Homogenization and optimal bounds in linear elasticity. (English) Zbl 0604.73013

Z. Hashin and S. Shtrikman’s derivation [J. Mech. Phys. Solids 11, 127-140 (1963; Zbl 0108.369)] of bounds on the bulk and shear moduli of a two phase isotropic elastic composite, and their optimality, are reconsidered in the context of homogenization using the techniques of ”H-convergence” and ”compensated compactness”. The bounds on the bulk modulus obtained coincide with those of Hashin and Shtrikman however they are not restricted to the case of well-ordered phases. The bounds on the shear modulus are claimed to be better than those obtained by B. Paul [Trans. ASME 218, 36-41 (1960)]. Also, it is shown that Hashin and Shtrikman bounds on the bulk and shear moduli are simultaneously achieved through a finite number of layering processes.
Reviewer: J.Ignaczak


74E05 Inhomogeneity in solid mechanics
74S30 Other numerical methods in solid mechanics (MSC2010)


Zbl 0108.369
Full Text: DOI


[1] G. E. Backus [1] ?Long-Wave Elastic Anisotropy Produced by Horizontal Layering?, J. Geophys. Res., 1962, V. 67, p. 4427-4440. · Zbl 1369.86005
[2] A. Bensoussan, J. L. Lions, & G. Papanicolaou [1] Asymptotic Analysis for Periodic Structures, North-Holland, Amsterdam, 1978. · Zbl 0404.35001
[3] R. M. Christensen [1] Mechanics of Composite Materials, Wiley Interscience, New York, 1979.
[4] P. Germain [1] Mécanique des Milieux Continus, Masson, Paris, 1973.
[5] K. Golden & G. Papanicolaou [1 ] ?Bounds for Effective Parameters of Heterogeneous Media by Analytic Continuation?, Commun. Math. Phys., 1983, V. 90, p. 473-491.
[6] Z. Hashin [1] ?Analysis of Composite Materials, A Survey?, J. Appl. Mech., 1983, V. 50, p. 481-505. · Zbl 0542.73092
[7] 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. · Zbl 0108.36902
[8] R. Hill [1] ?Elastic Properties of Reinforced Solids: Some Theoretical Principles?, J. Mech. Phys. Solids, 1963, V. 11, p. 357-372. · Zbl 0114.15804
[9] 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. · Zbl 0542.73003
[10] 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. · Zbl 0224.73016
[11] 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.
[12] 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.
[13] 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.
[14] G. W. Milton [2] Private communication, Oct. 1984.
[15] 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. · Zbl 0497.73016
[16] F. Murat [1] ?H-Convergence?, Séminaire d’Analyse Fonctionnelle et Numérique, 1977/1978, Univ. d’Alger, Multigraphed.
[17] F. Murat [2] ?Compacité par Compensation?, Ann. Sc. Norm. Sup. Pisa, 1978, V. 5, p. 489-507. · Zbl 0399.46022
[18] F. Murat [3] ?Control in Coefficients? in Systems and Control Encyclopaedia: Theory, Technology, Applications, Pergamon Press, Oxford, 1986, to appear.
[19] A. N. Norris [1] ?A Differential Scheme for the Effective Moduli of Composites?, Mech. of Materials, 1985, to appear.
[20] B. Paul [1] ?Prediction of Elastic Constants of Multiphase Materials?, Trans. A.S.M.E., 1960, V. 218, p. 36-41.
[21] E. Sanchez-Palencia [1] Non Homogeneous Materials and Vibration Theory, Springer Lecture Notes in Physics, V. 127, Berlin, Heidelberg, New York, 1980.
[22] L. Simon [1] ?On G-Convergence of Elliptic Operators?, Indiana Univ. Math. J., 1979, V. 28, p. 587-594. · Zbl 0433.35020
[23] S. Spagnolo [1] ?Sulla Convergenza di Soluzioni di Equazioni Paraboliche ed Ellitiche?, Ann. Sc. Norm. Sup. Pisa, 1968, V. 22, p. 577-597.
[24] 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.
[25] L. Tartar [2]Cours Peccot, Collège de France, 1977.
[26] 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.
[27] 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.
[28] 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.
[29] 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. · Zbl 0139.18701
[30] 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.
[31] 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. · Zbl 0445.35096
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.