Functionally Graded Nanobeam Research Articles

Wave propagation analysis of rotating thermoelastically-actuated nanobeams based on nonlocal strain gradient theory

This paper is concerned with the wave propagation behavior of rotating functionally graded (FG) temperature-dependent nanoscale beams subjected to thermal loading based on nonlocal strain gradient stress field. Uniform, linear and nonlinear temperature distributions across the thickness are investigated. Thermo-elastic properties of FG beam change gradually according to the Mori–Tanaka distribution model in the spatial coordinate. The nanobeam is modeled via a higher-order shear deformable refined beam theory which has a trigonometric shear stress function. The governing equations are derived by Hamilton's principle as a function of axial force due to centrifugal stiffening and displacement. The solution of these equations is provided employing a Galerkin-based approach which has the potential to capture various boundary conditions. By applying an analytical solution and solving an eigenvalue problem, the dispersion relations of rotating FG nanobeam are obtained. Numerical results illustrate that various parameters including temperature change, angular velocity, nonlocality parameter, wave number and gradient index have significant effects on the wave dispersion characteristics of the nanobeam under study. The outcome of this study can provide beneficial information for the next-generation research and the exact design of nano-machines including nanoscale molecular bearings, nanogears, etc.

Effects of neutral surface deviation on nonlinear resonance of embedded temperature-dependent functionally graded nanobeams

In this paper, forced vibration of functionally graded (FG) nanobeams resting on the nonlinear elastic foundations are investigated based on the nonlocal strain gradient theory. The material parameters of FG nanobeams are assumed to be temperature-dependent and change continuously along the thickness direction according to the power-law function (PFGM) or sigmoid function (SFGM). Based on the Euler–Bernoulli beam theory and von-Kármán geometric nonlinearity, the governing equations of motion are derived by considering the deviation between the geometrical and physical neutral surfaces. Closed-form approximate solution for nonlinear forced vibration of a FG nanobeam is derived by using multiple time scale method. The results show that decrease of non-homogeneity index and material length scale parameter, or increase of temperature variation and nonlocal parameter will increase the resonance frequencies of FG nanobeams. The effect of the in-coincidence of physical and geometrical neutral surfaces on the nonlinear resonance of the nanobeams could not be ignored, especially for SFGM nanobeams with larger non-homogeneity index and stronger size effects, embedded in a softer medium with obvious temperature variation.

Uncertainty modeling for vibration and buckling behaviors of functionally graded nanobeams in thermal environment

High accuracy modeling of nanostructure is a crucial issue in nano electromechanical systems. This study investigates the effect of nanomaterial uncertainties on vibration and buckling behaviors of functionally graded (FG) nanobeams in thermal environment. The size-dependent governing differential equation is derived based on the nonlocal Euler–Bernoulli beam theory with the thermal effect and the analytical formulations of the natural frequencies are deduced. To avoid the shortcoming of probabilistic method in the case of inadequate data, a non-probabilistic uncertainty modeling for the FG nanobeams is developed by quantifying nanomaterial uncertainties as interval parameters. Meanwhile, a hull iterative algorithm (HIA) for solving this model is presented to evaluate the bounds of the natural frequencies. After validation of HIA by comparing with Monte Carlo simulation and sensitivity based interval analysis method, the detailed parametric study is performed to understand the combined influences of nanomaterial uncertainties and size-dependent parameter, power-law index as well as temperature change on the natural frequencies. Additionally, the lower bound of the critical buckling temperature is discussed under different size-dependent parameters and power-law indices. The bounds of the natural frequencies and the critical buckling temperature obtained here are helpful for the safety and optimal design of nano electromechanical systems.

Nonlinear bending response of functionally graded nanobeams with material uncertainties

Material uncertainties caused by the defects in molecular structure and the inaccurate measurements at nano-scale have been reported. The influence of these uncertainties on mechanical behaviors of nanostructures is crucial to the potential application in the field of structural health monitoring where high accuracy modeling is required. This study focuses on the influence of material uncertainties on nonlinear bending response of nanobeam made of functionally graded (FG) materials. Based on the nonlocal strain gradient theory, the size-dependent equilibrium differential equation of the FG nanobeam is derived by using the minimum total potential energy principle and the analytical formulation of the nonlinear bending response is deduced, in which the geometric nonlinearity is considered. The effect of variation of material parameters on the nonlinear bending response is studied subsequently. Considering the drawback of probabilistic method in the case of insufficient sample data, a non-probabilistic uncertainty modeling for the FG nanobeam is developed by quantifying material uncertainties with interval parameters. Simultaneously, an iterative algorithm based interval analysis method (IA-IAM) is presented to solve this model. The combined effects of material uncertainties and size-dependent parameters as well as the power-law index of the FG material on the nonlinear bending response of the FG nanobeam are investigated in detail. Numerical results can provide helpful guidance for the reliability design of nanodevices.

Forced Vibration Analysis of Functionally Graded Nanobeams

In this paper, forced vibration responses of functionally graded (FG) nanobeams are presented for modified couple stress theory (MCST) with damping effect. The FG nanobeam is excited by a transverse triangular force impulse modulated by a harmonic motion. Mechanical properties of FG beam depends on the position. The Kelvin–Voigt model is considered in the damping effect. In solution of the dynamic problem, finite element method is used within Timoshenko beam theory. The obtained system of differential equations is reduced to a linear algebraic equation system and solved in the time domain by using Newmark average acceleration method. Influences of the geometry and material parameters on forced vibration responses of FG nanobeams are examined and discussed.

A modified nonlocal couple stress-based beam model for vibration analysis of higher-order FG nanobeams

ABSTRACTIn the present paper, nonlocal couple stress theory is developed to investigate free vibration characteristics of functionally graded (FG) nanobeams considering exact position of neutral axis. The theory introduces two parameters based on nonlocal elasticity theory and modified couple stress theory to capture the size effects much accurately. Therefore, a nonlocal stress field parameter and a material length scale parameter are used to involve both stiffness-softening and stiffness-hardening effects on responses of FG nanobeams. The FG nanobeam is modeled via a higher-order refined beam theory in which shear deformation effect is verified needless of shear correction factor. A power-law distribution is used to describe the graded material properties. The governing equations and the related boundary conditions are derived by Hamilton's principle and they are solved applying Galerkin's method, which satisfies various boundary conditions. A comparison study is performed to verify the present formulation with the provided data in the literature and a good agreement is observed. The parametric study covered in this paper includes several parameters, such as nonlocal and length scale parameters, power-law exponent, slenderness ratio, shear deformation, and various boundary conditions on natural frequencies of FG nanobeams in detail.

Nonlinear analysis of functionally graded nanoscale beams incorporating the surface energy and microstructure effects

A nonclassical size–dependent beam model is developed to study the geometrically nonlinear response of functionally graded (FG) nanoscale beams including the combined effect of surface elasticity, surface residual stress, microstructure and size-dependency. To this, the Gurtin–Murdoch surface elasticity theory in combination with the modified couple stress theory is employed to capture the simultaneous effects of surface energy and microstructure size phenomenon. Euler–Bernoulli and Timoshenko beam models considering the von Kármán geometric nonlinearity are adopted. The material properties of the bulk and surface of the FG beam are assumed to be changed continuously through the thickness according to a power law scheme and Mori–Tanaka homogenization technique. The geometrically nonlinear size–dependent governing equations and corresponding nonclassical boundary conditions are exactly obtained through Hamilton's principle. The generalized differential quadrature method (GDQM) is employed to discretize the nonlinear nonclassical governing differential equations and the nonclassical boundary conditions for various end supports. A good agreement between the results of the present model and those available in the literature is obtained. Selected numerical results show explicitly that the material length scale parameter, surface elastic modulus, surface residual stress, gradient index and material distribution have a significant influence on the geometrically nonlinear bending response of FG nanobeams.

Dynamic stability of functionally graded nanobeam based on nonlocal Timoshenko theory considering surface effects

Based on nonlocal Timoshenko beam theory, dynamic stability of functionally graded (FG) nanobeam under axial and thermal loading was investigated. Surface stress effects were implemented according to Gurtin-Murdoch continuum theory. Using power law distribution for FGM and von Karman geometric nonlinearity, governing equations were derived based on Hamilton's principle. The developed nonlocal models have the capability of interpreting small scale effects. Pasternak elastic medium was employed to represent the interaction of the FG nanobeam and the surrounding elastic medium. A parametric study was conducted to focus influences of the static load factor, temperature change, gradient index, nonlocal parameter, slenderness ratio, surface effect and springs constants of the elastic medium on the dynamic instability region (DIR) of the FG beam with simply-supported boundary conditions. It was found that differences between DIRs predicted by local and nonlocal beam theories are significant for beams with lower aspect ratio. Moreover, it was observed that in contrast to high temperature environments, at low temperatures, increasing the temperature change moves the origin of the DIR to higher excitation frequency zone and leads to further stability. Considering surface stress effects shifts the DIR of FG beam to higher frequency zone, also increasing the gradient index enhances the frequency of DIR.

An analytical solution for bending, buckling, and free vibration of FG nanobeam lying on Winkler-Pasternak elastic foundation using different nonlocal higher order shear deformation beam theories

In the present study, various Higher-order Shear Deformation beam Theories (HSDTs) are applied in order to achieve the exact analytical solution to bending, buckling, and free vibration of Functionally Graded (FG) nanobeam lying on the Winkler and Pasternak elastic foundations. HSDTs are those in which the eff ect of transverse shear strain is included. The displacement field of these theories involves a quadratic variationof transverse shear strains and stresses; hence, this hypothesis leads to the diminishing of transverse shear stresses at the top and bottom surfaces of a beam. Thus, necessarily, there is no need to use a shear correction factor in the HSDTs. Nanobeam has been made of FG materials in which the properties of these materials are changed through the thickness direction of nanobeam according to the power-law distribution. Hamilton's principle is used to derive the equation of motions and the related boundary conditions of simplysupported nanobeam. The present study shows that the stability and vibration behaviors of FG nanobeam are extremely dependent on the Winkler and Pasternak elastic foundation, gradient index, aspect ratio, and nonlocal parameter. The obtained results of the present study might be useful in the advanced fi eld of micro/nano electromechanical systems.

Forced vibration of sinusoidal FG nanobeams resting on hybrid Kerr foundation in hygro-thermal environments

ABSTRACTSize-dependent forced vibration behavior of functionally graded (FG) nanobeams subjected to an in-plane hygro-thermal loading and lateral concentrated and uniform dynamic loads is investigated via a higher-order refined beam theory, which captures shear deformation influences needless of any shear correction factor. The nanobeam is in contact with a three-parameter Kerr foundation consisting of upper and lower spring layers as well as a shear layer. Hygro-thermo-elastic material properties of the nanobeam are described via power-law distribution considering exact position of the neutral axis. Through nonlocal elasticity theory of Eringen and Hamilton's principle, the governing equations of higher-order FG nanobeams on Kerr foundation under dynamic loading are derived. These equations are solved for simply-supported and clamped-clamped boundary conditions. A detailed parametric study is performed to show the importance of moisture concentration rise, temperature rise, material composition, nonlocality, Kerr foundation parameters, and boundary conditions on forced vibration characteristics and resonance frequencies of FG nanobeams. As a consequence, Kerr foundation parameters lead to a significant delay in the occurrence of resonance frequencies.

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

ABSTRACTIn 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.

Investigating dynamic response of porous inhomogeneous nanobeams on hybrid Kerr foundation under hygro-thermal loading

This paper is concerned with the forced vibration analysis of imperfect functionally graded (FG) nanobeams resting on three-parameter Kerr foundation under hygro-thermal loading employing a higher order refined beam theory. The nanobeam is subjected to dynamic uniform distributed and concentrated point loads at upper surface. Hygro-thermo-elastic material properties of FGM nanobeam vary in thickness direction according to the power-law distribution. Implementing an analytical approach, the obtained nonlocal coupled governing equations from Hamilton’s principle according to refined beam model are solved for simply supported and clamped boundary conditions. The significance of moisture percentage, temperature change, material gradation, scale parameter, Kerr foundation coefficients and boundary conditions on forced vibration characteristics and resonance frequencies of FG nanobeam are examined.

Small-scale effects on the dynamic response of inhomogeneous nanobeams on elastic substrate under uniform dynamic load

In this article, forced vibration and resonance frequencies of functionally graded (FG) nanobeams resting on Winkler-Pasternak foundation are investigated employing a higher-order refined beam theory which captures shear deformation influences without the need for any shear correction factor. The nanobeam is subjected to a uniform dynamic load with a finite length. The two-parameter elastic medium consists of parallel springs and a shear layer. Gradation of the material properties of the nanobeam is described via the Mori-Tanaka distribution model. The governing equations of embedded higher-order FG nanobeams under dynamic loading are obtained by employing Eringen’s elastic differential law and Hamilton’s principle. These equations are solved for simply-supported-simply-supported (S-S) and clamped-clamped (C-C) boundary conditions. It is indicated that forced vibration characteristics and resonance frequencies of embedded FG nanobeams are significantly influenced by material composition, nonlocality, foundation parameters and boundary conditions. A nonlocal FG beam has lower resonance frequency compared with a local beam. However, the presence of elastic foundation leads to a significant delay in the occurrence of resonance frequencies.

Bending and stability analysis of size-dependent compositionally graded Timoshenko nanobeams with porosities

In this article, static deflection and buckling of functionally graded (FG) nanoscale beams made of porous material are carried out based on the nonlocal Timoshenko beam model which captures the small scale influences. The exact position of neutral axis is fixed, to eliminate the stretching and bending coupling due to the unsymmetrical material change along the FG nanobeams thickness. The material properties of FG beam are graded through the thickness on the basis of the power-law form, which is modified to approximate the material properties with two models of porosity phases. By employing Hamilton's principle, the nonlocal governing equations of FG nanobeams are obtained and solved analytically for simply-supported boundary conditions via the Navier-type procedure. Numerical results for deflection and buckling of FG nanoscale beams are presented and validated with those existing in the literature. The influences of small scale parameter, power law index, porosity distribution and slenderness ratio on the static and stability responses of the FG nanobeams are all explored.

Vibration analysis of embedded size dependent FG nanobeams based on third-order shear deformation beam theory

In this paper, free vibration characteristics of functionally graded (FG) nanobeams embedded on elastic medium are investigated based on third order shear deformation (Reddy) beam theory by presenting a Navier type solution for the first time. The material properties of FG nanobeam are assumed to vary gradually along the thickness and are estimated through the power-law and Mori-Tanaka models. A two parameters elastic foundation including the linear Winkler springs along with the Pasternak shear layer is in contact with beam. The small scale effect is taken into consideration based on nonlocal elasticity theory of Eringen. The nonlocal equations of motion are derived based on third order shear deformation beam theory through Hamilton's principle and they are solved applying analytical solution. According to the numerical results, it is revealed that the proposed modeling can provide accurate frequency results of the FG nanobeams as compared to some cases in the literature. The obtained results are presented for the vibration analysis of the FG nanobeams such as the influences of foundation parameters, gradient index, nonlocal parameter and slenderness ratio in detail.

Semi-analytical vibration analysis of functionally graded size-dependent nanobeams with various boundary conditions

In this paper, free vibration of functionally graded (FG) size-dependent nanobeams is studied within the framework of nonlocal Timoshenko beam model. It is assumed that material properties of the FG nanobeam, vary continuously through the thickness according to a power-law form. The small scale effect is taken into consideration based on nonlocal elasticity theory of Eringen. The non-classical governing differential equations of motion are derived through Hamilton

On the mechanics of functionally graded nanobeams with the account of surface elasticity

In this paper, size-dependent bending, buckling and free vibration responses of functionally graded (FG) nanobeams are investigated using an integrated nonclassical continuum model. The integrated model accounts for the simultaneous effects of the microstructure local rotation and the surface energy in the framework of the nonlocal elasticity. The proposed nonlocal-couple stress-surface elasticity (NLCSSE) model is formulated by incorporating the Eringen nonlocal elasticity theory, modified couple stress theory and surface elasticity theory into the classical Euler–Bernoulli beam model. It is assumed that the material properties of the bulk and surface of the FG nanobeam change continuously through the thickness according to a power law. The size-dependent equations of motion and corresponding boundary conditions are derived utilizing the Hamilton's principle. The proposed model is validated by comparing the obtained results with available benchmark results. Numerical results are presented to reveal the effects of nonlocal parameter, material length scale parameter, surface energy, gradient index, Poisson ratio, thickness and length-to-thickness ratio on the deflection, critical buckling load and natural frequency of a FG simply supported nanobeam.

Dynamic characteristics of temperature-dependent viscoelastic FG nanobeams subjected to 2D-magnetic field under periodic loading

In this study, natural frequency and dynamic stability of functionally graded (FG) viscoelastic nanobeams based on the Euler–Bernoulli theory on the visco-Pasternak foundation are predicted. The material characteristics of FG nanobeam are temperature-dependent, and vary according to power-law model along thickness. The FG viscoelastic nanobeam is located on a two-dimensional magnetic field which considers the effects of transverse and longitudinal magnetic field. The uniform, linear, and sinusoidal temperature fields are applied on the FG viscoelastic nanobeam. The governing equations are derived through Hamilton’s principle and Eringen’s nonlocal theory. The equations are solved by a Navier-type method and the Bolotin method for simply supported conditions. The effect of three different temperature fields on natural frequency and dynamic stability region of the nanobeam is analyzed. The importance of various parameters such as nonlocal parameter, gradient indexes, magnitude of magnetic field, angle of magnetic field, temperature changes, and aspect ratio on both natural frequency and dynamic stability region of the FG viscoelastic nanobeam is investigated.

A nonlocal quasi-3D theory for bending and free flexural vibration behaviors of functionally graded nanobeams

In this paper, size dependent bending and free flexural vibration behaviors of functionally graded (FG) nanobeams are investigated using a nonlocal quasi-3D theory in which both shear deformation and thickness stretching effects are introduced. The nonlocal elastic behavior is described by the differential constitutive model of Eringen, which enables the present model to become effective in the analysis and design of nanostructures. The present theory incorporates the length scale parameter (nonlocal parameter) which can capture the small scale effect, and furthermore accounts for both shear deformation and thickness stretching effects by virtue of a hyperbolic variation of all displacements through the thickness without using shear correction factor. The material properties of FG nanobeams are assumed to vary through the thickness according to a power law. The neutral surface position for such FG nanobeams is determined and the present theory based on exact neutral surface position is employed here. The governing equations are derived using the principal of minimum total potential energy. The effects of nonlocal parameter, aspect ratio and various material compositions on the static and dynamic responses of the FG nanobeam are discussed in detail. A detailed numerical study is carried out to examine the effect of material gradient index, the nonlocal parameter, the beam aspect ratio on the global response of the FG nanobeam. These findings are important in mechanical design considerations of devices that use carbon nanotubes.

Vibration analysis of FG nanobeams based on third-order shear deformation theory under various boundary conditions

In this study, free vibration of functionally graded (FG) micro/nanobeams based on nonlocal third-order shear deformation theory and under different boundary conditions is investigated by applying the differential quadrature method. Third-order shear deformation theory can consider the both small-scale effects and quadratic variation of shear strain and hence shear stress along the FG nanobeam thickness. The governing equations are obtained by using the Hamilton's principle, based on third-order shear deformation beam theory. The differential quadrature (DQ) method is used to discretize the model and attain the natural frequencies and mode shapes. The properties of FG micro/nanobeam are assumed to be chanfged along the thickness direction based on the simple power law distribution. The effects of various parameters such as the nonlocal parameter, gradient index, boundary conditions and mode number on the vibration characteristics of FG micro/nanobeams are discussed in detail.