Eur. Phys. J. Plus (2021) 136: 823
https://doi.org/10.1140/epjp/s13360-021-01826-w

Regular Article

Hybrid strain- and stress-driven integral non-local model

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

^b h_rouhi@guilan.ac.ir

Received: 2 June 2021
Accepted: 31 July 2021
Published online: 9 August 2021

Abstract

The non-local elasticity is widely utilized for considering the effect of non-locality on the behavior of nanomaterials. It has been revealed that the application of the differential strain-driven non-local elasticity leads to inconsistent predictions in some cases. Hence, several attempts have been made at capturing non-local effects within the frameworks of stress- and strain-driven integral formulation of this theory. In this article, the stress- and strain-driven non-local integral models are combined to develop the most comprehensive integral non-local model with the capability of considering the non-local effects in a general way. In the present non-local constitutive equation, the stress field at entire points of the domain is determined on the basis of the strain field of entire points of the domain. The proposed model has two non-local parameters by which the stiffening and softening non-local effects can be simultaneously captured. The Euler–Bernoulli beam is considered as the problem under investigation whose static bending is investigated. Also, a numerical solution approach is developed for solving the governing equations. In this approach, differential and integral quadrature-based operator matrices are employed for discretization. Based on the developed hybrid model, the non-local influences captured by the stress- and strain-driven parts on the bending response of nanoscale beams with arbitrary boundary conditions are shown and analyzed.