Free vibration analysis of nonlocal nanobeams: a comparison of the one-dimensional nonlocal integral Timoshenko beam theory with the two-dimensional nonlocal integral elasticity theory

Abstract

Beam theories such as the Timoshenko beam theory are in agreement with the elasticity theory. However, due to the different nonlocal averaging processes, they are expected to yield different results in their nonlocal forms. In the present work, the free vibration behavior of nonlocal nanobeams is studied using the nonlocal integral Timoshenko beam theory (NITBT) and two-dimensional nonlocal integral elasticity theory (2D-NIET) with different kernels and their results are compared. A new kernel, termed the compensated two-phase (CTP) kernel, is introduced, which entirely compensates for the boundary effects and does not suffer from the ill-posedness of previous kernels. Using the finite element method, the free vibration analysis is performed for different boundary conditions based on the first three natural frequencies. For both the NITBT and 2D-NIET with both the two-phase (TP) and CTP kernels, the nonlocal parameter has a softening effect on the natural frequencies for all the boundary conditions, without observing the paradoxical behaviors of the nonlocal differential theory. For both theories, the softening effect of the nonlocal parameter is more pronounced for the TP kernel compared to the CTP kernel. The sensitivity of the 2D-NIET to the nonlocal parameter is found to be higher than that of the NITBT. Also, the softening effects for different vibration modes are compared to each other for both theories and both kernels. The obtained results can be extended for various important beam problems with nonlocal effects and help obtain a better understanding of applicable nonlocal theories.

Nonlocal Timoshenko shear beam model for multilayer curved graphene nano-switches

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A gradient electromechanical theory for thin dielectric curved beams considering direct and converse flexoelectric effects

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Size-dependent nonlinear free vibration of magneto-electro-elastic nanobeams by incorporating modified couple stress and nonlocal elasticity theory

“…These include nonlocal elasticity theory [16][17][18][19][20], micropolar [21][22][23], strain gradient theory [24][25][26], surface elasticity theory [27,28]and modified couple stress theory [29][30][31][32][33]. Based on the aforementioned theories, significant attention has been dedicated to studying the mechanical behavior of Micro/Nano Structures, particularly in the context of bending [34][35][36], buckling [13,37,38], vibration [39][40][41][42], and wave propagation [43][44][45][46]. Among these studies, particular attention has been given to examining the free vibration behavior of nanoscale structures.…”