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.     

Analysis of Timoshenko nanobeams with a nonlocal formulation and meshless method

The nonlocal elasticity theory of Eringen is used to study bending, buckling and free vibration of Timoshenko nanobeams. A meshless method is used to obtain numerical solutions. Results are compared with available analytical solutions. Two different collocation techniques, global (RBF) and local (RBF-FD), are used with multi-quadrics radial basis functions.     

Non-linear vibration of nanobeams with various boundary condition based on nonlocal elasticity theory

In this study, nonlinear vibrations of Euler-Bernoulli nanobeams with various supports condition is investigated. The non-linear equations of motion including stretching of the neutral axis are derived. Forcing and damping effects are included in the analysis. Exact solutions for the mode shapes and frequencies are obtained for the linear part of the problem. For the non-linear problem approximate solutions using perturbation technique is applied to the equations of motion. The different of support cases are investigated and the cases analyzed in detail. The method of multiple time scale that is a perturbation technique is applied to the equations of motion. Natural frequencies and mode shapes for the linear problem are found for the nanobeam. Nonlinear frequencies are calculated; amplitude and phase modulation figures are presented for different cases. Frequency-response curves are drawn.     

Free vibration of magneto-electro-elastic nanobeams based on modified couple stress theory in thermal environment

In this paper, the size-dependent free vibration of magneto-electro-elastic (MEE) nanobeams in thermal environment is investigated. Size effects are taken into account using the modified couple stress theory, which is capable of accounting for higher-order electromechanical coupling, and the equations are developed on the basis of Euler–Bernoulli beam model and using von Karman nonlinear strain. The vibration of hinged–hinged nanobeams is investigated by way of example. Effects of various parameters such as temperature, thickness, and length on natural frequencies are demonstrated, and it is indicated that increased length and decreased thickness lead to decreased nanobeam natural frequencies.     

Nonlinear vibration of nanobeams under electrostatic force based on the nonlocal strain gradient theory

International Journal of Mechanics and Materials in Design - The nonlinear vibration of a nanobeam under electrostatic force is investigated through the nonlocal strain gradient theory. Using...     

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.     

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

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

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.     

Nonlinear flow-induced vibration of a SWCNT with a geometrical imperfection

Based on the nonlocal continuum theory, transverse vibration of a single-walled carbon nanotube (SWCNT) conveying fluid with immovable support conditions is investigated. Unlike previous similar studies, the SWCNT is assumed to be not perfectly straight and initially includes a slight geometrical curvature as an imperfection. The SWCNT is assumed to be embedded in a Pasternak-type foundation. Hamilton’s principle is applied to drive an efficient governing equation of motion, which covers stretching, large deformation, and imperfection nonlinearities. The perturbation method of multi scales (MMS) is applied and the nonlinear flow-induced frequency ratio is analytically calculated. The obtained results reveal that the imperfection of the nanotube at high flow velocities makes the model severely nonlinear, especially when considering the nonlocal effects. A noteworthy observation is that the nonlinear flow-induced frequency ratio is decreased as the imperfection of the nanotube increases. Whereas through a parametric study, the effects of the flow velocity, nonlocal parameter, the stiffness of the elastic foundation, and the boundary conditions (BCs) on this frequency reduction are calculated and discussed widely.     

Nonlinear vibration of smart circular functionally graded plates coupled with piezoelectric layers

Nonlinear vibration analysis of thin circular pre-stressed functionally graded (FG) plate integrated with two uniformly distributed piezoelectric actuator layers with an initial nonlinear large deformation are presented in this paper. Nonlinear governing equations of motion are derived based on classical plate theory (CPT) with von-Karman type geometrical large nonlinear deformations. A nonlinear static problem is solved first to determine the initial stress state and pre-vibration deformations of the plate that is subjected to in-plane forces and applied actuator voltage. By adding an incremental dynamic state to the pre-vibration state, the differential equations that govern the nonlinear vibration behavior of pre-stressed piezoelectrically actuated circular FG plate are derived. An exact series expansion method is used to model the nonlinear electro-mechanical vibration behavior of the structure. Control of the FG plate’s nonlinear deflections and natural frequencies using high control voltages are studied and their nonlinear effects are evaluated. In a parametric study the emphasis is placed on investigating the effect of varying the applied actuator voltage as well as gradient index of FG plate on the dynamic characteristics of the structure.     

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.     

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.     

Free vibration analysis of size-dependent functionally graded microbeams based on the strain gradient Timoshenko beam theory

Investigated herein is the free vibration characteristics of microbeams made of functionally graded materials (FGMs) based on the strain gradient Timoshenko beam theory. The material properties of the functionally graded beams are assumed to be graded in the thickness direction according to the Mori–Tanaka scheme. Using Hamilton’s principle, the equations of motion together with corresponding boundary conditions are obtained for the free vibration analysis of FGM microbeams including size effect. A detailed parametric study is performed to indicate the influences of beam thickness, dimensionless length scale parameter, and slenderness ratio on the natural frequencies of FGM microbeams. Moreover, a comparison between the various beam models on the basis of the classical theory (CT), modified couple stress theory (MCST), and strain gradient theory (SGT) is presented for different values of material property gradient index. It is observed that the value of gradient index play an important role in the vibrational response of the microbeams of lower slenderness ratios. It is further observed that by increasing the length-to-thickness ratio of the microbeam, the value of dimensionless natural frequency tends to decrease for all amounts of the gradient index.     

The thermal effect on vibration of zigzag single walled carbon nanotubes using nonlocal Timoshenko beam theory

Based on nonlocal theory of thermal elasticity mechanics, a nonlocal elastic Timoshenko beam model is developed for free vibration analysis of zigzag single-walled carbon nanotube (SWCNT) considering thermal effect. The nonlocal constitutive equations of Eringen are used in the formulations. The equivalent Young’s modulus and shear modulus for zigzag SWCNT are derived using an energy-equivalent model. Results indicate significant dependence of natural frequencies on the temperature change as well as the chirality of zigzag carbon nanotube. These findings are important in mechanical design considerations of devices that use carbon nanotubes.     

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.     

考虑压电效应时压电陶瓷圆形振子的耦合振动

在考虑压电效应的情况下，本文对压电陶瓷圆形振子的耦合振动进行了研究。根据压电陶瓷圆形振子的运动方程及压电方程，通过引入振子不同振动模式之间的机械耦合系数，分析了振子轴向及径向振动之间的耦合关系，得出了振子耦合振动的电导纳表达式，并导出了共振频率方程。与一维理论的结果相比，由本文理论得出的振子共振频率与实际测量值更加符合。     

Nonlinear free vibration of functionally graded carbon nanotube-reinforced composite beams

This paper investigates the nonlinear free vibration of functionally graded nanocomposite beams reinforced by single-walled carbon nanotubes (SWCNTs) based on Timoshenko beam theory and von Kármán geometric nonlinearity. The material properties of functionally graded carbon nanotube-reinforced composites (FG-CNTRCs) are assumed to be graded in the thickness direction and estimated though the rule of mixture. The Ritz method is employed to derive the governing eigenvalue equation which is then solved by a direct iterative method to obtain the nonlinear vibration frequencies of FG-CNTRC beams with different end supports. A detailed parametric study is conducted to study the influences of nanotube volume fraction, vibration amplitude, slenderness ratio and end supports on the nonlinear free vibration characteristics of FG-CNTRC beams. The results for uniformly distributed carbon nanotube-reinforced composite (UD-CNTRC) beams are also provided for comparison. Numerical results are presented in both tabular and graphical forms to investigate the effects of nanotube volume fraction, vibration amplitude, slenderness ratio, end supports and CNT distribution on the nonlinear free vibration characteristics of FG-CNTRC beams.     

Interlayer influences between double-layer graphene nanoribbons (shear and tensile-compressive) on free vibration using nonlocal elasticity theory

The shear and tensile-compressive effects of van der Waals (vdWs) bindings on the nonlocal vibrational behavior of bilayer graphene nanoribbons (BLGNRs) are simultaneously investigated in the present study. To idealize the structure of BLGNRs incorporating interlayer shear and tensile-compressive influences, a nonlocal sandwich beam (NSB) theory is employed to model the nanoribbon layers as faces and vdWs interactions as core of the NSB. The effects of interlayer moduli are investigated on the first four nonlocal natural frequencies of BLGNRs for various nonlocal parameters, considering that the small-scale effect causes mode shapes involving tensile-compressive effect of vdWs interactions be excited later.     

Nonlinear free and forced vibration behavior of functionally graded plate with piezoelectric layers in thermal environment

In the present study, finite element formulation based on higher order shear deformation plate theory is developed to analyze nonlinear natural frequencies, time and frequency responses of functionally graded plate with surface-bonded piezoelectric layers under thermal, electrical and mechanical loads. The von Karman nonlinear strain–displacement relationship is used to account for the large deflection of the plate. The material properties of functionally graded material (FGM) are assumed temperature-dependent. The temperature field has uniform distribution over the plate surface and varies in the thickness direction. The considered electric field only has non-zero-valued component E_z. Numerical results are presented to study effects of FGM volume fraction exponent, applied voltage in piezoelectric layers, thermal load and vibration amplitude on nonlinear natural frequencies and time response of FGM plate with integrated piezoelectric layers. In addition, nonlinear frequency response diagrams of the plate are presented and effects of different parameters such as FGM volume fraction exponent, temperature gradient, and piezoelectric voltage are investigated.     

An analytical method for free vibration analysis of Timoshenko beam theory applied to cracked nanobeams using a nonlocal elasticity model

This paper is concerned with the free transverse vibration of cracked nanobeams modeled after Eringen's nonlocal elasticity theory and Timoshenko beam theory. The cracked beam is modeled as two segments connected by a rotational spring located at the cracked section. This model promotes discontinuities in rotational displacement due to bending which is proportional to bending moment transmitted by the cracked section. The governing equations of cracked nanobeams with two symmetric and asymmetric boundary conditions are derived; then these equations are solved analytically based on concerning basic standard trigonometric and hyperbolic functions. Besides, the frequency parameters and the vibration modes of cracked nanobeams for variant crack positions, crack ratio, and small scale effect parameters are calculated. The vibration solutions obtained provide a better representation of the vibration behavior of short, stubby, micro/nanobeams where the effects of small scale, transverse shear deformation and rotary inertia are significant.