含界面裂纹的双相压电材料的动力反平面行为
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摘要
由于存在力和电的耦合作用,特别是在动力学问题中,压电材料的电致疲劳和电致裂纹情况时有发生,因而对压电材料中断裂行为的研究越来越受到重视。
本文采用Green函数方法研究了含界面裂纹的双相压电材料受SH波和面内电位移联合作用时的动力反平面问题。首要工作是构造一个适合于本文问题的位移Green函数和电位势Green函数,它们应当是各向同性压电材料的弹性半空间在其水平表面上任意一点承受时间谐和的出平面线源荷载作用时位移场的解答和在任意一点放置一沿z轴的正电荷线源时电位势函数的基本解。有了Green函数这一工具,就可以求解一类界面缺陷及复杂缺陷对SH波的动态响应问题。一般在求解过程中,将问题的模型视为“契合”问题:即将其剖分为两个压电介质的弹性半空间,分别在其剖分面上加置未知的出平面荷载,并在欲出现裂纹位置加置出平面反力使之产生可导通裂纹,利用Green函数写出界面位移的连续性条件,建立决定未知外力系的第一类Fredholm积分方程组。对于界面裂纹问题,裂纹尖端点的附加外力系可以代换成应力强度因子,这样采用直接离散的办法将定解积分方程组转化为线性代数方程组后,就可以直接求解出动应力强度因子的值。就典型问题给出了具体的算例结果,讨论了不同波数条件、不同外加电位移强度、不同材料常数组合对界面裂纹动应力强度因子的影响,部分计算结果与其他文献进行了比较。
As a result of the behaviors of electromechanical coupling, especially in the dynamic problems, fracture or failure often occurs prematurely in newly developed piezoelectric materials. More attention is paid to the investigation of the failure behaviors on piezoelectric materials.
This paper provides a comprehensive treatment of the dynamic character caused by an interface crack in piezoelectric bi-materials under steady-state inplane electrical and antiplane mechanical loads based on Green's function method. The first important step is to develop a couple of suitable Green's functions, which are fundamental solutions of elastic displacement field and electric potential field for a semi-infinite piezoelectric medium impacted by a pair of time-harmonic line force and line charge at the surface, respectively. Once the special Green's functions are obtained, various dynamic response problems of SH-waves by a kind of interface blemish or complex flaw can be solved. The model of the problem is composed of two semi-infinite piezoelectric media. Then additional anti-plane forces are applied at the interface in order to satisfy boundary continuity conditions. While the crack can be made in this method, which meets traction free condition at the crack's surface. A series of Fredholm integral equations of first kind for determining the unknown forces can be established by solving boundary value problem. The added force at the interface crack tip can be replaced by expression including dynamic stress intensity factor. The integral equations for determining the unknown forces can be transformed into algebraic equations. Some cases are calculated and the results are graphically presented. The calculation results show the influence of wave number, applied inplane electrical loads and combination of different materials and geometric parameters upon dynamic stress intensity factors. Some of them are compared with other published documents.
本文采用Green函数方法研究了含界面裂纹的双相压电材料受SH波和面内电位移联合作用时的动力反平面问题。首要工作是构造一个适合于本文问题的位移Green函数和电位势Green函数,它们应当是各向同性压电材料的弹性半空间在其水平表面上任意一点承受时间谐和的出平面线源荷载作用时位移场的解答和在任意一点放置一沿z轴的正电荷线源时电位势函数的基本解。有了Green函数这一工具,就可以求解一类界面缺陷及复杂缺陷对SH波的动态响应问题。一般在求解过程中,将问题的模型视为“契合”问题:即将其剖分为两个压电介质的弹性半空间,分别在其剖分面上加置未知的出平面荷载,并在欲出现裂纹位置加置出平面反力使之产生可导通裂纹,利用Green函数写出界面位移的连续性条件,建立决定未知外力系的第一类Fredholm积分方程组。对于界面裂纹问题,裂纹尖端点的附加外力系可以代换成应力强度因子,这样采用直接离散的办法将定解积分方程组转化为线性代数方程组后,就可以直接求解出动应力强度因子的值。就典型问题给出了具体的算例结果,讨论了不同波数条件、不同外加电位移强度、不同材料常数组合对界面裂纹动应力强度因子的影响,部分计算结果与其他文献进行了比较。
As a result of the behaviors of electromechanical coupling, especially in the dynamic problems, fracture or failure often occurs prematurely in newly developed piezoelectric materials. More attention is paid to the investigation of the failure behaviors on piezoelectric materials.
This paper provides a comprehensive treatment of the dynamic character caused by an interface crack in piezoelectric bi-materials under steady-state inplane electrical and antiplane mechanical loads based on Green's function method. The first important step is to develop a couple of suitable Green's functions, which are fundamental solutions of elastic displacement field and electric potential field for a semi-infinite piezoelectric medium impacted by a pair of time-harmonic line force and line charge at the surface, respectively. Once the special Green's functions are obtained, various dynamic response problems of SH-waves by a kind of interface blemish or complex flaw can be solved. The model of the problem is composed of two semi-infinite piezoelectric media. Then additional anti-plane forces are applied at the interface in order to satisfy boundary continuity conditions. While the crack can be made in this method, which meets traction free condition at the crack's surface. A series of Fredholm integral equations of first kind for determining the unknown forces can be established by solving boundary value problem. The added force at the interface crack tip can be replaced by expression including dynamic stress intensity factor. The integral equations for determining the unknown forces can be transformed into algebraic equations. Some cases are calculated and the results are graphically presented. The calculation results show the influence of wave number, applied inplane electrical loads and combination of different materials and geometric parameters upon dynamic stress intensity factors. Some of them are compared with other published documents.
引文
[1]张福学,王丽坤.现代压电学(中册).北京:科学出版社.2002:1-10.
[2]杨卫.电致失效力学.力学进展.1996,26(3):338-353.
[3]余寿文.固体力学与材料科学交缘的几个新课题.力学进展.1994,24(1):24-36.
[4]Parton VZ.Fracture mechanics of piezoelectric materials.Acta Astronautica.1976,3:671-683.
[5]Pak YE.Crack extension force in a piezoelectric material.J Appl Phys.1990,57:647-653.
[6]Dunn ML.The effects of cracks face boundary conditions on the fracture mechanics.Eng Frac Mech.1994,48:25-39.
[7]Sosa HA.Plane problems in piezoelectric media with defects.Int J Solids Structures.1991,28:491-505.
[8]Zhang TY,Tong P.Fracture mechanics for a mode Ⅲ crack in a piezoelectric material.Int J Solids Structures.1996,33:343-359.
[9]Park S,Sun CT.Fracture criteria for piezoelectric materials.J Am Ceram Soc.1995,78:1475-1480.
[10]王自强.压电材料裂纹顶端条状电饱和区模型的力学分析.力学学报.1999,31(3):311-319.
[11]Loboda,V.,Shevelova,A.Pre-fracture zones of an electrically permeable interface crack in a piezoelectric biomaterial.35th Solid Mechanics Conference(SolMech-06).2006,225-226.
[12]Ching-Hwei Chue,Wen-Bin Wei,Liu,T.J.-C..Antiplane electro-mechanical field of a bimaterial piezoelectric wedge under a pair of concentrated forces and surface charges.Journal of the Chinese Institute of Engineers.2006,29(3):467-80.
[13]Li,Q.,Chen,Y.H.Solution for a semi-permeable interface crack between two dissimilar piezoelectric materials.Journal of Applied Mechanics, Transactions ASME. 2007,74(5): 833-844.
[14] X. Wang, Z. Zhong, F.L. Wu. A moving conducting crack at the interface of two dissimilar piezoelectric materials. International Journal of Solids and Structures. 2003,40: 2381-2399.
[15]Xian-Fang Li, Bao-Lin Wang. Anti-plane shear crack normal to and terminating at the interface of two bonded piezoelectric ceramics. International Journal of Solids and Structures . 2007, 44: 3796-3810.
[16]Z.M. Xiao, J.F. Zhao. Electro-elastic stress analysis for a Zener-Stroh crack at the metal/piezoelectric bi-material interface. International Journal of Solids and Structures. 2004,41: 2501-2519.
[17]Hua-Dong Yong, You-He Zhou. A mode III crack in a functionally graded piezoelectric strip bonded to two dissimilar piezoelectric half-planes.Composite Structures. 2007,79: 404-410.
[18] Jian-Liang Sun, Zhen-Gong Zhou, Biao Wang. 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: 993-1005.
[19]X.-F. Li. Electroelastic analysis of a cracked piezoelectric ceramic strip sandwiched between two elastic dielectrics. European Journal of Mechanics A/Solids. 2005, 24: 69-80.
[20] F. Narita, M. Yoshida, Y. Shindo. Electroelastic effect induced by electrode embedded at the interface of two piezoelectric half-planes. Mechanics of Materials. 2004, 36: 999-1006.
[21]C.-H. Chue, T.J.-C. Liu. Magneto-electro-elastic antiplane analysis of a bimaterial BaTiO3-CoFe2O4 composite wedge with an interface crack. Theoretical and Applied Fracture Mechanics. 2005,44: 275-296.
[22]Yu-Guo Sun. Multi-interface cracks in a piezoelectric layer bonded to two half-piezoelectric spaces under anti-plane shear loading. Mechanics Research Communications. 2003, 30: 443-454.
[23]Z.C.Ou, Y.H. Chen. Near-tip stress fields and intensity factors for an interface crack in metal/piezoelectric bimaterials.International Journal of Engineering Science.2004,42:1407-1438.
[24]Zhen-Gong Zhou,Lin-Zhi Wu,Shan-Yi Du.Non-local theory solution for a Mode I crack in piezoelectric materials.European Journal of Mechanics A/Solids.2006,25:793-807.
[25]Shuling Hu,Shengping Shen,Toshihisa Nishioka.Numerical analysis for a crack in piezoelectric material under impact.International Journal of Solids and Structures.2007,44:8457-8492.
[26]K.P.Herrmann,A.V.Komarov,V.V.Loboda.On a moving interface crack with a contact zone in a piezoelectric biomaterial.International Journal of Solids and Structures.2005,42:4555-4573.
[27]V.B.Govorukha,V.V.Loboda,M.Kamlah.On the influence of the electric permeability on an interface crack in a piezoelectric bimaterial compound.International Journal of Solids and Structures.2006,43:1979-1990.
[28]Albert C.To,Shaofan Li,Steven D.Glaser.Propagation of a mode-Ⅲinterracial conductive crack along a conductive interface between two piezoelectric materials.Wave Motion.2006,43:368-386.
[29]Xiang-Fa Wu,Steve Cohn,Yuris A.Dzenis.Screw dislocation interacting with interfacial and interface cracks in piezoelectric bimaterials.International Journal of Engineering Science.2003,41:667-682.
[30]周振功,王彪.利用Schmidt方法研究双相压电材料Ⅰ-型界面裂纹问题.应用数学和力学.2006,27(7):765-774.
[31]王吉伟,匡震邦.双压电体界面上的电偶极子和裂纹.力学学报.2002,34(2):192-199.
[32]杜勇锋,王建国,李雪峰.条形压电材料和弹性材料Ⅲ型界面裂纹分析.合肥工业大学学报.2005,28(12):1574-1577.
[33]王海涛,杨笑梅.用裂纹单元分析双压电材料界面力电耦合奇异场.工程力学.2007,24(31:170-178.
[34]BIAN Wen-feng,WANG Biao.Dual equations and solutions of Ⅰ-type crack of dynamic problems in piezoelectric materials.Applied Mathematics and Mechanics(English Edition).2007,28(6):731-739.
[35]Qu Gui-min,ZHOU Zhen-gong,WANG Biao.Dynamic behavior of two collinear permeable cracks in a piezoelectric layer bonded to two half spaces.Applied Mathematics and Mechanics(English Edition).2005,26(10):1266-1276.
[36]Zhen-Gong Zhou,Pei-Wei Zhang,Lin-Zhi Wu.The closed form solution of a Mode-I crack in the piezoelectric/piezomagnetic materials.International Journal of Solids and Structures.2007,44:419-435.
[37]董辉,姜稚清,刘金喜.半无限平面压电双材料中导电界面边裂纹与螺位错的相互作用.石家庄铁道学院学报.2006,19(4):12-17.
[38]平学成,陈梦成,谢基龙,李强.基于新型裂尖杂交元的压电材料断裂力学研究.力学学报.2006,38(3):407-413.
[39]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.
[40]陆建飞,汪越胜,蔡兰.与两相材料界面接触的裂纹对SH波的散射.力学学报.2003,35(4):432-436.
[41]高存法,樊蔚勋.两压电介质之间的界面夹杂问题.应用数学和力学.2001,22(1):85-92.
[42]杨娟,李星.功能梯度压电带中裂纹对SH波的散射.力学季刊.2007,28(3):471-478.
[43]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.
[44]鲍亦兴,毛昭宙著.刘殿魁,苏先樾译.弹性波的衍射与动应力集中.北京:科学出版社.1993:72-202.
[45]张沛霖,张仲渊.压电测量.北京:国防工业出版社.1983:1-10.
[46]李远,秦自楷,周志刚.压电与铁电材料的测量.北京:科学出版社.1984:8-40.
[47]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.
[48]梁昆淼.数学物理方法(第三版).北京:高等教育出版社.1998:374.
[49]刘宏伟.界面圆孔和孔边裂纹对SH波散射问题的研究.博士学位论文,哈尔滨工程大学.1997:16-100.
[50]袁晓铭,廖振鹏.圆弧形凹陷地形对平面SH波散射问题的级数解答.地震程与工程震动.1993,13(2):1-14.
[51]Sih G.C.,ed.Mechanics of Fracture,vol.6,Cracks in Composite Materials.Leyden:Noordhoff.1981.
[52]张石生.积分方程.重庆:重庆出版社.1988.
[53]云天铨.积分方程及其在力学中的应用.广州:华南理工大学出版社.1990.
[54]Davis P.J.and P.Rabinowitz.Numerical Integration.Blaisdell.1967.
[55]Loeber J.F.and Sih G.C.Diffraction of Antiplane Shear Waves by a Finite Crack.Journal of the Acoustical Society of America.1968,44(1):90-98.
[56]X.D.Wang,S.A.Meguid.Dynamic antiplane behavior of interacting cracks in a piezoelectric medium.International Journal of Fracture.1998,91:391-403.
[57]X.D.Wang,S.A.Meguid.Effect of electromechanical coupling on the dynamic interaction of cracks in piezoelectric materials.Acta Mechanica.2000,143:1-15.
[2]杨卫.电致失效力学.力学进展.1996,26(3):338-353.
[3]余寿文.固体力学与材料科学交缘的几个新课题.力学进展.1994,24(1):24-36.
[4]Parton VZ.Fracture mechanics of piezoelectric materials.Acta Astronautica.1976,3:671-683.
[5]Pak YE.Crack extension force in a piezoelectric material.J Appl Phys.1990,57:647-653.
[6]Dunn ML.The effects of cracks face boundary conditions on the fracture mechanics.Eng Frac Mech.1994,48:25-39.
[7]Sosa HA.Plane problems in piezoelectric media with defects.Int J Solids Structures.1991,28:491-505.
[8]Zhang TY,Tong P.Fracture mechanics for a mode Ⅲ crack in a piezoelectric material.Int J Solids Structures.1996,33:343-359.
[9]Park S,Sun CT.Fracture criteria for piezoelectric materials.J Am Ceram Soc.1995,78:1475-1480.
[10]王自强.压电材料裂纹顶端条状电饱和区模型的力学分析.力学学报.1999,31(3):311-319.
[11]Loboda,V.,Shevelova,A.Pre-fracture zones of an electrically permeable interface crack in a piezoelectric biomaterial.35th Solid Mechanics Conference(SolMech-06).2006,225-226.
[12]Ching-Hwei Chue,Wen-Bin Wei,Liu,T.J.-C..Antiplane electro-mechanical field of a bimaterial piezoelectric wedge under a pair of concentrated forces and surface charges.Journal of the Chinese Institute of Engineers.2006,29(3):467-80.
[13]Li,Q.,Chen,Y.H.Solution for a semi-permeable interface crack between two dissimilar piezoelectric materials.Journal of Applied Mechanics, Transactions ASME. 2007,74(5): 833-844.
[14] X. Wang, Z. Zhong, F.L. Wu. A moving conducting crack at the interface of two dissimilar piezoelectric materials. International Journal of Solids and Structures. 2003,40: 2381-2399.
[15]Xian-Fang Li, Bao-Lin Wang. Anti-plane shear crack normal to and terminating at the interface of two bonded piezoelectric ceramics. International Journal of Solids and Structures . 2007, 44: 3796-3810.
[16]Z.M. Xiao, J.F. Zhao. Electro-elastic stress analysis for a Zener-Stroh crack at the metal/piezoelectric bi-material interface. International Journal of Solids and Structures. 2004,41: 2501-2519.
[17]Hua-Dong Yong, You-He Zhou. A mode III crack in a functionally graded piezoelectric strip bonded to two dissimilar piezoelectric half-planes.Composite Structures. 2007,79: 404-410.
[18] Jian-Liang Sun, Zhen-Gong Zhou, Biao Wang. 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: 993-1005.
[19]X.-F. Li. Electroelastic analysis of a cracked piezoelectric ceramic strip sandwiched between two elastic dielectrics. European Journal of Mechanics A/Solids. 2005, 24: 69-80.
[20] F. Narita, M. Yoshida, Y. Shindo. Electroelastic effect induced by electrode embedded at the interface of two piezoelectric half-planes. Mechanics of Materials. 2004, 36: 999-1006.
[21]C.-H. Chue, T.J.-C. Liu. Magneto-electro-elastic antiplane analysis of a bimaterial BaTiO3-CoFe2O4 composite wedge with an interface crack. Theoretical and Applied Fracture Mechanics. 2005,44: 275-296.
[22]Yu-Guo Sun. Multi-interface cracks in a piezoelectric layer bonded to two half-piezoelectric spaces under anti-plane shear loading. Mechanics Research Communications. 2003, 30: 443-454.
[23]Z.C.Ou, Y.H. Chen. Near-tip stress fields and intensity factors for an interface crack in metal/piezoelectric bimaterials.International Journal of Engineering Science.2004,42:1407-1438.
[24]Zhen-Gong Zhou,Lin-Zhi Wu,Shan-Yi Du.Non-local theory solution for a Mode I crack in piezoelectric materials.European Journal of Mechanics A/Solids.2006,25:793-807.
[25]Shuling Hu,Shengping Shen,Toshihisa Nishioka.Numerical analysis for a crack in piezoelectric material under impact.International Journal of Solids and Structures.2007,44:8457-8492.
[26]K.P.Herrmann,A.V.Komarov,V.V.Loboda.On a moving interface crack with a contact zone in a piezoelectric biomaterial.International Journal of Solids and Structures.2005,42:4555-4573.
[27]V.B.Govorukha,V.V.Loboda,M.Kamlah.On the influence of the electric permeability on an interface crack in a piezoelectric bimaterial compound.International Journal of Solids and Structures.2006,43:1979-1990.
[28]Albert C.To,Shaofan Li,Steven D.Glaser.Propagation of a mode-Ⅲinterracial conductive crack along a conductive interface between two piezoelectric materials.Wave Motion.2006,43:368-386.
[29]Xiang-Fa Wu,Steve Cohn,Yuris A.Dzenis.Screw dislocation interacting with interfacial and interface cracks in piezoelectric bimaterials.International Journal of Engineering Science.2003,41:667-682.
[30]周振功,王彪.利用Schmidt方法研究双相压电材料Ⅰ-型界面裂纹问题.应用数学和力学.2006,27(7):765-774.
[31]王吉伟,匡震邦.双压电体界面上的电偶极子和裂纹.力学学报.2002,34(2):192-199.
[32]杜勇锋,王建国,李雪峰.条形压电材料和弹性材料Ⅲ型界面裂纹分析.合肥工业大学学报.2005,28(12):1574-1577.
[33]王海涛,杨笑梅.用裂纹单元分析双压电材料界面力电耦合奇异场.工程力学.2007,24(31:170-178.
[34]BIAN Wen-feng,WANG Biao.Dual equations and solutions of Ⅰ-type crack of dynamic problems in piezoelectric materials.Applied Mathematics and Mechanics(English Edition).2007,28(6):731-739.
[35]Qu Gui-min,ZHOU Zhen-gong,WANG Biao.Dynamic behavior of two collinear permeable cracks in a piezoelectric layer bonded to two half spaces.Applied Mathematics and Mechanics(English Edition).2005,26(10):1266-1276.
[36]Zhen-Gong Zhou,Pei-Wei Zhang,Lin-Zhi Wu.The closed form solution of a Mode-I crack in the piezoelectric/piezomagnetic materials.International Journal of Solids and Structures.2007,44:419-435.
[37]董辉,姜稚清,刘金喜.半无限平面压电双材料中导电界面边裂纹与螺位错的相互作用.石家庄铁道学院学报.2006,19(4):12-17.
[38]平学成,陈梦成,谢基龙,李强.基于新型裂尖杂交元的压电材料断裂力学研究.力学学报.2006,38(3):407-413.
[39]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.
[40]陆建飞,汪越胜,蔡兰.与两相材料界面接触的裂纹对SH波的散射.力学学报.2003,35(4):432-436.
[41]高存法,樊蔚勋.两压电介质之间的界面夹杂问题.应用数学和力学.2001,22(1):85-92.
[42]杨娟,李星.功能梯度压电带中裂纹对SH波的散射.力学季刊.2007,28(3):471-478.
[43]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.
[44]鲍亦兴,毛昭宙著.刘殿魁,苏先樾译.弹性波的衍射与动应力集中.北京:科学出版社.1993:72-202.
[45]张沛霖,张仲渊.压电测量.北京:国防工业出版社.1983:1-10.
[46]李远,秦自楷,周志刚.压电与铁电材料的测量.北京:科学出版社.1984:8-40.
[47]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.
[48]梁昆淼.数学物理方法(第三版).北京:高等教育出版社.1998:374.
[49]刘宏伟.界面圆孔和孔边裂纹对SH波散射问题的研究.博士学位论文,哈尔滨工程大学.1997:16-100.
[50]袁晓铭,廖振鹏.圆弧形凹陷地形对平面SH波散射问题的级数解答.地震程与工程震动.1993,13(2):1-14.
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