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Structural Engineering and Mechanics Volume 69, Number 2, January10 2019 , pages 205-219 DOI: https://doi.org/10.12989/sem.2019.69.2.205 |
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Nonlinear vibration of functionally graded nano-tubes using nonlocal strain gradient theory and a two-steps perturbation method |
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Yang Gao, Wan-Shen Xiao and Haiping Zhu
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| Abstract | ||
| This paper analyzes nonlinear free vibration of the circular nano-tubes made of functionally graded materials in the framework of nonlocal strain gradient theory in conjunction with a refined higher order shear deformation beam model. The effective material properties of the tube related to the change of temperature are assumed to vary along the radius of tube based on the power law. The refined beam model is introduced which not only contains transverse shear deformation but also satisfies the stress boundary conditions where shear stress cancels each other out on the inner and outer surfaces. Moreover, it can degenerate the Euler beam model, the Timoshenko beam model and the Reddy beam model. By incorporating this model with Hamilton\'s principle, the nonlinear vibration equations are established. The equations, including a material length scale parameter as well as a nonlocal parameter, can describe the size-dependent in linear and nonlinear vibration of FGM nanotubes. Analytical solution is obtained by using a two-steps perturbation method. Several comparisons are performed to validate the present analysis. Eventually, the effects of various physical parameters on nonlinear and linear natural frequencies of FGM nanotubes are analyzed, such as inner radius, temperature, nonlocal parameter, strain gradient parameter, scale parameter ratio, slenderness ratio, volume indexes, different beam models. | ||
| Key Words | ||
| functionally graded material; nonlinear vibration; nonlocal strain gradient theory; nanotubes | ||
| Address | ||
| Yang Gao and Wan-Shen Xiao: 1) State Key Laboratory of Advanced Design and Manufacturing for Vehicle Body, Hunan University, Changsha 410082, China 2) College of Mechanical and Vehicle Engineering, Hunan University, Changsha 410082, China Haiping Zhu: School of Computing, Engineering and Mathematics, Western Sydney University, Locked, Bag 1797, Penrith, NSW 2751, Australia | ||