各向异性材料和压电材料奇异性场研究
|
摘要
各向异性材料和压电材料常常单独或与其它材料接合使用，材料中的裂纹、V型切口和夹杂角尖等部位均存在奇异性，很可能导致部件的功能失效和早期破坏。为了确保设备和结构的安全可靠性，全面准确地了解这些部位的奇异性场强度有着非常重要的意义。通常奇异性场可统一表示为∑(r，θ)=kr~λF(θ)，其中(r，θ)为原点设在奇异点的极坐标，特征值λ和特征角分布函数F(θ)为特征解，k为强度系数。为了确定奇异性场强度和建立断裂准则，不但要求解特征解，而且要确定强度系数k，为此，本文建立了几类新型奇异性单元，并与全域杂交元结合，用来求解各向异性材料和压电材料奇异性场强度。奇异单元建立的思路为：利用高次内插有限元特征法求解奇异性场的特征解，并以此为基础，通过广义Hellinger-Reissner变分原理建立围绕奇异点邻域的超级奇异性单元。将超级奇异单元与全域杂交元结合，用来求解各向异性材料和压电材料奇异性场强度。根据材料和结构形式的不同，主要包含以下研究内容：
1)建立各向异性材料裂纹尖端超级奇异单元，用于求解各向异性接合材料裂纹尖端场强度，确定应力强度因子的数值解。文中通过算例验证所建立的模型，并考察材料主轴走向、裂纹长度、裂纹间距、材料属性和结构形式等的影响。
2)建立面内极化压电材料的裂纹尖端超级奇异单元，用于求解奇异性电弹场强度。该单元考虑的裂纹面电边界条件包括：绝缘条件、部分导通条件和导通条件。通过典型算例，对方法进行了验证，并分析了各种电边界条件和极化方向等对应力和电位移强度因子的影响。
3)建立面外极化压电材料界面裂纹尖端超级奇异单元，用于求解层间裂纹的应力和电位移强度。考虑的接合材料种类包括：压电／压电、压电／弹性绝缘体以及压电／弹性导体。通过算例考察了电边界条件，层厚和外载等因素对强度因子和能量释放率的影响。
4)建立各向异性材料和压电材料超级界面端单元，用于求解接合材料界面端部的奇异性场强度。通过算例分析，考察了材料主轴走向，界面端角度，材料属性和结构尺度对应力强度和电位移强度的影响。
5)建立了各向异性材料和压电材料超级夹杂角尖单元，用于求解该处的奇异性场强度。算例考虑了夹杂角尖角度、夹杂尺寸和夹杂间距等的影响，据此判断界面的安全性。
Anisotropic materials and piezoelectric materials are often used by oneself or bonded with other materials. Unfortunately, strong singularities in the tips of the cracks, interface edges and inclusions may lead to function invalidation or even earlier destroy of components. To ensure the safety and reliability of the devices and structures, it is meaningful to understand completely and exactly the local singular fields. Commonly, the singular fields surrounding the singular points can be expressed as ∑ = k r~λ F(θ), in which (r,θ) is the polar coordinate system whose origin is set at the singular point, eigenvalue X and angular variation function F(θ) are eigensolutions, k is the magnified coefficient depending on the given loading, the geometry of the structure and material constants. To determine the singular fields and the fracture parameters, both the eigensolutions and the magnified coefficients should be solved. The main objective of this dissertation is to establish several kinds of super singular elements, and combine them with the standard hybrid element to solve the singular fields of anisotropic and piezoelectric materials. To establish the super singular elements, two steps are needed: Firstly, the eigensolutions of singular fields in the crack-tips of anisotropic materials are solved by a kind of ad hoc finite element eigenanalysis method; Secondly, based on generalized Hellinger-Reissner variational functional, a kind of super crack-tip singular element is established. In terms of materials and geometries in the researches, the main contents are described as following:
1) A kind of super crack-tip singular element is developed to investigate the singular fields around the crack-tips of bonded anisotropic materials, and the numerical solutions of stress intensity factors (SIFs) are obtained by the solved singular stresses. The new model is verified with avaible solutions. Other new numerical results are also given to investigate the influences of fibre orientation, cracks length, crack distances, material mismatches and material geometry.
2) A kind of super crack-tip element model is developed to investigate the singular electro-elastic fields around the crack-tips of inplane polarized piezoelectric materials, in which three kind of boundary conditions, i.e., impermeable condition, limited permeable and
1)建立各向异性材料裂纹尖端超级奇异单元，用于求解各向异性接合材料裂纹尖端场强度，确定应力强度因子的数值解。文中通过算例验证所建立的模型，并考察材料主轴走向、裂纹长度、裂纹间距、材料属性和结构形式等的影响。
2)建立面内极化压电材料的裂纹尖端超级奇异单元，用于求解奇异性电弹场强度。该单元考虑的裂纹面电边界条件包括：绝缘条件、部分导通条件和导通条件。通过典型算例，对方法进行了验证，并分析了各种电边界条件和极化方向等对应力和电位移强度因子的影响。
3)建立面外极化压电材料界面裂纹尖端超级奇异单元，用于求解层间裂纹的应力和电位移强度。考虑的接合材料种类包括：压电／压电、压电／弹性绝缘体以及压电／弹性导体。通过算例考察了电边界条件，层厚和外载等因素对强度因子和能量释放率的影响。
4)建立各向异性材料和压电材料超级界面端单元，用于求解接合材料界面端部的奇异性场强度。通过算例分析，考察了材料主轴走向，界面端角度，材料属性和结构尺度对应力强度和电位移强度的影响。
5)建立了各向异性材料和压电材料超级夹杂角尖单元，用于求解该处的奇异性场强度。算例考虑了夹杂角尖角度、夹杂尺寸和夹杂间距等的影响，据此判断界面的安全性。
Anisotropic materials and piezoelectric materials are often used by oneself or bonded with other materials. Unfortunately, strong singularities in the tips of the cracks, interface edges and inclusions may lead to function invalidation or even earlier destroy of components. To ensure the safety and reliability of the devices and structures, it is meaningful to understand completely and exactly the local singular fields. Commonly, the singular fields surrounding the singular points can be expressed as ∑ = k r~λ F(θ), in which (r,θ) is the polar coordinate system whose origin is set at the singular point, eigenvalue X and angular variation function F(θ) are eigensolutions, k is the magnified coefficient depending on the given loading, the geometry of the structure and material constants. To determine the singular fields and the fracture parameters, both the eigensolutions and the magnified coefficients should be solved. The main objective of this dissertation is to establish several kinds of super singular elements, and combine them with the standard hybrid element to solve the singular fields of anisotropic and piezoelectric materials. To establish the super singular elements, two steps are needed: Firstly, the eigensolutions of singular fields in the crack-tips of anisotropic materials are solved by a kind of ad hoc finite element eigenanalysis method; Secondly, based on generalized Hellinger-Reissner variational functional, a kind of super crack-tip singular element is established. In terms of materials and geometries in the researches, the main contents are described as following:
1) A kind of super crack-tip singular element is developed to investigate the singular fields around the crack-tips of bonded anisotropic materials, and the numerical solutions of stress intensity factors (SIFs) are obtained by the solved singular stresses. The new model is verified with avaible solutions. Other new numerical results are also given to investigate the influences of fibre orientation, cracks length, crack distances, material mismatches and material geometry.
2) A kind of super crack-tip element model is developed to investigate the singular electro-elastic fields around the crack-tips of inplane polarized piezoelectric materials, in which three kind of boundary conditions, i.e., impermeable condition, limited permeable and
引文
1. http://www.ourfrp.com/top/04/ourfrp 04 003.htm
2. httpA://peizo.blogdriver.com/peizo/
3. http://peizo.blogdriver.com/peizo/874088.html
4. http://www.cutech.edu.cn/ShowArticle.asp?ArticleID=14703
5. http://www.stcsm.gov.cn/learning/lesson/course/detail.asp?id=91&lessonnum=2&coursenum=31
6. http://www.sic.ac.cn/kpz2005/shuzi/16.htm
7．叶志明．各向异性材料与混凝土材料断裂力学引论．北京：中国铁道出版社，2000．
8．黄克智，杨卫，郑泉水．第19届国际理论与应用力学大会(ICTAM-19)．1996年8月25～31日，日本京都．
9. Williams M.L. Stress Singularities resulting from various boundary conditions in angular corners of plate extension. J.Appl.Mech. 1952,24:109-114.
10. Williams M.L. The stresses around a fault or crack in dissimilar media. Bull. Seis. Soc. Am. 1959,49: 199-204.
11. Rice J.R. Elastic fracture mechancs concepts for interface cracks. J. Appl. Mech. 1988, 55: 98-103.
12. Erdogan F. Stress distribution in a nonhomogeneous elastic plane with cracks. J. Appl. Mech. 1963, 30: 232-236.
13. Erdogan F. Stress distribution in bonded dissimilar materials with cracks. J. Appl. Mech. 1965, 32: 403-410.
14．许金泉，姜菊生．界面端附近裂纹的应力强度因子．上海力学，1998，19(3)：221-227．
15．朱哲明，谢和平．裂纹表面承受均布载荷时的应力强度因子及裂纹张开位移的边界配位法．计算力学学报，1997，14(2)：182-188．
16. Bogy D.B. On the plane elastostatic problem of a loaded crack terminating at a material interface. J. Appl. Mech. 1971,38:911-918.
17. Bogy D.B. The plane elastostatic problem for a symmetrically loaded crack in a strip composite. Int. J. Eng. Sci. 1973,11:985-996.
18. Hoenig A. Near tip behavior of a crack in a plane anisotropic elastic body. Eng. Fract. Mech. 1992,16:393-403.
19. Bassani J.L. and Erdogan 17. Sress intensity, factors in bonded half planes containing inclined cracks and subjected to antiplane shear loading. Int. J. Fract. 1979,15:145-158.
20. Ma C.C. and Hour B.L., Antiplane problems in composite anisotropic materials with an inclined crack terminating at bi-material interface. Int. J. Solids Structures 1990,26:1387～1400.
21. Lekhnitskii S.G. Theory of Elasticity of an Anisotropic Body. Holden-Day, San Francisco. 1963.
22. Eshelby J.D., Read W.T. and Shodkley W. Anisotropic elasticity with application to dislocation theory. Acta Metall. 1953,1:251-259.
23. Stroh A.N. Dislocations and cracks in anisotropic elasticity, Phil. Mag. 1958, 7:625-646.
24. Stroh A.N. Steady state problems in anisotropic elasticity. J. Math. Phys. 1963,41:77-103.
25. Sih G.C., Pairs RC., Irwin G.R. On cracks in rectilinearly anisotropic bodies. J. Fract. Mech. 1965, (1): 189-203.
26. Ting T.C.T. and Hoang RH. Singularities at the tip of a crack normal to the interface of an anisotropic layered composite. Int. J. Solids Structures 1984,20:439-454.
27. Ting T.C.T. Explicit solution and invariance of the singularities at an interface crack in anisotropic composites. Int. J. Solids Structures 1986,22:965-983.
28. Yang S. and Yuan F.G. Kinked crack in anisotropic bodies. Int. J. Solids Structures 2000,37(45): 6635-6682.
29. Suo Z. Singularities, interfaces and cracks in dissimilar anisotropic media. Proc. R. Soc. A 427,1990:331-358.
30．高存法，樊蔚勋．各向异性平面内双边半无限裂纹问题的基本解，工程力学，1997，14(2)：128-133．
31. Nazarov S.A. A crack at the interface of anisotropic bodies. Singularities of the elastic fields and a criterion for fracture when the crack surfaces are in contact. J. Appl. Math. Mech. 2004, 69(3): 473-483.
32. Yang L.H., Li Z.H. The interaction of mode I crack with multi-inclusions in a three-phase model. Int. J. Fract. 2004, 127: 193-200.
33．黄克智，余寿文，杨卫．裂纹尖端奇异场—清华大学工程力学系校友研究成果综述，清华大学工程力学与工程热物理学术会议论文集，清华大学出版社，1988：15-25．
34. Gao Y.C., Hwang KC. Exponential Stress and Strain Singularity at Tip of Mode I Growing Crack in Power Hardening Plastic Material. Theor. Appl. Fract. Mech. 1996,25:103-111.
35．唐立强，黄克智．理想正交各向异性材料平面应力条件下Ⅰ型静止裂纹尖端场，力学学报，1991，23：448-457．
36．刘一华，许金泉．纤维端部的界面裂纹分析．力学学报，1999，31(3)：385-390．
37. Kondo T., Sekine H. Anti-plane shear of kinked interface crack in bonded dissimilar anisotropic solid. Theor. Appl. Fract. Mech., 1993,18(3): 273-282.
38. Yetmez M., Gecit M.R. Finite strip with a central crack under tension. Int. J. Eng. Sci. 2005, 43:472-493.
39. Mahajan R., Erdogan F., Kilic B. and Madenci E. Cracking of an orthotropic substrate reinforced by an orthotropic plate. Int. J. Solids Structures 2003, 40: 6389-6415.
40. Tang R.J. and Erdogan F. A clamped rectangular plate containing a crack. Theor. Appl. Fract. Mech. 1985,4:328-339.
41．陈梦成，高闯，汤任基．三维界面裂纹的奇性应力场和应力强度因子分析，固体力学学报，1999，20：8-15．
42．陶伟明，郭乙木，曹志远，嵇醒．三维裂纹分析的主值型面力边界积分方程，固体力学学报，2001，22(3)：256-262．
43．张明，姚振汉，杜庆华，楼志文．双材料界面裂纹应力强度因子的边界元分析，应用力学学报，1999，16(1)：21-26．
44. Cho S.B., Lee K.R., Choy Y.S. and Yuuki R. Determination of stress intensity factors and boundary element analysis for interface cracks in dissimilar anisotropic materials. Eng. Fract. Mech. I992, 43(4): 603-614.
45．Yuuki R．，Xu J．Q．Stress intensity Factors for the interface crack between dissimilar orthotropicmaterials．日本机械学会論文集(A编)，1991，57(7)：1542-1549．
46．Xu J．Q，，Yuuki R．Stress intensity Factors for the interface crack between dissimilar orthotropic materials(The case where principal axes are not aligned with the interface，)．日本机械学会論文集(A编)，1994，60(9)：1943-1950．
47. Zhou W., Lira K.M., Lee K.H., Tay A.O.O. A new variable-order singular boundary element for calculating stress intensity factors in three-dimensional elasticity problems. Int. J. Solids Structures 2005, 42: 159-185.
48. Chyuan S.W., Lin J.H., Chen J.T. and Liu D.C. Dual boundary element analysis for fatigue behavior of missile structures. J. Chinese Institute of Engineering 2000, 23(3): 339-348.
49. Pan E. A general Boundary Element Analysis of 2-D Linear Elastic Fracture Mechanics. Int. J. Fract. 1997, 81(1): 41-59.
50. garsoum R.S. Cracks in anisotropic materials—an iterative solution of the eigenvalue. Int. J. Fract. 1986, 32: 59-67.
51. Bazant Z.R and Estenssord L.F. Surfaces singularity and crack propagation. Int. J. Solids Structures 1979,15:405-426.
52. Somaratna N. and Ting T.C.T. Three-dimensional stress singularities in anisotropic material and composites. Int. J. Eng. Sci. 1986,24:1115-1134.
53. Yamada Y. and Okumura H. Finite element analysis of stress and strain singularity eigenstate in homogenous media or composite materials. In Hybrid and Mixed Finite Methods (Edited by Atluri S.N., Gallagher R.H. and Zienkiewicz O.C.) 1983:325-343.
54. Pageau S.S., Joseph P.F. and Biggers S.B.Jr. Finite element analysis of anisotropic materials with singular inplane stress fields. Int. J. Solids Structures 1995,32 571-591.
55. Sze K.Y., Wang H.T. A simple finite element formulation for computing stress singularities at bimaterial interfaces. FE in Analysis and Design 2000, 35: 97-118.
56．平学成，陈梦成．楔形体尖端近似场的非协调有限元特征法．华东交通大学学报．2001，18(4)：6-11．
57. Kaddouri K., Belhouari M., Bouiadjra B.B. and Serier B. Finite element analysis of crack perendicular to bi-material interface. Comp. Mater. Sci. 2006, 35: 53-60.
58．卢炎麟，张淑佳，俞志清．焊趾表面裂纹应力强度因子计算的基本模式法，应用力学学报，1998(4)：73-81．
59. Seweryn A. Modeling of singular stress fields using finite element method. Int. J. Solids Structures 2002,39:4787-4804.
60. Chang J.H., Wu W.H. FE calculation of contour integrals in plane anisotropic elastic media with crack surface tractions. FE in Analysis and Design 2001, 37(9): 673-696.
61. Saxce Gery De, Kang C.H. Application of the hybrid mongrel displacement finite method to the computation of stress intensity factors in anisotropic material. Eng. Fract. Mech. 1992, 41(1): 71-83.
62. Song C.M., Wolf J.P. Semi-analytical representation of stress singularities as occurring in cracks in anisotropic multi-materials with the scaled boundary finite-element method. Computers & Structures 2002, 80: 183-197.
63. Lee J. and Gao H. A hybrid finite element analysis of interface cracks. Int. J. Num. Meth. Eng. 1995,38:2465-2482.
64. Pageau S.S. and Biggers S.B.Jr. Enrichment of finite elements with numerical solutions for singular stress fields. Int. J. Num. Meth. Eng. 1997,40:2693-2713
65. Chen M.C., Sze K.Y. and Wang H.T. Analysis of singular stresses in bounded bimaterial wedges by computed eigen solutions and bybrid element method. Commun. Num. Meth. Eng. 2001, 17: 495-507.
66. Chen M.C., Ping X.C. A novel hybrid element analysis for piezoelectric-parent material wedges. Comput. Mech. In press.
67. Barut A., Guven I. and Madenci E. Analysis of singular stress fields at junctions of multiple dissimilar materials under mechanical and thermal loading. Int. J. Solids Structures 2001: 38:9077-9109.
68. Tong P., Plan T.H.H., Lasry S.J. A hybrid element approach to crack problems in plane elasticity. International J. Num. Meth. Eng.1973, 7: 297-380.
69. Chert E,P. Finite element analysis of bimaterial interface crack. Theor. Appl. Fract. Mech. 1985, 3: 257-262.
70. Patton VZ. Fracture mechanics of piezoelectric materials. Acta Astronautica, 1976, 3:671-683
71．王自强，韩学礼．压电材料中心裂纹问题．固体力学学报，1999：20：95-103．
72. Barnett D.M., Lothe J. Dislocations and line charges in anisotropic piezoelectric insulators. Phys. Status. Solidi. 1975, 1367: 105-111.
73. Deeg W.E The analysis of dislocation, crack, and inclusion problems in piezoelectric solids (Ph.D Dissertation). Stanford Unversity, 1980, USA.
74. Pak Y.E. Linear electro-elastic fracture mechanics of piezoelectric materials. Int. J. Fract. 1992, 54: 79-100.
75. Kuo C.M., Barnett D.M. Stress singularities of interfacial cracks in bonded piezoelectric half-spaces. In: Wu J. J., Ting T.C.T., Barnett D.M. (Eds.), Modern Theory of Anisotropic Elasticity and Applications. SIAM, Philadelphia, 1991, pp. 33-50.
76. Suo Z., Kuo C. M., Barnett D.M. and Willis J.R. Fracture mechanics for piezoelectric ceramics. J.Mech. Phys. Solids 1992, 40: 739-765.
77. Sosa H.A. On the fracture mechanics of piezoelectric solids. Int. J. Solids Structures, 1992, 21: 2613-2622,
78. Sosa H.A., Khutoryansky, N., New developments concerning piezoelectric materials with defects. Int. J. Solids Structures 1996, 33: 3399-3414.
79. Park S. B. and Sun C. T. Effect of electric field on fracture of piezoelectric ceramics. Int. J. Fract. 1995, 70: 203-216.
80. Herrmann K., Loboda V. Fracture-mechanical assessment of electrically permeable interface cracks in piezoelectric bimaterial by consideration of various contact zone models. Arch. Appl. Mech. 2000, 70: 127-143.
81. Beom H.B., Atluri SN. Near-tip fields and intensity factors for interfacial cracks in dissimilar anisotropic piezoelectric media. Int, J, Fract. 1996, 75: 163-183.
82. Wang T.C,, Han X.L. Fracture mechanics of piezoelectric materials. Int. J. Fract., 1999, 98: 15-35.
83. Gao C.F., Wang M.Z. Green function of an interface crack between two dissimilar piezoelectric media. Int. J. Eng. Sci. 2000, 37: 4969-4986.
84. Hao T.H. Multiple collinear cracks in a piezoelectric material. Int. J. Solids Structures 2001: 38: 9201-9208.
85. Zhang T.Y., Zhao M., Tong R Fracture of piezoelectric ceramics. Adv. Appl. Mech., 2002, 38: 147-289.
86. Zhang T.Y., Gao C.F, Fracture behaviors of piezoelectric materials. Theor. Appl. Fract. Mech. 2004, 41: 339-379.
87．王旭，沈亚鹏．压电和多层材料中电极—陶瓷界面裂纹和椭圆夹杂．力学学报，2002，34(6)：881-894．
88. Ou Z.C. Singularity parameters for interface crack in transversely isotropic piezoelectric bimaterials, Int. J. Fract. 2003, 119: 41-46.
89. Ou Z.C., Chen Y.H. Near-tip stress fields and intensity factors for an interface crack in metal/piezoelectric bimaterials. Int. J. Eng. Sci. 2004,42: 1407-1438.
90. Ou Z.C., Chen Y.H. Interface crack-tip generalized stress field and stress intensity factors in transversely isotropic piezoelectric birnaterials, Mech. Res. Comm. 31(2004) 421-428.
91. Chue C.H., Weng S.M. Fracture analysis of piezoelectric materials with an arbitrarily oriented crack using energy density theory. Computers & Structures, 2005(83): 1251-1265.
92. Wang Z. Penny-shaped crack in transversely isotropic piezoelectric materials. Acta Mechanica Sinica, 1994, 10(1): 49-60.
93. Gao, H., Zhang, T.-Y., Tong, P. Local and global energy release rates for an electrically yielded crack in a piezoelectric ceramic. J. Mech. Phys. Solids 1997, 45: 491-510.
94. Fulton, C.C., Gao, H. Electrical non-linearity in fracture of piezoelectric ceramics. ASME Appl. Mech. Review 1997, 50: 56-63.
95. Dunn M.L. The effects of crack face boundary conditions on the fracture mechanics of piezoelectric solids. Eng. Fract. Mech. 1999, 48: 25-39.
96. McMeeking R.M. Towards a fracture mechanics for brittle piezoelectric and dielectric materials. Int. J. Fract. 2001, 108: 25-41.
97. Fulton, C.C., Gao, H. Effect of local polarization switching on piezoelectric fracture, J. Mech. Phys. Solids 2001, 49: 927-952.
98. Sosa H. Plane problems in piezoelectric media with defects. Int. J. Solids Structures 1991, 28: 491-505.
99. Gao C.F.和Fan W.X. Exact solutions for the plane problem in piezoelectric materials with an elliptic or a crack. Int. J. Solids Structures 1999, 36: 2527-2540.
100．杨卫。力电失效学。清华大学出版社，2001．
101. Hao T.H., Shen Z.Y. A new boundary conditions on the fracture mechanics of piezoelectric solids. Eng. Fract. Mech. 1994; 47: 793-802.
102．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1993．
103. Wang B.L. Noda N. Mixed mode crack initiation in piezoelectric ceramic strip. Theor. Appl. Fract. Mech. 2000, 34: 35-47.
104. Wang B.L., Sun Y.G. Intensity factors for some common piezoelectric fracture mechanics specimens with conducting cracks or electrodes. Int. J. Eng. Sci. 2004, 42:1129-1153.
105. Govorukha V., Kamlah M., Munz D. The interface crack problem for a piezoelectric semi-infinite strip under concentrated electromechanical loading. Eng. Fract. Mech. 2004, 71:1853-1871.
106. Wang B. Three-dimensional analysis of an ellipsoidal inclusion in a piezoelectric material. Int. J. Solids. Structures 1992, 29(3): 293-308.
107，刘金喜，王彪，杜善义。二维各向异性压电介质机电耦合场的基本解。应用数学和力学，1997，18(10)：885-891．
108．丁皓江，王国庆，陈伟球．压电材料平面问题的基本解．中国科学(E)，1997，27(3)：224-228
109. Huang Z.Y., Kuang Z.B. A first order perturbation analysis of a non-ideal crack. in a piezoelectric material. Int. J. Solids Structures 2001, 38: 726t-7281.
110. Huang Z.Y., Kuang Z.B. A mixed electric boundary value problem for a two- dimensional piezoelectric crack. Int. J. Solids Structures 2003, 40:1433-1453.
111. Chen M.C. Application of finite-part integrals to three-dimensional fracture problems for piezoelectric media. Part Ⅰ: Hypersingular integral equation and theoretical analysis. Int. J. Fract. 2003, 121: 133-148.
112. Chen M.C. Application of finite-part integrals to three-dimensional fracture problems for piezoelectric media. Part Ⅱ: Numerical analysis. Int. J. Fract. 2003, 121: 149-161.
113. Shindo Y., Watanabe K., Narita F. Electroelastic analysis of a piezoelectric ceramic strip with a central crack. J. Eng. Sci. 2000, 38,1-19.
114. Narita F., Shindo Y. Mode I crack growth rate for yield strip model of a marrow piezoelectric ceramic body. Theor. Appl. Fract. Mech. 2001,36: 73-85.
115. Garcia F., Saez Andres, Dominguez J. Anisotropic and piezoelectric materials fracture mechanics analysis by BEM. Computers & Structures 2005, 83: 804-820.
116. Khutoryansky N., Sosa H., Zu W. Approximate Green's function and a boundary element method for electro-elastic analyses of active materials. Computers & Structures 1998, 66: 289-299.
117. Groh U., Kuna M. Efficient boundary element analysis of cracks in 2D piezoelectric structures. Int. J. Solids Structures 2005, 42:2399-2416.
118. Shen N., Sze K.Y. Multi-region Trefftz boundary element method for fracture analysis in plane piezoelectricity. Comput. Mech. 2006, 37:381-393.
119．王国庆，刘正兴．压电材料平面问题应力强度因子的边界元计算．上海交通大学学报，1998，32(12)：74-76．
120. Wu C.C., Sze K.Y., Huang Y.Q. Numerical solutions on fracture of piezoelectric materials by hybrid element. Int. J. Solids Structures 2001, 38: 4315-4329.
121．刘鸣，吴长春．压电介质机电耦合裂尖场的杂交元数值分析，科学通报，1999，44：603-608
122. Sze K.Y., Wang H.T., Fan H. A finite element approach for computing edge singularities in piezoelectric materials. Int. J. Solids Structures 2001; 38:9233-9252.
123．陈梦成，平学成，朱剑军．压电材料中切口／接头端部平面电弹场奇异性有限元分析．固体力学学报，2005，26(2)：157-162．
124．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)：1-7．
125. Kuna M. Finite element analyses of crack problems in piezoelectric structures. Comp. Mater. Sci. 1998, 13: 67-80.
126. Abendroth M., Groh U., Kuna M., Ricoeur A. Finite element-computation of the electromechanical J-Integral for 2-D and 3-D crack analysis. Int. J. Fract. 2002:114: 359-378.
127. Shang F.L., Kuna M., Abendroth M. Finite element analyses of three-dimensional crack problems in piezoelectric structures. Eng. Fract. Mech. 2003, 70: 143-160.
128. Kumar S., Singh R.N. Influence of applied electric field and mechanical boundary condition on the stress distribution at the crack tip in piezoelectric materials. Mater. Sci. Eng. 1997, A231: 1-9.
129. Kumar S., Singh R.N. Effect of the mechanical boundary condition at the crack surfaces on the stress distribution at the crack tip in piezoelectric materials. Mater. Sci. Eng. 1997, A252: 64-77.
130. Wippler K., Ricoeur A., Kuna M. Towards the computation of electrically permeable cracks in piezoelectrics. Eng. Fract. Mech. 2004, 71: 2567-2586.
131. Scherzer M., Kuna M Asymptotic analysis of piezoelectric composite,materials. Proc. 1st Caesariurn "Smart Materials", Bonn 1999. In: Hoffmann K.H (eds) Smart Materials. Springer, Berlin Heidelberg New York, 2001, pp137-148.
132. Scherzer M., Kuna M. Combined analytical and numerical solution of 2D interface comer configurations between dissimilar piezoelectric materials. Int. J. Fract. 2004,127: 61-99.
133．李尧臣．压电材料平面裂纹问题的强度因子和能量释放率。同济大学学报，2002，30(4)：437-445．
134．周勇，王鑫伟．压电材料平面裂纹尖端场的杂交应力有限元分析．力学学报，2004，36(3)：354-358．
135. Govorukha V., Kamlah M. Asymptotic fields in the finite element analysis of electrically permeable interfacial cracks in piezoelectric bimaterials. Arch. Appl. Mech. 74, 92-101.
136. McMeeking R.M. Crack tip energy release rate for a piezoelectric compact tension specimen. Eng. Fract. Mech. 1999,64: 217-224.
137. Gruebner O., Kamlah K., Munz D. Finite element analysis of cracks in piezoelectric materials taking into account the permittivity of the crack medium. Eng. Fract. Mech. 2003, 70: 1399-1413.
138. Gruebner O., Kamlah K. Investigation of an interface crack with a contact zone in a piezoelectric bimaterial under limited permeable electric boundary conditions．Acta Mechanica 2005, 178: 85-99.
139. Pak Y.E. Crack extension force in piezoelectric material. J. Appl. Mech. 1990, 57: 863-869.
140. Pak Y.E, Goloubeva E. Electroelastic properties of cracked piezoelectric materials under longitudinal shear. Mech. Mater. 1996, 24: 287-303.
141. Zhang T.Y., Tong P. Fracture mechancs for a mode Ⅲ crack in a piezoelectric material. Int. J. Solids Structures 1996, 33: 343-359,
142．侯密山，梅甫良．不同压电材料反平面应变状态的电渗透型界面裂纹．科学通报，1998，43(2)：216-221．
143. Chen B.J., Liew K.M., Xiao Z.M. Green's functions for anti-plane problems in piezoelectric media with a finite crack. Int. J. Solids Structures 2004, 41:5285-5300.
144. Chen Z.T., Yu S.W., Karihaloo B.L. Anti-plane shear problem for a crack between two dissimilar piezoelectric materials. Int. J. Fract. 1997,86: 9-14.
145. Shindo Y., Narita F., Tanaka K. Electroelestic intensification near anti-plane shear crack in orthotropic piezoelectric ceramic strip. Theor. Appl. Fract. Mech. 1996, 254: 65-71.
146. Narita F., Shindo Y. Layered piezoelectric medium with interface crack under anti-plane shear. Theor. Appl. Fract. Mach. 1998, 30: 119-126.
147. Narita F. and Shindo Y. The interface crack problem for bonded piezoelectric and orthotropic layers under antiplane shear loading. Int. J. Fract. 1999,98:87-101
148. Sob 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. Eur. J. Mech. 2000,19:961-977.
149. Huang Z.Y., Kuang Z.B. A mixed electric boundary value problem for an anti-plane piezoelectric crack. Acta Mechanica Solida Sinica, 2003, 16(2): 110-115.
150. Li X.F., Tang G.J. Antiplane interface crack between two bonded dissimilar piezoelectric layers. Eur. J. Mech. A/Solids 2003, 22:231-242.
151. Li X.F. Electroelastic analysis of an interface anti-plane shear crack in a layered piezoelectric plate. Int. J. Eng. Sci. 2003, 41:1405-1422.
152. Li X.F. Electroelastic analysis of a cracked piezoelectric strip sandwiched between two elastic dielectrics. Eur. J. Mech. A/Solids 2005, 24:69 80.
153. Kwon S.M., Lee K.Y. Analysis of stress and electric fields in a rectangular piezoelectric body with a center crack under anti-plane shear loading. Int. J. Solids Structures 2000, 4859-4869.
154. Wang B.L, Han J.C., Du S.Y. Electroelastic fracture dynamics for multilayered piezoelectric materials under dynamics anti-plane shearing. Int. J. Solids Sturctures 2000, 38:5219-5231.
155. Chue C.H., Liu T.J.C.. Electro-elastic analysis of a bimaterial piezoelectric wedge with an interface crack under antiplane concentrated forces and inplane surface charges. Int. J. Solids Structures 2004, 41: 4179-4196.
156. Qin Q.H. Solving anti-plane problems of piezoelectric materials by the Trefftz finite element approach. Comput. Mech. 2003, 31:461-468.
157．陈梦成，平学成，朱剑军．压电—异材结合材料尖劈端部反平面奇异性问题．力学学报(已投稿)．
158. Williams M.L., Stress Singularities resulting from various boundary conditions in angular comers of plate extension. J. Appl. Mech. 1952,19:526-535.
159. Chen D.H., Nisitani H. Singular stress field near the corner of jointed dissimilar materials. J. Appl. Mech. 1993, 6(3): 607-613.
160. Dini D., Hills D.A. Asymptotic characterization of nearly-sharp notch root stress fields. Int. J. Fract. 2004, 130: 651-666.
161. Bogy D.B. Two edge-bonded elastic wedges of different materials and wedge angles under surface tractions. J. Appl. Mech. 1971,38:377-386.
162. Theocaris RS. The order of singularity at a multi-wedge comer of a composite plate. Int. J. Eng. Sci. 1974,12:107-120.
163. Ma C.C. and Hour B.L. Analysis of dissimilar anisotropic wedges subjected to antiplane shear deformation. Int. J. Solids Structures 1989,11 :1295-1309.
164. Wang S.S. and Choi I., Boundary-layer effects in composite laminates. Part 1: Free-edge stress singularities. J. Appl. Mech. 1982,49:541-548.
165. Zwiers R.I., Ting T.C. and Spilker R.L. On the logarithmic singularity of free-edge stress in laminated composites under uniform extension. J. Appl. Mech. 1982, 49:561-569.
166. Delale F. Stress singularities in bonded anisotropic materials. Int. J. Solids Structures, 1984, 20:31-40.
167. Xu X.L., Rajapakse R.K.N.D. Analytical solution for an arbitrarily oriented void/crack and fracture of piezoceramics. Acta. Mater. 1999, 47(6); 1735-1747
168. Xu X.L., Rajapakse R.K.N.D. On singularities in composite piezoelectric wedges and junctions. Int. J. Solids Structures 2000; 37: 3253-3275.
169. Chue C.H., Chen C.D. Decoupled formulation of piezoelectric elasticity under generalized plane deformation and its application to wedge problems. Int. J. Solids Structures 2002, 39:3131-3158.
170. Chen H.P. Stress singularities in anisotropic multi-material wedges and junctions. Int. J. Solids Structures 1998,11:1057-1073.
171. Huang T.F. and Chen W.H. On the free-edge stress singularity of general composite laminates under uniform axial strain. Int. J. Solids Structures 1994, 31:3139-3151.
172. MantiC V., Paris F. and CaNas J. Stress singularities in 2D orthotropic comers. Int. J. Fract. 1997, 83(1): 69-90.
173. Wu K.C. and Chen C.T. Stress analysis of anisotropic elastic V-notched bodies. Int. J. Solids Structures 1996,33(17): 2403-2416.
174. Zhang J.Q., Zhang B.N., Fan J.H. A coupled electromechanical analysis of a piezoelectric layer bonded to an elastic substrate: Part Ⅰ, development of governing equations. Int. J. Solids Structures 2003, 40:6781-6797.
175. Zhang B.N., Zhang J.Q., Fan J.H. A coupled electromechanical analysis of a piezoelectric layer bonded to an elastic substrate: Part Ⅱ, numerical solution and applications. Int. J. Solids Structures 2003, 40:6799-6812.
176．许金泉，王效贵，刘一华．振荡应力奇异性及其强度系数的数值分析方法．力学季刊，2000，21(2)：230-236．
177．唐亮，许金泉．直角结合异材界面端应力强度系数的经验公式．力学季刊，2005，25(1)：96-101．
178. Carpenter W.C. A caculation procedure for determining fracture mechanics parameters at a comer. The Ninth Canadian Congress of Applied Mechanics, Saskatchewan, Saskatoon, Canada, May 3 -June 3, 1983.
179. Sukumar N. and Kumosa M. Application of the finite element iterative method to cracks and sharp notches in orthotropic media. Int. J. Fract. 1992,58:177-192.
180. Gu L. and Belytschko T. A numerical study of stress singularities in a two-material wedge. Int. J. Solids Structures 1994,31:865-889.
181. Raju I.S. and Crews J.H.Jr. Interlaminar stress singularities at a straight free-edge in composite laminates. Computers & Structures 1981,4:21-28.
182. Artel J., Beeker W. Coupled and uncoupled analyses of piezoelectric free-edge effect in laminated plates. Composite Structures 2005, 69: 329-335.
183. Yeh J.R. and Tadjbakhsh I.G. Stress singularity in composite laminates by finite element method. J. Appl. Mech. 1982,49:541-548.
184．王效贵．界面端奇异行为的数值分析．浙江大学博士学位论文，2002．
185. Pageau S.S. and Biggers S.B.Jr. Finite element evaluation of free-edge singular stress fields in anisotropic materials. Int. J. Num. Meth. Eng. 1995,38:2225-2239.
186. Pageau S.S. and Biggers S.B.Jr. A finite element approach to three-dimensional singular stress states in anisotropic multi-material wedges and junctions. Int. J. Solids Structures 1996,33:33-47.
187. Zhang N.S., Joseph P.F. A nonlinear finite element eigenanalysis of singular stress fields in bimaterial wedges for plane strain. Int. J. Fract. 1998,94:299-319.
188. Sze K.Y., Wang H.T., Fan H, A finite element approach for computing edge singularities in piezoelectric materials. Int. J. Solids Structures 2001; 38: 9233-9252.
189．陈梦成，梁平英，平学成．各向异性结合材料缺口端部反平面奇异性问题研究．华东交通大学学报．2004，21(2)：6-9．
190．陈梦成，姜羡，朱剑军．复合压电材料尖劈端部奇异性问题有限元分析．华东交通大学学报，2004，21(1)：1-7．
191．平学成，谢基龙，陈梦成．一种复合材料尖劈和接头端部安全性评估的新模型，中国安全科学学报，2004，14(2)：105-108．
192．平学成，陈梦成，谢基龙．各向异性材料尖劈端部奇性场问题研究．力学与实践．2004，26(5)：36-40．
193．平学成，陈梦成，谢基龙，李强．各向异性两相材料界面端部的奇性应力指数．华东交通大学学报．2005，22(1)：7-10．
194．平学成，陈梦成，谢基龙，李强．各向异性复合材料尖劈和接头端部奇性场研究，北方交通大学学报，2004，28(1)：78-83．
195．平学成，陈梦成，谢基龙．三维各向异性复合材料尖劈和接头的奇性应力指数．应用力学学报，2004，21(3)：27-32．
196．平学成，谢基龙，陈梦成．各向异性两相材料尖劈奇性场的非协调元分析．力学学报．2005，37(1)：24-31
197. Ping X.C., Xie J.L., Chen M.C. Singular stress states in tips of anisotropic muti-material wedges under antiplane shear. ICHMM, June 21-26, 2004. Chongqing, China, Chongqing University Press.
198．平学成，陈梦成，谢基龙．复合材料尖劈和接头端部奇性场的反平面问题研究．固体力学学报．2005，26(2)：193-198．
199. Munz D. & Yang Y.Y. Stress Singularities at the Interface in Bonded Dissimilar Materials Under Mechanical and Thermal Loading. J. Appl. Mech. 1992, 59:857-861.
200. Tan M.A., Meguid S.A. Analysis of bimaterial wedges using a new singular finite element. Int. J. Fract. 1997, 88: 371-391.
201．杨晓翔．压电材料及界面断裂力学分析的某些问题．西安交通大学博士学位论文，1998．
202．傅向荣，龙驭球．解析试函数法分析平面切口问题．工程力学，2003，20(4)：33-38．
203. Ping X.C., Chert M.C., Xie J.L. Finite Element Analyses of Multi-Material Wedges and Junctions with Singular Antiplane Stress Fields, EPMESC X, Aug. 21-23, 2006, Sanya, Hainan, China, Tsinghua University Press & Springer-Verlag.
204. Inoue T. and Koguchi H. Influence of the intermediate material on the order of stress singularity in three-phase bonded structure. Int. J. Solids Sturctures 1996,33(3): 399-417.
205．Chen D．H．and NISITANI H．Stress Fields near the Comer of Jointed Dissimilar Materials．日本机械学会論文集(A编)，1991，57：366-372．
206．Chen D．H．and NISITANI H．Singular Stress Fields near a Comer Of Jointed Dissimilar Materials underAntiplane Loads．日本机械学会論文集(A编)，1991，57：2499-2503．
207．Chen D．H，and NISITANI H．Analysis of Singular Stress Fields around a Comer Tip of Inclusion．日本机械学会論文集(A编)，1991，57：2504-2508．
208．Chen D．H．and NISITANI H．Singular Stress Fields neaF a Corner of Jointed Dissimilar Materials．日本机械学会論文集(A编)，1991，57：2509-2515．
209. Chen D.H. Analysis of singular stress field around the inclusion comer tip. Eng. Fract. Mech. 1994, 49: 533-546.
210．王清，野田尚昭，上村仁誉．正方形夹杂角部的奇异应力场．山东工业大学学报，2000，30(1)：1-5．
211. Noda N.A., Takase Y., Chen M.C. Generalized stress intensity factors in the interaction between two fibers in matrix. Int. J. Fract. 2000, 103: 19-39.
212. Chen M.C., Ping X.C. Finite Element Analyses of Singular Inplane Stress Fields Around the Inclusion Comer Tip. EPMESC X, Aug. 21-23, 2006, Sanya, Hainan, China, Tsinghua University Press & Springer-Verlag.
213．平学成，陈梦成，谢基龙，李强．夹杂尖端奇异性场的新型杂交元分析．北京交通大学学报，2007，31(1)．(录用待发表)
214. Gradin P. A. A fracture criterion for edge bonded bimaterial bodies, J. Composite Materials, 1982, 448-456.
215. Groth H.L. A method to predict fracture in an adhesively bonded joint. Int. J. Adhesion and Adhesives 1985, 5: 19-22.
216. Groth H.L. Stress singuarities and fracture at interface comers in bonded joints. Int. J. Adhesion and Adhesives 1988, 5: 107-113.
217. Yamamoto T. Internal stress anisotropics induced by electric field in lanthanum modified PbTiO_3 ceramics. Ferroelectrics 1983, 50: 273-278.
218. Pisarenko G.G. Anisotropy of fracture toughness of piezoelectric ceramic. J. Am. Ceram. Soc. 1985, 68(5): 259-265.
219. Merhta K., Virkar A. V. Fracture mechanics in ferroelectric-ferroelastic lead Zirconate titamate (Zr: Ti=0.54: 0.46) ceramics. J. Am. Ceram. Soc. 1990, 73(3): 567-574.
220. Calderon-Moreno J.M. Fracture toughness anisotropy of PZT. Mater. Sci. Eng. 1997, a234: 1062-1066.
221. Park E.T. Anisotropic microhardness in single-crystal and polycrystalline BaTiO_3. J. Mater. Sci. 1998, 33: 669-673.
222. Sun C.T., Park S. Determination of fracture toughness of piezoelectrics under the influence of electric field using Vickers indentation. Proc. SPIE-Inter. Soc. Opt. Eng., 1995: 2441: 213-222.
223. Park S.B. and Sun C.T. Fracture criteria for piezoelectric ceramics. J. Am. Ceram. Soc. 1995, 78(6): 1475-1480.
224. Wang H., Singh R.N. Crack propagation in piezoelectric ceramics: effects of applied electric field. J. Appl. Phys. 1997, 81(11): 7471-7479.
225. Makiino H, Kamiya N. Effects of dc electric field on Mechanical properties of piezoelectric ceramics. Jpn. J. Appl. Phys., 1994, 33: 5323-5327.
226. Fu R., Zhang T.Y. Effects of an applied electric field on the modulus of rupture of poled lead zirconate titanate ceramics. J. Am. Ceram. Soc. 1998, 81(4): 1056-1060.
227. dos Santos e Lucatoa S.L., Bahrb H.A., Phamb V.B., Lupascua D.C., Balkeb H., ROdela J., Bahrc U. Electrically driven cracks in piezoelectric ceramics: experiments and fracture mechanics analysis. J. Mech. Phys. Solids, 2002, 50: 2333-2353.
228. Sob A.K., Lee K.L., Fang D.N. On the effects of an electric field on the fracture toughness of poled piezoelectric ceramics. Mater. Sci. Eng. 2003, A360:306-314.
229. Fang D.N., Zhang Z.K., Soh A.K., Lee K.L. Fracture criteria of piezoelectric ceramics with defects. Mech. Mater. 2004,36:917-928
230. Fang D.N., Liu Z.W., Xie H.M. Study on fracture behavior of ferroelectric ceramics under combined electromechanical loading by using a moir e interferometry technique. Acta Mechanica Sinica, 2004, 20(3): 263-269.
231．方岱宁，刘战伟，谢惠民，李尚，戴福隆，邴歧大。力电耦合载荷作用下铁电陶瓷破坏行为的 云纹干涉方法研究．试验力学，2003，18(2)：156-160．
232. Jelitto H., Kegler H., Schneider G.A., Balke H. Fracture behavior of poled piezoelectric PZT under mechanical and electrical loads. J. Eur. Ceram, Soc. 2005, 25: 749 757.
233. Heyer V., Schneider G.A. Balke H., Drescher B.H.A. A fracture criterion for conducting Cracks in homogeneously poled piezoelectric PZT-PICl51 ceramics. Acta Mater. 1998, 46(18): 6615-6622.
234. Pohanka R.C. Effect of the phase transformation on the fracture behavior of BaTiO_3. J. Am. Ceram. Soc. 1978:61 (1,2): 72-75.
235. Chung H., Shin B. and Kim H.G. Grain-size dependence of electrically induced microcracking in ferroelectric ceramics. J. Am. Ceram. Soc. 1989, 72(3): 327-329.
236. Kim S. and Kim D. Effect of grain size and poling on the fracture mode of lead zirconate titanate ceramics. J. Am. Ceram. Soe. 1990, 73(1): 161-163.
237. Zhang Z. and Raj R. Influence of grain on the ferroelectric toughening and piezoelectric behavior of lead Zirconate Titanzte, J. Am. Ceram. Soc. 1995: 78(12): 3363-3368.
238. Shin B. and Kim H.G. Partial discharge, microcracking and breakdown in BaTiO_3 ceramics. Ferroelectrics 1988, 77: 161-166.
239. Courant R. Variational method for solutions of problems of equilibrium and vibrations. Bull. Am. Math. Soc. 1943, 49: 1-23.
240. Turner M.J., Clough R.W., Martin H.C. and Topp L.C. Stiffness and deflection analysis of complex structures. J. Aero. Sci. 1956, 23: 805-823.
241. Clough R.W. The finite element method in plane stress analysis. Proc. 2nd ASCE Conference on Electronic Computation. Sept. 1960, Pittsburgh, RA., 345-378.
242．王勖成．有限单元法．北京：清华大学出版社，2003．
243. Washizu K. Variational methods in elasticity and plasticity, third ed. Pergamon Press, Oxford, 1982.
244. Pian T.H.H. Sumihara K. Rational approach for assumed stress finite elements. Int. J. Num. Meth. Eng. 1984, 20: 1685-1695.
245. Pian T.H.H., Wu C.C. A rational approach for choosing stress terms for hybrid finite element formulations. Int. J. Num, Meth. Eng. 1988, 26:2331-2343.
246. Wu C.C., Cheung Y.K. On optimization approaches of hybrid stress elements. FE in Analysis and Design 1995, 21: 111-128.
247. Sze K.Y. Efficient formulation of robust hybrid elements using orthogonal stress/strain interpolants and admissible matrix formulation. Int. J. Num. Meth. Eng. 1992, 35: 1-20.
248. EerNisse E.R Variational method for electroelastic vibration analysis. IEEE Trans. J. Sonics Ultrason. 1983, 14: 59-67.
249. Henshell R.D., Shaw K.D. Crack tip finite element are unnecessary. Int. J. Num. Meth. Eng. 1975, 9: 495-507.
250. Barsoum R.S. On the use of isoparametric finite elements in linear fracture mechanics. Int. J. Num. Meth. Eng. 1976, 10: 25-37.
251. Tracey D.M., Cook T. S. Analysis of power type singularities using finite element. Int. J. Num. Meth. Eng. 1989, 28: 2569-2581.
252. Lira W.K., Kim S.C., Further study to obtain a variable power singularity using quadratic isoparametric elements. Eng. Fract. Mech. 1994, 47: 223-228.
253．柳兆涛，刘一华，王炯华．平面V形切口双应力强度因子的光弹性实验研究．合肥工业大学学报，2002，25(2)：59-62．
254．谢惠民，赵兵，戴福隆，岸本哲，张维．纳米云纹法实验研究．光学技术，2000，26(3)：193-195．
255．亢一澜，陆桦，贾有权，邱宇，垂直界面裂纹断裂力学问题的实验研究．力学学报，1997，29(2)：242-247．
256. Park S.B., Park S,S., Carman G.P. Linear and nonlinear behaviour of piezoelectric materials SPIE, 1996, 2715: 366-375.
2. httpA://peizo.blogdriver.com/peizo/
3. http://peizo.blogdriver.com/peizo/874088.html
4. http://www.cutech.edu.cn/ShowArticle.asp?ArticleID=14703
5. http://www.stcsm.gov.cn/learning/lesson/course/detail.asp?id=91&lessonnum=2&coursenum=31
6. http://www.sic.ac.cn/kpz2005/shuzi/16.htm
7．叶志明．各向异性材料与混凝土材料断裂力学引论．北京：中国铁道出版社，2000．
8．黄克智，杨卫，郑泉水．第19届国际理论与应用力学大会(ICTAM-19)．1996年8月25～31日，日本京都．
9. Williams M.L. Stress Singularities resulting from various boundary conditions in angular corners of plate extension. J.Appl.Mech. 1952,24:109-114.
10. Williams M.L. The stresses around a fault or crack in dissimilar media. Bull. Seis. Soc. Am. 1959,49: 199-204.
11. Rice J.R. Elastic fracture mechancs concepts for interface cracks. J. Appl. Mech. 1988, 55: 98-103.
12. Erdogan F. Stress distribution in a nonhomogeneous elastic plane with cracks. J. Appl. Mech. 1963, 30: 232-236.
13. Erdogan F. Stress distribution in bonded dissimilar materials with cracks. J. Appl. Mech. 1965, 32: 403-410.
14．许金泉，姜菊生．界面端附近裂纹的应力强度因子．上海力学，1998，19(3)：221-227．
15．朱哲明，谢和平．裂纹表面承受均布载荷时的应力强度因子及裂纹张开位移的边界配位法．计算力学学报，1997，14(2)：182-188．
16. Bogy D.B. On the plane elastostatic problem of a loaded crack terminating at a material interface. J. Appl. Mech. 1971,38:911-918.
17. Bogy D.B. The plane elastostatic problem for a symmetrically loaded crack in a strip composite. Int. J. Eng. Sci. 1973,11:985-996.
18. Hoenig A. Near tip behavior of a crack in a plane anisotropic elastic body. Eng. Fract. Mech. 1992,16:393-403.
19. Bassani J.L. and Erdogan 17. Sress intensity, factors in bonded half planes containing inclined cracks and subjected to antiplane shear loading. Int. J. Fract. 1979,15:145-158.
20. Ma C.C. and Hour B.L., Antiplane problems in composite anisotropic materials with an inclined crack terminating at bi-material interface. Int. J. Solids Structures 1990,26:1387～1400.
21. Lekhnitskii S.G. Theory of Elasticity of an Anisotropic Body. Holden-Day, San Francisco. 1963.
22. Eshelby J.D., Read W.T. and Shodkley W. Anisotropic elasticity with application to dislocation theory. Acta Metall. 1953,1:251-259.
23. Stroh A.N. Dislocations and cracks in anisotropic elasticity, Phil. Mag. 1958, 7:625-646.
24. Stroh A.N. Steady state problems in anisotropic elasticity. J. Math. Phys. 1963,41:77-103.
25. Sih G.C., Pairs RC., Irwin G.R. On cracks in rectilinearly anisotropic bodies. J. Fract. Mech. 1965, (1): 189-203.
26. Ting T.C.T. and Hoang RH. Singularities at the tip of a crack normal to the interface of an anisotropic layered composite. Int. J. Solids Structures 1984,20:439-454.
27. Ting T.C.T. Explicit solution and invariance of the singularities at an interface crack in anisotropic composites. Int. J. Solids Structures 1986,22:965-983.
28. Yang S. and Yuan F.G. Kinked crack in anisotropic bodies. Int. J. Solids Structures 2000,37(45): 6635-6682.
29. Suo Z. Singularities, interfaces and cracks in dissimilar anisotropic media. Proc. R. Soc. A 427,1990:331-358.
30．高存法，樊蔚勋．各向异性平面内双边半无限裂纹问题的基本解，工程力学，1997，14(2)：128-133．
31. Nazarov S.A. A crack at the interface of anisotropic bodies. Singularities of the elastic fields and a criterion for fracture when the crack surfaces are in contact. J. Appl. Math. Mech. 2004, 69(3): 473-483.
32. Yang L.H., Li Z.H. The interaction of mode I crack with multi-inclusions in a three-phase model. Int. J. Fract. 2004, 127: 193-200.
33．黄克智，余寿文，杨卫．裂纹尖端奇异场—清华大学工程力学系校友研究成果综述，清华大学工程力学与工程热物理学术会议论文集，清华大学出版社，1988：15-25．
34. Gao Y.C., Hwang KC. Exponential Stress and Strain Singularity at Tip of Mode I Growing Crack in Power Hardening Plastic Material. Theor. Appl. Fract. Mech. 1996,25:103-111.
35．唐立强，黄克智．理想正交各向异性材料平面应力条件下Ⅰ型静止裂纹尖端场，力学学报，1991，23：448-457．
36．刘一华，许金泉．纤维端部的界面裂纹分析．力学学报，1999，31(3)：385-390．
37. Kondo T., Sekine H. Anti-plane shear of kinked interface crack in bonded dissimilar anisotropic solid. Theor. Appl. Fract. Mech., 1993,18(3): 273-282.
38. Yetmez M., Gecit M.R. Finite strip with a central crack under tension. Int. J. Eng. Sci. 2005, 43:472-493.
39. Mahajan R., Erdogan F., Kilic B. and Madenci E. Cracking of an orthotropic substrate reinforced by an orthotropic plate. Int. J. Solids Structures 2003, 40: 6389-6415.
40. Tang R.J. and Erdogan F. A clamped rectangular plate containing a crack. Theor. Appl. Fract. Mech. 1985,4:328-339.
41．陈梦成，高闯，汤任基．三维界面裂纹的奇性应力场和应力强度因子分析，固体力学学报，1999，20：8-15．
42．陶伟明，郭乙木，曹志远，嵇醒．三维裂纹分析的主值型面力边界积分方程，固体力学学报，2001，22(3)：256-262．
43．张明，姚振汉，杜庆华，楼志文．双材料界面裂纹应力强度因子的边界元分析，应用力学学报，1999，16(1)：21-26．
44. Cho S.B., Lee K.R., Choy Y.S. and Yuuki R. Determination of stress intensity factors and boundary element analysis for interface cracks in dissimilar anisotropic materials. Eng. Fract. Mech. I992, 43(4): 603-614.
45．Yuuki R．，Xu J．Q．Stress intensity Factors for the interface crack between dissimilar orthotropicmaterials．日本机械学会論文集(A编)，1991，57(7)：1542-1549．
46．Xu J．Q，，Yuuki R．Stress intensity Factors for the interface crack between dissimilar orthotropic materials(The case where principal axes are not aligned with the interface，)．日本机械学会論文集(A编)，1994，60(9)：1943-1950．
47. Zhou W., Lira K.M., Lee K.H., Tay A.O.O. A new variable-order singular boundary element for calculating stress intensity factors in three-dimensional elasticity problems. Int. J. Solids Structures 2005, 42: 159-185.
48. Chyuan S.W., Lin J.H., Chen J.T. and Liu D.C. Dual boundary element analysis for fatigue behavior of missile structures. J. Chinese Institute of Engineering 2000, 23(3): 339-348.
49. Pan E. A general Boundary Element Analysis of 2-D Linear Elastic Fracture Mechanics. Int. J. Fract. 1997, 81(1): 41-59.
50. garsoum R.S. Cracks in anisotropic materials—an iterative solution of the eigenvalue. Int. J. Fract. 1986, 32: 59-67.
51. Bazant Z.R and Estenssord L.F. Surfaces singularity and crack propagation. Int. J. Solids Structures 1979,15:405-426.
52. Somaratna N. and Ting T.C.T. Three-dimensional stress singularities in anisotropic material and composites. Int. J. Eng. Sci. 1986,24:1115-1134.
53. Yamada Y. and Okumura H. Finite element analysis of stress and strain singularity eigenstate in homogenous media or composite materials. In Hybrid and Mixed Finite Methods (Edited by Atluri S.N., Gallagher R.H. and Zienkiewicz O.C.) 1983:325-343.
54. Pageau S.S., Joseph P.F. and Biggers S.B.Jr. Finite element analysis of anisotropic materials with singular inplane stress fields. Int. J. Solids Structures 1995,32 571-591.
55. Sze K.Y., Wang H.T. A simple finite element formulation for computing stress singularities at bimaterial interfaces. FE in Analysis and Design 2000, 35: 97-118.
56．平学成，陈梦成．楔形体尖端近似场的非协调有限元特征法．华东交通大学学报．2001，18(4)：6-11．
57. Kaddouri K., Belhouari M., Bouiadjra B.B. and Serier B. Finite element analysis of crack perendicular to bi-material interface. Comp. Mater. Sci. 2006, 35: 53-60.
58．卢炎麟，张淑佳，俞志清．焊趾表面裂纹应力强度因子计算的基本模式法，应用力学学报，1998(4)：73-81．
59. Seweryn A. Modeling of singular stress fields using finite element method. Int. J. Solids Structures 2002,39:4787-4804.
60. Chang J.H., Wu W.H. FE calculation of contour integrals in plane anisotropic elastic media with crack surface tractions. FE in Analysis and Design 2001, 37(9): 673-696.
61. Saxce Gery De, Kang C.H. Application of the hybrid mongrel displacement finite method to the computation of stress intensity factors in anisotropic material. Eng. Fract. Mech. 1992, 41(1): 71-83.
62. Song C.M., Wolf J.P. Semi-analytical representation of stress singularities as occurring in cracks in anisotropic multi-materials with the scaled boundary finite-element method. Computers & Structures 2002, 80: 183-197.
63. Lee J. and Gao H. A hybrid finite element analysis of interface cracks. Int. J. Num. Meth. Eng. 1995,38:2465-2482.
64. Pageau S.S. and Biggers S.B.Jr. Enrichment of finite elements with numerical solutions for singular stress fields. Int. J. Num. Meth. Eng. 1997,40:2693-2713
65. Chen M.C., Sze K.Y. and Wang H.T. Analysis of singular stresses in bounded bimaterial wedges by computed eigen solutions and bybrid element method. Commun. Num. Meth. Eng. 2001, 17: 495-507.
66. Chen M.C., Ping X.C. A novel hybrid element analysis for piezoelectric-parent material wedges. Comput. Mech. In press.
67. Barut A., Guven I. and Madenci E. Analysis of singular stress fields at junctions of multiple dissimilar materials under mechanical and thermal loading. Int. J. Solids Structures 2001: 38:9077-9109.
68. Tong P., Plan T.H.H., Lasry S.J. A hybrid element approach to crack problems in plane elasticity. International J. Num. Meth. Eng.1973, 7: 297-380.
69. Chert E,P. Finite element analysis of bimaterial interface crack. Theor. Appl. Fract. Mech. 1985, 3: 257-262.
70. Patton VZ. Fracture mechanics of piezoelectric materials. Acta Astronautica, 1976, 3:671-683
71．王自强，韩学礼．压电材料中心裂纹问题．固体力学学报，1999：20：95-103．
72. Barnett D.M., Lothe J. Dislocations and line charges in anisotropic piezoelectric insulators. Phys. Status. Solidi. 1975, 1367: 105-111.
73. Deeg W.E The analysis of dislocation, crack, and inclusion problems in piezoelectric solids (Ph.D Dissertation). Stanford Unversity, 1980, USA.
74. Pak Y.E. Linear electro-elastic fracture mechanics of piezoelectric materials. Int. J. Fract. 1992, 54: 79-100.
75. Kuo C.M., Barnett D.M. Stress singularities of interfacial cracks in bonded piezoelectric half-spaces. In: Wu J. J., Ting T.C.T., Barnett D.M. (Eds.), Modern Theory of Anisotropic Elasticity and Applications. SIAM, Philadelphia, 1991, pp. 33-50.
76. Suo Z., Kuo C. M., Barnett D.M. and Willis J.R. Fracture mechanics for piezoelectric ceramics. J.Mech. Phys. Solids 1992, 40: 739-765.
77. Sosa H.A. On the fracture mechanics of piezoelectric solids. Int. J. Solids Structures, 1992, 21: 2613-2622,
78. Sosa H.A., Khutoryansky, N., New developments concerning piezoelectric materials with defects. Int. J. Solids Structures 1996, 33: 3399-3414.
79. Park S. B. and Sun C. T. Effect of electric field on fracture of piezoelectric ceramics. Int. J. Fract. 1995, 70: 203-216.
80. Herrmann K., Loboda V. Fracture-mechanical assessment of electrically permeable interface cracks in piezoelectric bimaterial by consideration of various contact zone models. Arch. Appl. Mech. 2000, 70: 127-143.
81. Beom H.B., Atluri SN. Near-tip fields and intensity factors for interfacial cracks in dissimilar anisotropic piezoelectric media. Int, J, Fract. 1996, 75: 163-183.
82. Wang T.C,, Han X.L. Fracture mechanics of piezoelectric materials. Int. J. Fract., 1999, 98: 15-35.
83. Gao C.F., Wang M.Z. Green function of an interface crack between two dissimilar piezoelectric media. Int. J. Eng. Sci. 2000, 37: 4969-4986.
84. Hao T.H. Multiple collinear cracks in a piezoelectric material. Int. J. Solids Structures 2001: 38: 9201-9208.
85. Zhang T.Y., Zhao M., Tong R Fracture of piezoelectric ceramics. Adv. Appl. Mech., 2002, 38: 147-289.
86. Zhang T.Y., Gao C.F, Fracture behaviors of piezoelectric materials. Theor. Appl. Fract. Mech. 2004, 41: 339-379.
87．王旭，沈亚鹏．压电和多层材料中电极—陶瓷界面裂纹和椭圆夹杂．力学学报，2002，34(6)：881-894．
88. Ou Z.C. Singularity parameters for interface crack in transversely isotropic piezoelectric bimaterials, Int. J. Fract. 2003, 119: 41-46.
89. Ou Z.C., Chen Y.H. Near-tip stress fields and intensity factors for an interface crack in metal/piezoelectric bimaterials. Int. J. Eng. Sci. 2004,42: 1407-1438.
90. Ou Z.C., Chen Y.H. Interface crack-tip generalized stress field and stress intensity factors in transversely isotropic piezoelectric birnaterials, Mech. Res. Comm. 31(2004) 421-428.
91. Chue C.H., Weng S.M. Fracture analysis of piezoelectric materials with an arbitrarily oriented crack using energy density theory. Computers & Structures, 2005(83): 1251-1265.
92. Wang Z. Penny-shaped crack in transversely isotropic piezoelectric materials. Acta Mechanica Sinica, 1994, 10(1): 49-60.
93. Gao, H., Zhang, T.-Y., Tong, P. Local and global energy release rates for an electrically yielded crack in a piezoelectric ceramic. J. Mech. Phys. Solids 1997, 45: 491-510.
94. Fulton, C.C., Gao, H. Electrical non-linearity in fracture of piezoelectric ceramics. ASME Appl. Mech. Review 1997, 50: 56-63.
95. Dunn M.L. The effects of crack face boundary conditions on the fracture mechanics of piezoelectric solids. Eng. Fract. Mech. 1999, 48: 25-39.
96. McMeeking R.M. Towards a fracture mechanics for brittle piezoelectric and dielectric materials. Int. J. Fract. 2001, 108: 25-41.
97. Fulton, C.C., Gao, H. Effect of local polarization switching on piezoelectric fracture, J. Mech. Phys. Solids 2001, 49: 927-952.
98. Sosa H. Plane problems in piezoelectric media with defects. Int. J. Solids Structures 1991, 28: 491-505.
99. Gao C.F.和Fan W.X. Exact solutions for the plane problem in piezoelectric materials with an elliptic or a crack. Int. J. Solids Structures 1999, 36: 2527-2540.
100．杨卫。力电失效学。清华大学出版社，2001．
101. Hao T.H., Shen Z.Y. A new boundary conditions on the fracture mechanics of piezoelectric solids. Eng. Fract. Mech. 1994; 47: 793-802.
102．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1993．
103. Wang B.L. Noda N. Mixed mode crack initiation in piezoelectric ceramic strip. Theor. Appl. Fract. Mech. 2000, 34: 35-47.
104. Wang B.L., Sun Y.G. Intensity factors for some common piezoelectric fracture mechanics specimens with conducting cracks or electrodes. Int. J. Eng. Sci. 2004, 42:1129-1153.
105. Govorukha V., Kamlah M., Munz D. The interface crack problem for a piezoelectric semi-infinite strip under concentrated electromechanical loading. Eng. Fract. Mech. 2004, 71:1853-1871.
106. Wang B. Three-dimensional analysis of an ellipsoidal inclusion in a piezoelectric material. Int. J. Solids. Structures 1992, 29(3): 293-308.
107，刘金喜，王彪，杜善义。二维各向异性压电介质机电耦合场的基本解。应用数学和力学，1997，18(10)：885-891．
108．丁皓江，王国庆，陈伟球．压电材料平面问题的基本解．中国科学(E)，1997，27(3)：224-228
109. Huang Z.Y., Kuang Z.B. A first order perturbation analysis of a non-ideal crack. in a piezoelectric material. Int. J. Solids Structures 2001, 38: 726t-7281.
110. Huang Z.Y., Kuang Z.B. A mixed electric boundary value problem for a two- dimensional piezoelectric crack. Int. J. Solids Structures 2003, 40:1433-1453.
111. Chen M.C. Application of finite-part integrals to three-dimensional fracture problems for piezoelectric media. Part Ⅰ: Hypersingular integral equation and theoretical analysis. Int. J. Fract. 2003, 121: 133-148.
112. Chen M.C. Application of finite-part integrals to three-dimensional fracture problems for piezoelectric media. Part Ⅱ: Numerical analysis. Int. J. Fract. 2003, 121: 149-161.
113. Shindo Y., Watanabe K., Narita F. Electroelastic analysis of a piezoelectric ceramic strip with a central crack. J. Eng. Sci. 2000, 38,1-19.
114. Narita F., Shindo Y. Mode I crack growth rate for yield strip model of a marrow piezoelectric ceramic body. Theor. Appl. Fract. Mech. 2001,36: 73-85.
115. Garcia F., Saez Andres, Dominguez J. Anisotropic and piezoelectric materials fracture mechanics analysis by BEM. Computers & Structures 2005, 83: 804-820.
116. Khutoryansky N., Sosa H., Zu W. Approximate Green's function and a boundary element method for electro-elastic analyses of active materials. Computers & Structures 1998, 66: 289-299.
117. Groh U., Kuna M. Efficient boundary element analysis of cracks in 2D piezoelectric structures. Int. J. Solids Structures 2005, 42:2399-2416.
118. Shen N., Sze K.Y. Multi-region Trefftz boundary element method for fracture analysis in plane piezoelectricity. Comput. Mech. 2006, 37:381-393.
119．王国庆，刘正兴．压电材料平面问题应力强度因子的边界元计算．上海交通大学学报，1998，32(12)：74-76．
120. Wu C.C., Sze K.Y., Huang Y.Q. Numerical solutions on fracture of piezoelectric materials by hybrid element. Int. J. Solids Structures 2001, 38: 4315-4329.
121．刘鸣，吴长春．压电介质机电耦合裂尖场的杂交元数值分析，科学通报，1999，44：603-608
122. Sze K.Y., Wang H.T., Fan H. A finite element approach for computing edge singularities in piezoelectric materials. Int. J. Solids Structures 2001; 38:9233-9252.
123．陈梦成，平学成，朱剑军．压电材料中切口／接头端部平面电弹场奇异性有限元分析．固体力学学报，2005，26(2)：157-162．
124．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)：1-7．
125. Kuna M. Finite element analyses of crack problems in piezoelectric structures. Comp. Mater. Sci. 1998, 13: 67-80.
126. Abendroth M., Groh U., Kuna M., Ricoeur A. Finite element-computation of the electromechanical J-Integral for 2-D and 3-D crack analysis. Int. J. Fract. 2002:114: 359-378.
127. Shang F.L., Kuna M., Abendroth M. Finite element analyses of three-dimensional crack problems in piezoelectric structures. Eng. Fract. Mech. 2003, 70: 143-160.
128. Kumar S., Singh R.N. Influence of applied electric field and mechanical boundary condition on the stress distribution at the crack tip in piezoelectric materials. Mater. Sci. Eng. 1997, A231: 1-9.
129. Kumar S., Singh R.N. Effect of the mechanical boundary condition at the crack surfaces on the stress distribution at the crack tip in piezoelectric materials. Mater. Sci. Eng. 1997, A252: 64-77.
130. Wippler K., Ricoeur A., Kuna M. Towards the computation of electrically permeable cracks in piezoelectrics. Eng. Fract. Mech. 2004, 71: 2567-2586.
131. Scherzer M., Kuna M Asymptotic analysis of piezoelectric composite,materials. Proc. 1st Caesariurn "Smart Materials", Bonn 1999. In: Hoffmann K.H (eds) Smart Materials. Springer, Berlin Heidelberg New York, 2001, pp137-148.
132. Scherzer M., Kuna M. Combined analytical and numerical solution of 2D interface comer configurations between dissimilar piezoelectric materials. Int. J. Fract. 2004,127: 61-99.
133．李尧臣．压电材料平面裂纹问题的强度因子和能量释放率。同济大学学报，2002，30(4)：437-445．
134．周勇，王鑫伟．压电材料平面裂纹尖端场的杂交应力有限元分析．力学学报，2004，36(3)：354-358．
135. Govorukha V., Kamlah M. Asymptotic fields in the finite element analysis of electrically permeable interfacial cracks in piezoelectric bimaterials. Arch. Appl. Mech. 74, 92-101.
136. McMeeking R.M. Crack tip energy release rate for a piezoelectric compact tension specimen. Eng. Fract. Mech. 1999,64: 217-224.
137. Gruebner O., Kamlah K., Munz D. Finite element analysis of cracks in piezoelectric materials taking into account the permittivity of the crack medium. Eng. Fract. Mech. 2003, 70: 1399-1413.
138. Gruebner O., Kamlah K. Investigation of an interface crack with a contact zone in a piezoelectric bimaterial under limited permeable electric boundary conditions．Acta Mechanica 2005, 178: 85-99.
139. Pak Y.E. Crack extension force in piezoelectric material. J. Appl. Mech. 1990, 57: 863-869.
140. Pak Y.E, Goloubeva E. Electroelastic properties of cracked piezoelectric materials under longitudinal shear. Mech. Mater. 1996, 24: 287-303.
141. Zhang T.Y., Tong P. Fracture mechancs for a mode Ⅲ crack in a piezoelectric material. Int. J. Solids Structures 1996, 33: 343-359,
142．侯密山，梅甫良．不同压电材料反平面应变状态的电渗透型界面裂纹．科学通报，1998，43(2)：216-221．
143. Chen B.J., Liew K.M., Xiao Z.M. Green's functions for anti-plane problems in piezoelectric media with a finite crack. Int. J. Solids Structures 2004, 41:5285-5300.
144. Chen Z.T., Yu S.W., Karihaloo B.L. Anti-plane shear problem for a crack between two dissimilar piezoelectric materials. Int. J. Fract. 1997,86: 9-14.
145. Shindo Y., Narita F., Tanaka K. Electroelestic intensification near anti-plane shear crack in orthotropic piezoelectric ceramic strip. Theor. Appl. Fract. Mech. 1996, 254: 65-71.
146. Narita F., Shindo Y. Layered piezoelectric medium with interface crack under anti-plane shear. Theor. Appl. Fract. Mach. 1998, 30: 119-126.
147. Narita F. and Shindo Y. The interface crack problem for bonded piezoelectric and orthotropic layers under antiplane shear loading. Int. J. Fract. 1999,98:87-101
148. Sob 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. Eur. J. Mech. 2000,19:961-977.
149. Huang Z.Y., Kuang Z.B. A mixed electric boundary value problem for an anti-plane piezoelectric crack. Acta Mechanica Solida Sinica, 2003, 16(2): 110-115.
150. Li X.F., Tang G.J. Antiplane interface crack between two bonded dissimilar piezoelectric layers. Eur. J. Mech. A/Solids 2003, 22:231-242.
151. Li X.F. Electroelastic analysis of an interface anti-plane shear crack in a layered piezoelectric plate. Int. J. Eng. Sci. 2003, 41:1405-1422.
152. Li X.F. Electroelastic analysis of a cracked piezoelectric strip sandwiched between two elastic dielectrics. Eur. J. Mech. A/Solids 2005, 24:69 80.
153. Kwon S.M., Lee K.Y. Analysis of stress and electric fields in a rectangular piezoelectric body with a center crack under anti-plane shear loading. Int. J. Solids Structures 2000, 4859-4869.
154. Wang B.L, Han J.C., Du S.Y. Electroelastic fracture dynamics for multilayered piezoelectric materials under dynamics anti-plane shearing. Int. J. Solids Sturctures 2000, 38:5219-5231.
155. Chue C.H., Liu T.J.C.. Electro-elastic analysis of a bimaterial piezoelectric wedge with an interface crack under antiplane concentrated forces and inplane surface charges. Int. J. Solids Structures 2004, 41: 4179-4196.
156. Qin Q.H. Solving anti-plane problems of piezoelectric materials by the Trefftz finite element approach. Comput. Mech. 2003, 31:461-468.
157．陈梦成，平学成，朱剑军．压电—异材结合材料尖劈端部反平面奇异性问题．力学学报(已投稿)．
158. Williams M.L., Stress Singularities resulting from various boundary conditions in angular comers of plate extension. J. Appl. Mech. 1952,19:526-535.
159. Chen D.H., Nisitani H. Singular stress field near the corner of jointed dissimilar materials. J. Appl. Mech. 1993, 6(3): 607-613.
160. Dini D., Hills D.A. Asymptotic characterization of nearly-sharp notch root stress fields. Int. J. Fract. 2004, 130: 651-666.
161. Bogy D.B. Two edge-bonded elastic wedges of different materials and wedge angles under surface tractions. J. Appl. Mech. 1971,38:377-386.
162. Theocaris RS. The order of singularity at a multi-wedge comer of a composite plate. Int. J. Eng. Sci. 1974,12:107-120.
163. Ma C.C. and Hour B.L. Analysis of dissimilar anisotropic wedges subjected to antiplane shear deformation. Int. J. Solids Structures 1989,11 :1295-1309.
164. Wang S.S. and Choi I., Boundary-layer effects in composite laminates. Part 1: Free-edge stress singularities. J. Appl. Mech. 1982,49:541-548.
165. Zwiers R.I., Ting T.C. and Spilker R.L. On the logarithmic singularity of free-edge stress in laminated composites under uniform extension. J. Appl. Mech. 1982, 49:561-569.
166. Delale F. Stress singularities in bonded anisotropic materials. Int. J. Solids Structures, 1984, 20:31-40.
167. Xu X.L., Rajapakse R.K.N.D. Analytical solution for an arbitrarily oriented void/crack and fracture of piezoceramics. Acta. Mater. 1999, 47(6); 1735-1747
168. Xu X.L., Rajapakse R.K.N.D. On singularities in composite piezoelectric wedges and junctions. Int. J. Solids Structures 2000; 37: 3253-3275.
169. Chue C.H., Chen C.D. Decoupled formulation of piezoelectric elasticity under generalized plane deformation and its application to wedge problems. Int. J. Solids Structures 2002, 39:3131-3158.
170. Chen H.P. Stress singularities in anisotropic multi-material wedges and junctions. Int. J. Solids Structures 1998,11:1057-1073.
171. Huang T.F. and Chen W.H. On the free-edge stress singularity of general composite laminates under uniform axial strain. Int. J. Solids Structures 1994, 31:3139-3151.
172. MantiC V., Paris F. and CaNas J. Stress singularities in 2D orthotropic comers. Int. J. Fract. 1997, 83(1): 69-90.
173. Wu K.C. and Chen C.T. Stress analysis of anisotropic elastic V-notched bodies. Int. J. Solids Structures 1996,33(17): 2403-2416.
174. Zhang J.Q., Zhang B.N., Fan J.H. A coupled electromechanical analysis of a piezoelectric layer bonded to an elastic substrate: Part Ⅰ, development of governing equations. Int. J. Solids Structures 2003, 40:6781-6797.
175. Zhang B.N., Zhang J.Q., Fan J.H. A coupled electromechanical analysis of a piezoelectric layer bonded to an elastic substrate: Part Ⅱ, numerical solution and applications. Int. J. Solids Structures 2003, 40:6799-6812.
176．许金泉，王效贵，刘一华．振荡应力奇异性及其强度系数的数值分析方法．力学季刊，2000，21(2)：230-236．
177．唐亮，许金泉．直角结合异材界面端应力强度系数的经验公式．力学季刊，2005，25(1)：96-101．
178. Carpenter W.C. A caculation procedure for determining fracture mechanics parameters at a comer. The Ninth Canadian Congress of Applied Mechanics, Saskatchewan, Saskatoon, Canada, May 3 -June 3, 1983.
179. Sukumar N. and Kumosa M. Application of the finite element iterative method to cracks and sharp notches in orthotropic media. Int. J. Fract. 1992,58:177-192.
180. Gu L. and Belytschko T. A numerical study of stress singularities in a two-material wedge. Int. J. Solids Structures 1994,31:865-889.
181. Raju I.S. and Crews J.H.Jr. Interlaminar stress singularities at a straight free-edge in composite laminates. Computers & Structures 1981,4:21-28.
182. Artel J., Beeker W. Coupled and uncoupled analyses of piezoelectric free-edge effect in laminated plates. Composite Structures 2005, 69: 329-335.
183. Yeh J.R. and Tadjbakhsh I.G. Stress singularity in composite laminates by finite element method. J. Appl. Mech. 1982,49:541-548.
184．王效贵．界面端奇异行为的数值分析．浙江大学博士学位论文，2002．
185. Pageau S.S. and Biggers S.B.Jr. Finite element evaluation of free-edge singular stress fields in anisotropic materials. Int. J. Num. Meth. Eng. 1995,38:2225-2239.
186. Pageau S.S. and Biggers S.B.Jr. A finite element approach to three-dimensional singular stress states in anisotropic multi-material wedges and junctions. Int. J. Solids Structures 1996,33:33-47.
187. Zhang N.S., Joseph P.F. A nonlinear finite element eigenanalysis of singular stress fields in bimaterial wedges for plane strain. Int. J. Fract. 1998,94:299-319.
188. Sze K.Y., Wang H.T., Fan H, A finite element approach for computing edge singularities in piezoelectric materials. Int. J. Solids Structures 2001; 38: 9233-9252.
189．陈梦成，梁平英，平学成．各向异性结合材料缺口端部反平面奇异性问题研究．华东交通大学学报．2004，21(2)：6-9．
190．陈梦成，姜羡，朱剑军．复合压电材料尖劈端部奇异性问题有限元分析．华东交通大学学报，2004，21(1)：1-7．
191．平学成，谢基龙，陈梦成．一种复合材料尖劈和接头端部安全性评估的新模型，中国安全科学学报，2004，14(2)：105-108．
192．平学成，陈梦成，谢基龙．各向异性材料尖劈端部奇性场问题研究．力学与实践．2004，26(5)：36-40．
193．平学成，陈梦成，谢基龙，李强．各向异性两相材料界面端部的奇性应力指数．华东交通大学学报．2005，22(1)：7-10．
194．平学成，陈梦成，谢基龙，李强．各向异性复合材料尖劈和接头端部奇性场研究，北方交通大学学报，2004，28(1)：78-83．
195．平学成，陈梦成，谢基龙．三维各向异性复合材料尖劈和接头的奇性应力指数．应用力学学报，2004，21(3)：27-32．
196．平学成，谢基龙，陈梦成．各向异性两相材料尖劈奇性场的非协调元分析．力学学报．2005，37(1)：24-31
197. Ping X.C., Xie J.L., Chen M.C. Singular stress states in tips of anisotropic muti-material wedges under antiplane shear. ICHMM, June 21-26, 2004. Chongqing, China, Chongqing University Press.
198．平学成，陈梦成，谢基龙．复合材料尖劈和接头端部奇性场的反平面问题研究．固体力学学报．2005，26(2)：193-198．
199. Munz D. & Yang Y.Y. Stress Singularities at the Interface in Bonded Dissimilar Materials Under Mechanical and Thermal Loading. J. Appl. Mech. 1992, 59:857-861.
200. Tan M.A., Meguid S.A. Analysis of bimaterial wedges using a new singular finite element. Int. J. Fract. 1997, 88: 371-391.
201．杨晓翔．压电材料及界面断裂力学分析的某些问题．西安交通大学博士学位论文，1998．
202．傅向荣，龙驭球．解析试函数法分析平面切口问题．工程力学，2003，20(4)：33-38．
203. Ping X.C., Chert M.C., Xie J.L. Finite Element Analyses of Multi-Material Wedges and Junctions with Singular Antiplane Stress Fields, EPMESC X, Aug. 21-23, 2006, Sanya, Hainan, China, Tsinghua University Press & Springer-Verlag.
204. Inoue T. and Koguchi H. Influence of the intermediate material on the order of stress singularity in three-phase bonded structure. Int. J. Solids Sturctures 1996,33(3): 399-417.
205．Chen D．H．and NISITANI H．Stress Fields near the Comer of Jointed Dissimilar Materials．日本机械学会論文集(A编)，1991，57：366-372．
206．Chen D．H．and NISITANI H．Singular Stress Fields near a Comer Of Jointed Dissimilar Materials underAntiplane Loads．日本机械学会論文集(A编)，1991，57：2499-2503．
207．Chen D．H，and NISITANI H．Analysis of Singular Stress Fields around a Comer Tip of Inclusion．日本机械学会論文集(A编)，1991，57：2504-2508．
208．Chen D．H．and NISITANI H．Singular Stress Fields neaF a Corner of Jointed Dissimilar Materials．日本机械学会論文集(A编)，1991，57：2509-2515．
209. Chen D.H. Analysis of singular stress field around the inclusion comer tip. Eng. Fract. Mech. 1994, 49: 533-546.
210．王清，野田尚昭，上村仁誉．正方形夹杂角部的奇异应力场．山东工业大学学报，2000，30(1)：1-5．
211. Noda N.A., Takase Y., Chen M.C. Generalized stress intensity factors in the interaction between two fibers in matrix. Int. J. Fract. 2000, 103: 19-39.
212. Chen M.C., Ping X.C. Finite Element Analyses of Singular Inplane Stress Fields Around the Inclusion Comer Tip. EPMESC X, Aug. 21-23, 2006, Sanya, Hainan, China, Tsinghua University Press & Springer-Verlag.
213．平学成，陈梦成，谢基龙，李强．夹杂尖端奇异性场的新型杂交元分析．北京交通大学学报，2007，31(1)．(录用待发表)
214. Gradin P. A. A fracture criterion for edge bonded bimaterial bodies, J. Composite Materials, 1982, 448-456.
215. Groth H.L. A method to predict fracture in an adhesively bonded joint. Int. J. Adhesion and Adhesives 1985, 5: 19-22.
216. Groth H.L. Stress singuarities and fracture at interface comers in bonded joints. Int. J. Adhesion and Adhesives 1988, 5: 107-113.
217. Yamamoto T. Internal stress anisotropics induced by electric field in lanthanum modified PbTiO_3 ceramics. Ferroelectrics 1983, 50: 273-278.
218. Pisarenko G.G. Anisotropy of fracture toughness of piezoelectric ceramic. J. Am. Ceram. Soc. 1985, 68(5): 259-265.
219. Merhta K., Virkar A. V. Fracture mechanics in ferroelectric-ferroelastic lead Zirconate titamate (Zr: Ti=0.54: 0.46) ceramics. J. Am. Ceram. Soc. 1990, 73(3): 567-574.
220. Calderon-Moreno J.M. Fracture toughness anisotropy of PZT. Mater. Sci. Eng. 1997, a234: 1062-1066.
221. Park E.T. Anisotropic microhardness in single-crystal and polycrystalline BaTiO_3. J. Mater. Sci. 1998, 33: 669-673.
222. Sun C.T., Park S. Determination of fracture toughness of piezoelectrics under the influence of electric field using Vickers indentation. Proc. SPIE-Inter. Soc. Opt. Eng., 1995: 2441: 213-222.
223. Park S.B. and Sun C.T. Fracture criteria for piezoelectric ceramics. J. Am. Ceram. Soc. 1995, 78(6): 1475-1480.
224. Wang H., Singh R.N. Crack propagation in piezoelectric ceramics: effects of applied electric field. J. Appl. Phys. 1997, 81(11): 7471-7479.
225. Makiino H, Kamiya N. Effects of dc electric field on Mechanical properties of piezoelectric ceramics. Jpn. J. Appl. Phys., 1994, 33: 5323-5327.
226. Fu R., Zhang T.Y. Effects of an applied electric field on the modulus of rupture of poled lead zirconate titanate ceramics. J. Am. Ceram. Soc. 1998, 81(4): 1056-1060.
227. dos Santos e Lucatoa S.L., Bahrb H.A., Phamb V.B., Lupascua D.C., Balkeb H., ROdela J., Bahrc U. Electrically driven cracks in piezoelectric ceramics: experiments and fracture mechanics analysis. J. Mech. Phys. Solids, 2002, 50: 2333-2353.
228. Sob A.K., Lee K.L., Fang D.N. On the effects of an electric field on the fracture toughness of poled piezoelectric ceramics. Mater. Sci. Eng. 2003, A360:306-314.
229. Fang D.N., Zhang Z.K., Soh A.K., Lee K.L. Fracture criteria of piezoelectric ceramics with defects. Mech. Mater. 2004,36:917-928
230. Fang D.N., Liu Z.W., Xie H.M. Study on fracture behavior of ferroelectric ceramics under combined electromechanical loading by using a moir e interferometry technique. Acta Mechanica Sinica, 2004, 20(3): 263-269.
231．方岱宁，刘战伟，谢惠民，李尚，戴福隆，邴歧大。力电耦合载荷作用下铁电陶瓷破坏行为的 云纹干涉方法研究．试验力学，2003，18(2)：156-160．
232. Jelitto H., Kegler H., Schneider G.A., Balke H. Fracture behavior of poled piezoelectric PZT under mechanical and electrical loads. J. Eur. Ceram, Soc. 2005, 25: 749 757.
233. Heyer V., Schneider G.A. Balke H., Drescher B.H.A. A fracture criterion for conducting Cracks in homogeneously poled piezoelectric PZT-PICl51 ceramics. Acta Mater. 1998, 46(18): 6615-6622.
234. Pohanka R.C. Effect of the phase transformation on the fracture behavior of BaTiO_3. J. Am. Ceram. Soc. 1978:61 (1,2): 72-75.
235. Chung H., Shin B. and Kim H.G. Grain-size dependence of electrically induced microcracking in ferroelectric ceramics. J. Am. Ceram. Soc. 1989, 72(3): 327-329.
236. Kim S. and Kim D. Effect of grain size and poling on the fracture mode of lead zirconate titanate ceramics. J. Am. Ceram. Soe. 1990, 73(1): 161-163.
237. Zhang Z. and Raj R. Influence of grain on the ferroelectric toughening and piezoelectric behavior of lead Zirconate Titanzte, J. Am. Ceram. Soc. 1995: 78(12): 3363-3368.
238. Shin B. and Kim H.G. Partial discharge, microcracking and breakdown in BaTiO_3 ceramics. Ferroelectrics 1988, 77: 161-166.
239. Courant R. Variational method for solutions of problems of equilibrium and vibrations. Bull. Am. Math. Soc. 1943, 49: 1-23.
240. Turner M.J., Clough R.W., Martin H.C. and Topp L.C. Stiffness and deflection analysis of complex structures. J. Aero. Sci. 1956, 23: 805-823.
241. Clough R.W. The finite element method in plane stress analysis. Proc. 2nd ASCE Conference on Electronic Computation. Sept. 1960, Pittsburgh, RA., 345-378.
242．王勖成．有限单元法．北京：清华大学出版社，2003．
243. Washizu K. Variational methods in elasticity and plasticity, third ed. Pergamon Press, Oxford, 1982.
244. Pian T.H.H. Sumihara K. Rational approach for assumed stress finite elements. Int. J. Num. Meth. Eng. 1984, 20: 1685-1695.
245. Pian T.H.H., Wu C.C. A rational approach for choosing stress terms for hybrid finite element formulations. Int. J. Num, Meth. Eng. 1988, 26:2331-2343.
246. Wu C.C., Cheung Y.K. On optimization approaches of hybrid stress elements. FE in Analysis and Design 1995, 21: 111-128.
247. Sze K.Y. Efficient formulation of robust hybrid elements using orthogonal stress/strain interpolants and admissible matrix formulation. Int. J. Num. Meth. Eng. 1992, 35: 1-20.
248. EerNisse E.R Variational method for electroelastic vibration analysis. IEEE Trans. J. Sonics Ultrason. 1983, 14: 59-67.
249. Henshell R.D., Shaw K.D. Crack tip finite element are unnecessary. Int. J. Num. Meth. Eng. 1975, 9: 495-507.
250. Barsoum R.S. On the use of isoparametric finite elements in linear fracture mechanics. Int. J. Num. Meth. Eng. 1976, 10: 25-37.
251. Tracey D.M., Cook T. S. Analysis of power type singularities using finite element. Int. J. Num. Meth. Eng. 1989, 28: 2569-2581.
252. Lira W.K., Kim S.C., Further study to obtain a variable power singularity using quadratic isoparametric elements. Eng. Fract. Mech. 1994, 47: 223-228.
253．柳兆涛，刘一华，王炯华．平面V形切口双应力强度因子的光弹性实验研究．合肥工业大学学报，2002，25(2)：59-62．
254．谢惠民，赵兵，戴福隆，岸本哲，张维．纳米云纹法实验研究．光学技术，2000，26(3)：193-195．
255．亢一澜，陆桦，贾有权，邱宇，垂直界面裂纹断裂力学问题的实验研究．力学学报，1997，29(2)：242-247．
256. Park S.B., Park S,S., Carman G.P. Linear and nonlinear behaviour of piezoelectric materials SPIE, 1996, 2715: 366-375.
© 2004-2018 中国地质图书馆版权所有 京ICP备05064691号 京公网安备11010802017129号 地址：北京市海淀区学院路29号 邮编：100083 电话：办公室：(+86 10)66554848；文献借阅、咨询服务、科技查新：66554700 |