Hostname: page-component-77c89778f8-m42fx Total loading time: 0 Render date: 2024-07-18T17:56:58.199Z Has data issue: false hasContentIssue false
  • English
  • Français

Two-component composites whose effective conductivities are power means of the local conductivities

Published online by Cambridge University Press:  01 October 2008

ANNETTE MEIDELL ,
RALPH HØIBAKK ,
DAG LUKKASSEN  and
GUY BEERI
ANNETTE MEIDELL
Affiliation:
Narvik University College, P.O. Box 385 N-8505 Narvik, Norway
RALPH HØIBAKK
Affiliation:
Narvik University College, P.O. Box 385 N-8505 Narvik, Norway
DAG LUKKASSEN
Affiliation:
Narvik University College, P.O. Box 385 N-8505 Narvik, Norway Norut Narvik, P.O. Box 250, N-8504 Narvik, Norway
GUY BEERI
Affiliation:
Narvik University College, P.O. Box 385 N-8505 Narvik, Norway
Get access Rights & Permissions [Opens in a new window]

Abstract

We study a scale of two-component composite structures of equal proportions with infinitely many microlevels. The structures are obtained recursively and we find that their effective conductivities are power means of the local conductivities.

Type
Papers
Information
European Journal of Applied Mathematics , Volume 19 , Issue 5 , October 2008 , pp. 507 - 517
Copyright
Copyright © Cambridge University Press 2008

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Bensoussan, A., Lions, J. L. & Papanicolaou, G. C. (1978) Asymptotic Analysis for Periodic Structures, North Holland, Amsterdam.Google Scholar
[2]Braides, A. & Lukkassen, D. (2000) Reiterated homogenization of integral functionals. Math. Mod. Meth. Appl. Sci. 10 (1), 47–71.CrossRefGoogle Scholar
[3]Craster, R. V. & Obnosov, Y. V. (2001) Four phase checkerboard composites. SIAM J. Appl. Math. 61 (6), 1839–1856.CrossRefGoogle Scholar
[4]Dal Maso, G. (1993) An Introduction to Γ -Convergence, Birkhäuser, Boston.CrossRefGoogle Scholar
[5]De Giorgi, E. & Spagnolo, S. (1973) Sulla convergenza degli intehrali dell'energia per operaori ellittici del secondo ordine. Boll. Un. Mat. Ital. 8, 391–411.Google Scholar
[6]Dychne, A. M. (1970) Conductivity of a two-phase two-dimensional system. J. Exp. Theor. Phys. 59 (7), 110–115.Google Scholar
[7]Jikov, V. V., Kozlov, S. M. & Oleinik, O. A. (1994) Homogenization of Differential Operators and Integral Functionals, Springer-Verlag, Berlin.CrossRefGoogle Scholar
[8]Keller, J. B. (1964) A theorem on the conductivity of a composite medium. J. Math. Phys. 5 (4), 548–549.CrossRefGoogle Scholar
[9]Lions, J.-L., Lukkassen, D., Persson, L.-E. & Wall, P. (2000) Reiterated homogenization of monotone operators. C. R. Acad. Sci., Paris, Ser. I, Math. 330 (8), 675–680.CrossRefGoogle Scholar
[10]Lions, J.-L., Lukkassen, D., Persson, L.-E. & Wall, P. (2001) Reiterated homogenization of nonlinear monotone operators. Chin. Ann. Math., Ser. B. 22 (1), 1–14.CrossRefGoogle Scholar
[11]Lukkassen, D. (1999) A new reiterated structure with optimal macroscopic behaviour. SIAM J. Appl. Math. 59 (5), 1825–1842.CrossRefGoogle Scholar
[12]Lukkassen, D. & Meidell, A. Effective properties of some new self-similar structures, to appear in Applicable Anal.Google Scholar
[13]Lukkassen, D., Meidell, A. & Wall, P. Multiscale homogenization of monotone operators, to appear in Discrete Contin. Dynam. Syst.–Ser. A.Google Scholar
[14]Lukkassen, D. & Milton, G. W. (2002) On hierarchical structures and reiterated homogenization. In Proceedings of the Conference on Function Spaces, Interpolation Theory and Related Topics in Honour of Jaak Peetre on his 65th Birthday, 17–22 August 2000, Walter de Gruyter, Berlin, pp. 311–324.Google Scholar
[15]Milton, G. W. (2001) Proof of a conjecture on the conductivity of checkerboards. J. Math. Phys. 42 (10), 4873–4882.CrossRefGoogle Scholar
[16]Milton, G. W. (2002) The Theory of Composites. Cambridge Monographs on Applied and Computational Mathematics, 6. Cambridge University Press, Cambridge, UK, 2002.CrossRefGoogle Scholar
[17]Obnosov, Y. V. (1996) Exact solution of a boundary-value problem for a rectangular checkerboard field. Proc. R. Soc. Lond., Ser. A, Math. Phys. Sci. 452, 2423–2442.Google Scholar
[18]Obnosov, Y. V. (1999) Periodic heterogeneous structures: New explicit solutions and effective characteristics of refraction of an imposed field. SIAM J. Appl. Math. 59, 1267–1287.CrossRefGoogle Scholar