Functionally Graded Nanobeam Research Articles

Buckling analysis of functionally graded nanobeams via surface stress-driven model

The manuscript investigates the buckling behaviour of Bernoulli-Euler nanobeams composed of Functionally-Graded (FG) materials with different cross-sectional shapes. This analysis is conducted using the surface stress-driven model of elasticity. The nonlocal governing equations for the elastostatic buckling problem are derived employing the principle of virtual work. The study also includes a parametric investigation, presenting and discussing the main results while varying the nonlocal parameter, material gradient index, the cross-sectional shapes and the constraints at the ends of the FG nanobeams. Critical loads are numerically calculated and compared with those obtained by other authors using the classical stress-driven model elasticity.

Investigation of accelerated moving load on dynamic response of FG Timoshenko nanobeam in thermal environment based on nonlocal strain gradient theory

For the first time, this paper investigates the forced vibrations of a functionally graded (FG) nanobeam with an accelerating moving load in a thermal environment. There is no exact coupling solution for the vibrations of nanobeam with an accelerated moving load, so this paper aims to provide a method to obtain an accurate solution for nanoscale structures with an accelerated moving load. The equations of motion are derived using Timoshenko's beam theory and the nonlocal strain gradient theory (NSGT). The Laplace method has been used to solve the coupling and exact differential equations. Then, by inverting Laplace from the coupled equations, an exact solution of the temporal response for FG nanobeam with constant acceleration and initial velocity in a thermal environment was obtained. The natural frequency was compared with previous works for validity and had acceptable results. Finally, the effect of parameters such as changes in acceleration and velocity of moving force, negative acceleration, power law index of FG material, different temperatures, nonlocal parameter, and longitudinal scale parameter on the maximum dynamic displacement of nanobeam is investigated. These results can be used better to design FG nanostructures with accelerated moving loads. Considering the sensitivity of data analysis in nano dimensions, it is necessary to provide an analytical solution method in these dimensions to reduce the error percentage in nano dimensions to zero. which is presented in this work for the first time.

Thermal Buckling of FG Nanobeams via an Indeterminate Integral Variable with Trigonometric Displacement Models in Conjunction with the Gradient Elasticity Theory

This contribution presents a new methodology and a new indeterminate integral variable of displacement field using trigonometric deformation in conjunction with gradient elasticity theory. The aim, for the first time, is to explore the thermal buckling behavior of functionally graded (FG) nanobeam thus elastic gradient theory takes into account the size effect on the critical thermal buckling load of FG nanobeams. Secondly, the impact of various factors, such as nonlocal coefficient, porosity parameter, material index, thermal loading type, and aspect ratio on the critical thermal buckling load of FG nanobeams. The material characteristics are considered to vary in thickness as the power law varies and the pore network is assumed to be empty or filled with low-pressure air.

Investigating hygrothermal bending behavior of FG nanobeams via local/nonlocal stress gradient theory of elasticity with general boundary conditions

The bending response of functionally graded (FG) nanobeams under hygrothermal loading was investigated to emphasize the different scenarios that arise when using simplified and original boundary conditions. The governing equations were derived by using the principle of virtual work on the basis of the local/nonlocal stress gradient theory of elasticity. A Wolfram language code in Mathematica was written by the authors to develop a numerical investigation for different values of the material gradient index, the gradient length parameter, the nonlocal parameter, and considering two distinct types of thermal loading, that is, uniform temperature rise and heat conduction across the thickness of FG nanobeam cross-section.

A size-dependent quasi-3D model for bending and buckling of porous functionally graded curved nanobeam

In this work, we provide an analysis of the bending and buckling properties of functionally graded (FG) nanobeams with trigonometric function-dependent porosity distribution. Herein, a high-order curved nanobeam model considering both shear deformations and thickness stretching effect is established. Two-phase local/nonlocal stress-driven gradient theory is utilized to capture small-scale effects. Hamilton's principle is adopted to obtain the governing equation and boundary conditions, and the analytical solutions, such as exact formulas for lateral displacement and critical buckling load are obtained by employing Navier's method for simply supported boundary conditions. Furthermore, the effects of local volume fraction, nonlocal parameter, material length scale parameter, shear deformation, material distribution, and porosity distribution pattern in stress-driven models are analyzed in detail. The parametric studies indicate that the thickness stretching effect has a different influence on bending and buckling. In addition, the thickness stretching effect and the opening angle show different coupling mechanisms during bending and buckling. On the other hand, the higher-order buckling mode leads to a more significant scale effect and thickness tensile effect, but reduces the influence of the opening angle. Finally, we find that due to the coupling effect between the pore distribution pattern and the orientation bias of the FG nanobeam, the variation of the material gradient index leads to a different order of the dimensionless critical buckling loads predicted by the three pore distributions.

Bending Responses of Bi-Directional Advanced Composite Nanobeams Using Higher Order Nonlocal Strain Gradient Theory

The bending response of two-dimensional (2D) functionally graded (FG) nonlocal strain gradient nanobeams is explored analytically in this work. The longitudinal and transverse orientations vary in material gradation and material characteristics. Kinematic relations of nanobeams are proposed according to hybrid hyperbolic-parabolic functions. The virtual work principle obtains the equilibrium equations, which are then solved using Navier's method. The accuracy and dependability of the suggested analytical model are demonstrated by comparing the results to predictions made in the literature. A thorough parametric study also determines how sensitive the material distribution, the nonlocal length-scale parameter, the strain gradient microstructure-scale parameter, and the geometry are to how the bending response and stresses of 2D FG nanobeams. The results obtained provide benchmark results, which can be used in the design of composite structures.

Bending analysis of functionally graded nanobeams based on stress-driven nonlocal model incorporating surface energy effects

The bending response of Bernoulli-Euler nanobeams made of a functionally graded (FG) material with different cross-sectional shapes is investigated in this manuscript by a stress-driven model incorporating surface energy effects. In particular, the FG nanobeam is composed of a bulk volume and a surface layer regarded as a membrane of zero thickness perfectly adhered to the bulk continuum. The bulk material is made of a mixture of metal and ceramic, whose distributions spatially vary from the bottom to the top surface of the FG nanobeams. The nonlocal governing equations of the elastostatic bending problem are derived by using the virtual work principle. The main results of a parametric investigation are also presented and discussed varying the nonlocal parameter, the material gradient index and the boundary conditions at the ends of the nanobeams. They show how the proposed model is able to study the bending behavior of inflected FG nanobeams including surface effects.

An eigenvalue solution for nonlocal vibration of guide supported perfect/imperfect functionally graded power‐law and sigmoid nanobeams on one‐parameter elastic foundation

AbstractTo the best of the authors' knowledge, this study for the first time, proposes a finite element (FE) model to investigate the size‐dependent vibrational responses of guide supported imperfect functionally graded (FG) nonlocal beams embedded in an elastic foundation in the context of nonlocal elasticity theory (NET). The main objective of this paper is to present a nonlocal FE solution for vibration analysis of imperfect FG nanobeams including nonlocal effect, power‐law distribution function, sigmoid distribution function, even porosity model, uneven porosity model, elastic foundation parameter, and guide support condition. It is considered that perfect/imperfect FG nanobeams are made of silicon carbide (SiC) and aluminum (Al) and that beam properties vary along the height direction. In addition, the nanobeams rest on a one‐parameter elastic foundation and this foundation provides the interaction between the structure and the soil with elastic springs. The frequency values of perfect/imperfect FG power‐law and sigmoid nanobeams obtained by using a nonlocal FE method are shown via tables and figures.

The deflection platform phenomenon of functionally graded flexoelectric simply supported nanobeam

Direct flexoelectricity appears in dielectrics as a kind of coupling effect of electric field and elastic strain gradient. Its strong size dependence makes it effective in nanostructures. In this paper, based on the flexoelectric theory and mixed finite element method (FEM), the mechanical properties of the two-dimensional functionally graded (FG) flexoelectric nanobeam are studied, and the phenomenon of the deflection platform is found. By applying various types of loads to FG nanobeams, the influence of geometric parameters, length scale, and material distribution on the phenomenon of the deflection platform is discussed. According to the simulation results, it is found that the aspect ratio and the length scale of the beam will change the shape of the deflection platform, making it change regularly among various shapes. The geometric size will greatly increase the height of the platform, but it has little effect on the dimensionless deflection. Reasonable material distribution can enhance the internal coupling effect, increase the height of the platform and reduce the required load. Our results suggest that when studying nano-scale flexoelectric models, the deflection platforms or similar phenomena should be considered.

Investigating parametric homogenization models for natural frequency of FGM nano beams

<abstract> <p>This research focuses on exploring the free vibration behavior of functionally graded (FG) nano-beams. To calculate the effective properties of the FG nano-beam, which varies solely in the thickness direction, the four homogenization schemes Mori-Tanaka, Tamura, Reuss and Voigt are employed. This study employs high-order shear deformation nano-beam theory and derives the governing equations of motion using nonlocal differential constitutive relations of Eringen. Hamilton's principle is utilized in conjunction with the refined three variables beam theory. The consideration of a length scale parameter accounts for small-scale effects. Analytical solutions are obtained for a simply supported FG nano-beam and compared with existing literature solutions. The research also investigates the influence of different homogenization schemes, the nonlocal parameter, beam aspect ratio and various material compositions on the dynamic response of the FG nano-beam.</p> </abstract>

Functionally graded nonlocal thermoelastic nanobeam with memory-dependent derivatives

The purpose of this study is to investigate vibrations in 2D functionally graded nanobeams (FGN) with memory-dependent derivatives. A sinusoidal variation of temperature is assumed. The dimensionless expressions for axial displacement, thermal moment, lateral deflection, strain and temperature distribution are found in the transformed domain using Laplace Transforms, and the expressions in the physical domain are derived by numerical inversion techniques. The nanobeam is simply supported at the both ends and have constant temperatures. The FGN is a non-homogenous composite structure with constant structural variations along with the layer thickness, changing from ceramic at the bottom to metal at the top. Adding non-local MDD to thermoelastic models opens up new possibilities for the study of thermal deformations in solid mechanics. The effect of different kernel functions and periodic frequency of thermal vibration is illustrated graphically for lateral deflection, axial displacement, strain, temperature, and thermal moment.Article highlightsA novel model of vibrations in a functionally graded nanobeams is presented.The medium is subjected to sinusoidal variation of temperature.Dynamic response of memory dependent derivative theory of thermoelasticity and non-local parameter is investigated.The effects of kernel functions and periodic frequency of thermal vibration on all physical fields are investigated and shown graphically.

Nonlinear dynamic buckling and vibration of thermally post-buckled temperature-dependent FG porous nanobeams based on the nonlocal theory

In this paper, nonlinear dynamic snap-through buckling and vibration behavior of the thermally post-buckled functionally graded (FG) porous nanobeams subjected to static and sudden mechanical loads are investigated utilizing the nonlocal elasticity theory. The physical properties of the nanobeam are considered to be functions of temperature based on the Touloukian model. In addition, to describe the FG porous materials, two different patterns of porosity distribution are adopted using trigonometric functions through the thickness of the nanobeam. The equations of motion in conjunction with the von-Kármán nonlinear assumption are established in the framework of Hamilton’s principle. By employing the Chebyshev-Ritz procedure, the nonlinear equations are discretized for three types of edge supports. Following that, the cylindrical arc-length technique is employed to assess the vibrational responses of the post-buckled nanobeam during static snap-through buckling. To evaluate the nonlinear dynamic buckling of the graded nanobeam under a sudden dynamic load, the Newmark time integration scheme together with the Newton-Raphson iterative method are utilized. Next, by means of the Budiansky-Roth criterion and the phase-plane approach, the dynamic snap-through loads are identified. After validating the developed mathematical model, a comprehensive investigation is carried out to determine the role of various physical and geometrical parameters on the dynamic snap-through buckling and vibration characteristics of the post-buckled FG nanobeams.

Application of the Higher-Order Hamilton Approach to the Nonlinear Free Vibrations Analysis of Porous FG Nano-Beams in a Hygrothermal Environment Based on a Local/Nonlocal Stress Gradient Model of Elasticity.

Nonlinear transverse free vibrations of porous functionally-graded (FG) Bernoulli–Euler nanobeams in hygrothermal environments through the local/nonlocal stress gradient theory of elasticity were studied. By using the Galerkin method, the governing equations were reduced to a nonlinear ordinary differential equation. The closed form analytical solution of the nonlinear natural flexural frequency was then established using the higher-order Hamiltonian approach to nonlinear oscillators. A numerical investigation was developed to analyze the influence of different parameters both on the thermo-elastic material properties and the structural response, such as material gradient index, porosity volume fraction, nonlocal parameter, gradient length parameter, mixture parameter, and the amplitude of the nonlinear oscillator on the nonlinear flexural vibrations of metal–ceramic FG porous Bernoulli–Euler nano-beams.

Modelling, analyzing and simulating the complex dynamics of mass sensors based on a functionally graded nanobeam model

This paper proposes a theoretical, miniature, structural model, a functionally graded nanobeam (FGNB), to investigate the complex dynamic mechanism and the responses of particle mass sensors to an electric load in practical applications. The Kelvin-Voigt model and the nonlocal strain gradient theory (NSGT) describe the viscoelastic and size-dependent effects. A generalized Hamilton's principle is used to obtain a mathematical model of FGNB with attached particles. The multiple-scale method and global perturbation method of an infinite-dimensional system are applied to derive the threshold of the homoclinic chaos phenomenon. Then, the differential quadrature method (DQM) is employed to numerically test the validity of the proposed model and the theoretical predication. Numerical results highlight the FG index, size-dependent parameters, and viscoelastic coefficient to determine the influencing vibration characteristics of the FGNB. The results are useful for studying the vibration responses of sensitive and high-efficiency sensors.

Nonlinear Vibration and Stability Analysis of Functionally Graded Nanobeam Subjected to External Parametric Excitation and Thermal Load

Analysis of vibration stability of simply supported Euler-Bernoulli functionally graded (FG) nanobeam embedded in viscous elastic medium with thermal effect under external parametric excitation is presented in this work. An attempt has been made for the first time is investigating the effect of thermal load on dynamic behavior, amplitude response, instability region and bifurcation points of functionally graded nanobeam. Thermal loads are supposed to be uniform, linear or nonlinear distribution along the thickness direction. Nonlocal continuum theory and the principle of the minimum total potential energy are applied to derive the governing equations. The partial differential equations (PDE) are transported to the ordinary differential equations (ODE) by using the Petrov-Galerkin method and the multiple time scales method are manipulated to solve the motion equation. To study the effect of external parametric excitation and thermal effect, different temperature distributions along the thickness such as uniform, linear, and nonlinear distribution are considered. Moreover, stable and unstable regions and bifurcation points are determined. It is obtained that the thermal load can affect the amplitude response of FG nanobeam. Also, it is observed that the instability of the system is affected by the detuning parameter and the parametric excitation amplitude plays great role in the instability of system. Nanobeams are used in many devices like nanoresonators, nanosensors and nanoswitches. This paper is helpful for designing and manufacturing nanoscale structures specially nanoresonators under different thermal loads.

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

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

Closed-form expressions for bending and buckling of functionally graded nanobeams by the Laplace transform

This paper presents analytical solutions for bending and buckling of nonlocal functionally graded (FG) Euler–Bernoulli (EB) nanobeams. Material gradation along the thickness direction could be defined by a power function (P-FG), a sigmoidal function (S-FG), and an exponential function (E-FG). Laplace transform is applied to the differential form of the equation of motion of the nonlocal elasticity theory. Closed-form expressions for bending deflection and critical buckling load of FG nanobeams are derived. Effects of material gradations as well as the nonlocal parameter are examined. It is found that bending displacements and critical buckling loads could be controlled by an appropriate choice of material distribution parameter for P-FG nanobeams. The presented results also demonstrate the influences of factors such as the choice of material gradation, power-law index, and nonlocal parameter on bending and buckling behavior.

Buckling analysis of functionally graded beams with periodic nanostructures using doublet mechanics theory

Buckling analysis of functionally graded (FG) nanobeams is examined using doublet mechanics theory. The material properties of FG nanobeams change with the thickness coordinate. A periodic nanostructure model is considered in FG nanobeams which has a simple crystal square lattice type and Euler–Bernoulli beam theory is used in the formulation. Softening or hardening material behaviour has been observed by changing chiral angle of the considered FG periodic nanobeams in the present doublet mechanics theory. Unlike other size dependent theories such as nonlocal stress gradient elasticity theory, couple stress theory, strain gradient theory, this mechanical response (softening or hardening) is seen for the first time in doublet mechanics theory. Mechanical material responses are directly affected by the atomic structure of the considered material in the doublet mechanics theory. Firstly, micro-stress and micro-strain relations are obtained for the considered nanostructure model in doublet mechanics theory. Then, these microequations are transformed to macroequations in the present doublet mechanics theory. Thus, more physical and accurate mechanical results can be obtained in nanostructures using the doublet mechanics theory. After developing the mathematical formulations of FG periodic nanobeams, Ritz method is applied to obtain the critical buckling loads for different boundary conditions. Comparison of example studies with the present doublet mechanics model is presented for verification, and effects of chiral angle on stability response of periodic FG nanobeams are discussed.

Hygro-Thermal Vibrations of Porous FG Nano-Beams Based on Local/Nonlocal Stress Gradient Theory of Elasticity.

In this manuscript the dynamic response of porous functionally-graded (FG) Bernoulli–Euler nano-beams subjected to hygro-thermal environments is investigated by the local/nonlocal stress gradient theory of elasticity. In particular, the influence of several parameters on both the thermo-elastic material properties and the structural response of the FG nano-beams, such as material gradient index, porosity volume fraction, nonlocal parameter, gradient length parameter, and mixture parameter is examined. It is shown how the proposed approach is able to capture the dynamic behavior of porous functionally graded Bernoulli–Euler nano-beams under hygro-thermal loads and leads to well-posed structural problems of nano-mechanics.

Nonlocal finite element model for the bending and buckling analysis of functionally graded nanobeams using a novel shear deformation theory

In this study, an efficient nonlocal finite element model is developed to investigate the bending and buckling behavior of functionally graded (FG) nanobeams. New two-node beam element with eight degrees of freedom is formulated based on the recently refined higher order shear deformation theory proposed by the authors. The present theory can provide an accurate parabolic distribution of transverse shear stress through the thickness direction satisfying the traction free boundary conditions needless of any shear correction factor. In order to capture the small size effect, Eringen’s nonlocal elasticity theory is incorporated. The material properties of the FG nanobeams are assumed to vary continuously through the thickness direction according to the power-law form. The performance and reliability of the proposed model is demonstrated by comparing the author’s results with those available in the literature. The numerical results show that the present element model is free of shear locking and has a high accuracy and fast rate of convergence. Moreover, a detailed numerical study is carried out to examine the effects of several parameters such as boundary conditions, power-law index, nonlocal parameter and length-to-height ratio on the deflection and critical buckling load of the FG nanobeams. Many new results are also reported in the current study, which will serve as a benchmark for future research.