Analysis of Interpolating Element-free Galerkin Scaled Boundary Method for Piezoelectric Cracks

doi:10.3901/JME.2017.06.053

Abstract

Abstract:

In order to obtain a more effective and accurate method to study the fracture behavior of the piezoelectric materials, an interpolating element-free Galerkin scaled boundary method (IEFG-SBM) is proposed for two-dimensional fracture analysis of piezoelectric material based on the improved interpolating moving least-squares (IIMLS) method. This method allows the stress and electric displacement intensity factors to be calculated directly from their definitions. Only the boundary of the computational domain requires to be discretized by the element-free Galerkin (EFG) method and thus the spatial dimension is reduced by one. However, in contrast to the boundary element method, no fundamental solution is required. The solution in the radial direction is analytical, therefore the simulation precision of this method is relatively high. In the IIMLS method, the shape functions satisfy Kronecker delta property and the weight function involved is nonsingular. Moreover, the number of unknown coefficients in the trial function of the IIMLS method is less than that of the conventional moving least-squares (MLS) approximation. At last, numerical examples are presented to demonstrate the effectiveness and correctness of the proposed method for fracture analysis of piezoelectric material.

Key words: element-free Galerkin method, intensify factors, scaled boundary method, piezoelectric material

CHEN Shenshen, WANG Juan. Analysis of Interpolating Element-free Galerkin Scaled Boundary Method for Piezoelectric Cracks[J]. Journal of Mechanical Engineering, 2017, 53(6): 53-59.

References

[1] PARK S，SUN C T. Fracture criteria for piezoelectric ceramics[J]. Journal of the American Ceramic Society，1995，78(6)：1475-1480.
[2] 平学成，陈梦成，谢基龙，等. 基于新型裂尖杂交元的压电材料断裂力学研究[J]. 力学学报，2006，38(3)：407-412.
PING Xuecheng，CHEN Mengcheng，XIE Jilong，et al. Fracture mechanics researches on piezoelectric materials based on a novel crack-tip hybrid finite element method[J]. Acta Mechanica Sinica，2006，38(3)：407-412.
[3] 雷钧，刘和国，杨庆生. 基于相互作用积分法的压电裂纹边界元分析[J]. 北京工业大学学报，2016，42(2)：161-166.
LEI Jun，LIU Heguo，YANG Qingsheng. Analysis of boundary element for piezoelectric cracks using an interaction integral method[J]. Journal of Beijing University of Technology，2016，42(2)：161-166．
[4] LI C，MAN H，SONG C M，et al. Fracture analysis of piezoelectric materials using the scaled boundary finite element method[J]. Engineering Fracture Mechanics，2013，97：52-71.
[5] 刘钧玉，林皋，范书立，等. 裂纹面受荷载作用的应力强度因子的计算[J]. 计算力学学报，2008，25(5)： 621-626.
LIU Junyu，LIN Gao，FAN Shuli，et al. The calculation of stress intensity factor including the effects of surface tractions[J]. Chinese Journal of Computational Mechanics，2008，25(5)：621-626.
[6] 姜潮，龙湘云，韩旭，等. 一种随机裂纹结构的裂纹扩展路径分析方法[J]. 固体力学学报，2014，35(1)：30-38.
JIANG Chao，LONG Xiangyun，HAN Xu，et al. A crack propagation path analysis method of random cracked structure [J]. Chinese Journal of Solid Mechanics，2014，35(1)：30-38.
[7] DEEKS A J，WOLF J P. Semi-analytical solution of Laplace equation in non-equilibrating unbounded problems[J]. Computers & Structures，2003，81(15)：1525-1537.
[8] 胡志强，林皋，王毅，等. 基于Hamilton体系的弹性力学问题的比例边界有限元法[J]. 计算力学学报，2011，28(4)：510-516.
HU Zhiqiang，LIN Gao，WANG Yi，et al. A Hamiltonian based derivation of scaled boundary finite element method for elasticity [J]. Chinese Journal of Computational Mechanics，2011，28(4)：510-516.
[9] 戴保东，程玉民. 改进的无网格局部边界积分方程方法[J]. 机械工程学报，2008，44(10)：108-113.
DAI Baodong，CHENG Yumin. Improved meshless local boundary integral equation[J]. Chinese Journal of Mechanical Engineering，2008，44(10)：108-113.
[10] 秦义校，程玉民. 温度场分析的重构核粒子边界无单元法[J]. 机械工程学报，2008，44(6)：95-100.
QIN Yixiao，CHENG Yumin. Reproducing kernel particle boundary element-free method for temperature field problems[J]. Chinese Journal of Mechanical Engineering，2008，44(6)：95-100.
[11] 张清东，张勃洋，李博，等. 板带轧制过程二维流函数-无网格伽辽金求解方法及其应用[J]. 机械工程学报，2015，51(10)：48-54.
ZHANG Qingdong，ZHANG Boyang，LI Bo，et al. Element free Galerkin method coupled flow function for plane strain strip rolling process[J]. Journal of Mechanical Engineering，2015，51(10)：48-54.
[12] LABIBZADEH M. Voronoi based discrete least squares meshless method for heat conduction simulation in highly irregular geometries[J]. Chinese Journal of Mechanical Engineering，2016，29(1)：98-111.
[13] BELYTSCHKO T，LU Y Y，GU L. Element-free Galerkin method[J]. International Journal for Numerical Methods in Engineering，2005，64：1610-1627.
[14] ATLURI S N，ZHU T. A new meshless local Petrov-Galerkin (MLPG) approach in computational mechanics[J]. Computational Mechanics，1998，22：117-127.
[15] HE Y Q，YANG H T，DEEKS A J. An element-free Galerkin (EFG) scaled boundary method[J]. Finite Elements in Analysis and Design，2012，62：28-36.
[16] HE Y Q，YANG H T，DEEKS A J. An element-free Galerkin scaled boundary method for steady-state heat transfer problems[J]. Numerical Heat Transfer，Part B， 2013，64：199-217.
[17] LANCASTER P，SALKAUSKAS K. Surface generated by moving least squares methods[J]. Mathematics of Computation，1981，37：141-158.
[18] REN H P，CHENG Y M. The interpolating element-free Galerkin (IEFG) method for two-dimensional potential problems[J]. Engineering Analysis with Boundary Elements，2012，36：873-880.
[19] 任红萍，程玉民，张武. 改进的移动最小二乘插值法研究[J]. 工程数学学报，2010，27(6)：1021-1029.
REN Hongping，CHENG Yumin，ZHANG Wu. Researches on the improved interpolating moving least-squares method[J]. Chinese Journal of Engineering Mathematics，2010，27(6)：1021-1029.
[20] WANG J F，SUN F X，CHENG Y M. An improved interpolating element-free Galerkin method with a nonsingular weight function for two-dimensional potential problems[J]. Chinese Physics B，2012，21：090204.
[21] WANG J F，WANG J F，SUN F X，et al. An interpolating boundary element-free method with nonsingular weight function for two-dimensional potential problems[J]. International Journal of Computational Methods，2013，10：1350043.
[22] SONG C M. A matrix function solution for the scaled boundary finite element equation in statics[J]. Computer Methods in Applied Mechanics and Engineering，2004，193：2325-2356.
[23] SUNG K H，CHARLES K，CHANG F K. Finite element analysis of composite structures containing distributed piezoceramic sensors and actuator[J]. AIAA Journal，1992，30：772780.
[24] WANG B L，MAI Y W. A piezoelectric material strip with a crack perpendicular to its boundary surfaces[J]. International Journal of Solids and Structures，2002，39(17)：4501-4524.
[25] SHENG N，SZE K Y，CHEUNG Y K. Trefftz solutions for piezoelectricity by Lekhnitskiis and boundary collocation method[J]. International Journal for Numerical Methods in Engineering，2006，65(13)：2113-2138.
[26] PAN E. A BEM analysis of fracture mechanics in 2D anisotropy piezoelectric solids[J]. Engineering Analysis with Boundary Elements，1999，23：67-76.