压电材料Ⅰ型裂纹动态问题的对偶方程组及其求解

Dual Equations and Solutions of Ⅰ-Type Crack of Dynamic Problems in Piezoelectric Materials

首先引入势函数,用势函数表示压电材料的基本微分方程,并采用Laplace变换、半无限对称Fourier正弦变换和Fourier余弦变换,对微分方程进行变换和初步求解;然后通过Fourier反演和引入边界条件,建立了二维压电材料动态裂纹问题的对偶方程组;再根据Bessel函数性质,利用Abel型积分方程及其反演,将对偶方程组化为第二类Fredholm积分方程组.结果表明,方法是可行的,可以成为研究此类问题的一种有效方法.

Firstly, basic differential equations of piezoelectric materials expressed in terms of the potential functions, which are introduced in the very beginning, were worked out. Then these equations were primarily solved through Laplace transformation, semi-infinite Fourier sine transformation and cosine transformation.After that, the dual equations of dynamic cracks problem in the 2D piezoelectric materials were founded with the help of Fourier reverse transformation and the introduction of boundary conditions. Finally, according to the character of the Bessel function and by making ful use of Abel integral equation and its reverse transform, the dual equations were changed into the second type of Fredholm integral equations. The investigation indicates that the study approach taken is feasible and has potential to be an effective method to do research on issues of this kind.