Abstract
We study the macroscopic mechanical behavior of materials with microscopic holes or hard inclusions. Specifically, we deal with the effective elastic moduli of composites whose microgeometry consists of either soft or hard isolated inclusions surrounded by an elastic matrix. We approach this problem by taking the stiffness of the inclusion phase to be a complex variable, which we eventually evaluate at the soft or hard limits. Our main result states that there is a certain class of non-physical, negative-definite values of the elastic moduli of the inclusion phase for which the effective tensor does not have infinities or become otherwise singular.
We present applications of this result to the estimation of effective moduli and to homogenization theorems. The first application involves using complexanalytic methods to obtain rigorous and accurate bounds on the effective moduli of the high-contrast composites under consideration. We also discuss the variational estimates of Rubenfeld & Keller, which yield a complementary set of bounds on these moduli. The best bounds are given by a combination of the analytical and variational results. As a second application, we show that certain known theorems of homogenization for materials with holes are simple consequences of our main result, and in this connection we establish corresponding new theorems for materials with hard inclusions. While our rederivation of the homogenization theorems for materials with holes can be closely related to other known constructions, it appears that certain elements provided by our main result are essential in the proof of homogenization for the hard-inclusion case.
Similar content being viewed by others
References
Ahlfors, L. V., Complex analysis, 3rd ed., McGraw-Hill, New York (1979).
Akhiezer, N. I. & Glazman, I. M., Theory of linear operators in Hilbert space, vol. 1, Pitman, Boston (1981).
Allaire, G., Homogenization of the Navier-Stokes equations in open sets perforated with tiny holes I. Abstract framework, a volume distribution of holes, Arch. Rational Mech. Anal. 113, 209–259 (1991).
Allaire, G., Homogenization of the Navier-Stokes equations in open sets perforated with tiny holes II. Non-critical sizes of the holes for a volume distribution and a surface distribution of holes, Arch. Rational Mech. Anal. 113, 261–298 (1991).
Bensoussan, A., Lions, J. L. & Papanicolaou, G., Asymptotic analysis for periodic structures, North-Holland, Amsterdam (1978).
Beran, M., Use of the variational approach to determine bounds for the effective permittivity in random media, Il Nuovo Cimento 38, 771–782 (1965).
Bergman, D. J., The dielectric constant of a composite material-A problem in classical physics, Physics Reports 43, 378–407 (1978).
Bruno, O. P., The effective conductivity of strongly heterogeneous composites, Proc. R. Soc. Lond. A 433, 353–381 (1991).
Christensen, R. M., Mechanics of composite materials, Wiley, New York (1979).
Cioranescu, D. & Murat, F., Une terme étrange venu d'ailleurs, I, II, in Nonlinear partial differential equations and their applications, Vols II, III, H. Brezis & J. L. Lions, eds., Research Notes in Mathematics 60, 70, Pitman, Boston (1982).
Cioranescu, D. & Paulin, J. S. J., Homogenization in open sets with holes, J. Math. Anal. Appl. 71, 590–607 (1979).
Damlamian, A., Hager, W. W. & Rostamian, R., Homogenization for degenerate equations of elasticity, Asymptotic Analysis 1, 283–302 (1988).
De Giorgi, E. & Spagnolo, S., Sulla convergenza delli integrali dell'energia per operatori ellittici del secondo ordine, Boll. Un. Mat. Ital. 8, 291–411 (1978).
Dell'Antonio, G. F., Figari, R. & Orlandi, E., An approach through orthogonal projections to the study of inhomogeneous or random media with linear response, Ann. Inst. Henri Poincare 44, 1–28 (1986).
Flewitt, P. E. J., Microstructural characterization of metals and alloys, Institute of Metals, London (1985).
Fung, Y. C., Foundation of solid mechanics, Prentice Hall Inc., Englewood Cliffs, New Jersey (1965).
Golden, K. & Papanicolaou, G., Bounds for effective parameters of heterogeneous media by analytic continuation, Comm. Math. Phys. 90, 473–491 (1983).
Hashin, Z. & Shtrikman, S., A variational approach to the theory of the elastic behavior of multiphase materials, J. Mech. Phys. Solids 11, 127–140 (1963).
Jackson, J. D., Classical electrodynamics, 2nd ed., John Wiley and Sons, New York (1975).
Kantor, Y. & Bergman, D., Elastostatic resonances — a new approach to the calculation of the effective elastic constants of composites, J. Mech. Phys. Solids 30, 355–376 (1982).
Kantor, Y. & Bergman, D., Improved rigorous bounds on the effective elastic moduli of a composite material, J. Mech. Phys. Solids 32, 41–62 (1984).
Keller, J. B., Rubenfeld, L. & Molyneux, J., Extremum principles for slow viscous flows with applications to suspensions, J. Fluid Mech. 30, 97–125 (1962).
Lions, J. L., Some methods in the mathematical analysis of systems and their control, Gordon and Breach, New York (1981).
Maxwell, J. C., Electricity and magnetism, 1st ed., Clarendon Press, Oxford (1873).
Milton, G. W. & Kohn, R. V., Variational bounds on the effective moduli of anisotropic composites, J. Mech. Phys. Solids 36, 597–629 (1988).
Morse, P. M. & Feshbach, H., Methods of theoretical physics, vol. 2, McGraw-Hill, New York (1953).
Murat, F. & Tartar, L., Calcul des variations et homogénéisation, in Les méthodes de l'homogénéisation: Théorie et applications en physique, Coll. de la Dir. des Etudes et Recherches de Electricité de France, Eyrolles, Paris, 319–370 (1985).
Nečas, J., Les méthodes directes en théorie des équations elliptiques, Masson, Paris (1967).
Oleinik, O. A., On homogenization problems, in Trends and applications of pure mathematics to mechanics, P. G. Ciarlet & M. Roseau, eds., Springer Lecture Notes in Physics 195, 248–272 (1984).
Oleinik, O. A., Iosifyan, G. A. & Panasenko, G. P., Asymptotic expansion of solutions of the system of elasticity theory in perforated domains, Math USSR Sb. 48, 19–39 (1984).
Rubenfeld, L. A. & Keller, J. B., Bounds on elastic moduli of composite media, SIAM J. Appl. Math. 17, 495–510 (1969).
Rubinstein, J. & Torquato, S., Diffusion controlled reactions: Mathematical formulation, variational principles, and rigorous bounds, J. Chem. Phys. 88, 6372–6380 (1988).
Rudin, W., Functional analysis, McGraw-Hill, New York (1973).
Sangani, A. S. & Acrivos, A., The effective conductivity of a periodic array of spheres, Proc. R. Soc. Lond. A 386, 263–275 (1983).
Sokolnikov, I. S., Mathematical theory of elasticity, McGraw-Hill, New York (1956).
Tartar, L., Cours Peccot, College de France (1977).
Torquato, S. & Rubinstein, J., Improved bounds on the effective conductivity of high-contrast suspensions, J. Appl. Phys. 69, 7118–7125 (1991).
Walpole, L. J., On bounds for the overall elastic moduli of inhomogeneous systems-I, J. Mech. Phys. Solids 14, 151–162 (1966).
Author information
Authors and Affiliations
Additional information
Communicated by D. Kinderlehrer
Rights and permissions
About this article
Cite this article
Bruno, O.P., Leo, P.H. On the stiffness of materials containing a disordered array of microscopic holes or hard inclusions. Arch. Rational Mech. Anal. 121, 303–338 (1993). https://doi.org/10.1007/BF00375624
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF00375624