RSS feeds
РусскийEnglish
Issue 4, 2024
Volume year page
Issues Authors PACS Subscription For authors

Issues

 / 

1996

 / 

October

←  →

Reviews of topical problems


Macroscopic conductivity of random inhomogeneous media. Calculation methods


National Research University of Electronic Technology (MIET), sq. Shokina 1, Zelenograd, Moscow, 124498, Russian Federation

The macroscopic conductivity of nonhomogeneous media such as polycrystals, composites, etc., is discussed. One of the major global parameters of a randomly inhomogeneous medium (RIM), the effective (macroscopic) conductivity tensor (ECT), is considered. Based on functional analysis ideas, particularly the projector and orthogonal reduced field concepts, a method for obtaining bounds on ECT components is developed. A criterion is given by which a position of each iteration solution relative to the preceding one is determined. It is shown that previous model ECT results fall within, and are in fact special cases, of the scheme proposed by the author. The importance of the auxiliary, zero-fluctuation parameter σc in constructing convergent series for quantities of interest is established. Extended versions of the Hashin-Shtrikman variational principles and of Keller’s theorem are obtained. The structural parameters introduced for describing an RIM are expressed, in the proposed method, in terms of the n-point probabilities (n-point interactions for the random local conductivity field). The piecewise uniform ’polarised’ field approximation is combined with classical energy theorems to obtain the ECT bounds best achievable within the n-point RIM description.

Fulltext pdf (803 KB)
Fulltext is also available at DOI: 10.1070/PU1996v039n10ABEH000173
PACS: 02.90.+p, 72.90.+y (all)
DOI: 10.1070/PU1996v039n10ABEH000173
URL: https://ufn.ru/en/articles/1996/10/d/
A1996VV43300003
Citation: Fokin A G "Macroscopic conductivity of random inhomogeneous media. Calculation methods" Phys. Usp. 39 1009–1032 (1996)
BibTexBibNote ® (generic)BibNote ® (RIS)MedlineRefWorks

Оригинал: Фокин А Г «Макроскопическая проводимость случайно-неоднородных сред. Методы расчета» УФН 166 1069–1093 (1996); DOI: 10.3367/UFNr.0166.199610d.1069

References (117) ↓ Cited by (23) Similar articles (20)

  1. Fricke H Phys. Rev. 24 575 (1924)
  2. Bruggeman D A G Ann. Phys. 24 636 (1935)
  3. Osborn J A Phys. Rev. 67 351 (1945)
  4. Lifshits I M, Rosentsveîg L N Zh. Eksp. Teor. Fiz. 16 967 (1946) [Revisions: Zh. Eksp. Teor. Fiz. 21 1184 (1951)]
  5. Pekeris C L Phys. Rev. 71 268 (1947)
  6. Lifshits I M, Parkhomovskiî G D Zh. Eksp. Teor. Fiz. 20 175 (1950)
  7. Landauer R J. Appl. Phys. 23 779 (1952)
  8. Brown W F (Jr) J. Chem. Phys. 23 1514 (1955)
  9. Hashin Z, Shtrikman S J. Appl. Phys. 33 3125 (1962)
  10. Hashin Z, Shtrikman S Phys. Rev. 130 129 (1963)
  11. Finkel’berg V M Zh. Tekh. Fiz. 34 509 (1964) [Sov. Phys. Tech. Phys. 9 (1964)]
  12. Finkel’berg V M Zh. Eksp. Teor. Fiz. 46 725 (1964) [Sov. Phys. JETP 19 494 (1964)]
  13. Keller J B J. Math. Phys. 5 548 (1964)
  14. Beran M Nuovo Cimento 38 771 (1965)
  15. Frisch H L Trans. Soc. Rheol. 9 293 (1965)
  16. Ryzhov Yu A Izv. Vyssh. Uchebn. Zaved. Radiofiz. 9 39 (1966) [Sov. Radiophys. 9 25 (1966)]
  17. Walpole L J J. Mech. Phys. Solids 14 151 (1966)
  18. Walpole L J J. Mech. Phys. Solids 14 289 (1966)
  19. Miller M N J. Math. Phys. 10 1988 (1969)
  20. Dykhne A M Zh. Eksp. Teor. Fiz. 59 110 (1970) [Sov. Phys. JETP 32 63 (1971)]
  21. Fokin A G Zh. Tekh. Fiz. 41 1073 (1971) [Sov. Phys. Tech. Phys. 16 849 (1971)]
  22. Beran M J, Silnutzer N R J. Composite Mater. 5 246 (1971)
  23. Fokin A G Zh. Tekh. Fiz. 43 71 (1973) [Sov. Phys. Tech. Phys. 18 44 (1973)]
  24. Kirkpatrick S Rev. Mod. Phys. 45 574 (1973)
  25. Mendelson K S J. Appl. Phys. 46 917 (1975)
  26. Schulgasser K J. Math. Phys. 17 382 (1976)
  27. Fokin A G Zh. Tekh. Fiz. 47 1121 (1977) [Sov. Phys. Tech. Phys. 22 645 (1977)]
  28. Shklovskiî B I Zh. Eksp. Teor. Fiz. 72 288 (1977) [Sov. Phys. JETP 45 152 (1977)]
  29. Fokin A G Prikl. Mat. Mekh. 42 546 (1978)
  30. Rytov C M, Kravtsov Yu A, Tatarskiî V I Vvedenie v Statisticheskuyu Radiofiziku Chast’ 2 Sluchaînye Polya (Introduction into Statistical Radiophysics Part II Random Fields, Moscow: Nauka, 1978)
  31. Beran M J, in Kompozitsionnye Materialy Tom 2 Mekhanika Kompozitsionnykh Materialov [Composite Materials Vol. 2 Mechanics of Composite Materials (Ed. J Sendetski)] (Moscow: Mir, 1978) p. 242
  32. Willemse M W M Ph. D. Thesis. Tech. Univ. Twente, 1979
  33. Willemse M W M, Caspers W J J. Math. Phys. 20 1824 (1979)
  34. Fokin A G Izv. Vyssh. Uchebn. Zaved. Fiz. 24 (7) 117 (1981)
  35. Fokin A G Phys. Status Solidi B 111 281 (1982)
  36. Fokin A G Phys. Status Solidi B 115 509 (1983)
  37. Fokin A G Phys. Status Solidi B 119 741 (1983)
  38. Hashin Z Trans. ASME J. Appl. Mech. 50 481 (1983)
  39. Shvidler M I Zh. Eksp. Teor. Fiz. 84 1185 (1983) [Sov. Phys. JETP 57 688 (1983)]
  40. Balagurov B Ya Zh. Eksp. Teor. Fiz. 89 1796 (1985) [Sov. Phys. JETP 62 1036 (1985)]
  41. Rozhnov G V Zh. Eksp. Teor. Fiz. 98 1737 (1990) [Sov. Phys. JETP 71 975 (1990)]
  42. Fokin A G Zh. Eksp. Teor. Fiz. 101 67 (1992) [Sov. Phys. JETP 74 36 (1992)]
  43. Bergman D J Phys. Rep. 43 377 (1978)
  44. Milton G W J. Mech. Phys. Solids 30 177 (1982)
  45. Milton G W, Golden K, in Thermal Conductivity 18 (Eds T Ashworth, D R Smith, New York: Plenum Press, 1985) p. 571
  46. Milton M Phys. Rev. B 38 11296 (1988)
  47. Beran M, Molyneux J Nuovo Cimento 30 1406 (1963)
  48. Kohler W, Papanicolaou G C, in Macroscopic Properties of Disordered Media (Eds R Burridge, S Childress, G Papanicolaou, Berlin, Heidelberg, New York: Springer, 1982) p. 111
  49. Willis J R J. Mech. Phys. Solids 25 185 (1977)
  50. Willis J R J. Mech. Phys. Solids 39 73 (1991)
  51. Torquato S Rev. Chem. Eng. 4 151 (1987)
  52. Weng G J Int. J. Engug. Sci. 28 1111 (1990)
  53. Cook H F Nature (London) 168 247 (1951)
  54. Odelevskiî V I Zh. Tekh. Fiz. 21 667 (1951)
  55. Odelevskiî V I Zh. Tekh. Fiz. 21 678 (1951)
  56. Mitoff S P Adv. Mater. Res. Vol. 3 (New York, London, Sydney, Toronto, 1968) p. 305
  57. Cohen M H, Jortner J Phys. Rev. Lett. 30 696 (1973)
  58. Hale D K J. Mater. Sci. 11 2105 (1976)
  59. Kerner E H Proc. Phys. Soc. London B 69 802 (1956)
  60. Carmona F, El Amarti A Phys. Rev. B 35 3284 (1987)
  61. Cohen M H, Jortner J Phys. Rev. Lett. 30 699 (1973)
  62. Shklovskiî B I, Éfros A L Usp. Fiz. Nauk 117 401 (1975) [Sov. Phys Usp. 18 845 (1975)]
  63. Shklovskii B I Phys. Status Solidi B 85 K111 (1978)
  64. Landauer R, in Electrical, Transport and Optical Properties of Inhomogeneous Media (Eds J C Garland, D B Tanner, New York: AIP, 1978) p. 2
  65. Glushkov E A, Reztsov V V Fiz. Tekh. Poluprovodn. 13 1469 (1979) [Sov. Phys. Semicond. 13 859 (1979)]
  66. Shpinar L I, Yaskovets I I Fiz. Tverd. Tela (Leningrad) 26 1725 (1984) [Sov. Phys. Solid State 26 1045 (1984)]
  67. Neîmark A V Zh. Eksp. Teor. Fiz. 98 611 (1990) [Sov. Phys. JETP 71 341 (1990)]
  68. Juretschke H J, Landauer R, Swanson J A J. Appl. Phys. 27 838 (1956)
  69. Herring C J. Appl. Phys. 31 1939 (1960)
  70. Dykhne A M Zh. Eksp. Teor. Fiz. 59 641 (1970) [Sov. Phys. JETP 32 348 (1971)]
  71. Dreîzin Yu A, Dykhne A M Zh. Eksp. Teor. Fiz. 63 242 (1972) [Sov. Phys. JETP 36 127 (1973)]
  72. Stroud D Phys. Rev. B 12 3368 (1975)
  73. Emets Yu P Élektricheskie Kharakteristiki Kompozitsionnykh Materialov s Regulyarnoî Strukturoî (Electrical Properties of Composite Materials with Regular Structure, Kiev: Naukova Dumka, 1986)
  74. Kuchis E V Gal’vanomagnitnye Éffekty i Metody ikh Issledovaniya (Galvanomagnetic Effects and Methods of Their Study, Moscow: Radio i Svyaz’, 1990)
  75. Barabanenkov Yu N et al. Usp. Fiz. Nauk 102 3 (1970) [Sov. Phys. Usp. 13 551 (1971)]
  76. Volkov S D, Stavrov V P Statisticheskaya Mekhanika Kompozitnykh Materialov (Statistical Mechanics of Composite Materials, Minsk: Izd. Belorus. Gos. Univ., 1978)
  77. Mekhontseva D M, Rybalko F P, Volkov S D Fiz. Metallov Metallovedenie 22 135 (1966) [Phys. Metals Metallography 22 (1, 1967)]
  78. Christensen R M Mechanics of Composite Materials [New York: Wiley, 1979) [Translated into Russian (Moscow: Mir, 1982)]
  79. Az’muko A A et al. Zavodskaya Laboratorya (12) 26 (1985)
  80. Papadakis E, in Primenenie Fizicheskoî Akustiki v Kvantovoî Fizike i Fizike Tverdogo Tela Tom IVB Fizicheskaya Akustika [Application of Physical Acoustics in Quantum Physics and Physics of Solids Vol. IVB Physical Acoustics (Ed. W M Mason)] (Moscow: Mir, 1970) p. 317
  81. Hershey A V J. Appl. Mech. 21 236 (1954)
  82. Batchelor G K Ann. Rev. Fluid Mech. 6 227 (1974)
  83. Fokin A G Zh. Eksp. Teor. Fiz. 104 3178 (1993) [JETP 77 492 (1993)]
  84. Fonshteîn N M, Bortsov A N, Chernyavskiî K S Zavodskaya Laboratorya (12) 34 (1985)
  85. Lurie K A, Cherkaev A V Usp. Mech. 9 (2) 3 (1986)
  86. Kleîza V V, Kleîza I V Dokl. Akad. Nauk 325 711 (1992)
  87. Demishev S V et al. Usp. Fiz. Nauk 164 195 (1994) [Phys. Usp. 37 185 (1994)]
  88. Polyakov V V, Egorov A V Fiz. Metallov Metallovedenie 77 (6) 41 (1994)
  89. Yakubov I T Usp. Fiz. Nauk 163 (5) 35 (1993) [Phys. Usp. 36 365 (1993)]
  90. Geselowitz D B Proc. IEEE 77 857 (1989)
  91. Semenov M A et al. Biofiz. 39 628 (1994) [Biophys. 39 631 (1994)]
  92. Belyaev A Yu Dokl. Akad. Nauk SSSR 304 575 (1989) [Sov. Phys. Dokl. 34 30 (1989)]
  93. Korn G, Korn T Spravochnik po Matematike dlya Nauchnykh Rabotnikov i Inzhenerov (Mathematical Handbook for Scientists and Engineers, Moscow: Nauka, 1974) Ch. 14, Sec. 4; Ch. 15 [In English (New York: McGraw-Hill, 1968)]
  94. Alimov Yu I, Kravtsov Yu A Usp. Fiz. Nauk 162 (7) 149 (1992) [Sov. Phys. Usp. 35 606 (1992)]
  95. Lifshits I M, Gredeskul S A, Pastur L A Vvedenie v Teoriyu Neuporyadochennykh Sistem (Introduction to the Theory of Disorded Systems, Moscow: Nauka, 1982) [Translated into English (New York: Wiley, 1988)]
  96. Gredeskul S A, Freîlikher V D Usp. Fiz. Nauk 160 (2) 239 (1990) [Sov. Phys. Usp. 33 134 (1990)]
  97. Klyatskin V I, Saichev A I Usp. Fiz. Nauk 162 (3) 161 (1992) [Sov. Phys. Usp. 35 231 (1992)]
  98. Dotsenko Vik. S Usp. Fiz. Nauk 165 481 (1995) [Phys. Usp. 38 457 (1995)]
  99. Riesz F, Sz.-Nagy B Lecons de’Analyse Fonctionnelle (Budapest: Akademia’i Kiado, 1972) [Translated into Russian (Moscow: Mir, 1979)]
  100. Kantorovich L V, Akilov G P Funktsional’nyî Analis (Functional Analysis, Moscow: Nauka, 1977) [Translated into English (Oxford, New York: Pergamon Press, 1982)]
  101. Aleksandrov K S Dokl. Akad. Nauk SSSR 176 295 (1967) [Sov. Phys. Dokl. 12 872 (1968)]
  102. Landau L D, Lifshits E M Élektrodinamika Sploshnykh Sred (Electrodynamics of Continuous Media, Moscow: Nauka, 1992) [Translated into English (New York, London: Pergamon, 1984)]
  103. Gel’fand I M, Shilov G E Obobshchennye Funktsii i Deîstviya nad nimi Vyp. 1 (Generalised Functions and Operations on them. Iss. 1, Moscow: Nauka, 1958)
  104. Fokin A G, in Sbornik Nauchnykh Trudov po Problemam Mikroélektroniki. Ser. Fiz. Mat. [Collection of Proceedings on Microelectronics (Ed. A N Vasil’ev)] Ser. Phis. Math. (Moscow: MIÉT, 1969) p. 141
  105. Fokin A G Prikl. Mat. Tekh. Fiz. (1) 98 (1972)
  106. Monecke J Phys. Status Solidi B 140 279 (1987)
  107. Hill R, in Progr. in Appl. Mech., The Prager Anniversary Volume (New York, London, 1963) p. 99
  108. Prager V Vvedenie v Mekhaniku Sploshnykh Sred (Introduction into Mechanics of Continuous Media, Moscow: Inostr. Liter., 1963)
  109. Novatskiî V Teoriya Uprugosti (Theory of Elasticity, Moscow: Mir, 1975) Ch. 4
  110. Kröner E Z. Phys. 151 504 (1958)
  111. Eshelby J D Proc. R. Soc. London Ser. (A) 241 376 (1957)
  112. Balagurov B Ya Zh. Eksp. Teor. Fiz. 81 665 (1981) [Sov. Phys. JETP 54 355 (1981)]
  113. Mendelson K S J. Appl. Phys. 46 4740 (1975)
  114. Schulgasser K J. Phys. C: Solid State Phys. 10 407 (1977)
  115. Schulgasser K J. Math. Phys. 17 378 (1976)
  116. Picinbono M B C.R. Acad. Sci. 240 2206 (1955)
  117. Brown W F (Jr) J. Math. Phys. 15 1516 (1974)
© 1918–2024 Uspekhi Fizicheskikh Nauk
Email: ufn@ufn.ru Editorial office contacts About the journal Terms and conditions