功能梯度矩形板的近似理论与解析解

提出了压电功能梯度矩形板在竖向载荷作用下的近似理论与解析解. 引入了板理论的Kirchhoff假设、Reissner-Mindlin假设和提出的补充假设, 并假设材料常数在板厚方向按指数规律变化. 推导了板在周边简支同时又接地情况下中性层法线转角的解和用Fourier级数表示的电势解. 该解在形式上比精确解简单得多, 进行数值计算时也相当方便与快捷. 计算结果与ANSYS软件用三维实体单元的有限元计算结果进行了比较, 证实了该方法即使在厚板情况下仍然具有很高的精度. 

APPROXIMATE THEORY AND ANALYTICAL SOLUTION FOR FUNCTIONALLY GRADED PIEZOELECTRIC RECTANGULAR PLATES

Functionally graded piezoelectric material (FGPM) has electric-mechanically coupled property. Furthermore, its material property parameters vary continuously in some direction so that the stress concentration effect due to temperature change can be greatly decreased. Probably, some prospective properties can also be obtained by adjusting the varying gradient of the material property. Thus, it has a good prospect in application. Recently, study of FGPM is focused on the analyses of thermal response, static and dynamic response of the material, buckling behavior, fracture behavior of structures, optimal design and parameter identification of structures, etc. The theory and approaches for FGPM plates and shells includes simplified model method, laminated model method, asymptotic method, exact solution, finite element method, etc. But, the work for FGPM plates and shells based on approximate theory is rarely found. Although some exact solutions and FEM solutions have been worked out, simplified approximate solutions based on some assumptions with satisfactory precision are also attractive. In this work, several assumptions, such as Kirchhoff assumption, Reissner-Mindlin assumption and some other assumptions proposed by the authors are introduced. The first order shearing theory for plates is employed. Exponential gradient for material properties across the thickness of the plates is prescribed. Using the governing equations of FGPM and the pertinent boundary conditions, the approximate theory for the FGPM plate is established. The solutions of deflection slopes and electric potential for simply supported rectangular plate with its periphery grounded and electric-mechanical transverse loading applied are obtained. The electric potential solution is expressed in the form of double Fourier series. The solution is typically much simpler than the exact solutions, and its numerical computation is proved to be quite easy. The numerical results of this solution are given and compared with the 3D finite element solution by ANSYS. It is shown that this solution is in good agreement with the 3D finite element solution. It is found that the present solution has high precision even for thick plates. Finally, the limitation of this theory and the analytical solution is discussed.