Vibration analysis of piezoelectrically actuated curved nanosize FG beams via a nonlocal strain-electric field gradient theory

In this article, nonlocal free vibration analysis of curved functionally graded piezoelectric (FGP) nanobeams is conducted using a Navier-type solution method. The model contains a nonlocal stress field parameter and also a nonlocal strain-electric field gradient parameter to capture the size effects. Inclusion of these nonlocal parameters introduces both stiffness-softening and stiffness-hardening effects in the analysis of curved nanobeams. Nonlocal governing equations of curved FGP nanobeam are obtained from Hamilton's principle based on the Euler–Bernoulli beam model. The results are validated with those of curved FG nanobeams available in the literature. Finally, the influences of electric voltage, length scale parameter, nonlocal parameter, opening angle, material composition, and slenderness ratio on vibrational characteristics of nanosize curved FG piezoelectric beams are explored. These results may be useful in accurate analysis and design of smart nanostructures constructed from piezoelectric materials.     

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.     

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

In 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.     

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.     

Electro-mechanical vibration of smart piezoelectric FG plates with porosities according to a refined four-variable theory

Here, free vibration analysis of functionally graded piezoelectric (FGP) plates with porosities is carried out based on refined four-unknown plate theory. The present plate theory captures shear deformation impacts needless of shear correction factor. A modified power-law model is adopted to describe the graded material properties of a functionally graded piezoelectric plate. Implementing an analytical approach, which satisfies different boundary conditions, governing equations derived from Hamilton's principle are solved. The obtained results are compared with those provided in the literature. The impacts of applied voltage, porosity distribution, material graduation, plate geometrical parameters, and boundary conditions on vibration of porous FGP plate are discussed.     

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.     

Nonlinear vibration of the piezoelectric nanobeams based on the nonlocal theory

This paper investigates the nonlinear vibration of the piezoelectric nanobeams based on the nonlocal theory and Timoshenko beam theory. The piezoelectric nanobeam is subjected to an applied voltage and a uniform temperature change. The nonlinear governing equations and boundary conditions are derived by using the Hamilton principle and discretized by using the differential quadrature (DQ) method. A direct iterative method is employed to determine the nonlinear frequencies and mode shapes of the piezoelectric nanobeams. A detailed parametric study is conducted to study the influences of the nonlocal parameter, temperature change and external electric voltage on the size-dependent nonlinear vibration characteristics of the piezoelectric nanobeams.     

Longitudinal varying elastic foundation effects on vibration behavior of axially graded nanobeams via nonlocal strain gradient elasticity theory

In this research, vibration characteristics of axially functionally graded nanobeams resting on variable elastic foundation are investigated based on nonlocal strain gradient theory. This nonclassical nanobeam model contains a length scale parameter to explore the influence of strain gradients and also a nonlocal parameter to study the long-range interactions between the particles. The present model can degenerate into the classical models if the material length scale parameter and the nonlocal stress field parameter are both taken to be zero. Elastic foundation consists of two layers: a Winkler layer with variable stiffness and a Pasternak layer with constant stiffness. Linear, parabolic and sinusoidal variations of Winkler foundation in longitudinal direction are considered. Material properties are graded axially via a power-law distribution scheme. Hamilton's principle is employed to derive the governing equations that are solved applying a Galerkin-based solution for different boundary edges. Comparison study is also performed to verify the present formulation with those of previous papers. Results are presented to investigate the influences of the nonlocal and length scale parameters, various material compositions, elastic foundation parameters, type of foundation and various boundary conditions on the vibration frequencies of AFG nanobeams in detail.     

A unified nonlocal formulation for bending,buckling and free vibration analysis of nanobeams

Abstract

A unified nonlocal formulation is developed for the bending, buckling, and vibration analysis of nanobeams. Theoretical formulations of eighteen nonlocal beam theories are presented by using unified formulation. Small scale effect is considered based on the nonlocal differential constitutive relations of Eringen. The governing equations of motion and associated boundary conditions of the nanobeam are derived using Hamilton's principle. Closed form solutions are presented for a simply supported boundary condition using Navier's solution technique. Numerical results for axial and transverse shear stress are first time presented in this study which will serve as a benchmark for the future research.     

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

Size-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.     

Flapwise bending vibration analysis of rotary tapered functionally graded nanobeam in thermal environment

In this article, transverse vibration of rotary functionally graded size-dependent tapered Bernoulli–Euler nanobeam in thermal environment at low temperature has been investigated based on Eringen's nonlocal theory for cantilever and propped cantilever boundary conditions. Material properties of FG nanobeam are supposed to be temperature dependant and vary continuously along the thickness according to the power-law form. The axial force is also included in the model as the true spatial variation due to the rotation. The nonlocal equations of motion are derived through Hamilton's principle and they are solved by the differential quadrature method. Validations are done by comparing available literatures and obtained results, which reveal the accuracy of the applied method. 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 angular velocity, material distribution profile, different boundary conditions, small-scale parameter and rate of cross-section change on the first three nondimensional natural frequencies of the rotary FG nanobeam in detail. Numerical results are presented to serve as benchmarks for the application and the design of nanoelectronic, nanodrive devices and nanomotor, in which nanobeams act as basic elements. They can also be useful as valuable sources for validating other approaches and approximate methods.The results of this article are suitable in designation of micromachines, such as micromotors and micro-rotors.     

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.     

Nonlinear vibration of piezoelectric nanoplates using nonlocal Mindlin plate theory

ABSTRACT

This article investigates the nonlinear vibration of piezoelectric nanoplate with combined thermo-electric loads under various boundary conditions. The piezoelectric nanoplate model is developed by using the Mindlin plate theory and nonlocal theory. The von Karman type nonlinearity and nonlocal constitutive relationships are employed to derive governing equations through Hamilton's principle. The differential quadrature method is used to discretize the governing equations, which are then solved through a direct iterative method. A detailed parametric study is conducted to examine the effects of the nonlocal parameter, external electric voltage, and temperature rise on the nonlinear vibration characteristics of piezoelectric nanoplates.     

Size-dependent free flexural vibrational behavior of functionally graded nanobeams using semi-analytical differential transform method

In this paper nonlocal Euler–Bernoulli beam theory is employed for vibration analysis of functionally graded (FG) size-dependent nanobeams by using Navier-based analytical method and a semi analytical differential transform method. Two kinds of mathematical models, namely, power law and Mori-Tanaka models are considered. The nonlocal Eringen theory takes into account the effect of small size, which enables the present model to become effective in the analysis and design of nanosensors and nanoactuators. Governing equations are derived through Hamilton's principle and they are solved applying semi analytical differential transform method (DTM). It is demonstrated that the DTM has high precision and computational efficiency in the vibration analysis of FG nanobeams. The good agreement between the results of this article and those available in literature validated the presented approach. 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 small scale effects, different material compositions, mode number and thickness ratio on the normalized natural frequencies of the FG nanobeams in detail. It is explicitly shown that the vibration of a FG nanobeams is significantly influenced by these effects. Numerical results are presented to serve as benchmarks for future analyses of FG nanobeams.     

Nonlinear analysis of forced vibration of nonlocal third-order shear deformable beam model of magneto–electro–thermo elastic nanobeams

This paper deals with the forced vibration behavior of nonlocal third-order shear deformable beam model of magneto–electro–thermo elastic (METE) nanobeams based on the nonlocal elasticity theory in conjunction with the von Kármán geometric nonlinearity. The METE nanobeam is assumed to be subjected to the external electric potential, magnetic potential and constant temperature rise. Based on the Hamilton principle, the nonlinear governing equations and corresponding boundary conditions are established and discretized using the generalized differential quadrature (GDQ) method. Thereafter, using a Galerkin-based numerical technique, the set of nonlinear governing equations is reduced into a time-varying set of ordinary differential equations of Duffing type. The pseudo-arc length continuum scheme is then adopted to solve the vectorized form of nonlinear parameterized equations. Finally, a comprehensive study is conducted to get an insight into the effects of different parameters such as nonlocal parameter, slenderness ratio, initial electric potential, initial external magnetic potential, temperature rise and type of boundary conditions on the natural frequency and forced vibration characteristics of METE nanobeams.     

Torsion of functionally graded nonlocal viscoelastic circular nanobeams

The elastostatic problem of functionally graded circular nanobeams under torsion, with nonlocal elastic behavior proposed by Eringen, is preliminarily formulated. Exact solutions are detected for nanobeams with arbitrary axial gradations of elastic properties and radially quadratic distributions of shear moduli. Extension of the treatment to nonlocal viscoelastic composite circular nanobeams is then performed. An effective solution procedure based on Laplace transform is developed, providing a new correspondence principle in nonlocal viscoelasticity for functionally graded materials. Displacements, shear strains and stresses are established for nonlocal viscoelastic nanobeams made of periodic fiber-reinforced materials, with polymeric matrix described by a Maxwell model connected in series with a Voigt model.     

Surface effects on nonlinear vibration of embedded functionally graded nanoplates via higher order shear deformation plate theory

In this paper, free vibration behavior of functionally nanoplate resting on a Pasternak linear elastic foundation is investigated. The study is based on third-order shear deformation plate theory with small scale effects and von Karman nonlinearity, in conjunction with Gurtin–Murdoch surface continuum theory. It is assumed that functionally graded (FG) material distribution varies continuously in the thickness direction as a power law function and the effective material properties are calculated by the use of Mori–Tanaka homogenization scheme. The governing and boundary equations, derived using Hamilton's principle are solved through extending the generalized differential quadrature method. Finally, the effects of power-law distribution, nonlocal parameter, nondimensional thickness, aspect of the plate, and surface parameters on the natural frequencies of FG rectangular nanoplates for different boundary conditions are investigated.     

Surface effects on free vibration of piezoelectric functionally graded nanobeams using nonlocal elasticity

Free vibration of functionally graded material (FGM) nanobeams is investigated by considering surface effects including surface elasticity, surface stress, and surface density as well as the piezoelectric field using nonlocal elasticity theory. The balance conditions between the nanobeam bulk and its surfaces are satisfied assuming a cubic variation for the normal stress, ${\sigma_{zz}}$ , through the piezoelectric FG nanobeam thickness. Accordingly, the surface density is introduced into the governing equation of the free vibration of nanobeams. The results are obtained for various gradient indices, voltage values of the piezoelectric field, nanobeam lengths, and mode numbers. It is shown that making changes to voltage values and modifying mechanical properties of piezoelectric FGM nanobeams are two main approaches to achieve desired natural frequencies.     

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

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