Cross-section effect on the deflections of simply supported Timoshenko nanobeams using nonlocal elasticity (original) (raw)
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VIBRATION ANALYSIS OF SIZE-DEPENDENT NANOBEAMS BASED ON NONLOCAL TIMOSHENKO BEAM THEORY
In the present paper, the semi-analytical differential transformation method (DTM) is employed for vibration analysis of size-dependent nanobeams based on nonlocal Timoshenko beam theory (TBT). The governing motion equations of nanobeam with different edge conditions are derived by the Hamilton's principle. DTM is applied to discretize the governing equations and boundary conditions, which are then solved to obtain the frequency parameters of nanobeam. In the numerical examples, the good agreements between the present results and existing literature verified the validity and accuracy of the present solution method. The detailed mathematical derivations are presented and numerical investigations are performed while the emphasis is placed on investigating the effect of small scale parameters, mode number, aspect ratios and edge conditions on the normalized natural frequencies of the nanobeams. It is explicitly shown that the vibration of a nanobeam is significantly influenced by these effects.
On the static stability of nonlocal nanobeams using higher-order beam theories
Advances in nano research, 2016
This paper investigates the effects of thermal load and shear force on the buckling of nanobeams. Higher-order shear deformation beam theories are implemented and their predictions of the critical buckling load and post-buckled configurations are compared to those of Euler-Bernoulli and Timoshenko beam theories. The nonlocal Eringen elasticity model is adopted to account a size-dependence at the nano-scale. Analytical closed form solutions for critical buckling loads and post-buckling configurations are derived for proposed beam theories. This would be helpful for those who work in the mechanical analysis of nanobeams especially experimentalists working in the field. Results show that thermal load has a more significant impact on the buckling behavior of simply-supported beams (S-S) than it has on clamped-clamped (C-C) beams. However, the nonlocal effect has more impact on CC beams that it does on S-S beams. Moreover, it was found that the predictions obtained from Timoshenko beam theory are identical to those obtained using all higher-order shear deformation theories, suggesting that Timoshenko beam theory is sufficient to analyze buckling in nanobeams.
ScienceDirect, 2019
In this paper, bending and buckling behavior of nanobeam utilizing different beam theories including Timoshenko, Euler-Bernoulli, and higher-order beam theories are developed to investigate. The governing equations are derived based on nonlocal strain gradient theory incorporating surface effects. In order to solve the governing equation by a numerical solution, the Navier's method is utilized, and the simply supported boundary condition is imposed. Critical buckling load and maximum deflection are on the main concerns of this study. Obtained results represent the effect of surface, nonlocal, and length scale parameters. Moreover, various beam theories are evaluated, and their discrepancies are discussed. Results disclose that the Timoshenko and higher-order beam theories with negligible diversions are the critical ones which predict the lowest critical buckling load and highest maximum deflection compared to Euler-Bernoulli beam theory. As a primary result, residual surface stress and surface Young's modulus magnitude reveal a direct relation with material stiffness. Finally, as the small scale parameter increases the material stiffness decreases whereas increasing the length scale parameter stiffens the material structure.
Doublet mechanical analysis of bending of Euler-Bernoulli and Timoshenko nanobeams
ZAMM - Journal of Applied Mathematics and Mechanics / Zeitschrift für Angewandte Mathematik und Mechanik, 2018
By taking the effect of the scale parameter explicitly into account, bending behavior of Euler-Bernoulli and Timoshenko nanobeams is studied using doublet mechanics. In addition, the effect of chirality on the softening or hardening behavior of an Euler-Bernoulli nanobeam in bending is studied by taking the effect of the chiral angle explicitly into account. For the bending of the Timoshenko nanobeam the effect of the scale parameter and chirality on the axial and shear stresses is studied and the results are compared with the gradient and Eringen-like models. It is shown that as the chiral angle increases, a simply supported Timoshenko nanobeam under a uniform loading changes from a softening to a hardening nanobeam indicating that chirality has a pronounced effect on the bending response of the nanobeams. K E Y W O R D S bending of Euler-Bernoulli nanobeam, bending of Timoshenko nanobeam, chirality, doublet mechanics, scale parameter 1 INTRODUCTION With recent advances in science and technology, materials with microstructural features at micro and nano levels are discovered and developed which exhibit strikingly different characteristics compared to the conventional homogeneous materials. As such, consideration of discrete microstructure of new materials becomes of paramount importance in analyzing their static and dynamic response. Examples of such materials include carbon-based nanobeams, nanotubes, and nanospheres with different nano-level microstructures. A nano-sized beam is generally called a nanobeam when one or more of its dimensions are in the range of 1nm to 100nm. In Euler-Bernoulli nanobeams, the cross-sections remain perpendicular to the neutral axis after the deformation, but this restriction is not made for Timoshenko nanobeams. Among the nano-structures, nanobeams are found to have many important technological applications like nanosensors, biosensors, actuators and nanowires in a variety of circumstances such as in nanoelectromechanical devices, medical instruments and metal-, polymer-, and silicon-based composites. [1,2] Accordingly, studying the static and dynamic response of nanobeams is both of theoretical and practical importance. The microstructure of nanomaterials reveals a discrete medium having small scale parameters, like, interatomic distances among atoms or molecules which do not appear in classical local continuum theories. As a result, non-local theories have been developed in recent years wherein the state of stress at a point is a function of the strain field in all points of the continuum in order to incorporate the effect of small scale parameters, as expounded in Eringen. [3] Accordingly, a number of researchers have investigated the use of non-local theories in analyzing the static deflection of the Euler-Bernoulli and Timoshenko nanobeams. [4-8] In addition, a new version of the non-local theory, known as the higher order Eringen model, [9] is used to solve non-local differential equations and higher-order boundary conditions for Euler-Bernoulli nanobeams using a thermodynamic approach. Unlike most of the non-local beam theories that have only one scale parameter, one version of the non-local theories, known as
Composites Part B: Engineering, 2018
Size-dependent vibrational behavior of functionally graded (FG) Timoshenko nano-beams is investigated by strain gradient and stress-driven nonlocal integral theories of elasticity. Hellinger-Reissner's variational principle is preliminarily exploited to establish the equations governing the elastodynamic problem of FG strain gradient Timoshenko nano-beams. Differential and boundary conditions of dynamical equilibrium of FG Timoshenko nanobeams, with nonlocal behavior described by the stress-driven integral theory, are formulated. Free vibrational responses of simple structures of technical interest, associated with nonlocal stress-driven and strain gradient strategies, are analytically evaluated and compared in detail. The stress-driven nonlocal model for FG Timoshenko nano-beams provides an effective tool for dynamical analyses of stubby composite parts of Nano-Electro-Mechanical Systems.
This paper aims to develop a new non-classical Bernoulli-Euler model, taking into account the effects of a set of size dependent factors which ignored by the classical continuum mechanics. Among those factors are the microstructure local rotation, long-range interactions between a particle and the other particles of the continuum and the surface energy effects. The model used the modified couple-stress theory to study the effect of the local rotational degree of freedom of a specific particle. Furthermore, the surface elasticity model developed by Gurtin and Murdoch has been used to determine the surface energy effects on the behavior of the particle. The effects of the local rotation and surface energy are investigated in the framework of nonlocal elasticity theory, which is employed to study the nonlocal and long-range interactions between the particles. In addition, Poisson's effect incorporated in the newly developed beam model. The equations of equilibrium and complete boundary conditions of the new beam are derived using the principle of virtual work.
Composite Structures, 2013
In this paper, static bending and buckling of a functionally graded (FG) nanobeam are examined based on the nonlocal Timoshenko and Euler-Bernoulli beam theory. This non-classical (nonlocal) nanobeam model incorporates the length scale parameter (nonlocal parameter) which can capture the small scale effect. The material properties of the FG nanobeam are assumed to vary in the thickness direction. The governing equations and the related boundary conditions are derived using the principal of the minimum total potential energy. The Navier-type solution is developed for simply-supported boundary conditions, and exact formulas are proposed for the deflections and the buckling load. The effects of nonlocal parameter, aspect ratio, various material compositions on the static and stability responses of the FG nanobeam are discussed. Some illustrative examples are also presented to verify the present formulation and solutions. Good agreement is observed. The results show that the new nonlocal beam model produces larger deflection and smaller buckling load than the classical (local) beam model.
Computers & Structures, 2017
The paper deals with a general method to obtain a closed-form analytical solution of the problem of bending of a shear deformable beam resting on an elastic medium. Within a well posed analytical framework, the basic equations governing the interaction problem, can be obtained in strong form by a differential formulation based on both the constitutive equation of the Timoshenko beam and the direct and inverse constitutive equation of the supporting local elastic medium. A general finite element is then derived for shear deformable beams with or without a continuous Winkler type elastic support. The obtained analytical results are discussed in the light of nonlocal elasticity of Eringen differential type, applied to an Euler-Bernoulli beam model. As a result the stiffness-matrix and equivalent nodal loads of an Euler-Bernoulli nonlocal elastic beam, can be defined in analogy to those of a first order shear deformable beam. This conclusion allows handling the elastostatic problem of nanobeams, modelled according to Eringen's nonlocal elasticity, by slight modifications of the existing computational tools for the solution of the elastostatic problem of a local shear deformable beam.
Physica E: Low-dimensional Systems and Nanostructures, 2015
This article presents a simplified three-unknown shear and normal deformations nonlocal beam theory for the bending analysis of nanobeams in thermal environment. Eringen's nonlocal constitutive equations are considered in the analysis. Governing equations are derived according to the present refined theory using Hamilton's principle. Central deflections of nanobeams under uniform and point loads are given and compared with the available ones in the literature. Additional results of displacement and stresses are presented for future comparison. The effects of nonlocality, temperature parameters, length of beam, length-to-depth ratio as well as shear and normal strains are all investigated.
2016
Out-of-plane static behavior of circular nanobeams with point loads are investigated. Inclusion of small length scales such as lattice spacing between atoms, surface properties, grain size etc. are considered in the analysis by employing Eringen’s nonlocal elasticity theory in the formulations. The nonlocal equations are arranged in cylindrical coordinates and applied to the beam theory. The effect of shear deformation is considered. The governing differential equations are solved exactly by using the initial value method. The displacements, rotation angle about the normal and tangential axes and the force resultants are established.