Early short time transient response of finite L-shaped Mindlin plate

The early short time transient response of an L-shaped plate is investigated by the method of reverberation ray matrix (MRRM) and the generalized ray method (GRM). The MRRM and GRM have been successfully used to study the transient elastic wave transmitting in the beams, planar trusses, space frames and infinite layered media. In the present paper, these two methods are extended to investigate the flexural wave propagation in a finite L-shaped Mindlin plate. The reverberation-ray matrix representing the multi-reflected and scattered waves in the L-shaped plate is presented. The rays in the GRM propagating in the L-shaped plate are determined by the shock source and the boundary conditions. From the numerical results, it indicates that the early short time transient responses of L-shaped plate calculated by the MRRM are the same as those by the GRM. The early short time transient acceleration responses are very large, while the early short time transient displacement responses are very small for the L-shaped plate subjected to the unit impulse. Furthermore, the effects of the different shock signals and the plate thickness on the early short time transient responses of the finite L-shaped Mindlin plates are also analyzed.     

Computation of the stress resultants of a floating Mindlin plate in response to linear wave forces

In this paper, we focus on the computation of stress resultants of a floating elastic plate using the Mindlin plate theory. The proposed method makes use of the linear wave theory and shallow-draft assumption. However, the usual Kirchhoff theory is replaced by the Mindlin theory for the plate. For a single frequency, the coupled water-plate problem is solved using a higher-order-coupled finite element–boundary element method. The solutions for the stress-resultants computed using the proposed method are more satisfactory than these based on the Kirchhoff plate theory.     

ANALYSIS AND EXPERIMENT ON TRANSIENT DYNAMIC RESPONSE IN FINITE MINDLIN PLATE

The transient wave propagation in the finite rectangular Mindlin plate is investigated by the analytical and experimental methods. The generalized ray method (GRM) which has been successfully applied to study the transient responses of beams, planar trusses, space frames and infinite layered media is extended to investigate the transient wave propagation and early short time transient response in finite Mindlin plate. Combining the wave solution, the shock source and the boundary conditions, the ray groups transmitted in the finite rectangular plate can be determined. Numerical simulations and experiments are performed and compared with each other. The results show that the transient wave propagation and early short time transient responses in the finite plate can be studied using the GRM. The early short time transient accelerations are very large for the finite plate subjected to the unit impulse, while the early short time transient displacements are very small. The early short time transient accelerations under the unit impulse are much larger than those under the unit step impulse. The thickness and material characteristics have remarkable effects on the early short time transient responses.     

基于厚板理论分析深水域中弹性浮板的水波响应

基于线性水波理论和Mindlin厚板动力学理论,采用Wiener-Hopf 方法,研究了不同水深水面上弹性浮板在不同入射波数水波作用下的动力学响应问题。首先推导了无限深水域中弹性浮板水波响应的解析解,并将本文分析计算结果与采用其他方法(经典薄板理论)得到的计算结果进行了对比和分析;其次,采用本文方法研究了大型浮板在三种入射波数的水波作用下动弯矩幅值的分布情况;最后,根据其他文献的方法计算了不同水深(有限水深)情况下浮板的动响应,并与本文的计算结果进行了对比分析。     

基于应变梯度中厚板单元的石墨烯振动研究

基于应变梯度理论建立了单层石墨烯等效明德林(Mindlin) 板动力学方程,推导了四边简支明德林中厚板自由振动固有频率的解析解. 提出了一种考虑应变梯度的4 节点36 自由度明德林板单元,利用虚功原理建立了单层石墨烯的等效非局部板有限元模型. 通过对石墨烯振动问题的研究,验证了应变梯度有限元计算结果的收敛性. 运用该有限元法研究了尺寸、振动模态阶数以及非局部参数对石墨烯振动特性的影响. 研究表明,这种单元能够较好地适用于研究考虑复杂边界条件石墨烯的尺度效应问题. 基于应变梯度理论的明德林板所获得石墨烯的固有频率小于基于经典明德林板理论得到的结果. 尺寸较小、模态阶数较高的石墨烯振动尺度效应更加明显. 无论采用应变梯度理论还是经典弹性本构关系,考虑一阶剪切变形的明德林板模型预测的固有频率低于基尔霍夫(Kirchho) 板所预测的固有频率.      

ACTIVE VIBRATION CONTROL OF FINITE L-SHAPED BEAM WITH TRAVELLING WAVE APPROACH

In this paper, the disturbance propagation and active vibration control of a finite L-shaped beam axe studied. The dynamic response of the structure is obtained by the travelling wave approach. The active vibration suppression of the finite L-shaped beam is performed based on the structural vibration power flow. In the numerical calculation, the influences of the near field effect of the error sensor and the small error of the control forces on the control results are all considered. The simulation results indicate that the structural vibration response in the medium and high frequency regions can be effectively computed by the travelling wave method. The effect of the active control by controlling the power flow is much better than that by controlling the acceleration in some cases. And the control results by the power flow method are slightly affected by the locations of the error sensor and the small error of the control forces.     

Vibration band gap properties of periodic Mindlin plate structure using the spectral element method

In this paper, vibration band gap properties of periodic Mindlin plate structure with two simply supported opposite edges are studied using the spectral element method (SEM). Based on the Mindlin plate theory (MPT), the spectral Mindlin plate stiffness matrix is deduced. The spectral equation of the whole periodic structure is further established. Frequency responses are calculated by the spectral equations to illustrate band gap properties. Compared with the results based on the classic plate theory (CPT), the effects of the shear and rotary inertia on the band gap properties are analyzed. The results are also compared with those calculated by the finite element method (FEM). It can be observed that the SEM is more accurate in high frequency ranges. Furthermore, the influences of the material properties and structure dimension on the band gap behaviors are investigated.     

Higher-order shear deformable theories for flexure of sandwich plates—Finite element evaluations

A simple isoparametric finite element formulation based on a higher-order displacement model for flexure analysis of multilayer symmetric sandwich plates is presented. The assumed displacement model accounts for non-linear variation of inplane displacements and constant variation of transverse displacement through the plate thickness. Further, the present formulation does not require the fictitious shear correction coefficient(s) generally associated with the first-order shear deformable theories. Two sandwich plate theories are developed: one in which the free shear stress conditions on the top and bottom bounding planes are imposed and another, in which such conditions are not imposed. The validity of the present development(s) is established through, numerical evaluations for deflections/stresses/stress-resultants and their comparisons with the available three-dimensional analyses/closed-form/other finite element solutions. Comparison of results from thin plate. Mindlin and present analyses with the exact three-dimensional analyses yields some important conclusions regarding the effects of the assumptions made in the CPT and Mindlin type theories. The comparative study further establishes the necessity of a higher-order shear deformable theory incorporating warping of the cross-section particularly for sandwich plates.     

基于Mindlin板理论的偏移损伤成像数值仿真研究

提出了一种应用散射Lamb波的偏移技术对板结构中多部位损伤进行实时识别. 基于Mindlin板理论,推导了板结构中弥散性弯曲波频率-波数域的快速偏移方法. 首先对由线性传感器阵列激励和接收到的入射和散射波场在波数-频率域分别进行延拓,然后根据Huygens原理,结合波场延拓的时间一致性原理施加成像条件,对损伤进行成像识别. 数值仿真研究采用基于Mindlin板理论的有限差分法模拟结构中含不同形状及尺寸损伤时的散射波场. 对模拟散射波场进行偏移成像的结果表明该方法不仅能够识别多部位损伤的位置,还具有识别损伤程度的能力,其快速计算的优点满足在线结构健康监测系统对实时性的要求.      

ANLYTICAL SOLUTION OF THERMOELASTIC DAMPING IN RECTANGULAR MINDLIN MICRO PLATES 1)

首次给出了四边简支的 Mindlin 矩形微板热弹性阻尼的解析解. 基于考虑一阶剪切变形的 Mindlin 板理论和单向耦合热传导理论建立了微板热弹性耦合自由振动控制微分方程. 忽略温度梯度在面内的变化,在上下表面绝热边界条件下求得了用变形几何量表示的温度场的解析解. 进一步将包含热弯曲内力的结构振动方程转化为只包含挠度振幅的四阶偏微分方程. 利用特征值问题之间在数学上的相似性,在四边简支条件下给出了用无阻尼 Kirchhoff 微板的固有频率表示的 Mindlin 矩形微板的复频率解析解,从而利用复频率法求得了反映热弹性阻尼水平的逆品质因子. 最后,通过数值结果定量地分析了剪切变形、材料以及几何参数对热弹性阻尼的影响 规律. 结果表明,Mindlin 板理论预测的热弹性阻尼小于 Kirchhoff 板理论预测的热弹性阻尼. 两种理论预测的热弹性阻尼之间的差值在临界厚度附近十分显著. 另外,随着微板的边/厚比增大,Mindlin 微板的热弹性阻尼最大值单调增大,而 Kirchhoff 微板的热弹性阻尼最大值却保持不变.     

Wave scattering by a floating porous elastic plate of arbitrary shape: A semi-analytical study

In this paper, a semi-analytical model based on linear potential flow theory and an eigenfunction expansion method is developed to study wave scattering by a porous elastic plate with arbitrary shape floating in water of finite depth. The water domain is divided into the interior and exterior regions, corresponding to the domain beneath the plate and the rest extending towards infinite distance horizontally, respectively. The unknown coefficients in the potential expressions are determined by satisfying the continuity conditions for pressure and velocity at the interface of the two regions, together with the conditions for the motion/force at the edge of the plate, where the Fourier series expansion method is employed to deal with the terms associated with the radius function. A plate with three shapes – circular, cosine and elliptical – and three edge conditions are considered. We find that the largest deflection of the plate with a simply supported edge and a clamped edge is more sensitive to the change in porosity when the porosity is small. The power dissipated by an elliptical plate with its major axis perpendicular to the incident wave direction is the largest among the case studies for the majority of the porosity values tested.     

Mindlin 矩形微板的热弹性阻尼解析解

首次给出了四边简支的 Mindlin 矩形微板热弹性阻尼的解析解. 基于考虑一阶剪切变形的 Mindlin 板理论和单向耦合热传导理论建立了微板热弹性耦合自由振动控制微分方程. 忽略温度梯度在面内的变化,在上下表面绝热边界条件下求得了用变形几何量表示的温度场的解析解. 进一步将包含热弯曲内力的结构振动方程转化为只包含挠度振幅的四阶偏微分方程. 利用特征值问题之间在数学上的相似性,在四边简支条件下给出了用无阻尼 Kirchhoff 微板的固有频率表示的 Mindlin 矩形微板的复频率解析解,从而利用复频率法求得了反映热弹性阻尼水平的逆品质因子. 最后,通过数值结果定量地分析了剪切变形、材料以及几何参数对热弹性阻尼的影响 规律. 结果表明,Mindlin 板理论预测的热弹性阻尼小于 Kirchhoff 板理论预测的热弹性阻尼. 两种理论预测的热弹性阻尼之间的差值在临界厚度附近十分显著. 另外,随着微板的边/厚比增大,Mindlin 微板的热弹性阻尼最大值单调增大,而 Kirchhoff 微板的热弹性阻尼最大值却保持不变.      

应变梯度Mindlin板边值问题的讨论

实验和分子动力学计算结果表明,当材料/结构的特征尺寸降为微纳米量级时,他们将表现出明显的尺度效应,因此能否建立精确表征其力学行为的连续介质力学模型具有重要的理论和现实意义.尽管现有文献对非经典Mindlin板的力学行为进行了大量研究,但该模型的变分自洽的边值问题是近年来未攻克的科学问题之一.基于简化的应变梯度理论给出了各向同性Mindlin板应变能的表达式,通过变分原理和张量分析,得到了Mindlin板变分自洽的边值问题及其对应角点条件的位移微分表达式.本文Mindlin板模型的边值问题可退化为相应的Timoshenko梁和Kirchhoff板模型的边值问题,验证了本文结果的有效性.研究结果发现,该Mindlin板模型的控制方程是一个解耦后横向振动具有12阶的偏微分方程,因此需要每个板边提供6个边界条件.角点条件由双应力(double stress)产生,并与经典的剪力、弯矩和扭矩沿截面的法向梯度有关.本文首次澄清了应变梯度Mindlin板存在角点条件这一事实,所得的变分结果有望为其有限元法和伽辽金法等数值方法提供理论依据.     

On canonical bending relationships for plates

In recent years, a series of papers have appeared on algebraic relationships between the solutions (e.g., deflections, buckling loads and frequencies) of a given higher-order plate theory and the classical plate theory. The bending relationships, for example, can be used to generate the transverse deflection of a plate according to the particular higher-order theory from the known deflection of the same problem according to the classical plate theory. In the present study relationships between the bending solutions of several higher-order plate theories and the classical plate theory are obtained in a canonical form (i.e., one set of relationships contain several theories and they can be specialized to a specific theory by assigning values to the constants appearing in the relationships). Numerical examples of bending solutions for rectangular plates with various boundary conditions are presented to show how the relations can be used to determine the deflections and bending moments for various theories. The relationships are validated by comparing the numerical results obtained using the relationships for the Mindlin plate theory against those computed using the ABAQUS finite element program.     

A simple and accurate mixed FE-DQ formulation for free vibration of rectangular and skew Mindlin plates with general boundary conditions

A simple and accurate mixed finite element-differential quadrature formulation is proposed to study the free vibration of rectangular and skew Mindlin plates with general boundary conditions. In this technique, the original plate problem is reduced to two simple bar (or beam) problems. One bar problem is discretized by the finite element method (FEM) while the other by the differential quadrature method (DQM). The mixed method, in general, combines the geometry flexibility of the FEM and high accuracy and efficiency of the DQM and its implementation is more easier and simpler than the case where the FEM or DQM is fully applied to the problem. Moreover, the proposed formulation is free of the shear locking phenomenon that may be encountered in the conventional shear deformable finite elements. A simple scheme is also presented to exactly implement the mixed natural boundary conditions of the plate problem. The versatility, accuracy and efficiency of the proposed method for free vibration analysis of rectangular and skew Mindlin plates are tested against other solution procedures. It is revealed that the proposed method can produce highly accurate solutions for the natural frequencies of rectangular and skew Mindlin plates with general boundary conditions.     

含孔软铁磁材料Mindlin板中弹性波散射问题

基于考虑磁弹相互作用的Mindlin板弯曲波动方程，采用波函数展开法，分析研究 了含孔软铁磁材料Mindlin板中弹性波散射与动应力集中问题，给出了问题的分析 解和数值算例. 通过分析发现：磁感应强度对动弯矩集中系数和动剪力集中系数有 增加的作用，特别是在低频的情况下.     

功能梯度材料明德林矩形微板的热弹性阻尼

功能梯度材料微板谐振器热弹性阻尼的建模和预测是此类新型谐振器热?弹耦合振动响应的新课题. 本文采用数学分析方法研究了四边简支功能梯度材料中厚度矩形微板的热弹性阻尼. 基于明德林中厚板理论和单向耦合热传导理论建立了材料性质沿着厚度连续变化的功能梯度微板热弹性自由振动控制微分方程. 在上下表面绝热边界条件下采用分层均匀化方法求解变系数热传导方程, 获得了用变形几何量表示的变温场的解析解. 从而将包含热弯曲内力的结构振动方程转化为只包含挠度振幅的偏微分方程. 然后,利用特征值问题在数学上的相似性,求得了四边简支条件下功能梯度材料明德林矩形微板的复频率解析解, 进而利用复频率法获得了反映谐振器热弹性阻尼水平的逆品质因子. 最后, 给出了材料性质沿板厚按幂函数变化的陶瓷?金属组分功能梯度矩形微板的热弹性阻尼数值结果. 定量地分析了横向剪切变形、材料梯度变化以及几何参数对热弹性阻尼的影响规律. 结果表明, 采用明德林板理论预测的热弹性阻尼值小于基尔霍夫板理论的预测结果, 而且两者的差别随着相对厚度的增大而变得显著.      

Mathematical construction of a Reissner–Mindlin plate theory for composite laminates

A Reissner–Mindlin theory for composite laminates without invoking ad hoc kinematic assumptions is constructed using the variational-asymptotic method. Instead of assuming a priori the distribution of three-dimensional displacements in terms of two-dimensional plate displacements as what is usually done in typical plate theories, an exact intrinsic formulation has been achieved by introducing unknown three-dimensional warping functions. Then the variational-asymptotic method is applied to systematically decouple the original three-dimensional problem into a one-dimensional through-the-thickness analysis and a two-dimensional plate analysis. The resulting theory is an equivalent single-layer Reissner–Mindlin theory with an excellent accuracy comparable to that of higher-order, layer-wise theories. The present work is extended from the previous theory developed by the writer and his co-workers with several sizable contributions: (a) six more constants (33 in total) are introduced to allow maximum freedom to transform the asymptotically correct energy into a Reissner–Mindlin model; (b) the semi-definite programming technique is used to seek the optimum Reissner–Mindlin model. Furthermore, it is proved the first time that the recovered three-dimensional quantities exactly satisfy the continuity conditions on the interface between different layers and traction boundary conditions on the bottom and top surfaces. It is also shown that two of the equilibrium equations of three-dimensional elasticity can be satisfied asymptotically, and the third one can be satisfied approximately so that the difference between the Reissner–Mindlin model and the second-order asymptotical model can be minimized. Numerical examples are presented to compare with the exact solution as well as the classical lamination theory and the first-order shear-deformation theory, demonstrating that the present theory has an excellent agreement with the exact solution.     

基于平面偶应力-Reissner/Mindlin板比拟的偶应力有限元

偶应力理论的有限元列式面临本质性的C1连续性困难. 平面偶应力理论和Reissner/Mindlin板弯曲理论之间的比拟关系表明这两个理论系统的有 限元的同一性，而R/M板有限元并不存在C1连续性困难. 因此，研究将R/M板单元转化为具有一般位移自由度的平面偶应力单元的一般方法. 根据这一方法，将典型的8节点Serendipity型R/M板单元Q8S转化为一个4节点12 自由度的四边形平面偶应力单元，数值结果表明该单元具有良好的精度和收敛性     

An advanced higher-order theory for laminated composite plates with general lamination angles

This paper proposes a higher-order shear deformation theory to predict the bending response of the laminated composite and sandwich plates with general lamination configurations.The proposed theory a priori satisfies the continuity conditions of transverse shear stresses at interfaces.Moreover,the number of unknown variables is independent of the number of layers.The first derivatives of transverse displacements have been taken out from the inplane displacement fields,so that the C 0 shape functions are only required during its finite element implementation.Due to C 0 continuity requirements,the proposed model can be conveniently extended for implementation in commercial finite element codes.To verify the proposed theory,the fournode C 0 quadrilateral element is employed for the interpolation of all the displacement parameters defined at each nodal point on the composite plate.Numerical results show that following the proposed theory,simple C 0 finite elements could accurately predict the interlaminar stresses of laminated composite and sandwich plates directly from a constitutive equation,which has caused difficulty for the other global higher order theories.