On Flexure of Shear Deformable Isotropic Rectangular Nanobeams

Abstract

In this paper, newly developed single variable new first-order shear deformation nonlocal beam theory is utilized for performing flexure of shear deformable isotropic rectangular nanobeams under the action of sinusoidally distributed transverse load. This theory is applicable for flexure of linear isotropic nanobeams undergoing small deformations. Displacement functions of this theory give rise to constant transverse shear strain through the beam thickness. Hence similar to nonlocal Timoshenko beam theory, this theory requires shear correction factor. In this theory, nonlocal differential stress-strain constitutive relations of Eringen are utilized in order to take into account size-dependent effects which govern mechanical behavior of nanobeams. These nonlocal differential constitutive relations relate not only the beam axial stress with the beam axial strain but also the beam transverse shear stress with the beam transverse shear strain. Governing differential equation of this theory is obtained by utilizing beam gross equilibrium equations and it has a strong resemblance with governing differential equation of nonlocal Bernoulli-Euler beam theory. Profiles of non-dimensional beam transverse displacement for various values of nonlocal parameter of Eringen and beam thickness-to-length ratio in case of simply supported, clamped, and cantilever isotropic rectangular nanobeams under the action of sinusoidally distributed transverse load are presented. Effect of nonlocal parameter and beam thickness-to-length ratio on maximum non-dimensional beam transverse displacement for abovementioned cases of beams is presented. Obtained results are compared with corresponding results obtained by utilizing nonlocal Timoshenko beam theory so as to demonstrate the efficacy of single variable new first-order shear deformation nonlocal beam theory.