压电薄板屈曲有限元分析及DKQ单元

THE BUCKLING ANALYSIS OF PIEZOELECTRIC THIN PLATES USING FINITE ELEMENT METHOD AND DKQ ELEMENT

在机电耦合本构方程基础上，利用Ｈａｍｉｌｔｏｎ原理推导了压电薄板屈曲分析的有限元特征方程和机电耦合的内力计算公式，在有限元实现中选择了基于Ｋｉｒｃｈｈｏｆｆ薄板假定的四边形薄板单元（ＤＫＱ单元），并给出该单元的几何刚度阵及其数值积分方法．在大型通用有限元分析和优化设计软件系统ＪＩＦＥＸ中实现了该方法．给出的数值验证了 ＤＫＱ单元在屈曲分析和压电薄板静力分析中具有较高精度和收敛性，通过机械荷载和电荷载联合作用下的临界荷载计算，表明压电耦合效应能够影响结构的稳定性，可以通过改变外加电压对结构稳定性进行控制．

To meet the requirements of the space engineering, especially aircraft, satellites and robot manufacture, the intelligent structures become an active research field in the recent years. The piezoelectric material, as an important part of intelligent structure materials, has not only the ability of carrying load, but also the mechanical-electric coupling property can serve as actuator and sensor. Because of mechanical-electric coupling effect, the analysis of the piezoelectric intelligent structures is more complicated than conventional materials. The exact solution of the piezoelectric structure is difficult to obtain and the research work of the numerical algorithm for the piezoelectric structures is necessary. The passing research works about piezoelectric intelligent structure are mainly focus on the static and dynamic characteristic. But it is not sufficient with the increasing engineering application of the piezoelectric material, the buckling stability should be considered. Considering mechanical-electric coupling effect, the paper derives the piezoelastic finite element equation of the buckling analysis for piezoelectric thin plate structuires with Hamilton pie. The DKQ element, based on Kirchhoff theory, is employ6d to form the piezoelectric element, because the DKQ element has a good behavior in the static analysis of the conventional material. Then the paper simply describes the formulation of the DKQ element, derives geometric stiffness matrix of DKQ element and detailed discusses different numerical integration technique between the elastic stiffness matrix and the geometry stiffness matrix. On the basis of above works, the stress of the piezoelectric thin plate is obtained and the characteristic equation of piezoelectric thin plate buckling is given. mom the equation we can find the extern load changes with the extern voltage. When the voltage brings tense stress, the structure buckling stability can be increased and when the voltage brings compressive stress, the structure buckling stability can be decreased. An FEM code is also implemented in the JIFEX, a general-purpose software for the finite element analysis and design optimization. By comparing the calculation results employing DKQ element with those calculated using the other methods, this paper firstly checks the high accuracy and convergence of the DKQ element in the thin plate elastic buckling and static piezoelectric analysis. Numerical example of a thin plat subjected to the electric and mechanical load shows that mechanical-electric coupling effect can affect the buckling stability of thin plates and regulating external voltages can control stability of piezoelectric thin plate. Numerical examples given in the paper also demonstrate the effectiveness of algorithm and the program in this paper.