压电体的混合变分原理及叠层板的自由振动分析

建立了具有机一电耦合效应的压电材料修正后的Hellinger—Reissner(H—R)混合变分原理，并推导了压电材料的Hamilton正则方程，即压电材料自由振动的控制微分方程；根据矩阵分析理论给出了带有压电材料层的叠层矩形板自由振动的精确求解方法，文中没有引入任何位移模式或应力模式假设，实例分析得到了压电混合叠层板正逆效应两种情况自由振动的低阶频率，并与已有文献结果进行了比较。本文提出的压电材料修正后的H—R混合变分原理将有利于压电材料动力问题的有限元法或半解析法的推导。

Modified Mixed Variational Principle for Piezoelectric Materials and Free Vibration Analysis of Laminated Plates

Modified Hellinger-Reissner (H-R) mixed variational principle for the free vibration problem of three-dimensional piezoelectric material with mechanical-electric coupling effect is established; the Hamilton canonical equation of there-dimensional piezoelectric bodies is derived from the variational principle, namely, the control differential equation for free vibration of piezoelectric material. Based on the matrix theory, without any assumption about displacements and stresses model, the exact solution of free vibration of the laminated piezoelectric rectangular plates with simple support is presented. Numerical example shows that the low natural frequencies of direct and inverse piezoelectric effect are in good agreement with those of published reference. The variational principle established in this paper benefits to derive the finite element method or semi-analytical solution for piezoelectric material dynamics problem.