Arabian Journal for Science and Engineering

, Volume 44, Issue 2, pp 763–776 | Cite as

Micromechanical Constitutive Equations for the Effective Thermoelastic Properties of Carbon Nanotube-Reinforced Composites

  • Yasser M. ShabanaEmail author
  • T. Morimoto
  • F. Ashida
Research Article - Mechanical Engineering


Predicting thermomechanical properties of composites containing carbon nanotubes (CNTs) is significantly depending on the assumed microstructural parameters (MSPs) of CNTs and CNT/matrix morphology. These MSPs include geometry, dispersion and orientation. On the other hand, CNT/matrix morphology refers to two microstructural observations. The first is whether or not an interphase exists between CNTs and matrix, whereas the second is whether or not voids exist due to, for example, debonding of CNTs. In this work, the aim is to propose micromechanical constitutive equations, which are based on the micromechanics principles of Eshelby and Mori-Tanaka models, for considering all of these MSPs altogether in addition to the other well-known MSPs. Accordingly, these equations can be used for modeling realistic nanocomposites to predict their effective thermomechanical properties in different directions. The obtained computational results are compared with other results of both experimental and theoretical investigations found in the literature, and good agreement is obtained.


Micromechanical constitutive equations Composite materials Carbon nanotubes Microstructural parameters Thermomechanical properties 


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  1. 1.
    Lau, K.T.; Lu, M.; Liao, K.: Improved mechanical properties of coiled carbon nanotubes reinforced epoxy nanocomposites. Compos. Part A: Appl. Sci. Manuf. 37, 1837–1840 (2006)CrossRefGoogle Scholar
  2. 2.
    Andrews, R.; Weisenberger, M.C.: Carbon nanotube polymer composites. Curr. Opin. Solid State Mater Sci. 8, 31–37 (2004)CrossRefGoogle Scholar
  3. 3.
    Matveeva, A.Y.; Pyrlin, S.V.; Ramos, M.D.; Böhm, H.J.; Hattum, F.J.: Influence of waviness and curliness of fibres on mechanical properties of composites. Comput. Mater. Sci. 87, 1–11 (2014)CrossRefGoogle Scholar
  4. 4.
    Rafiee, R.: Influence of carbon nanotube waviness on the stiffness reduction of CNT/polymer composites. Compos. Struct. 97, 304–309 (2013)CrossRefGoogle Scholar
  5. 5.
    Shi, D.L.; Feng, X.Q.; Huang, Y.Y.: Hwang, K.C.; Gao, H.: The effect of nanotube waviness and agglomeration on the elastic property of carbon nanotube-reinforced composites. ASME 126, 250–257 (2004)Google Scholar
  6. 6.
    Joshi, U.A.; Sharma, S.C.; Harsha, S.P.: Effect of carbon nanotube orientation on the mechanical properties of nanocomposites. Compos. Part B: Eng. 43, 2063–2071 (2012)CrossRefGoogle Scholar
  7. 7.
    Tserpes, K.I.; Chanteli, A.: Parametric numerical evaluation of the effective elastic properties of carbon nanotube-reinforced polymers. Compos. Struct. 99, 366–374 (2013)CrossRefGoogle Scholar
  8. 8.
    Bradshaw, R.D.; Fisher, F.T.; Brinson, L.C.: Fiber waviness in nanotube-reinforced polymer composites–II: modeling via numerical approximation of the dilute strain concentration tensor. Compos. Sci. Technol. 63, 1705–1722 (2003)CrossRefGoogle Scholar
  9. 9.
    Fisher, F.; Bradshaw, R.; Brinson, L.: Fiber waviness in nanotube-reinforced polymer composites–I: Modulus predictions using effective nanotube properties. Compos. Sci. Technol. 63, 1689–1703 (2003)CrossRefGoogle Scholar
  10. 10.
    Zhang, J.; Tanaka, M.: Systematic study of thermal properties of CNT composites by the fast multipole hybrid boundary node method. Eng. Anal. Bound. Elements 31, 388–401 (2007)CrossRefzbMATHGoogle Scholar
  11. 11.
    Yuan, Z.; Lu, Z.: Numerical analysis of elastic-plastic properties of polymer composite reinforced by wavy and random CNTs. Comput. Mater. Sci. 95, 610–619 (2014)CrossRefGoogle Scholar
  12. 12.
    Kundalwal, S.I.; Ray, M.C.: Improved thermoelastic coefficients of a novel short fuzzy fiber-reinforced composite with wavy carbon nanotubes. J. Mech. Mater. Struct. 9, 1–25 (2014)CrossRefGoogle Scholar
  13. 13.
    Odegard, G.M.; Gates, T.S.; Wise, K.E.; Park, C.; Siochi, E.J.: Constitutive modeling of nanotube-reinforced polymer composites. Compos. Sci. Technol. 63, 1671–1687 (2003)CrossRefGoogle Scholar
  14. 14.
    Karevan, M.; Pucha, R.V.; Bhuiyan, M.A.; Kalaitzidou, K.: Effect of interphase modulus and nanofiller agglomeration on the tensile modulus of graphite nanoplatelets and carbon nanotube reinforced polypropylene nanocomposites. Carbon Lett. 11, 325–331 (2010)CrossRefGoogle Scholar
  15. 15.
    Peng, R.D.; Zhou, H.W.; Wang Jr., H.W.; LM, : Modeling of nano-reinforced polymer composites: microstructure effect on Young’s modulus. Comput. Mater. Sci. 60, 19–31 (2012)Google Scholar
  16. 16.
    Bhuiyan, M.A.; Pucha, R.V.; Worthy, J.; Karevan, M.; Kalaitzidou, K.: Defining the lower and upper limit of the effective modulus of CNT/polypropylene composites through integration of modeling and experiments. Compos. Struct. 95, 80–87 (2013)CrossRefGoogle Scholar
  17. 17.
    Shao, L.H.; Luo, R.Y.; Bai, S.L.; Wang, J.: Prediction of effective moduli of carbon nanotube-reinforced composites with waviness and debonding. Compos. Struct. 87, 274–281 (2009)CrossRefGoogle Scholar
  18. 18.
    Tohgo, K.; Cho, Y.: Theory of reinforcement damage in discontinuously-reinforced composites and its application. JSME Int. J. Ser. A 42, 521–529 (1999)CrossRefGoogle Scholar
  19. 19.
    Yasser, M.S.: Development of constitutive laws for thermo-mechanical behaviors of composites containing multi-type ellipsoidal reinforcements. Int. J. Solids Struct. 46, 824–836 (2009)CrossRefzbMATHGoogle Scholar
  20. 20.
    Mesbah, A.; Zairi, F.; Boutaleb, S.; Gloaguen, J.M.; Nait-Abdelaziz, M.; Xie, S.; Boukharouba, T.; Lefebvre, J.M.: Experimental characterization and modeling stiffness of polymer/clay nanocomposites within a hierarchical multiscale framework. J. Appl. Polym. Sci. 114, 3274–3291 (2009)CrossRefzbMATHGoogle Scholar
  21. 21.
    Nam, T.H.; Goto, K.; Yamaguchi, Y.; Premalal, E.A.; Shimamura, Y.; Inoue, Y.; Naito, K.; Ogihara, S.: Effects of CNT diameter on mechanical properties of aligned CNT sheets and composites. Compos. Part A: Appl. Sci. Manuf. 76, 289–298 (2015)CrossRefGoogle Scholar
  22. 22.
    Shirasu, K.; Yamamoto, G.; Tamaki, I.; Ogasawara, T.; Shimamura, Y.; Inoue, Y.; Hashida, T.: Negative axial thermal expansion coefficient of carbon nanotubes: experimental determination based on measurements of coefficient of thermal expansion for aligned carbon nanotube reinforced epoxy composites. Carbon 95, 904–909 (2015)CrossRefGoogle Scholar
  23. 23.
    Dominkovics, Z.; Hári, J.; Kovács, J.; Fekete, E.; Pukánszky, B.: Estimation of interphase thickness and properties in PP/layered silicate nanocomposites. Eur. Polym. J. 47, 1765–1774 (2011)CrossRefGoogle Scholar
  24. 24.
    Seidel, G.D.: Micromechanics modeling of the multifunctional nature of carbon nanotube-polymer nanocomposites. PhD thesis, Texas A&M University (2007)Google Scholar
  25. 25.
    Lan, T.; Pinnavaia, T.J.: Clay-reinforced epoxy nanocomposites. Chem. Mater. 6, 2216–2219 (1994)CrossRefGoogle Scholar
  26. 26.
    Fornes, T.D.; Paul, D.R.: Modeling properties of nylon 6/clay nanocomposites using composite theories. Polymer 44, 4993–5013 (2003)CrossRefGoogle Scholar
  27. 27.
    Wang, X.; Jiang, Q.; Xu, W.; Cai, W.; Inoue, Y.; Zhu, Y.: Effect of carbon nanotube length on thermal, electrical and mechanical properties of CNT/bismaleimide composites. Carbon 53, 145–152 (2013)CrossRefGoogle Scholar
  28. 28.
    Odegard, G.M.; Clancy, T.C.; Gates, T.S.: Modeling of the mechanical properties of nanoparticle/polymer composites. Polymer 46, 553–562 (2005)CrossRefGoogle Scholar
  29. 29.
    Ray, M.C.; Kundalwal, S.I.: Effect of carbon nanotube waviness on the load transfer characteristics of short fuzzy fiber-reinforced composite. J. Nanomech. Micromech. 4, A4013010 (2014)CrossRefGoogle Scholar
  30. 30.
    Kundalwal, S.I.; Ray, M.C.: Shear lag analysis of a novel short fuzzy fiber-reinforced composite. Acta Mech. 225, 2621–2643 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Kundalwal, S.I.; Ray, M.C.; Meguid, S.: Shear lag model for regularly staggered short fuzzy fiber reinforced composite. ASME J. Appl. Mech. 81, 091001 (2014)CrossRefGoogle Scholar
  32. 32.
    Kundalwal, S.I.; Ray, M.C.: Effect of carbon nanotube waviness on the elastic properties of the fuzzy fiber reinforced composites. ASME J. Appl. Mech. 80, 021010 (2013)CrossRefGoogle Scholar
  33. 33.
    Kundalwal, S.I.; Kumar, S.: Multiscale modeling of stress transfer in continuous microscale fiber reinforced composites with nano-engineered interphase. Mech. Mater. 102, 117–131 (2016)CrossRefGoogle Scholar
  34. 34.
    Schelling, P.K.; Keblinski, P.: Thermal expansion of carbon structures. Phys. Rev. B 68, 035425 (2003)CrossRefGoogle Scholar
  35. 35.
    Jiang, H.; Liu, B.; Huang, Y.; Hwang, K.C.: Thermal expansion of single wall carbon nanotubes. J. Eng. Mater. Technol. 126, 265–270 (2004)CrossRefGoogle Scholar

Copyright information

© King Fahd University of Petroleum & Minerals 2018

Authors and Affiliations

  1. 1.Mechanical Design Department, Faculty of Engineering, El-MatariaHelwan UniversityCairoEgypt
  2. 2.Interdisciplinary Graduate School of Science and EngineeringShimane UniversityMatsueJapan

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