A theoretical analysis of piezoelectric/composite laminate with larger-amplitude deflection effect,Part II: Hermite differential quadrature method and application

Incorporating the effects of larger-amplitude deflection and electro-elastical properties of piezoelectric lamina, the Hamilton’s variation principle was used to deduce the fundamental formulations of smart anisotropic composite plate in Part I in terms of Reddy’s simple higher-order theory. In order to solve the five highly coupled nonlinear partial differential equations with complicated overlapping boundary conditions, a novel numerical method-Hermite differential quadrature (HDQ) method was developed to implement the differential equations with complicated overlapping boundary conditions. Based on the presently developed HDQ method, any orders derivatives of the unknown functions or any boundary conditions can be point-collocation-based discretized by a set of point-values along x- and y-direction. Then, a system of complete algebraic nonlinear equations can be constructed to calculate out the final point-values of the mid-plane displacements by using the governing equations and relative boundary conditions with HDQ method. Finally, some detailed numerical examples for the anisotropic piezoelectric/composite laminate with the distributed poling directions of piezoelectric layer and fiber orientations of composite layers were studied to validate the developed theoretical analysis model and HDQ numerical method.     

Stress analyses of a smart composite pipe joint integrated with piezoelectric composite layers under torsion loading

In order to improve the joint failure strength, an adhesively bonded smart composite pipe joint system has been developed by integrating electromechanical coupling piezoelectric layers with the connection coupler. It has been validated that the integrated piezoelectric ceramic layers can smartly reduce stress concentration in the adhesive layer bond-line under bending or axial tension loads. In this study, piezoelectric particle/fiber reinforced polymer composite was utilized to construct adhesively bonded smart composite pipe joint systems, in order to overcome the brittle characteristic of the piezoelectric ceramic layers and to facilitate joint construction. Since torsion is one of the dominating loading conditions in practice, the behavior of the newly developed smart pipe joint system subjected to torsion loading was investigated in-detail to evaluate the effect of the integrated piezoelectric reinforced polymer composite layer on the joint performance. Firstly, based on the first-order shear deformation theory, the fundamental equations with relevant boundary and continuity conditions were developed to theoretically model the smart pipe joint system subjected to torsion loading. Further, the analytical solutions for the mid-plane displacements and the shear and peel stresses in the adhesive layer were obtained by using the Levy solution and the state-space method. Finally, some numerical examples were presented to evaluate the detailed effect of the stacking sequence and thickness of the integrated composite piezoelectric layers in the connection coupler on reducing the stress concentration in the adhesive layer; the effect of the applied electric fields on shear and peel stresses in the adhesive layer was also illustrated.     

Nonlinear vibrations of a sandwich piezo-beam system under piezoelectric actuation

The paper describes the use of active structures technology for deformation and nonlinear free vibrations control of a simply supported curved beam with upper and lower surface-bonded piezoelectric layers, when the curvature is a result of the electric field application. Each of the active layers behaves as a single actuator, but simultaneously the whole system may be treated as a piezoelectric composite bender. Controlled application of the voltage across piezoelectric layers leads to elongation of one layer and to shortening of another one, which results in the beam deflection. Both the Euler–Bernoulli and von Karman moderately large deformation theories are the basis for derivation of the nonlinear equations of motion. Approximate analytical solutions are found by using the Lindstedt–Poincaré method which belongs to perturbation techniques. The method makes possible to decompose the governing equations into a pair of differential equations for the static deflection and a set of differential equations for the transversal vibration of the beam. The static response of the system under the electric field is investigated initially. Then, the free vibrations of such deformed sandwich beams are studied to prove that statically pre-stressed beams have higher natural frequencies in regard to the straight ones and that this effect is stronger for the lower eigenfrequencies. The numerical analysis provides also a spectrum of the amplitude-dependent nonlinear frequencies and mode shapes for different geometrical configurations. It is demonstrated that the amplitude–frequency relation, which is of the hardening type for straight beams, may change from hard to soft for deformed beams, as it happens for the symmetric vibration modes. The hardening-type nonlinear behaviour is exhibited for the antisymmetric vibration modes, independently from the system stiffness and dimensions.

     

Active vibration control of thin constrained composite damping plates with double piezoelectric layers

Vibrations and the damping behaviour of thin constrained composite plates with double piezoelectric layers are analytically explored by using Fourier transformation and classical laminated plate theory. Electric potential equations in the double piezoelectric layers are solved with respect to closed and open circuit boundary conditions, an exterior dielectric slab and active control. The natural frequencies and loss factors of the constrained smart composite plates with passive control methods are not notably changed in comparison with those of the constrained composite plates without piezoelectric effects since vibrational energy does not efficiently convert to electrical energy. The loss factors of the composite plates with active constrained damping increase and the natural frequencies have significant variations as the proportional derivative gains increase. Transverse displacement power spectra of the piezoelectric composite plates with active control are compared with those of the piezoelectric composite plates with passive control showing that active control has the best suppression performance of vibrations for the constrained laminated plates with double piezoelectric layers. Radial power spectral density, phase angles and cylindrical-wave power spectral density are calculated. Interesting patterns of wave propagation are explained when plane wave expansion is used to obtain Bessel cylindrical waves.     

Design and analysis of a smart composite pipe joint system integrated with piezoelectric layers under bending

Incorporating with the high electro-mechanical coupling performance of piezoelectric materials, design and analysis of an adhesively bonded smart composite pipe joint system were conducted. In this joint system, piezoelectric layers were integrated into the joint coupler in order to reduce stress concentration in the joint adhesive layer. To theoretically verify the composite action and efficiency of the integrated piezoelectric layers, an electro-mechanical model based on the first-order shear deformation theory was established. This model was able to clarify the energetic characteristics of the proposed joint system on the improvement in the joint strength, which was under the action of a bending moment at the joint ends. The state-space method was utilized to obtain the final analytical solutions, including the peel and shear stress distributions in the adhesive layer. Finally, some numerical examples were calculated to evaluate the effect of the detailed stacking sequence and size of the integrated piezoelectric layers on reducing the stress concentration in the adhesive layer as well as the applied electric fields. These numerical results validated the integrity of the developed adhesively bonded smart composite pipe joint system.     

旋转功能梯度智能结构的刚-柔耦合动力学分析

基于一次耦合模型理论建立了中心刚体-压电层-功能梯度材料智能梁系统的刚柔耦合动力学模型．研究了开环状态下将压电材料作为传感器的压电效应和质量刚度效应对系统动力学特性的影响．通过仿真算例与另两种不同建模理论（传统零次近似耦合模型、一次近似耦合模型）作了对比．随着中心刚体外驱动力矩的增大,零次近似耦合模型和一次近似耦合模型计算结果逐渐发散,而本文的一次耦合模型的计算结果始终保持收敛,较其他近似耦合模型具有一定优势．对三种不同的结构的计算结果表明,压电材料的压电效应对系统的动力学特性影响显著,压电材料的质量刚度效应也会影响智能梁的动力学行为,前者比后者的影响大得多．此外,功能梯度材料功能梯度指数对系统动力学特性的影响也较大．     

Nonlinear active control of damaged piezoelectric smart laminated plates and damage detection

Considering mass and stiffness of piezoelectric layers and damage effects of composite layers, nonlinear dynamic equations of damaged piezoelectric smart laminated plates are derived. The derivation is based on the Hamilton＇s principle, the higher- order shear deformation plate theory, von Karman type geometrically nonlinear straindisplacement relations, and the strain energy equivalence theory. A negative velocity feedback control algorithm coupling the direct and converse piezoelectric effects is used to realize the active control and damage detection with a closed control loop. Simply supported rectangular laminated plates with immovable edges are used in numerical computation. Influence of the piezoelectric layers＇ location on the vibration control is in- vestigated. In addition, effects of the degree and location of damage on the sensor output voltage are discussed. A method for damage detection is introduced.     

Development of a smart composite pipe joint integrated with piezoelectric layers under tensile loading

To actively reduce the stress concentration effect in adhesive layers, a novel smart adhesively bonded composite pipe joint system was developed by integrating piezoelectric layers as sensor/actuator in the connection coupler. In the presently developed smart pipe joint system, the mechanical loading induced structural deformation can be detected and monitored by integrated sensing piezoelectric layers, and then the signal is fed back to the integrated actuating piezoelectric layers to adaptively produce additional forces and moments so as to decrease the maximum peel and shear stresses in the adhesive layer. In order to theoretically predict the efficiency of the developed smart pipe joint system, an electro-mechanical theoretical analytical model was established to investigate the characteristics of the joint system under end tension load in terms of first-order shear deformation theory. Simultaneously, the state-space method was utilized to deduce the final analytical solutions, including the peel and shear stress distributions in the adhesive layer. Finally, some detailed numerical results were obtained to demonstrate the optimal design method of such smart pipe joint system and further validate the integrity of this joint system.     

带剪切型压电激励器的板的高阶剪切变形理论

利用压电材料固有的正，逆压电效应可以对结构变形和振动进行控制。与外加电场与极化方向平行于板厚度的压电材料的拉伸作动机制相比，外加电场与极化方向垂直的压电材料的剪切作动机制可以在作动器内产生较小的应力，从而降低作动器边界产生分层破坏的危险。本文对于压电材料的剪切作动机制进行研究，应用三阶剪切变形理论建立带剪切型压电激励器的智能层合板模型。采用哈密顿原理导出带剪切型压电激励器的层合板的控制方程。采用空间法得到了各种边界条件组合条件下板的解析解。数值算例对一三层板采用高阶和一阶剪切变形理论进行计算，结果表明两种理论所得的变形曲线很相似。但对于厚度剪切型激励器而言，由于激励器是引起板的剪切变形，而高阶剪切变形理论比一阶剪切变形理论能更好地反映结构的剪切应变能，因此高阶剪切变形理论可以提供板变形的更为精确的解。因此，对于厚度剪切型激励器，剪切变形理论的选取对于板变形结果的好坏有重要的作用。     

压电复合材料梁和板的变形控制问题

由于具有良好的结构、力学性能,复合材料层合板在现代飞行器上大量应用;而压电复合材料,作为一种新兴的智能材料,由于其独特的力电耦合性能得到了人们更多的关注。本文研究含有压电片的复合材料梁和板在电场作用下的变形控制问题。基于经典的梁理论和层合板理论,分别研究了下列问题:(1)双压电片布置的悬臂梁的变形;(2)含有压电层的层合板变形控制问题;(3)含有一对压电片的层合板的变形控制问题。针对上述问题,分别给出了理论解和数值解,并进行了相关讨论分析。结果表明压电材料可对结构进行精确控制,因此本文的结果可对复合材料梁和板在电场作用下的变形控制问题提供工程参考。     

Finite element formulation for active functionally graded thin-walled structures

For the analysis and design process of smart structures with integrated piezoelectric patches, the finite element method provides an effective simulation approach. In this paper, an attempt on modeling and simulation of the behavior of hybrid active structures is carried out using developed Kirchhoff-type-four-node shell element.The finite element results are compared with reference solutions taking into account the electromechanical responses of smart structures with various geometries, and the results show very high agreement. The main aspect of the application of the proposed element is to predict the behavior of FGM shells containing piezoelectric layers. A set of numerical analyses is performed in order to highlight the applicability and effectiveness of the present finite element model, notably for smart FGM structures. A comprehensive parametric study is conducted to show the influence of material composition, the placement and the thickness of the piezoelectric layers on the deformation of the laminated structure.     

The effective properties of piezoelectric composite materials with transversely isotropic spherical inclusions

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压电层合圆板传感器的非线性静力耦合分析

本文基于von Karman薄板非线性理论，压电基本方程和经典层合板理论，对上，下表面粘有压电传感膜层的圆薄板压力传感器的力-电耦合特性进行了定理分析。首先，针对压电层合圆板传感器件建立了考虑几何非线性变形的力-意耦合模型。给出了压电薄膜层感应电量的计算公式；然后，采用级数解法进行了求解，得到了非线性静力耦合问题的幂级数精确解，给出了传感器静态测量过程中所关心的压力-电量特征曲线和中心挠度，电量特征曲线；同时，定量讨论了压电二次效应对感应电量和层合结构中心挠度的影响。通过所得结果，在压电式压力传感器设计中，为采用电信号直接输出有关结构的变形和外界载荷等待测参量提供了依据。     

Three-dimensional solution for a hybrid cylindrical shell under axisymmetric thermoelectric load

Summary An exact axisymmetric piezothermoelastic solution is presented for a simply-supported hybrid cylindrical shell made of cross-ply composite laminate and piezoelectric layers. Numerical results for hybrid shells are presented for sinusoidal and central band thermal and electrical loads. The effect of the loading, the radius-to-thickness ratio, the span-to-radius ratio and the number of layers of the substrate on the response is investigated. The interface between the substrate and the actuated piezoelectric layer has been found to be subjected to high shear stress. It has been shown that the maximum values of the deflection and the stresses, due to thermal load, can be appreciably reduced by appropriate application of actuation potential. Accepted for publication 13 October 1996     

压电板屈曲和后屈曲的有限元分析

采用板的一附剪切理论,并考虑结构的几何非线性,基于Ｔｏｔａｌ Ｌａｇｒａｎｇｅ方法,建立了在弱非线性耦合假设下压电智能的有限元控制方法,分析了没边界条件下压电板的屈曲和后屈曲,计算结果表明,在单向压力作用下,材料的压电效应和外加电压对板的屈曲载荷及后屈曲影响很小;而电场对位移加载简支板的屈曲影响较显著。该文为压电材料的工程应用提供了理论指导,同时提供了一种有效的有限元分析方法。     

Assessment of improved zigzag and smeared theories for smart cross-ply composite cylindrical shells including transverse normal extensibility under thermoelectric loading

The improved zigzag theory recently developed by the authors for smart, piezoelectric, and laminated cylindrical shells is assessed for the response of finite-length cross-ply shells and shell panels under mechanical, potential, and thermal loading, in direct comparison with the exact three-dimensional (3D) piezothermoelasticity solution. This theory has the unique features of including the transverse normal strain due to thermoelectric loading without introducing additional deflection variables, capturing the nonlinear potential field and actual temperature profile across laminate thickness, accounting for the layerwise (zigzag) variation of inplane displacements, and satisfying the conditions on transverse shear stresses at the layer interfaces and at the inner and outer surfaces. For the assessment, new results are obtained for the 3D exact solution for smart cylindrical shells having a test laminate with widely different material properties across layers, a piezo-composite laminate and a piezo-sandwich laminate. To ascertain the contribution of the layerwise terms in the inplane displacements, the theory is compared with its smeared counterpart with the same number of primary variables. The effect of inclusion of transverse normal extensibility in these theories is established by comparing with their conventional counterparts that assume constant deflection across the thickness. The effect of span angle (for shell panels), length, and thickness parameters on the error of the 2D theories is illustrated.     

Magnetoelectroelastic shear body waves in periodically layered metalized ferrite–piezoelectric media

A method of deriving the dispersion equations for magnetoelectroelastic shear waves in periodically layered media is proposed. The media are formed by uniting identical metalized laminates, each consisting of two dissimilar piezoelectric layers separated by a layer with the properties of ferrite. A numerical analysis is carried out and the propagation of body waves in different structures consisting of ZnO and GaYtC layers is described within wide ranges of frequencies and wave numbers. The influence of the physical, mechanical, and geometrical parameters of the layers on the structure of the transmission and suppression zones and the effect of the piezoelectric effect of the position of transmission edges are examined     

压电智能结构拓扑优化研究进展

压电材料因其变形精度高、反应速度快、易于制作成小型化元件已经被广泛应用于精密驱动、振动控制、精确定位等领域。改变压电智能结构中压电元器件的位置、大小、形状等参数能够有效地改善系统的力学性能,因而吸引了许多学者和工程师的关注和研究。拓扑优化作为有效的优化工具,已经成功应用于压电智能结构的优化设计中。本文首先阐述了压电智能结构拓扑优化的背景和意义,简要回顾了压电智能结构主动控制及分析方法,并综述了面向结构静变形控制的压电智能结构优化、面向振动控制的压电智能结构优化、压电俘能器的设计与优化等三个方面的研究进展。最后,简单归纳压电智能结构拓扑优化研究中值得关注的几个问题。     

压电智能结构拓扑优化研究进展

压电材料因其变形精度高、反应速度快、易于制作成小型化元件已经被广泛应用于精密驱动、振动控制、精确定位等领域。改变压电智能结构中压电元器件的位置、大小、形状等参数能够有效地改善系统的力学性能,因而吸引了许多学者和工程师的关注和研究。拓扑优化作为有效的优化工具,已经成功应用于压电智能结构的优化设计中。本文首先阐述了压电智能结构拓扑优化的背景和意义,简要回顾了压电智能结构主动控制及分析方法,并综述了面向结构静变形控制的压电智能结构优化、面向振动控制的压电智能结构优化、压电俘能器的设计与优化等三个方面的研究进展。最后,简单归纳压电智能结构拓扑优化研究中值得关注的几个问题。     

具有叉指式电极的1-3型压电复合材料层合板力电耦合行为的三维精确分析

基于三维弹性理论和压电理论,导出了含有1-3型压电复合材料层的有限长矩形层合简支板的静力平衡方程和边界条件,给出了该层合板在叉指式电极和外力共同作用下力电耦合特性的三维精确解.数值算例的计算结果与有限元解进行了对比,取得了很好的一致性.研究了压电矩阵各向异性和刚度矩阵各向异性以及电势等因素对其挠曲面扭率最大值的影响.数值结果表明层合板扭率最大值的绝对值随压电矩阵各向异性系数Rd的增大而增大并随刚度矩阵各向异性系数Rc的减小而增加.