Axial buckling analysis of single-walled carbon nanotubes in thermal environments via the Rayleigh–Ritz technique

Axial buckling characteristics of single-walled carbon nanotubes (SWCNTs) including thermal environment effect are studied in this paper. Eringen’s nonlocal elasticity equations are incorporated into the classical Donnell shell theory to establish a nonlocal elastic shell model which takes small-scale effects into account. The Rayleigh–Ritz technique is implemented in conjunction with the set of beam functions as modal displacement functions to consider the four commonly used boundary conditions namely as simply supported–simply supported, clamped–clamped, clamped–simply supported, and clamped-free in the buckling analysis. Selected numerical results are presented to demonstrate the influences of small scale effect, aspect ratio, thermal environment effects and boundary conditions in detail. It is found that the value of aspect ratio has different effects on the critical axial buckling loads of SWCNTs in low and high temperature environments. Also, it is observed that the difference between the thermal axial buckling responses of SWCNTs relevant to various boundary conditions is more prominent for higher values of nonlocal elasticity constant.     

Nonlocal anisotropic elastic shell model for vibrations of single-walled carbon nanotubes with arbitrary chirality

A nonlocal anisotropic elastic shell model is developed to study the effect of small scale on shell-like vibration of single-walled carbon nanotubes (SWCNTs) with arbitrary chirality. Anisotropic elastic shell model is reformulated using the nonlocal differential constitutive relations of Eringen. The equations of motion are derived and analytical solution for the vibration of anisotropic SWCNTs is presented by using the Flügge shell theory and complex method. The suggested model is justified by a good agreement between the results given by the present model and available data in literature. Furthermore, the model is used to elucidate the effect of small scale on the vibration of zigzag, armchair and chiral SWCNTs. Our results show that small scale is essential for vibration of SWCNTs when the axial wave-length is not extremely long. Moreover, the results show that local model substantially overestimates vibrational frequencies of almost all modes.     

Thermo-mechanical vibration of a single-walled carbon nanotube embedded in an elastic medium based on nonlocal elasticity theory

A single-elastic beam model has been developed to analyze the thermal vibration of single-walled carbon nanotubes (SWCNT) based on thermal elasticity mechanics, and nonlocal elasticity theory. The nonlocal elasticity takes into account the effect of small size into the formulation. Further, the SWCNT is assumed to be embedded in an elastic medium. A Winkler-type elastic foundation is employed to model the interaction of the SWCNT and the surrounding elastic medium. Differential quadrature method is being utilized and numerical solutions for thermal-vibration response of SWCNT is obtained. Influence of nonlocal small scale effects, temperature change, Winkler constant and vibration modes of the CNT on the frequency are investigated. The present study shows that for low temperature changes, the difference between local frequency and nonlocal frequency is comparatively high. With embedded CNT, for soft elastic medium and larger scale coefficients (e₀a) the nonlocal frequencies are comparatively lower. The nonlocal model-frequencies are always found smaller than the local model-frequencies at all temperature changes considered.     

Dynamic stability analysis of multi-walled carbon nanotubes with arbitrary boundary conditions based on the nonlocal elasticity theory

Dynamic stability of embedded multi-walled carbon nanotubes (MWCNTs) in an elastic medium and thermal environment and subjected to an axial compressive force is studied based on the nonlocal elasticity and Timoshenko beam theory. The developed nonlocal beam model has the capability to consider the small scale effects. The generalized differential quadrature method is employed to discretize the dynamic governing differential equations of MWCNTs with various end supports. A parametric study is conducted to investigate the influences of static load factor, temperature change, nonlocal parameter, slenderness ratio, and spring constant of elastic medium on the dynamic stability characteristics of MWCNTs.     

Studying nonlinear thermomechanical wave propagation in a viscoelastic layer based upon the Lord-Shulman theory

Abstract

Based on the Lord–Shulman (L-S) theory, the thermally nonlinear coupled thermo-viscoelasticity of a layer is analyzed using a numerical approach. Using the Kelvin-Voigt theory of viscoelasticity in conjunction with the L-S theory, the coupled governing equations are first derived in variational form. Then, the variational differential quadrature (VDQ) method is applied to solve the problem numerically. Parametric studies are presented in order to reveal the effects of viscoelastic parameter, intensity of heat flux and relaxation time on the propagation of displacement, temperature and stress waves. Furthermore, the influence of considering nonlinearity in the energy equation is analyzed.     

基于非局部应变梯度欧拉梁模型的充流单壁碳纳米管波动分析

将非局部弹性理论和应变梯度理论结合,再根据流体滑移边界理论,建立了考虑流体和固体小尺度效应的充流单壁碳纳米管(SWCNT)流固耦合动力学模型,分别以非局部应力效应、应变梯度效应和流体滑移边界效应模拟微观小尺度效应对系统的影响,推导得出充流单壁碳纳米管的Euler-Bernoulli梁波动控制方程。通过对控制方程的求解,分析材料不同类型尺度效应对充流碳纳米管的振动和波动特性影响。结果显示,应变梯度效应和流体边界效应对低频波动起促进作用,对高频波动起阻尼作用,应力非局部效应则对波动始终产生阻尼作用。三种尺度效应对低流速系统的振动有促进作用,而对高流速系统产生阻尼作用。     

Nonlocal continuum model and molecular dynamics for free vibration of single-walled carbon nanotubes

Free transverse, longitudinal and torsional vibrations of single-walled carbon nanotubes (SWCNTs) are investigated through nonlocal beam model, nonlocal rod model and verified by molecular dynamics (MD) simulations. The nonlocal Timoshenko beam model offers a better prediction of the fundamental frequencies of shorter SWCNTs, such as a (5, 5) SWCNT shorter than 3.5 nm, than local beam models. The nonlocal rod model is employed to study the longitudinal and torsional vibrations of SWCNT and found to enable a good prediction of the MD results for shorter SWCNTs. Nonlocal and local continuum models provide a good agreement with MD results for relatively longer SWCNTs, such as (5, 5) SWCNTs longer than 3.5 nm. The scale parameter in nonlocal beam and rod models is estimated by calibrations from MD results.     

On the influence of pyroelectricity upon thermally induced vibrations of piezothermoelastic plates

Summary. We study thermally induced vibrations of laminated plates with attached or embedded piezoelectric layers. A model governing the bending motion under changing thermal environments is developed. The modeling is based on the kinematic assumption of Kirchhoff; consistently, the thickness distribution of the electric potential is considered being of second order. The electric potential distribution is found as a function of mechanical strain and temperature distribution; mechanical strain accounting for the direct piezoelectric effect and temperature entering via the pyroelectric effect. The plate theory incorporates the direct piezoelectric effect by means of effective plate stiffness and the pyroelectric effect by means of effective thermal loading. The theory is capable to model piezoelectric layers with the electrodes left open by using nonlocal constitutive relations. Moreover, nonlocal terms enter the effective thermal loading. These nonlocal relations introduce a new aspect into thin plate theory.Dedicated to Prof. Dr. h. c. Franz Ziegler; this paper is an extended version of a paper presented at the 5th Int. Congress on Thermal Stresses and Related Topics (special mini-symposium to honor Franz Ziegler), Blacksburg, VA, June 8–11, 2003.     

The effects of agglomerated CNTs as reinforcement on the size-dependent vibration of embedded curved microbeams based on modified couple stress theory

Vibration of an embedded nanocomposite curved microbeam is investigated based on the modified couple stress theory and Timoshenko beam model. The microbeam is reinforced by single-walled carbon nanotubes (SWCNTs) considering agglomeration effects. The structure is subjected to axial magnetic field. Applying Hamilton's principle, the motion equations of structure are derived. The generalized differential quadrature method is applied to obtain the frequency of the nanocomposite curved microbeam. The effects of the SWCNTs' volume percent, agglomeration of SWCNTs, elastic medium, magnetic field, boundary conditions, size effects, and central angle of microbeam on the frequency of the structure are shown.     

Nonlocal beam model for nonlinear analysis of carbon nanotubes on elastomeric substrates

Postbuckling, nonlinear bending and nonlinear vibration analyses are presented for single-wall carbon nanotubes (SWCNTs) resting on a two-parameter elastic foundation in thermal environments. The SWCNT is modeled as a nonlocal nanobeam which contains small scale effects. The elastomeric substrate with finite depth is modeled as a two-parameter elastic foundation. The thermal effects are included and the material properties of both SWCNTs and the substrate are assumed to be temperature-dependent. The governing equation that includes beam–foundation interaction is solved by a two-step perturbation technique. The numerical results reveal that the small scale parameter e₀a reduces the postbuckling equilibrium paths, the static large deflections and natural frequencies of SWCNTs resting on an elastic foundation. The results also reveal that the effect of the small scale parameter is significant for compressive buckling, but less pronounced for static bending and marginal for free vibration of SWCNTs resting on an elastic foundation.     

Electro-thermo-mechanical buckling of DWBNNTs embedded in bundle of CNTs using nonlocal piezoelasticity cylindrical shell theory

Axial buckling analysis of double-walled Boron Nitride nanotubes (DWBNNTs) embedded in an elastic medium under combined electro-thermo-mechanical loadings is presented in this article. Virtual displacement method based on nonlocal cylindrical piezoelasticity continuum shell theory is employed to derive the equilibrium equations. Boron Nitride nanotube (BNNT) is assumed to be surrounded by a bundle of carbon nanotubes (CNTs) as elastic medium for reinforcement. The elastic medium is simulated as Winkler–Pasternak foundation, and adjacent layers interactions are assumed to have been coupled by van der Walls (vdW) force evaluated based on the Lennard–Jones model. The effects of parameters such as electric and thermal loads, elastic medium and small scale are investigated on the buckling behavior of the DWBNNTs. The electric field and its direction are found to have affected the magnitude of the critical buckling load. Moreover, an analysis is carried out to estimate the nonlocal critical electro-thermo-mechanical load for the axial buckling of embedded DWBNNTs.     

Effect of various thermal loadings on buckling and vibrational characteristics of nonlocal temperature-dependent functionally graded nanobeams

In this article, the thermal effects on buckling and free vibrational characteristics of functionally graded (FG) size-dependent nanobeams subjected to various types of thermal loading are investigated by presenting a Navier-type solution for the first time. Temperature-dependent material properties of FG nanobeams vary continuously along the thickness according to the power-law form. The small-scale effect is taken into consideration based on Eringen's nonlocal elasticity theory. The nonlocal equations of motion are derived through Hamilton's principle and they are solved applying an analytical solution. It is revealed that the proposed modeling can provide accurate frequency results of the FG nanobeams.     

Small scale effect on the thermal buckling of orthotropic arbitrary straight-sided quadrilateral nanoplates embedded in an elastic medium

As a first endeavor, the small scale effect on the thermal buckling characteristic of orthotropic arbitrary straight-sided quadrilateral nanoplates embedded in an elastic medium is investigated. The surrounding elastic medium is modeled as the two-parameter elastic foundation. The formulation is derived using the classical plate theory (CPT) in conjunction with the nonlocal elasticity theory. The solution procedure is based on the transformation of the governing equations from physical domain to computational domain and then discretization of the spatial derivatives by employing the differential quadrature method (DQM) as an efficient and accurate numerical tool. The fast rate of convergence of the method is shown and the results are compared against existing results in literature. Then, the influence of small scale parameter in combination with the elastic medium parameters, geometrical shape and the boundary conditions on the thermal buckling load of the nanoplates is investigated.     

Effect of nonlocal elasticity on the performance of a flexoelectric layer as a distributed actuator of nanobeams

The paper is concerned with the development of finite element model for the static analysis of smart nanobeams integrated with a flexoelectric layer on its top surface, using nonlocal elastic theory. The flexoelectric layer acts as a distributed actuator of the nanobeam. A layerwise displacement theory has been used to derive the element stiffness matrices from variational principles incorporating nonlocal effects. The finite element model for nonlocal response of the beams has been validated with the exact solution for the case of a simply supported standalone flexoelectric layer. Also, the finite element model of the simply supported smart beam has been validated with exact solutions and numerical models for the local elastic case. The performance of the flexoelectric actuator has been compared for different values of nonlocal parameters and different combinations of nonlocal and local elastic substrate and flexoelectric layer. Further, the model developed has been utlized for investigating the performance of the active flexoelectric layer in case of cantilever beam, for which the exact solutions are not available.     

Temperature effects on wave propagation in nanoplates

In this paper, the thermal effects on the ultrasonic wave propagation characteristics of a nanoplate are studied based on the nonlocal continuum theory. The nonlocal governing equations are derived for the nanoplate under thermal environment. The axial stress caused by the thermal effects is considered. The wave propagation analysis is carried out using spectral analysis. The influences of the nonlocal small scale coefficient, the room or low temperature, the high temperature and the axial half wave numbers on the wave dispersion properties of nanoplate are also discussed. Numerical results show that the small scale effects and the thermal effects are significant for larger half wavenumbers. The results are qualitatively different from those obtained based on the local plate theory and thus, are important for the development of graphene-based nanodevices such as strain sensor, mass and pressure sensors, atomic dust detectors, and enhancer of surface image resolution.     

Thermo-mechanical vibration analysis of nonlocal temperature-dependent FG nanobeams with various boundary conditions

In this paper, the thermal effect on free vibration characteristics of functionally graded (FG) size-dependent nanobeams subjected to an in-plane thermal loading are investigated by presenting a Navier type solution and employing a semi analytical differential transform method (DTM) for the first time. Material properties of FG nanobeam are supposed to vary continuously along the thickness according to the power-law form. The small scale effect is taken into consideration based on nonlocal elasticity theory of Eringen. The nonlocal equations of motion are derived through Hamilton's principle and they are solved applying DTM. According to the numerical results, it is revealed that the proposed modeling and semi analytical approach can provide accurate frequency results of the FG nanobeams as compared to analytical results and also some cases in the literature. The detailed mathematical derivations are presented and numerical investigations are performed while the emphasis is placed on investigating the effect of the several parameters such as thermal effect, material distribution profile, small scale effects, mode number and boundary conditions on the normalized natural frequencies of the temperature-dependent FG nanobeams in detail. It is explicitly shown that the vibration behavior of an FG nanobeams is significantly influenced by these effects. Numerical results are presented to serve as benchmarks for future analyses of FG nanobeams.     

Nonlocal and surface effects on the buckling behavior of flexoelectric sandwich nanobeams

In this article, an analytical approach is presented to study the surface and flexoelectric effects on the buckling characteristics of an embedded piezoelectric sandwich nanobeam. According to the nonlocal elasticity theory, the flexoelectricity is believed to be authentic for size-dependent properties in nanostructures. The boundary conditions and the governing equations are derived by Hamilton's principle and are solved by Navier method. The results obtained from the present work show that the nonlocal term has an important reduction on the critical load and also the flexoelectricity shows an increasing influence on the buckling loads of the sandwich nanobeam, especially at lower thicknesses.     

Surface and nonlocal effects on the nonlinear free vibration of non-uniform nanobeams

The surface and nonlocal effects on the nonlinear flexural free vibrations of elastically supported non-uniform cross section nanobeams are studied simultaneously. The formulations are derived based on both Euler–Bernoulli beam theory (EBT) and Timoshenko beam theory (TBT) independently using Hamilton’s principle in conjunction with Eringen’s nonlocal elasticity theory. Green’s strain tensor together with von Kármán assumptions are employed to model the geometrical nonlinearity. The differential quadrature method (DQM) as an efficient and accurate numerical tool in conjunction with a direct iterative method is adopted to obtain the nonlinear vibration frequencies of nanobeams subjected to different boundary conditions. After demonstrating the fast rate of convergence of the method, it is shown that the results are in excellent agreement with the previous studies in the limit cases. The influences of surface free energy, nonlocal parameter, length of non-uniform nanobeams, variation of nanobeam width and elastic medium parameters on the nonlinear free vibrations are investigated.     

A new nonlocal elasticity theory with graded nonlocality for thermo-mechanical vibration of FG nanobeams via a nonlocal third-order shear deformation theory

In the present research, free vibration study of functionally graded (FG) nanobeams with graded nonlocality in thermal environments is performed according to the third-order shear deformation beam theory. The present nanobeam is subjected to uniform and nonlinear temperature distributions. Thermo-elastic coefficients and nonlocal parameter of the FG nanobeam are graded in the thickness direction according to power-law form. The scale coefficient is taken into consideration implementing nonlocal elasticity of Eringen. The governing equations are derived through Hamilton's principle and are solved analytically. The frequency response is compared with those of nonlocal Euler–Bernoulli and Timoshenko beam models, and it is revealed that the proposed modeling can accurately predict the vibration frequencies of the FG nanobeams. The obtained results are presented for the thermo-mechanical vibrations of the FG nanobeams to investigate the effects of material graduation, nonlocal parameter, mode number, slenderness ratio, and thermal loading in detail. The present study is associated to aerospace, mechanical, and nuclear engineering structures that are under thermal loads.     

Time discretization effect on the nonlinear vibration of embedded SWBNNT conveying viscous fluid

In this study, the effect of time discretization on the nonlinear transverse vibration and instability of single-walled boron nitride nanotube (SWBNNT) conveying viscous fluid is investigated based on the nonlocal piezoelasticity theory. SWBNNT is considered as an Euler–Bernoulli beam and is subjected to combined mechanical loading, thermal changes and electrical field. The elastic medium is simulated as Winkler and Pasternak foundation. The interaction between the inner viscous fluid and SWBNNT is obtained using Navier–Stokes equation. The axial inertia is neglected and a new approach is introduced to decouple the mechanical and electrical fields considering charge equation. Motion equations are derived by Hamilton’s principle using the Von-Kármán nonlinearity theory. In the first approach, time and space domains are discretized using the method of multiple scale (MMS) and Galerkin procedure respectively, and in the second one differential quadrature method (DQM) is utilized to space discretization. Good agreement is shown between the results of first and second approach. Numerical results indicate the significant effects of aspect ratio, elastic medium and nonlocality on the frequency and instability of the SWBNNT.