摘要
压电材料作为一种新型的智能材料,已经被广泛应用到国防建设、工业生产以及和人民生活密切相关的许多领域中,其力电耦合特性得到了广大学者的普遍重视,而含各类缺陷压电材料的断裂力学分析更是人们关注的焦点。本文在线性压电理论的框架下,对压电介质、双相压电介质以及直角域压电介质中含圆孔、裂纹、夹杂或复合缺陷的反平面动力学问题进行了研究,得到了一些有价值的结论。
论文的主要工作可总结为以下三部分：
第一部分求解了压电介质和双相压电介质中圆孔边径向裂纹的反平面动态问题。首先给出满足本问题边界条件的Green函数,得到不含裂纹情况下介质中的总位移场和总电位势场,然后利用契合思想和裂纹切割技术将问题转化为求解一组第一类Fredholm定解积分方程,采用直接数值积分方法求解,最后作为算例,给出了裂纹长度、圆孔半径、入射波频率以及材料参数等因素对裂纹尖端动应力强度因子的影响规律。结果表明,孔边裂纹尖端的动应力强度因子在动态情况下并不总是小于将圆孔看作裂纹的一部分而得到的直线型裂纹尖端的动应力强度因子,而是围绕其动应力强度因子曲线呈现一定的波动性。
第二部分研究了双相压电介质中界面附近单圆孔和双圆孔以及圆孔与界面裂纹相互作用的动态反平面问题。利用满足问题边界条件的Green函数,根据契合思想得到求解界面上附加外力系和外电场的两组第一类Fredholm积分方程,采用直接数值积分法进行求解,得到附加外力系和外电场在各离散点上的值,从而写出圆孔周边动应力集中和电场强度集中的表达式。对于界面裂纹问题,同样采用契合思想和裂纹切割技术将问题简化为求解一组第一类Fredholm积分方程,并通过代换使其直接包含裂纹尖端的动态应力强度因子。作为算例,给出了圆孔半径、圆孔与界面之间的距离、入射波频率、材料参数以及裂纹长度对圆孔处的力电场集中和裂纹尖端动应力强度因子的影响规律。
第三部分分析了直角域各向同性压电介质中含有单一圆柱形缺陷的动力反平面问题。利用镜像法构造出满足控制方程以及两自由表面处边界条件的力电波场,然后采用复变函数法、多极坐标移动技术以及叠加原理构造出满足各边界条件的散射力电波场,最后写出缺陷内部的力电波场并根据缺陷处的边界条件对表达式中的各未知系数进行求解,从而得到缺陷附近应力集中和电场强度集中的解析表达式。给出了缺陷半径、缺陷与两自由表面之间的距离、材料参数以及入射波频率对缺陷附近力电场分布的影响规律。
本文关于含缺陷压电材料的反平面动力学问题的研究,希望能够对工程设计、生产和安全使用提供一定的参考。
As a kind of new intelligent materials, the piezoelectric material, which has been widely used in national defense construction, industrial production and many areas which closely relates to the people's lives, attracts extensive attention of many researchers due to its properties of electromechanical coupling. And more attention is paid to the investigation of the fracture behaviors on piezoelectric materials with various kinds of defects. Dynamic antiplane problems of the piezoelectric medium, piezoelectric bimaterials and quarter-infinite piezoelectric medium which contain circular cavity, crack, inclusion and composite defects are investigated in this article using the linear piezoelectric theory. Some useful conclusions are obtained.
The main work in present paper can be summarized into three parts as follows:
In part one, dynamic antiplane behaviors of the radial cracks at the edge of the circular cavity are solved in piezoelectric medium and piezoelectric bimaterials. Firstly, the Green's functions which satisfy the boundary conditions of this problem are given. The total displacement field and the total electric potential field are also obtained in piezoelectric medium without any cracks. Secondly, the problems are reduced to a set of the first kind of Fredholm integral equations by the conjunction method and the crack-division technique. And the integral equations are solved by the direct numerical integration method. Finally, the numerical results reveal the effects of the length of the crack, the circular cavity radius, the incident wave frequency and the material parameters on dynamic stress intensity factors of the crack-tip. It can be concluded that the values of radial crack-tip DSIFs on a circular cavity aren't invariably less then those on a reduced straight crack with an effective length at the dynamic situation. The oscillating phenomenon can be seen around the straight crack DSIFs.
Then, dynamic antiplane behaviors of one cavity and two cavities near the interface are investigated in piezoelectric bimaterials. And the dynamic interaction between a circular cavity and a interface crack is also analyzed in the present paper. Based on the Green's functions which are agreed with the boundary conditions, a pair of the first kind of Fredholm integral equations for solutions of the unknown stresses and the unknown electric fields at the interface can be established according to the conjunction method. The values of the additional stress and electric fields on the discrete points are obtained to solve the integral equations by the direct numerical integration method. And the expressions of dynamic stress concentration and electric intensity concentration at the edge of the circular cavity are also given. The interface crack problem is also reduced to solve a set of the first kind of Fredholm integral equations by the conjunction method and the crack-division technique. Dynamic stress intensity factors of the crack-tip can be contained in the integral equations by a substitution. As a numerical example, the effects of the circular cavity radius, the distance between the hole and the interface, the incident wave frequency, the material parameters and the length of the crack on dynamic stress concentration, electric field intensity concentration at the edge of the circular cavity and dynamic stress intensity factors at the crack-tip are given.
Dynamic antiplane problems in the quarter-infinite piezoelectric medium with a single cylindrical defect are studied in part three. Firstly, the displacement field and the electric potential field, which satisfy the governing equations and the boundary conditions at the two free surfaces, are structured by the image method. Secondly, the scattering fields of displacement and electric potential are constructed by the complex variable function method, multi-polar coordinate transformation and the superposition theorem. And the scattering fields satisfy all the boundary conditions. Finally, the expressions of displacement field and electric potential field in the defect are given. And the unknown coefficients in the expressions can be solved according to the boundary conditions at the edge of the defect. The analytical expressions of the stress concentration and the electric field intensity concentration at the edge of the defect are achieved. The effects of the defect radius, the distances between the defect and the two free surfaces, material parameters and the incident wave frequency on the distributions of stress and electric field at the edge of the defect are drawn.
It is expected that the investigations on dynamic antiplane behaviors of piezoelectric material with various kinds of defects can provide some references for the project designing, manufacture and application of piezoelectric material.
引文
[1]王保林.压电材料及其结构的断裂力学.北京：国防工业出版社.2003：1-4页
[2]方岱宁,刘金喜.压电与铁电体的断裂力学.北京：清华大学出版社.2008：1-5页,97-98页,125-126页,139-150页
[3]许金泉.界面力学.北京：科学出版社.2006：1-7页
[4]Parton V.Z. Fracture mechanics of piezoelectric materials. Acta Astronautica.1976,3: 671-683P
[5]Pak Y.E. Crack extension force in a piezoelectric material. Journal of Applied Mechanics.1990,57:647-653P
[6]Pak Y.E. Linear electroelastic fracture-mechanics of piezoelectric materials. International Journal of Fracture.1992,54:79-100P
[7]Sosa H., Pak Y.E. Three-dimensional eigenfunction analysis of a crack in a piezoelectric material. International Journal of Solids and Structures.1990,26:1-15P
[8]Sosa H. Plane problems in piezoelectric media with defects. International Journal of Solids and Structures.1991,28:491-505P
[9]Suo Z., Kuo C.M., Barnett D.M., Willis J.R. Fracture mechanics for piezoelectric ceramics. Journal of the Mechanics and Physics of Solids.1992,40:739-765P
[10]Wang B. Three-dimensional analysis of a flat elliptical crack in a piezoelectric material. International Journal of Engineering Science.1992,30:781-791P
[11]Zhang T.Y., Hack J.E. Mode-Ⅲ cracks in piezoelectric materials. Journal of Applied Physics.1992,71:5865-5870P
[12]刘宏伟.界面圆孔和孔边裂纹对SH波散射问题的研究.哈尔滨工程大学工学博士学位论文.1998,67页
[13]Yan X.Q. A numerical analysis of cracks emanating from an elliptical hole in a 2-D elasticity plate. European Journal of Mechanics A/Solid.2006,25:142-153P
[14]闫相桥.方孔分支裂纹的一种边界元分析.哈尔滨工业大学学报.2008,140(7)：1085-1088页
[15]闫相桥.内部压力作用下矩形板中源于椭圆孔的分支裂纹应力强度因子的一种数值分析.计算力学学报.2005,22(6)：711-715页
[16]闫相桥.平面弹性裂纹分析的一种有效边界元方法.应用数学与力学.2005,26(6)：749-756页
[17]闫相桥.双向荷载作用下无限大板中源于正方形的分支裂纹的应力强度因子.哈尔 滨工业大学学报.2006,38(8)：1224-1313页
[18]闫相桥.双轴载荷作用下源于椭圆孔的分支裂纹的一种边界元分析.固体力学学报.2004,25(4)：430-434页
[19]闫相桥.无限大板椭圆孔的分支裂纹的边界元分析.哈尔滨工业大学学报.2007,39(7)：1084-1087页
[20]Yan X.Q. Cracks emanating from a rhombus hole in infinite and finite plate subjected to internal pressure. Engineering Failure Analysis.2007,14:548-556P
[21]Tweed J., Melrose G. Cracks at the edge of an elliptic hole in out of plane shear. Engineering Fracture Mechanics.1989,34(3):743-747P
[22]Chen Y.Z., Lin X.Y., Wang Z.X. A semi-analytic solution for multiple curved cracks emanating from circular hole using singular integral equation. Applied Mathematics and Computation.2009,213:389-404P
[23]Abdelmoula R., Semani K., Li J. Analysis of cracks originating at the boundary of a circular hole in an infinite plate by using a new conformal mapping approach. Applied Mathematics and Computation.2007,188:1891-1896P
[24]Gai B.Z., Wang L.Q. Effect of crack face contact on dynamic stress intensity factors for a hole-edge crack. Journal of Harbin Institute of Technology (New Series).2009, 16(2):194-197P
[25]Chung N.Y., Song C.H. Effects of interface cracks emanating from a circular hole on stress intensity factors in bonded dissimilar materials. International Journal of Automotive Technology.2005,6(3):293-303P
[26]Prasad P.B.N., Norio Hasebe, Wang X.F., Shirai Y. Green's Function of a bimaterial problem with a cavity on the interface-Part I:Theory. Journal of Applied Mechanics. 2005,72:389-393P
[27]Sharma D.S., Ukadgaonker V.G. Stress Intensity Factors for Cracks Emanating from a Circular Hole in Laminated Composite Infinite Plate Under Different Loading Conditions. First International Conference on Emerging Trends in Engineering and Technology.2008:781-786P
[28]Malezhik M.P., Malezhik O.P., Zirka A.I., Chernyshenko I.S. Stress wave fields in plates weakened by curvilinear hole with edge cracks. International Applied Mechanic. 2006,42(2):192-195P
[29]郭俊宏,刘官厅.带双裂纹的椭圆孔口问题的应力分析.力学学报.2007,39(5)：699-703页
[30]刘殿魁,刘宏伟.孔边裂纹对SH波的散射及其动应力强度因子.力学学报.1999, 31(3)：292-299页
[31]刘殿魁,陈志刚.椭圆孔边裂纹对SH波的散射及其动应力强度因子.应用数学与力学.2004,25(9)：958-966页
[32]Garcia-Sanchez F., Saez Andres, Dominguez J. Anisotropic and piezoelectric materials fracture analysis by BEM. Computers and Structures.2005,83:804-820P
[33]Garcia-Sanchez F., Rojas-Diaz R., Saez A., Zhang Ch. Fracture of magneto-electroelastic composite materials using boundary element method (BEM). Theoretical and Applied Fracture Mechanics.2007,47:192-204P
[34]Wang Y.J., Gao C.F. The mode Ⅲ cracks originating from the edge of a circular hole in a piezoelectric solid. International Journal of Solids and Structures.2008,45:4590-4599P
[35]王勇健,高存法.压电体内孔边裂纹的应力强度因子.力学季刊.2008,29(2)：205-209页
[36]Guo J.H., Lu Z.X., Han H.T., Yang Z.Y. Exact solutions for anti-plane problem of two asymmetrical edge cracks emanating from an elliptical hole in a piezoelectric material. International Journal of Solids and Structures.2009,46:3799-3809P
[37]Guo J.H., Lu Z.X., Han H.T., Yang Z.Y. The behavior of two non-asymmetrical permeable cracks emanating from an elliptical hole in a piezoelectric solid. European Journal of Mechanics A/Solids.2009,46:3799-3809P
[38]Xu X.L., Rajapakse R.K.N.D. Boundary element analysis of piezoelectric solid with defects. Composites Part B.1998,29B:655-669P
[39]Zhao M.H., Wang H., Yang F., Liu T. A magnetoelectroelastic medium with an elliptical cavity under combined mechanical-electric-magnetic loading. Theoretical and Applied Fracture Mechanics.2006,45:227-237P
[40]Suzuki T., Sasaki T., Kimura K., Yoshino N. Analyses of isotropic piezoelectric materials with multilayered circular inclusions under out-of-plane shear loadings and their numerical examples. Nippon Kikai Gakkai Ronbunshu, A Hen/Transactions of the Japan Society of Mechanical Engineers, Part A.2003,69(3):579-584P
[41]Suzuki T., Sasaki T., Hirashima K., etc. Analyses of isotropic piezoelectric materials with multilayered elliptical inclusion under out-of-plane shear loadings. Acta Mechanica.2005,179:211-225P
[42]Chen F.M., Shen M.H., Hung S.Y. Circularly cylindrical layered media in antiplane piezoelectricity. Institute of Physics Publishing.2006,39:4250-4256P
[43]Shodja H.M., Ghazisaeidi M. Effects of couple stresses on anti-plane problems of piezoelectric media with inhomogeneities. European Journal of Mechanics A/Solids. 2007,26:647-658P
[44]Kirilyuk V.S., Levchuk O.I. Electroelastic stress state of a piezoceramic body with a paraboloidal cavity. International Applied Mechanics.2006,42(9):1011-1020P
[45]Pak Y.E. Elliptical inclusion problem in antiplane piezoelectricity:Implications for fracture mechanics. International Journal of Engineering Science.2010,48:209-222P
[46]Gao C.F., Fan W.X. Exact solutions for the plane problem in piezoelectric materials with an elliptic or a crack. International Journal of Solids and Structures.1999,25: 2527-2540P
[47]Lee K.L., Soh A.K., Fang D.N., etc. Fracture behavior of inclined elliptical cavities subjected to mixed-mode I and II electro-mechanical loading. Theoretical and Applied Fracture Mechanics.2004,41:125-135P
[48]Gao C.F. Influence of mechanical stresses on partial discharge in a piezoelectric solid containing cavities. Engineering Fracture Mechanics.2008,75:4920-4924P
[49]Yang B.H., Gao C.F., Noda N. Interactions between N circular cylindrical inclusions in a piezoelectric matrix. Acta Mechanica.2008,197(1-2):31-42P
[50]Sosa Horacio, Khutoryansky Naum. New Developments Concerning Piezoelectric Materials with Defects. International Journal of Solids and Structures.1996,33(23): 3399-3414P
[51]Meguid S.A., Zhong Z. On the elliptical inhomogeneity problem in piezoelectric materials under antiplane shear and inplane electric field. International Journal of Engineering Science.1998,36(3):329-344P
[52]Chen F.M., Shen M.H., Chen S.N. Piezoelastic study on singularities interacting with circular and straight interfaces. International Journal of Solids and Structures.2006,43: 5541-5554P
[53]Chung M.Y., Ting T.C.T. Piezoelectric solid with an elliptic inclusion or hole. International Journal of Solids and Structures.1996,33(23):3343-3361P
[54]Shen M.H., Chen F.M., Hung S.Y. Piezoelectric study for a three-phase composite containing arbitrary inclusion. International Journal of Mechanical Sciences.2010,52: 561-571P
[55]Yang B.H., Gao C.F. Plane problems of multiple piezoelectric inclusions in a non-piezoelectric matrix. International Journal of Engineering Science.2010,48:518-528P
[56]杨丽敏,柳春图,曾晓辉.含圆孔压电板弯曲问题.机械强度.2005,27(1)：85-94页
[57]杨新华,曾国伟,陈传尧.含周期性分布导电夹杂压电陶瓷的力电损伤分析.机械强 度.2008,30(5)：844-847页
[58]周志东,赵社戌,匡震邦.任意点载荷下含椭圆孔压电介质中广义应力和位移分析.上海交通大学学报.2004,38(8)：1403-1407页
[59]戴隆超,郭万林.压电体椭圆孔边的力学分析.力学学报.2004,36(2)：224-228页
[60]Shodja H.M., Kargamovin M.H., Hashemi R. Electroelastic fields in interacting piezoelectric inhomogeneities by the electromechanical equivalent inclusion method. Smart Materials and Structures.2010,19:1-12P
[61]赵永茂,刘进.含椭圆刚性夹杂压电材料反平面问题的电弹分析.石家庄铁道学院学报.1999,12(2)：38-41页
[62]王祥琴,刘金喜.含椭圆夹杂压电材料反平面问题的基本解.工程力学增刊.1997,410-413页
[63]侯密山,高存法.含椭圆形刚性夹杂的压电材料平面问题.计算力学学报.1997,14(2)：189-195页
[64]侯密山,杨其俊.含椭圆形夹杂的压电材料平面问题.应用数学和力学.1998,19(2)：137-144页
[65]戴隆超,郭万林,佘崇民.含椭圆形夹杂的压电体平面应变问题.应用数学和力学.2005,26(12)：1463-1469页
[66]高存法,樊蔚勋.两压电介质之间的界面夹杂问.应用数学和力学.2001,22(1)：85-92页
[67]侯密山,高存法.压电材料反平面应变状态的任意形状夹杂问题.应用数学和力学.1997,14(1)：135-141页
[68]王旭,王子昆.压电材料反平面应变状态的椭圆夹杂及界面裂纹问题.上海力学.1993,14(4)：26-34页
[69]仲政.压电材料椭圆夹杂界面局部脱粘问题的分析.应用数学和力学.2004,25(4)：405-416页
[70]仲政.压电材料椭圆夹杂界面开裂问题的电弹性耦合解.上海力学.1998,19(1)：9-14页
[71]王旭,沈亚鹏.压电复合材料中的Eshelby夹杂问题.力学学报.2003,35(1)：26-32页
[72]王旭,沈亚鹏.压电基体中部分脱开的刚性导体椭圆夹杂分析.应用数学和力学.2001,22(1)：32-46页
[73]刘金喜,姜稚清,冯文杰.压电螺位错与椭圆夹杂的电弹相互作用.应用数学和力学.2000,21(11)：1185-1190页
[74]Shindo Y., Moribayashi H., Narita F. Scattering of antiplane shear waves by a circular piezoelectric inclusion embedded in a piezoelectric medium subjected to a steady-state electrical load. ZAMM Zeitschrift fur Angewandte Mathematik und Mechanik.2002, 82(1):43-49P
[75]Du J.K., Shen Y.P., Wang X. Scattering of anti-plane shear waves by a partially debonded piezoelectric circular cylindrical inclusion. Acta Mechanica.2002,158:169-182P
[76]Feng W., Wang L., Jiang Z., Zhao Y. Shear wave scattering from a partially debonded piezoelectric cylindrical inclusion. Acta Mechanica Solida Sinica.2004,17(3):258-269P
[77]宋天舒,刘殿魁,于新华.SH波在压电材料中的散射和动应力集中.哈尔滨工程大学学报.2002,23(1)：120-123页
[78]宋天舒,刘殿魁,付国庆.含刚性圆柱夹杂压电介质的动力反平面特性.哈尔滨工程大学学报.2003,24(5)：574-577页
[79]Song T.S., Li D., Sun L.L. Dynamic anti-plane behaviors of interacting circular cavities in an infinite piezoelectric medium. ASME International Mechanical Engineering Congress and Exposition, Proceedings.2009,11:199-204P
[80]Singh B.M., Rokne J., Dhaliwal R.S., Vrbik J. Antiplane crack at the interface of two boned dissimilar graded piezoelectric materials. European Journal of Mechanics A/Solids.2008,27:346-364P
[81]Bakirov V.F., Kim T.W. Analysis of a crack at the piezoceramic-metal interface and estimates of adhesion fracture energy. International Journal of Engineering Science. 2009,47:793-804P
[82]Ding S.H., Li X., Guo L.F. Analysis of bonded piezoelectric materials with a crack perpendicular to the interface subjected to in-plane loading. Computational Materials Science.2010,47:977-984P
[83]Govorukha V., Kamlah M. On contact zone models for an electrically limited permeable interface crack in a piezoelectric bimaterial. International Journal of Fracture.2010,164(1):133-146P
[84]Ru C.Q. A hybrid complex-variable solution for piezoelectric/isotropic elastic interfacial cracks. International Journal of Fracture.2008,152(2):169-178P
[85]Li Y.S., Feng W.J., Xu Z.H. A penny-shaped interface crack between a functionally graded piezoelectric layer and a homogeneous piezoelectric layer. Meccanica.2009, 44(4):377-387P
[86]Li Q., Chen Y.H. Analysis of a permeable interface crack in elastic dielectric/piezoelectric bimaterials. Acta Mechanica Sinica.2007,23(6):681-687P
[87]Zhou Z.G., Wang B. An interface crack between two dissimilar functionally graded piezoelectric/piezomagnetic material half infinite planes subjected to the harmonic anti-plane shear stress waves. International Journal of Applied Electromagnetics and Mechanics.2008,27(1-2):117-132P
[88]Li Y.D., Lee K.Y. Anti-plane crack intersecting the interface in a bonded smart structure with graded magnetoelectroelastic properties. Theoretical and Applied Fracture Mechanics.2008,50:235-242P
[89]Li Y.D., Lee K.Y. Anti-plane fracture analysis for the weak-discontinuous interface in a non-homogeneous piezoelectric bi-material structure. European Journal of Mechanics A/Solids.2009,28:241-247P
[90]Chen X.H., Ma C.C., Ing Y.S., Tsai C.H. Dynamic interfacial crack propagation in elastic-piezoelectric bi-materials subjected to uniformly distributed loading. International Journal of Solids and Structures.2008,45:959-997P
[91]Singh B.M., Rokne J., Dhaliwal R.S. Electroelastic problem of two anti-plane collinear cracks at the interface of two bonded dissimilar piezoelectric layers. SDHM Structural Durability and Health Monitoring.2008,4(2):95-105P
[92]Li Y.D., Lee K.Y. Fracture analysis on a piezoelectric sensor with a viscoelastic interface. European Journal of Mechanics A/Solids.2009,28:738-743P
[93]Natroshvili D., Stratis I.G., Zazashvili S. Interface crack problems for metallic-piezoelectric composite structures. Mathematical Methods in the Applied Sciences. 2010,33(4):539-562P
[94]Singh B.M., Rokne J., Dhaliwal R.S., Vrbik J. Scattering of anti-plane shear waves By an interface crack between two bonded dissimilar functionally graded piezoelectric materials. Proceeding of the Royal Society A.2009,465:1249-1269P
[95]Li X., Liu J. Scattering of the SH wave from a crack in a piezoelectric substrate bonded to a half-space of functionally graded materials. Acta Mechanica.2009,208(3-4):299-308P
[96]Nazarov S.A., Specovius-Neugebauer M. Singularities at the tip of a crack on the interface of piezoelectric bodies. Journal of Mathematical Sciences.2009,159(4):524-540P
[97]Li Q., Chen Y.H. Solution for a semi-permeable interface crack in elastic dielectric/piezoelectric bimaterials. Journal of Applied Mechanics.2008,75:011010-1-13P
[98]Li Q., Chen Y.H. Analysis of crack-tip singularities for an interfacial permeable crack in metal/piezoelectric bimaterials. Acta Mechanica Solida Sinica.2007,20(3):247-257P
[99]吕念春,程云虹,田修波,程靳.Ⅲ型界面裂纹Dugdale模型的动态扩展问题.应用数学和力学.2005,26(9)：1105-1113P
[100]Wang X., Zhong Z., Wu F.L. A moving conducting crack at the interface of two dissimilar piezoelectric materials. International Journal of Solids and Structures.2003, 40:2381-2399P
[101]Li X.F., Fan T.Y., Wu X.F. A moving mode-Ⅲ crack at the interface between two dissimilar piezoelectric materials. International Journal of Engineering Science.2000, 38:1219-1234P
[102]Qin Q.H., Yu S.W. An arbitrarily-oriented plane crack terminating at the interface between dissimilar piezoelectric materials. International Journal of Solids and Structures.1997,34(5):581-590P
[103]Soh A.K., Fang D.N., Lee K.L.. Analysis of a bi-piezoelectric ceramic layer with an interfacial crack subjected to anti-plane shear and in-plane electric loading. European Journal of Mechanics A/Solids.2000,19:961-977P
[104]Zhou Z.G., Wang B., Cao M.S.. Two collinear anti-plane cracks in a piezoelectric layer bonded to dissimilar half spaces. European Journal of Mechanics A/Solids.2001,20: 213-226P
[105]Li X.F., Tang G.J.. Antiplane interface crack between two bonded dissimilar piezoelectric layers. European Journal of Mechanics A/Solids.2003,22:231-242P
[106]Liu S.H., Zou Z.Z., Xu B.Q., Zhang Z.G.. Anti-plane interface edge crack between two dissimilar piezoelectric blocks. Key Engineering Materials.2004,261-263(1):471-476P
[107]Li X.F., Wang B.L.. Anti-plane shear crack normal to and terminating at the interface of two bonded piezoelectric ceramics. International Journal of Solids and Structures. 2007,44:3796-3810P
[108]Tian W.Y., Chau K.T.. Arbitrarily oriented crack near interface in piezoelectric bimaterials. International Journal of Solids and Structures.2003,40:1943-1958P
[109]Govorukha V., Kamlah M.. Asymptotic fields in the finite element analysis of electrically permeable interface cracks in piezoelectric bimaterials. Archive of Applied Mechanics.2004,74:92-101P
[110]Zhou Z.G., Wu L.Z.. Basic solutions of two parallel mode-I cracks or four parallel mode-I cracks in the piezoelectric materials. Engineering Fracture Mechanics.2007,74: 1413-1435P
[111]Gao C.F., Wang M.Z.. Collinear permeable cracks between dissimilar piezoelectric materials. International Journal of Solids and Structures.2000,37(36):4969-4986P
[112]Kulikov A.A., Nazarov S.A.. Cracks in piezoelectric and electroconductive bodies. Journal of Applied and Industrial Mathematics.2007,1(2):201-216P
[113]Sun J.L., Zhou Z.G., Wang B.. Dynamic behavior of unequal parallel permeable interface multi-cracks in a piezoelectric layer bonded to two piezoelectric materials half planes. European Journal of Mechanics A/Solids.2004,23:993-1005P
[114]Feng W.J., Su R.K.L., Zou Z.Z.. Dynamic response of multiple coplanar interface cracks between two dissimilar piezoelectric materials. Key Engineering Materials. 2004,261-263(1):477-482P
[115]Gu B., Yu S.W., Feng X.Q.. Elastic wave scattering by an interface crack between a piezoelectric layer and an elastic substrate. International Journal of Fracture.2002, 116(2):29-34P
[116]Li X.F.. Electroelastic analysis of an internal interface crack in a half-plane consisting of two bonded dissimilar piezoelectric quarter-planes. Meccanica.2003,38:309-323P
[117]Gao C.F., Tong P., Zhang T.Y.. Interaction of a dipole with an interfacial crack in piezoelectric media. Composites Science and Technology.2005,65:1354-1362P
[118]Ou Z.C., Chen Y.H.. Interface crack problem in elastic dielectric/piezoelectric bimaterials. International Journal of Fracture.2004,130:427-454P
[119]Ou Z.C., Chen Y.H.. Interface crack -tip generalized stress field and stress intensity factors in transversely isotropic piezoelectric bimaterials. Mechanics Research Communications.2004,31:421-428P
[120]Gu B., Yu S.W., Feng X.Q., et al. Scattering of love waves by an interface crack between a piezoelectric layer and an elastic substrate. Acta Mechanica Solida Sinica. 2002,15:111-118P
[121]Wang X.D.. On the dynamic behavior of interacting interfacial cracks in piezoelectric media. International Journal of Solids and Structures.2001,38:815-831P
[122]Wang X.Y., Yu S.W.. Scattering of SH waves by an arc-shape crack between a cylindrical piezoelectric inclusion and matrix-Ⅱ: Far fields. International Journal of Fracture.1999,100:35-40P
[123]Huang H.M., Shi H.J., Yin Y.J.. Multi-cracks problem for piezoelectric materials strip subjected to dynamic loading. Mechanics Research Communications.2002,29:413- 424P
[124]Zhao X.H., Meguid S.A.. On the dynamic behaviour of a piezoelectric laminate with multiple interfacial collinear cracks. International Journal of Solids and Structures. 2002,39:2477-2494P
[125]Huang G.L., Wang X.D.. On the dynamic behaviour of interfacial cracks between a piezoelectric layer and an elastic substrate. International Journal of Fracture.2006,141: 63-73P
[126]Meguid S.A., Zhao X.. The interface crack problem of bonded piezoelectric and elastic half-space under transient electromechanical loads. Journal of Applied Mechanics. 2002,69:244-253P
[127]孙建亮,周振功,王彪.压电材料中两平行不相等界面裂纹的动态特性研究.应用数学和力学.2005,26(2)：145-154页
[128]周振功,王彪.压电压磁复合材料中界面裂纹对弹性波的散射.应用数学和力学.2005,26(1)：16-24页
[129]张沛霖,张仲渊.压电测量.北京：国防工业出版社.1983：1-10页
[130]孙慷,张福学.压电学(上册).北京：国防工业出版社.1984：301-311页
[131]Bleustein J.L.. A new surface wave in piezoelectric materials. Applied Physics letter. 1968,13:412-413P
[132]王保林,韩杰才,杜善义.压电材料中裂纹面电边界条件的适用性.固体力学学报.2004,25(4)：399-403页
[133]李星.积分方程.北京：科学出版社.2008：92-100页,151-152页
[134]沈以淡.积分方程(第二版).北京：北京理工大学出版社.2002：142-150页
[135]Sih G.C.. Stress distribution near internal crack tips for longitudinal shear problems. Journal of Applied Mechanics.1965,32(1):51-58P
[136]Loeber J.F., Sih G.C.. Diffraction of anti-plane shear waves by a finite crack. Journal of the Acoustical Society of America.1968,44(1):90-98P
[137]Loeber J.F., Sih G.C.. Transmission of anti-plane shear waves past and interface crack in dissimilar media. International Journal of Engineering Fracture Mechanics.1973,5: 699-725P
[138]刘殿魁,林宏.SH波对双相介质界面附近圆形孔洞的散射.固体力学学报.2003,24(2)：197-204页
[139]Pao Y.H., Mow C.C.. Diffraction of elastic waves and dynamic stress concentrations. Crane and Russak, New York.1973.
[140]Liu D.K., Lin H.. Scattering of SH-waves by an interacting interface linear crack and a circular cavity near biomaterial interface. Acta Mechanica Sinica.2004,20(3):317-326P
[141]Achenbach J. D.. Wave propagation in elastic solids. Amsterdam:North-Holland Publishing Company.1973:110-114P
[142]折勇,齐辉,杨在林.SH波对直角平面区域内圆形孔洞的散射与地震动.应用力学学报.2008,25(3)：392-397页
[143]Song T.S., Wang S.L.. Dynamic stress concentration on a semi-infinite piezoelectric medium with a circular cavity near the surface. Proceeding of the ASME International Design Engineering Technical Conferences.2008,1:1741-1746P
