The study of the heat transfer properties of heterogeneous materials has been a world-wide research focus, for example in sectors related to thermal management in aerospace, architecture, and geology. In this paper, the emergent behavior and thermal conduction characteristics of three-dimensional and three-phase heterogeneous materials thermal networks are studied using the finite element method. The results show that the existence of percolation paths of each phase has a strong impact on the effective properties of the network when the contrast in thermal conductivities of each phase is high, and percolation also affects the effective thermal conductivity of the whole thermal network system. However, when the contrast in thermal conductivity between the two phases is low, the thermal networks exhibit a more consistent and “emergent” behavior, and the effective thermal conductivity of thermal networks at the same volume fraction changes to a lower extent from network to network. This paper also demonstrates that a logarithmic mixing rule can predict the effective thermal conductivity in the low contrast emergent region in three-dimensional networks, and the modeling method provides new approaches for the design of multi-phase composites and prediction of their thermal conduction properties.

Original languageEnglish
JournalAdvances in Mechanical Engineering
Issue number8
Publication statusPublished - 23 Aug 2023

Bibliographical note

Funding Information:
The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: The present work is financially supported by the National Natural Science Foundation of China (No. 11302159), the Principal Foundation of Xi’an Technological University (No. XGPY200213), Basic Research Plan of Natural Science in Shaanxi Province (No. 2021JQ-650), and Scientific Research Program Funded by Shaanxi Provincial Education Department (No. 19JK0412).


  • effective thermal conductivity
  • emergent
  • Heterogeneous
  • thermal networks

ASJC Scopus subject areas

  • Mechanical Engineering


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