Generalized Clausius-Mossotti formula for random composite with circular fibers

Journal of Statistical Physics

Volumn 102, Issue 1-2, 2001, Pages 115-145

  Berlyand, L ^a,c   Mityushev, V ^b  

^a PENNSYLVANIA STATE UNIVERSITY   (United States)

^b POMERANIAN UNIVERSITY IN S UPSK   (Poland)

^c The University of Akron   (United States)

Author keywords

Effective conductivity: Homogenization; Functional equation; Random composite material

Indexed keywords

EID: 0035530273     PISSN: 00224715     EISSN: None     Source Type: Journal    
DOI: 10.1023/A:1026512725967     Document Type: Article

References (32)

1. 1. M. Ablowitz and A. Fokas, Complex Variables (Cambridge University Press, 1997).
2. 2. A. Bensoussan, J.-L. Lions, and Papanicolaou, Asymptotic Analysis for Periodical Structures (North Holland, Amsterdam, 1978).
3. 3. M. J. Beran, Statistical Continuum Theory (Wiley, New York, 1968).
4. 4. V. L. Berdichevskij, Variational Principles of Continuum Mechanics (Moscow, Nauka, 1983).
5. 5. D. J. Bergman and K. J. Dunn, Bulk effective dielectric constant of a composite with periodic micro-geometry, Phys. Rev. Ser. B 45:13262-13271 (1992).
6. 6. L. Berlyand and A. Kolpakov, Asymptotics in the Relative Interparticle Distance of the Effective Properties of a High Contrast Random Composite (preprint) (1999).
7. 7. H. Cheng and L. Greengard, A method of images for the evaluation of electrostatic fields in systems of closely spaced conducting cylinders, SIAM J. Appl. Math. 58:122-141 (1998).
8. 8. P. Furmanski, Heat conduction in composites: Homogenization and macroscopic behavior, Appl. Mech. Rev. 50(61:327-356 (1997).
9. 9. V. V. Jikov, S. M. Kozlov, and O. A. Olejnik, Homogenization of Differential Operators and Integral Functionals (Springer-Verlag, Berlin, 1994).
10. 10. F. D. Gakhov, Boundary Value Problems (Moscow, Nauka, 1977).
11. 11. G. M. Golusin, Solution to the plane problem of steady heat conduction for a multiply connected domain which are bounded by circles. Matem. Sbornik 42:191-198 (1935).
12. 12. E. I. Grigolyuk and L. A. Fil'shtinskij, Periodical Piecewise Homogeneous Elastic Structures (Moscow, Nauka, 1992).
13. 13. A. Hurwitz and R. Courant, Theory of Functions (Moscow, Nauka, 1968).
14. 14. S. M. Kozlov, Averaging of random operators. Matem. Sbornik 109(2):188-203 (1980) (1979) (English Transl.: Math USSR, Sb. 37(2):167 180).
15. 14. S. M. Kozlov, Averaging of random operators. Matem. Sbornik 109(2):188-203 (1980) (1979) (English Transl.: Math USSR, Sb. 37(2):167 180).
16. 15. S. M. Kozlov, Geometrical aspects of averaging, Russian Math. Surveys 44(2):91-144 (1989).
17. 16. D. Landauer, Electrical conductivity in inhomogeneous media, in Electrical Transport and Optical Properties of Inhomogeneous Media, J. C. Garland and D. B. Tanner, eds., (American Institute of Physics, New York, 1978), pp. 2-43.
18. 17. L. Rayleigh, On the influence of obstacles arranged in rectangular order upon the properties of medium, Phil. Mag. 34:481-502 (1892).
19. 18. R. C. McPhedran and D. R. McKenzie, The conductivity of lattices of spheres. I. The simple cubic lattice, Proc. Roy. Soc. Lond. Sci: A 359:45-52 (1978).
20. 19. R. C. McPhedran, Transport properties of cylinder pairs and of the square array of cylinders. Proc. Roy. Soc. Lond. Ser. A 408:31-43 (1986).
21. 20. G. W. Milton, Mechanics of Composites (Camdridge University Press, 2000) (to appear).
22. 21. G. W. Milton, Bounds on the electromagnetic, elastic, and other properties of two-component composites, Phys. Rev. Lett. 46:542 (1981).
23. 22. L. G. Mikhajlov, A New Class of Singular Integral Equations, (Wolters-Noordhoff Publ., Groningen, 1970).
24. 23. V. Mityushev, Application of Functional Equations to the Effective Thermal Conductivity of Composite Materials (WSP Publisher, Slupsk, 1996) [in Polish].
25. 24. V. Mityushev, A functional equation in a class of analytic functions and composite materials, Demostratio Math. 30:63-70 (1997).
26. 25. V. Mityushev, Transport properties of regular arrays of cylinders, ZAMM 77:115-120 (1997).
27. 26. V. Mityushev and S. V. Rogosin, Constructive Methods for Linear and Nonlinear Boundary Value Problems of the Analytic Functions Theory (Chapman & Hall/CRC, Boca Raton, 1999).
28. 27. G. Papanicolaou and S. Varadan, Boundary Value Problems with Rapidly Oscillating Random Coefficients (Colloquia Mathematics, Societatis Janos Bolyai 27, Radon Fields, Esztergom, Hungary, 1979), pp. 835-873.
29. 28. A. S. Sangani and C. Yao, Transport processes in random arrays of cylinders. I. Thermal conduction. Phys. Fluids 31:2426-2434 (1988).
30. 29. J. F. Thovert, I. C. Kim, S. Torquato, and A. Acrivos, Bounds on the effective properties of polydispersed suspensions of spheres: An evaluation of two relevant morphological parameters, J. Appl. Phyx. 67:6088-6098 (1990).
31. 30. S. Torquato, Random heterogeneous media: Microstructure and improved bounds on effective properties, Appl. Mech. Rev. 44:37-76 (1991).
32. 31. A. Weil, Elliptic Functions According to Eisenstein and Kronecker (Springer-Verlag, Berlin, 1976).