Nonlocal Timoshenko Beam Model Research Articles

Transversely heterogeneous nonlocal Timoshenko beam theory: A reduced-order modeling via distributed-order fractional operators

Nonlocal effects are widely identified in micro/nano systems and have been studied extensively in recent decades. Nevertheless, existing studies mostly focus on the modeling of homogeneous nonlocal systems and as a result, cannot model micro/nanostructures with heterogeneously distributed nonlocal effects. This study introduces a reduced-order nonlocal Timoshenko beam model that facilitates precise modeling of micro-elastic beams with layer-wise heterogeneous nonlocal effects. First, a fully-resolved two-dimensional (2D) plane-strain fractional-order nonlocal elastic framework is developed to provide reference solutions for 2D nonlocal beam problems. Building upon this general nonlocal elasticity theory, an equivalent one-dimensional (1D) nonlocal Timoshenko model is then developed. The layer-wise heterogeneous nonlocal information is captured by using distributed-order (DO) operators, and the impacts arising from the nonlocal heterogeneity are further characterized by introducing two auxiliary parameters. Both 1D and 2D approaches are applied to simulate the mechanical responses of nonlocal beams. Direct comparisons of numerical simulations produced by either the 1D or the fully-resolved 2D model confirm that the DO Timoshenko beam formulation (together with the two auxiliary parameters) can capture not only the overall beam deflection but also the additional shear effect induced by the heterogeneous nonlocality. Computational cost assessment also indicates the proposed approach’s superior performance. The proposed DO Timoshenko model enables model-order reduction without compromising the heterogeneous nonlocal description of the material hence leading to an efficient and accurate reduced-order nonlocal modeling approach that addresses the limitations of prior nonlocal microstructural theories and extends applicability to a broader range of heterogeneous nano/microstructures.

Nanostructure modeling for thermal properties of zigzag carbon nanotubes

In the nanotube design application, the case of dynamic damage is key problem attracted many engineering researchers. The thermal effect on the axial buckling of zigzag double walled carbon nanotubes (DWCNTs) embedded in an elastic medium is investigated in this article based on the nonlocal Timoshenko beam model. The effects of the van der Waals (vdW) forces between inner and outer tubes and the surrounding elastic medium are taken into account based on a Winkler model. Moreover, a modified molecular mechanics model is developed to study the effect of thermal environment on mechanical properties of zigzag double carbon nanotubes, in which the covalent bonds are traded as a truss. It is found that the axial buckling properties of zigzag DWCNTs increase with the increase of the temperature at room or low temperature, while the axial buckling properties of zigzag DWCNTs decrease with the increase of the temperature at high temperature. Moreover, the buckling strain considering the thermal effect is larger than the buckling strain ignoring the effect of temperature change, and increases with the increase of temperature change, which is contrary to the case of high temperature.

Stress-driven nonlocal Timoshenko beam model for buckling analysis of carbon nanotubes constrained by surface van der Waals interactions

Stress-driven nonlocal Timoshenko beam model for buckling analysis of carbon nanotubes constrained by surface van der Waals interactions

Application of the modified Fourier series method and the genetic algorithm for calibration of small‐scale parameters in the nonlocal strain gradient nanobeams

In recent years, nonlocal strain gradient theory (NSGT) has been widely applied by researchers for the analysis of nanostructures. In this theory, the appropriate selection of small‐scale parameters and the type of high‐order boundary conditions is of great importance. In the current paper, free vibrations of carbon nanotubes are studied using a nonlocal strain gradient Timoshenko beam model, and the small‐scale parameters are calibrated based on the molecular dynamics (MD) results. The calibration process is conducted by defining an optimization problem, and the genetic algorithm is applied for obtaining the best values of the small‐scale parameters. The governing equations are derived based on Hamilton's principle and solved analytically using the modified Fourier series method. The variational consistent high‐order boundary conditions are obtained applying the weighted residual approach. The small‐scale parameters are calibrated for the first three natural frequencies considering two different boundary conditions. Moreover, the type of high‐order boundary conditions that is more consistent with the MD results is determined. The outcomes of this paper provide an appropriate reference for researchers who are using the NSGT for the analysis of micro/nanobeams.

Wave propagation analysis of magnetic nanotubes conveying nanoflow

According to the nonlocal strain gradient theory, wave propagation in magnetic nanotubes conveying magnetic nanoflow under longitudinal magnetic field is inspected. The nonlocal strain gradient Timoshenko beam model is coupled with magnetic nanoflow considering slip boundary condition to model fluid structure interaction. By applying Hamilton’s principle, the size-dependent governing equations of motion have been obtained. Calculation of the wave frequency as well as phase velocity has been carried out based on the harmonic solution. The influences of strain gradient length scale, nonlocal parameter, Knudsen number, longitudinal magnetic field and magnetic nanoflow on nanotubes’ wave propagation behavior have been examined. According to analytical results, the magnetic intensity related to the longitudinal magnetic field contributes significantly to increasing nanotubes’ wave frequency as well as phase velocity. Besides, the magnetic nanotubes conveying magnetic nanoflow predict the highest phase velocity and wave frequency. Also, the wave frequency decrease when the nonlocal parameter increases or the strain gradient length scale decreases. Moreover, an increase in fluid velocity reduces the wave frequency and phase velocity.Article highlightsThe nonlocal strain gradient Timoshenko beam model is considered.Wave propagation in magnetic nanotubes conveying magnetic nanoflow is studied.Longitudinal magnetic field and magnetic nanoflow with considering slip boundary condition is inspected.Wave frequency decrease when the nonlocal parameter increases or the strain gradient length scale decreases.Increase in fluid velocity reduces the wave frequency and phase velocity.

Free vibration of a single-walled carbon nanotube based on the nonlocal Timoshenko beam model

Abstract Free vibration of a single-walled carbon nanotube (SWCNT) embedded in an elastic medium is studied on the basis of the nonlocal Timoshenko beam model. Influences of the slenderness ratios, the boundary conditions, the atomic structures and the stiffness of the embedded medium on the natural frequencies and mode shapes of SWCNT are examined. The nonlocal effect is significant for the higher modes of SWCNT with a small slenderness ratio embedded in a soft elastic medium, and it softens the SWCNT except for the fundamental frequency of the clamped–free SWCNT.

Effect of boundary conditions and constitutive relations on the free vibration of nonlocal beams

The free vibrations of nonlocal Euler and Timoshenko beams have been studied extensively, but there still remain some problems concerning boundary conditions and constitutive relations. For cantilever beams, a counterintuitive stiffening phenomenon is widely observed. And to the best of the authors’ knowledge, there is no work concerning about the Timoshenko beam with nonlocal shear force and local bending moment. This work reconsiders and explains these problems based on nonlocal differential theory. After obtaining the governing equation for nonlocal Euler beam, it is observed that the governing equation has the same form as that for the vibration of local beam with axial compressive force. Thus by comparing the nonlocal effects to the effects of end axial force, the stiffening phenomenon of cantilever beams is found to be caused by a meaningless term in boundary bending moment. This explanation is also suitable for nonlocal Timoshenko cantilever beams. Besides, for Timoshenko beams, the nonlocal effects on tensile and shear stress constitutive relations are considered separately. Three kinds of nonlocal Timoshenko beam models are solved. Numerical comparison shows that the effects of nonlocal tensile are bigger than that of nonlocal shear stress. Especially, for Timoshenko beams with nonlocal shear force only, the nonlocal effects lower the natural frequency for all boundary conditions. This finding can verify our explanation for the stiffening phenomenon of cantilever beams. In addition, it is also observed that, for both nonlocal beam theories, if all nonlocal internal forces are included in the governing equations, the eigenvalue equations and natural frequencies corresponding to local and nonlocal boundary conditions are the same.

Bending Analysis of Bidirectional FGM Timoshenko Nanobeam Subjected to Mechanical and Magnetic Forces and Resting on Winkler–Pasternak Foundation

Bending of bidirectional functionally graded nanobeams under mechanical loads and magnetic force was investigated. The nanobeam is assumed to be resting on the Winkler–Pasternak foundation. Eringen’s nonlocal elasticity theory and Timoshenko beam model are utilized to describe the mechanical behavior of the nanobeam. Material properties of the functionally graded beam are assumed to vary in the thickness and length of the nanobeam. Hamilton’s principle is employed to derive the governing equation and related boundary conditions. These equations are solved using the generalized differential quadrature method. The obtained results are compared with the results presented in other studies, to ensure the validity and versatility of this method. This comparison shows a good agreement between the results. Results are presented and discussed for different values of functionally graded materials indices, different aspect ratios, and different boundary conditions. The effect of the magnetic field and elastic foundation on buckling load has also been studied. The difference in nanobeam behavior for different values of the size-effect parameter is clearly shown.

Investigating vibrations of viscoelastic fluid-conveying carbon nanotubes resting on viscoelastic foundation using a nonlocal fractional Timoshenko beam model

On the basis of fractional viscoelasticity, the size-dependent free-vibration response of viscoelastic carbon nanotubes conveying fluid and resting on viscoelastic foundation is studied in this article. To this end, a nonlocal Timoshenko beam model is developed in the context of fractional calculus. Hamilton’s principle is applied in order to obtain the fractional governing equations including nanoscale effects. The Kelvin–Voigt viscoelastic model is also used for the constitutive equations. The free-vibration problem is solved using two methods. In the first method, which is limited to the simply supported boundary conditions, the Galerkin technique is employed for discretizing the spatial variables and reducing the governing equations to a set of ordinary differential equations on the time domain. Then, the Duffing-type time-dependent equations including fractional derivatives are solved via fractional integrator transfer functions. In the second method, which can be utilized for carbon nanotubes with different types of boundary conditions, the generalized differential quadrature technique is used for discretizing the governing equations on spatial grids, whereas the finite difference technique is used on the time domain. In the results, the influences of nonlocality, geometrical parameters, fractional derivative orders, viscoelastic foundation, and fluid flow velocity on the time responses of carbon nanotubes are analyzed.

Stability analysis of nanobeams in hygrothermal environment based on a nonlocal strain gradient Timoshenko beam model under nonlinear thermal field

AbstractThis article is dedicated to analyzing the buckling behavior of nanobeam subjected to hygrothermal environments based on the principle of the Timoshenko beam theory. The hygroscopic environment has been considered as a linear stress field model, while the thermal environment is assumed to be a nonlinear stress field based on the Murnaghan model. The size-dependent effect of the nanobeam is captured by the nonlocal strain gradient theory (NSGT), and the governing equations of the proposed model have been derived by implementing a variational principle. The critical buckling loads have been calculated for the hinged–hinged boundary condition by incorporating the Navier approach and considering other elasticity theories such as classical elasticity theory, Eringen nonlocal elasticity theory, and strain gradient theory along with the NSGT. The present model is also validated with the pre-existing model in exceptional cases. Further, a parametric investigation has been performed to report the influence of various scaling parameters like hygroscopic environment, thermal environment, length-to-diameter ratio, small scale parameter, and length scale parameter on critical buckling loads by considering both the linear and nonlinear temperature distributions.

Vibration of fluid-conveying nanotubes subjected to magnetic field based on the thin-walled Timoshenko beam theory

In this article, the flutter vibrations of fluid-conveying thin-walled nanotubes subjected to magnetic field is investigated. For modeling fluid structure interaction, the nonlocal strain gradient thin-walled Timoshenko beam model, Knudsen number and magnetic nanoflow are assumed. The Knudsen number is considered to analyze the slip boundary conditions between the fluid-flow and the nanotube's wall, and the average velocity correction parameter is utilized to earn the modified flow velocity of nano-flow. Based on the extended Hamilton's principle, the size-dependent governing equations and associated boundary conditions are derived. The coupled equations of motion are transformed to a general eigenvalue problem by applying extended Galerkin technique under the cantilever end conditions. The influences of nonlocal parameter, strain gradient length scale, magnetic nanoflow, longitudinal magnetic field, Knudsen number on the eigenvalues and critical flutter velocity of the nanotubes are studied.

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.

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.

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.

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.

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.

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.

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.

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.

Thermal and Small-Scale Effects on Vibration of Embedded Armchair Single-Walled Carbon Nanotubes

The non-local Timoshenko beam theory has been implemented to investigate the free vibration of armchair single-walled carbon nanotubes embedded in elastic medium including the thermal effects. The mechanical properties of nano-composite (carbon nanotubes and polymer matrix) are treated as functions of temperature change and the analytical solution is derived according to the governing equations of non-local Timoshenko beam models. The equivalent Young’s modulus and shear modulus for armchair single-walled carbon nanotubes are derived using an energy-equivalent model. Influence of small-scale coefficient, vibrational mode number, matrix of nano-composite and aspect ratio on the frequency ratio of the armchair single-walled carbon nanotubes including the thermal effect are studied and discussed. The research work reveals the significance of the small-scale coefficient, the vibrational mode number, the elastic medium and the aspect ratio on the frequency ratio. It is also demonstrated that some properties of free vibrations of single-walled carbon nanotubes are dependent on the change of temperature.