Journal of Mathematical Chemistry

, Volume 53, Issue 3, pp 828–843 | Cite as

Hashin–Shtrikman bounds on the effective thermal conductivity of a transversely isotropic two-phase composite material

  • Carmen Calvo-Jurado
  • William J. Parnell
Original Paper


This paper is concerned with the estimation of the effective thermal conductivity of a transversely isotropic two phase composite. We describe the general construction of the Hashin–Shtrikman bounds from first principles in the conductivity setting. Of specific interest in composite design is the fact that the shape of the inclusions and their distribution can be specified independently. This case covers a multitude of composites used in applications by taking various limits of the spheroid aspect ratio, including both layered media and unidirectional composites. Furthermore the expressions derived are equally valid for a number of other effective properties due to the fact that Laplace’s equation governs a significant range of applications, e.g. electrical conductivity and permittivity, magnetic permeability and many more. We illustrate the implementation of the scheme with several examples.


Hashin–Shtrikman bounds Conductivity Transport problem Hill tensor 

Mathematics Subject Classification

74Q20 74Q15 



This work has been partially supported by the project MTM 2011-24457 of the “Ministerio de Ciencia e Innovación” of Spain and the research group FQM-309 of the “Junta de Andalucía”.


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Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  1. 1.Dpto. de Matemáticas, Escuela PolitécnicaUniversity of ExtremaduraCáceresSpain
  2. 2.School of MathematicsUniversity of ManchesterManchesterUK

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