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Purpose: Purpose of this paper is to study the effective thermal conductivity of infinite unidirectional circular cylinders arbitrary distributed in a uniform host. Design/methodology/approach: Applying the method of complex potentials we present the heat flux as a complex analytic function in the section of the fibre composite perpendicular to the direction of the cylinders. Findings: The flux is written in the form of a series. Special attention is paid to the question of polydispersity, when many cylinders of different radii are distributed in the periodicity cell. Research limitations/implications: The proposed exact formulas hold for fiber composites. Practical implications: The deduced formulas are useful in prediction and estimation of the effective thermal conductivity of fiber composites. Originality/value: The obtained formulas for the effective conductivity are new and were not known before.
Wydawca
Rocznik
Tom
Strony
45--52
Opis fizyczny
Bibliogr. 23 poz.
Twórcy
autor
- Department of Techniques, Pedagogical University, ul. Podchorążych 2, 30-084 Kraków, Poland, natalia.rylko@ap.krakow.pl
Bibliografia
- [1] L. Berlyand, V. Mityushev, Increase and decrease of the effective conductivity of a two phase composites due to polydispersity, Journal of State Physic 118 (2005) 481-509.
- [2] V. Mityushev, Transport properties of doubly periodic arrays of circular cylinders and optimal design problems, Applied Mathematic Optimization 44 (2001) 17-31.
- [3] V. Golubeva, V. Mityushev, Steady heat conduction of composites with large number of inclusions, Izvestia vuzov. Energetika 6 (1994) 86-90 (in Russian).
- [4] V. Mityushev, S.V. Rogosin, Constructive Methods for Linear and Nonlinear Boundary Value Problems for Analytic Functions. Theory and Applications, Chapman & Hall / CRC, Monographs and Surveys in Pure and Applied Mathematics 1999.
- [5] V. Mityushev, Steady heat conduction of the material with an array of cylindrical holes in the non-linear case, IMA Journal of Applied Mathematic 61 (1998) 91-102.
- [6] V. Mityushev, Transport properties of two-dimensional composite materials with circular inclusions, Proceedings of Royal Society, London A455 (1999) 2513-2528.
- [7] V. Mityushev, P.M. Adler, Longitudial permeability of a doubly periodic rectangular array of circular cylinders, Journal of Applied Mathematics and Mechanics 82 (2002) 335-345.
- [8] V. Mityushev, R-linear problem on torus and its application to composites, Complex Variables 50 (2005) 621-630.
- [9] V. Mityushev, Exact solution of the R-linear problem for a disk in a class of doubly periodic functions, Journal of Applied Functional Analysis 2 (2007) 115-127.
- [10] V. Mityushev, Representative cell in mechanics of composites and generalized Eisenstein-Rayleigh sums, Complex Variables 51 (2006) 1033-1045.
- [11] D.J. Bergman, The dielectric constants of a composite material - a problem in classical physics, Physic Reports C43 (1978) 377-407.
- [12] D. J. Bergman, Bulk physical properties of composite media, in: Les methodes de l'homogeneisation: theorie et applications en physique, pp. 1-128. Volume 57 of Collection de la Direction des etudes et recherches d' Electricite de France, Session qui s'est tenue au Centre du Breau-sans-Nappe du 27 juin au 13 juillet 1983, Eyrolles, Paris, France 1985.
- [13] J. B. Keller, Conductivity of a Medium Containing a Dense Array of Perfectly Conducting Spheres or Cylinders or Nonconducting Cylinders, Journal of Applied Physic 34 (1963) 991-993.
- [14] J.A. Kolodziej, T. Strek, Analytical approximation of the shape factors for conductive heat flow in circular and regular polygonal cross-sections, International Journal Heat and Mass Transfer 44 (2001) 999-1012.
- [15] R.C. McPhedran, Transport properties of cylinder pairs and of the square array of cylinders, Proceedings of Royal Society, London A408 (1986 ) 31-43.
- [16] R.C. McPhedran, G.W. Milton, Transport properties touching of cylinder pairs and of the square array of touching cylinders, Proceedings of Royal Society, London A411 (1987 ) 313-326.
- [17] R.C. McPhedran, L. Poladian, G.W., Asymptotic Studies of Closely Spaced, Highly Conducting Cylinders, Proceedings of Royal Society, London A415 (1988 ) 185-196.
- [18] Lord Rayleigh, On the influence of obstacles arranged in rectangular order upon the properties of medium, Philosophy Magazine 34 (1892) 481-502.
- [19] N. Rylko, Transport properties of a rectangular array of highly conducting cylinders, Journal of Engineering Mathematics 38 (2000) 473-474.
- [20] P.L. Walker, Elliptic Functions: A Constructive Approach, New York: Wiley, 1996.
- [21] A. Weil, Elliptic Functions According to Eisenstein and Kronecker, Berlin, Springer-Verlag, 1976.
- [22] M. Bilewicz, J.C. Viana, L.A. Dobrza ski, Self reinforced polymer-polymer composites, Journal of Achievements in Materials and Manufacturing Engineering 24/2 (2007) 43-46.
- [23] L.A. Dobrza ski, Z. Brytan, M. Rosso, Sinter-hardening process applicable to stainless steels, Journal of Achievements in Materials and Manufacturing Engineering 24/2 (2007) 11-18.
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Bibliografia
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bwmeta1.element.baztech-article-BSL9-0029-0009