On effective resistivity and related parameters for periodic checkerboard composites

Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences

Volumn 456, Issue 2003, 2000, Pages 2741-2754

  Craster, R V ^a  

^a IMPERIAL COLLEGE LONDON   (United Kingdom)

Author keywords

Complex analysis; Composite materials; Effective parameters

Indexed keywords

EID: 0008243316     PISSN: 13645021     EISSN: None     Source Type: Journal    
DOI: 10.1098/rspa.2000.0637     Document Type: Article

References (27)

1. Abramowitz, M. & Stegun, I. A. 1969 Handbook of mathematical functions. New York: Dover.
2. Bcrdichevski, V. L. 1985 The thermal conductivity of chess structures. VestnMosk. Univ. Mat. Mech. 40, 56-63 (in Russian).
3. Borcea, L. & Papanicolaou, G. C. 1998 Network approximation for transport properties of high contrast materials. SIAM J. Appl. Math. 58, 501-539.
4. Bruggeman, D. A. G. 1935 Berechnung verschiedener physikalischer Konstanten von heterogenen Substanzen. Ann. Phys. Lpz. 24, 636-679 (in German).
5. Clark, K. E. & Milton, G. W. 1995 Optimal bounds correlating electric, magnetic and thermal properties of two-phase, two-dimensional composites. Proc. R. Soc. Land. A 448, 161-190.
6. Craster, R. V. 1997 The solution of a class of free boundary problems. Proc. R. Soc. Land. A 453, 607-630.
7. Craster, R. V. & Obnosov, Y. V. 2000 Four phase periodic composites. SIAM J. Appl. Math. (Submitted.)
8. Dykhne, A. M. 1971 Conductivity of a two-dimensional two-phase system. Sov. Phys. JETP 32, 63-65.
9. Gautesen, A. K. 1988 The effective conductivity of a composite-material with a periodic rectangular geometry. SIAM J. Appl. Math. 48, 393-404.
10. Helsing, J. 1991 Transport properties of two-dimensional tilings with corners. Phys. Rev. B 44, 11667-11682.
11. Keller, J. B. 1963 Conductivity of a medium containing a dense array of perfectly conducting spheres or cylinders or nonconducting cylinders. J. Appl. Phys. 34, 991-993.
12. Keller, J. B. 1964 A theorem on the conductivity of a composite medium. J. Math. Phys. 5, 548-549.
13. Keller, J. B. 1987 Effective conductivity of periodic composites composed of two very unequal conductors. J. Math. Phys. 28, 2516-2520.
14. Lawden, D. F. 1989 Elliptic functions and applications. Springer.
15. Maxwell, J. C. 1904 A treatise on electricity and magnetism. Oxford University Press.
16. Mendelson, K. S. 1975 A theorem on the effective conductivity of a two-dimensional heterogeneous medium. J. Appl. Phys. 46, 4740-4741.
17. Milton, G. W. & Golden, K. 1990 Representations for conductivity functions of multicomponent composites. Commun. Pure Appl. Math. 43, 647-671.
18. Milton, G. W., McPhedran, R. C. & McKenzie, D. R. 1981 Transport properties of intersecting cylinders. Appl. Phys. 25, 23-30.
19. Mityushev, V. V. 1997 Transport properties of double-periodic arrays of circular cylinders. Z. Angew. Math. Mech. 77, 115-120.
20. Nicorovici, N. A., McPhedran, R. C. &: Milton, G. W. 1993 Transport properties of a three-phase composite material: the square array of coated cylinders. Proc. R. Soc. Land. A 442, 599-620.
21. Obnosov, Y. V. 1992 Solution of certain boundary problems of 7£-linear conjugation with piecewise constant coefficients. RUSS. Math. (Izv. VUZ) 36, 38-47.
22. Obnosov, Y. V. 1996 Exact solution of a boundary-value problem for a rectangular checkerboard field. Proc. R. Soc. Land. A 452, 2423-2442.
23. Obnosov, Y. V. 1999 Periodic heterogeneous structures: new explicit solutions and effective characeristics of refraction of an imposed field. SIAM J. Appl. Math. 59, 1267-1287.
24. Obnosov, Y. V. 1999& Seepage refraction in a semicircular lens located at the boundary of two porous massifs. J. Appl. Math. Mech. 62, 749-762.
25. Rayleigh, J. W. 1892 On the influence of obstacles arranged in rectangular order upon the properties of the medium. Phil. Mag. 34, 481-502.
26. Talbot, D. R. S. & Willis, J. R. 1994 Upper and lower bounds for the overall properties of a nonlinear composite dielectric. I. Random microgeometry. Proc. R. Soc. Land. A 447, 365-384.
27. Torquato, S. 1991 Random heterogeneous media: microstructure and improved bounds on effective properties. Appl. Mech. Rev. 44, 37-76.