Bounds on the effective thermal conductivity of a dispersion of fully penetrable spheres

S. Torquato, G. Stell

Research output: Contribution to journalArticlepeer-review

31 Scopus citations


We evaluate upper and lower bounds on the effective thermal conductivity Ke of a model of two-phase composite materials in which one of the phases consists of spherical inclusions (or voids) of conductivity K2 and volume fraction φ2, dispersed randomly throughout a matrix phase of conductivity K1 and volume fraction φ1. Our evaluations compare third-order bounds of Beran and of Brown, which utilize the three-point matrix probability function of the model, with bounds of De Vera and Strieder (which apply only to the aforementioned "fully penetrable sphere" model) and of Hashin and Shtrikman. The comparisons are made over extended ranges of values of both φ2 and α = K2 K1 and reveal that the best bounds among those we have tested (generally those of Beran) are sharp enough to give quantitatively useful estimates of Ke for 0.1 ≤ K2 K1 ≤ 10 over a wide range of φ2 values. They are sharp at high φ2 values (i.e., φ2 = 0.9) and very sharp at low φ2 values (e.g. φ2 = 0.1) where they remain useful for K2 K1 ≈ 100. They are less sharp at intermediate values (e.g. φ2 = 0.5). As is well known, such results immediately translate into equivalent results for the electrical conductivity, dielectric constant, or magnetic permeability of composites.

Original languageEnglish (US)
Pages (from-to)375-383
Number of pages9
JournalInternational Journal of Engineering Science
Issue number3
StatePublished - 1985
Externally publishedYes

All Science Journal Classification (ASJC) codes

  • Mechanics of Materials
  • General Engineering
  • Mechanical Engineering
  • General Materials Science


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