摘要
本文基于文献调研的基础上,简介了有限单元法的发展历史以及压电复合材料的应用和研究现状。引入了压电材料的空间等参数单元,并且从弹性力学与电学基础上,根据三维有限元法,建立了压电材料的能量泛函。由于最小势能原理,建立了单元刚度矩阵和压电材料的整体刚度方程。并且推导了压电层合板的边界条件。
在此基础上,利用了Matlab语言编制了具有一定工程意义的计算机程序,对于各种边界条件以及各种形状的压电层合板和非压电层合板的位移、应力、电势、电位移以及电场强度等做了计算与分析。其中一部分结果与相应的解析解做了详细的比较,且证明了本文结果的合理性。本文的研究不仅扩大了解决问题的范围,同时也为求解该问题的解析解或半解析解提供了理想的比较结果。
Based on the theory of elasticity and piezoelectricity, formulas for the three-dimensional finite element analysis of laminated piezoelectric plates are presented. According to the fundamental equations of elastic and electric fields, total potential energy for the electro-elastic laminate under general boundary conditions is established. By using 3-D isoparametric element, the global stiffness matrix equation for the piezoelectric laminate is obtained on the bases of minimum principle of total potential energy. The boundary conditions related to electric variables are discussed and this facilitates the solution of global stiffness equation. Numerical examples are given at the end of the paper to calculate laminated plates with various kinds of elastic and electric boundary conditions. Some results are compared with analytical ones.
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