Dual Equations and Solutions of Ⅰ-Type Crack of Dynamic Problems in Piezoelectric Materials
-
摘要: 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．Abstract: 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,seiminfinite 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.
-
[1] WANG Biao. Three dimensional analysis of an ellipsoidal inclusion in a piezoelectric material[J].Internat J Solids Struct，1992，29(3)：293-308. doi: 10.1016/0020-7683(92)90201-4 [2] WANG Biao. Three dimensional analysis of a flat elliptical crack in a piezoelectric material[J].Internat J Engrg Sci,1992，30(6):781-791. doi: 10.1016/0020-7225(92)90107-R [3] ZHOU Zhen-gong，SUN Jian-liang，WANG Biao. Investigation of the behavior of a crack in a piezoelectric material subjected to a uniform tension loading by use of the non-local theory[J].Internat J Engrg Sci，2004，42(19/20)：2041-2063. doi: 10.1016/j.ijengsci.2004.08.004 [4] Pak Y E.Crack extension force in a piezoelectric material[J].J Appl Mech,1990，57(3)：647-653. doi: 10.1115/1.2897071 [5] Khutoryansky H M,Sosa H.Dynamic representation formulas and fundamental solutions for piezoelectricity[J].Internat J Solids Structures,1995，32(22)：3307-3325. doi: 10.1016/0020-7683(94)00308-J [6] 侯密山，边文凤. 反平面电弹性断裂动力问题的拟应力解[J]. 机械强度，2001,23(3):326-328. [7] Chen Z T，Yu S W. Anti-plane Yoffe crack problem in piezoelectric materials[J].Internat J Fracture，1997，84(3): L41-L45. [8] Erdélyi A.高级超越函数[M].第二册.张致中 译.上海：科学技术出版社, 1958. [9] 王竹溪，郭敦仁.特殊函数概论[M].北京：科学出版社, 1979. [10] 张石生. 积分方程[M].重庆：重庆出版社， 1988. [11] 边文凤.电弹性问题的势函数解和辛解[D].博士论文.哈尔滨: 哈尔滨工业大学，2006. [12] 边文凤，王彪，贾宝贤.动态裂纹积分变换法中的数学问题[J].应用数学和力学，2004，25(3)：228-232. -
点击查看大图
计量
- 文章访问数: 2547
- HTML全文浏览量: 54
- PDF下载量: 703
- 被引次数: 0
下载:
