本文旨在研究雙層壓電材料界面完全結合而裂紋位在界面附近之問題，界面附近之裂紋則考慮為界面下含單一裂紋或界面上下皆含裂紋之問題，利用廣義Stroh理論及廣義差排基本解，將分析之問題化為以廣義差排密度為未知函數之耦合奇異積分方程式。推演之耦合奇異積分方程式，其核函數為複數之形式，且適用於任何異向性壓電材料，推得之耦合奇異積分方程式則以數值方法求解，而彈電耦合效應則以廣義應力強度因子表示。

對於雙層壓電材料界面下含單一裂紋之問題，文中進一步針對橫向等向性壓電材料，將複數形式之核函數化為實數形式，由實數核函數，可以瞭解彈電間的交互作用關係。而當彈電耦合效應消失時，對應於彈性效應的核函數則與文獻結果相同，由實數形式之核函數也可以瞭解上層材料勁度與介電常數参數變化時，上層材料参數對下層材料行為之影響，文中討論了上層材料勁度趨於無窮大或零與介電常數趨於無窮大或零等情況下之實數核函數，且可考慮上層材料消失，雙層壓電材料成為半平面四種不同邊界條件之問題。對於半平面廣義邊界條件的問題，文中另外推演了對應之實數核函數，而此廣義邊界條件之實數核函數包含了由雙層材料退化為半平面問題之核函數。由界面下含單一裂紋問題之研究結果顯示，有關彈電效應之間的交互行為，半平面壓電材料在不同邊界條件下，不論裂紋為平行或是垂直，在裂紋面受到不同均佈外力作用時，彈電耦合之應力強度因子或電位移強度因子可以直接視為由彈電分離材料受純彈性機械力或電位移作用所求得。然而，在上層材料存在時，即考慮雙層壓電材料界面下含單一裂紋之問題(或各含單一裂紋之問題)，其彈電耦合之廣義應力強度因子則須考慮壓電常數效應的影響，才能得到正確的廣義應力強度因子。

對於雙層壓電材料界面上下皆含裂紋之問題，本文所討論之複數形式之核函數適用於上下裂紋角度可為任意之問題，對此任意角度裂紋問題雖亦可將複數形式之核函數化為實數核函數，但文中僅列出兩裂紋皆為垂直共線且當上下層材料皆為彈電分離時所對應之實數核函數，至於兩裂紋為其他角度之情況，則僅藉由複數形式之核函數及利用數值方法探討界面上下裂紋交互作用之關係及彈電效應之間的交互行為。研究成果顯示，當兩裂紋皆為水平時，且下層裂紋面分別受均佈單位壓曳力作用所產生之 、剪曳力作用所產生之 與電位移作用所產生之 皆小於雙層壓電材料界面下含單一水平裂紋之結果。然而，當兩裂紋皆為垂直時，下層裂紋之上尖端的廣義應力強度因子則大於雙層壓電材料界面下含單一垂直裂紋之上尖端結果。另外，當下層裂紋維持水平位置，而上層裂紋以不同角度變化時，所得結果顯示裂紋角度變化對於上層裂紋尖端之廣義應力強度因子影響較大，而對於下層裂紋尖端之廣義應力強度因子則影響較小。

In this paper, the problems of a cracked anisotropic piezoelectric bimaterial with perfectly bonded interface are considered. Either one crack embedded in the lower-half material or two cracks embedded in either side of the interface is considered. Based on the generalized Stroh formalism, the problems are formulated in terms of systems of singular integral equations. The kernel functions developed are in complex form for general anisotropic piezoelectric materials. The systems of singular integral equations are solved numerically and the coupled electro- mechanical phenomena are presented by the generalized stress intensity factors.

For the problem of a crack embedded in the lower-half material, the kernel functions are further developed in real forms for transversely isotropic piezoelectric materials. The obtained coupled mechanical and electric real kernel functions may be reduced to those kernel functions for purely elastic problems when the electric effects disappear. The real kernel functions for stiffness or dielectric constants of lower material taking infinite or zero values are further investigated. Moreover, by considering the vanishing of the upper material, the problem of a half-plane cracked material with four different types of boundary conditions is also considered. Crack embedded in a half-plane anisotropic piezoelectric material with generalized boundary conditions is also treated where the kernel functions for different boundary conditions in a half-plane anisotropic piezoelectric material are included as special cases in our results. It is found that the evaluations of the mechanical stress intensity factors (electric displacement intensity factor) under mechanical loadings (electric displacement loading) for coupled mechanical and electric problems may be evaluated directly by considering the corresponding decoupled elastic (electric) problem irrespective of what electric boundary is applied on the boundary. However, investigations of a sub-interface piezoelectric bimaterial composed by PZT-4/PZT-6B show that, purely elastic bimaterial analysis or purely electric bimaterial analysis is inadequate for the determination of the generalized stress intensity factors, instead, for better accuracy of the generalized stress intensity factors to be achieved, both elastic and electric properties of the bimaterial’s constants should be taken simultaneously into account.

The kernel functions corresponding to the problem for two cracks near the interface are developed in complex forms and are valid for arbitrary crack’s orientations. These kernel functions may be expressed in real forms, however, only results for elastic-electric decoupled problems with cracks perpendicular to the interface are exploited. In the studies when cracks are in arbitrary positions, kernel functions in complex forms are used. Analyses of the problem for two cracks near the interface show that when both two cracks are in horizontal position relative to the interface, the stress intensity factor under uniform pressure loading, the stress intensity factor under uniform shear loading, and the electric displacement intensity factor under uniform electric displacement loading are all smaller than those obtained for the same problem but with only one sub-interface crack near the interface, while, for the problem when two cracks are in vertical positions, it is found that the results of generalized stress intensity factors for the crack tip in the lower material nearest to the interface are larger than that of the problem of only one single vertical crack embedded in the lower material. Besides, as the orientation of the upper crack varies, it is found that the effects of the orientation of the upper crack has a profound influence on the generalized stress intensity factors for the crack tip in the upper material while the effects on the crack tip in lower material are not significant.

Barnett, D.M., Asaro, R.J., 1972. The fracture mechanics of slit-like cracks in anisotropic elastic media. Journal of the Mechanics and Physics of Solids 20, 353-366.
Beom, H.G., Jeong, K.M., Kim, Y.H., 2003. Intensity factors for subinterface cracks in dissimilar anisotropic piezoelectric media. Archive of Applied Mechanics 73 (3-4), 184-198.
Berlincourt, D.A., Curran, D.R., Jaffe, H., 1964. Piezoelectric and piezomagnetic materials and their function in transducers. Physical Acoustics I-A, W.P.(Ed.), Academic Press, New York.
Chadwick, P., Ting, T.C.T., 1987. On the structure and the Barnett-Lothe tensors. Quarterly of Applied Mathematics 45, 419-427.
Chen, B.J., Xiao, Z.M., and Liew, K.M., 2004. A line dislocation interacting with a semi-infinite crack in piezoelectric solid. International Journal of Engineering Science 42, 1-11.
Chen, D.H., 1997. Stress intensity factor for a crack normal to an interface between two orthotropic materials. International Journal of Fracture 88, 19-39.
Chung, M.Y., Ting, T.C.T., 1996. Piezoelectric solid with and elliptic inclusion or hole. International Journal of Solids and Structures 33, pp. 3343-3361.
Cook, T.S., Erdogan, F., 1972. Stresses in bonded materials with a crack perpendicular to the interface. International Journal of Engineering Science 10, 677-697.
Deeg, W.F., 1980. The analysis of dislocation, crack, and inclusion problems in piezoelectric solids. PhD thesis, Stanford University.
Erdogan, F., Biricikoglu, V., 1973. Two bonded half planes with a crack going through the interface. International Journal of Engineering Science 11, 745-766.
Eshelby, J.D., Read, W.T., Shockley, W., 1953. Anisotropic elasticity with application to dislocation theory. Acta Metall 1, 251-259.
Fan, H., Sze, K.Y., Yang, W., 1996. Two-dimension contact on a piezoelectric half-space. International Journal of Solids and Structures 33, 1305-1315.
Gao, C.F., Fan, W.X., 1998. Green’s functions for the plane problem in a half-infinite piezoelectric medium. Mechanics Research Communications 25, 69-74.
Gao, C.F., Fan, W. X., 1999. Exact solutions for the plane problem in piezoelectric materials with an elliptic or a crack. International Journal of Solids and Structures 36, 2527-2540.
Gao, C.F., Noda, N., 2004. Green’s functions of a half-infinite piezoelectric body:Exact solutions. Acta Mechanica. 172, 169-179.
Gao, C.F., Wang, M.Z., 1999. Periodical cracks in piezoelectric media. Mechanics Research Communications 26, 427-432.
Gerasoulis, A., 1982. The use of piecewise quadratic polynomials for the solution of singular integral equations of Cauchy type. Computational Mathematics with Applications 8, 15-22.
Gandhi, M.V., Thompson, B.S., 1992. Smart Materials and Structures. Chapman & Hall, London.
Hao, T.H., Shen Z.Y., 1994. A new electric boundary condition of electric fracture mechanics and its applications. Engineering Fracture Mechaincs 47, 793-802.
Hao, T.H., 2001. Multiple collinear cracks in a piezoelectric material. International of Solids and Structures 38, 9201-9208.
Hu, Y., Huang, Y., Chen, C., Zhong, Weifang, 2000. Green’s functions for a piezoelectric semi-infinite body with a fixed conductor surface. Mechanics Research Communications 27, 333-338.
Ikeda, T., 1990. Fundamentals of piezoelectricity. Oxford University Press, New York.
Jaffe, B., Cook, W.R., Jaffe, H., 1971. Piezoelectric Ceramics. Academic Press Limited, India.
Kraus, J.D., Fleisch, D.A., 1999. Electromagnetic with Application. Fifth edition, McGraw-Hill, New York.
Krenk, S., 1979. On the elastic constants of plane orthotropic elasticity. Journal of Composite Materials 13, 108-116.
Lekhnitskii, S.G., 1963. Theory of Elasticity of an Anisotropic Elastic Body, Holden-Day, San Francisco, California.
Li, X.F., Tang, G.J., 2003. Electroelastic analysis of an interface antiplane shear crack in a layered piezoelectric plate. International Journal of Engineering Science 41, 1405-1422.
Li, X.F., Lee, K.Y., 2004. Fracture analysis of cracked piezoelectric materials. International Journal of Solids and Structures 41, 4137-4161.
Liou, J.Y., Sung, J.C., 2007. On the generalized Barnett-Lothe tensors for monoclinic piezoelectric materials. International Journal of Solids and Structures 44, 5208-5221.
Liu, B., Fnag, D.N., Soh, A.K., Hwang, K.C., 2001. An approach for analysis of poled/depolarized piezoelectric materials with a crack. International Journal of Fracture 111, 395-407.
Liu, J., Wang, B., Du, S., 1997. Electro-elastic Green’s functions for a piezoelectric half-space and their application. Applied Mathematics and Mechanics 18, 1073-1043.
Lu, P., Williams, F.W., 1998. Green Functions of piezoelectric material with an elliptic hole or inclusion. International Journal of Solids and Structures 35, 651-664.
Muskhelishvili, N.I., 1953. Some Basic Problems of the Mathematical Theory of Elasticity, tansl. by J.R.M. Radok, Noordhoff, Gronigen.
Newnhan, R.E., 1993. Smart, very smart, and intelligent material. MSR Bulletin 18 (4), 24-26.
Nye, J.F., 1985. Physical Properties of Crystals. Clarendon Press, Oxford.
Ou, Z.C., Chen, Y.H., 2003. Explicit expressions of eigenvalues and eigenvectors for transversely isotropic piezoelectric materials. Acta Mechanics. 162, 213-219.
Ou, Z.C., Chen, Y.H, 2003. Discussion of the crack face electric boundary condition in piezoelectric fracture mechanics. International Journal of Fracture 123, 151-155.
Ou, Z.C., Chen, Y.H., 2004. Near-tip stress fields and intensity factors for an interface crack in metal/piezoelectric bimaterials. International Journal of Engineering Science 42, 1407-1438.
Pak, Y.E., 1990a. Crack extension force in a piezoelectric material. Transactions of the ASME, Journal of Applied Mechanics 57, 647-653.
Pak, Y.E., 1990b. Force on a piezoelectric screw dislocation, Transactions of the ASME, Journal of Applied Mechanics 57, 863-869.
Pak, Y.E., 1992. Linear electro-elastic fracture mechanics of piezoelectric materials. International Journal of Fracture 54, 79-100.
Pan, E., 2002. Mindlin’s problem for an anisotropic piezoelectric half-space with general boundary conditions. Proceedings of the Royal Society A458, 181-208.
Park, S.B., Sun, C.T., 1995. Effect of electric field on fracture of piezoelectric ceramics. International Journal of Solids and Structures 70, 203-216.
Parton, V.Z., 1976. Fracture mechanics of piezoelectric materials. Acta Astronaut. 3, 671-683.
Parton, V.Z., Kudryavtsev, B.A., 1988. Electromagnetoelasticity. Gordon and Breach Science Publishers.
Qin, Q.H., Zhang, X., 2000. Crack deflection at an interface between dissimilar piezoelectric materials. International Journal of Fracture 102, 355-370.
Rice, J.R., Sih, G.C., 1965. Plane problems of cracks in dissimilar media. Transactions of the ASME, Journal of Applied Mechanics 32, 418-423.
Ricketts, D., 1981. Model for a piezoelectric polymer flexural plate Hydrophone. Journal of the Acoust Society of America 70 (4), 929-935.
Shen, S., Nishioka, T., 2000. Fracture of piezoelectric materials: energy density criterion. Theoretical and Applied Fracture Mechanics 33, 57-65.
Shindo, Y., Watanabe, K., Narita, F., 2000. Electroelastic analysis of a piezoelectric ceramic strip with a central crack. International Journal of Engineering Science 38, 1-19.
Sosa, H., 1991. Plane problems in piezoelectric media with defects. International Journal of Solids and Structures 28, 491-505.
Sosa, H., 1992. On the fracture mechanics of piezoelectric solids. International Journal of Solids and Structures 29, 2613-2622.
Sosa, H., Khutoryansky, N., 1996. New developments concerning piezoelectric materials with defects. International Journal of Solids and Structures 33, 3399-3414.
Sosa, H.A., Castro, M.A., 1994, “On concentrated loads at the boundary of a piezoelectric half-plane,” Journal of the Mechanics and Physics of Solids 42, 1105-1122.
Stroh, A.N., 1958. Dislocations and cracks in anisotropic elasticity. Philosophical Magazine 7, 625-646.
Suo, Z., Kuo, C.M., Barnett, D.M., Willis, J.R., 1992. Fracture mechanics for piezoelectric ceramics. Journal of the Mechanics and Physics of Solids 40, 739-765.
Sung, J.C., Liou, J.Y., 1995a. Analysis of a crack embedded in a linear elastic half-plane solid. Transactions of the ASME, Journal of Applied Mechanics 62, 78-86.
Sung, J.C., Liou, J.Y., 1995b. Subinterface crack in dissimilar orthotropic materials. International Journal of Solids and Structures 32 (19), 2873-2889.
Sung, J. C., Liou, J.Y., 1995c. An Internal Crack in a Half-Plane Solid with Clamped Boundary. Computer Methods in Applied Mechanics and Engineering 121, 361-372.
Sung, J.C., Liou, J.Y., Lin, Y.Y., 1996. Some phenomena of cracks perpendicular to an interface between dissimilar orthotropic materials. Transactions of the ASME, Journal of Applied Mechanics 63, 190-203.
Tian, W.Y., Chau, K.T., 2003. Arbitrarily oriented crack near interface in piezoelectric bimaterials. International Journal of Solids and Structures 40 (8), 1943-1958.
Tian, W.Y., Chen, Y.H., 2000. Interaction between an interface crack and subinterface microcracks in metal/piezoelectric bimaterials. International Journal of Solids and Structures 37 (52), 7743-7757.
Tian, W.Y., Chen, Y.H., 2002. Interaction between subinterface cracks and interface in metal/piezoelectric ceramic bimaterials. Science in China Series E-Technological Sciences 45 (1), 10-18.
Tian, W.Y., Gabbert, U., 2005. Parallel crack near the interface of magnetoelectroelastic bimaterials. Computational Materials Scienece 32 (3-4), 562-567.
Ting, T.C.T., 1988. Some identities and the structure of Ni in the Stroh formalism of anisotropic elasticity. Quarterly of Applied Mathematics 46, 109-120.
Ting, T.C.T., 1996. Anisotropic Elasticity: Theory and Application. Oxford University Press.
Ueda, S., 2007. Electromechanical impact of an impermeable parallel crack in a functionally graded piezoelectric strip. European Journal of Mechanics A/Solids 26, 123-136.
Wang, B.L., Mai, Y.W., 2004. Impermeable crack and permeable crack assumptions, which one is more realistic. Transactions of the ASME, Journal of Applied Mechanics 71, 575-578.
Wang, X.D., Jiang, L.Y., 2002. Fracture behaviour of cracks in piezoelectric media with electromechanically coupled boundary conditions. Proceedings of the Royal Society A458, 2545-2560.
Zhao, L.G., Chen, Y.H., 1997. Further investigation of subinterface cracks. Archive of Applied Mechanics 67, 393-406.
Zhang, T.Y., Qian, C.F., Tong, P., 1998. Linear electro-elastic analysis of a cavity or a crack in a piezoelectric material. International Journal of Solids and Structures 35, 2121-2149.
Zhang, T.Y., Zhao, M., Tong, P., 2002. Fracture of piezoelectric ceramics. Advances in Applied Mechanics 38, 148-298.
Zhang, T.Y., Wang T., Zhao M., 2003. Failure behavior and failure criterion of conductive cracks (deep notches) in thermally depoled PZT-4 ceramics. Acta Materialia 51, 4881-4895.
Zhang, T.Y., Gao, C.F., 2004. Fracture behaviors of piezoelectric materials. Theoretical and Applied Fracture Mechanics 41, 339-379.
Zuo, J.Z., Sih, G.C., 2000. Energy density theory formulation and interpretation of cracking behavior for piezoelectric ceramics. Theoretical and Applied Fracture Mechanics 34, 17-33.
劉鈞耀，1994，雙層異向性材料界面附近裂紋之分析，國立成功大學土木工程研究所博士論文。