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2014 | Vol. 66, nr 4 | 287--301
Tytuł artykułu

Cross-properties of the effective conductivity of the regular array of ideal conductors

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Języki publikacji
EN
Abstrakty
EN
We present an accurate expression for the effective conductivity of a regular square-lattice arrangement of ideally conducting cylinders, valid for arbitrary concentrations. The formula smoothly interpolates between the two asymptotic expressions derived for low and high concentrations of the cylinders. Analogy with critical phenomena is suggested and taken to the extent of calculating the superconductivity critical exponent and the particle-phase threshold from the very long expansions in concentration. The obtained formula is valid for all concentrations including touching cylinders, hence it completely solves with high accuracy the problem of the effective conductivity for the square array.
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Rocznik
Strony
287--301
Opis fizyczny
Bibliogr. 36 poz., rys.
Twórcy
autor
  • Department of Computer Sciences and Computer Methods Pedagogical University ul. Podchorążych 2 30-084 Kraków, Poland, mityu@up.krakow.pl
  • Department of Computer Sciences and Computer Methods Pedagogical University ul. Podchorążych 2 30-084 Kraków, Poland, wnawalaniec@gmail.com
Bibliografia
  • 1. I.V. Andrianov, J. Awrejcewicz, New trends in asymptotic approaches: summation and interpolation methods, Appl. Mech. Rev., 54, 69–92, 2001.
  • 2. I.V. Andrianov, Two-point Padé approximants in the mechanics of solids, ZAMM, 74, 121–122, 1994.
  • 3. I.V. Andrianov, V.V. Danishevskyy, A.L. Kalamkarov, Analysis of the effective conductivity of composite materials in the entire range of volume fractions of inclusions up to the percolation threshold, Composites: Part B Engineering, 41, 503–507, 2010.
  • 4. I.V. Andrianov, G.A. Starushenko, V.V. Danishevskyy, S. Tokarzewski, Homogenization procedure and Padé approximants for effective heat conductivity of composite materials with cylindrical inclusions having square cross-section, Proc. R. Soc. Lond. A, 455, 3401–3413, 1999, doi: 10.1098/rspa.1999.0457.
  • 5. I. Andrianov, V. Danishevskyy, S. Tokarzewski, Quasifractional approximants in the theory of composite materials, Acta Applicandae Mathematicae, 61, 29–35, 2000.
  • 6. G.A. Baker, P. Graves-Moris, Padé Approximants, Cambridge Univ. Press, Cambridge 1996).
  • 7. C.M. Bender, S. Boettcher, Determination of f(∞) from the asymptotic series for f(x) about x = 0, J. Math. Phys., 35, 1914-1921, 1994.
  • 8. L. Berlyand, A. Kolpakov, Network approximation in the limit of small interparticle distance of the effective properties of a high contrast random dispersed composite, Arch. Ration. Mech. Anal. 159, 179–227, 2001.
  • 9. T.C. Choy, Effective Medium Theory. Principles and Applications, Clarendon Press, Oxford 1999.
  • 10. R. Czapla, W. Nawalaniec, V. Mityushev, Effective conductivity of random twodimensional composites with circular non-overlapping inclusions, Comput. Mat. Sci., 63, 118–126, 2012.
  • 11. S. Gluzman, V. Mityushev, Series, index and threshold for random 2D composite, 2014, arXiv:1405.1016v1.
  • 12. S. Gluzman, V. Yukalov, Unified approach to crossover phenomena, Physical Review E, 58, 4197–4209, 1998.
  • 13. S. Gluzman, V.I. Yukalov, Self-similar extrapolation from weak to strong coupling, J. Math. Chem., 48, 883–913, 2010.
  • 14. S. Gluzman, V.I. Yukalov, D. Sornette, Self-similar factor approximants, Physical Review E, 67, 2, art. 026109, DOI: 10.1103, 2003.
  • 15. J.B. Keller, Conductivity of a medium containing a dense array of perfectly conducting spheres or cylinders or nonconducting cylinders, J. Appl. Phys., 34, 991–993, 1963.
  • 16. J.B. Keller, A Theorem on the Conductivity of a composite medium, J. Math. Phys. 5, 548–549, 1964; doi: 10.1063/1.1704146.
  • 17. R.C. McPhedran, Transport properties of cylinder pairs and of the square array of cylinders, Proc.R. Soc. Lond. A, 408, 31–43, 1986, doi: 10.1098/rspa.1986.0108,
  • 18. R.C. McPhedran, L. Poladian, G.W. Milton, Asymptotic studies of closely spaced, highly conducting cylinders, Proc. R. Soc. A, 415, 185–196, 1988.
  • 19. J.C. Maxwell, Electricity and Magnetism, 1st ed., Oxford, Clarendon Press, 1873, p. 365.
  • 20. V.V. Mityushev, E. Pesetskaya, S.V. Rogosin, Analytical Methods for Heat Conduction in Composites and Porous Media, [in:] Cellular and Porous Materials: Thermal Properties Simulation and Prediction, A. Öchsner, G.E. Murch, M.J.S. de Lemos [eds.], Wiley, 2008, 121–164.
  • 21. V. Mityushev, Steady heat conduction of a material with an array of cylindrical holes in the nonlinear case, IMA Journal of Applied Mathematics, 61, 91–102, 1998.
  • 22. V. Mityushev, Exact solution of the R-linear problem for a disk in a class of double periodic functions, J. Appl. Functional Analysis, 2, 115–127, 2007.
  • 23. W.T. Perrins, D.R. McKenzie, R.C. McPhedran, Transport properties of regular array of cylinders, Proc. R. Soc. A, 369, 207–225, 1979.
  • 24. Lord Rayleigh, On the influence of obstacles arranged in rectangular order upon the properties of medium, Phil. Mag., 34, 481–502, 1892.
  • 25. N. Rylko, Transport properties of the regular array of highly conducting cylinders, J. Engrg. Math. 38, 1–12, 2000.
  • 26. N. Rylko, Structure of the scalar field around unidirectional circular cylinders, Proc. R. Soc. A, 464, 391–407, 2008.
  • 27. H.E. Stanley, Scaling, universality, and renormalization: Three pillars of modern critical phenomena, Reviews of Modern Physics, 71, 358–366, 1999.
  • 28. S.P. Suetin, Padé approximants and efficient analytic continuation of a power series, Russian Mathematical Surveys, 57, 43–141, 2002.
  • 29. S. Tokarzewski, J. Blawzdziewicz, I. Andrianov, Effective conductivity for densely parked highly condenced cylinders, Appl. Physics A, 59, 601–604, 1994.
  • 30. S. Tokarzewski, J.J. Telega, M. Pindor, J. Gilewicz, Basic inequalities for multipoint Padé approximants to Stieltjes functions, Arch. Mech., 54, 141–153, 2002.
  • 31. S. Torquato, Random Heterogeneous Materials: Microstructure and Macroscopic Properties, Springer, New York 2002.
  • 32. S. Torquato, F.H. Stillinger, Jammed hard-particle packings: From Kepler to Bernal and beyond, Reviews of Modern Physica, 82, 2634–2672, 2010.
  • 33. V.I. Yukalov, S. Gluzman, Critical indices as limits of control functions, Phys. Rev. Lett., 79, 333-336, 1997.
  • 34. V.I. Yukalov, S. Gluzman, D. Sornette, Summation of power series by self-similar factor approximants, Physica A, 328, 409–438, 2003.
  • 35. E.P. Yukalova, V.I. Yukalov, S. Gluzman, Extrapolation and interpolation of asymptotic series by self-similar approximants, Journal of Mathematical Chemistry, 47, 959–983, 2010, doi: 10.1007/s 10910-009-9618-1.
  • 36. V.I. Yukalov, S. Gluzman, Optimization of self-similar factor approximants, Molecular Physics, 107, 2237–2244, 2009.
Typ dokumentu
Bibliografia
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