Out-of-Plane Static Analysis of Nanoarches Using Eringen’s Nonlocal Elasticity Theory (original) (raw)
Related papers
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.
A Nonlocal Elasticity Approach for the In-Plane Static Analysis of Nanoarches
Proceedings of the 2nd World Congress on New Technologies, 2016
Eringen's nonlocal elasticity theory is incorporated into classical beam model considering the effects of axial extension and the shear deformation to capture unique static behavior of the nanobeams under continuum mechanics theory. The governing differential equations are obtained for curved beams and solved exactly by using the initial value method. Circular uniform beam with concentrated loads are considered. The effects of shear deformation, axial extension, geometric parameters and small scale parameter on the displacements and stress resultants are investigated.
A general nonlocal beam theory: Its application to nanobeam bending, buckling and vibration
2009
In the present study, a generalized nonlocal beam theory is proposed to study bending, buckling and free vibration of nanobeams. Nonlocal constitutive equations of Eringen are used in the formulations. After deriving governing equations, different beam theories including those of Euler-Bernoulli, Timoshenko, Reddy, Levinson and Aydogdu [Compos. Struct., 89 ] are used as a special case in the present compact formulation without repeating derivation of governing equations each time. Effect of nonlocality and length of beams are investigated in detail for each considered problem. Present solutions can be used for the static and dynamic analyses of single-walled carbon nanotubes.
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.
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.
Analytical Solutions for Vibration of Simply Supported Nonlocal Nanobeams with an Axial Force
International Journal of Structural Stability and Dynamics, 2011
This paper presents exact, analytical solutions for the transverse vibration of simply supported nanobeams subjected to an initial axial force based on nonlocal elasticity theory. Classical continuum theory is inherently size independent while nonlocal elasticity exhibits size dependence. The latter has significant effects on bending moment, which results in a conceptually different definition of a new effective nonlocal bending moment with respect to the corresponding classical bending moment. A sixth-order partial differential governing equation is subsequently obtained. The effects of nonlocal nanoscale on the vibration frequencies and mode shapes are considered and analytical solutions are solved. Effects of the nonlocal nanoscale and dimensionless axial force including axial tension and axial compression on the first three mode frequencies are presented and discussed. It is found that the nonlocal nanoscale induces higher natural frequencies and stiffness of the nano structures.
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.
International Journal of Computational Materials Science and Engineering, 2018
In this paper, the free vibration and buckling behaviors of nanoscale beams with different boundary conditions are analyzed using the integral formulation of Eringen’s nonlocal elasticity theory. To this end, both strain- and stress-driven nonlocal integral models are employed. The nanobeams are modeled according to the Euler–Bernoulli beam theory. Moreover, a novel numerical approach is proposed for solving the obtained governing equations. By this numerical method, which uses matrix differential and integral operators, the integral governing equation is directly solved and the difficulties related to converting the integral governing equation into the differential one are bypassed. Comparisons are made between the predictions of strain and stress-driven models about the vibration and buckling responses of nanobeams subject to various end conditions. The results indicate that based on the stress-driven model, the frequency and critical buckling load increase with increasing the non...
Journal of Characterization, 2022
To know the mechanical behavior of such materials has been become important due to the wide use of onedimensional and different cross-sectional nanomaterials in nano-electro-mechanical systems (NEMS) technology. One-dimensional nanomaterials can be modelled with longitudinal mechanical elements like beams. Also, in the mechanics of nanostructures, it should be specified that formulations based on size effect are used since classical beam models do not give realistic results. With this motivation, this study examines the effect of section type on the size-dependent static deflection behavior of nano scaled beams. The nonlocal elasticity theory is used to consider the atomic size effect. In order to study the mechanical behavior of nanobeam more realistically, Timoshenko beam theory is formulated because it includes the shear effect. According to this, the differential equation of static analysis is solved for simply supported nanobeams under different load types such as uniform and concentrated. Then, deflection values and deflection ratios of nanobeams with square, circular, box, and hoop section are presented. Finally, numerical results are discussed in detail.
Composite Structures, 2018
In the present work, vibration analysis of curved nanobeams is investigated using nonlocal elasticity approach based on Eringen formulation coupled with a higher-order shear deformation theory accounting for through thickness stretching effect. The formulation developed here is general in the sense that it can be deduced to examine the influence of different structural theories and analyses of nanobeams. The governing equations derived are solved employing finite element method by introducing a 3-nodes curved beam element. The formulation is validated considering problems for which solutions are available. A comparative study is made using various structural models. The effects of various structural and material parameters such as thickness ratio, beam length, rise of the curved beam, boundary conditions, and size-dependent or nonlocal parameter are brought out on the vibration behaviours of curved nanobeams.