Mixing Rules of Young's Modulus, Thermal Expansion Coefficient and Thermal Conductivity of Solid Material with Particulate Inclusion

  • Hirata, Yoshihiro ;
  • Shimonosono, Taro
  • Received : 2015.12.21
  • Accepted : 2016.01.22
  • Published : 2016.01.31


This analyzed a Young's modulus (E), a thermal expansion coefficient (TEC, ${\beta}$) and a thermal conductivity (${\kappa}$) of the material with simple cubic particulate inclusion using two model structures: a parallel structure and a series structure of laminated layers. The derived ${\beta}$ equations were applied to calculate the ${\beta}$ value of the W-MgO system. The accuracy was higher for the series model structure than for the parallel model structure. Young's moduli ($E_c$) of sintered porous alumina compacts were theoretically related to the development of neck growth of grain boundary between sintered two particles and expressed as a function of porosity. The series structure model with cubic pores explained well the increased tendency of $E_c$ with neck growth rather than the parallel structure model. The thermal conductivity of the three phase system of alumina-mullite-pore was calculated by a theoretical equation developed in this research group, and compared with the experimental results. The pores in the sintered composite were treated as one phase. The measured thermal conductivity of the composite with 0.5-25% porosity (open and closed pores) was in accordance with the theoretical prediction based on the parallel structure model.


Sintering;Porous ceramics;Thermal conductivity;Thermal expansion


  1. Y. Hirata, "Thermal Conduction Model of Metal and Ceramics," Ceram. Inter., 35 3259-68 (2009).
  2. M. Wang and N. Pan, "Predictions of Effective Physical Properties of Complex Multiphase Materials," Mater. Sci. Eng., R63 1-30 (2008).
  3. J. Wang, J. K. Carson, M. F. North, and D. J. Cleland, "A New Structural Model of Effective Thermal Conductivity for Heterogeneous Materials with Co-Continuous Phases," Inter. J. Heat Mass Transfer, 51 2389-97 (2008).
  4. A. Bouchair, "Steady State Theoretical Model of Fired Clay Hollow Bricks for Enhanced External Wall Thermal Insulation," Building and Environment, 43 1603-18 (2008).
  5. Y. Hirata, "Representation of Thermal Conductivity of Solid Material with Particulate Inclusion," Ceram. Inter., 35 2921-26 (2009).
  6. Y. Hirata, N. Matsunaga, J. Yoshitomi, and T. Kayama, "Theoretical Analysis of Thermal Conductivity of Graphite-Containing Refractory Brick," J. Tech. Assoc. Refract. Japan, 31 [3] 156-63 (2011).
  7. Y. Hirata, "Representation of Thermal Expansion Coefficient of Solid Material with Particulate Inclusion," Ceram. Inter., 41 2706-13 (2015).
  8. W. D. Kingery, H. K. Bowen, and D. R. Uhlmann, Introduction to Ceramics; pp. 583-624, Second Ed., John Wiley & Sons, New York, 1976.
  9. Y. Hirata and T. Shimonosono, "Theoretical Prediction of Compressive Strength, Young's Modulus and Strain at Fracture of Sintered Porous Alumina Compacts," Ceram. Inter., 42 3014-18 (2016).
  10. D. Hull and T. W. Clyne, An Introduction to Composite Materials; pp. 237-44, Second Ed., Cambridge University Press, Cambridge, 1996.
  11. K. Hata, Chemical Handbook, Basic Part II; pp. 21-22, 3rd ed., The Chemical Society of Japan, Maruzen, Tokyo, 1984.
  12. Y. Hirata, T. Shimonosono, T. Sameshima, and S. Sameshima, "Compressive Mechanical Properties of Porous Alumina Powder Compacts," Ceram. Inter., 40 2315-22 (2014).
  13. Y. Hirata, T. Shimonosono, S. Sameshima, and H. Tominaga, "Sintering of Alumina Powder Compacts and Their Compressive Mechanical Properties," Ceram. Inter., 41 11449-55 (2015).
  14. Y. Hirata, "Theoretical Analyses of Thermal Shock and Thermal Expansion Coefficients of Metals and Ceramics," Ceram. Inter., 41 1145-53 (2015).
  15. K. Hata, Chemical Handbook, Basic Part II; pp. 72-7, 3rd Ed., The Chemical Society of Japan, Maruzen, Tokyo, 1984.
  16. R. F. Davis and J. A. Pask, "Mullite,", pp. 37-75 in High temperature oxide, part IV. Ed. by A.M. Alper, Academic Press, New York, 1971.
  17. S. Itoh, Y. Hirata, T. Shimonosono, and S. Sameshima, "Theoretical and Experimental Analyses of Thermal Conductivity of the Alumina-Mullite System," J. Eur. Ceram. Soc., 35 605-12 (2015).