Nonlocal Euler-Bernoulli Beam Research Articles

Chebyshev–Ritz and Navier's methods for hygro‐magneto vibration of Euler–Bernoulli nanobeam resting on Winkler–Pasternak elastic foundation

AbstractThis research employs Chebyshev–Ritz method along with Navier's method to investigate the vibration characteristics of a nanobeam subject to a longitudinal magnetic field and linear hygroscopic environment. The nanobeam is characterized by a Winkler–Pasternak elastic foundation and follows the nonlocal Euler–Bernoulli beam theory. The governing equation of motion is derived using Hamilton's principle, and non‐dimensional frequency parameters are computed for Simply Supported‐Simply Supported (SS), Clamped‐Clamped (CC), and Clamped‐Free (CF) boundary conditions. The motivation behind this study is to provide a comprehensive and efficient analytical framework for understanding the dynamic behavior of nanobeams in complex environments. By investigating the influence of magnetic and hygroscopic factors on the vibration characteristics of nanobeams, this research aims to offer valuable insights for the design and optimization of nanoscale structures. Employing shifted Chebyshev polynomials as shape functions in Chebyshev–Ritz method offers several advantages in the proposed model. Firstly, these polynomials possess orthogonal properties, which can significantly enhance computational efficiency. The orthogonality of shifted Chebyshev polynomials allow for simpler and more streamlined numerical computations compared to non‐orthogonal basis functions. Additionally, the orthogonality ensures that the resulting system of equations is well‐conditioned, even for higher‐order polynomial approximations. A closed‐form solution for SS boundary condition is obtained through Navier's method. Convergence analysis is performed to validate the accuracy and effectiveness of the proposed model against existing models. The non‐dimensional frequency parameters obtained using both Navier's method and Chebyshev–Ritz method demonstrate strong agreement, further validating the proposed nanobeam model. Additionally, a comprehensive parametric study evaluates the impact of various characteristics, including the small‐scale parameter, Winkler modulus, shear modulus, magnetic parameter, and hygroscopic parameter. The findings contribute to a nuanced understanding of nanobeam vibrations under the influence of a magnetic field and hygroscopic environment, providing valuable insights for the design and optimization of nanoscale structures in practical applications.

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.

Nonlinear forced vibration of a nanobeam resting on Winkler-Pasternak elastic foundation and subjected to a mechanical impact

The nonlinear governing equations of nanobeam taking into consideration its curvature, resting on an elastic Winkler-Pasternak foundation and based on non-local Euler-Bernoulli beam theory is analyzed. The equation of motion and the boundary conditions are modeled within the framework of a simple supported nanobeam which accounts the presence of a mechanical impact and nonlinear von-Karman strain. The resulting nonlinear differential equations are reduced to only one differential equation which is studied by means of the Optimal Auxiliary Functions Method (OAFM). An explicit analytical solution is proposed for a complex problem. The main quality of our technique consists in the existence of some auxiliary functions derived from the expressions of the solution for the initial linear equation and the form of nonlinear term calculated from the above solution of the linear equation. The convergence-control parameters present in the auxiliary functions are evaluated by a rigorous mathematical procedure. The obtained solutions are in very good agreement with the numerical solution.

Nonlinear phenomena in vibrations of embedded carbon nanotubes conveying viscous fluid

Various nonlinear phenomena such as bifurcations and chaos in the responses of carbon nanotubes (CNTs) are recognized as being major contributors to the inaccuracy and instability of nanoscale mechanical systems. Therefore, the main purpose of this paper is to predict the nonlinear dynamic behavior of a CNT conveying viscous fluid and supported on a nonlinear elastic foundation. The proposed model is based on nonlocal Euler–Bernoulli beam theory. The Galerkin method and perturbation analysis are used to discretize the partial differential equation of motion and obtain the frequency-response equation, respectively. A detailed parametric study is reported into how the nonlocal parameter, foundation coefficients, fluid viscosity, and amplitude and frequency of the external force influence the nonlinear dynamics of the system. Subharmonic, quasi-periodic, and chaotic behaviors and hardening nonlinearity are revealed by means of the vibration time histories, frequency-response curves, bifurcation diagrams, phase portraits, power spectra, and Poincaré maps. Also, the results show that it is possible to eliminate irregular motion in the whole range of external force amplitude by selecting appropriate parameters.

Nonlinear nonlocal damped free and forced vibrations of piezoelectric SWCNTs under longitudinal magnetic field due to surface effects using a two steps perturbation method

In the present work, damped free and forced vibrations of single-walled piezoelectric carbon nanotubes under longitudinal magnetic field due to surface effects surrounded on a non-linear viscoelastic medium using the nonlocal Euler-Bernoulli beam theory and multiple time scales method are investigated. Lorentz force equation is used to obtain the vertical force due to the applied voltage to the system. The surface effects as well as a combinational non-linear viscoelastic foundation are considered, and finally, the dynamic equilibrium equations are used, and non-linear equations of motion are extracted. In the following, the Galerkin and multiple time scales methods are used, and finally, analytical solutions are extracted as the non-linear free and forced vibrational responses of the system. The relevant coefficients of the extracted analytical solutions are discovered for two both simple support (S-S) and clamped (C-C) boundary conditions. In the following, , and the effects of the different parameters such as non-local parameter as well as electric-magnetic fields, effect of hardness-linear damping parameters of nonlinear considered viscoelastic foundation, applied magnetic field, base modes for different forms considering surface effects, and etc. will be studied. As one the results of this study, the presence of a non-local parameter has increased the curvature deviation to the right and the stiffening effect. In other words, the non-local parameter is a factor to increase the nonlinear effect of the system. Also, it is predictable that as the load affect position moves away from the center of the single-walled piezoelectric carbon nanotube toward the supports, the amplitude of the dynamic response decreases significantly, and this relative reduction is greater for the C-C boundary condition than for the S-S boundary condition. It is also important to note that the location of the load has no effect on the rate of deviation of the curve peak, and the degree of nonlinearity of the vibrational response of the system.

Investigation of fluid viscosity and density on vibration of nano beam submerged in fluid considering nonlocal elasticity theory

Predicting the dynamical behavior of nanostructures in contact with fluid is of paramount importance owing to their vast range of applications in civil engineering, aerospace, etc. Therefore, the free vibration of a nonlocal Euler-Bernoulli beam in contact with fluid is studied in this research. Using the Navier-Stokes relation, the interaction forces between the fluid and nanobeam are obtained. The equation of motion is solved using the Galerkin method of weighted residuals, and the natural frequency values in different fluids are presented, considering the viscosity effect. Further, the fluid-solid interaction model on a macroscopic scale is compared with the finite element analysis results. By considering the fluid parameters to be zero, the results of the nonlocal beam have been validated with past research. According to the obtained results, when studying the vibration of nanobeams, while density has a remarkable impact on the reduction of nanobeam natural frequencies, the fluid viscosity can be neglected. The fluid has the greatest impact on lower modes, whereas the nonlocal parameter exerts the biggest influence on higher modes. Moreover, in all vibration modes, with the increase of the nonlocal parameter, the effect of the fluid on the reduction of the natural frequencies increases, and in the nanoscale, it has a more significant role in changing the vibrational behavior of the system compared to the classical condition.

Correction to: Nonlocal Euler–Bernoulli beam theories with material nonlinearity and their application to single-walled carbon nanotubes

Correction to: Nonlocal Euler–Bernoulli beam theories with material nonlinearity and their application to single-walled carbon nanotubes

Nonlocal Euler–Bernoulli beam theories with material nonlinearity and their application to single-walled carbon nanotubes

Although the small-scale effect and the material nonlinearity significantly impact the mechanical properties of nanobeams, their combined effects have not attracted the interest of researchers. The present paper proposes two new nonlinear nonlocal Euler–Bernoulli theories to model mechanical properties corresponding to extensible or inextensible nanobeams. Two new theories consider the material nonlinearity and the small-scale effect induced by the nonlocal effect. The new models are used to analyze the static bending and the forced vibrations for single-walled carbon nanotubes (SWCNTs). The results indicate that the material nonlinearity and the nonlocal effect significantly impact SWCNT’s mechanical properties. Therefore, neglecting the two factors may cause qualitative mistakes.

On the Exact Solution of Nonlocal Euler–Bernoulli Beam Equations via a Direct Approach for Volterra-Fredholm Integro-Differential Equations

First, we develop a direct operator method for solving boundary value problems for a class of nth order linear Volterra–Fredholm integro-differential equations of convolution type. The proposed technique is based on the assumption that the Volterra integro-differential operator is bijective and its inverse is known in closed form. Existence and uniqueness criteria are established and the exact solution is derived. We then apply this method to construct the closed form solution of the fourth order equilibrium equations for the bending of Euler–Bernoulli beams in the context of Eringen’s nonlocal theory of elasticity (two phase integral model) under a transverse distributed load and simply supported boundary conditions. An easy to use algorithm for obtaining the exact solution in a symbolic algebra system is also given.

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.

A novel numerical approach for the stability of nanobeam exposed to hygro‐thermo‐magnetic environment embedded in elastic foundation

AbstractIn this paper, a novel numerical technique, namely, shifted Chebyshev polynomials based Rayleigh‐Ritz method has been employed to analyze the buckling characteristics of the nanobeam. The main advantage of the shifted Chebyshev polynomials is that, due to the orthogonality of the polynomials, ill‐conditioning of the system is being avoided with higher number of terms in the approximation. The nanobeam is exposed to both hygroscopic and thermal environments while being subjected to a longitudinal magnetic field. Further, the beam is modeled with Winkler‐Pasternak elastic foundation and nonlocal Euler–Bernoulli beam theory. The governing equation of motion of the proposed model has been derived using the Hamilton's principle, and critical buckling loads for Hinged‐Hinged (HH), Clamped‐Hinged (CH), and Clamped‐Clamped (CC) boundary conditions have been computed. The proposed model is validated against the existing model in special cases, exhibiting excellent agreement. Then a convergence study is performed to ensure the correctness and effectiveness of the method. Furthermore, a comprehensive parametric study has been carried out to determine the impact of various parameters such as the small scale parameter, Winkler modulus, shear modulus, magnetic field intensity parameter, hygroscopic parameter, and temperature parameter.

Lateral vibration of an axially moving thermoelastic nanobeam subjected to an external transverse excitation

<abstract> <p>This paper gives a mathematical formulation for the transverse resonance of thermoelastic nanobeams that are simply supported and compressed with an initial axial force. The nonlocal elasticity concept is used to analyze the influence of length scale with the dual-phase-lag (DPL) heat transfer theory. The nanobeam is due to a changing thermal load and moves in one direction at a constant speed. The governing motion equation for the nonlocal Euler-Bernoulli (EB) beam hypothesis can also be derived with the help of Hamilton's principle and then solved by means of the Laplace transform technique. The impacts of nonlocal nanoscale and axial velocity on the different responses of the moving beam are investigated. The results reveal that phase delays, as well as the nonlocal parameter and external excitation load, have a substantial impact on the system's behavior.</p> </abstract>

Chaotic vibrations of carbon nanotubes subjected to a traversing force considering nonlocal elasticity theory

The existence of chaos in the lateral vibration of the carbon nanotube (CNT) can contribute to source of instability and inaccuracy within the nano mechanical systems. So, chaotic vibrations of a simply supported CNT which is subjected to a traversing harmonic force are studied in this paper. The model of the system is formulated by using nonlocal Euler–Bernoulli beam theory. The equation of motion is solved using the Rung–Kutta method. The effects of the nonlocal parameter, velocity and amplitude of the traversing harmonic force on the nonlinear dynamic response of the system are analyzed by the bifurcation diagrams, phase plane portrait, power spectra analysis, Poincaré map and the maximum Lyapunov exponent. The results indicate that the nonlocal parameter, velocity and amplitude of the traversing harmonic force have considerable effects on the bifurcation behavior and can be used as effective control parameters for avoiding chaos.

Computational analysis of high precision nano-sensors for diagnosis of viruses: Effects of partial antibody layer

By using these high precision nano-bio-sensors, the slightest mechanical changes are also recognizable, as its detection limit is one hundred times more than conventional methods such as electrochemical sensing. In this research, a silicon nano-bio-sensor is modeled based on a novel modified nonlocal Euler–Bernoulli beam theory. On this basis, shift of resonant frequencies are determined due to adsorption of fine biological particles on the antibody layer. In this regard, the effects of stiffness and mass of partial antibody layer in addition to surface tensions, nonlocal parameter and rotary inertia are investigated, simultaneously. Consequently, the precise resonant frequency of a cantilever biological nano-sensor is determined for diagnosis of viruses. Totally, the results show that resonant frequency of nano-sensor is reduced by taking these effects into account, which should not be ignored at nano-scale. Finally, for the validation of numerical results, the presented results are compared with existing literatures and show complete agreement. This research can create proper insights into the analysis and design of precise resonator bio-nano-sensors based on computational methods.

Thermal effect on forced vibration analysis of FG nanobeam subjected to moving load by Laplace transform method

In this study, dynamic response of functionally graded (FG) nanobeam under a moving load in a thermal environment has been investigated. By considering the size effect, nonlocal Euler–Bernoulli beam theory has been used to model a FG nanobeam. By applying Hamilton’s principle, governing equations of motion and boundary conditions, have been derived. Analytical solution by using Laplace transform method has been achieved. Presented results have been verified by comparing with the results found in the literature, which this comparison shows good agreement between them. Influences of the nonlocal parameter, material distribution index, velocity of the moving load, and temperature changes on the response of the FG nanobeam have been investigated. The results show that increasing the nonlocal parameter and temperature changes leads to raising the dynamic deflection of the FG nanobeam.

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.

Nonlocal viscoelastic Euler-Bernoulli beam model: a Bayesian approach for parameter estimation using the delayed rejection adaptive metropolis algorithm

The present work presents a model for nonlocal and viscoelastic Euler-Bernoulli beams and aspects of its calibration are addressed. The nonlocal feature of the model is described by the nonlocal elasticity theory proposed by Eringen and its viscoelastic behaviour is modelled by means of internal variables. Parametric analyses are performed to determine the impact of the nonlocal and viscoelastic parameters on the modal properties of the system. Inverse analyses are performed under the Bayesian framework and samples of the posterior density function are obtained by means of the Delayed Rejection Adaptive Metropolis (DRAM). The inverse analyses consider a nonlocal viscoelastic beam model with one internal variable and they address three aspects, namely: the impact of a misspecification of the beam diameter, the impact of modelling the beam diameter as an unknown but uninteresting model parameter and the model calibration when synthetic experimental data comes from a model containing two internal variables. The model parameters were chosen such that the system resembles a Single-Walled Carbon Nanotube (SWCNT).

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.

Nonlinear random vibrations of functionally graded porous nanobeams using equivalent linearization method

In the present study, the nonlinear random vibration of a porous functionally graded nanobeam supported by a viscoelastic foundation has been investigated by making use of the statistical linearization method for the first time. Nonlocal Euler-Bernoulli beam theory with von Kármán type nonlinearity is utilized to derive the governing equation of the motion of the FG porous nanobeam. Two types of porosity distribution have been considered for the FG beam. To discretize the governing equation Galerkin's method is applied. The mean square value of the response of the resulted Duffing's equation in the presence of a random input with nonzero mean value has been extracted using statistical linearization method. The results of the present study are compared with the analytical results of the FPK approach and the results of the perturbation method for the classical limit case of the homogeneous beam. It is revealed through detailed numerical analysis that the statistical linearization method (in contrast to the perturbation method) gives reliable results and there is an excellent agreement between its results and the analytical results. The effect of various parameters on the mean value of the response is also investigated numerically and illustrated graphically. It is found that the mean square value of vibrations increases when porosity, power-law index, nonlocal parameter, or the mean value of the input increases.