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.     

Scale effects on buckling analysis of orthotropic nanoplates based on nonlocal two-variable refined plate theory

This article presents the buckling analysis of orthotropic nanoplates such as graphene using the two-variable refined plate theory and nonlocal small-scale effects. The two-variable refined plate theory takes account of transverse shear effects and parabolic distribution of the transverse shear strains through the thickness of the plate, hence it is unnecessary to use shear correction factors. Nonlocal governing equations of motion for the monolayer graphene are derived from the principle of virtual displacements. The closed-form solution for buckling load of a simply supported rectangular orthotropic nanoplate subjected to in-plane loading has been obtained by using the Navier’s method. Numerical results obtained by the present theory are compared with first-order shear deformation theory for various shear correction factors. It has been proven that the nondimensional buckling load of the orthotropic nanoplate is always smaller than that of the isotropic nanoplate. It is also shown that small-scale effects contribute significantly to the mechanical behavior of orthotropic graphene sheets and cannot be neglected. Further, buckling load decreases with the increase of the nonlocal scale parameter value. The effects of the mode number, compression ratio and aspect ratio on the buckling load of the orthotropic nanoplate are also captured and discussed in detail. The results presented in this work may provide useful guidance for design and development of orthotropic graphene based nanodevices that make use of the buckling properties of orthotropic 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.     

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.     

Vibration of rotating functionally graded Timoshenko nano-beams with nonlinear thermal distribution

The vibration analysis of rotating, functionally graded Timoshenko nano-beams under an in-plane nonlinear thermal loading is studied for the first time. The formulation is based on Eringen's nonlocal elasticity theory. Hamilton's principle is used for the derivation of the equations. The governing equations are solved by the differential quadrature method. The nano-beam is under axial load due to the rotation and thermal effects, and the boundary conditions are considered as cantilever and propped cantilever. The thermal distribution is considered to be nonlinear and material properties are temperature-dependent and are changing continuously through the thickness according to the power-law form.     

Transverse vibration of rotary tapered microbeam based on modified couple stress theory and generalized differential quadrature element method

The transverse vibration of a rotary tapered microbeam is studied based on a modified couple stress theory and Euler–Bernoulli beam model. The governing differential equation and boundary conditions are derived according to Hamilton's principle. The generalized differential quadrature element method is then used to solve the governing equation for cantilever and propped cantilever boundary conditions. The effect of the small-scale parameter, beam length, rate of cross-section change, hub radius, and nondimensional angular velocity on the vibration behavior of the microbeam is presented.     

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.     

Temperature effects on wave propagation in nanoplates

In this paper, the thermal effects on the ultrasonic wave propagation characteristics of a nanoplate are studied based on the nonlocal continuum theory. The nonlocal governing equations are derived for the nanoplate under thermal environment. The axial stress caused by the thermal effects is considered. The wave propagation analysis is carried out using spectral analysis. The influences of the nonlocal small scale coefficient, the room or low temperature, the high temperature and the axial half wave numbers on the wave dispersion properties of nanoplate are also discussed. Numerical results show that the small scale effects and the thermal effects are significant for larger half wavenumbers. The results are qualitatively different from those obtained based on the local plate theory and thus, are important for the development of graphene-based nanodevices such as strain sensor, mass and pressure sensors, atomic dust detectors, and enhancer of surface image resolution.     

Thermoelastic analysis of rotating laminated functionally graded cylindrical shells using layerwise differential quadrature method

An accurate and efficient solution procedure based on the elasticity theory is employed to investigate the thermoelastic behavior of rotating laminated functionally graded (FG) cylindrical shells in thermal environment. The material properties are assumed to be temperature dependent and graded in the thickness direction. In order to accurately model the variation of the field variables across the thickness, the shell is divided into a set of mathematical layers. The differential quadrature method (DQM) is adopted to discretize the governing differential equations of each layer together with the related boundary and compatibility conditions at the interface of two adjacent layers. Using the DQM enables one to accurately and efficiently discretize the partial differential equations, especially along the graded direction, and also implement the boundary and compatibility conditions in their strong forms. After demonstrating the convergence and accuracy of the presented approach, the effects of material and geometrical parameters and also temperature dependence of material properties on the stresses and displacement components of rotating laminated FG cylindrical shells are studied.     

Thermal buckling of a nanoplate with small-scale effects

In this paper, thermal buckling properties of a nanoplate with small-scale effects are studied. Based on the nonlocal continuum theory, critical temperatures for the nonlocal Kirchhoff and Mindlin plate theories are derived. The thermal buckling characteristics are presented with different models. The influences of the scale coefficients, half-wave numbers, width ratios, and the ratios of the width to the thickness are discussed. From this work, it can be observed that the small-scale effects are significant for the thermal buckling properties. Both the half-wave number and width ratio have influence. The nonlocal Kirchhoff plate theory is valid for the thin nanoplate, and the nonlocal Mindlin plate theory is more appropriate for simulating the mechanical behaviors of the thick nanoplate.     

Thermal buckling of cross-ply laminated composite beams

Summary Thermal buckling of thick, moderately thick and thin cross-ply laminated beams subjected to uniform temperature distribution are analyzed. Exact analytical solutions of refined beam theories are developed to obtain the critical buckling temperature of cross-ply beams with various boundary conditions. The state space concept in conjunction with Jordan canonical form will be used to solve exactly the governing equations of the thermal buckling problems. The effects of length to thickness ratio, modulus ratio, thermal expansion coefficients ratio, various boundary conditions and number of layers on the critical buckling temperature are investigated.     

Free vibration analysis of rotating functionally graded cylindrical shells in thermal environment

The free vibration analysis of rotating functionally graded (FG) cylindrical shells subjected to thermal environment is investigated based on the first order shear deformation theory (FSDT) of shells. The formulation includes the centrifugal and Coriolis forces due to rotation of the shell. The material properties are assumed to be temperature-dependent and graded in the thickness direction. The initial thermo-mechanical stresses are obtained by solving the thermoelastic equilibrium equations. The equations of motion and the related boundary conditions are derived using Hamilton’s principle. The differential quadrature method (DQM) as an efficient and accurate numerical tool is adopted to discretize the thermoelastic equilibrium equations and the equations of motion. The convergence behavior of the method is demonstrated and comparison studies with the available solutions in the literature are performed. Finally, the effects of angular velocity, Coriolis acceleration, temperature dependence of material properties, material property graded index and geometrical parameters on the frequency parameters of the FG cylindrical shells with different boundary conditions are investigated.     

不同边界条件下转动锥壳的参激失稳特性分析

采用Haar小波方法结合Floquet指数法对不同边界条件下转动锥壳的参激振动稳定性进行了分析。基于Love一阶近似壳体理论,给出了周期性载荷作用下转动锥壳的动力学控制微分方程,采用Haar小波离散方法将其转化为具有周期性时变系数的Mathieu-Hill型常微分方程组。考虑到Bolotin法不能应用于陀螺系统的参激失稳特性分析,以及多尺度法受限于小参数情形的事实,该研究采用了对参激系统普遍适用的Floquet指数法对转动锥壳的参激振动稳定性进行分析。通过与其他文献结果的对比,验证了所采用模型及稳定性分析方法的正确性。在此基础上,讨论了固支-固支、简支-简支、固支-简支和简支-固支等几种不同边界条件下转速和半顶角对转动锥壳不稳定区的影响。     

Three-dimensional stagnation point flow of a second grade fluid towards a moving plate

The problem dealing with the steady three-dimensional flow of a second grade fluid near the stagnation point of an infinite plate moving parallel to itself with constant velocity has been investigated. By using the appropriate transformations for the velocity components and temperature, the basic equations governing flow and heat transfer have been reduced to a set of ordinary differential equations. These equations have been solved approximately subject to the relevant boundary conditions by employing a numerical technique. The effect of a nondimensional elastic parameter on the velocity components, wall shear stress, temperature and heat transfer has been examined carefully.     

Reflection of plane waves in thermo-diffusion elasticity without dissipation under the effect of rotation

The governing equations of rotating thermoelastic solids with diffusion are solved in context of the Green–Naghdi theory (Type II) to show the numerical existence of four coupled plane waves. The required boundary conditions at the stress free and thermally insulated surface of the model are satisfied to obtain the reflection coefficients of various reflected waves for an incident plane wave. These coefficients are found to depend on rotation parameter, the angle of incidence of striking waves, thermo-diffusion parameters, and other material constants. Complex absolute values of speeds and reflection coefficients are computed for a particular material representing the model. Graphical representation of numerical results show the effects of rotation and diffusion on speed and reflection coefficients of plane waves.     

湿热环境下复合材料层板的几何非线性分析

根据Reddy的高阶剪切变形理论,在应力应变本构关系中考虑湿热环境的影响,用虚位移原理推导出以位移形式表达的复合材料层板的几何非线性控制方程及相应的边界条件。选定的边界条件为弹性转动约束。选择不同的弹性转动系数,可以得到介于简支到固支之间的不同边界条件。假设材料的物性参数不随湿热环境而变化。用Galerkin方法求解控制方程。数值结果所用的边界条件为四边固支。最后讨论了温度和湿度、长厚比、长宽比和铺层数等各种参数变化的影响。计算和分析的结果均表明温度的升高对层合板的弯曲产生非常不利的影响;而加湿对层合板的弯曲影响是非常有限的。在其它几何条件不变的情况下,增加铺层数,不仅能降低挠度,也能明显地降低弯矩或弯曲应力。     

超急速传热过程中热弹性响应的解析分析

考虑超急速传热过程中诱发的热冲击效应,基于L-S广义热弹性理论,建立了温度突变加热条件下热弹性响应的控制方程组。借助于Laplace正逆变换,在适当简化的条件下推导了一维超急速传热问题热弹性响应的解析表达式。通过对温度场、位移场及应力场的解析求解,给出了超急速传热过程中热波和热弹性波在弹性体内的传递规律,并指出在超急速传热条件下,由于热波和热弹性波的相互叠加作用削弱了热作用产生的热冲击效应。     

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.     

The generalized differential quadrature method for frequency analysis of a rotating conical shell with initial pressure

Using the generalized differential quadrature (GDQ) method which is an improved version of the differential quadrature (DQ) method, this paper examines the influence of initial pressure load on the free vibration of a rotating thin truncated circular isotropic conical shell with different boundary conditions. The present governing equations of motion include the influence of initial stress field and the effects of initial hoop tension and also the centrifugal and coriolis accelerations due to rotation. The influence of initial pressure on the frequency characteristics is discussed in detail for various conditions. To validate the present analysis, frequency comparisons are made with those available in published works, and very good agreements are obtained. Copyright © 2000 John Wiley & Sons, Ltd.     

Generalized two-variable plate theory for multi-layered graphene sheets with arbitrary boundary conditions

In the present study, the free vibration, mechanical buckling and thermal buckling analyses of multi-layered graphene sheets (MLGSs) are investigated. Eringen’s nonlocal elasticity equations are incorporated in new two-variable plate theories accounting for small-scale effects. The MLGSs are taken to be perfectly bonded to the surrounding medium. The governing differential equations of this model are solved analytically under various boundary conditions and taking into account the effect of van der Waals forces between adjacent layers. New functions for the displacements are proposed here to satisfy the different boundary conditions. Comparison of the results with those being in the open literature is made. This comparison illustrates that the present scheme yields very accurate results. Furthermore, the influences of nonlocal coefficient, moduli of the surrounding elastic medium and aspect ratio on the frequencies and buckling of the embedded MLGSs are examined.