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Improved variational bounds for conductive periodic composites with 3D microstructures and nonuniform thermal resistance

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Abstract

Improved variational bounds for the effective conductivity of a matrix-inclusion conductive periodic composite are obtained. The studied composite is macroscopically anisotropic with nonuniform interfacial thermal resistance between isotropic phases. The homogenization theory is applied to a three-dimensional heat conduction problem which is stated in terms of nondimensional parameters. The Biot number is explicitly given in the variational formulation of the local problems and in the related minimization problems. The approach is based on the Lipton–Vernescu variational principles which allow to derive narrower bounds by incorporating more detailed morphological information. The bounds depend on the concentration and the conductivity of each phase, the periodic distribution and the shape of the inclusions, the Biot number and the nonuniform interfacial resistance.

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Authors and Affiliations

  1. Facultad de Economía, Universidad de La Habana, L y 21. Vedado, CP 10400, La Habana, Cuba

    G. López-Ruiz

  2. Facultad de Matemática y Computación, Universidad de La Habana, L y San Lázaro. Vedado, CP 10400, La Habana, Cuba

    J. Bravo-Castillero, R. Guinovart-Díaz & R. Rodríguez-Ramos

  3. Sorbonne Universités, UPMC Univ Paris 06, CNRS, UMR 7190, Institut Jean le Rond d’Alembert, 75005, Paris, France

    R. Brenner

  4. Universidade Federal do Rio de Janeiro, Engenharia Mecânica/Politécnica/COPPE, CT, CP 68503, Rio de Janeiro, RJ, 21941-972, Brazil

    M. E. Cruz

  5. Departamento de Matemática e Estatística, Instituto de Física e Matemática, Universidade Federal de Pelotas, CP 354, Pelotas, CEP, 96010-971, Brazil

    L. D. Pérez-Fernández

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  1. G. López-Ruiz
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  2. J. Bravo-Castillero
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  3. R. Brenner
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  4. M. E. Cruz
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  5. R. Guinovart-Díaz
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  6. L. D. Pérez-Fernández
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  7. R. Rodríguez-Ramos
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Correspondence to J. Bravo-Castillero.

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López-Ruiz, G., Bravo-Castillero, J., Brenner, R. et al. Improved variational bounds for conductive periodic composites with 3D microstructures and nonuniform thermal resistance. Z. Angew. Math. Phys. 66, 2881–2898 (2015). https://doi.org/10.1007/s00033-015-0540-z

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  • DOI: https://doi.org/10.1007/s00033-015-0540-z

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