Nonlinear Bending Analysis of Nanobeams Based on the Nonlocal Strain Gradient Model Using an Isogeometric Finite Element Approach

Abstract

In this paper, the static bending of nanoscale beams is studied in the nonlinear regime. For this purpose, a size-dependent Timoshenko beam model is developed by which nonlocal and strain gradient effects are simultaneously captured. The most comprehensive nonlocal strain gradient model without any simplification is used herein. The strain gradient influences are considered based upon the most general form of strain gradient theory which can accommodate simpler theories such as the modified strain gradient and couple stress theories. Moreover, to take the nonlocal effects into account, the original integral form of Eringen’s nonlocal elasticity is employed. The governing equations are derived using the minimum total potential energy principle. Also, the formulation of model is represented in matrix–vector form with the aim of using in numerical approaches, especially in finite element or isogeometric analyses. To solve the governing equations of the developed integral nonlocal strain gradient model, a non-classical isogeometric analysis is proposed. The simultaneous effects of nonlocal and small-scale parameters on the nonlinear bending behavior of simply supported, clamped and clamped-free nanobeams are studied in the numerical results. Furthermore, the results obtained based on the differential and integral nonlocal models are presented for the comparison goal.