Propagation of flexural waves and localized vibrations in the strip plate with a layer using Hamilton system

Based on the theory of laminated plates and applying the method in Hamiltonian state space, the propagation of flexural waves and vibrations in the strip plate covered with a layer are investigated. The boundaries at the two lateral sides are free of traction. According to the character of solar panel, the existence of all kinds of localized vibration modes and wave propagation modes is analyzed. By using eigenfunction expansion method, the dispersion relations of waveguide modes in the strip plate covered with a layer are derived. Through the numerical examples of solar panel, the existence of all kinds of vibration modes and propagating modes is analyzed. The dispersion curves of the strip plate covered with a layer under different parameters are presented and analyzed. The effects of the properties of the covering layer on the propagation of flexural waves are also examined.     

波导结构频散分析的特征频率法及在板条结构中的应用

提出一种求解波导结构频散特性的有限元特征频率法,该方法基于振动问题的特征频率计算理论,根据模态振型识别波数与模态类型,建立了相速度及群速度的求解方法。该方法可适用于任意波导结构的频散关系求解。首先分析满足收敛精度要求的最大网格单元尺寸与最小模型长度,并用该方法对简支板条结构的频散特性进行了计算。结果表明,有限元特征频率法适合求解波动频散关系,板条结构中模态受边界影响会产生同阶高次模态,边界尺寸决定新模态的截止频率;随频率的增大,同阶低次反对称模态会趋于一致;对称模态能量分布受边界影响较大。本文也为板条类结构导波实验结果的分析提供了理论依据。     

FLEXURAL WAVE PROPAGATION IN NARROW MINDLIN''''S PLATE

Appling Mindlin's theory of thick plates and Hamilton system to propagation of elastic waves under free boundary condition, a solution of the problem was given. Dispersion equations of propagation mode of strip plates were deduced from eigenfunction expansion method. It was compared with the dispersion relation that was gained through solution of thick plate theory proposed by Mindlin. Based on the two kinds of theories, the dispersion curves show great difference in the region of short waves, and the cutoff frequencies are higher in Hamiltonian systems. However, the dispersion curves are almost the same in the region of long waves.     

Equilibrium of a Prestressed Strip Reinforced with Elastic Plates

The contact problem for a prestressed elastic strip reinforced with equally spaced elastic plates is considered. The Fourier integral transform is used to construct an influence function of a unit concentrated force acting on the infinite elastic strip with one edge constrained. The transmission of forces from the thin elastic plates to the prestressed strip is analyzed. On the assumption that the beam bending model and the uniaxial stress model are valid for an elastic plate subjected to both vertical and horizontal forces, the problem is mathematically formulated as a system of integro-differential equations for unknown contact stresses. This system is reduced to an infinite system of algebraic equations solved by the reduction method. The effect of the initial stresses on the distribution of contact forces in the strip under tension and compression is studied     

GUIDED THERMOELASTIC WAVE PROPAGATION IN LAYERED PLATES WITHOUT ENERGY DISSIPATION

In this paper, the propagation of guided thermoelastic waves in laminated orthotropic plates subjected to stress-free, isothermal boundary conditions is investigated in the context of the Green-Naghdi (GN) generalized thermoelastic theory (without energy dissipation). The coupled wave equations and heat conduction equation are solved by the Legendre orthogonal polynomial series expansion approach. The validity of the method is confirmed through a comparison. The dispersion curves of thermal modes and elastic modes are illustrated simultaneously. Dispersion curves of the corresponding pure elastic plate are also shown to analyze the influence of the thermoelasticity on elastic modes. The displacement and temperature distributions are shown to discuss the differences between the elastic modes and thermal modes.     

FLEXURAL WAVE PROPAGATION IN NARROW MINDLIN＇S PLATE

Appling Mindlin＇s theory of thick plates and Hamilton system to propagation of elastic waves under free boundary condition, a solution of the problem was given. Dispersion equations of propagation mode of strip plates were deduced from eigenfunction expansion method. It was compared with the dispersion relation that was gained through solution of thick plate theory proposed by Mindlin. Based on the two kinds of theories, the dispersion curves show great difference in the region of short waves, and the cutoff frequencies are higher in Hamiltonian systems. However, the dispersion curves are almost the same in the region of long waves.     

On bending of strain gradient elastic micro-plates

Bending of strain gradient elastic thin plates is studied, adopting Kirchhoff’s theory of plates. Simple linear strain gradient elastic theory with surface energy is employed. The governing plate equation with its boundary conditions are derived through a variational method. It turns out that new terms are introduced, indicating the importance of the cross-section area in bending of thin plates. Those terms are missing from the existing strain gradient plate theories; however, they strongly increase the stiffness of the thin plate.     

Transient wave propagation of isotropic plates using a higher-order plate theory

Transient wave propagation of isotropic thin plates using a higher-order plate theory is presented in this paper. The aim of this investigation is to assess the applicability of the higher-order plate theory in describing wave behavior of isotropic plates at higher frequencies. Both extensional and flexural waves are considered. A complete discussion of dispersion of isotropic plates is first investigated. All the wave modes and wave behavior for each mode in the low and high-frequency ranges are provided in detail. Using the dispersion relation and integral transforms, exact integral solutions for an isotropic plate subjected to pure impulse load and a number of wave excitations based on the higher-order theory are obtained and asymptotic solutions which highlight the physics of waves are also presented. The axisymmetric three-dimensional analytical solutions of linear wave equations are also presented for comparison. Results show that the higher-order theory can predict the wave behavior closely with exact linear wave solutions at higher frequencies.     

Tensile and compressive buckling of plates weakened by cracks

Plates are susceptible to buckling under compression when the thickness dimension becomes sufficiently small. Such mode of structure failure can prevail even if the plates were extended in tension. Wribkling of stretched thin sheets is a commonly observed phenomenon that leads to complex deflection patterns, particularly in regions close to crack-like imperfections. Predictions of the buckled displacement modes for plates weakened by cracks will be made on the basis of a theory formulated by application of variational calculus. Finite element method is used such that defects of other shapes can also be analyzed. Various buckled displacement modes of a center-cracked plate are determined and displayed graphically. The critical buckling loads are found to decrease with increasing crack size. Moreover, local wrinkling of the plate surface becomes less pronounced for the higher buckling modes. The method of solution applies equally well to plates that are initially curved.     

Dynamics of cantilever plates and its hybrid vibration control

Applying Lagrange–Germain’s theory of elastic thin plates and Hamiltonian formulation, the dynamics of cantilever plates and the problem of its vibration control are studied, and a general solution is finally given. Based on Hamiltonian and Lagrangian density function, we can obtain the flexural wave equation of the plate and the relationship between the transverse and the longitudinal eigenvalues.Based on eigenfunction expansion, dispersion equations of propagation mode of cantilever plates are deduced. By satisfying the boundary conditions of cantilever plates, the natural frequencies of the cantilever plate structure can be given.Then, analytic solution of the problem in plate structure is obtained. An hybrid wave/mode control approach, which is based on both independent modal space control and wave control methods, is described and adopted to analyze the active vibration control of cantilever plates. The low-order(controlled by modal control) and the high-order(controlled by wave control) frequency response of plates are both improved. The control spillover is avoided and the robustness of the system is also improved. Finally, simulation results are analyzed and discussed.     

Lamb波理论及层合板冲击损伤的实验研究

从理论上分析了板中Lamb波信号的传播特性,并给出Lamb波在板中传播的频散方程。理论分析及实验均表明,Lamb波的频散特性与复合材料结构损伤有着直接的联系,而且最低阶的对称和反对称Lamb波模态对层合板的损伤比较敏感,但应用Lamb波的频散效应监测结构的损伤在检测技术上还难以实现。根据板中导波形成Lamb波的共振原理,板中应力波的幅频特性很大程度上反映了Lamb波的谐振特征。因此,利用压电元件的压电阻抗谱分析应力波的各阶模态频率及振幅对结构损伤的变化,能够反映材料内部损伤与Lamb波的频散特性。文中针对表面粘贴压电元件的层合板智能结构,建立了包含Lamb波谐振模式的压电阻抗计算模型。冲击损伤试件的实验表明,由于结构损伤的出现压电阻抗谱中的模态频率及其阻抗幅值等特征信息将发生变化。因此,引入应力波损伤因子可以对结构冲击损伤的存在和程度进行初步评价。该方法基于结构的机-电动态阻抗特性,不受结构的几何形状限制,测试用的压电元件成本低,方法简单可行,有望在智能结构的健康诊断方面获得应用。     

SHEAR-HORIZONTAL WAVES IN A ROTATED Y-CUT QUARTZ PLATE WITH AN ISOTROPIC ELASTIC LAYER OF FINITE THICKNESS

We study shear-horizontal (SH) waves in a rotated Y-cut quartz plate carrying an isotropic elastic layer of finite thickness.The three-dimensional theories of anisotropic elasticity and isotropic elast...     

Closed-form solutions for free vibration of rectangular FGM thin plates resting on elastic foundation

This article presents closed-form solutions for the frequency analysis of rectangular functionally graded material(FGM) thin plates subjected to initially in-plane loads and with an elastic foundation. Based on classical thin plate theory, the governing differential equations are derived using Hamilton's principle. A neutral surface is used to eliminate stretching–bending coupling in FGM plates on the basis of the assumption of constant Poisson's ratio. The resulting governing equation of FGM thin plates has the same form as homogeneous thin plates. The separation-ofvariables method is adopted to obtain solutions for the free vibration problems of rectangular FGM thin plates with separable boundary conditions, including, for example, clamped plates. The obtained normal modes and frequencies are in elegant closed forms, and present formulations and solutions are validated by comparing present results with those in the literature and finite element method results obtained by the authors. A parameter study reveals the effects of the power law index n and aspect ratio a/b on frequencies.     

Experimental and theoretical study of the buckling of narrow thin plates on an elastic foundation under compression

The paper describes the processes of elastic deformation of thin films under mechanical loading. The film is modeled longitudinally by a compressed plate arranged on an elastic foundation. A computer model of the buckling of the narrow thin plate with a delamination portion located on an elastic foundation is constructed. This paper also touches upon the supercritical behavior of the plate–substrate system. The experiments on the axial compression of a metal strip adhered to a rubber plate are performed, and 2 to 7 buckling modes are obtained therein. The critical loads and buckling modes obtained in the numerical calculations are compared with the experimental data. It is shown that there is the possibility of progressive delamination of the metal plate from the foundation if the critical load is exceeded. It is found that the use of the proposed approach, which, in contrast to other approaches, accounts for the elastic deformation of the substrate, causes the dependence between the critical bending stress and the stiffness of the foundation.     

冲击载荷作用下矩形薄板的弹性动力屈曲

为了研究冲击载荷作用下考虑应力波效应弹性矩形薄板的动力屈曲,根据动力屈曲发生瞬间的能量转换和守恒准则,导出板的屈曲控制方程和波阵面上的补充约束条件,真实的屈曲位移应同时满足控制方程和波阵面上的附加约束条件。满足上述条件,建立了该问题的完整数值解法,对屈曲过程中冲击载荷、屈曲模态和临界屈曲长度之间的关系进行研究,定量计算了横向惯性效应对提高薄板动力屈曲临界应力的贡献。研究表明:板的厚宽比一定时,临界屈曲长度随冲击载荷的增大而减小;由于屈曲时的横向惯性效应,应力波作用下薄板一阶临界力参数是相应边界板的静力失稳临界力参数的1.5倍;随着边界约束逐渐减弱,板临界力参数逐渐减小,动力特征参数逐渐增大。     

梯度弹性基础上正交异性薄板的屈曲分析

基于双参数弹性基础模型,研究了梯度弹性基础上正交异性薄板的屈曲问题. 首先,基于能量法与变分原理,给出了梯度弹性基础上正交异性薄板的屈曲控制方程,并得到了梯度弹性基础刚度系数K₁ 与K₂的计算式;进而,通过将位移函数采用三角函数展开的方法,给出了单向压缩载荷作用下、四边简支正交异性弹性基础板屈曲载荷的计算式;在算例中,通过将该文的解退化到单纯的正交异性板,并与经典弹性解比较,证明了理论的正确性;最后,求解了弹性模量在厚度方向上呈幂律分布的梯度基础上的薄板屈曲问题,分析了基础上下表层材料弹性模量比与体积分数指数对屈曲载荷的影响.     

Hydroelastic interaction between water waves and thin elastic plate floating on three-layer fluid

The wave-induced hydroelastic responses of a thin elastic plate floating on a three-layer fluid, under the assumption of linear potential flow, are investigated for two-dimensional cases. The effect of the lateral stretching or compressive stress is taken into account for plates of either semi-infinite or finite length. An explicit expression for the dispersion relation of the flexural-gravity wave in a three-layer fluid is analytically deduced. The equations for the velocity potential and the wave elevations are solved with the method of matched eigenfunction expansions. To simplify the calculation on the unknown expansion coefficients, a new inner product with orthogonality is proposed for the three-layer fluid, in which the vertical eigenfunctions in the open-water region are involved. The accuracy of the numerical results is checked with an energy conservation equation, representing the energy flux relation among three incident wave modes and the elastic plate. The effects of the lateral stresses on the hydroelastic responses are discussed in detail.     

Viscoelastic shear horizontal wave in graded and layered plates

In this paper, viscoelastic shear horizontal (SH) wave propagation in functionally graded material (FGM) plates and laminated plates are investigated. The controlling differential equation in terms of displacements is deduced based on the Kelvin–Voigt viscoelastic theory. The SH wave characteristics is controlled by two elastic constants and their corresponding viscous coefficients. By the Legendre polynomial series method, the asymptotic solutions are obtained. In order to verify the validity of the method, a homogeneous plate is calculated to make a comparison with available data. Through three different graded plates, the influences of gradient shapes on dispersion and attenuation are discussed. The viscous effects on the displacement and stress shapes are illustrated. The different boundary conditions are analyzed. The influential factors of the viscous effect are analyzed. Finally, two multilayered (two layer and five layer) viscoelastic plates that are composed of the same material volume fraction are calculated to show their differences from the graded plate.     

Mechanical buckling and free vibration of thick functionally graded plates resting on elastic foundation using the higher order B-spline finite strip method

In this study, the mechanical buckling and free vibration of thick rectangular plates made of functionally graded materials (FGMs) resting on elastic foundation subjected to in-plane loading is considered. The third order shear deformation theory (TSDT) is employed to derive the governing equations. It is assumed that the material properties of FGM plates vary smoothly by distribution of power law across the plate thickness. The elastic foundation is modeled by the Winkler and two-parameter Pasternak type of elastic foundation. Based on the spline finite strip method, the fundamental equations for functionally graded plates are obtained by discretizing the plate into some finite strips. The results are achieved by the minimization of the total potential energy and solving the corresponding eigenvalue problem. The governing equations are solved for FGM plates buckling analysis and free vibration, separately. In addition, numerical results for FGM plates with different boundary conditions have been verified by comparing to the analytical solutions in the literature. Furthermore, the effects of different values of the foundation stiffness parameters on the response of the FGM plates are determined and discussed.     

Photoelastic stress analysis of orthtropic materials by using an isotropic plate

Based on the two-dimensional theory of elasticity for an orthotropic plate, relations between stress components produced in two different orthotropic plates are considered and the conditions to realize the similar stress fields in different orthotropic plates are studied. On the basis of the similarity law, a convenient photoelastic method to analyze stress fields in an orthotropic plate, using an isotropic plate, is presented. Two examples are treated. One deals with the stress-concentration problem around a circular hole in a strip with edges parallel to the symmetric axis of elasticity. In the second example, the edges of the strip are assumed to be inclined by 30 deg to the elastically symmetric axes. The experimental results are compared with theoretical calculations. Paper was presented at the SEM VII International Congress on Experimental Mechanics held in Las Vegas, NV on June 8–11, 1992.