problem of Formation of an Optimal Isotropic Multicomponent Composite”, Preprint A. F. Ioffe Physical Technical Institute, Academy of Sciences of the U.S.S.R., Leningrad, 1984, N∘ 895.

  • G. W. Milton [1] “Modelling the Properties of Composites by Laminates”, in Proceedings of the Workshop on Homogenization and Effective Moduli of Materials and Media (Minneapolis, Oct. 84), to appear.

  • G. W. Milton [2] Private communication, Oct. 1984.

  • G. W. Milton & N. Phan-Thien [1] “New Bounds on Effective Elastic Moduli of Two-Component Materials”, Proc. R. Soc. Lond. A, 1982, V. 380, p. 305–331.

    Google Scholar 

  • F. Murat [1] “H-Convergence”, Séminaire d'Analyse Fonctionnelle et Numérique, 1977/1978, Univ. d'Alger, Multigraphed.

    Google Scholar 

  • F. Murat [2] “Compacité par Compensation”, Ann. Sc. Norm. Sup. Pisa, 1978, V. 5, p. 489–507.

    Google Scholar 

  • F. Murat [3] “Control in Coefficients” in Systems and Control Encyclopaedia: Theory, Technology, Applications, Pergamon Press, Oxford, 1986, to appear.

  • A. N. Norris [1] “A Differential Scheme for the Effective Moduli of Composites”, Mech. of Materials, 1985, to appear.

  • B. Paul [1] “Prediction of Elastic Constants of Multiphase Materials”, Trans. A.S.M.E., 1960, V. 218, p. 36–41.

    Google Scholar 

  • E. Sanchez-Palencia [1] Non Homogeneous Materials and Vibration Theory, Springer Lecture Notes in Physics, V. 127, Berlin, Heidelberg, New York, 1980.

  • L. Simon [1] “On G-Convergence of Elliptic Operators”, Indiana Univ. Math. J., 1979, V. 28, p. 587–594.

    Google Scholar 

  • S. Spagnolo [1] “Sulla Convergenza di Soluzioni di Equazioni Paraboliche ed Ellitiche”, Ann. Sc. Norm. Sup. Pisa, 1968, V. 22, p. 577–597.

    Google Scholar 

  • L. Tartar [1] “Problème de Contrôle des Coefficients dans des Equations aux Dérivées Partielles”, in Control Theory, Numerical Methods and Computer Systems Modelling, Ed. A. Bensoussan & J. L. Lions, Springer Lecture Notes in Economics and Mathematical Systems, V. 107, Berlin, Heidelberg, New York, 1975, p. 420–426.

  • L. Tartar [2]Cours Peccot, Collège de France, 1977.

  • L. Tartar [3] “Estimation de Coefficients Homogénéisés” in Computing Methods in Applied Sciences and Engineering, 1977, I, Ed. R. Glowinskj & J. L. Lions, Springer Lecture Notes in Mathematics, V. 704, Berlin, Heidelberg, New York, 1979 p. 364–373.

  • L. Tartar [4] “Compensated Compactness and Applications to Partial Differential Equations” in Non Linear Mechanics and Analysis, Heriot-Watt Symposium, Volume IV, Ed. R. J. Knops, Pitman Research Notes in Mathematics, V. 39, Boston, 1979, p. 136–212.

  • L. Tartar [5] “Estimations Fines de Coefficients Homogénéisés”, in Ennio De Giorgi Colloquium, Ed. P. Krée, Pitman Research Notes in Mathematics, V. 125, Boston, 1985, p. 168–187.

  • L. J. Walpole [1] “On Bounds for the Overall Elastic Moduli of Inhomogeneous Systems, I”, J. Mech. Phys. Solids, 1966, V. 14, p. 151–162.

    Google Scholar 

  • J. Willis [1] “Elasticity Theory of Composites”, in Mechanics of Solids, the Rodney Hill 60th Anniversary Volume, Ed. H. G. Hopkins & M. J. Sewell, Pergamon Press, Oxford, 1982, p. 353–386.

    Google Scholar 

  • V. V. Zhikov, S. M. Kozlov, O. A. Oleinik, & Kha T'en Ngoan [1] “Averaging and G-Convergence of Differential Operators”, Russian Math. Surveys, 1979, V. 34, p. 69–147.

    Google Scholar 

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Communicated by J. M. Ball

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Francfort, G.A., Murat, F. Homogenization and optimal bounds in linear elasticity. Arch. Rational Mech. Anal. 94, 307–334 (1986). https://doi.org/10.1007/BF00280908

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