基于三阶剪切变形理论的压电功能梯度板静力学等几何分析

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振动与冲击 ›› 2021, Vol. 40 ›› Issue (1) : 73-85.

论文

刘涛1，2，李朝东1，汪超2，蒋雅芬3，刘庆运2

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Static iso-geometric analysis of piezoelectric functionally graded plate based on third-order shear deformation theory

LIU Tao1，2, LI Chaodong1, WANG Chao2, JIANG Yafen3, LIU Qingyun2

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摘要

针对压电功能梯度板的静力学问题，建立了一种基于三阶剪切变形理论的等几何分析求解方法。其中，定义功能梯度板的材料属性为板厚方向的幂函数分布，并假设压电功能梯度板中的机械位移场与电势场相互独立。首先，利用压电材料的第二类本构方程以及哈密顿变分原理，推导出压电功能梯度板的相关等几何有限元方程。其次，在压电功能梯度板的自由振动分析中，研究了各类机械边界条件的等几何数值方法的收敛性及精度问题。并分析了开短、路状电学边界条件、功能梯度指数n、功能梯度层的宽厚比、压电层与功能梯度层的厚度比对其固有频率的影响。最后，分析了机械载荷、电载荷以及机电耦合情况下，压电功能梯度板的静态弯曲行为，并利用位移反馈控制规律实现了压电功能梯度板的闭环变形控制。通过算例及相关文献对比，表明了本文求解方法的精确性和可靠性。

Abstract

Aiming at the static problem of a piezoelectric functionally graded plate (PFGP), an iso-geometric analysis method based on the third order shear deformation theory was proposed. The material properties of PFGP were assumed to be a power function distribution along plate thickness, and its mechanical displacement field and electric potential field were assumed to be independent of each other. Firstly, using the second type constitutive equation of piezoelectric materials and Hamilton principle, the iso-geometric finite element equations of a PFGP were derived. Secondly, in its free vibration analysis, convergence and accuracy of the proposed method for PFGP’s various mechanical boundary conditions were studied, and the effects of electrical boundary conditions, functional gradient index n, width to thickness ratio of functional gradient layer, and thickness ratio of functional gradient layer to piezoelectric layer on PFGP’s natural frequencies were analyzed. Finally, static bending behaviors of PFGP under mechanical load, electrical load and electro-mechanical coupling condition were analyzed, respectively and a displacement feedback control law was used to realize a close-loop deformation control of PFGP. Several examples’ results obtained with the proposed method were compared with those published in relevant literatures to demonstrate the reliability and accuracy of the proposed method.

导出引用

刘涛1，2，李朝东1，汪超2，蒋雅芬3，刘庆运2. 基于三阶剪切变形理论的压电功能梯度板静力学等几何分析[J]. 振动与冲击, 2021, 40(1): 73-85

LIU Tao1，2, LI Chaodong1, WANG Chao2, JIANG Yafen3, LIU Qingyun2. Static iso-geometric analysis of piezoelectric functionally graded plate based on third-order shear deformation theory[J]. Journal of Vibration and Shock, 2021, 40(1): 73-85

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