Vibration analysis of orthotropic circular and elliptical nano-plates embedded in elastic medium based on nonlocal Mindlin plate theory and using Galerkin method

Abstract

In the present study a continuum model based on the nonlocal elasticity theory is developed for free vibration analysis of embedded orthotropic thick circular and elliptical nano-plates rested on an elastic foundation. The elastic foundation is considered to behave like a Pasternak type of foundations. Governing equations for vibrating nano-plate are derived according to the Mindlin plate theory in which the effects of shear deformations of nano-plate are also included. The Galerkin method is then employed to obtain the size dependent natural frequencies of nano-plate. The solution procedure considers the entire nano-plate as a single super-continuum element. Effect of nonlocal parameter, lengths of nano-plate, aspect ratio, mode number, material properties, thickness and foundation on circular frequencies are investigated. It is seen that the nonlocal frequencies of the nano-plate are smaller in comparison to those from the classical theory and this is more pronounced for small lengths and higher vibration modes. It is also found that as the aspect ratio increases or the nanoplate becomes more elliptical, the small scale effect on natural frequencies increases. Further, it is observed that the elastic foundation decreases the influence of nonlocal parameter on the results. Since the effect of shear deformations plays an important role in vibration analysis and design of nano-plates, by predicting smaller values for fundamental frequencies, the study of these nano-structures using thick plate theories such as Mindlin plate theory is essential.

Free vibration analysis of graded porous circular micro/nanoplates with various boundary conditions based on the nonlocal elasticity theory

“…Inserting Equations ( 23) and (24) into Equation ( 22), integrate the spatial coordinate variables by parts, and classify the coefficients of 𝛿𝜓 and 𝛿𝑤 to obtain the two motion governing differential equations of the graded porous circular nanoplate as follows…”

“…The results show that if the nonlocal effect is ignored for nanoplates, there is a large error in the research results. On the bases of ENET, Anjomshoa and Tahani [24] studied the free vibration of Orthotropic thick circular nanoplates on elastic foundation, and applied this method to the free vibration analysis of elliptical nanoplates. Studies had shown that compared with the classical theory, the nonlocal frequency of nanoplates is smaller, which is more obvious in smaller length and higher vibration mode.…”

Free vibration analysis of a composite elliptical plate made of a porous core and two piezoelectric layers

“…Further, Anjomshoa 9 investigated the application of Ritz functions in the buckling analysis of micro-or nano-elliptical and circular orthotropic plates, based on both nonlocal elasticity theory and the Galerkin method. 10 These authors found that the critical buckling load is highly dependent on the non-localization of micro-or nano-elliptical plates, especially as it relates to the microscopic dimensions. In the same context, Anjomshoa et al 11 presented the frequency analysis of micro-or nano-orthotropic circular and elliptical plates, using the principle of nonlocal variation based on the nonlocal classical plates theory (NCPT).…”