Nonlocal Beam Model Research Articles

Temperature-related study on buckling properties of double-walled carbon nanotubes

In this paper, the influence of finite temperature on the axial buckling of double walled carbon nanotubes (DWCNTs) is investigated based on the non-local Timoshenko beam model. The double walled carbon nanotube with different nonlocal effects is considered as an elastic structure model of mechanics, which is composed of the Van der Waals' force between the two-layers/two walls of carbon nanotubes. Molecular structure mechanics model incorporating thermal temperature is developed to study the relation between Young's modulus and temperature. The temperature dependent Young's modulus is applied to study the mechanical property of double walled carbon nanotubes. It is found that the axial buckling properties of DWCNTs decrease with the increase of the finite temperature at high temperature, however, the axial buckling properties of DWCNTs increase with the increase of the finite temperature at low or room temperature. Moreover, it is noted that the continuous temperature has litter effect on the axial buckling property of DWCNTs at the low temperature. However, the influence of continuous temperature on the mechanical properties of DWCNTs can not be neglected at the high temperature.

Flexoelectric effects on wave propagation responses of piezoelectric nanobeams via nonlocal strain gradient higher order beam model

This paper is devoted to investigate the effects of the flexoelectric phenomenon on wave dispersion characteristics of piezoelectric nanobeams. Modified constitutive relations containing the flexoelectric effects are taken into account. The problem is formulated utilizing Reddy’s higher order shear deformation beam theory in conjunction with the nonlocal strain gradient theory (NSGT). Additionally, Hamilton’s principle is employed to derive the governing equations of the problem. Thereafter, considering harmonic wave dispersion, the solution of equations of motion is expressed as complex form. Applying this solution to the governing equations gives rise to the eigenvalue problem. Afterwards, the eigenvalue problem is solved analytically to achieve the wave frequencies and phase velocities of the system. Totally, the role of various involved parameters namely the flexoelectric effects and applied electric voltage on the wave dispersion curves is studied in detail. In addition, the size-dependency of the flexoelectric phenomenon on wave dispersion behavior of the nanobeam is explored. Due to the fact that adequate knowledge about wave dispersion characteristics of structures is of great value especially in nanotechnology field, it is hoped that the results presented here may be helpful for exact behavior determination of piezoelectric nano-structures.

Elastic waves in uniformly infinite-periodic jungles of single-walled carbon nanotubes under action of longitudinal magnetic fields

Exploring applicable ways to control characteristics of transverse waves in periodic jungles of single-walled carbon nanotubes (SWCNTs) has been of interest to nanotechnologists and applied mechanics community. Herein, the theoretical-mechanical aspects of the influence of the longitudinal magnetic field on such highly conductive nanosystems are going to be examined. Using nonlocal Rayleigh and Timoshenko beam models, the discrete and continuous versions of equations of motion of magnetically affected nanosystems are derived. Commonly, the discrete models suffer from both labor costs and computational efforts for highly populated nanosystems. To conquer these special deficiencies of discrete nanosystems, appropriate continuous models have been established and their efficiency in capturing frequencies of discrete models is proved. The roles of wavenumber, radius of SWCNTs, magnetic field strength, nonlocality, and intertube distance in flexural and shear frequencies as well as their corresponding phase and group velocities are displayed and discussed. The obtained results confirm this fact that the longitudinal magnetic field could be employed as an efficient way to control characteristics of both flexural and shear waves in periodic jungles of SWCNTs.

Nonlocal Hardening-Damage Beam Model and Its Application to a Force-Based Element Formulation

AbstractThis paper introduces a novel nonlocal hardening-damage model for beams and integrates it within a flexibility-based frame element formulation. The proposed formulation eliminates strain si...

Nonlocal Effect on the Pull-in Instability Analysis of Graphene Sheet Nanobeam Actuator

ABSTRACTIn this study, the pull-in phenomenon of a Nano-actuator is investigated employing a nonlocal Bernoulli-Euler beam model with clamped-clamped conditions. The model accounts for viscous damping, residual stresses, the van der Waals (vdW) force and electrostatic forces with nonlocal effects. The hybrid differential transformation/finite difference method (HDTFDM) is used to analyze the nonlocal effects on a graphene sheet nanobeam, which is electrostatically actuated under the influence of the coupling effect, the von Kármán nonlinear strains and the fringing field effect. The pull-in voltage as calculated by the presented model deviates by no more than 0.29% from previous literature, verifying the validity of the HDTFDM. Furthermore, the nonlocal nonlinear behavior of the electrostatically actuated nanobeam is investigated, and the effects of viscous damping, residual stresses, and length-gap ratio are examined in detail. Overall, the results reveal that small scale effects significantly influence the characteristics of the graphene sheet nanobeam actuator.

Flexural wave propagation analysis of single-walled carbon nanotubes using molecular structural mechanics approach

In the present study, the flexural wave propagation along single-walled carbon nanotubes (SWCNT) is investigated using molecular structural mechanics approach. The dispersion curves associated to the wave propagation characteristics (i.e., phase velocity versus wave number) of (5,5) and (10,10) armchair test tubes are obtained and compared with the results of the Molecular Dynamics (MD), non-local elastic Timoshenko beam model (NT) and non-local Euler beam model (NE). In addition to the advantages of the present approach such as low computational cost against MD technique and its independency to any extra parameters which have to be experimentally evaluated such as unknown length scales in NT and NE methods, the results show good agreement with the MD results, especially at the high wave number (small wavelength) regimes, which the NT and NE techniques fail to achieve an acceptable predictions.

Nonlocal strain gradient beam model for nonlinear secondary resonance analysis of functionally graded porous micro/nano-beams under periodic hard excitations

Functionally graded porous materials (FGPMs) have a wide range of applications as hollow members in biomedical and aeronautical engineering. In the FGPMs, the porosity is varied over the material volume because of the density change of pores. In the present work, an analytical treatment on the size-dependent nonlinear secondary resonance of FGPM micro/nano-beams subjected to periodic hard excitations is proposed in the simultaneous presence of the nonlocality and strain gradient size dependencies. Based upon the closed-cell Gaussian-random field scheme, the mechanical properties of the FGPM micro/nano-beams are extracted corresponding to the uniform and three different functionally graded patterns of the porosity dispersion. The nonlocal strain gradient theory of elasticity is applied to the classical beam theory to formulate a newly combined size-dependent beam model. Thereafter, an analytical solving methodology based on the multiple time-scales together with the Galerkin technique is adopted to achieve the nonlocal strain gradient frequency–response and amplitude–response curves associated with the subharmonic and superharmonic external excitations. For the subharmonic excitation, it is observed that the nonlocality causes to shift the junction point of the stable and unstable branches to the higher value of the detuning parameter. However, the strain gradient size dependency plays an opposite role. For the superharmonic one, it is illustrated that the nonlocal size effect makes an increment in the height of jump phenomenon and shifts the peak to higher value of the detuning parameter. However, the strain gradient small scale effect leads to decrease the height of the jump phenomenon and shifts the peak to lower value of the detuning parameter.

Application of hetero junction CNTs as mass nanosensor using nonlocal strain gradient theory: An analytical solution

This paper aims to propose an analytical solution for dynamic analysis of the hetero junction carbon nanotubes (HJCNTs)-based mass nanosensors using a nonlocal strain gradient Timoshenko beam model. To have a more precise nanosensor, it is necessary to have deep information about the vibration characteristics of the nanostructure. So, two main goals are followed in this paper. At first, the vibration of HJCNTs with general (elastic) boundary conditions and without attached mass are studied using a proposed analytical solution. Afterward, the HJCNT is applied as a cantilever mass sensor for sensing light as well as heavy masses attached to its tip. For the large and heavy masses, the rotary inertia of the attached mass is also considered in the analysis. The governing differential equations are derived based on the Hamilton's principle and solved by an analytical method, which is based on the modified Fourier series. The weighted residual method is employed for obtaining the variationally consistent boundary conditions using the known equations of motion of the structure. The field quantities are obtained in the closed forms. The convergence and accuracy of the proposed solution are validated through some special cases available in the literature. The effects of small scale parameters and the elastic boundaries on the frequency and mode shapes of HJCNTs are studied. Moreover, the factors that affect the frequency shift of HJCNT-mass sensor are discussed. The obtained results introduce HJCNTs as new mass nanosensors that can operate more efficiently than uniform CNTs. This paper can be greatly useful in designing HJCNT-mass sensors and may serve as a benchmark for the future research in this field.

Size-dependent nonlinear forced oscillation of self-assembled nanotubules based on the nonlocal strain gradient beam model

Self-assembled protein micro/nanotubule has been aroused considerable interest as a self-assembled supramolecular structure with bioactive properties. The prime aim of this study is to predict size dependency in the nonlinear forced oscillation of the self-assembled nanotubules embedded in an elastic biomedium. To accomplish this end, the nonlocal strain gradient elasticity theory including both softening and stiffening features of size effect is applied to the refined hyperbolic shear deformable beam model. By using the principle of Hamilton, unconventional governing differential equations of motion have been extracted. Subsequently, generalized differential quadrature method in conjunction with the Galerkin technique is employed to solve the nonclassical problem numerically. The nonlocal strain gradient frequency response and amplitude response relevant to the primary resonance of the self-assembled nanotubules are obtained corresponding to different types of boundary conditions. It is anticipated that the nonlocal size effect causes to decrease the excitation amplitudes associated with both bifurcation points, but its effect on the first one is more considerable. However, the strain gradient size dependency has an opposite influence and leads to increase them. Furthermore, it is found that by changing the end supports from simply supported to clamped one, the influence of the nonlocality on the excitation amplitude associated with the bifurcation points increases, but the influence of the strain gradient size dependency decreases.

Buckling Analysis of Nanobeams Based on Nonlocal Timoshenko Beam Model by the Method of Initial Values

Investigated herein is the buckling of nanobeams based on a nonlocal Timoshenko beam model by the method of initial values within the framework of nonlocal elasticity. Since the nonlocal Timoshenko beam theory is of higher order than the nonlocal Euler–Bernoulli beam theory, it is known to be superior in predicting the small-scale effect. The buckling determinants and critical loads for bars with various kinds of supports are presented. The Carry-Over matrix (Transverse Matrix) is presented and the priorities of the method of initial values are depicted. To the best of the researchers’ knowledge, this is the first work that investigates the buckling of nonlocal Timoshenko beam with the method of initial values.

Size-Dependent Forced Vibration Analysis of Three Nonlocal Strain Gradient Beam Models with Surface Effects Subjected to Moving Harmonic Loads

The forced vibration behaviors are examined for nonlocal strain gradient nanobeams with surface effects subjected to a moving harmonic load travelling with a constant velocity in terms of three beam models namely, the Euler-Bernoulli, Timoshenko and modified Timoshenko beam models. The modification for nonlocal strain gradient Timoshenko nanobeams is exerted to the constitutive equations by exclusion of the nonlocality in the shear constitutive relation. Some analytical closed-form solutions for three nonlocal strain gradient beam models with simply supported boundary conditions are derived by using the Galerkin discretization method in conjunction with the Laplace transform method. The effects of the three beam models, the nonlocal and material length scale parameters, the velocity and excitation frequency of the moving harmonic load on the dynamic behaviors of nanobeams are discussed in some detail. Specifically, the critical velocities are examined in some detail. Numerical results have shown that the aforementioned parameters are very important factors for determining the dynamic behavior of the nanobeams accurately.

Size-dependent nonlinear secondary resonance of micro-/nano-beams made of nano-porous biomaterials including truncated cube cells

Porous biomaterials have been utilized in cellular structures in order to mimic the function of bone as a branch of tissue engineering approach. With the aid of nanoporous biomaterials in which the pore size is at nanoscale, the capability of biological molecular isolation becomes more efficient. In the present study, firstly the mechanical properties of nanoporous biomaterials are estimated on the basis of a truncated cube cell model including a refined hyperbolic shear deformation for the associated lattice structure. After that, based upon a nonlocal strain gradient beam model, the size-dependent nonlinear secondary resonance of micro/nano-beams made of the nanoporous biomaterial is predicted corresponding to the both of subharmonic and superharmonic excitations. The non-classical governing differential equation of motion is constructed via Hamilton's principle. By employing the Galerkin technique together with the multiple time-scales method, the nonlocal strain gradient frequency-response and amplitude-response of the nonlinear oscillation of micro/nano-beams made of a nanoporous biomaterial under hard excitation are achieved. It is shown that in the superharmonic case, increasing the pore size leads to enhance the nonlinear hardening spring-type behavior of jump phenomenon and the height of limit point bifurcations. In the subharmonic case, higher pore size causes to increase the gap between two branches associated with the high-frequency and low-frequency solutions.

Three-dimensional dynamics of beam-like nanorotors on the basis of newly developed nonlocal shear deformable mode shapes

The longitudinal, chordwise, and flapwise vibrations of nanorotors are going to be examined via nonlocal Euler-Bernoulli and Timoshenko beam models. By exploiting Hamilton’s principle on the basis of the nonlocal constitutive relations, the novel-nonlocal equations of motion that display three-dimensional vibrations of the beam-like nanorotors are constructed and solved for natural frequencies using the Galerkin-based assumed mode methodology. For capturing the frequencies more accurately, the newly developed-size-dependent mode shapes are employed in which the true nonlocal boundary conditions at the free end are satisfied exactly. Accounting for dynamic couplings of the longitudinal and chordwise vibrations, the roles of the angular velocity, nonlocality, length of the nanorotor, and radius of the nanoshaft on the longitudinal, chordwise, and flapwise frequencies are displayed and discussed. Further, the critical angular velocity of the nanorotor by considering the nonlocality is derived and its influential factors are displayed. The crucial role of the shear deformation on the obtained results is also explained methodically.

Nonlocal Analysis of Natural Vibrations of Carbon Nanotubes

When the atomic level is considered, the dynamic properties have a major influence on nanostructures behavior because of their ultrahigh or very high natural frequencies. Vibrations are essential in the analysis of resonators, oscillators, and sensors for carbon nanotube-based devices. The present paper is dedicated to the eigenfrequencies analysis of single-walled carbon nanotubes. In the analysis, the small-scale coefficient is introduced, and the nonlocal elasticity theory is applied in the modal analysis of the carbon nanotubes. The correlation between nonlocal small-scale parameter and the vibrational behavior of the carbon nanotubes is studied. The determination of nonlocal parameter is based on the structural finite element model of the nanotube. A detailed parametric study is realized to investigate the effects of the length-to-diameter ratio L/D, nonlocal coefficient, the influence of boundary conditions on eigenfrequencies of carbon nanotubes. The main contribution of this work is the evaluation of the nonlocal parameter for single-walled carbon nanotubes based on the finite element approximation in modeling the dynamic behavior instead of commonly used MD simulations. It was clearly displayed that the application of the nonlocal beam models involves analysis of the small-scale parameter, whose value determines the final eigenfrequencies. The highest values of eigenfrequencies were observed for the local beam model. It was presented that the eigenfrequencies for carbon nanotubes decrease when the nanotube length increases for all analyzed modes. On the other hand, when the diameter of carbon nanotubes increases, the eigenfrequency values rise. The numerical validation of the nonlocal parameter revealed the high dependence on the boundary conditions and the mode number. However, the influence of the mode number and the boundary conditions is vanishing for long single-walled carbon nanotubes.

Coupling effect of thickness and shear deformation on size-dependent bending of micro/nano-scale porous beams

• A unified nonlocal strain gradient beam model with the thickness effect is developed. • Analytical solutions for the bending of hinged-hinged beams are obtained. • Size effect can be not only length-dependent but also thickness-dependent. • The generalized Young’s modulus depends on applied load types. • The coupling of the shear and thickness effects needs be drawn attention. A unified nonlocal strain gradient beam model with the thickness effect is developed to investigate the static bending behavior of micro/nano-scale porous beams. Size-dependent governing equations and corresponding analytical solutions for the bending of hinged-hinged beams are obtained by employing minimum total potential energy principle, the Navier solution method as well as the variational-consistent boundary conditions. For nonlocal strain gradient theory (NSGT) with thickness effect, virtual strain energy function of shear beams can contain additional nonlocal shear stress and high-order nonlocal shear stress related to the thickness direction in comparison with that of Euler–Bernoulli beam, so the coupling of the shear and thickness effects should be drawn huge attention. By means of detailed numerical analysis, it is found that, the stiffness-hardening effect is underestimated in NSGT without the thickness effect, and the stiffness-hardening and stiffness-softening effects of NSGT with the thickness effect can be not only length-dependent but also thickness-dependent. Interestingly, the generalized Young’s modulus depends on half-wave number, which means that the generalized Young’s modulus may be different due to applied load types. In the context of NSGT with the thickness effect, the deflection of Euler–Bernoulli beam predicted is smaller than that of shear beam, especially for thick beams. Furthermore, porosities distributed in the top or bottom of beams can possess a greater influence on the decrease of overall stiffness of beam than those distributed in the vicinity of the middle plane of beams.

Free vibration analysis of viscoelastic nanotubes under longitudinal magnetic field based on nonlocal strain gradient Timoshenko beam model

In this paper, the free vibration of viscoelastic nanotube under longitudinal magnetic field is investigated. The governing equation is formulated by utilizing Timoshenko beam model and Kelvin-Voigt model based on the nonlocal strain gradient theory. The local adaptive differential quadrature method (LADQM) is applied in the analyzing procedure. We also investigated the influences of the nonlocal parameter, structural damping coefficient, material length scale parameter and the longitudinal magnetic field on the natural frequencies of the system. The results of this research may be helpful for understanding the potential applications of nanotubes in Nano-Electromechanical System.

Free vibration of a fluid-conveying nanotube constructed by carbon nanotube and boron nitride nanotube

The free vibration of a fluid-conveying nanotube composed of carbon nanotube and boron nitride (BN) nanotube is studied in this paper. The nonlocal Euler-Bernoulli beam model is adopted to describe the transverse displacement of the hybrid nanotube, and the governing equation is segmented because of the material properties' change at the junction. Then the dynamic stiffness method is used to solve the governing equation, and we investigated the effects of length ratio and nonlocal parameter on the vibration behavior and stability of the fluid-conveying hybrid nanotube. It can be concluded that the instability of the nanotube is significantly dependent on the length ratio and the nonlocal parameter. The increase of length ratio or nonlocal parameter will both make the system more unstable, in particularly, the nonlocal parameter has a great effect on the second-mode divergence. And we also find that the extra support at the junction will lead a great increment to the first two order frequencies and critical flow velocity.

Analytical analysis for the forced vibration of CNT surrounding elastic medium including thermal effect using nonlocal Euler-Bernoulli theory

This article studies the free and forced vibrations of the carbon nanotubes CNTs embedded in an elastic medium including thermal and dynamic load effects based on nonlocal Euler- Bernoulli beam. A Winkler type elastic foundation is employed to model the interaction of carbon nanotube and the surrounding elastic medium. Influence of all parameters such as nonlocal small-scale effects, high temperature change, Winkler modulus parameter, vibration mode and aspect ratio of short carbon nanotubes on the vibration frequency are analyzed and discussed. The non-local Euler-Bernoulli beam model predicts lower resonance frequencies. The research work reveals the significance of the small-scale coefficient, the vibrational mode number, the elastic medium and the temperature change on the non-dimensional natural frequency.

Flow-induced vibration and stability analysis of carbon nanotubes based on the nonlocal strain gradient Timoshenko beam theory

A nonlocal strain gradient Timoshenko beam model is developed to study the vibration and instability analysis of the carbon nanotubes conveying nanoflow. The governing equations of motion and boundary conditions are derived by employing Hamilton’s principle, including the effects of moving fluid, material length scale and nonlocal parameters, Knudsen number and gravity force. The material length scale and nonlocal parameters are considered, in order to take into account the size effects. Also, to consider the small-size effects on the flow field, the Knudsen number is used as a discriminant parameter. The Galerkin approach is chosen to analyze the governing equations under clamped–clamped, clamped–hinged and hinged–hinged boundary conditions. It is found that the natural frequency and critical fluid velocity can be decreased by increasing the nonlocal parameter or decreasing the material length scale parameter. Furthermore, it is revealed that the critical flow velocity does not affected by two size-dependent parameters and various boundary conditions in the free molecular flow regime.

A research into bi-Helmholtz type of nonlocal elasticity and a direct approach to Eringen’s nonlocal integral model in a finite body

There are exhaustive reports revolving around the nonlocal differential beam models of micro- and nano-electromechanical systems (MEMS–NEMS), carbon nanotubes (CNTs), nanomaterials, etc., since the nonlocal continuum theory is considered a viable option for shedding light on size effect phenomena. A constitutive equation with a second-order differential operator, Helmholtz type, is employed in these studies. However, these models do not produce quadratic, self-adjoint energy functionals and give rise to paradoxes in static and dynamical problems. The transformation of Eringen’s integral constitutive equation into a differential one is not an injective process in a finite domain and that is probably responsible for the inconsistencies and paradoxes raised. This work researches into the adequacy of a higher-order differential operator, bi-Helmholtz type, applied to engineering problems. The bi-Helmholtz-type operator is more effective for describing wave dispersion of atomic models than the Helmholtz type. Eringen’s nonlocal integral stress model is also looked into beams for both types of kernels. The nonlocal integral model’s key benefit is the energy consistent formulas produced, whereas its main drawback lies in the handling of the 1st kind Fredholm governing equation. Exploiting the modified nonlocal attenuation function (kernel) normalized in a finite domain, the 1st kind Fredholm integral equation is directly transformed into a 2nd kind one. Furthermore, the modified kernel successfully handles the physical inconsistencies in a finite domain, such as the reversion of the nonlocal integral stress model to the classic-local model when the nonlocal parameter tends to zero. Our research objective is to explore bi-Helmholtz operator’s application and the corresponding kernel in the nonlocal integral model. The results deduced and the conclusions reached can be considered adequate for the nonlocal integral form’s suitability for dealing with problems in nanoscale.