Nonlocal Beam Theory Research Articles

Abnormal nonlocal scale effect on static bending of single-layer MoS2

The nonlocal scale parameter of nonlocal Euler–Bernoulli beam theory is evaluated for the static bending of single-layer molybdenum disulfide (SLMoS2) without predetermined bending rigidity. The evaluation is performed by matching the fitted curve between the maximum deflection and the beam length obtained from molecular mechanics simulations. It was observed that the fitted curves have an abnormal sign in the second-order term of the maximum deflection for SLMoS2, opposite to that for graphene and regardless of the interatomic interaction potentials used. Based on the nature of ‘nonlocal’ and the phenomenological point of view, a modified nonlocal constitutive relation with a positive sign in front of the higher-order term is suggested for SLMoS2. The nonlocal parameter and the bending rigidity of SLMoS2 are finally extracted, and the effect of the nonlocal scale parameter on the bending response for SLMoS2 is found to be significant for beam length less than a critical length, depending on both the interatomic interaction potentials and the boundary conditions. Our new perspective should be useful for researchers who are interested in the engineering application of graphene-like quasi-two-dimensional nanostructures using nonlocal beam theories.

Dynamic characteristics of curved nanobeams using nonlocal higher-order curved beam theory

Here, an analytical approach for the dynamic analysis, viz., free and forced vibrations, of curved nanobeams using nonlocal elasticity beam theory based on Eringen formulation coupled with a higher-order shear deformation accounting for through thickness stretching is investigated. The formulation is general in the sense that it can be deduced to analyse the effect of various structural theories pertaining to curved nanobeams. It also includes inplane, rotary and coupling inertia terms. The governing equations derived, using Hamiltons principle, are solved in conjunction with Naviers solutions. The free vibration results are obtained employing the standard eigenvalue analysis whereas the displacement/stress responses in time domain for the curved nanobeams subjected to rectangular pulse loading are evaluated based on Newmarks time integration scheme. The formulation is validated considering problems for which solutions are available. A comparative study is done here by different theories obtained through the formulation. The effects of various structural parameters such as thickness ratio, beam length, rise of the curved beam, loading pulse duration, and nonlocal scale parameter are brought out on the dynamic behaviours of curved nanobeams.

Buckling temperature of a single-walled boron nitride nanotubes using a novel nonlocal beam model

In this paper, the critical buckling temperature of single-walled Boron Nitride nanotube (SWBNNT) is estimated using a new nonlocal first-order shear deformation beam theory. The present model is capable of capturing both small scale effect and transverse shear deformation effects of SWBNNT and is based on assumption that the inplane and transverse displacements consist of bending and shear components, in which the bending components do not contribute toward shear forces and, likewise, the shear components do not contribute toward bending moments. Results indicate the importance of the small scale effects in the thermal buckling analysis of Boron Nitride nanotube.

Interlayer influences between double-layer graphene nanoribbons (shear and tensile-compressive) on free vibration using nonlocal elasticity theory

ABSTRACTThe shear and tensile-compressive effects of van der Waals (vdWs) bindings on the nonlocal vibrational behavior of bilayer graphene nanoribbons (BLGNRs) are simultaneously investigated in the present study. To idealize the structure of BLGNRs incorporating interlayer shear and tensile-compressive influences, a nonlocal sandwich beam (NSB) theory is employed to model the nanoribbon layers as faces and vdWs interactions as core of the NSB. The effects of interlayer moduli are investigated on the first four nonlocal natural frequencies of BLGNRs for various nonlocal parameters, considering that the small-scale effect causes mode shapes involving tensile-compressive effect of vdWs interactions be excited later.

Surface effects on vibration analysis of elastically restrained piezoelectric nanobeams subjected to magneto-thermo-electrical field embedded in elastic medium

In the present study, a generalized nonlocal beam theory is utilized to study the magneto–thermo–mechanical vibration characteristic of piezoelectric nanobeam by considering surface effects rested in elastic medium for various elastic boundary conditions. The nonlocal elasticity of Eringen as well as surface effects, including surface elasticity, surface stress and surface density are implemented to inject size-dependent effects into equations. Using the Hamilton’s principle and Euler–Bernoulli beam theory, the governing differential equations and associated boundary conditions will be obtained. The differential transformation method (DTM) is used to discretize resultant motion equations and related boundary conditions accordingly. The natural frequencies are obtained for the various elastic boundary conditions in detail to show the significance of nonlocal parameter, external voltage, temperature change, surface effects, elastic medium, magnetic field and length of nanobeam. Moreover, it should be noted that by changing the spring stiffness at each end, the conventional boundary conditions will be obtained which are validated by well-known literature.

Assessment of various nonlocal higher order theories for the bending and buckling behavior of functionally graded nanobeams

In this paper, various nonlocal higher-order shear deformation beam theories that consider the size dependent effects in Functionally Graded Material (FGM) beam are examined. The presented theories fulfill the zero traction boundary conditions on the top and bottom surface of the beam and a shear correction factor is not required. Hamilton's principle is used to derive equation of motion as well as related boundary condition. The Navier solution is applied to solve the simply supported boundary conditions and exact formulas are proposed for the bending and static buckling. A parametric study is also included to investigate the effect of gradient index, length scale parameter and length-to-thickness ratio (aspect ratio) on the bending and the static buckling characteristics of FG nanobeams.

A new refined nonlocal beam theory accounting for effect of thickness stretching in nanoscale beams

In this paper, a simple and refined nonlocal hyperbolic higher-order beam theory is proposed for bending and vibration response of nanoscale beams. The present formulation incorporates the nonlocal scale parameter which can capture the small scale effect, and it considers both shear deformation and thickness stretching effects by a hyperbolic variation of all displacements across the thickness without employing shear correction factor. The highlight of this formulation is that, in addition to modeling the displacement field with only two unknowns, the thickness stretching effect (ez=/0) is also included in the present model. By utilizing the Hamilton\'s principle and the nonlocal differential constitutive relations of Eringen, the equations of motion of the nanoscale beam are reformulated. Verification studies demonstrate that the developed theory is not only more accurate than the refined nonlocal beam theory, but also comparable with the higher-order shear deformation theories which contain more number of unknowns. The theoretical formulation proposed herein may serve as a reference for nonlocal theories as applied to the static and dynamic responses of complex-nanobeam-system such as complex carbon nanotube system.

Buckling and vibration of Hencky bar-chain with internal elastic springs

This paper presents the Hencky bar-chain model (HBM) that allows for the presence of internal elastic springs. With the development of this general HBM, one can analyse beam-like structures with repetitive cells with allowance for internal damaged sections or bracing members. In this paper, we have derived formulas for the internal rotational and lateral spring stiffnesses of HBMs based on the finite difference model (FDM). In addition, the analytical buckling loads and vibration frequencies of the HBM and FDM with internal elastic springs are presented for the first time. These analytical solutions converge to the exact solutions for continuum beams when the HBM segmental number approaches infinity. The solutions can also be used to calibrate the Eringen's small length scale coefficient e0 in the nonlocal beam theory based on their phenomenological similarities.

Thermal Buckling Analysis of Size-Dependent FG Nanobeams Based on the Third-Order Shear Deformation Beam Theory

In this paper, the thermal effects on the buckling of functionally graded (FG) nanobeams subjected to various types of thermal loading including uniform, linear and non-linear temperature changes are investigated based on the nonlocal third-order shear deformation beam theory. The material properties of FG nanobeam are supposed to vary gradually along the thickness direction according to the power-law form. The governing equations are derived through Hamilton’s principle and solved analytically. Comparison examples are performed to verify the present results. Obtained results are presented for thermal buckling analysis of FG nanobeams such as the effects of the power-law index, nonlocal parameter, slenderness ratio and thermal loading in detail.

Nonlocal nonlinear first-order shear deformable beam model for postbuckling analysis of magneto-electro-thermo elastic nanobeams

In this study, the size-dependent postbuckling behavior of magneto-electro-thermo-elastic (METE) nanobeams with different edge supports is investigated. Based on the nonlocal first-order shear deformation beam theory and considering the von Karman hypothesis, a size-dependent nonlinear METE nanobeam model is developed, in which the effects of small scale parameter and thermo-electro-magnetic-mechanical loadings are incorporated. A numerical solution procedure based on the generalized differential quadrature (GDQ) and pseudo arc-length continuation methods is utilized to describe the size-dependent postbuckling behavior of METE nanobeams under various boundary conditions. The effects of different parameters such as nonlocal parameter, external electric voltage, external magnetic potential and temperature rise on the postbuckling path of METE nanobeams are explored. The results indicate that increasing the non-dimensional nonlocal parameter, imposed positive voltage, negative magnetic potential and temperature rise decrease the critical buckling load and post-buckling load-carrying capacity of METE nanobeams, whilean increase in the negative voltage, positive magnetic potential lead to a considerable increase of critical buckling load as well as postbuckling strength of the METE nanobeams.

On boundary conditions for buckling and vibration of nonlocal beams

In this paper, we argue that the boundary conditions for Eringen's nonlocal beam theory have to take on the discrete forms similar to those for the first order central finite difference beam model for beam buckling and vibration problems. The latter finite difference beam model has been found to be analogous to the physical Hencky bar-chain and therefore these two models can be regarded as one discrete beam model. Based on the phenomenological similarities between this discrete beam model and the Eringen's nonlocal beam theory, one may calibrate the Eringen's small length scale coefficient e0 which is expected to be a constant appropriate to each material. When using the classical continuous nonlocal boundary conditions, we find that e0 is not a constant but it depends on the cases of boundary conditions (such as pinned-pinned, clamped-clamped, clamped-pinned ends). However, this conundrum may be resolved if we use the discrete boundary conditions instead. By adopting the discrete boundary conditions, analytical solutions for e0 may be obtained and the e0 values finally converge to 0.289 for buckling problems and 0.408 for free vibration problems regardless of the cases of boundary conditions.

Evaluation of nonlocal higher order shear deformation models for the vibrational analysis of functionally graded nanostructures

ABSTRACTSmall scale effects in the functionally graded beam are investigated by using various nonlocal higher-order shear deformation beam theories. The material properties of a beam are supposed to vary according to power law distribution of the volume fraction of the constituents. The nonlocal equilibrium equations are obtained and an exact solution is presented for vibration analysis of functionally graded (FG) nanobeams. The accuracy of the present model is discussed by comparing the results with previous studies and a parametric investigation is presented to study the effects of power law index, small-scale parameter, and aspect ratio on the vibrational behavior of FG nanostructures.

A simplified shear and normal deformations nonlocal theory for bending of functionally graded piezomagnetic sandwich nanobeams in magneto-thermo-electric environment

A simplified three-unknown shear and normal deformations nonlocal beam theory for thermo-electro-magneto mechanical bending analysis of a nanobeam with a functionally graded material core and two functionally piezomagnetic layers is studied in this paper. The assumed structure is subjected to mechanical, thermal, electrical, and magnetic loads. An initial applied voltage and magnetic load is considered on the functionally graded piezomagnetic material layers. Eringen’s nonlocal constitutive equations are considered in the analysis. Governing equations are derived according to the present refined theory using the principle of virtual displacements. The numerical results including the deflection, electric, and magnetic potential distribution are calculated in terms of important parameters of the problem such as applied electric and magnetic potentials, two parameters of temperature distribution, and nonlocal parameter. The numerical results indicate that increase in applied electric potential increases the deflection unlike the applied magnetic potential that decreases the deflection. Furthermore, it can be concluded that increasing the nonlocal parameter leads to increase in the deflection.

Nonlocal modeling for dynamic stability of spinning nanotube under axial load

Spinning nanotubes are the main part of different high precision nano-machines such as nano sensors, nano motors, and nano pumps. Nanotubes experience various types of loads in such devices. In this paper, different type of instability for a spinning nanotube under tangential load is investigated. For this purpose, Euler and Timoshenko nonlocal beam theory is used. Equations of motion are obtained by applying Hamilton’s principle on the energy terms of nanotube by considering rotational velocity and compressive tangential load. Numerical results are exactly calculated for different parameters and are checked by comparing them with literature. Results show that there are two type of instability for simply supported spinning nanotubes as divergent and flutter, generally. Also, effects of different nonlocal parameter, geometry and rotational speed on these instabilities are investigated, comprehensively. It is approved that nonlocal parameter has significant effect on both of the instabilities.

Size-dependent mechanical behavior of functionally graded trigonometric shear deformable nanobeams including neutral surface position concept

A nonlocal trigonometric shear deformation beam theory based on neutral surface position is developed for bending, buckling, and vibration of functionally graded (FG) nanobeams using the nonlocal differential constitutive relations of Eringen. The present model is capable of capturing both small scale effect and transverse shear deformation effects of FG nanobeams, and does not require shear correction factors. The material properties of the FG nanobeam are assumed to vary in the thickness direction. The equations of motion are derived by employing Hamilton\'s principle, and the physical neutral surface concept. Analytical solutions are presented for a simply supported FG nanobeam, and the obtained results compare well with those predicted by the nonlocal Timoshenko beam theory.

Differential quadrature method for nonlocal nonlinear vibration analysis of a boron nitride nanotube using sinusoidal shear deformation theory

ABSTRACTThis article presents a nonlocal sinusoidal shear deformation beam theory (SDBT) for the nonlinear vibration of single-walled boron nitride nanotubes (SWBNNTs). The surrounding elastic medium is simulated based on nonlinear Pasternak foundation. Based on the nonlocal differential constitutive relations of Eringen, the equations of motion of the SWBNNTs are derived using Hamilton's principle. Differential quadrature method (DQM) for the nonlinear frequency is presented, and the obtained results are compared with those predicted by the nonlocal Timoshenko beam theory (TBT). The effects of nonlocal parameter, vibrational modes, length, and elastic medium on the nonlinear frequency of SWBNNTs are considered.

Nonlocal beam theory for nonlinear vibrations of a nanobeam resting on elastic foundation

In the present study, nonlinear vibrations of an Euler-Bernoulli nanobeam resting on an elastic foundation is studied using nonlocal elasticity theory. Hamilton’s principle is employed to derive the governing equations and boundary conditions. The nonlinear equation of motion is obtained by including stretching of the neutral axis that introduces cubic nonlinearity into the equations. Forcing and damping effects are included in the equations of the motion. The multiple scale method, a perturbation technique for deriving the approximate solutions of the equations, is applied to the nonlinear systems. Natural frequencies and mode shapes for the linear problem are found and also nonlinear frequencies are found for a nonlocal Euler-Bernoulli nanobeam resting on an elastic foundation. In the numerical calculation, frequency-response curves are drawn for various parameters like nonlocal parameters, elastic foundation, and boundary conditions. The effects of the different nonlocal parameters (γ) and elastic foundation parameters (κ) as well as the effects of different boundary conditions on the vibrations are discussed.

Circumferential nonlocal effect on the buckling and vibration of nanotubes

The nonlocal beam theories are widely used to study the mechanics of cylindrical nanotubes (NTs). The one-dimensional models however are unable to account for the nonlocal effect in the circumferential direction, which may substantially affect the applicability of the nonlocal beam models. To address the issue this letter examines the circumferential nonlocal effect (CNE) on the buckling and vibration of the NTs. Here the CNE is characterized by the difference between the nonlocal beam model considering the axial nonlocal effect only and the nonlocal shell model with both axial and circumferential nonlocal effects. The aspect ratio and radius-dependence of the CNE are calculated for the singlewall carbon NTs selected as a typical example. The results show that the CNE is substantial for the buckling and vibration of the NTs with small radius (e.g., <1 nm) and aspect ratio (e.g., <15). It however decreases with the rising radius and the aspect ratio, and turns out to be small for relatively wide and long NTs. The nonlocal beam theories thus may overestimate the buckling load and vibration frequency for the thin and short NTs.

The effect of calibrated nonlocal constant on the modal parameters and stability of axially compressed CNTs

Nowadays investigating the vibration behavior of carbon nanotubes (CNTs) has drawn considerable attention due to the superior mechanical properties of the CNTs. One of the powerful theoretical methods to study the vibration behavior of CNTs is implementing the nonlocal theory. Most of studies on the vibration behavior of CNTs have assumed a fixed value for small scale parameter for all vibration modes, however, this value is mode-dependent. Therefore, in this paper, the small scale parameter is calibrated for a single-walled carbon nanotube (SWCNT) with respect to each vibration mode. For this propose, the governing equation of motion based on the nonlocal beam theory is extracted by applying the Hamilton's principle. Then, by using the power series method, an eigenvalue problem is defined to derive the calibrated value of small scale constant and nonlocal mode shapes of the CNT. By using the expansion theory, the equation of motion is discretized, and the effect of nonlocality on the modal parameters and stability of the CNT under compressive force is investigated. Finally, the possibility of estimating nonlocal parameter based on simulated frequency domain response of the system by using modal analysis methods is studied. The results show that the calibration of small scale constant is important and the critical axial force is highly sensitive to this value.

Investigation of Vibration and Thermal Buckling of Nanobeams Embedded in An Elastic Medium Under Various Boundary Conditions

AbstractAnalyses of free vibration and thermal buckling of nanobeams using nonlocal shear deformation beam theories under various boundary conditions are precisely illustrated. The present beam is restricted by vertically distributed identical springs at the top and bottom surfaces of the beam. The equations of motion are derived using the dynamic version of Hamilton's principle. The governing equations are solved analytically when the edges of the beam are simply supported, clamped or free. Thermal buckling solution is formulated for two types of temperature change through the thickness of the beam: Uniform and linear temperature rise. To validate the accuracy of the results of the present analysis, the results are compared, as possible, with solutions found in the literature. Furthermore, the influences of nonlocal coefficient, stiffness of Winkler springs and span-to-thickness ratio on the frequencies and thermal buckling of the embedded nanobeams are examined.