Nonlocal Timoshenko Beam Theory 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.

Vibration and Buckling Analysis of Nano-Scale Beams via Nonlocal Elasticity and Timoshenko Beam Theory : A Differential Quadrature Approach

A single-elastic beam model is being developed based on Eringens nonlocal elasticity and Timoshenko beam theory. Nonlocal parameter takes into account the small scale effects in the analysis of nano-size structures. Derived herein is a decoupled sixth-order nonlocal governing differential equations for the frequency and stability analysis of nano-scale short beams. The effect of shear deformation is thereby included in the small scale analysis. A Differential Quadrature (DQ) approach is being applied and its elaborate method of solution is illustrated. The higher order DQ approach is found to be a good numerical technique for the convenient and rapid solution of the aforementioned nonlocal problems. Frequency and buckling results for various scale-based nonlocal parameters are shown. Effect of number of interpolation points on the accuracy of the results is also investigated. It is seen that there is a significant effect of aspect ratio and nonlocal parameter on the nonlocal frequency and buckling loads of nano-scale beams. Also, the present study could open up a new approach of solution technique for the analysis of nano-scale structures based on nonlocal Timoshenko theory.

Forced Vibration Study of Carbon Nanotubes Using Non-Local Elasticity and Timoshenko Beam Theories

Forced vibrations of the single wall carbon nanotubes (SWCNTs) using the non-local elasticity theory and Timoshenko beam theory is studied. Governing equations for the vibration response of the carbon nanotubes are derived employing Eringen’s nonlocal elasticity theory. Analysis is carried out for both EulerBernoulli beam theory and Timoshenko beam theory. Present forced vibration results for Euler-Bernoulli are in good agreement with those reported in literature. Effects of (i) nonlocal parameter (ii) length of the carbon nanotube and (iii) modes of vibration on the vibration response of the carbon nanotubes are investigated. The effect of nonlocal parameter on the vibration response are significant for small size carbon nanotube and higher modes of vibration.

On the Static Flexural Response of Shear Deformable Isotropic Rectangular Nanobeams under the Parabolic Loading

It is well known that size-dependent effects, which are insignificant for macroscopic structures, become significant for small-scale structures like nanobeams. In addition, effects of the beam transverse shear deformation, which are insignificant for slender beams, become significant for shear deformable beams. Hence the need arises for a beam theory which is simple-to-use as well as which accounts for just-mentioned features of shear deformable nanobeams. Recently, the authors have developed a single variable new first-order shear deformation nonlocal beam theory (NFSDNBT, doi: 10.1007/s40430-019-2128-6). The NFSDNBT is applicable for the flexure of linear isotropic nanobeams undergoing small deformations. Displacement functions of the NFSDNBT give rise to the constant transverse shear strain through the beam thickness. Hence similar to the nonlocal Timoshenko beam theory (NTBT), the NFSDNBT also requires a shear correction factor. In the NFSDNBT, nonlocal differential stress-strain constitutive relations of Eringen are utilized, through which size-dependent effects have been taken into account. These relations relate not only the beam axial stress with the beam axial strain but also the beam transverse shear stress with the beam transverse shear strain. The governing equation of the NFSDNBT is obtained by utilizing beam gross equilibrium equations and it has a strong resemblance with the governing equation of the nonlocal Bernoulli-Euler beam theory (NBEBT). In this paper, the NFSDNBT is utilized for finding the static flexural response of shear deformable isotropic rectangular nanobeams under the action of parabolically distributed transverse loading. For the simply-supported, clamped-clamped and cantilever nanobeams, effects of variations in values of the nonlocal parameter of Eringen and beam thickness-to-length ratio on the maximum non-dimensional beam transverse displacement are presented. Obtained results are compared with corresponding results obtained by utilizing the NTBT and NBEBT so as to demonstrate the efficacy of the NFSDNBT. Along-the-length profiles of the non-dimensional beam transverse displacement for the just-mentioned cases of nanobeams, for various values of the nonlocal parameter and beam thickness-to-length ratio are also presented.

Free vibration of multiple-nanobeam system with nonlocal Timoshenko beam theory for various boundary conditions

The free vibration of multiple-nanobeam system is studied for various edge boundary conditions and number of coupled nanobeams. The size effect is captured using the Eringen's nonlocal elasticity theory. The governing equations of the beams which are coupled in the multiple-nanobeam system are obtained by Timoshenko beam theory. The vibration of multiple-nanobeam system with the number of m coupled nanobeams is described by 2 m coupled partial differential equations. A meshless formulation is presented to discretize the governing equations to a set of ordinary differential equations in time domain. The number of nanobeams of multiple-nanobeam system and the boundary conditions of each nanobeam can be arbitrary. The accuracy of results is examined by comparison of the predictions with available analytical results in the literature for especial cases, and good agreement is seen. In the numerical results the free vibration frequencies and mode shapes of multiple-nanobeam system for various edge boundary conditions are presented and the effects of parameters such as coupling stiffness, nonlocal parameters and number of nanobeams of multiple-nanobeam system are investigate. The presented method can be very useful for analysis of multiple-nanobeam system with arbitrary number of nanobeams, arbitrary boundary conditions, coupling stiffness and length to thickness ratio.

Nonlocal Timoshenko modeling effectiveness for carbon nanotube-based mass sensors

Carbon nanotubes (CNTs) have been considered for a wide range of nanotechnology applications due to their unique mechanical, thermal, and electrical properties. One of the most notable nanotechnology applications is the frequency shift-based carbon nanotube mass sensor, which has motivated the development of reliable reduced-order models. As such, this research effort focuses on a carbon nanotube-based mass sensor that utilizes Timoshenko beam theory to extend the usability of these models to short and stout structures. Additionally, Eringen's nonlocal elasticity is considered to account for certain nanoscale phenomena. To the authors' knowledge, this is the first time a CNT-based mass sensor modeled with the nonlocal Timoshenko beam theory with a complex-shaped deposited particle has been studied. Considering Timoshenko beam theory and Eringen's nonlocal theory, the Hamilton's principle is utilized to derive the governing equations of motion, boundary conditions, and continuity conditions accounting for a single deposited nanoparticle. Then, it is shown that the obtained frequency shift of the sensor is dependent on the rotary inertia, shear effects, nonlocal parameter, and particle mass, geometry, and location. The variation of each of these parameters leads to a holistic characterization of the proposed system. The discrepancies and limits of applicability between Timoshenko and Euler-Bernoulli models are deeply explored and discussed, particularly for short and stout structures. It is shown that the Euler-Bernoulli model leads to an overprediction of the frequency shifts compared to Timoshenko beam theory, especially for higher modes. The nonlocal Timoshenko-based model is valuable because short and stout structures were found to be preferred for mass sensing applications, since they lead to a higher sensitivity. Other researchers can utilize these findings for the design, modeling, and analysis of nanoscale sensors and resonators.

Transverse wave propagation analysis in single-walled and double-walled carbon nanotubes via higher-order doublet mechanics theory

ABSTRACT In the present study, transverse wave propagation in single-walled and double-walled carbon nanotubes is investigated based on doublet mechanics theory. The Euler–Bernoulli and Timoshenko beam theories are considered to study the wave dispersion relations of single-walled and double-walled carbon nanotubes (CNTs) with the set of relevant governing equations are achieved by use of the variational Hamilton’s principle. Obtained governing equations are presented for zigzag and armchair CNT models. Using these models, scale dependent wave frequency, phase velocity and group velocity of CNTs are obtained. It has been observed that doublet mechanics results have so called van Hove singularity. This property is the first seen among the scale-dependent continuum theories. The effectiveness and validity of the present method are approved by comparing the predicted numerical results with molecular dynamics simulation and nonlocal Timoshenko beam theory. An excellent validation is obtained between the doublet mechanics based on Timoshenko beam theory and molecular dynamics simulation for phase velocities of (5,5) and (10,10) armchair single-walled CNTs. Also, this work clarifies the importance of the length scale effect on transverse wave dispersion in multi-walled CNTs.

Free vibration analysis of nonlocal nanobeams: a comparison of the one-dimensional nonlocal integral Timoshenko beam theory with the two-dimensional nonlocal integral elasticity theory

Beam theories such as the Timoshenko beam theory are in agreement with the elasticity theory. However, due to the different nonlocal averaging processes, they are expected to yield different results in their nonlocal forms. In the present work, the free vibration behavior of nonlocal nanobeams is studied using the nonlocal integral Timoshenko beam theory (NITBT) and two-dimensional nonlocal integral elasticity theory (2D-NIET) with different kernels and their results are compared. A new kernel, termed the compensated two-phase (CTP) kernel, is introduced, which entirely compensates for the boundary effects and does not suffer from the ill-posedness of previous kernels. Using the finite element method, the free vibration analysis is performed for different boundary conditions based on the first three natural frequencies. For both the NITBT and 2D-NIET with both the two-phase (TP) and CTP kernels, the nonlocal parameter has a softening effect on the natural frequencies for all the boundary conditions, without observing the paradoxical behaviors of the nonlocal differential theory. For both theories, the softening effect of the nonlocal parameter is more pronounced for the TP kernel compared to the CTP kernel. The sensitivity of the 2D-NIET to the nonlocal parameter is found to be higher than that of the NITBT. Also, the softening effects for different vibration modes are compared to each other for both theories and both kernels. The obtained results can be extended for various important beam problems with nonlocal effects and help obtain a better understanding of applicable nonlocal theories.

Derivation of nonlocal FEM formulation for thermo-elastic Timoshenko beams on elastic matrix

Free thermal vibration analysis of nanobeams surrounded by an elastic matrix is examined via nonlocal elasticity and Timoshenko beam theories in this article. Elastic matrix is formulated with a two-parameter elastic foundation modelled by combining Winkler and Pasternak assumptions. The equation of motion for free vibration is reached via Hamilton’s principle and solved by analytical method (separation of variable). However, since the separation of variable cannot be applied for boundary conditions other than simply supported nanobeams, a weighted residue-based finite element formulation is developed. Within the scope of numerical results, firstly, it is understood from the comparisons given for nondimensional frequencies that the nonlocal finite element formulation has a high accuracy and then, using this formulation, nondimensional frequencies of nanobeams with different boundary conditions are computed under different parameters. Additionally, the detailed discussions of the numerical results are presented. Finally, by giving some general results, the effect of size dependency and environmental factors on the dynamic behavior of nanobeams is expressed.

A comparative study of 1D nonlocal integral Timoshenko beam and 2D nonlocal integral elasticity theories for bending of nanoscale beams

In this paper, the bending behavior of nanoscale beams is studied using the 1D nonlocal integral Timoshenko beam theory (NITBT) and the 2D nonlocal integral elasticity theory (2D-NIET) using two types of nonlocal kernels, i.e., the two-phase kernel and a modified kernel which compensates the boundary effects. The governing equations are solved using the finite element method and the COMSOL code. Mesh sensitivity study and numerical verifications are presented. The main differences and similarities in both theories at the nanoscale are revealed. For both theories and both kernels, a softening behavior is found by increasing the nonlocal parameter and decreasing the phase parameter, for different boundary and load conditions. In contrast to the differential theory, no paradoxical behavior for the cantilever conditions is found for both theories. The sensitivity of the 2D-NIET to the nonlocal parameter is found higher than that of the NITBT. The normalized transverse deflection for the 2D-NIET is found independent of the boundary and load conditions. Also, the normalized transverse deflection varies linearly versus the normalized nonlocal parameter for both theories with the two-phase kernel and any condition except for the simply supported beam under distributed load condition in the NITBT. The boundary effect, resulting in a different softening near the boundaries, reduces by increasing the phase parameter. The modified kernel in the 2D-NIET is found sensitive to the pinned not to the free and fixed boundaries. It is in detail shown that for the 2D-NIET especially with the modified kernel, by increasing the nonlocal parameter, the deflection increases with almost the same ratio through the entire length of the beam and for all the boundary conditions. The obtained results can be used for modeling of various beam problems with nonlocal effects at the nanoscale.

Nonlocal Timoshenko representation and analysis of multi-layered functionally graded nanobeams

The vibration properties of functionally graded multiple nanobeams is studied using Eringen’s nonlocal elasticity theory. Beam layers are considered to be continuously connected by layer of linear springs and the nonlocal Timoshenko beam theory is used to model each layer of beam which applies the size dependent effects in FG beam. The material behaviors of FG nanobeams are assumed to vary over the thickness based to the power law. The Hamilton’s principle is used to derive the governing differential equations of motion according to Eringen nonlocal theory and a Chebyshev spectral collocation method is employed to convert the coupled equations of motion into algebraic equations. The discretized boundary conditions are applied to adjust the Chebyshev differentiation matrices, and the system of equations is then expressed in the matrix–vector form. Next, the coupled natural frequencies and corresponding mode shapes are obtained by solving the standard eigenvalue problem. The model is confirmed by comparing the obtained results with benchmark results existing in the literature. Next, a parametric study is carried out to determine the influences of the material gradation, length scale, and stiffness parameters on the vibration properties of multiple functionally graded nanobeams. It is demonstrated that these parameters are vital in examination of the free vibration of a multi-layered FG nanobeam.

Vibration Analysis of Fluid Conveying Carbon Nanotubes Based on Nonlocal Timoshenko Beam Theory by Spectral Element Method.

In this work, we applied the spectral element method (SEM) to analyze the dynamic characteristics of fluid conveying single-walled carbon nanotubes (SWCNTs). First, the dynamic equations for fluid conveying SWCNTs were deduced based on the nonlocal Timoshenko beam theory. Then, the spectral element formulation was established for a free/forced vibration analysis of fluid conveying SWCNTs by introducing discrete Fourier transform. Furthermore, the proposed method was validated using several comparison examples. Finally, the natural frequencies and dynamic responses of a simply-supported fluid conveying SWCNTs were calculated by the SEM, considering different internal fluid velocities and small-scale parameters (SSPs). The effects of fluid velocity and SSPs on the dynamic characteristics of SWCNTs conveying fluid were revealed by the numerical results. Compared with other methods, the SEM shows high accuracy and efficiency.

A single-variable first-order shear deformation nonlocal theory for the flexure of isotropic nanobeams

In this paper, a displacement-based single-variable first-order shear deformation nonlocal theory for the flexure of isotropic nanobeams is presented. In the present theory, the beam axial displacement consists of a bending component, whereas the beam transverse displacement consists of a bending component and a shearing component. Bending components do not take part in the cross-sectional shearing force, and the shearing component does not take part in the cross-sectional bending moment. The present theory utilizes linear strain–displacement relations. The displacement functions of the present theory give rise to the constant transverse shear strain through the beam thickness. Hence as is the case with the nonlocal Timoshenko beam theory, the present theory also requires a shear correction factor. In the present theory, nonlocal differential stress–strain constitutive relations of Eringen are utilized in order to take into account the size-dependent behaviour of nanobeams. In the present theory, beam gross equilibrium equations are utilized to derive the governing differential equation. As against other first-order shear deformation nonlocal beam theories, the present theory involves only one governing differential equation of fourth order and has only one unknown function. There exists a strong resemblance between expressions of the cross-sectional shearing force, cross-sectional bending moment and governing differential equation of the present theory and corresponding expressions of the nonlocal Bernoulli–Euler beam theory based on the nonlocal elasticity theory of Eringen. For the present theory, boundary conditions are described. The present theory also describes two distinct and physically meaningful clamped boundary conditions. Illustrative examples pertaining to the flexure of shear deformable isotropic nanobeams demonstrate the efficacy of the present theory.

Vibration performance evaluation of smart magneto-electro-elastic nanobeam with consideration of nanomaterial uncertainties

This study is devoted to examining the vibration behaviors of magneto-electro-elastic nanobeams with consideration of nanomaterial uncertainties induced by the atom defect and manufacturing deviation. Based on the nonlocal Timoshenko beam theory, the governing equations of a magneto-electro-elastic nanobeam resting on a Winkler–Pasternak foundation and subjected to electric and magnetic potentials are derived. The material properties of the magneto-electro-elastic nanobeam are treated as uncertain parameters with well-defined bounds to overcome the extensive information required in probabilistic evaluation. The range of natural frequency of the magneto-electro-elastic nanobeam is predicted via a non-probabilistic evaluation methodology, which is validated by comparing with Monte Carlo simulation and probabilistic evaluation methodology. Then, the parametric analyses are performed to reveal the coupling effects of nanomaterial uncertainties, and nonlocal parameter, as well as elastic foundation parameters on the vibration performance of magneto-electro-elastic nanobeams. It is demonstrated that the nanomaterial uncertainties affect the mechanical behaviors of magneto-electro-elastic nanostructures significantly and the present model can be degenerated into the deterministic model as the nanomaterial uncertainty is eliminated.

DQEM analysis of free transverse vibration of rotating non-uniform nanobeams in the presence of cracks based on the nonlocal Timoshenko beam theory

This paper deals with the free transverse vibration characteristics of a rotating non-uniform nanocantilever with multiple open cracks. Employing Eringen’s nonlocal elasticity and the Timoshenko beam theory, the non-dimensional governing differential equations for the above-mentioned problem are derived. The cracked beam is divided into intact sub-beams between two subsequent cracks connected by linear and rotational springs. Differential quadrature element method is utilized to solve the established governing equations of motion of each segment, along with the corresponding boundary conditions and compatibility conditions at the cracked sections. The frequency parameters and vibration modes of the rotating cracked beam for different crack positions and severities under various nonlocal, geometric and dynamic conditions are studied, and the relevant graphs are plotted. Since rotating nanocantilevers are found mostly as blades of rotating nanodevices, the results can provide useful guidance for the study and design of the next generations of nanoturbines, nanogears etc.

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.

Free Vibration Analysis of a Rotating Non-uniform Nanocantilever Carrying Arbitrary Concentrated Masses Based on the Nonlocal Timoshenko Beam Using DQEM

In this paper, a differential quadrature element method (DQEM) is proposed for free vibration analysis of rotating non-uniform nanocantilevers carrying multiple concentrated masses. Employing Hamilton’s principle, the governing equations of rotating nanoblades, modeled by the nonlocal Timoshenko beam theory, are derived. The differential quadrature (DQ) analogs of the governing equations of motion, compatibility conditions at the positions of masses, and related boundary conditions are established, and then, an eigen-value problem is obtained. The vibration characteristics of the problem are investigated with various conditions of number, positions, and magnitudes of the masses, while the cross section, rotational velocity, hub radius, and nonlocal parameters are arbitrary. The accuracy of the results is confirmed by the exact data available in the literature. In comparison with the other applicable methods, DQEM is less time-consuming and also enables the researcher to analyze the problem under arbitrary conditions, which are complex or even sometimes impossible to solve. The studies show that rotation rates, geometric properties, and masses conditions can contribute significantly in the dynamic characteristics of rotating N/MEMS devices such as nanoturbines, nanoscale molecular bearings, shaft and gear, and nanosensors.

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.

Vibration of FG magneto-electro-viscoelastic porous nanobeams on visco-Pasternak foundation

In this paper, the size-dependent vibrational behaviors of functionally graded (FG) magneto-electro-viscoelastic nanobeams in the presence of porosities and embedded in the viscoelastic medium are studied based on the nonlocal Timoshenko beam theory in conjunction with the Kelvin-Voigt viscoelastic model. The viscoelastic medium is modeled as a visco-Pasternak foundation with consideration of both shear modulus and medium damping coefficient. The FG material properties are supposed to vary along the thickness direction in a power-law exponent form, and two types of porosity distributions are considered. The present model is validated by comparison with several existing theories and a good agreement is achieved. Parametric studies are carried out to investigate the coupling influences of the nonlocal parameter, FG power-law index, porosity volume fraction, porosity distribution, boundary condition, structural damping coefficient and viscoelastic foundation parameters, as well as electric voltage and magnetic potential on the vibrational performance of FG magneto-electro-viscoelastic porous nanobeams. The results are helpful for the design and applications of nano-electro-mechanical systems (NEMS).

Uncertainty analysis for wave dispersion behavior of carbon nanotubes embedded in Pasternak-type elastic medium

At nano-scale, the molecular defects and manufacturing tolerances of carbon nanotubes (CNTs) both would introduce inevitable uncertainties. In this paper, a theoretical framework based on interval analysis method is presented to study the wave-dispersion behavior of CNTs embedded in a Pasternak-type elastic medium with uncertain material properties. Different from probabilistic analysis approach, the material properties of CNTs are considered as uncertain-but-bounded variables, which can overcome the massive information required to set up the probabilistic distribution function. Based on the nonlocal Timoshenko beam theory, the wave-dispersion relationship of the embedded CNTs is derived. The upper and lower bounds of the wave frequencies are predicted under different elastic foundation and size scale parameters. It is demonstrated that the uncertain material properties have considerable effect on the wave-dispersion behavior of CNTs, and the size scale and elastic medium will also affect the wave-dispersion response of CNTs significantly. The Monte Carlo simulation is also presented to validate the proposed interval analysis method. The numerical results presented in this paper provide useful guidelines for employing CNTs as reinforced fibers to enhance the conventional composite materials.