Nonlocal Size Dependence of a Softness Nanobeam with Large Axial Tension under Various Boundary Conditions

2012 ◽  
Vol 490-495 ◽  
pp. 3226-3230
Author(s):  
Cheng Li ◽  
Wei Guo Huang
Keyword(s):  
Natural Frequency ◽  
Scale Parameter ◽  
Separation Of Variables ◽  
Nonlocal Elasticity Theory ◽  
Small Scale ◽  
Problem Model ◽  
Various Boundary Conditions ◽  
Size Dependent ◽  
Axial Tension ◽  
Dependent Theory

The transverse dynamical behaviors of softness Euler-Bernoulli nanobeams subjected to a biggish initial axial force based on nonlocal elasticity theory are investigated in this paper. The size-dependent theory is considered and a small intrinsic length scale parameter unavailable in classical continuum mechanics is adopted into the problem model as a size parameter. The linear partial differential governing equation is derived from the Newton’s second law and the ordinary equation and its dispersion relation are gained from by the method of separation of variables. Five sets of supporting conditions are presented respectively including simply supported, fully clamped, flexible fixed ends, sliding supports ends and completely free ends. Vibration frequencies are obtained approximately and correlations between the natural frequency and the dimensionless small scale parameter are also analyzed and discussed in detail. It shows that an increase in small scale parameter and dimensionless initial axial tension causes natural frequency to increase, while an increase in the dimensionless stiffness of nanostructures causes natural frequency to decrease, or the nanostructural bending stiffness is enhanced when nonlocal effects are considered.

Micro Cantilever Beam Theory for Transverse Dynamics Using a Continuum Mechanics Model

2011 ◽  
Vol 415-417 ◽  
pp. 760-763
Author(s):  
Cheng Li ◽  
Wei Guo Huang ◽  
Lin Quan Yao
Keyword(s):  
Scale Parameter ◽  
Beam Theory ◽  
Nonlocal Elasticity Theory ◽  
Small Scale ◽  
Beam Model ◽  
Cantilever Beams ◽  
Mechanics Model ◽  
Axial Tension ◽  
Vibrational Characteristics ◽  
Micro Cantilever

The vibrational characteristics of cantilever beams with initial axial tension were studied using a nonlocal continuum Euler-Bernoulli beam model. Small size effects are essential to nanotechnology and it can not be ignored in micro or nano scale. Nonlocal elasticity theory has been proved to work well in nanomechanics and it is considered into the governing equation which can be transformed into a fourth-order ordinary differential equation together with a dispersion relation. Boundary conditions are applied so as to determine the analytical solutions of vibrational mode shape and transverse deformation through a numerical method. Relations between natural frequency and the small scale parameter are obtained, including the fundamental and the second order frequencies. It is found that both the small scale parameter and dimensionless initial axial tension play remarkable roles in dynamic behaviors of micro cantilever beams and their effects are analyzed and discussed in detail.

Size-Dependent Dynamic Behavior and Instability Analysis of Nano-Scale Rotational Varactor in the Presence of Casimir Attraction

2016 ◽  
Vol 08 (02) ◽  
pp. 1650018 ◽  
Author(s):  
Hamid M. Sedighi ◽  
Meisam Moory-Shirbani ◽  
Mohammad Shishesaz ◽  
Ali Koochi ◽  
Mohamadreza Abadyan
Keyword(s):  
Dynamic Behavior ◽  
Casimir Force ◽  
Scale Parameter ◽  
Dynamic Instability ◽  
Couple Stress Theory ◽  
Modified Couple Stress Theory ◽  
Small Scale ◽  
Stress Theory ◽  
Size Dependent ◽  
Casimir Attraction

When the size of structures approaches to the sub-micron scale, physical responses of such systems become size-dependent, hence, classic theories may not be able to predict the behavior of the miniature structures. In the present article, the modified couple stress theory (MCST) is employed to account for the effect of the size-dependency on the dynamic instability of torsional nano-electromechanical systems (NEMS) varactor. By incorporating the Coulomb, Casimir and damping forces, the dimensionless governing equations are derived. The influences of Casimir force, applied voltage and length scale parameter on the dynamic behavior and stability of fixed points are investigated by plotting the phase portrait and bifurcation diagrams. It is found that the Casimir force reduces the instability threshold of the systems and the small-scale parameter enhances the torsional stability. The pull-in instability phenomenon shows the saddle-node bifurcation for torsional nano-varactor.

Effect of Flexoelectricity on Band Structures of One-Dimensional Phononic Crystals

2013 ◽  
Vol 81 (5) ◽  
Author(s):  
Chenchen Liu ◽  
Shuling Hu ◽  
Shengping Shen
Keyword(s):  
Elastic Wave ◽  
Band Gap ◽  
Periodic Structures ◽  
Scaling Law ◽  
Small Scale ◽  
Filling Fraction ◽  
Size Dependent ◽  
Flexoelectric Effect ◽  
Dependent Theory ◽  
Band Gap Structure

As a size-dependent theory, flexoelectric effect is expected to be prominent at the small scale. In this paper, the band gap structure of elastic wave propagating in a periodically layered nanostructure is calculated by transfer matrix method when the effect of flexoelectricity is taken into account. Detailed calculations are performed for a BaTiO3-SrTiO3 two-layered periodic structure. It is shown that the effect of flexoelectricity can considerably flatten the dispersion curves, reduce the group velocities of the system, and decrease the midfrequency of the band gap. For periodic two-layered structures whose sublayers are of the same thickness, the width of the band gap can be decreased due to flexoelectric effect. It is also unveiled from our analysis that when the filling fraction is small, wider gaps at lower frequencies will be acquired compared with the results without considering flexoelectric effect. In addition, the band gap structures will approach the classical result as the total thickness of the unit cell increases. Our results indicate that the scaling law does not hold when the sizes of the periodic structures reach the nanoscale dimension. Therefore, the consideration of flexoelectric effect on the band structure of a nanosized periodic system is significant for precise manipulation of elastic wave propagation and its practical application.

Natural Frequency and Buckling of Orthotropic Nanoplates Resting on Two-Parameter Elastic Foundations with Various Boundary Conditions

2014 ◽  
Vol 30 (5) ◽  
pp. 443-453 ◽  
Author(s):  
M. Sobhy
Keyword(s):  
Boundary Conditions ◽  
Natural Frequency ◽  
Plate Theory ◽  
Equations Of Motion ◽  
Nonlocal Parameter ◽  
Nonlocal Elasticity Theory ◽  
Graphene Sheets ◽  
Elastic Foundations ◽  
Various Boundary Conditions ◽  
Mechanics Of Composite Materials

AbstractIn this article, the analyses of the natural frequency and buckling of orthotopic nanoplates, such as single-layered graphene sheets, resting on Pasternak's elastic foundations with various boundary conditions are presented. New functions for midplane displacements are suggested to satisfy the different boundary conditions. These functions are examined by comparing their results with the results obtained by using the functions suggested by Reddy (Reddy JN. Mechanics of Composite Materials and Structures: Theory and Analysis. Boca Raton, FL: CRC Press; 1997). Moreover, these functions are very simple comparing with Reddy's functions, leading to ease of calculations. The equations of motion of the nonlocal model are derived using the sinusoidal shear deformation plate theory (SPT) in conjunction with the nonlocal elasticity theory. The present SPT are compared with other plate theories. Explicit solution for buckling loads and vibration are obtained for single-layered graphene sheets with isotropic and orthotropic properties; and under biaxial loads. The formulation and the method of the solution are firstly validated by executing the comparison studies for the isotropic nanoplates with the results being in literature. Then, the influences of nonlocal parameter and the other parameters on the buckling and vibration frequencies are investigated.

Determination of Transverse Amplitude of Microbeam Vibrating Accounting for Nonlinear Effects

2012 ◽  
Vol 446-449 ◽  
pp. 1190-1193
Author(s):  
Cheng Li ◽  
Wei Guo Huang
Keyword(s):  
Nonlinear Partial Differential Equation ◽  
Nonlinear Effects ◽  
Nonlocal Theory ◽  
Basic Element ◽  
Small Scale ◽  
Problem Model ◽  
Axial Tension ◽  
Small Scale Effect ◽  
Mechanical Devices

The nonlinear dynamics of a microbeam with initial axial tension is presented. The nonlocal theory with a small scale effect is applied to the problem model. Considering the axial protraction due to the transverse deformation of the microbeam, a nonlinear partial differential equation that governs the dynamic motion is derived. Explicit solution of transverse amplitude is obtained by the method of multiscale analysis. Both nonlinear and nonlocal effects are found to play significant roles in the vibration behaviors of a microbeam. The results may be helpful for the application and design of various micro-electronic-mechanical devices, where a microbeam acts as a basic element.

scholarly journals Nonlocal FEM Formulation for Vibration Analysis of Nanowires on Elastic Matrix with Different Materials

2019 ◽  
Vol 24 (2) ◽  
pp. 38 ◽  
Author(s):  
Büşra Uzun ◽  
Ömer Civalek
Keyword(s):  
Boundary Conditions ◽  
Nonlocal Elasticity Theory ◽  
Silver Nanowire ◽  
Small Scale ◽  
Weighted Residual Method ◽  
Simply Supported ◽  
Scale Parameters ◽  
Various Boundary Conditions ◽  
Winkler Elastic Foundation ◽  
Foundation Model

In this study, free vibration behaviors of various embedded nanowires made of different materials are investigated by using Eringen’s nonlocal elasticity theory. Silicon carbide nanowire (SiCNW), silver nanowire (AgNW), and gold nanowire (AuNW) are modeled as Euler–Bernoulli nanobeams with various boundary conditions such as simply supported (S-S), clamped simply supported (C-S), clamped–clamped (C-C), and clamped-free (C-F). The interactions between nanowires and medium are simulated by the Winkler elastic foundation model. The Galerkin weighted residual method is applied to the governing equations to gain stiffness and mass matrices. The results are given by tables and graphs. The effects of small-scale parameters, boundary conditions, and foundation parameters on frequencies are examined in detail. In addition, the influence of temperature change on the vibrational responses of the nanowires are also pursued as a case study.

Effect of External Moving Torque on Dynamic Stability of Carbon Nanotube

2020 ◽  
Vol 61 ◽  
pp. 118-135 ◽  
Author(s):  
Seyyed Amirhosein Hosseini ◽  
Farshad Khosravi ◽  
Majid Ghadiri
Keyword(s):  
Boundary Condition ◽  
Carbon Nanotube ◽  
Aspect Ratio ◽  
Natural Frequency ◽  
Nonlocal Parameter ◽  
Nonlocal Elasticity Theory ◽  
Small Scale ◽  
Torsional Mode ◽  
Angular Displacements ◽  
The Galerkin Method

In nano-dimension, the strength of the material is considerable, and the failure is unavoidable in a torsional mode. Because of this reason, the free and forced torsional vibrations of single-walled carbon nanotube (SWCNT) are investigated in this paper. For dynamic analysis, the moving harmonic torsional load is exerted to SWCNT. The related boundary condition and equation of motion are derived by Hamilton’s principle, and the equation is discretized by the Galerkin method. In order to demonstrate the nonlocality and small–scale effect, Eringen’s theory based on nonlocal elasticity theory is applied. A clamped-clamped (C-C) boundary condition is fitted for the end supports. The influences of the aspect ratio and mode number on the free natural frequency are investigated. Furthermore, the dynamic effects of nonlocal parameter, velocity, thickness, length, and excitation-to-natural frequencies on dimensional and nondimensional angular displacements are indicated. Moreover, the natural frequency was investigated due to the variation of the aspect ratio.

Vibration of Nonlocal Euler Beams Using Chebyshev Polynomials

2011 ◽  
Vol 471-472 ◽  
pp. 1016-1021 ◽  
Author(s):  
Seyyed Amir Mahdi Ghannadpour ◽  
Bijan Mohammadi
Keyword(s):  
Natural Frequencies ◽  
Beam Theory ◽  
Nonlocal Elasticity Theory ◽  
Small Scale ◽  
Polynomial Functions ◽  
Euler Beam ◽  
Various Boundary Conditions ◽  
Small Scale Effect ◽  
Euler Beam Theory ◽  
Comparison Of The Results

This paper is concerned with the free vibration problem for micro/nano beams modelled after Eringen’s nonlocal elasticity theory and Euler beam theory. The small scale effect is taken into consideration in the former theory. The natural frequencies are obtained using the Hamilton’s principle and Chebyshev polynomial functions. The present method, which uses Rayleigh–Ritz technique in this paper, provides an efficient and extremely accurate vibration solution of micro/nano beams where the effects of small scale are significant. Numerical results for a variety of some micro/nano beams with various boundary conditions are given and compared with the available results wherever possible. Through the comparison of the results and the appropriate discussion, the knowledge of the level of capability of the developed method is promoted.

Size-Dependent Buckling and Postbuckling Analyses of First-Order Shear Deformable Magneto-Electro-Thermo Elastic Nanoplates Based on the Nonlocal Elasticity Theory

2017 ◽  
Vol 17 (01) ◽  
pp. 1750014 ◽  
Author(s):  
R. Ansari ◽  
R. Gholami
Keyword(s):  
Boundary Conditions ◽  
Elasticity Theory ◽  
Shear Deformation ◽  
Nonlocal Elasticity ◽  
Nonlocal Parameter ◽  
Nonlocal Elasticity Theory ◽  
Small Scale ◽  
First Order ◽  
Size Dependent ◽  
Shear Deformable

This paper presents a nonlocal nonlinear first-order shear deformable plate model for investigating the buckling and postbuckling of magneto-electro-thermo elastic (METE) nanoplates under magneto-electro-thermo-mechanical loadings. The nonlocal elasticity theory within the framework of the first-order shear deformation plate theory along with the von Kármán-type geometrical nonlinearity is used to derive the size-dependent nonlinear governing partial differential equations and associated boundary conditions, in which the effects of shear deformation, small scale parameter and thermo-electro-magneto-mechanical loadings are incorporated. The generalized differential quadrature (GDQ) method and pseudo arc-length continuation algorithm are used to determine the critical buckling loads and postbuckling equilibrium paths of the METE nanoplates with various boundary conditions. Finally, the influences of the nonlocal parameter, boundary conditions, temperature rise, external electric voltage and external magnetic potential on the critical buckling load and postbuckling response are studied.

Buckling Analysis of Micro- and Nano-Rods/Tubes Based on Nonlocal Timoshenko Beam Theory Using Chebyshev Polynomials

2010 ◽  
Vol 123-125 ◽  
pp. 619-622 ◽  
Author(s):  
Seyyed Amir Mahdi Ghannadpour ◽  
Bijan Mohammadi
Keyword(s):  
Scale Effects ◽  
Beam Theory ◽  
Nonlocal Elasticity Theory ◽  
Small Scale ◽  
Buckling Loads ◽  
Various Boundary Conditions ◽  
Total Potential ◽  
Nonlocal Timoshenko Beam Theory ◽  
Nano Rods ◽  
Comparison Of The Results

This paper presents the elastic buckling behavior of nonlocal micro- and nano- Timoshenko rods/tubes based on Eringen’s nonlocal elasticity theory. The critical buckling loads are obtained using the theorem of minimum total potential energy and Chebyshev polynomial functions. The present method, which uses Rayleigh–Ritz technique in this paper, provides an efficient and extremely accurate buckling solution. Numerical results for a variety of some micro- and nano-rods/tubes with various boundary conditions are given and compared with the available results wherever possible. Through the comparison of the results and the appropriate discussion, the knowledge of the level of capability of the developed method is promoted. The small scale effects on the buckling loads of rods/tubes are determined and discussed.

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