A comprehensive investigation into the impact of nonlocal strain gradient and modified couple stress models on the rates of surface energy layers of BiTiO3-CoFe2O4 nanoplates: a vibration analysis

According to the nonlocal strain gradient and modified couple stress theories, a unified size-dependent platemodel is proposed for the vibration analysis of rectangular magneto-electrothermo-elastic (METE) nanoplates to analyze the rates of its surface energy layers. The METE nanoplate is composed of BiTiO3-CoFe2O4 materilas. The developed model can simultaneously catch nonlocal, strain gradient, and modified couple stress models, in addition to surface energy effects. Using the refined plate theory, we have developed the non-classical governing equations based on the principle of minimum potential energy. The refined plate theory is employed for the first time to get the governing equations through the modified couple stress theory. Analytical solutions are obtained for the natural frequency of rectangular METE nanoplates under the influence of various external in-plane loadings. Verification of the proposed model is done based on the comparison of the degenerated results with those previously accessible in the literature. It is demonstrated that the nonlocal scale parameter exerts an opposite impact on the degree of surface layers, as compared with those of the strain gradient and modified couple stress scale parameters. So, by enhancing the nonlocal scale parameter, it would be possible to increase the effects of surface energy layers on the natural frequencies ratio of METE nanoplate, while the influence of the surface energy layers could be reduced with the enhancement of the strain gradient and modified couple stress scale parameters.