Three-dimensional asymptotic nonlocal elasticity theory for the free vibration analysis of embedded single-walled carbon nanotubes

Chih Ping Wu, Yen Jung Chen, Yung Ming Wang

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3 Citations (Scopus)


Within the framework of three-dimensional (3D) nonlocal elasticity theory, the authors develop an asymptotic theory to investigate the free vibration characteristics of simply supported, single-walled carbon nanotubes (SWCNTs) non-embedded or embedded in an elastic medium using the multiple time scale method. Eringen's nonlocal constitutive relations are adopted to account for the small length scale effect in the formulation. The interactions between the SWCNT and its surrounding medium are modeled as a two-parameter Pasternak foundation model. After performing a series of mathematical processes, including nondimensionalization, asymptotic expansion, and successive integration, etc., the authors obtain recurrent sets of motion equations for various order problems. The nonlocal classical shell theory (CST) is derived as a first-order approximation of the current 3D nonlocal elasticity problem, and the equations of motion for higher-order problems retain the same differential operators as those of the nonlocal CST, although with different nonhomogeneous terms. The current asymptotic solutions for the natural frequency parameters of non-embedded or embedded SWCNTs and their corresponding through-thickness modal stress and displacement component distributions are obtained to assess the accuracy of various nonlocal shell and beam theories available in the literature.

Original languageEnglish
Pages (from-to)161-182
Number of pages22
JournalComputers and Mathematics with Applications
Issue number1
Publication statusPublished - 2020 Jul 1

All Science Journal Classification (ASJC) codes

  • Modelling and Simulation
  • Computational Theory and Mathematics
  • Computational Mathematics

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