Eur. Phys. J. Plus (2019) 134: 527
https://doi.org/10.1140/epjp/i2019-12863-y

Regular Article

Nonlinear bending and postbuckling analysis of FG nanoscale beams using the two-phase fractional nonlocal continuum mechanics

¹ Faculty of Mechanical Engineering, University of Guilan, P.O. Box 3756, Rasht, Iran
² Department of Engineering Science, Faculty of Technology and Engineering, East of Guilan, University of Guilan, P.C. 44891-63157, Rudsar-Vajargah, Iran

^* e-mail: h_rouhi@guilan.ac.ir

Received: 12 October 2018
Accepted: 1 July 2019
Published online: 23 October 2019

Abstract

The geometrically nonlinear bending and postbuckling of nanoscale beams are investigated herein according to the two-phase fractional nonlocal continuum model. It is considered that the beams have been made from functionally graded materials (FGMs), and the Bernoulli-Euler beam model is employed for their modeling. The variational form of the governing fractional equation is obtained first by means of an energy approach. Thereafter, a novel numerical solution method is proposed named as fractional variational differential quadrature method (FVDQM). In FVDQM, which is applied to the variational statement of the problem in a direct way, a combination of the differential quadrature method and matrix operators is utilized. The efficiency of the proposed fractional nonlocal model is evaluated by molecular dynamics (MD) simulations. Selected numerical results are given to explore the influences of fractional order, nonlocality, length-to-thickness ratio and FG index on the nonlinear bending and postbuckling responses of FG nanobeams with various types of boundary conditions.