Thermo-mechanical vibration of orthotropic cantilever and propped cantilever nanoplate using generalized differential quadrature method

In this article, the vibration frequency of an orthotropic nanoplate under the effect of temperature change is investigated. Using nonlocal elasticity theory, governing equations are derived. Based on the generalized differential quadrature method for cantilever and propped cantilever boundary conditions, the frequencies of orthotropic nanoplates are considered and the obtained results are compared with valid reported results in the literature. The effects of temperature variation, small scale, different boundary conditions, aspect ratio, and length on natural nondimensional frequencies are studied. The present analysis is applicable for the design of rotating and nonrotating nano-devices that make use of thermo-mechanical vibration characteristics of nanoplates.     

An analytical solution for thermal vibration of compositionally graded nanoplates with arbitrary boundary conditions based on physical neutral surface position

In this article, an analytical method is presented for thermo-mechanical vibration analysis of functionally graded (FG) nanoplates with different boundary conditions under various thermal loadings including uniform, linear, and nonlinear temperature rise via a four-variable plate theory considering neutral surface position. The temperature-dependent material properties of FG nanoplate vary gradually along the thickness according to the Mori-Tanaka homogenization scheme. The exactness of solution is confirmed by comparing obtained results with those provided in the literature. A parametric study is performed investigating the effects of nonlocal parameter, temperature fields, gradient index, and boundary conditions on vibration behavior of FG nanoplates.     

Nonlocal free vibration of orthotropic non-prismatic skew nanoplates

The free vibration of orthotropic non-prismatic skew nanoplate based on the first-order shear deformation theory (FSDT) in conjunction with Eringen’s nonlocal elasticity theory is presented. As a simple, accurate and low computational effort numerical method, the differential quadrature method (DQM) is employed to solve the related differential equations. For this purpose, after deriving the equations of motion and the related boundary conditions, they are transformed from skewed physical domain to rectangular computational domain of DQM and accordingly discretized. After validating the formulation and method of solution, the effects of nonlocal parameter in combination with geometrical parameters and boundary conditions on the natural frequencies of the orthotropic skew nanoplates 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.     

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.     

Nonlocal buckling and vibration analysis of thick rectangular nanoplates using finite strip method based on refined plate theory

The buckling and vibration of thick rectangular nanoplates is analyzed in this article. A graphene sheet is theoretically assumed and modeled as a nanoplate in this study. The two-variable refined plate theory (RPT) is applied to obtain the differential equations of the nanoplate. The theory accounts for parabolic variation of transverse shear stress through the thickness of the plate without using a shear correction factor. Besides, the analysis is based on the nonlocal theory of elasticity to take the small-scale effects into account. For the first time, the finite strip method (FSM) based on RPT is employed to study the vibration and buckling behavior of nanoplates and graphene sheets. Hamilton’s principle is employed to obtain the differential equations of the nanoplate. The stiffness, stability and mass matrices of the nanoplate are formed using the FSM. The displacement functions of the strips are evaluated using continuous harmonic function series which satisfy the boundary conditions in one direction and a piecewise interpolation polynomial in the other direction. A matrix eigenvalue problem is solved to find the free vibration frequency and buckling load of the nanoplates subjected to different types of in-plane loadings including the uniform and nonuniform uni-axial and biaxial compression. Comparison studies are presented to verify the validity and accuracy of the proposed nonlocal refined finite strip method. Furthermore, a number of examples are presented to investigate the effects of various parameters (e.g., boundary conditions, nonlocal parameter, aspect ratio, type of loading) on the results.     

Exact solution for nonlocal vibration of double-orthotropic nanoplates embedded in elastic medium

An analytical approach for free vibration analysis of all edges simply-supported double-orthotropic nanoplates is presented. The two nanoplates are assumed to be bonded by an internal elastic medium and surrounded by external elastic foundation. The governing equations are derived based on the nonlocal theory and the expressions of the natural frequencies are proposed in an explicit form. The suggested model is justified by a good agreement between the results given by present model and available data in literature. The model is used to study the vibration of double-orthotropic nanoplates for three typical deformation modes. The influences of small scale coefficient, stiffness of the external and internal mediums and aspect ratio on the frequencies of the double-orthotropic nanoplates are also elucidated.     

Analysis of flexoelectric response in nanobeams using nonlocal theory of elasticity

This paper is concerned with the derivation of exact solutions for the static responses of simply supported nonlocal flexoelectric nanobeams. Considering both the direct and the converse flexoelectric effects, and employing the nonlocal theory of elasticity, the governing equations and the associated boundary conditions of the beams are derived to obtain the exact solutions for the displacements, nonlocal stresses and the electric potential in the beams. Both the direct and the converse flexoelectric effects are influenced by the nonlocal parameter. Active beams significantly counteract the applied mechanical load by virtue of the converse flexoelectric effect. The normal and the transverse shear deformations in the beams are affected by the converse flexoelectric effect resulting in the coupling of bending and stretching deformations in the beams even if the beams are homogeneous. Because of the consideration of the nonlocal theory of elasticity, the nonlocal stiffness of the beam appears to be less than the classical stiffness of the beam. The nonlocal elasticity does not influence the stresses of the passive beam while the nonlocal stresses in the active beam due to converse flexoeletric effect are less than the local or classical stresses in the active beam. The benchmark results presented here may be useful for verifying the numerical model and experimental results for nonlocal flexoelectric nanobeams. The present study suggests that the flexoelectric nanobeams may be effectively exploited for developing advanced smart nanosensors and nanoactuators. The research work carried out here also conveys that the nonlocal theory of elasticity must be employed for accurate analysis of flexoelectric solids.     

Small scale effect on the free vibration of orthotropic arbitrary straight-sided quadrilateral nanoplates

As a first endeavor, the free vibration of orthotropic arbitrary straight-sided quadrilateral nanoplates is investigated using the nonlocal elasticity theory. The formulation is derived based on the first order shear deformation theory (FSDT). The solution procedure is based on the transformation of the governing equations from physical domain to computational domain and then discretization of the spatial derivatives by employing the differential quadrature method (DQM) as an efficient and accurate numerical tool. The formulation and the method of the solution are firstly validated by carrying out the comparison studies for the isotropic and orthotropic rectangular plates against existing results in literature. Then, the effects of nonlocal parameter in combination with the geometrical shape parameters, thickness-to-length ratio and the boundary conditions on the frequency parameters of the nanoplates are investigated.     

Vibration analysis of orthotropic graphene sheets using nonlocal elasticity theory and differential quadrature method

The small scale effect on the vibration analysis of orthotropic single layered graphene sheets (SLGS) is studied. Elastic theory of the graphene sheets is reformulated using the nonlocal differential constitutive relations of Eringen. The equations of motion of the nonlocal theories are derived for the graphene sheets. Differential quadrature method (DQM) is employed to solve the governing differential equations for various boundary conditions. Nonlocal theories are employed to bring out the small scale effect of the nonlocal parameter on the natural frequencies of the orthotropic graphene sheets. Further, effects of (i) nonlocal parameter, (ii) size of the graphene sheets, (iii) material properties and (iv) boundary conditions on nondimensional vibration frequencies are investigated.     

Hygrothermal deformation of orthotropic nanoplates based on the state-space concept

This paper deals with the investigation of the effect of hygrothermal conditions on the bending of nanoplates using Levy type solution model employing the state-space concept. The nanoplates are assumed to be subjected to a hygrothermal environment. The two-unknown function plate theory is used to derive the governing differential equations on the basis of Eringen's nonlocal elasticity theory. The governing equations contain the small scale effect as well as hygrothermal and mechanical effects. These equations are converted into a set of first-order linear ordinary differential equations with constant coefficients. Analytical solution of bending response for nanoplates under combinations of simply supported, clamped and free boundary conditions is obtained. Comparison of the results with those being in the open literature is made. The influences played by small scale parameter, temperature rise, the degree of moisture concentration, boundary conditions, plate aspect ratio and side-to-thickness ratio are studied.     

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.     

Nonlocal and surface effects on the buckling behavior of flexoelectric sandwich nanobeams

In this article, an analytical approach is presented to study the surface and flexoelectric effects on the buckling characteristics of an embedded piezoelectric sandwich nanobeam. According to the nonlocal elasticity theory, the flexoelectricity is believed to be authentic for size-dependent properties in nanostructures. The boundary conditions and the governing equations are derived by Hamilton's principle and are solved by Navier method. The results obtained from the present work show that the nonlocal term has an important reduction on the critical load and also the flexoelectricity shows an increasing influence on the buckling loads of the sandwich nanobeam, especially at lower thicknesses.     

Vibration analysis of orthotropic graphene sheets embedded in Pasternak elastic medium using nonlocal elasticity theory and differential quadrature method

In this paper, the small scale effect on the vibration analysis of orthotropic single layered graphene sheets embedded in elastic medium is studied. Elastic theory of the graphene sheets is reformulated using the nonlocal differential constitutive relations of Eringen. Both Winkler-type and Pasternak-type foundation models are employed to simulate the interaction between the graphene sheet and surrounding elastic medium. Using the principle of virtual work the governing differential equations are derived. Differential quadrature method is employed to solve the governing differential equations for various boundary conditions. Nonlocal theories are employed to bring out the small scale effect of the nonlocal parameter on the natural frequencies of the orthotropic graphene sheets embedded in elastic medium. Further, effects of (i) nonlocal parameter, (ii) size of the graphene sheets, (iii) stiffness of surrounding elastic medium and (iv) boundary conditions on non-dimensional vibration frequencies are investigated.     

Effect of nonlocal elasticity on the performance of a flexoelectric layer as a distributed actuator of nanobeams

The paper is concerned with the development of finite element model for the static analysis of smart nanobeams integrated with a flexoelectric layer on its top surface, using nonlocal elastic theory. The flexoelectric layer acts as a distributed actuator of the nanobeam. A layerwise displacement theory has been used to derive the element stiffness matrices from variational principles incorporating nonlocal effects. The finite element model for nonlocal response of the beams has been validated with the exact solution for the case of a simply supported standalone flexoelectric layer. Also, the finite element model of the simply supported smart beam has been validated with exact solutions and numerical models for the local elastic case. The performance of the flexoelectric actuator has been compared for different values of nonlocal parameters and different combinations of nonlocal and local elastic substrate and flexoelectric layer. Further, the model developed has been utlized for investigating the performance of the active flexoelectric layer in case of cantilever beam, for which the exact solutions are not available.     

Semi-analytical solution for free transverse vibrations of Euler–Bernoulli nanobeams with manifold concentrated masses

In the present study, transverse vibrations of nanobeams with manifold concentrated masses, resting on Winkler elastic foundations, are investigated. The model is based on the theory of nonlocal elasticity in the presence of concentrated masses applied to Euler–Bernoulli beams. A closed-form expression for the transverse vibration modes of Euler–Bernoulli beams is presented. The proposed expressions are provided explicitly as the function of two integrated constants, which are determined by the standard boundary conditions. The utilization of the boundary conditions leads to definite terms of natural frequency equations. The natural frequencies and vibration modes of the concerned nanobeams with different numbers of concentrated masses in different positions under some typical boundary conditions (simply supported, cantilevered, and clamped–clamped) have been analyzed by means of the proposed closed–form expressions in order to show their efficiency. It is worth mentioning that the effect of various nonlocal length parameters and Winkler modulus on natural frequencies and vibration modes are also discussed. Finally, the results are compared with those corresponding to a classical local model.     

考虑小尺度效应的微圆轴扭转振动

在微机电系统中,微纳米构件常常表现出尺度效应。基于非局部弹性理论,建立了微圆轴的扭转振动模型,并结合3种常见的边界条件,给出了具体的算例。结果表明:对比于经典连续力学,非局部弹性理论预言的圆轴扭转振动固有频率下降,并且微圆轴的外特征尺度即横截面半径越小,二者相差越大;振动频率的阶数越高,影响也越明显。随着截面半径的增加,振动频率下降并且非局部尺度效应逐渐消失。同时考察了扭转振动的模态函数和相对转角,发现前者与经典弹性理论结果一致。此外还讨论了材料内禀尺度的选取问题,以数值算例证明了内禀尺度与材料晶格常数非常接近,晶格常数可近似用作微纳米力学中材料的内禀尺度参数。     

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 free vibration of embedded double-walled carbon nanotubes with layerwise boundary conditions

This paper investigates the large-amplitude free vibration of a double-walled carbon nanotube (DWCNT) surrounded by an elastic medium in the presence of temperature change. Based on continuum mechanics, a nonlocal elastic beam model is employed in which nanotubes are coupled together via the van der Waals (vdW) interlayer interactions. The Pasternak foundation model and a nonlinear vdW model are utilized to describe the surrounding elastic medium effect and the vdW interlayer interactions, respectively. DWCNTs with different boundary conditions are analyzed utilizing the Timoshenko beam theory that considers the shear deformation and rotary inertia effects. The governing equations are derived from Hamilton’s principle; the Galerkin method is utilized to discretize the governing equations. The influences of the nonlocal parameter, spring constant, carbon nanotube aspect ratio, and temperature change on the nonlinear free vibration characteristics of a double-walled carbon nanotube with different boundary conditions are thoroughly investigated. It is deduced that the nonlocal parameter, spring constant, and the aspect ratio play significant roles for the value of the nonlinear frequency. Also, the temperature change and the type of boundary conditions have an effect on the nonlinear frequency.     

Axisymmetric free and forced vibrations of initially stressed circular nanoplates embedded in an elastic medium

As a first endeavor, the axisymmetric free and forced vibrations of circular single- and double-layered nanoplates under initial in-plane radial stresses and embedded in an elastic medium are investigated. The governing equations are derived by decoupling the nonlocal constitutive equations of the Eringen theory in polar coordinates in conjunction with the classical plate theory. The elastic medium is modeled as a two-parameter elastic foundation (Pasternak type). Galerkin’s method is employed to solve the resulting equation for vibration frequencies and dynamic response. The effects of small scale together with the other parameters such as initial in-plane load, Winkler and shear elastic foundation coefficients and the radius of the nanoplate are investigated. It is shown that the corresponding natural frequencies obtained by nonlocal elasticity theory are very different from those predicted by classical elasticity theory when the radius of the nanoplate is less than an approximate limit value.