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Bending behaviors of the in-plane bidirectional functionally graded piezoelectric material plates
Mechanics of Advanced Materials and Structures ( IF 2.8 ) Pub Date : 2020-11-17 , DOI: 10.1080/15376494.2020.1846100
Pengchong Zhang 1, 2 , Chengzhi Qi 1, 2 , Xu Sun 1 , Hongyuan Fang 3 , Yesheng Huang 4
Affiliation  

Abstract

The transverse bending behaviors of in-plane bidirectional functionally graded piezoelectric material (FGPM) plates are semi-analytically investigated by the scaled boundary finite element method (SBFEM) in association with the precise integration method (PIM). The proposed scheme is able to explore the structural characteristics of FGPM plates with the material coefficients obeying arbitrary form of mathematical functions according to the in-plane coordinates. The present methodology selects only four quantities consisting of three translational displacement components and the electric potential as the fundamental unknowns. Additionally, variations of the four primary variables across the thickness direction are expressed as an analytical exponential matrix. In the developed approach, plates are regarded as a kind of three dimensional structure. But, an arbitrary in-plane surface discretized with two-dimensional spectral elements is set as the research domain. The practice is conducive to cut down the computational effort and increase the calculation efficiency. The SBFEM governing equations are formulated from the three-dimensional basic equations of piezoelectric materials without any assumptions on the plate kinematics and distributions of electromechanical components. By means of the scaled boundary coordinate system and the dual vector methodology, the key partial differential equations of piezoelectric materials are conveniently converted into the easily-solved first order ordinary differential SBFEM governing equation. The stiffness matrix is constructed from the analytical exponential matrix aided by the highly accurate PIM to predict the changing patterns of mechanical and electric quantities. To further improve the precision, the technology of dividing the plate into two parts with equal thickness is exploited. Finally, numerical exercises of square, rectangular and triangular piezoelectric plates are provided to validate the accuracy and fast convergence of the developed technique and reveal the effect of geometrical shapes, gradient functions, types of external loadings and thickness-to-span ratios on the static flexure of FGPM plates owning the in-plane bidirectional stiffness.



中文翻译:

平面内双向功能梯度压电材料板的弯曲行为

摘要

采用比例边界有限元法 (SBFEM) 结合精确积分法 (PIM) 对面内双向功能梯度压电材料 (FGPM) 板的横向弯曲行为进行了半解析研究。所提出的方案能够根据面内坐标探索材料系数服从任意形式的数学函数的FGPM板的结构特征。本方法仅选择由三个平移位移分量和电势组成的四个量作为基本未知数。此外,四个主要变量在厚度方向上的变化表示为解析指数矩阵。在开发的方法中,板被视为一种三维结构。但是,以二维光谱元素离散的任意平面内表面为研究域。实践有利于减少计算量,提高计算效率。SBFEM 控制方程是由压电材料的三维基本方程制定的,没有对板运动学和机电元件的分布进行任何假设。借助尺度边界坐标系和对偶向量法,将压电材料的关键偏微分方程方便地转化为易于求解的一阶常微分SBFEM控制方程。刚度矩阵由解析指数矩阵构建,并借助高精度 PIM 来预测机械量和电量的变化模式。为了进一步提高精度,开发了将板分成等厚的两部分的技术。最后,提供了方形、矩形和三角形压电板的数值练习,以验证所开发技术的准确性和快速收敛性,并揭示几何形状、梯度函数、外部载荷类型和厚跨比对静力的影响。具有面内双向刚度的 FGPM 板的弯曲。

更新日期:2020-11-17
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