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Linear and nonlinear torsional free vibration of functionally graded micro/nano-tubes based on modified couple stress theory

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Abstract

The linear and nonlinear torsional free vibration analyses of functionally graded micro/nano-tubes (FGMTs) are analytically investigated based on the couple stress theory. The employed non-classical continuum theory contains one material length scale parameter, which can capture the small scale effect. The FGMT model accounts for the through-radius power-law variation of a two-constituent material. Hamilton’s principle is used to develop the non-classical nonlinear governing equation. To study the effect of the boundary conditions, two types of end conditions, i.e., fixed-fixed and fixed-free, are considered. The derived boundary value governing equation is of the fourthorder, and is solved by the homotopy analysis method (HAM). This method is based on the Taylor series with an embedded parameter, and is capable of providing very good approximations by means of only a few terms, if the initial guess and the auxiliary linear operator are properly selected. The analytical expressions are developed for the linear and nonlinear natural frequencies, which can be conveniently used to investigate the effects of the dimensionless length scale parameter, the material gradient index, and the vibration amplitude on the natural frequencies of FGMTs.

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Authors and Affiliations

  1. Faculty of Mechanical and Aerospace Engineering, Shiraz University of Technology, Shiraz, 71555, Iran

    A. R. Setoodeh, M. Rezaei & M. R. Zendehdel Shahri

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  1. A. R. Setoodeh
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  2. M. Rezaei
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  3. M. R. Zendehdel Shahri
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Correspondence to A. R. Setoodeh.

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Setoodeh, A.R., Rezaei, M. & Zendehdel Shahri, M.R. Linear and nonlinear torsional free vibration of functionally graded micro/nano-tubes based on modified couple stress theory. Appl. Math. Mech.-Engl. Ed. 37, 725–740 (2016). https://doi.org/10.1007/s10483-016-2085-6

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  • DOI: https://doi.org/10.1007/s10483-016-2085-6

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Chinese Library Classification

2010 Mathematics Subject Classification

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