A single-variable first-order shear deformation nonlocal theory for the flexure of isotropic nanobeams

Abstract

In this paper, a displacement-based single-variable first-order shear deformation nonlocal theory for the flexure of isotropic nanobeams is presented. In the present theory, the beam axial displacement consists of a bending component, whereas the beam transverse displacement consists of a bending component and a shearing component. Bending components do not take part in the cross-sectional shearing force, and the shearing component does not take part in the cross-sectional bending moment. The present theory utilizes linear strain-displacement relations. The displacement functions of the present theory give rise to the constant transverse shear strain through the beam thickness. Hence as is the case with the nonlocal Timoshenko beam theory, the present theory also requires a shear correction factor. In the present theory, nonlocal differential stress-strain constitutive relations of Eringen are utilized in order to take into account the size-dependent behaviour of nanobeams. In the present theory, beam gross equilibrium equations are utilized to derive the governing differential equation. As against other first-order shear deformation nonlocal beam theories, the present theory involves only one governing differential equation of fourth order and has only one unknown function. There exists a strong resemblance between expressions of the cross-sectional shearing force, cross-sectional bending moment and governing differential equation of the present theory and corresponding expressions of the nonlocal Bernoulli-Euler beam theory based on the nonlocal elasticity theory of Eringen. For the present theory, boundary conditions are described. The present theory also describes two distinct and physically meaningful clamped boundary conditions. Illustrative examples pertaining to the flexure of shear deformable isotropic nanobeams demonstrate the efficacy of the present theory.

On Effects of Shear Deformation on the Static Pull-in Instability Behaviour of Narrow Rectangular Timoshenko Microbeams

“…In the first step, the developed TBT-SFE is utilized to find out the maximum non-dimensional beam transverse displacement (w * max = (100 wmax E I) / (qo L 4 )) of the shear deformable propped-cantilever rectangular beam having a thickness-to-length ratio of 0.20, shear correction factor of 5 / 6 and Poisson's ratio of 0.3, with different combinations of number of nodes per element and total number of elements. These values are then compared with the corresponding analytical solution (w * max = 0.6910, Pakhare et al [12]). Following four cases have been considered to arrive at the best combination of number of nodes per element and total number of elements for the subsequent electrostatic-elastic analysis:…”

“…For this purpose, authors have first calculated results of the maximum beam transverse displacement, for a shear deformable propped-cantilever microbeam under the action of uniformly distributed transverse load, obtained by utilizing the developed TBT-SFE with different combinations of number of nodes per element and total number of elements. These results are then compared with corresponding analytical results of Pakhare et al [8] to arrive at the best combination of number of nodes per element and total number of elements for the electrostaticelastic analysis. In the second step, the finalized TBT-SFE is utilized to determine static pull-in instability parameters of narrow microbeams with various fixity conditions and beam thickness-to-length ratios.…”

On the Static Flexural Response of Shear Deformable Isotropic Rectangular Nanobeams under the Parabolic Loading

“…Based on the physical understanding of the nanobeam deformation, Pakhare et al [5] have specified physically meaningful beam end conditions of the NFSDNBT. These end conditions with regard to the beam end x = 0, and can be defined at the beam end x = L based on a similar logic, for an illustrative purpose are as follows:…”

“…From the above-mentioned discussion, it is evident that size-dependent effects as well as transverse shear deformation effects of at least first-order must be accounted for, so as to effectively predict the deformation behaviour of shear deformable nanobeams. Pakhare et al [5] have recently proposed a single variable new first-order shear deformation nonlocal beam theory (NFSDNBT), by taking a cue from the work reported by Shimpi et al [6], for the flexure of linear isotropic nanobeams undergoing small deformations. Unlike the nonlocal Timoshenko beam theory (NTBT) which involves two coupled governing equations with two primary unknowns (Wang et al [7]), the NFSDNBT involves only one governing equation with one primary unknown.…”

Free torsional vibration of triangle microwire based on modified couple stress theory

“…Numanoğlu et al 41 carried out the axial free vibrational behavior of nanorod models for various boundary conditions within the framework of nonlocal elasticity theory. Pakhare et al 42 worked on the flexural behavior of the isotropic nanobeams using nonlocal differential stress–strain constitutive relations of Eringen for two different boundary conditions. Civalek and Demir 43 employed the classical elasticity theory and finite element method to analyze the buckling behavior of protein microtubules, and ultimately to assess the geometric along with the elastic foundation on the responses.…”