1. Nonlinear vibration of functionally graded nonlocal nanobeam with thermal effect: analytical model versus finite element approach.
- Author
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Tang, Yuan, Bian, PeiLiang, and Qing, Hai
- Abstract
This work presents nonlocal strain gradient integral model to analyze linear and nonlinear free vibration of functionally graded (FG) nanobeam under thermal effect. The integral constitutive relations are transformed into the equivalent differential forms equipped with two constitutive constraints. The linear vibration frequency and mode shape are derived explicitly through the Laplace transform technique, and the nonlinear vibration frequency can be calculated based on linear vibration mode shape with different boundary conditions by employing Ritz-Galerkin method. Finite element method (FEM) together with Newton's iterative process is utilized to obtain the numerical results for linear and nonlinear vibration frequency. The weak form is derived through Hamilton's principle, and the internal forces are converted into the external forces at the boundaries. Meanwhile, the external forces satisfy the constitutive boundary conditions by eliminating higher-order variables. A finite element formulation is established with the aid of Lagrange multiplier method (LMM). Lower computational cost, conversion of higher-order variables, simple shape functions, flexibility of degrees of freedom and well convergence are advantages of the present nonlocal strain gradient finite element model. Several comparisons are used to verify the efficiency and accuracy of the present results. Numerical results indicate the importance of applying size dependency on the thermal load. It also reveals that the linear vibration frequency increases consistently with the decrease of nonlocal parameters and the increase of gradient parameters. The nonlinear frequency ratio increases consistently with the increase of nonlocal parameters for clamped–clamped (CC) and clamped-hinged (CH) boundary conditions and with the decrease of gradient parameters for clamped–clamped (CC), clamped-hinged (CH) and hinged-hinged (HH) boundary conditions. Meanwhile, the frequency ratio of the local model is higher than those of corresponding nonlocal strain gradient models for HH boundary condition. Moreover, the effect of functional gradation of material is also investigated. [ABSTRACT FROM AUTHOR]
- Published
- 2025
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