A semi-analytical model for evaluation of effective thermal conductivity of composites with periodic microstructure

  • Eduardo Nobre Lages
  • Severino Pereira Cavalcanti MarquesEmail author
Technical Paper


This paper presents a semi-analytical micromechanical model for evaluation of effective thermal conductivity of composite materials with periodic microstructure. The model is based on the equivalent inclusion thermal problem and utilizes Fourier series for representation of periodic functions involved in the material homogenization approach. Two main objectives can be highlighted in the work. The first of them is the derivation of the thermal micromechanical model, which consists in an extension of a formulation originally derived for homogenization of elastic heterogeneous solids. The second objective consists in a detailed investigation on the performance of the model, considering convergence of results and efficiency of strategies employed for the approximate solution of the thermal homogenization problem. Analyses on the complexity of transformation temperature gradient functions are also included in this investigation. The results obtained for two examples of periodic composites with different microstructural architectures are presented and discussed in detail.


Thermal conductivity Periodic composites Fourier series Homogenization 



The authors acknowledge the support provided by the Brazilian National Council for Scientific and Technological Development - CNPq to develop this work.


  1. 1.
    Oppelt T, Urbaneck T, Böhme H, Platzer B (2017) Numerical investigation of effective thermal conductivity for two-phase composites using a discrete model. Appl Therm Eng 115:1–8. CrossRefGoogle Scholar
  2. 2.
    Gou J-J, Gong C-L, Gu L-X, Li S, Tao W-Q (2017) Unit cells of composites with symmetric structures for the study of effective thermal properties. Appl Therm Eng 126:602–619. CrossRefGoogle Scholar
  3. 3.
    Hatta H, Taya M (1986) Equivalent inclusion method for steady state heat conduction in composites. Int J Eng Sci 24(7):1159–1172. CrossRefzbMATHGoogle Scholar
  4. 4.
    Benveniste Y (1987) Effective thermal conductivity of composites with a thermal contact resistance between the constituents: nondilute case. J Appl Phys 61(8):2840–2843. CrossRefGoogle Scholar
  5. 5.
    Lee Y-M, Yang R-B, Gau S-S (2006) A generalized self-consistent method for calculation of effective thermal conductivity of composites with interfacial contact conductance. Int Commun Heat Mass Transf 33(2):142–150. CrossRefGoogle Scholar
  6. 6.
    Bonfoh N, Dreistadt C, Sabar H (2017) Micromechanical modeling of the anisotropic thermal conductivity of ellipsoidal inclusion-reinforced composite materials with weakly conducting interfaces. Int J Heat Mass Transf 108:1727–1739. CrossRefGoogle Scholar
  7. 7.
    Bonfoh N, Sabar H (2018) Anisotropic thermal conductivity of composites with ellipsoidal inclusions and highly conducting interfaces. Int J Heat Mass Transf 118:498–509. CrossRefGoogle Scholar
  8. 8.
    Xiao J, Xu Y, Zhang F (2018) An analytical method for predicting the effective transverse thermal conductivity of nano coated fiber composites. Compos Struct 189:553–559. CrossRefGoogle Scholar
  9. 9.
    Torquato S (2002) Random heterogeneous materials: microstructure and macroscopic properties. Springer, New YorkCrossRefGoogle Scholar
  10. 10.
    Ostoja-Starzewski M (2006) Material spatial randomness: from statistical to representative volume element. Probab Eng Mech 21(2):112–132. MathSciNetCrossRefGoogle Scholar
  11. 11.
    Drago A, Pindera M-J (2007) Micro-macromechanical analysis of heterogeneous materials: macroscopically homogeneous versus periodic microstructures. Compos Sci Technol 67(6):1243–1263. CrossRefGoogle Scholar
  12. 12.
    Rolfes R, Hammerschmidt U (1995) Transverse thermal conductivity of CFRP laminates: a numerical and experimental validation of approximation formulae. Compos Sci Technol 54(1):45–54. CrossRefGoogle Scholar
  13. 13.
    Sihn S, Roy AK (2011) Micromechanical analysis for transverse thermal conductivity of composites. J Compos Mater 45(11):1245–1255. CrossRefGoogle Scholar
  14. 14.
    Liu K, Takagi H, Osugi R, Yang Z (2012) Effect of physicochemical structure of natural fiber on transverse thermal conductivity of unidirectional abaca/bamboo fiber composites. Compos Part A 43(8):1234–1241. CrossRefGoogle Scholar
  15. 15.
    Kushch VI, Sevostianov I (2004) Effective elastic properties of the particulate composite with transversely isotropic phases. Int J Solids Struct 41(3–4):885–906. CrossRefzbMATHGoogle Scholar
  16. 16.
    Michel JC, Moulinec H, Suquet P (1999) Effective properties of composite materials with periodic microstructure: a computational approach. Comput Methods Appl Mech Eng 172(1–4):109–143. MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Yvonnet J, He Q-C, Toulemonde C (2008) Numerical modelling of the effective conductivities of composites with arbitrarily shaped inclusions and highly conducting interface. Compos Sci Technol 68(13):2818–2825. CrossRefGoogle Scholar
  18. 18.
    Cavalcante MAA, Marques SPC, Pindera M-J (2007) Parametric formulation of the finite-volume theory for functionally graded materials. Part I: analysis. J Appl Mech 74(5):935–945. CrossRefGoogle Scholar
  19. 19.
    Cavalcante MAA, Pindera M-J, Khatam H (2012) Finite-volume micromechanics of periodic materials: past, present and future. Compos Part B 43(6):2521–2543. CrossRefGoogle Scholar
  20. 20.
    Escarpini Filho RS, Marques SPC (2014) A model for evaluation of effective thermal conductivity of periodic composites with poorly conducting interfaces. Mat Res 17(5):1344–1355. CrossRefGoogle Scholar
  21. 21.
    Nemat-Nasser S, Iwakuma T, Hejazi M (1982) On composites with periodic structures. Mech Mater 1(3):239–267. CrossRefGoogle Scholar
  22. 22.
    Iwakuma T, Nemat-Nasser S (1983) Composites with periodic microstructure. Comput Struct 16(1–4):13–19. CrossRefzbMATHGoogle Scholar
  23. 23.
    Luciano R, Barbero EJ (1994) Formulas for the stiffness of composites with periodic microstructure. Int J Solids Struct 31(21):2933–2944. CrossRefzbMATHGoogle Scholar
  24. 24.
    Caporale A, Luciano R, Penna R (2013) Fourier series expansion in non-orthogonal coordinate system for the homogenization of linear viscoelastic periodic composites. Compos Part B 54:241–245. CrossRefGoogle Scholar
  25. 25.
    Caporale A, Feo L, Luciano R (2015) Eigenstrain and Fourier series for evaluation of elastic local fields and effective properties of periodic composites. Compos Part B 81:251–258. CrossRefGoogle Scholar
  26. 26.
    Le Quang H, Phan T-L, Bonnet G (2011) Effective thermal conductivity of periodic composites with highly conducting imperfect interfaces. Int J Therm Sci 50(8):1428–1444. CrossRefGoogle Scholar
  27. 27.
    Eshelby JD (1957) The determination of the elastic field of an ellipsoidal inclusion, and related problems. Proc R Soc Lond A 241:376–396. MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Nemat-Nasser S, Hori M (1999) Micromechanics: overall properties of heterogeneous materials, 2nd edn. Elsevier Science Publishers, AmsterdamzbMATHGoogle Scholar
  29. 29.
    Barbero EJ, Damiani TM, Trovillion J (2005) Micromechanics of fabric reinforced composites with periodic microstructure. Int J Solids Struct 42(9–10):2489–2504. CrossRefzbMATHGoogle Scholar
  30. 30.
    Khatam H, Pindera M-J (2010) Plasticity-triggered architectural effects in periodic multilayers with wavy microstructures. Int J Plast 26(2):273–287. CrossRefzbMATHGoogle Scholar
  31. 31.
    Mirtich B (1996) Fast and accurate computation of polyhedral mass properties. J Graph Tools 1(2):31–50. CrossRefGoogle Scholar

Copyright information

© The Brazilian Society of Mechanical Sciences and Engineering 2019

Authors and Affiliations

  1. 1.Laboratory of Scientific Computing and Visualization, Center of TechnologyFederal University of AlagoasMaceióBrazil

Personalised recommendations