Eur. Phys. J. Plus (2020) 135: 206
https://doi.org/10.1140/epjp/s13360-019-00018-x

Regular Article

An integral nonlocal model for the free vibration analysis of Mindlin nanoplates using the VDQ method

Faculty of Mechanical Engineering, University of Guilan, P.O. Box 3756, Rasht, Iran

^* e-mail: R_ansari@guilan.ac.ir
^** e-mail: Jalal.torabii@gmail.com

Received: 21 January 2019
Accepted: 17 September 2019
Published online: 3 February 2020

Abstract

Eringen’s nonlocal theory (ENT) provides an efficient tool to consider the size dependency for the structural analysis of small-scale structures. Recently, it has been indicated that in some cases, inconsistent results may be obtained using the Eringen differential model (EDM) and Eringen integral model (EIM) should be employed. In this regard, the vibration of nanoplates resting on the elastic medium is investigated on the basis of the EIM using variational differential quadrature (VDQ) method. On the basis of the first-order shear deformation theory (FSDT), the EIM and EDM formulations are given. An effective numerical technique is applied in the framework of the energy method to find the natural frequencies. Comprehensive results are reported to compare the EIM and EDM. The results show that using the proposed integral model, the paradox related to cantilever nanoplates is resolved.