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.      

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.     

A comparative analysis of Mindlin and Kirchhoff bending solutions for nonlinearly tapered annular plate with free edges

In this study, we consider the problem of nonlinearly tapered annular plate with a free edge. The supported edge may be simply supported, clamped or elastically restrained against rotation. Exact expressions of deflection, moment-resultants, and stresses are presented for nonuniform thickness. We compare the results of the Kirchhoff plate theory and the Mindlin plate theory. It is shown that the Kirchhoff plate theory and the Mindlin plate theory provide approximately the same results for the positive values of the thickness factor, but the difference between the deflections diverges as the thickness increases at the inner edge. We also propose that the Kirchhoff plate theory may be used in the region of −0.4 ≤ α < 1 and the Mindlin plate theory must be used for α < −0.4.     

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.      

基于各向异性修正偶应力理论的Reddy型层合板的自由振动

本文基于各向异性修正偶应力理论建立了只含一个尺度参数的Reddy型复合材料层合板的自由振动模型。同见诸于文献的细观尺度Kirchhoff薄板偶应力模型相比,本文提出的新模型能够更精确的预测细观尺度下的中、厚层合板的自振频率。基于Hamilton原理推导了细观尺度下Reddy型复合材料层合板的运动微分方程以及边界条件,并以正交铺设的四边简支复合材料层合方板为例进行了解析求解,分析了尺度参数对自振频率的影响并对比了Kirchhoff、Mindlin和Reddy等三种板模型计算结果的异同。算例结果表明本文所给出的模型能够捕捉到复合材料层合板自由振动问题的尺度效应。另外,在细观尺度下Kirchhoff板模型所预测的自振频率相对于Mindlin板模型和Reddy板模型总是过高,且越接近厚板三者的差别就越大,这与经典理论中三种板模型的对比情况是一致的。     

基于新修正偶应力理论的Mindlin层合板稳定性分析

基于新的各向异性修正偶应力理论提出一个Mindlin复合材料层合板稳定性模型。该理论包含纤维和基体两个不同的材料长度尺度参数。不同于忽略横向剪切应力的修正偶应力Kirchhoff薄板理论,Mindlin层合板考虑横向剪切变形引入两个转角变量。进一步建立了只含一个材料细观参数的偶应力Mindlin层合板工程理论的稳定性模型。计算了正交铺设简支方板Mindlin层合板的临界载荷。计算结果表明该模型可以用于分析细观尺度层合板稳定性的尺寸效应。     

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.     

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.     

Closed form solutions for free vibrations of rectangular Mindlin plates

A new two-eigenfunctions theory, using the amplitude deflection and the generalized curvature as two fundamental eigenfunctions, is proposed for the free vibration solutions of a rectangular Mindlin plate. The three classical eigenvalue differential equations of a Mindlin plate are reformulated to arrive at two new eigenvalue differential equations for the proposed theory. The closed form eigensolutions, which are solved from the two differential equations by means of the method of separation of variables are identical with those via Kirchhoff plate theory for thin plate, and can be employed to predict frequencies for any combinations of simply supported and clamped edge conditions. The free edges can also be dealt with if the other pair of opposite edges are simply supported. Some of the solutions were not available before. The frequency parameters agree closely with the available ones through pb-2 Rayleigh-Ritz method for different aspect ratios and relative thickness of plate.     

Bending of annular sectorial Mindlin Plates using Kirchhoff results

This study is concerned with the elastic bending problem of a class of annular sectorial plates whose radial edges are simply supported. Exact bending relationships between the Mindlin plate results and the corresponding Kirchhoff plate solutions have been derived based on the concept of load equivalence. These bending relationships facilitate the deduction of thick (Mindlin) plate results from the corresponding classical thin (Kirchhoff) plate solutions, thus bypassing the need to solve the more complicated governing equations of thick plates. The correctness of the relationships is established by solving the bending problem of annular sectorial plates under a uniformly distributed load and comparing the results with existing thick plate solutions.     

A Bending-Gradient model for thick plates,Part II: Closed-form solutions for cylindrical bending of laminates

In the first part (Lebée and Sab, 2010a) of this two-part paper we have presented a new plate theory for out-of-plane loaded thick plates where the static unknowns are those of the Kirchhoff–Love theory (3 in-plane stresses and 3 bending moments), to which six components are added representing the gradient of the bending moment. The new theory, called Bending-Gradient plate theory is an extension to arbitrarily layered plates of the Reissner–Mindlin plate theory which appears as a special case when the plate is homogeneous. Moreover, we demonstrated that, in the general case, the Bending-Gradient model cannot be reduced to a Reissner–Mindlin model. In this paper, the Bending-Gradient theory is applied to laminated plates and its predictions are compared to those of Reissner–Mindlin theory and to full 3D (Pagano, 1969) exact solutions. The main conclusion is that the Bending-Gradient gives good predictions of deflection, shear stress distributions and in-plane displacement distributions in any material configuration. Moreover, under some symmetry conditions, the Bending-Gradient model coincides with the second-order approximation of the exact solution as the slenderness ratio L/h goes to infinity.     

An eight noded composite plate element for the dynamic analysis of spatial mechanism systems

The development of a shear-deformable laminated plate element, based on the Mindlin plate theory, for use in large reference displacement analysis is presented. The element is sufficiently general to accept an arbitrary number of layers and an arbitrary number of orthotrophic material property sets. Coordinate mapping is utilized so that non-rectangular elements may be modeled. The Gauss quadrature method of numerical integration is utilized to evaluate volume integrals. A comparative study is done on the use of full Gauss quadrature, reduced Gauss quadrature, mixed Gauss quadrature, and closed form integration techniques for the element. Dynamic analysis is performed on the RSSR (Revolute-Spherical-Spherical-Revolute) mechanism, with the coupler modeled as a flexible plate. The results indicate the differences in the dynamic response of the transverse shear deformable eight-noded element as compared to a four-noded plate element. Dynamically induced stresses are examined, with the results indicating that the primary deformation mode of the eight-noded Mindlin plate model being bending.     

A Bending-Gradient model for thick plates. Part I: Theory

This is the first part of a two-part paper dedicated to a new plate theory for out-of-plane loaded thick plates where the static unknowns are those of the Kirchhoff–Love theory (3 in-plane stresses and 3 bending moments), to which six components are added representing the gradient of the bending moment. The new theory, called the Bending-Gradient plate theory is described in the present paper. It is an extension to arbitrarily layered plates of the Reissner–Mindlin plate theory which appears as a special case of the Bending-Gradient plate theory when the plate is homogeneous. However, we demonstrate also that, in the general case, the Bending-Gradient model cannot be reduced to a Reissner–Mindlin model. In part two (Lebée and Sab, 2011), the Bending-Gradient theory is applied to multilayered plates and its predictions are compared to those of the Reissner–Mindlin theory and to full 3D Pagano’s exact solutions. The main conclusion of the second part is that the Bending-Gradient gives good predictions of both deflection and shear stress distributions in any material configuration. Moreover, under some symmetry conditions, the Bending-Gradient model coincides with the second-order approximation of the exact solution as the slenderness ratio L/h goes to infinity.     

A new twelve DOF quadrilateral element for analysis of thick and thin plates

A simple quadrilateral 12 DOF plate bending element based on Reissner–Mindlin theory for analysis of thick and thin plates is presented in this paper. This element is constructed by the following procedure:

• 1.the variation functions of the rotation and shear strain along each side of the element are determined using Timoshenko's beam theory; and
• 2.the rotation, curvature and shear strain fields in the domain of the element are then determined using the technique of improved interpolation.

The proposed element, denoted by ARS-Q12, is robust and free of shear locking and, thus, it can be employed to analyze very thin plate. Numerical examples show that the proposed element is a high performance element for thick and thin plates.     

Shear deformable bending solutions for nonuniform beams and plates with elastic end restraints from classical solutions

Summary The relationships of bending solutions between Timoshenko beams and Euler-Bernoulli beams are derived for uniform and non-uniform beams with elastic rotationally restrained ends. Extensions of these relationships for the cylindrical bending of Mindlin and Kirchhoff plates and for the bending of symmetrically laminated beams are also discussed. The new set of general relationships is useful because the more complex Timoshenko beam and Mindlin plate solutions may be readily obtained from their simpler Euler-Bernoulli beam and Kirchhoff plate solutions respectively, without much tedious mathematics. Received 16 March 1997; accepted for publication 26 November 1997     

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.      

A homogenized Reissner–Mindlin model for orthotropic periodic plates: Application to brickwork panels

This paper describes a new procedure for the homogenization of orthotropic 3D periodic plates. The theory of Caillerie [Caillerie, D., 1984. Thin elastic and periodic plates. Math. Method Appl. Sci., 6, 159–191.] – which leads to a homogeneous Love–Kirchhoff model – is extended in order to take into account the shear effects for thick plates. A homogenized Reissner–Mindlin plate model is proposed. Hence, the determination of the shear constants requires the resolution of an auxiliary 3D boundary value problem on the unit cell that generates the periodic plate. This homogenization procedure is then applied to periodic brickwork panels.A Love–Kirchhoff plate model for linear elastic periodic brickwork has been already proposed by Cecchi and Sab [Cecchi, A., Sab, K., 2002b. Out-of-plane model for heterogeneous periodic materials: the case of masonry. Eur. J. Mech. A-Solids 21, 249–268 ; Cecchi, A., Sab, K., 2006. Corrigendum to A comparison between a 3D discrete model and two homogenised plate models for periodic elastic brickwork [Int. J. Solids Struct., vol. 41/9–10, pp. 2259–2276], Int. J. Solids Struct., vol. 43/2, pp. 390–392.]. The identification of a Reissner–Mindlin homogenized plate model for infinitely rigid blocks connected by elastic interfaces (the mortar thin joints) has been also developed by the authors Cecchi and Sab [Cecchi A., Sab K., 2004. A comparison between a 3D discrete model and two homogenised plate models for periodic elastic brickwork. Int. J. Solids Struct. 41/9–10, 2259–2276.]. In that case, the identification between the 3D block discrete model and the 2D plate model is based on an identification at the order 1 in the rigid body displacement and at the order 0 in the rigid body rotation.In the present paper, the new identification procedure is implemented taking into account the shear effect when the blocks are deformable bodies. It is proved that the proposed procedure is consistent with the one already used by the authors for rigid blocks. Besides, an analytical approximation for the homogenized shear constants is derived. A finite elements model is then used to evaluate the exact shear homogenized constants and to compare them with the approximated one. Excellent agreement is found. Finally, a structural experimentation is carried out in the case of masonry panel under cylindrical bending conditions. Here, the full 3D finite elements heterogeneous model is compared to the corresponding 2D Reissner–Mindlin and Love–Kirchhoff plate models so as to study the discrepancy between these three models as a function of the length-to-thickness ratio (slenderness) of the panel. It is shown that the proposed Reissner–Mindlin model best fits with the finite elements model.     

Wave Propagation Characteristics in Thick Conventional and Auxetic Cellular Plates

Based on Mindlin plate models and Kirchhoff plate models,this study was concerned with the wave propagation characteristics in thick conventional and auxetic cellular structures,with the objective to clarify the effects of negative Poisson's ratio,shear factor and orthotropic mechanical properties on the dynamic behaviors of thick plates.Numerical results revealed that the predictions using variable shear factor in Mindlin plate models resulted in high wave frequencies,which were more significant for plates with negative values of Poisson's ratio.The present study can be useful for the design of critical applications by varying the values of Poisson's ratio.     

Thick Lévy plates re-visited

This paper is concerned with the bending problem of Lévy plates which are simply supported on two opposite edges with any combination of simply supported, clamped or free conditions at the remaining two edges. This study attempts to solve thick Lévy plate problems in a novel way by establishing bending relationships that allow the prediction of Mindlin plate results using the corresponding Kirchhoff solutions. Based on the concept of load equivalence, these relationships obviate the need for complicated thick plate analyses that involve significant computation time and effort. Numerical plate solutions are then determined from these relationships and the validity of these results is verified using other known results and those generated using the abaqus software. It is through this study that the only analytical Mindlin plate solutions by Cooke and Levinson (Int. J. Mech. Sci. 25 (1983) 207) are found to contain errors. In this study, it is found that there are important distinctions between the Mindlin and Reissner plate theories. These differences will also be substantiated by numerical comparison.     

Statics and dynamics of simply supported polygonal Reissner-Mindlin plates by analogy

Summary Free and forced vibrations of moderately thick, transversely isotropic plates loaded by lateral forces and hydrostatic (isotropic) in-plane forces are analyzed in the frequency domain. Influences of shear, rotatory inertia, transverse normal stress and of a two-parameter Pasternak foundation are taken into account. First-order shear-deformation theories of the Reissner–Mindlin type are considered. These theories are written in a unifying manner using tracers to account for the various influencing parameters. In the case of a general polygonal shape of the plate and hard-hinged support conditions, the Reissner-Mindlin deflections are shown to coincide with the results of the classical Kirchhoff theory of thin plates. The background Kirchhoff plate, which has effective (frequency-dependent) stiffness and mass, is loaded by effective lateral and in-plane forces and by imposed fictitious “thermal” curvatures. These deflections are further split into deflections of linear elastic prestressed membranes with effective stiffness, mass and load. This analogy for the deflections is confirmed by utilizing D'Alembert's dynamic principle in the formulation of Lagrange, which yields an integral equation. Furthermore, the analogy is extended in order to include shear forces and bending moments. It is shown that in the static case, with no in-plane prestress taken into account, the stress resultants for certain groups of Reissner-type shear-deformable plates are identical with those resulting from the Kirchhoff theory of the background. Finally, results taken from the literature for simply supported rectangular and polygonal Mindlin plates are yielded and verified by analogy in a quick and simple manner. Received 29 September 1998; accepted for publication 22 June 1999