In this paper, a general third-order beam theory that accounts for nanostructure-dependent size effects and two-constituent material variation through the nanobeam thickness, i.e., functionally graded material (FGM) beam is presented. The material properties of FG nanobeams are assumed to vary through the thickness according to the power law. A detailed derivation of the equations of motion based on Eringen nonlocal theory using Hamilton's principle is presented, and a closedform solution is derived for buckling behavior of the new model with various boundary conditions. The nonlocal elasticity theory includes a material length scale parameter that can capture the size effect in a functionally graded material. The proposed model is efficient in predicting the shear effect in FG nanobeams by applying third-order shear deformation theory. The proposed approach is validated by comparing the obtained results with benchmark results available in the literature. In the following, a parametric study is conducted to investigate the influences of the length scale parameter, gradient index, and length-to-thickness ratio on the buckling of FG nanobeams and the improvement on nonlocal third-order shear deformation theory comparing with the classical (local) beam model has been shown. It is found out that length scale parameter is crucial in studying the stability behavior of the nanobeams.
In this study, free vibration analysis of magneto-electro-thermo-elastic (METE) nanobeams resting on a Pasternak foundation is investigated based on nonlocal theory and Timoshenko beam theory. Coupling effects between electric, magnetic, mechanical and thermal loading are considered to derive the equations of motion and distribution of electrical potential and magnetic potential along the thickness direction of the METE nanobeam. The governing equations and boundary conditions are obtained using the Hamilton principle and discretized via the differential quadrature method (DQM). Numerical results reveal the effects of the nonlocal parameter, magneto-electro-thermomechanical loading, Winkler spring coefficients, Pasternak shear coefficients and height-to-length ratio on the vibration characteristics of METE nanobeams. It is observed that the natural frequency is dependent on the magnetic, electric, temperature, elastic medium, small-scale coefficient, and height-to-length ratio. These results are useful in the mechanical analysis and design of smart nanostructures constructed from magneto-electro-thermo-elastic materials.
In the present study, an exact solution for free vibration analysis of piezoelectric nanobeams based on the nonlocal theory is obtained. The Euler beam model for a long and thin beam structure is employed, together with the electric potential satisfying the surface free charge condition for free vibration analysis. The governing equations and the boundary conditions are derived using Hamilton's principle. These equations are solved analytically for the vibration frequencies of beams with various end conditions. The model has been verified with the previously published works and found a good agreement with them. A detailed parametric study is conducted to discuss the influences of the nonlocal parameter, on the vibration characteristics of piezoelectric nanobeams. The exact vibration solutions should serve as benchmark results for verifying numerically obtained solutions based on other beam models and solution techniques.
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