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Articles

Improved algorithm for analytical solution of the heat conduction problem in doubly periodic 2D composite materials

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Pages 1-23 | Received 25 Aug 2013, Accepted 04 Dec 2013, Published online: 05 Feb 2014

References

  • Berlyand L, Mityushev VV. Generalized Clausius-Mossotti formula for random composite with circular fibers. J. Stat. Phys. 2001;102:115–145.
  • King JL. A simple continuum model of a layered composite material. J. Strain Anal. Eng. Des. 1972;7:146–150.
  • Markov KZ. Elementary micromechanics of heterogeneous media. In: Markov KZ, Preziosi L, editors. Heterogeneous media: modelling and simulation. Boston: Birkhauser; 1999. p. 1–162.
  • Suen WM, Wong SP, Young K. The lattice model of heat conduction in a composite material. J. Phys. D: Appl. Phys. 1979;12:1325–1338.
  • Allaire G. Shape optimization by the homogenization method. Berlin: Springer Verlag; 2002.
  • Cherkaev AV. Variational methods for structural optimization. New York (NY): Springer Verlag; 2000.
  • Kalamkarov AL, Kolpakov AG. Analysis, design and optimization of composite structures. Chichester: John Wiley & Sons; 1997.
  • Manevitch LI, Andrianov IV, Oshmyan VG. Mechanics of periodically heteregeneous structures. Foundations of engineering mechanics. Berlin: Springer; 2002.
  • Milton GW. The theory of composites. Cambridge monographs on applied and computational mathematics. Cambridge: Cambridge University Press; 2002.
  • Movchan AV, Movchan NV, Poulton CG. Asymptotic models of fields in dilute and densely packed composites. London: Imperial College Press; 2002.
  • Bakhvalov NS, Panasenko GP. Homogenization: averaging processes in periodic media. Moscow: Nauka; 1984. Russian; English transl., Dordrecht/Boston/London: Kluwer; 1989.
  • Jikov VV, Kozlov SM, Oleinik OA. Homogenization of differential operators and integral functionals. Berlin: Springer; 1994.
  • Cherednichenko KD, Smyshlyaev VP. On full two-scale expansion of the solutions of nonlinear periodic rapidly oscillating problems and higher-order homogenised variational problems. Arch. Ration. Mech. Anal. 2004;174:385–442.
  • Cherednichenko KD, Smyshlyaev VP, Zhikov VV. Non-local homogenized limits for composite media with highly anisotropic periodic fibres. Proc. R. Soc. Edin. 2006;136:87–114.
  • Kushch VI. Micromechanics of composites: multipole expansion approach. Amsterdam: Butterworth–Heinemann; 2013.
  • Mityushev V, Rogosin S. Constructive methods for linear and nonlinear boundary value problems for analytic functions. Theory and applications. Monographs and surveys in pure and applied mathematics. Boca Raton (FL): Chapman & Hall/CRC Press; 1999.
  • Obnosov YV. Boundary value problems of the theory of heterogeneous media: multiphase media, separated by second order curves. Kazan: Kazan State University; 2009. Russian.
  • Zohdi TI, Wriggers P. Introduction to computational micromechanics. Berlin:Springer; 2005.
  • Andrianov IV, Bolshakov VI, Danishevs’kyy VV, Weichert D. Higher order asymptotic homogenization and wave propagation in periodic composite materials. Proc. R. Soc. London. 2008;464:1181–1201.
  • Fiedler T, Pesetskaya E, Oechsner A, Grácio J. On the determination of the effective thermal conductivity of composite materials. In: Proceedings of the second Workshop on Advanced Computational Engineering Mechanics; Erlangen, Germany; 2005. p. 187–194.
  • Kachanov M, Tsukrov I, Shafiro B. Effective moduli of solids with cavities of various shapes. Appl. Mech. Rev. 1994;47:S151–S174.
  • Kanaun SK, Levin VM. Self-consistent methods for composites. Dordrecht:Springer; 2008.
  • Mityushev VV, Pesetskaya EV, Rogosin SV. Analytical methods for heat conduction in composites and porous media. In: Öchsner A, Murch GE, de Lemos MJS, editors. Thermal properties of cellular and porous materials. Amsterdam: WILEY-VCH; 2007. p. 124–167.
  • Pesetskaya E. Effective conductivity of composite materials with random positions of cylindrical inclusions: finite number inclusions in the cell. Appl. Anal. 2005;84:843–865.
  • Kachanov M, Sevostianov I. On quantitative characterization of microstructures and effective properties. Int. J. Solids Struct. 2005;42:309–336.
  • Sevostianov I, Kachanov M. Connections between elastic and conductive properties of heterogeneous materials. In: Aref H, van der Giessen E, editors. Advances in applied mechanics. San Diego (CA): Academic Press Vol. 42;2008. p. 69-252.
  • Mityushev V. Transport properties of doubly periodic arrays of circular cylinders and optimal design problems. Appl. Math. Optim. 2001;44:17–31.
  • Weil A. Elliptic functions according to Eisenstein and Kronecker. Berlin: Springer-Verlag; 1976.
  • Mityushev V, Adler P. Darcy flow around a two-dimensional lens. J. Phys. A: Math. Gen. 2006;39:3545–3560.
  • Milton GW. Mechanics of composites. Cambridge: Cambridge University Press; 2000.
  • Mityushev V, Rylko N. Maxwell’s approach to effective conductivity and its limitations. Q. J. Mech. Appl. Math. 2013;66:241–251.
  • Perrins WT, McKenzie DR, McPhedran RC. Transport properties of regular arrays of cylinders. Proc. R. Soc. Lond. A. 1979;369:207–225.
  • Hurwitz A. Lectures on general function theory and elliptic functions. Berlin: Springer-Verlag; 1964. German.

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