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  • Open Access

Exact homogenization method for heat conduction

Gal Shmuel*

John R. Willis†

  • Department of Applied Mathematics and Theoretical Physics, Centre for Mathematical Sciences, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, United Kingdom

  • *Contact author: meshmuel@technion.ac.il
  • †Contact author: jrw1005@cam.ac.uk
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Phys. Rev. Applied 25, 014072 – Published 29 January, 2026

DOI: https://doi.org/10.1103/rgg3-19pc

Abstract

We determine the macroscopic features of thermal transport in heterogeneous conductors by generalizing an exact, source-driven homogenization method originally developed for waves. The formulation accommodates random or periodic media of finite or infinite extent, with or without pores. Our homogenization shows that the effective heat flux and entropy are spatiotemporally nonlocal functions of both the effective temperature and its gradient, and that the emergent bianisotropic cross-couplings form an adjoint pair when the microscopic relations are self-adjoint. A spatially local approximation highlights how the homogenized diffusion equation can become hyperbolic due to temporal nonlocality, and that the medium’s thermal impedance can become direction dependent, as captured by the bianisotropic terms. In addition, we develop a retrieval method for one-dimensional deterministic composites, whose results reinforce our conclusions.

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References (57)

  1. Zvi Hashin, Analysis of composite materials—a survey, J. Appl. Mech. 50, 481 (1983).
  2. Graeme W. Milton, The Theory of Composites (Cambridge University Press, Cambridge, 2002).
  3. G. Papanicolaou, A. Bensoussan, and J. L. Lions, Asymptotic Analysis for Periodic Structures, Studies in Mathematics and its Applications (Elsevier Science, Amsterdam, 1978).
  4. T. Antonakakis, R. V. Craster, and S. Guenneau, Asymptotics for metamaterials and photonic crystals, Proc. R. Soc. London A Math. Phys. Eng. Sci. 469, 20120533 (2013).
  5. H. Nassar, Q.-C. He, and N. Auffray, On asymptotic elastodynamic homogenization approaches for periodic media, J. Mech. Phys. Solids 88, 274 (2016).
  6. J. R. Willis, Variational principles for dynamic problems for inhomogeneous elastic media, Wave Motion 3, 1 (1981).
  7. J. R. Willis, Continuum Micromechanics. Chapter Dynamics of Composites (Springer-Verlag New York, Inc., New York, NY, USA, 1997), p. 265.
  8. J. R. Willis, Effective constitutive relations for waves in composites and metamaterials, Proc. R. Soc. London A Math. Phys. Eng. Sci. 467, 1865 (2011).
  9. John R. Willis, The construction of effective relations for waves in a composite, Comptes Rendus Mécanique 340, 181 (2012), Recent Advances in Micromechanics of Materials.
  10. Shixu Meng and Bojan B. Guzina, On the dynamic homogenization of periodic media: Willis’ approach versus two-scale paradigm, Proc. R. Soc. London A Math. Phys. Eng. Sci. 474, 20170638 (2018).
  11. H. Nassar, Q.-C. He, and N. Auffray, Willis elastodynamic homogenization theory revisited for periodic media, J. Mech. Phys. Solids 77, 158 (2015).
  12. Michael B. Muhlestein, Caleb F. Sieck, Andrea Alù, and Michael R. Haberman, Reciprocity, passivity and causality in Willis materials, Proc. R. Soc. A 472, 20160604 (2016).
  13. Caleb F. Sieck, Andrea Alù, and Michael R. Haberman, Origins of Willis coupling and acoustic bianisotropy in acoustic metamaterials through source-driven homogenization, Phys. Rev. B 96, 104303 (2017).
  14. René Pernas-Salomón and Gal Shmuel, Symmetry breaking creates electro-momentum coupling in piezoelectric metamaterials, J. Mech. Phys. Solids 134, 103770 (2020).
  15. R. Pernas-Salomón and G. Shmuel, Fundamental principles for generalized Willis metamaterials, Phys. Rev. Appl. 14, 064005 (2020).
  16. René Pernas-Salomón, Michael R. Haberman, Andrew N. Norris, and Gal Shmuel, The electromomentum effect in piezoelectric Willis scatterers, Wave Motion 106, 102797 (2021).
  17. Alan Muhafra, Majd Kosta, Daniel Torrent, René Pernas-Salomón, and Gal Shmuel, Homogenization of piezoelectric planar Willis materials undergoing antiplane shear, Wave Motion 108, 102833 (2022).
  18. K. Muhafra, M. R. Haberman, and G. Shmuel, Discrete one-dimensional models for the electromomentum coupling, Phys. Rev. Appl. 20, 014042 (2023).
  19. G. Shmuel and J. R. Willis, Thermally bianisotropic metamaterials induced by spatial asymmetry, Phys. Rev. Lett. 135, 116303 (2025).
  20. D. Torrent, O. Poncelet, and J. C. Batsale, Nonreciprocal thermal material by spatiotemporal modulation, Phys. Rev. Lett. 120, 125501 (2018).
  21. L. Xu, G. Xu, J. Li, Y. Li, J. Huang, and C. W. Qiu, Thermal Willis coupling in spatiotemporal diffusive metamaterials, Phys. Rev. Lett. 129, 155901 (2022).
  22. L. Xu, G. Xu, J. Huang, and C. W. Qiu, Diffusive Fizeau drag in spatiotemporal thermal metamaterials, Phys. Rev. Lett. 128, 145901 (2022).
  23. Chris Fietz and Gennady Shvets, Current-driven metamaterial homogenization, Physica B: Condensed Matter 405, 2930 (2010), Proceedings of the Eighth International Conference on Electrical Transport and Optical Properties of Inhomogeneous Media.
  24. Andrea Alù, First-principles homogenization theory for periodic metamaterials, Phys. Rev. B 84, 075153 (2011).
  25. G. W. Milton and J. R. Willis, On modifications of Newton’s second law and linear continuum elastodynamics, Proc. R. Soc. London A Math. Phys. Eng. Sci. 463, 855 (2007).
  26. J. R. Willis, Exact effective relations for dynamics of a laminated body, Mech. Mater. 41, 385 (2009).
  27. W. Kaminski, Hyperbolic heat conduction equation for materials with a nonhomogeneous inner structure, J. Heat Transfer 112, 555 (1990).
  28. S. L. Sobolev, Heat conduction equation for systems with an inhomogeneous internal structure, J. Eng. Phys. Thermophys. 66, 436 (1994).
  29. K. Mitra, S. Kumar, A. Vedavarz, and M. K. Moallemi, Experimental evidence of hyperbolic heat conduction in processed meat, J. Heat Transfer 117, 568 (1995).
  30. Carlo Cattaneo, Sulla conduzione del calore, Atti Sem. Mat. Fis. Univ. Modena 3, 83 (1948).
  31. Gianfranco Capriz, Krzysztof Wilmanski, and Paolo Maria Mariano, Exact and approximate Maxwell-Cattaneo-type descriptions of heat conduction: A comparative analysis, Int. J. Heat Mass Transf. 175, 121362 (2021).
  32. Andrea Alù, Restoring the physical meaning of metamaterial constitutive parameters, Phys. Rev. B 83, 081102(R) (2011).
  33. Graeme W. Milton, New metamaterials with macroscopic behavior outside that of continuum elastodynamics, New J. Phys. 9, 359 (2007).
  34. With a slight abuse of notation, we use the same symbols for functions and their Laplace transforms.
  35. J. R. Willis, A comparison of two formulations for effective relations for waves in a composite, Mech. Mater. 47, 51 (2012).
  36. Conversely, the structure of Pernas-Salomón and Shmuel [14] generalizes Willis’ result [8], by accounting for additional differential operations and field variables present in the electroelastic problem.
  37. The purpose of these apparently redundant definitions (such as f=r earlier) is to link this structure to the general structure in Ref. [14].
  38. A formulation employing a comparison linear medium that lends itself to approximation will be presented elsewhere.
  39. Michael B. Muhlestein, Caleb F. Sieck, Preston S. Wilson, and Michael R. Haberman, Experimental evidence of Willis coupling in a one-dimensional effective material element, Nat. Commun. 8, 15625 EP (2017).
  40. In our companion paper [19], s=iω was adopted.
  41. R. Pérez-Álvarez, René Pernas-Salomón, and V. R. Velasco, Relations between transfer matrices and numerical stability analysis to avoid the ω problem, SIAM J. Appl. Math. 75, 1403 (2015).
  42. C. A. Bennett and R. R. Patty, Thermal wave interferometry: A potential application of the photoacoustic effect, Appl. Opt. 21, 49 (1982).
  43. D. R. Smith, S. Schultz, P. Markoš, and C. M. Soukoulis, Determination of effective permittivity and permeability of metamaterials from reflection and transmission coefficients, Phys. Rev. B 65, 195104 (2002).
  44. V. Fokin, M. Ambati, C. Sun, and X. Zhang, Method for retrieving effective properties of locally resonant acoustic metamaterials, Phys. Rev. B 76, 144302 (2007).
  45. Xiaoshi Su and Andrew N. Norris, Retrieval method for the bianisotropic polarizability tensor of Willis acoustic scatterers, Phys. Rev. B 98, 174305 (2018).
  46. Equation (8) in Ref. [19] is slightly different: there we used iω instead of s and defined the components of the state vector differently.
  47. M. Kohmoto, L. P. Kadanoff, and C. Tang, Localization problem in one dimension: Mapping and escape, Phys. Rev. Lett. 50, 1870 (1983).
  48. John Lekner, Light in periodically stratified media, J. Opt. Soc. Am. A 11, 2892 (1994).
  49. Ben Lustig and Gal Shmuel, On the band gap universality of multiphase laminates and its applications, J. Mech. Phys. Solids 117, 37 (2018).
  50. The bars over variables introduced above are now omitted.
  51. Relative to the chosen dimensionless formulation, l=1.
  52. J. R. Willis, The nonlocal influence of density variations in a composite, Int. J. Solids Struct. 21, 805 (1985).
  53. Ying Li, Wei Li, Tiancheng Han, Xu Zheng, Jiaxin Li, Baowen Li, Shanhui Fan, and Cheng-Wei Qiu, Transforming heat transfer with thermal metamaterials and devices, Nat. Rev. Mater. 6, 488 (2021).
  54. Zeren Zhang, Liujun Xu, Teng Qu, Min Lei, Zhi-Kang Lin, Xiaoping Ouyang, Jian-Hua Jiang, and Jiping Huang, Diffusion metamaterials, Nat. Rev. Phys. 5, 218 (2023).
  55. Muamer Kadic, Tiemo Bückmann, Robert Schittny, and Martin Wegener, Metamaterials beyond electromagnetism, Rep. Prog. Phys. 76, 126501 (2013).
  56. G. Shmuel and J. R. Willis, Multiphysics bianisotropy in wave-diffusion metamaterials, in preperation.
  57. John R. Willis, Data for “Exact homogenization method for heat conduction”, 2025, https://zenodo.org/records/18048366.

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Phys. Rev. Applied 25, 014072– Published 29 January, 2026
  • Received 21 September 2025
  • Revised 8 December 2025
  • Accepted 23 December 2025
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