A comparative study of 1D nonlocal integral Timoshenko beam and 2D nonlocal integral elasticity theories for bending of nanoscale beams

Abstract

In this paper, the bending behavior of nanoscale beams is studied using the 1D nonlocal integral Timoshenko beam theory (NITBT) and the 2D nonlocal integral elasticity theory (2D-NIET) using two types of nonlocal kernels, i.e., the two-phase kernel and a modified kernel which compensates the boundary effects. The governing equations are solved using the finite element method and the COMSOL code. Mesh sensitivity study and numerical verifications are presented. The main differences and similarities in both theories at the nanoscale are revealed. For both theories and both kernels, a softening behavior is found by increasing the nonlocal parameter and decreasing the phase parameter, for different boundary and load conditions. In contrast to the differential theory, no paradoxical behavior for the cantilever conditions is found for both theories. The sensitivity of the 2D-NIET to the nonlocal parameter is found higher than that of the NITBT. The normalized transverse deflection for the 2D-NIET is found independent of the boundary and load conditions. Also, the normalized transverse deflection varies linearly versus the normalized nonlocal parameter for both theories with the two-phase kernel and any condition except for the simply supported beam under distributed load condition in the NITBT. The boundary effect, resulting in a different softening near the boundaries, reduces by increasing the phase parameter. The modified kernel in the 2D-NIET is found sensitive to the pinned not to the free and fixed boundaries. It is in detail shown that for the 2D-NIET especially with the modified kernel, by increasing the nonlocal parameter, the deflection increases with almost the same ratio through the entire length of the beam and for all the boundary conditions. The obtained results can be used for modeling of various beam problems with nonlocal effects at the nanoscale.