Determining effective conductivities of fractal objects

V. R. Voller, Fábio D.A. Aarão Reis

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We develop formulas to approximate the effective thermal conductivity of composite materials consisting of a matrix with nominal dimensionless conductivity Kmat=1 containing inclusions (holes) with a conductivity Khole≤1 laid out in fractal patterns: 2D Sierpinski carpets and 3D Menger sponges. Direct simulation of steady state heat conduction in finite stages of construction of these objects provide benchmark conductivity values. Recent theory on random walks in fractal media provides conductivity estimates for Sierpinski carpets for the case where Khole=0. Recognizing that the estimate in the fractal generators is related to the classic Rayleigh model, we obtain an expression for the conductivity in the more general case 0≤Khole≤1. Comparison with direct simulated values shows a relative error below 6% when the conductivity contrast is high (Khole≈0) and below 1% when the contrast is low. This significantly improves the accuracy of estimates obtained from simple resistor networks, which are interpreted as upper and lower bounds. The extension of this approach to Menger sponges gives a conductivity estimator related to the classic Maxwell conductivity model. This estimator, accounting for the effects of pattern correlations, has relative errors in the same range as the carpet estimator. A conjecture for the random walk dimension in sponges is also derived. The effective conductivities for 2D and 3D patterns, with low volume fractions of the matrix and high conductivity contrast, also represent a significant improvement over the classic formulas for uncorrelated distributions of inclusions.

Original languageEnglish (US)
Article number106577
JournalInternational Journal of Thermal Sciences
StatePublished - Jan 1 2021

Bibliographical note

Funding Information:
FDAAR acknowledges support from the Brazilian agencies Faperj ( E-26/202.881/2018 ), CAPES ( 88887.310427/2018-00 ), and CNPq ( 305391/2018-6 ). VRV acknowledges support from the James L. Record Professorship and CAPES ( 88887.363427/2019-00 ). The authors also thank the reviewers for their insightful and helpful comments and suggestions.


  • Composites
  • Conductivity
  • Fractals
  • Random walks


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