正交压电复合材料层板各类边界的解析解

压电复合材料层板由压电片与纤维层叠合而成。基于等效单层理论的位移场和电势场,针对正交铺层压电复合材料层板柱面弯曲问题,建立了力电耦合平衡方程,获得了一般边界的解析解。解析解由特解和通解两部分组成,特解对应于简支边界条件,通解由其他各类边界条件确定。平衡方程的变量仅4个,且不随层数变化。如采用相应的位移和电势分布函数,可以得到一阶理论、高阶理论、指数型理论等多种理论的解析解。算例中给出了各种边界条件下位移、应力和电势的解,讨论了各种理论的精度,观察到了固支边界的应力奇异现象。     

复合材料层板层间缺陷分析——剪切滑移

根据三维弹性平衡方程和层间剪切滑移条件,导出了一个复合材料层板层间剪切滑移模型。本文模型具有一般形式的二维板壳理论的位移场及其平衡方程,但因引入了能反映层板界面粘贴情况及板面条件的剪切变形函数,模型因而简单又精确。层板的弯曲问题和屈曲问题被考虑,层间弱粘贴的影响被讨论。数值结果与精确解比较,表明了本文模型的高精度。      

含压电作动器和传感器的复合材料层板静变形有限元分析和形状控制研究

本文研究了含压电作动器和传感器层的复合材料层板理论,建立了位移和电自由度的四节点有限元素,利用总势能最小原理推导了静力平衡方程,实现和验证了含压电作动器/传感器复合材料层板静态分析有限元程序。并对该类复合材料层板进行了形状控制研究。      

剪切型压电层合圆柱壳的精确解

对剪切型压电层合圆柱壳在电场和机械力作用下的应力场、位移场和电势分布进行了分析。在每层的局部坐标系下对该层的应力场、位移场和电势进行傅立叶展开, 然后采用状态空间方法得到变系数的8 维一阶常微分方程组, 采用Frobenius 方法得到应力场、位移场和电势的解。数值算例显示了电场和机械力作用下, 壳体厚度对应力场和位移场分布的不同影响。得到的3 维精确解为其他剪切型压电弹性层合壳的壳体理论提供理论依据和验证。      

含压电作动器和传感器的复合材料层板静变形有限元分析和形状控制研究

本文研究了含压电作动器和传感器层的复合材料层板理论，建立了位移和电自由度的四节点有限元素，利用总势能最小原理推导了静力平衡方程，实现和验证了含压电作动器／传感器复合材料层板静态分析有限元程序。并对该类复合材料层板进行了形状控制研究。     

压电梯度薄壳的高阶理论解

压电功能梯度执行器能产生较大的位移、降低应力峰值并避免了粘结层带来的问题,压电梯度超声换能器能拓展频带宽度。本文作者提出了一个简单而有效的求解压电梯度薄壳力、电行为特性的高阶理论。设定位移分量为壳厚的线性函数,而电势沿厚度方向为二次分布。考虑了压电作动元的驱动信号不同时所具有的不同形式的电荷平衡方程。应用Fourier级数法得到压电系数沿厚度坐标变化的梯度壳的力电耦合的解析解。所得结果可退化至梁、板等多种特殊情况。利用所得方程分析了一非均匀简支压电层合板,并与三维精确结果作了对比,两者吻合得很好,表明该理论的正确性。最后具体求解了压电梯度圆柱壳的力、电特性,给出了位移、应力、电势沿厚度方向的变化规律。     

压电复合材料层板弱界面力-电-热多场耦合研究

压电复合材料层合结构界面缺陷的物理描述呈现出多场耦合现象。界面弹性场由位移跃变与横向应力的关系来描述;界面温度场由温度跃变与横向位移跃变的经验公式来描述;界面电场分别由电可导通假设、电不可导通假设和电半可导通假设来描述,获取了界面位移、温度和电势跃变的线性表达。基于这些界面假设,得到了柱面弯曲简支压电复合材料层板的力-电-热多场耦合解。算例表明,不同的界面电学假设对弹性场的影响有限,对电场的影响显著,其中电不可导通假设在各种载荷下均产生最大的界面电势跃变。算例提供的数值结果可作为后续研究比较之用。     

机电耦合载荷下的压电层合板瞬态响应分析

针对压电层合板在机电耦合激振下的瞬态响应问题, 提出一种高效混合数值计算方法。经过位移场、 电势场在厚度方向的离散, 利用机电耦合理论和哈密顿原理, 推导出结构的运动方程。引入傅里叶变换, 得到波数域内运动控制方程。应用模态分析方法求解波数域内的位移场和电势场, 对结果进行傅里叶逆变换, 得到空间域内的瞬态响应。以PZT-5A/0° PVDF铺层两相材料复合压电层合板为算例, 分析了力、 电耦合线载荷激励下, 位移场和电势场的瞬态响应历程与分布规律, 计算结果给出了该结构的动力学基本特征。该方法结合了有限元法、 傅里叶变换和模态分析法, 计算高频载荷激振下的压电层合板瞬态响应较一般有限元法大幅减少了单元的划分。该方法可推广至分析任意机电载荷下的各类铺层材料压电层合板瞬态响应问题。      

复合材料夹层板壳一种等效分析方法

本文研究了复合材料夹层板壳的一种等效分析方法,首先由壳体理论导出了夹层板壳的内力和位移表达式,通过与各向异性单层板壳理论的比较,提出了二者等效的条件,从而把夹层板壳问题简化为一个等效各向异性单层板壳问题,这用有限元法很容易求解,并对一个工程实例进行了计算,证实了该方法的有效性。      

复合材料夹层板壳一种等效分析方法

本文研究了复合材料夹层板壳的一种等效分析方法,首先由壳体理论导出了夹层板壳的内力和位移表达式,通过与各向异性单层板壳理论的比较,提出了二者等效的条件,从而把夹层板壳问题简化为一个等效各向异性单层板壳问题,这用有限元法很容易求解,并对一个工程实例进行了计算,证实了该方法的有效性。     

Free vibration of laminated piezoelectric composite plates based on an accurate theory

An accurate theory for laminated piezoelectric composite plates in cylindrical bending is developed for free vibration analysis. The displacement and electric potential fields are depicted approximately by the accurate displacement and electric potential distribution functions through thickness, respectively. The two functions are formulated according to particular solutions to the three-dimensional elasticity equilibrium equations and the electrostatics charge equation. The complicated electromechanical coupling relations and the interfacial continuity conditions are enforced. Accordingly the two functions are coupled and make the displacement and potential fields coupled. The governing equations use only four displacement and potential variables, the number of which is independent of the number of layers involved. A corresponding finite element model is also developed. Natural frequencies of piezoelectric laminates subjected to different sets of boundary conditions are given and parameter studies are conducted in numerical examples. The high accuracy of this theory is demonstrated by comparing the present results with the existing exact three-dimensional solutions.     

HSDT-layerwise analytical solution for rectangular piezoelectric laminated plates

The present paper develops a formulation for laminated plates with extensional distributed piezoelectric sensors/actuators. This formulation is based on linear electroelasticity, and an equivalent single layer is used for the mechanical displacement field, applying a Higher-Order Shear Deformation Theory (HSDT), whereas a layerwise discretization is used in the thickness direction for the electric potential. The electric and mechanical local equilibrium equations and local constitutive equations for the problem are identified. The Principle of Virtual Work is used to derive the dynamic equilibrium equations in terms of generalized forces and the consistent boundary conditions. The piezoelectric laminate constitutive equations are built and used to write the equations of motion in terms of generalized displacements. Finally, analytical solutions for simply supported square laminates with piezoelectric layers are developed. The entire laminate, composed of the base structure and piezoelectric layers, can be arbitrary orthotropic. The solution is adequate for an arbitrary number of piezoelectric layers and stacking positions. Moreover, the solution takes into account all material coefficients, whether mechanical, piezoelectric or dielectric. Analytical results are obtained for static bending, both in sensor and actuation modes, and for free vibration of symmetric cross-ply laminates with piezoelectric layers externally bonded to the plate.     

A sub-layer model for a thick piezoelectric patch bonded on elastic substrate

Summary. In this paper, the two-dimensional electromechanical coupling problems that a piezoelectric patch of finite size bonded to an elastic substrate are considered. A subdivision model that the single physical piezoelectric layer is mathematically divided into a number of thinner layers is proposed to analyze the electromechanical responses of the structures. Within each virtual sub-layer of the piezoelectric patch the electric displacement and normal stress in the axial direction are assumed to be linear functions of the thickness coordinate. Hellinger-Reissner variational principle for elasticity is extended to the systems of piezoelectric multi-materials. The governing equations that comprise one-dimensional differential equations and integro-differential equations are rigorously derived from the stationary conditions of the variational functional along with substitution of the assumed electromechanical fields. The subdivision model satisfies all mechanical and electric continuity conditions across the virtual interfaces and the physical interface of piezoelectrics/substrate. The numerical solutions of the governing equations are conducted, and the convergence of the subdivision model is demonstrated.     

Modeling of functionally graded smart plates with gradient elasticity effects

ABSTRACT

In this article, the equations of motion for functionally graded plates with surface-mounted piezoelectric layers, while accounting for the gradient elasticity through the modified couple stress model and linear piezoelectricity, are derived using Hamilton’s principle. The formulation includes the coupling between mechanical deformations and the charge equations of electrostatics. The mathematical model developed herein is an equivalent single layer theory for mechanical displacement field and the potential functions. The in-plane displacements are assumed to vary as cubic functions of the thickness coordinate while the transverse displacement is assumed to vary as a quadratic function of the thickness coordinate through plate thickness. The potential function is assumed as the combination of half cosine variation of electric potential and linear variation of applied voltage on outer surfaces. The approach described here is that standard plate models can be enhanced to include the coupling between the charge equations and the mechanical deformations as well as the size dependent effect of micro- and nano-scale structures. An analytical solution of the developed model is presented using the Navier solution technique. A parametric study is performed to study the effect of material variation through thickness of plates, length scale parameters to capture the size-dependent effects, and thickness ratio between piezoelectric layers and the whole plate.     

压电复合材料层板弱界面的四种力电热物理模型

基于对压电复合材料层合结构弱界面弹性场、电场、温度场的宏观和细观尺度分析, 构建了四种界面力-电-热模型, 分别为非耦合型、耦合型和混合型。依据这些界面模型, 得到了柱面弯曲简支压电复合材料层板的力-电-热多场耦合解。结果表明, 损伤界面的多场耦合物理描述介于宏观和细观尺度之间, 需要多尺度考虑, 且基于不同尺度的描述也可获得相似的规律。在热载和力载下, 不同的界面模型对弹性场影响趋同、差异甚小, 而对电场和温度场, 四种模型的影响各异、但规律相似; 在电载作用下, 不同的界面模型无论对弹性场还是电场, 都有显著的影响。     

An equivalent single-layer model for magnetoelectroelastic multilayered plate dynamics

An equivalent single-layer model for the dynamic analysis of magnetoelectroelastic laminated plates is presented. The electric and magnetic fields are assumed to be quasi-static and the first-order shear deformation theory is used. The formulation of the model provides for a preliminary fulfillment of the electro-magnetic governing equations, which allows to determine the electric and magnetic potential as functions of the mechanical variables. Then, by using this result, the equations of motion are written leading to the problem governing equations. They involve the same terms of the elastic dynamic problem weighted by effective stiffness coefficients, which take the magneto-electro-mechanical couplings into account. Additional terms, exclusively arising in force of the piezoelectric and piezomagnetic behavior, appear. The electromagnetic inputs are treated as equivalent external distributed axial forces and bending moments. Free and forced vibrations solutions for simply-supported plates are presented to validate the model by comparing the present results with benchmark solutions found in the literature.     

Assessment of coupled 1D models for hybrid piezoelectric layered functionally graded beams

A unified formulation is presented for coupled efficient zigzag, third order, consistent third order and first order models for hybrid piezoelectric layered FGM beams under thermoelectromechanical load. The transverse as well as inplane electric fields are considered. The potential and thermal fields are discretised sublayerwise as piecewise linear which can adequately approximate the actual distributions of these fields across the thickness. The zigzag theory accounts for the non-uniform deflection across the thickness due to the potential and thermal fields and satisfies exactly the conditions on transverse shear stress at the top, bottom and layer interfaces. The governing equilibrium equations have been derived from a variational principle. For the first time, the accuracy of the 1D models is assessed in direct comparison with the exact 2D solutions for elastic as well as piezoelectric hybrid layered FGM beams for mechanical, potential and thermal loads. The effects of the inhomogeneity parameter, span-to-thickness ratio and electric boundary conditions are investigated. It is established that the zigzag theory results are superior to those of other 1D models, even though they have the same number of displacement variables.     

Mesh‐free models for static analysis of smart laminated composite beams

This paper is concerned with the development of mesh‐free models for the static analysis of smart laminated composite beams. The overall smart composite beam is composed of a laminated substrate composite beam and a piezoelectric layer attached partially or fully at the top surface of the substrate beam. The piezoelectric layer acts as the distributed actuator layer of the smart beam. A layer‐wise displacement theory and an equivalent single‐layer theory have been used to derive the models. Several cross‐ply substrate beams are considered for presenting the numerical results. The responses of the smart composite beams computed by the present new mesh‐free model based on the layer‐wise displacement theory excellently match with those obtained by the exact solutions. The mesh‐free model based on the equivalent single‐layer theory cannot accurately compute the responses due to transverse actuation by the piezoelectric actuator. The models derived here suggest that the mesh‐free method can be efficiently used for the numerical analysis of smart structures. Copyright © 2016 John Wiley & Sons, Ltd.     

On refined nonlinear theories of laminated composite structures with piezoelectric laminae

In this paper geometrically nonlinear theories of laminated composite plates with piezoelectric laminae are developed. The formulations are based on thermopiezoelectricity, and include the coupling between mechanical deformations, temperature changes, and electric displacements. Two different theories are presented: one based on an equivalent-single-layer third-order theory and the other based on the layerwise theory, both of which were developed by the senior author for composite laminates without piezoelectric laminae. In the present study, they are extended to include piezoelectric laminae. In both theories, the electric field is expanded layerwise through the laminate thickness. The dynamic version of the principle of virtual displacements (or Hamilton’s principle) is used to derive the equations of motion and associated boundary conditions of the two theories. These theories may be used to accurately determine the response of laminated plate structures with piezoelectric laminae and subjected to thermomechanical loadings.