Nonlocal Euler-Bernoulli Beam Model Research Articles

Chaotic vibration of a curved CNT conveying magnetic fluid in the thermo-magnetic field considering the surface effects

The nonlinear chaotic vibration of curved single-walled carbon nanotube (CSWCNT) conveying magnetic fluid is studied based on the nonlocal Euler-Bernoulli beam model. The governing equation of CSWCNT is presented considering the axial thermo-magnetic load and the surface effect. By employing the Galerkin decomposition approximation method along with the admissible beam shape function satisfying the cantilevered beam boundary conditions, the nonlinear partial differential equation of the system is reduced to a nonlinear ordinary differential equation and numeric integration procedures are used to solve it. By considering the non-dimensional damping, cubic and quadratic terms, and amplitude of external force as controlling parameters, the bifurcation diagram and the largest Lyapunov exponent are employed to distinguish the chaotic, periodic, and quasi-periodic critical parameters of the curved CNT dynamics and validate the predicted results. Also, the phase plane and Poincare map for these critical parameters are provided. The results show that decreasing values of damping coefficient and quadratic term leads to quasi-periodic or chaotic motion, while system has periodic behaviour for smaller values of the external force and nonlinear cubic term. Also, magnetic field intensity and length of the CSWCNT help preventing the chaotic motion. Furthermore, adverse effects of higher values of temperature change, flow velocity and its correction factor are seen based on the obtained results.

Application of the Green's function method for static analysis of nonlocal stress-driven and strain gradient elastic nanobeams

This paper focuses on the static analysis of a generally restrained Euler-Bernoulli nanobeam (GRNB), which is subjected to arbitrary distributed or concentrated loads. The small length scale effect is introduced through a strongly nonlocal integral elastic law called the stress-driven nonlocal Euler-Bernoulli beam model, and a weakly nonlocal elastic law called the strain-gradient Euler-Bernoulli beam model. The stress-driven nonlocal theory and the strain gradient elasticity theory are both employed, to formulate the differential equation governing the Euler-Bernoulli nanobeams. Both theories are ruled by a similar 6th order differential equation, with distinct higher-order boundary conditions. The paper demonstrates that the strain gradient theory can be also reformulated as a stress-driven nonlocal theory with unsymmetrical kernel, in contrast to the initial stress-driven nonlocal theory which has symmetrical exponential kernel. Additionally, some theoretical backgrounds about both nonlocal elasticity theorems are provided. Exact solutions for both nanobeam models are presented, using Green's function method. The parameters of the Green's functions are derived for each beam theory, and then utilized to establish the static displacement function of both nanobeam theories. Exact solutions are provided for the bending of both nanobeams with various end boundary conditions, that account for the small length scale effects. The paper also compares the responses of the stress-driven Euler-Bernoulli beam and the strain gradient Euler-Bernoulli beam, showing close deflections and similar effects of increasing the nonlocal parameter on beam stiffness. A major difference highlighted between the two theories is that the strain gradient theory predicts a homogeneous strain state for homogeneous stress state, as opposed to the stress-driven theory.

Effect of chiral angle and chiral index on the vibration of single-walled carbon nanotubes using nonlocal Euler-Bernoulli beam model

The effect of various physical phenomena on the vibration characteristics of CNTs has attracted much interest. In this work, the effect of chiral index and chiral angle dependent from diameter on vibration of single-walled carbon nanotubes (SWCNTs) is investigated. The model of nonlocal Euler-Bernoulli beam is used to study the free vibration characteristics of SWCNT. Based on the nonlocal continuum theory, the governing equation of motion is formulated and derived. The influences of small-scale parameter, chirality of SWCNTs, vibrational mode number, aspect ratio (i.e. length–to-diameter ratio) of SWCNTs, chiral index and chiral angle on the vibration characteristics of SWCNTs are examined and discussed. The results indicate significant dependence of natural frequencies on the nonlocal parameter, chiral angle, chiral index, aspect ratio and chirality of SWCNTs. From numerical results, it is concluded that the vibration characteristics of SWCNT have been influenced by the chiral angle and chiral index. This work is supposed to be useful reference to researches who are designing nanodevices.

Free Transverse Vibration of Nickel Coated Carbon Nanotubes

Transverse vibration of nickel coated carbon nanotubes is investigated by using molecular dynamics simulations. The simulations are carried out for armchair and zig-zag carbon nanotubes with various lengths. Uncoated and nickel coated carbon nanotubes having same lengths are analyzed and their vibrational behaviors are compared. Free transverse vibrations of nickel coated carbon nanotubes are modelled by using a two-phase local–nonlocal Euler–Bernoulli beam model and solved by finite element method. Nonlocal parameter of the beam model is calibrated based on molecular dynamics simulation results. It is seen that for the same length diameter ratio, the nickel coated carbon nanotubes have similar vibrational characteristics with the uncoated carbon nanotubes but their natural frequencies are smaller than the uncoated ones. Also, it is shown that by using proper nonlocal parameters for each radius length ratio, the two-phase local–nonlocal Euler–Bernoulli beam model can successfully predict the natural frequencies of both short and long nanotubes. Besides natural frequencies and mode shapes, the clustering of nickel atoms depend on simulation temperature which is discussed during oscillation of nickel coated carbon nanotubes.

Cross-section effect on mechanics of nonlocal beams

In the current practice, the one-dimensional nonlocal constitutive relations are always employed to model beam-type structures, regardless of the inherent three-dimensional interactions between atoms, resulting in inaccurately predicted nonlocal structural behaviors. To improve modeling, the present work first reveals and establishes how the nonlocal interactions in beams’ width and height directions substantially affect the bending behaviors of nanobeams. Based on the new concept of a general three-dimensional nonlocal constitutive relation, a three-dimensional nonlocal Euler–Bernoulli beam model is developed for bending analysis. Closed-form solutions for the deflections of simply supported and cantilevered beams are derived. The cross-section effect on the nonlocal stress, the bending moment and the deflection is explored. Moreover, an effective calibration method is proposed for the determination of the intrinsic characteristic length based on molecular dynamics simulations. It has been found that the actually nonlocal physical dimension of beam-type structures is not in agreement with that from the one-dimensional geometric direction of classical beam mechanics, and the beams’ width and height must be taken into consideration, instead of just employing the beam’s length, a practice which has been somewhat blindly undertaken in the current practice without indeed much theoretical support and understanding. A beam problem at nanoscale has to be viewed and treated as a “three-dimensional” geometric and physical problem due to its geometric dimension in length and its physical dimensions in both the thickness and the width. When its intrinsic characteristic length is comparable to its width or thickness, a rigidity-softening effect of a beam can be observed. Further, it is established that the nonlocal effect is more sensitive in the thickness direction, as compared with the width direction.

Thermal stress and magnetic effects on nonlinear vibration of nanobeams embedded in nonlinear elastic medium

Nonlinear vibration of nanobeams embedded in the linear and nonlinear elastic materials under magnetic and temperature effects is investigated in this study. Von Karman’s strain–displacement relation is applied to a nonlocal Euler–Bernoulli beam model. Equation of motion is derived using Hamilton’s principle. Galerkin’s method is applied to decompose the nonlinear partial differential equation into a nonlinear ordinary differential equation (NODE). The NODE is solved using He’s method. The nanobeams are embedded in the Winkler, Pasternak, and nonlinear elastic media. The effects of low and high temperatures, nonlocal parameter, magnetic force, amplitude, and linear and nonlinear elastic materials are examined.

Fluid structure interaction of cantilever micro and nanotubes conveying magnetic fluid with small size effects under a transverse magnetic field

Micro and nanotubes have found major application in fluidic systems as channels for conveying fluid. In some micro and nanofluidic applications such as drug delivery, a transverse magnetic field can be used to guide the fluid flow by generating an axial force in the flow direction. An important issue in the design of micro and nanofluidic systems is the structural vibration caused by the fluid flow. In the current study, we investigate the effect of transverse magnetic field on the vibration of cantilever micro and nanotubes conveying fluid by considering the small size effects. We couple the nonlocal Euler–Bernoulli beam model with Navier–Stokes theory to determine a fluid structure interaction (FSI) model for the vibration analysis of the system. We modify the FSI governing equation by driving a velocity correction factor to consider the effect of transverse magnetic field on the fluid flow’s pattern through the tube. Then, we use the Galerkin’s method to obtain the frequency diagrams for the instability analysis of the system. We show that the transverse magnetic field can have a substantial effect on the dynamics of tube conveying fluid by increasing the system’s natural frequencies and critical flow velocity which contributes to the flutter instability. We also discover that although the transverse magnetic field plays a crucial role on dynamics of microstructures, its effect on the dynamics of nanotubes is not significant and can be ignored.

Exact analysis of antibody-coated silicon biological nano-sensors (SBNSs) to identify viruses and bacteria

In this paper, the vibration analysis of a Silicon Biological Nano-sensor (SBNS) with full coverage of Myosin as biologically adsorbent layer is investigated based on modified nonlocal Euler–Bernoulli beam model. This SBNS works based on calculating the shift of resonant frequency in the presence of Myosin layer and adsorbed viruses and bacteria. For this end, the effects of surface stresses, nonlocal parameter, and rotary inertia as well as the mass and stiffness of the adsorbent layer are taken into account, which can play a major role in changing the resonant frequency and the precision of SBNSs at nano-scale. The results illustrate that the effects of adsorbent layer, surface stresses, nonlocal parameter and rotary inertia may reduce resonant frequency of SBNS, which is significant especially at nano-scale. Finally, for the purpose of verification assessment, the numerical results were compared with the results of other studies and showed complete agreement. The present study can provide helpful insights for the design and characterization of accurate biological Nano-sensors.

Chaotic vibrations of a harmonically excited carbon nanotube with consideration of thermomagnetic filed and surface effects

In the studies of the dynamic response of carbon nanotubes, the stability, predictable, and unpredictable chaotic vibrations are fundamental characteristics. In this paper, we investigate the chaotic and periodic vibrations of a single-walled carbon nanotube resting on the viscoelastic foundation, based on the nonlocal Euler–Bernoulli beam model. It is assumed that the single-walled carbon nanotube is subjected to an external harmonic excitation. The axial thermomagnetic field and the surface effect on the governing equation of single-walled carbon nanotube are taken into account. We also solve the nonlinear governing equation by using the Galerkin decomposition method along with the fourth-order Rung–Kutta numerical integration scheme. Furthermore, we analyze the effects of amplitude and frequency of excitation on the formation of chaotic and periodic regions using bifurcation diagrams and largest Lyapunov exponents. Moreover, we present the phase portrait, Poincare maps, and time history to observe the periodic and chaotic responses of the system. The results show that the nonlinear dynamic response of single-walled carbon nanotube is much more sensitive to both amplitude and frequency of excitation.

Dynamic stability of harmonically excited nanobeams including axial inertia

This article is concerned with the dynamic stability problem of a nanobeam under a time-varying axial loading. The nonlocal Euler–Bernoulli beam model has been used for the continuum modeling of the nanobeam structure. This problem leads to a time-dependent Mathieu-Hill equation and has been solved by using the Lindstedt–Poincaré perturbation expansion method. The effect of a small-scale parameter on the dynamic displacement and critical dynamic buckling load of nanobeams has been investigated. Stability regions have been obtained from the local and nonlocal elasticity theories. The effect of the longitudinal vibration of nanobeams on instability regions has been included in the present analysis. Amplitudes of an arbitrary point of a nanobeam due to harmonic loads have been determined. Nonlocal and longitudinal vibration effects reduce the area of the instability region and increase amplitudes.

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 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.

Free vibration analysis of chiral double-walled carbon nanotube embedded in an elastic medium using non-local elasticity theory and Euler Bernoulli beam model

The transverse free vibration of chiral double-walled carbon nanotube (DWCNTs) embedded in elastic medium is modeled by the non-local elasticity theory and Euler Bernoulli beam model. The governing equations are derived and the solutions of frequency are obtained. According to this study, the vibrational mode number, the small-scale coefficient, the Winkler parameter and chirality of double-walled carbon nanotube on the frequency ratio (xN) of the (DWCNTs) are studied and discussed. The new features of the vibration behavior of (DWCNTs) embedded in an elastic medium and the present solutions can be used for the static and dynamic analyses of double-walled carbon nanotubes.

Analysis of boundary conditions effects on vibration of nanobeam in a polymeric matrix

In this study, we investigate the vibration of single-walled carbon nanotubes embedded in a polymeric matrix using nonlocal elasticity theories with account arbitrary boundary conditions effects. A Winkler type elastic foundation is employed to model the interaction of nanobeam and the surrounding elastic medium. Influence of all parameters such as nonlocal small-scale effects, Winkler modulus parameter, vibration mode and aspect ratio of nanobeam on the vibration frequency are analyzed and discussed. The mechanical properties of carbon nanotubes and polymer matrix are treated and an analytical solution is derived using the governing equations of the nonlocal Euler-Bernoulli beam models. Solutions have been compared with those obtained in the literature and The results obtained show that the non-dimensional natural frequency is significantly affected by the small-scale coefficient, the vibrational mode number and the elastic medium.

Application of Homotopy Analysis Method for the Pull-In and Nonlinear Vibration Analysis of Nanobeams Using a Nonlocal Euler–Bernoulli Beam Model

Abstract In this investigation, the homotopy analysis method (HAM) is utilized for the pull-in and nonlinear vibration analysis of nanobeams based on the stress-driven model (SDM) of nonlocal elasticity theory. The physical properties of nanobeams are assumed not to vary through the thickness. The nonlinear equation of motion and the corresponding boundary condition are derived on the basis of the Euler–Bernoulli beam theory. For the solution purpose, the Galerkin method is employed for reducing the nonlinear partial differential equation to a nonlinear ordinary differential equation in the time domain, and then, the resulting equation is analytically solved using the HAM. In the results section, the influences of different parameters, including nonlocal parameter, electrostatic and intermolecular van der Waals forces and fringing field effect changes on the pull-in and nonlinear vibration response are investigated.

Quantum effects on thermal vibration of single-walled carbon nanotubes conveying fluid

In this paper, we investigate the microfluid induced vibration of a nanotube in thermal environment. Attention is focused on a special case that the law of energy equipartition is unreliable unless the quantum effect is taken into account. A nonlocal Euler–Bernoulli beam model is used to model the transverse vibration of a single-walled nanotube (SWCNT). Results reveal that the root of mean squared (RMS) amplitude of thermal vibration of the fluid-conveying SWCNT predicted from the quantum theory is lower than that predicted from the law of energy equipartition. The quantum effect on the thermal vibration of the fluid-conveying SWCNT is more significant for the cases of higher-order modes, lower flow velocity, lower temperature, and lower fluid density.

Axial dynamic buckling analysis of embedded single-walled carbon nanotube by complex structure-preserving method

• Load on CNT depends on spatial coordinate and isn't perpendicular to the CNT axis. • A complex structure-preserving method is proposed for oscillation model of CNT. • Axial dynamic buckling is more likely to occur with increase of axial load. • Axial dynamic buckling is more likely to occur with frequency of load close to MHz. The occurrence of the axial dynamic buckling in the carbon nanotube may cause the failure of some nano-sensors, which is difficult to be investigated by the experimental approach for the rigorous measuring accuracy requirement. Thus, a complex structure-preserving method is proposed to investigate the axial dynamic buckling properties of an embedded single-walled carbon nanotube in this paper. The nonlocal Euler–Bernoulli beam model of the embedded single-walled carbon nanotube under a concentrated excitation with a small included angle in the acting direction of the excitation against the cross-section of the carbon nanotube is presented. Introducing several intermediate variables, the multi-symplectic form of the oscillation model is obtained, which is a structure-preserving order-reduce process. To investigate the axial dynamic buckling of the carbon nanotube numerically with an acceptable step length, a complex structure-preserving method that combines the precise integration method for the grids near the acting position of the excitation and the Preissman multi-symplectic scheme for other grids is proposed. Considering two typical excitations, the oscillation of the carbon nanotube is simulated by the complex structure-preserving method and the axial dynamic buckling properties of the nanotube are investigated. From the numerical results, it can be concluded that the axial dynamic buckling phenomenon in the carbon nanotube is more likely to occur when the included angle increase or the frequency of the excitation is close to MHz with the given parameters, which gives guidance for the design of the excitation of the nano-oscillator.

Vibration analysis of viscoelastic single-walled carbon nanotubes resting on a viscoelastic foundation

Vibration responses were investigated for a viscoelastic Single-walled carbon nanotube (visco-SWCNT) resting on a viscoelastic foundation. Based on the nonlocal Euler-Bernoulli beam model, velocity-dependent external damping and Kelvin viscoelastic foundation model, the governing equations were derived. The Transfer function method (TFM) was then used to compute the natural frequencies for general boundary conditions and foundations. In particular, the exact analytical expressions of both complex natural frequencies and critical viscoelastic parameters were obtained for the Kelvin-Voigt visco-SWCNTs with full foundations and certain boundary conditions, and several physically intuitive special cases were discussed. Substantial nonlocal effects, the influence of geometric and physical parameters of the SWCNT and the viscoelastic foundation were observed for the natural frequencies of the supported SWCNTs. The study demonstrates the efficiency and robustness of the developed model for the vibration of the visco-SWCNT-viscoelastic foundation coupling system.

Static behavior of nonlocal Euler-Bernoulli beam model embedded in an elastic medium using mixed finite element formulation

The size-dependent behavior of single walled carbon nanotubes (SWCNT) embedded in the elastic medium and subjected to the initial axial force is investigated using the mixed finite element method. The SWCNT is assumed to be Euler- Bernoulli beam incorporating nonlocal theory developed by Eringen. The mixed finite element model shows its great advantage of dealing with nonlocal behavior of SWCNT subjected to a concentrated load owing to the existence of two coefficients alpha1 and alpha2. This is the first numerical approach to deal with a puzzling fact of nonlocal theory with concentrated load. Numerical examples are performed to show the accuracy and efficiency of the present method. In addition, parametric study is carefully carried out to point out the influences of nonlocal effect, the elastic medium, and the initial axial force on the behavior of the carbon nanotubes.