压电材料三维V形切口端部力电耦合奇异场分析

Singularity analysis of 3D electromechanical field of V-notch structures in piezoelectric materials

对于压电材料元器件,往往由于裂纹/切口而导致结构存在应力奇异性和电奇异性的现象。本文主要研究压电材料在力电耦合情形下的三维柱向V形切口问题。针对压电材料,结合Maxwell方程和Williams渐近展开式,推导并建立了三维柱向V形切口在力电耦合情形下的特征微分方程组,并采用插值矩阵法获得了其在不同边界条件下的奇性指数,算例表明,本文方法具有良好的计算精度。本文方法也适用于裂纹尖端的奇异性分析,且具有程序操作方便,前处理工作量小等优点。

Piezoelectrics has been extensively used in sensors, actuators, resonators and intelligent structures, in which the crack and notch problems are always encountered. Due to the discontinuities of material or geometry at a notch/crack tip, the mechanical failure or dielectric breakdown may initiate from the notch apex. In this paper, a new way is proposed to evaluate the singularity at the vertex of three dimensional(3D) column-shaped V-notch encountered in piezoelectric structures. Based on the asymptotic assumption for the physical field near the notch tip, the singular functions of 3D electromechanical field near the V-notch are expressed by the eigenfunction expansion approach. The characteristic differential equations with the singular parameters of the notch are built from 3D equilibrium equations and Maxwell equations. And the mechanical and electric boundary conditions are expressed by the combination of the singular orders and characteristic functions. Thus, the evaluation of the notched singular orders is transformed into the boundary value problem of the ordinary differential equations. Then by applying the interpolating matrix method to solve the characteristic differential equations, the singular orders and the corresponding characteristic functions near the V-notch tip are achieved. The numerical results show that the computed results from the present method have very high accuracy compared with the existing solutions.