Uncertainty Analysis of Mechanical Behavior of Functionally Graded Carbon Nanotube Composite Materials

  • E. García-MacíasEmail author
  • R. Castro-Triguero
  • Michael I. Friswell
  • A. Sáez-Pérez
  • R. Gallego
Conference paper
Part of the Conference Proceedings of the Society for Experimental Mechanics Series book series (CPSEMS)


The remarkable mechanical and sensing properties of carbon nanotubes (CNTs) suggest that they are ideal candidates for high performance and self-sensing cementitious composites. However, there is still a lack of deeper knowledge of the uncertainty associated with their incorporation, concretely in functionally graded composite materials (FGM). The influence of these uncertainties can be critical for future applications in the field of Structural Health Monitoring (SHM), techniques that usually require high accuracy modeling. Most researches restrict the aim of their studies to the analysis of composite materials with uniform or linear grading profiles. This study throws light on the basis of stochastic representation of the grading profiles and analyzes the propagation of its uncertainty into the response of some structural elements. The finite element method (FEM) is employed to study the individual and interactive effects of the mechanical properties (Young’s modulus, density, Poisson’s ratio and CNT’s waviness) and grading profiles. The effects of stochastic uncertainties on the overall properties of the composite material are represented using the probability theory. Numerical results show the influence of these variables in several benchmark cases such as cylindrical, spherical and doubly curved shells, in terms of their static and dynamic characteristics by performing modal analysis.


Stochastic analysis Kriging metamodel RS-HDMR metamodel Carbon nanotube composites  Hu-Washizu principle 



This research was supported by Spanish ministry of economy and competitively under the Project Ref: DPI2014-53947-R. E. G-M was also supported by a FPU contract-fellowship from the Spanish Ministry of Education Ref: FPU13/04892.


  1. 1.
    Iijima, S.: Helical microtubules of graphitic carbon. Nature 354(6348), 56–58 (1991)CrossRefGoogle Scholar
  2. 2.
    Gibson, R.F.: A review of recent research on mechanics of multifunctional composite materials and structures. Compos. Struct. 92(12), 2793–2810 (2010). ISSN: 0263-8223CrossRefGoogle Scholar
  3. 3.
    Esawi, A.M.K., Farag, M.M.: Carbon nanotube reinforced composites: potential and current challenges. Mater. Des. 28(9), 2394–2401 (2007). ISSN: 0261-3069CrossRefGoogle Scholar
  4. 4.
    Udupa, G., Shrikantha Rao, S., Gangadharan, K.V.: Functionally graded composite materials: an overview. Proc. Mater. Sci. 5, 1291–1299 (2014). ISSN: 2211-8128CrossRefGoogle Scholar
  5. 5.
    Shen, H.-S.: Nonlinear bending of functionally graded carbon nanotube-reinforced composite plates in thermal environments. Compos. Struct. 91(1), 9–19 (2009). ISSN: 0263-8223CrossRefGoogle Scholar
  6. 6.
    Cressie, N.: The origins of Kriging. Math. Geol. 22(3), 239–252 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Rabitz, H., Aliç, Ö.F.: General foundations of high-dimensional model representations. J. Math. Chem. 25(2–3), 197–233 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Lophaven, S.N., Nielsen H.B., Søndergaard, J.: DACE-A Matlab Kriging toolbox, version 2.0, Technical Report (2002)Google Scholar
  9. 9.
    Ziehn, T., Tomlin, A.S.: GUI-HDMR. A software tool for global sensitivity analysis of complex models. Environ. Model Softw. 24(7), 775–785 (2009). ISSN: 1364-8152CrossRefGoogle Scholar
  10. 10.
    Wempner, G., Talaslidis, D.: Mechanics of Solids and Shells. CRC Press, Boca Raton (2003). ISBN 0-8493-9654-9zbMATHGoogle Scholar
  11. 11.
    Spencer, A.J.M.: The formulation of constitutive equation for anisotropic solids. In: Mechanical Behavior of Anisotropic Solids/Comportment Méchanique des Solides Anisotropes, pp. 3–26. Springer, Netherlands (1982)Google Scholar
  12. 12.
    Lubarda, V., Chen, M.: On the elastic moduli and compliances of transversely isotropic and orthotropic materials. J. Mech. Mater. Struct. 3(1), 153–171 (2008). ISSN: 1559-3959CrossRefGoogle Scholar
  13. 13.
    Efraim, E., Eisenberger, M.: Exact vibration analysis of variable thickness thick annular isotropic and FGM plates. J. Sound Vib. 299(4), 720–738 (2007). ISSN: 0022-460XCrossRefGoogle Scholar
  14. 14.
    Ziehn, T., Tomlin, A.S.: A global sensitivity study of sulfur chemistry in a premixed methane flame model using HDMR. In. J. Chem. Kinet. 40(11), 742–753 (2008). ISSN: 1097-4601CrossRefGoogle Scholar
  15. 15.
    Shi, D.-L., et~al.: The effect of nanotube waviness and agglomeration on the elastic property of carbon nanotube-reinforced composites. J. Eng. Mater. Technol. 126(3), 250–257 (2004). ISSN: 0094-4289CrossRefGoogle Scholar
  16. 16.
    Fidelus, J.D., et~al.: Thermo-mechanical properties of randomly oriented carbon/epoxy nanocomposites. Compos. A: Appl. Sci. Manuf. 36(11), 1555–1561 (2005). ISSN: 1359-835XCrossRefGoogle Scholar
  17. 17.
    Sobhani Aragh, B., Nasrollah Barati, A.H., Hedayati, H.: Eshelby-Mori-Tanaka approach for vibrational behavior of continuously graded carbon nanotube-reinforced cylindrical panels. Compos. Part B 43(4), 1943–1954 (2012). ISSN: 1359-8368CrossRefGoogle Scholar
  18. 18.
    Shen, H.-S., Zhang, C.-L.: Thermal buckling and postbuckling behavior of functionally graded carbon nanotube-reinforced composite plates. Mater. Des. 31(7), 3403–3411 (2010). ISSN: 0261-3069MathSciNetCrossRefGoogle Scholar
  19. 19.
    Han, Y., Elliott, J.: Molecular dynamics simulations of the elastic properties of polymer/carbon nanotube composites. Comput. Mater. Sci. 39(2), 315–323 (2007). ISSN: 0927-0256CrossRefGoogle Scholar

Copyright information

© The Society for Experimental Mechanics, Inc. 2016

Authors and Affiliations

  • E. García-Macías
    • 1
    Email author
  • R. Castro-Triguero
    • 2
  • Michael I. Friswell
    • 3
  • A. Sáez-Pérez
    • 1
  • R. Gallego
    • 4
  1. 1.University of SevilleSevillaSpain
  2. 2.University of CordobaCórdobaSpain
  3. 3.University of SwanseaSwanseaUK
  4. 4.University of GranadaGranadaSpain

Personalised recommendations