Functional series and Hashin–Shtrikman type bounds on the effective conductivity of random media

European Journal of Applied Mathematics

Volumn 6, Issue 6, 1995, Pages 611-629

  Markov, K Z ^a   Zvyatkov, Kr D ^b  

^a UNIVERSITY OF SOFIA   (Bulgaria)

^b SHUMEN UNIVERSITY   (Bulgaria)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 84974306808     PISSN: 09567925     EISSN: 14694425     Source Type: Journal    
DOI: 10.1017/S0956792500002096     Document Type: Article

References (34)

1. Beasley J. D. & Torquato S. 1986 Bounds on the conductivity of a suspension of random impenetrable spheres. J. Appl. Phys. 60, 3576–3581.
2. Beran M. 1965 Use of a variational approach to determine bounds for the effective permittivity of a random medium. Nuovo Cimento 38, 771–782.
3. Beran M. 1968 Statistical Continuum Theories. Wiley.
4. Brown W. F. 1955 Solid mixture permittivities. J. Chein. Physics 23, 1514–1517.
5. Christov C. I. 1985 A further development of the concept of random density function with application to Volterra-Wiener expansions. C. R. Acad. Bulg. Sci. 38 (1), 35–38.
6. Christov C. I. & Markov K. Z. 1985 Stochastic functional expansion for random media with perfectly disordered constitution. SIAM J. Appl. Math. 45, 289–311.
7. Felderhof B. U. 1982 Bounds on the effective dielectric constant of a suspension of uniform spheres. J. Phys. C15, 3953–3966.
8. Felderhof B. U., Ford G. W. & Cohen E. G. D. 1982 Two-particle cluster integral in the expansion of the dielectric constant. J. Stat. Phys. 28, 649–672.
9. Finkel'berg V. M. 1964 Dielectric permeability of a mixture. J. Tekhn. Fiziki 34 (3), 509–518. (in Russian.) (English translation in Techn. Phys. 9, 396–402.)
10. Jeffrey D. J. 1973 Conduction through a random suspension of spheres. Proc. Roy. Soc. London A335, 355–367.
11. Jeffrey D. J. 1978 The physical significance of non-convergent integrals in expressions for bulk quantities. In Continuum Models and Discrete Systems (ed. Provan J. ), University of Waterloo Press, Ontario, pp. 653–674.
12. Hashin Z. & Shtrikman S. 1962 A variational approach to the theory of the effective magnetic permeability of multiphase materials. J. Appl. Phys. 33, 3125–3131.
13. Hill R. 1963 New derivations of some elastic extremum principles. In Progress in Applied Mechanics, Prager Anniversary Volume, Macmillan, pp. 99–106.
14. Kröner E. 1977 Bounds for effective elastic moduli of disordered materials. J. Mech. Phys. Solids 25, 137–155.
15. Markov K. Z. 1987 Application of Volterra-Wiener series for bounding the overall conductivity of heterogeneous media. I. General procedure. II. Suspensions of equi-sized spheres. SIAM J. Appl. Math. 47, 831–849, 850–870.
16. Markov K. Z. 1989 On the heat propagation problem for random dispersions of spheres. Math. Balkanica (New Series) 3, 399–417.
17. Markov K. Z. 1991 On the factorial functional series and their application to random media. SIAM J. Appl. Math. 51, 172–186.
18. Markov K. Z. & Christov C. I. 1992 On the problem of heat conduction for random dispersions of spheres allowed to overlap. Math. Models & Methods in Appl. Sci. 2, 249–269.
19. Markov K. Z. & Zvyatkov Kr. D. 1988 c2- bounds on the elastic moduli of random dispersions of spheres. C. R. Acad. Bulg. Sci. 41 (7), 21–24.
20. Markov K. Z. & Zvyatkov Kr. D. 1990 New c2-bounds on the effective properties of random dispersions of spheres. In Proc. 6th Bulg. Congress on Mechanics, Vol. 2, Bulg. Acad. Sci., Sofia, pp. 313–316.
21. Markov K. Z. & Zvyatkov Kr. D. 1991a On the optimal third-order bounds on the effective conductivity of random dispersions of spheres. J. Theor. Applied. Mech., Bulg. Acad. Sci. 22 (3), 107–116.
22. Markov K. Z. & Zvyatkov Kr. D. 1991b Optimal third-order bounds on the effective properties of some composite media, and related problems. Adv. in Mechanics (Warsaw) 14 (4), 3–46.
23. Markov K. Z. & Zvyatkov Kr. D. 1994 Functional series and Hashin-Shtrikman's type bounds on the effective properties of random media. In: Recent Advances in Mathematical Modelling of Composite Materials (ed. Markov K. Z. ), pp. 59–106. World Scientific.
24. Milton G. W. & Phan-Thien N. 1982 New bounds on effective elastic moduli of two-components material. Proc. Roy. Soc. London A380, 305–331.
25. Phan-Thien N. & Milton G. W. 1982 New bounds on effective thermal conductivity of N-phase materials. Proc. Roy. Soc. London A380, 333–348.
26. Prager S. 1963 Diffusion and viscous flow in concentrated suspensions. Physica 29, 129–138.
27. Stratonovich R. L. 1967 Topics in Theory of Random Noises, Vol. 1. Gordon and Breach.
28. Schetzen D. L. 1980 The Volterra and Wiener Theories of Nonlinear Systems. Wiley.
29. Torquato S. 1986 Bulk properties of two-phase disordered media. III. New bounds on the effective conductivity of dispersions of penetrable spheres. J. Chem. Phys. 84, 6345–6359.
30. Torquato S. & Stell G. 1985 Bounds on the effective thermal conductivity of a dispersion of fully penetrable spheres. Lett. Appl. Engng Sci. 23, 375–383.
31. Willis J. R. 1978 Variational principles and bounds for the overall properties of composites. In: Continuum Models and Discrete Systems (ed. Provan J. ), University of Waterloo Press, Ontario, pp. 185–215.
32. Willis J. R. 1980 Relationship between derivations of the overall properties of composites by perturbation expansions and variational principles. In: Variational Methods in Mechanics of Solids (ed. Nemat-Nasser S. ), Pergamon Press, pp. 59–66.
33. Willis J. R. 1981 Variational and related methods for the overall properties of composites. Adv. Appl. Mechanics 21, 1–78.
34. Zvyatkov Kr. D. 1992 On certain Hashin-Shtrikman's type bounds for random dispersion. C. R. Acad. Bulg. Sci. 45 (9), 21–23.