Eur. Phys. J. Plus (2019) 134: 191
https://doi.org/10.1140/epjp/i2019-12575-4

Regular Article

Nonlinear strain gradient analysis of nanoplates embedded in an elastic medium incorporating surface stress effects

¹ Faculty of Mechanical Engineering, K. N. Toosi University of Technology, P. O. Box: 19395-1999, Tehran, Iran
² Department of Engineering “Enzo Ferrari”, University of Modena and Reggio Emilia, Via Pietro Vivarelli 10/1, 41125, Modena, Italy

^* e-mail: asgari@kntu.ac.ir

Received: 13 August 2018
Accepted: 17 February 2019
Published online: 7 May 2019

Abstract

Nonlinear vibration of nano graphene plates with considering surface effects is studied in this paper based on the nonlocal strain gradient theory and von Kármán geometric nonlinearity. The isotropic nanoplate is assumed to lie on an elastic foundation with the simply supported boundary conditions. Both Winkler-type and Pasternak-type models are utilized to simulate the interaction of the nano graphene with a surrounding elastic medium. Due to the increase in the surface-to-volume ratios at smaller scales, the surface elasticity theory of Gurtin and Murdoch is developed to study the effects of surface properties which are the basis for size-dependent behaviors. The governing equation of motion can be obtained by von Kármán nonlinear strain-displacement relationship and the nonlinear frequency is obtained analytically using the perturbation approach. Moreover, two moveable and immoveable in-plane conditions are analyzed. The presented method is verified by comparing the results with their counterparts reported in the open literature and a good agreement is observed for two different boundary conditions. Finally, the effects of various parameters such as nonlocal parameter, material characteristic parameter, residual surface tension, mode number, temperature change and elastic medium coefficients for two kinds of in-plane conditions are discussed.