Three-dimensional solution of axisymmetric bending of functionally graded circular plates

Based on three-dimensional theory, this paper investigates the axisymmetric bending of transversely isotropic and functionally graded circular plates subject to arbitrarily transverse loads using the direct displacement method. The material properties can arbitrarily vary along the thickness of the plate. The transverse load is expanded in the Fourier–Bessel series and superposition principle is then used to obtain the total response based on the results of each item of the series. For one item of the series of the load, we assume the distributions of the displacements in the radial direction and therefore only the distributions of the displacements in thickness direction are required to find. If the material properties vary in an exponential law, the exact solutions can be obtained for elastic simple support and rigid slipping support, which are satisfied on the every point of the boundaries. Moreover, the analytical solutions are also presented for simply supported and clamped conditions, which are satisfied using Saint Venant principle. Simultaneously, through the layerwise method a semi-analytical solution is proposed for the case of arbitrary variation of the material properties. Finally the numerical examples are presented to verify the proposed method.     

Large deflections and large amplitude vibrations of axisymmetric circular plates

This paper is concerned with conventional and mixed finite element analysis of large deflection bending and free vibration of axisymmetric circular plates including shear deformation. Conventional and mixed finite element formulations are presented. Numerical results are included for isotropic as well as orthotropic, circular and annular plates under various edge conditions and loadings.     

Dynamics of circular three-layer plates with elastic base under axisymmetric loading

We consider axisymmetric transverse vibrations of a circular three-layer plate with light filler on the elastic base subjected to the action of suddenly applied and pulsed surface loads. The reaction of the base is described by the Winkler model. For the analysis of the kinematics of a sandwich asymmetric over the thickness, we use the hypotheses of broken normal. The analytic solutions are obtained by using a system of Heaviside functions and the Dirac δ-functions. The numerical calculations and comparative analysis of the obtained solutions are performed.     

Analytical solutions of functionally graded piezoelectric circular plates subjected to axisymmetric loads

Based on the three-dimensional theory of piezoelectricity, this work analytically investigates the axisymmetric bending of circular plates whose material properties vary along the thickness. The transverse loads are expanded in terms of the Fourier–Bessel series, and the solutions corresponding to each item of the series are derived by a semi-inverse method. The overall results are then obtained through the superposition principle. The exact solutions are obtained for two unusual boundary conditions if the material properties obey an exponential law. Meanwhile, a layerwise model is employed for the case of arbitrary and independent distribution of each component of material properties and the approximate solutions are proposed for simply supported and clamped boundaries. Finally, the numerical examples are illustrated, and the results are compared with those of the finite element method to demonstrate the present method.     

热环境中功能梯度材料圆板的自由振动

基于von Kaman经典板理论,建立了功能梯度材料圆板在升温场内的大挠度动力学控制方程.将控制方程的响应分解为热过屈曲静态解和振动解两部分,得到了功能梯度材料圆板在热过屈曲平衡构形附近小振幅线性自由振动的微分方程.采用打靶法同时求热过屈曲和振动问题的控制方程,得到了随温度载荷变化的热过屈曲平衡路径以及前三阶固有频率的数值解.分析和讨论了板的材料梯度参数、温度场分布参数、边界条件等因素对过屈曲变形和振动响应的影响.分析中考虑了功能梯度板的组份材料的物性参数对温度的依赖性.     

Free vibration of laminated composite plates with a central circular hole

This paper presents finite element results and some experimental data on the free vibration of symmetric laminated composite plates containing a central hole. The varying parameters in the study are the hole sizes, the boundary conditions along the edges of the plate and the aspect ratio of the plate. Hole diameters ranging from 0.1 to 0.9 times the width of the plate and the aspect ratio varying from 1 to 4 were analysed in the numerical study. The other parameters such as the stacking sequence and the lamination materials are kept constant. The stacking sequence and material properties chosen for the study are those typically found in aircraft structural applications. The close correlation between numerical and experimental data provides some confidence in the finite element method used.     

Null-field integral equation approach for free vibration analysis of circular plates with multiple circular holes

In this paper, a semi-analytical approach for the eigenproblem of circular plates with multiple circular holes is presented. Natural frequencies and modes are determined by employing the null-field integral formulation in conjunction with degenerate kernels, tensor rotation and Fourier series. In the proposed approach, all kernel functions are expanded into degenerate (separable) forms and all boundary densities are represented by using Fourier series. By uniformly collocating points on the real boundary and taking finite terms of Fourier series, a linear algebraic system can be constructed. The direct searching approach is adopted to determine the natural frequency through the singular value decomposition (SVD). After determining the unknown Fourier coefficients, the corresponding mode shape is obtained by using the boundary integral equations for domain points. The result of the annular plate, as a special case, is compared with the analytical solution to verify the validity of the present method. For the cases of circular plates with an eccentric hole or multiple circular holes, eigensolutions obtained by the present method are compared well with those of the existing approximate analytical method or finite element method (ABAQUS). Besides, the effect of eccentricity of the hole on the natural frequency and mode is also considered. Moreover, the inherent problem of spurious eigenvalue using the integral formulation is investigated and the SVD updating technique is adopted to suppress the occurrence of spurious eigenvalues. Excellent accuracy, fast rate of convergence and high computational efficiency are the main features of the present method thanks to the semi-analytical procedure.     

Nonlinear theories of axisymmetric bending of functionally graded circular plates with modified couple stress

A microstructure-dependent nonlinear theory for axisymmetric bending of circular plates, which accounts for through-thickness power-law variation of a two-constituent material, is developed using the principle of virtual displacements. The formulation is based on a modified couple stress theory, power-law variation of the material, temperature-dependent properties, and the von Kármán geometric nonlinearity. Classical and first-order shear deformation theories are considered in the study. The modified couple stress theory contains a material length scale parameter that can capture the size effect in a functionally graded material plate. The theories presented herein can be used to develop analytical solutions of bending, buckling, and free vibration for the linear case and finite-element models for the nonlinear case to determine the effect of the geometric nonlinearity, power-law index, and microstructure-dependent constitutive relations on linear and nonlinear response of axisymmetric analysis of circular plates.     

热过屈曲正交异性圆(环)板的自由振动响应

基于vonKármán薄板几何非线性理论,建立了以中面位移为基本未知量的加热极正交异性圆板轴对称大挠度动力学控制方程。然后,将控制方程的响应分解为热过屈曲静态解和振动解两部分,并在小振幅振动假设下得到了板在热过屈曲静平衡构形附近自由振动的线性微分方程。最后,采用打靶法获得了板在热过屈曲前后的固有频率与升温参数之间的特征关系曲线。结果表明,周边面内约束板的前两阶固有频率在热屈曲前随着温度升高而降低,而一旦板进入过屈曲平衡状态,前两阶频率都随着升温而单调增加。     

Investigations of the transverse vibration characteristics of piezoceramic circular plates with V-notches

This article pertains to experimental validation of the study presented in a previous paper by the same authors , in which the transverse vibration characteristics of piezoceramic circular plates with V-notches were investigated theoretically using Ritz’s method incorporating defined equivalent constants. For experimental measurement, two optical methods – amplitude-fluctuation electronic speckle pattern interferometry (AF-ESPI) and laser Doppler vibrometry (LDV) – were employed to conceptualize the analytical results. The AF-ESPI method demonstrates the advantages of combining high fringe sensitivity and noise reduction, which are achieved using the time-averaged and subtraction electronic speckle pattern interferometry methods, respectively. Both vibration mode shapes and resonant frequencies of V-notched piezoceramic plates were obtained using the AF-ESPI method; in addition, the resonant frequencies were also determined quickly using the LDV method. By varying the corner angles and notch–depth ratios, various types of piezoceramic plate (V-notched, sectorial, semicircular, segmented, and sharp-notched) were used for experimental measurements. Numerical calculations were performed using the finite element method, and the results were compared with the experimental measurements and theoretical predictions. It was shown that the resonant frequencies and mode shapes obtained using the two experimental methods agreed fairly closely with the theoretical and numerical results.     

考虑非局部应变梯度效应的轴对称压电纳米圆板热-力-电耦合振动EI北大核心CSCD

基于非局部应变梯度理论和Mindlin板理论,研究了热‐力‐电多场耦合下轴对称压电纳米圆板的振动特性。通过Hamilton原理推导了非局部应变梯度本构框架内的运动方程,采用微分求积法数值求解了理论模型微分方程组,分析了压电纳米圆板的振动固有频率受内尺度参数与外场参数的影响。压电纳米圆板的固有频率随着非局部参数的增大而减小,随着应变梯度特征参数的增大而增大。当非局部参数小于应变梯度特征参数时,纳米圆板表现出刚度硬化行为;当非局部参数大于应变梯度特征参数时,表现出刚度软化行为。当非局部参数等于应变梯度特征参数时,纳米圆板的刚度退化为相应的经典连续介质理论结果。此外,固有频率随着径向压力和正电压的增大而减小,随着径向拉力和负电压的增大而增大,随着温差的增加而小幅减小。特别地,研究发现当径向载荷和电压增大到一定程度时,纳米圆板出现了振动失稳现象,并分析了非局部参数与应变梯度特征参数对失稳临界径向载荷及临界电压的影响。     

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.     

均匀热载荷作用下功能梯度圆板的非线性振动

基于经典板理论,研究了热载荷作用下功能梯度圆板的大幅振动问题.在经典板理论下利用物理中面概念,导出了功能梯度圆板的非线性运动方程.利用Ritz?Kantorovich方法消去时间变量,将非线性运动方程转换成了一组关于空间变量的非线性常微分方程.采用打靶法数值求解所得方程,并利用数值结果研究了热载荷作用下功能梯度圆板静态...     

Theoretical analysis and experimental measurement for resonant vibration of piezoceramic circular plates

Based on the electroelastic theory for piezoelectric plates, the vibration characteristics of piezoceramic disks with free-boundary conditions are investigated in this work by theoretical analysis, numerical simulation, and experimental measurement. The resonance of thin piezoceramic disks is classified into three types of vibration modes: transverse, tangential, and radial extensional modes. All of these modes are investigated in detail. Two optical techniques, amplitude-fluctuation electronic speckle pattern interferometry (AF-ESPI) and laser Doppler vibrometer (LDV), are used to validate the theoretical analysis. Because the clear fringe patterns are shown only at resonant frequencies, both the resonant frequencies and the corresponding mode shapes are obtained experimentally at the same time by the proposed AF-ESPI method. Good quality of the interferometric fringe patterns for both the transverse and extensional vibration mode shapes are demonstrated. The resonant frequencies of the piezoceramic disk also are measured by the conventional impedance analysis. Both theoretical and experimental results indicate that the transverse and tangential vibration modes cannot be measured by the impedance analysis, and only the resonant frequencies of extensional vibration modes can be obtained. Numerical calculations based on the finite element method also are performed, and the results are compared with the theoretical analysis and experimental measurements. It is shown that the finite element method (FEM) calculations and the experimental results agree fairly well for the resonant frequencies and mode shapes. The resonant frequencies and mode shapes predicted by theoretical analysis and calculated by finite element method are in good agreement, and the difference of resonant frequencies for both results with the thickness-to-diameter (h/D) ratios, ranging from 0.01 to 0.1, are presented.     

The effects of large vibration amplitudes on the stresses of thin circular functionally graded plates

Variations of stresses of a thin circular functionally grade plate, due to large vibration amplitudes, are developed in this paper. The plate thickness is constant and temperature dependent functionally graded material (FGM) properties vary through the thickness of the plate. For harmonic vibrations, governing equations are solved. Corresponding results are illustrated in the case of steady-state free vibration. The results show that the variation of volume fraction index is considerable in FGM properties and the amount of stresses.     

Asymmetric flexural vibration and thermoelastic stability of FGM circular plates using finite element method

Here, asymmetric free vibration characteristics and thermoelastic stability of functionally graded circular plates are investigated using finite element procedure. A three-noded shear flexible plate element based on the field-consistency principle is used. Temperature field is assumed to be a uniform distribution over the plate surface and varied in thickness direction only. Material properties are graded in the thickness direction according to simple power law distribution. For the numerical illustrations, aluminum/alumina is considered as functionally graded material. The variation in critical buckling load is highlighted considering gradient index, temperature, radius-to-thickness ratios, circumferential wave number and boundary condition of the plate.     

嵌入形状记忆合金圆板在升温场内的自由振动

基于形状记忆合金Brinson一维热力学本构关系和圆板小振幅振动理论,研究了径向嵌入SMA丝的圆板在均匀升温场作用下的自由振动响应,获得了周边不可移简支和夹紧圆板的前三阶固有频率随温度变化的特征关系曲线。结果表明,形状记忆合金丝在从马氏体到奥氏体相变的过程中产生的回复力对板的固有频率具有显著增强作用。由此可见,通过嵌入SMA纤维丝和施加升温载荷可以主动而有效地调节圆板的固有频率。     

Inelastic response of axisymmetric plates with cracks

The behavior of stepped annular plates made of a fiber reinforced composite material is studied making use of the method of mode form motions. The plates having stable cracks of prescribed deepness at the re-entrant corners of steps are subjected to an initial impact and the subsequent motion is due to inertia. The designs of stepped plates are established which minimize maximal residual deflections under given material consumption. Numerical results are presented for the plate with single step.     

Analytical layerwise free vibration analysis of circular/annular composite sandwich plates with auxetic cores

In the present research, free vibration of circular and annular sandwich plates with auxetic (negative Poisson’s ratio) cores and isotropic/orthotropic face sheets is investigated for different combinations of the boundary conditions. To ensure that the results are accurate and reliable, a global–local layerwise plate theory is employed instead of the traditional equivalent single-layer theories. The governing equations are derived based on Hamilton’s principle and solved using a Taylor transform whose center is located at the outer radius of the plate. Due to this hint, the resulting semi-analytical solution can be employed for both circular and annular sandwich plates. After investigation of vibration behavior of a single-layer annular auxetic plate, a comprehensive parametric study including evaluation of effects of the auxeticity for sandwich plates with isotropic and orthotropic face sheets, symmetric and asymmetric layups, different core to sheet thickness, radius to thickness, and inner to outer radius ratios, and various boundary conditions, is carried out. Results show that unlike the single-layer auxetic plates that exhibit a transition state, the auxeticity may considerably increase the natural frequencies and rigidities of the circular/annular sandwich plates, especially when the boundary conditions induce higher rigidity in the plate or when the fibers are along the radial direction. Accuracy of results of the employed layerwise theory and the proposed semi-analytical solution is verified by comparing the results with those of the three-dimensional theory of elasticity extracted from the ABAQUS software.