Functionally Graded Nanobeam Research Articles

An uncoupled theory of FG nanobeams with the small size effects and its exact solutions

Due to unphysical coupling induced by the material inhomogeneity, FG (functionally graded) nanobeam problems were formulated in a very complex way so that they cannot be analytically solved. In this paper, an uncoupled theory is proposed for FG nanobeams considering their small size effects. First, with the aid of the neutral axis, the axial displacement is expressed in terms of generalized displacements for FG nanobeams. Based on the nonlocal strain gradient theory, the generalized stresses and strains are accordingly defined and uncoupled constitutive relations are derived. Based on the principle of virtual work, an uncoupled theory is eventually established, including governing equations and boundary conditions. Within the present framework, analytical solutions to FG nanobeams are obtained for the first time for general boundary conditions. These solutions not only re-evaluate the previous results but shed light on the small size effects of FG nanobeams.

Analyzing dynamic response of nonlocal strain gradient porous beams under moving load and thermal environment

This research presents dynamic response analysis of a porous functionally graded (FG) nanobeam under a moving point load. The nanobeam formulation has been established with the use of a higher-order refined beam model and nonlocal strain gradient theory (NSGT) including two scale factors named nonlocal and strain gradient factors. The porous FG material has been modeled via modified power-law functions which contain porosity volume according to even or uneven porosity dispersions. Moreover, graded nonlocality has been considered in order to provide a better modeling of size effects for FG nano-size structures. The governing equations of the nanobeam have been discretized with the use of differential quadrature method (DQM) and inverse Laplace transform approach has been utilized to calculate the dynamic deflections. The main findings of the present research indicate the influences of nonlocal strain gradient factors, moving load speed, pore amount, porosity distribution and elastic medium on dynamic deflection of FG nanobeams.

Investigation of thermal preloading and porosity effects on the nonlocal nonlinear instability of FG nanobeams with geometrical imperfection

The aim of the present research is twofold. The first is to present a study on nonlinear thermo-mechanical bending and thermal postbuckling analysis of functionally graded (FG) porous perfect/imperfect nanobeam according to the nonlocal elasticity theory. The second, concurrent aim is to address the snap-through phenomenon in the thermally preloaded graded porous nanobeams due to lateral mechanical load. Two types of thermal loading including, uniform temperature rise and heat conduction through the thickness as well as two cases of porosity distribution, including uniform and uneven, are considered. Geometrically imperfect FG porous nanobeam with four different types of immovable boundary conditions is also taken into account. Thermo-mechanical material properties of the porous nanobeam are assumed to be position-dependent according to the modified rule of mixture. The nonlinear equilibrium equations are constructed using the Euler–Bernoulli beam model and von-Kármán type of geometrical nonlinearity. Chebyshev polynomial based Ritz method is utilized into the principle of virtual displacement to obtain the matrix representation of the nonlocal governing equations. Two different strategies, including the Newton–Raphson technique and direct displacement control scheme, are first proposed to extract the nonlinear bending and postbuckling curves of the graded nanobeam. Afterwards, to capture the snapping behavior and trace the path beyond the limit loads of the thermally preloaded nanobeam, the cylindrical arch-length technique is adopted. Numerical results are provided to explore the effect of different parameters such as the power-law index, nonlocal parameter, boundary conditions, porosity distribution, thermal loading, and imperfection amplitude on the nonlinear thermo-mechanical bending and instability of the FG nanobeam.

Vibration of FG nano-sized beams embedded in Winkler elastic foundation and with various boundary conditions

In the current study, vibration analysis of functionally graded (FG) nano-sized beams resting on a elastic foundation is presented via a finite element method. The elastic foundation is simulated by using one-parameter Winkler type elastic foundation model. Euler-Bernoulli beam theory and Eringen’s nonlocal elasticity theory are utilized to model the functionally graded nano-sized beams with various boundary conditions such as simply supported at both ends (S-S), clamped-clamped (C-C) and clamped-simply supported (C-S). Material properties of functionally graded nanobeam vary across the thickness direction according to the power-law distribution. The vibration behaviors of functionally graded nanobeam composed of alumina (Al2O3) and steel are shown using nonlocal finite element formulation. The importance of this paper is the utilize of shape functions and the Eringen's nonlocal elasticity theory to set up the stiffness matrices and mass matrices of the functionally graded nano-sized beam resting on Winkler elastic foundation for free vibration analysis. Bending stiffness, foundation stiffness and mass matrices are obtained to realize the solution of vibration problem of the FG nanobeam. The influences of power-law exponent (k), dimensionless nonlocal parameters (e0a/L), dimensionless Winkler foundation parameters (KW), mode numbers and boundary conditions on frequencies are investigated via several numerical examples and shown by a number of tables and figures.

Dynamic analysis of functionally graded beams with periodic nanostructures

It is known from experimental studies that softening or hardening material behaviours of nanostructures change with the microstructure of the considered material. However, most of the size dependent continuum models (nonlocal stress gradient, strain gradient and couple stress theories) predict only softening or hardening material behaviour. Except of these size dependent theories doublet mechanics predicts both softening and hardening responses of the material like experiments in micro/nano-structures. In the present study, free vibration analysis of functionally graded (FG) periodic structure nanobeams are investigated via doublet mechanics theory. Periodic FG nanobeams are modelled as a simple crystal lattice type. Micro strains and stresses are expanded in Taylor series and obtained micro relations transformed to macro stress-strain relations. Thus, by use of bottom-up approach yields the more physical and accurate analysis of nanostructures in the doublet mechanics model. After deriving the mathematical formulation of a periodic FG nanobeam, vibration problem is examined for general boundary conditions. Ritz method is used in the solution. Adjusting of softening and hardening responses of the material gives a beneficial optimization and design of nanostructures.

Vibration and thermal buckling analysis of rotating nonlocal functionally graded nanobeams in thermal environment

A new nonlocal model for free vibration and thermal buckling analysis of rotating temperature-dependent functionally graded (FG) nanobeams in thermal environment is developed by introducing an axial nonlinear second-order coupling deformation based upon the Eringen's nonlocal elasticity theory (ENET) and Euler-Bernoulli beam theory (EBT). Material properties of FG nanobeams are assumed to be through-thickness symmetric and the temperature distribution is uniform in the thickness direction. The Hamilton's principle is utilized to derive the governing equations and associated boundary conditions of the nonlocal rotating FG nanobeam model considering thermal, small-scale and centrifugal stiffening effects. The nonlocal governing differential equations are transformed into algebraic eigenvalue equations evaluating the vibration and buckling properties through the Galerkin method. The validity and accuracy of the present nonlocal model are verified by convergence and comparative studies. Numerical results are presented to investigate the influences of dimensionless angular velocity, hub radius ratio, temperature change, material gradient index, slenderness ratio and nonlocal parameter on the natural frequencies and critical buckling temperatures of rotating FG nanobeams. The obtained results clearly show that these effects significantly affect the thermal buckling and vibrational behaviors of rotating FG nanobeams.

Considering Bending and Vibration of Homogeneous Nanobeam Coated by a FG Layer

In this research static deflection and free vibration of homogeneous nanobeams coated by a functionally graded (FG) layer is investigated according to the nonlocal elasticity theory. A higher order beam theory is used that does not need the shear correction factor. The equations of motion (equilibrium equations) are extracted by using Hamilton’s principle. The material properties are considered to vary in the thickness direction of FG coated layer. This nonlocal nanobeam model incorporates the length scale parameter (nonlocal parameter) that can capture the small scale effects. In the numerical results section, the effects of different parameters, especially the ratio of thickness of FG layer to the total thickness of the beam are considered and discussed. The results reveal that the frequency is maximum for a special value of material power index. Also, increasing the ratio of thickness of FG layer to the total thickness of the beam increases the static deflection and decreases the natural frequencies. These results help with the understanding such coated structures and designing them carefully. The results also show that the new nonlocal FG nanobeam model produces larger vibration and smaller deflection than homogeneous nonlocal nanobeam.

Size-dependent dynamics of a FG Nanobeam near nonlinear resonances induced by heat

This study explores heat-induced nonlinear vibration of a functionally graded (FG) capacitive nanobeam within the framework of nonlocal strain gradient theory (NLSGT). The elastic FG beam, which is firstly deflected by a DC voltage, is driven to vibrate about its deflected position by a periodic heat load. The nano-structure, which consists of a clamped-clamped nanobeam, is modeled assuming Euler–Bernoulli beam assumption which accounts for the nonlinear von-Karman strain and the electrostatic and intermolecular forcing. To simulate the static and dynamic responses, a model reduction procedure is carried out by employing the Galerkin method. The method of Averaging as a regular semi-analytic perturbation method is applied to obtain governing equations of the steady-state responses. With the purpose of establishing the validity of the solution, a Shooting technique in conjunction with the Floquet theory is used to capture the periodic motions and then examine their stability. The nonlinear resonance frequency of the FG nanobeam near its fundamental natural frequency (primary resonance) and near principal parametric resonance is investigated while the emphasis is placed on studying the effect of various parameters including DC voltage, amplitude of the periodic heat source, material index, damping ratio, and small scale parameters. The main objective of this study is to model a miniature structure which can be used as either a sensitive remote temperature sensor or a high-efficiency thermal energy harvester.

Nonlinear thermal behavior of shear deformable FG porous nanobeams with geometrical imperfection: Snap-through and postbuckling analysis

Present paper deals with the nonlinear thermal bending response, thermal postbuckling behavior, and snap-through phenomenon due to lateral mechanical load in a thermally preloaded functionally graded (FG) porous perfect/imperfect nanobeam subjected to two types of thermal loading, including heat conduction across the thickness and uniform temperature rise. Heterogenous material properties of the porous nanobeams are assumed to be position/temperature-dependent, where dependency is obtained according to the modified rule of mixture and Touloukian formulation. Two types of porosity distribution and also geometrical imperfection of the nanobeam are considered. Assuming the Timoshenko beam model and von-Karman nonlinearity, the nonlinear equilibrium equations are extracted on the basis of the nonlocal theory of elasticity. With the establishment of the principle of virtual displacement and Chebyshev polynomial of the first kind as the basic functions, the Ritz method is utilized to obtain the matrix form of the nonlocal governing equations. Two different strategies, including the Newton-Raphson technique and direct displacement control scheme, are first proposed to extract the nonlinear bending and postbuckling trajectories of the graded porous nanobeam. Afterwards, to trace the snapping behavior beyond limit loads of the graded porous nanobeam with thermal preloading, the cylindrical arch-length scheme as a path-following method is adopted. Numerical parametric investigations are given to discuss the influences of the power-law index, two types of porosity distribution and thermal loads, size-dependent nonlocal parameter, imperfection amplitude, and boundary conditions on the snap-through behavior and the nonlinear thermal stability of the FG nanobeam.

Comprehensive investigation of vibration of sigmoid and power law FG nanobeams based on surface elasticity and modified couple stress theories

Based on the modified couple stress theory and Gurtin–Murdoch surface elasticity theory, a size-dependent Timoshenko beam model is developed for investigating the nonlinear vibration response of functionally graded (FG) micro-/nanobeams. The model is capable of capturing the simultaneous effects of microstructure couple stress, surface energy, and von Karman’s geometric nonlinearity. Sigmoid function and power law homogenization schemes are used to model the material gradation of the beam. Hamilton’s principle is exploited to establish the nonclassical nonlinear governing equations and corresponding higher-order boundary conditions. To account for the nonhomogeneity in boundary conditions, the solution of the problem is split into two parts: the nonlinear static response with the nonhomogeneous boundary conditions and the nonlinear dynamic response. The resulting boundary conditions for the dynamic response are homogeneous, and so Galerkin’s approach is applied to reduce the set of PDEs to a nonlinear system of ODEs. The generalized differential quadrature method in terms of spatial variables is applied to obtain the static response and linear vibration mode. Considering the nonlinear system of ODEs in terms of time-related variables, both pseudo-arclength continuation and Runge–Kutta methods are used to obtain the nonlinear free vibration behavior of FG Timoshenko micro-/nanobeams with simply supported and clamped ends. Verification of the proposed model and solution procedure is performed by comparing the obtained results with those available in the open literature. The effects of the nonhomogeneous boundary conditions, surface elasticity modulus, surface residual stress, material length scale parameter, gradient index, and thickness on the characteristics of linear and nonlinear free vibrations of sigmoid function and power law FG micro-/nanobeams are discussed in detail.

On vibration of bi-directional functionally graded nanobeams under magnetic field

In this article, the transverse vibrations of bi-directional functionally graded (FG) nanobeams subjected to a longitudinal magnetic field are investigated. The nonlocal elasticity theory is exploited to consider the small-scale effect. Material properties of FG nanobeam are assumed to vary continuously along the thickness and length with arbitrary function. The equations of motion and boundary conditions are obtained using Hamilton’s principle. These equations are solved by employing the generalized differential quadrature method. The effect of different parameters such as bi-directional FG parameters, size-dependent parameter, and value of the magnetic field on vibration are investigated. It is concluded that different factors as notated, play significant roles in the vibrational response of bi-directional FG nanobeams, and by using the magnetic field, the vibration frequencies of nanostructures can be controlled. Communicated by Jie Yang.

Effect of Microstructure and Surface Energy on the Static and Dynamic Characteristics of FG Timoshenko Nanobeam Embedded in an Elastic Medium

This paper presents an investigation of the size-dependent static and dynamic characteristics of functionally graded (FG) Timoshenko nanobeams embedded in a double-parameter elastic medium. Unlike existing Timoshenko nanobeam models, the combined effects of surface elasticity, residual surface stress, surface mass density and Poisson’s ratio, in addition to axial deformation, are incorporated in the newly developed model. Also, the continuous gradation through the thickness of all the properties of both bulk and surface materials is considered via power law. The Navier-type solution is developed for simply supported FG nanobeam in the form of infinite power series for bending, buckling and free vibration. The obtained results agree well with those available in the literature. In addition, selected numerical results are presented to explore the effects of the material length scale parameter, surface parameters, gradient index, elastic medium, and thickness on the static and dynamic responses of FG Timoshenko nanobeams.

Free vibration of FG nanobeam using a finite‐element method

In this work, a non-local finite-element formulation is developed to analyse free vibration of functionally graded (FG) nanobeams considering power-law variation of material through thickness of the nanobeam. The Euler–Bernoulli beam theory based on Eringen's non-local elasticity theory with one length scale parameter is used to model the FG nanobeam. To this end, two types of FG nanobeams composed of two different materials are analysed by using the developed non-local finite-element formulation. First FG nanobeam is made of alumina (Al2O3) and steel, whereas second one is composed of silicon carbide (SiC) and stainless steel (SUS304). Numerical results are presented to show the effect of power-law exponent (k) and nanostructural length scale (e 0 a/L) on the free vibration of FG nanobeams.

Bending, Buckling and Free Vibration Analysis of Size-Dependent Nanoscale FG Beams Using Refined Models and Eringen’s Nonlocal Theory

In this study, a theoretical unification of twenty-one nonlocal beam theories are presented by using a unified nonlocal beam theory. The small-scale effect is considered based on the nonlocal differential constitutive relations of Eringen. The present unified theory satisfies traction free boundary conditions at the top and bottom surface of the nanobeam and hence avoids the need of shearing correction factor. Hamilton’s principle is employed to derive the equations of motion. The present unified nonlocal formulation is applied for the bending, buckling and free vibration analysis of functionally graded (FG) nanobeams. The elastic properties of FG material vary continuously by gradually changing the volume fraction of the constituent materials in the thickness direction. Closed-form analytical solutions are obtained by using Navier’s solution technique. Non-dimensional displacements, stresses, natural frequencies and critical buckling loads for FG nanobeams are presented. The numerical results presented in this study can be served as a benchmark for future research.

Nonlinear thermal stability and snap-through buckling of temperature-dependent geometrically imperfect graded nanobeams on nonlinear elastic foundation

In this research, thermal postbuckling and nonlinear thermal bending of size-dependent functionally graded (FG) perfect/imperfect nanobeams in-contact with a nonlinear elastic foundation and exposed to thermal loading are first scrutinized. The second objective of this research is to investigate snap-through instability in a thermally postbuckled FG nanobeam due to uniform lateral load. The geometrical imperfection of the nanobeam is taken into account. Thermo-mechanical material properties are temperature-dependent and are assumed to vary continuously throughout the nanobeam thickness on the basis of the power-law model. The nonlinear equilibrium equations are derived according to the Euler-Bernoulli beam theory in conjunction with the nonlinear von-Karman assumptions based on the nonlocal elasticity theory. Using Chebyshev polynomial of the first kind, Ritz method is utilized into the principle of virtual displacement to form nonlocal governing equations. Three different methodologies, including direct displacement control approach, Newton-Raphson iterative method, and cylindrical arch-length scheme are utilized to investigate the nonlinear thermal stability curves and snap-through phenomenon through limit points of a thermally postbuckled FG nanobeam. The primary purpose of this research is to explore the influences of the nonlocal parameter, nonlinear elastic foundation coefficients, power-law index, imperfection amplitude as well as the different types of boundary conditions on the nonlinear thermal stability and snap-through phenomenon of the FG nanobeam.

On thermal snap-buckling of FG curved nanobeams

In the present work, we perform the thermal snap-buckling analysis of functionally graded (FG) curved nanobeams. The curved FG nanobeam is subjected to uniform temperature distributions across the thickness. The material properties of the nanobeams are temperature-dependent. Nonlocal strain gradient theory is employed to capture the size effects. Nonlinear governing equations of the FG curved nanobeams with clamped ends are derived via Hamilton’s principle and Akavci’s beam theory. In the end, the effects of thermal loadings, nonlocal parameter, strain gradient parameter, power law index on the snap-buckling of the nanobeams are discussed.

Finite Element Model of Functionally Graded Nanobeam for Free Vibration Analysis

In the present study, free vibration of functionally graded (FG) nanobeam is investigated. The variation of material properties is assumed in the thickness direction according to the power law. FG nanobeam is modeled as Euler-Bernoulli beam with different boundary conditions and investigated based on Eringen’s nonlocal elasticity theory. Governing equations are derived via Hamilton principle. Frequency values are found by using finite element method. FG nanobeam is composed of silicon carbide (SiC) and stainless steel (SUS304). The effects of dimensionless small-scale parameters (e0a/L), power law exponent (k) and boundary conditions on frequencies are examined for FG nanobeam.

A novel meshless particle method for nonlocal analysis of two-directional functionally graded nanobeams

Based on the Reddy–Bickford beam theory (RBBT), a comprehensive study on high-order bending, buckling and free vibration of two-directional functionally graded (FG) nanobeams is presented. The variation of the material properties is assumed to obey arbitrary power-law form in both axial and thickness directions. The equations of motion are derived using Hamilton’s principle, and small-scale effects are captured by nonlocal elasticity theory of Eringen. The RBBT formulation leads to complicated governing equations, and 2D FGM introduces additional stiffness terms. Accordingly, the governing equations cannot be solved by classical analytical schemes, and thus, symmetric smoothed particle hydrodynamics (SSPH) meshless method is adopted as an efficient numerical solution approach. The revised super-Gauss function is used as the kernel function. To validate the developed SSPH code, benchmark problems are studied and the results are compared to the analytical solutions. In this context, excellent agreement is observed. Several numerical examples are included to illustrate the effects of gradient indexes, boundary conditions, size scale parameters, aspect and elastic modulus ratios on static and dynamic responses of two-directional FG nanobeams.

On dynamic instability of magnetically embedded viscoelastic porous FG nanobeam

Dynamic instability of viscoelastic porous functionally graded (FG) nanobeam embedded on visco-Pasternak medium subjected to an axially oscillating loading as well as magnetic field is presented in this research. Porosity-dependent material properties of the porous nanobeam are described via a modified power-law function. Viscoelasticity of the nanostructure is considered based on the Kelvin–Voigt model. Employing Eringen's differential law in conjunction with Timoshenko beam theory (TBT), the motion equations are derived via Hamilton's variational principle. Navier's solution as well as Bolotin's approach are utilized to obtain the dynamic instability region of viscoelastic porous FG nanobeam. Some benchmark results related to the effects of structural damping, length to thickness ratio, foundation type, nonlocal parameter (NP), static load factor, power-law index, porosity volume index and magnetic field on the instability region of porous FG nanobeam are evaluated. The results reveal that with increasing power-law index and structural damping, the pulsation frequency decreases and so, instability region shifts to left side while as magnetic field magnifies, the dynamic instability moves to right side. Also, it is represented that the porosity effect on the dynamic stability of FG nanobeam depends significantly on the values of power-low index and magnetic field.

Vibration of FG viscoelastic nanobeams due to a periodic heat flux via fractional derivative model

In this work, the vibrations of viscoelastic functionally graded Euler–Bernoulli nanostructure beams are investigated using the fractional-order calculus. It is assumed that the functionally graded nanobeam (FGN) is due to a periodic heat flux. FGN can be considered as nonhomogenous composite structures; with continuous structural changes along the thick- ness of the nanobeam usually, it changes from ceramic at the bottom of the metal at the top. Based on the nonlocal model of Eringen, the complete analytical solution to the problem is established using the Laplace transform method. The effects of different parameters are illustrated graphically and discussed. The effects of fractional order, damping coefficient, and periodic frequency of the vibrational behavior of nanobeam was investigated and discussed. It also provides a conceptual idea of the FGN and its distinct advantages compared to other engineering materials. The results obtained in this work can be applied to identify of many nano-structures such as nano-electro mechanical systems (NEMS), nano-actuators, etc.