Eur. Phys. J. Plus (2018) 133: 336
https://doi.org/10.1140/epjp/i2018-12183-x

Regular Article

Stress-driven nonlocal and strain gradient formulations of Timoshenko nanobeams

¹ Department 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: r_ansari@guilan.ac.ir

Received: 26 January 2018
Accepted: 10 July 2018
Published online: 24 August 2018

Abstract

In this paper, three size-dependent formulations are developed for the analysis of Timoshenko nanobeams with various end conditions based on the nonlocal and strain gradient theories. The nonlocal governing equations are presented based on the stress-driven model of Eringen’s theory. First, a strain gradient Timoshenko beam model is developed. The governing equations of the integral stress-driven model, and then those of differential stress-driven model together with associated constitutive boundary conditions are obtained in the next step. With the aim of addressing the static bending and free vibration problems, the nonlocal governing equations in integral form are directly solved by constructing matrix differential and integral operators. Furthermore, the governing equations in differential form together with constitutive boundary conditions are discretized and solved via the mentioned operators. It is shown that there is a good agreement between the results obtained from solving the integral and differential governing equations of stress-driven nonlocal models. Several comparative studies are also conducted for the bending and vibration analyses of nanobeams based on the strain gradient and stress-driven nonlocal models. The results reveal that in both models, increasing the nonlocal/length scale parameter has a stiffening effect on the response of the system. However, the stiffening effect corresponding to the strain gradient model is more pronounced than that corresponding to the stress-driven nonlocal model.