本文第一部份針對沿z軸極化之半平面壓電陶瓷，表面有極薄之實圓電極，探討其軸對稱之電彈場問題。根據線彈性壓電力學，利用漢可轉換可推導出對偶積分方程式，並求得閉合解。第二部份探討有限厚度之壓電陶瓷，表面有極薄之實圓電極, 探之軸對稱之電彈場問題。經複雜的數學推導，將求得之對偶積分方程式化簡為第二類弗雷德霍姆積分方程式。針對第二類雷德霍姆積分方程式，討論當中所遇到困難和未來可能工作。

The first part of the thesis derives analytically the axisymmetric electro-elastic field of a piezoelectric ceramic half plane with circular surface electrode. The piezoelectric material is polarized along the z-axis. The dual integral equations are derived by using Hnakel Transform and solved by transforming functions into the Fredholm integral equation of the second kind. The second part deals with the finite thickness disk. Due to mathematical difficulty encountered, specific results are obtained in the thesis. A further studies for solving the dual integral equations are needed.

[1] Aleksandrov, V.M. and Chebakov, M.I., “On a method of solving dual integral equations”, Journal of Applied Mathematics and Mechanics, Vol. 37, No. 6, pp. 1087-1097, 1973.
[2] Aleksandrov, V.M., “On a method of reducing dual integral equations and dual series equations to infinite algebraic systems”, Journal of Applied Mathematics and Mechanics, Vol. 39, No. 2, pp. 324-332, 1975.
[3] Boresi, A.P. and Chong, K.P., Elasticity in Engineering Mechanics. Wiley, New York, 2000.
[4] Cooke, J.C., “Triple integral equations”, Quarterly Journal of Mechanics and Applied Mathematics, Vol. 16, pt. 2, pp. 193-203, 1963.
[5] Ikeda, T., Fundamentals of Piezoelectricity, Oxford University Press, New York, 2000.
[6] Jain, D.L. and Kanwal, R.P., “An integral equation method for solving mixed boundary value problems”, Journal on Applied Mathematics, Vol. 20, No. 4, pp. 642-658, 1971.
[7] Kokunov, V.A., Kudryavtsev, B.A. and Senik, N.A., “The plane problem of electroelasticity for a piezoelectric layer with a periodic system of electrodes at the surfaces”, PMM, Vol. 49, No. 3, pp. 374-379, 1985.
[8] Kokunov, V.A. and Parton, V.Z., “Axisymmetric problem of electroelasticity for a piezoelectric layer with annular electrodes”, Translated from problemy Prochnosti, No. 5, pp. 84-88 ,1988.
[9] Lebedev, N.N. and Skalskaia, I.P., “Dual integral equations related to the Kontorovich-Lebedev transform”, Journal of Applied Mathematics and Mechanics, Vol. 38, No. 6, pp. 1090-1097, 1974.
[10] Li, X.F. and Lee, K.Y., “Electric and elastic behaviors of a piezoelectric ceramic with a charged surface electrode”, Smart Materials and Structures, Vol. 13, pp. 424-432, 2004.
[11] Muskhelishvili, N.I., Some Basic Problems of the Mathematical Theory of Elasticity, Noordhoff, The Netherlands, 1963.
[12] Maleknejad, K., Aghazadeh, N. and Rabbani, M., “Numerical solution of second kind Fredholm integral equations system by using a Taylor-series expansion method”, Applied Mathematics and Computation, Vol. 175, pp. 1229-1234, 2006.
[13] Malits, P., “Indentation of an incompressible inhomogeneous layer by a rigid circular indenter”, Quarterly Journal of Mechanics and Applied Mathematics, Vol. 59, No. 3, pp. 343-358, 2006.
[14] Nobel, B., “The solution of Bessel function dual integral equations by a multiplying-factor method”, Proceedings of the Cambridge Philolgical Society, Vol. 59, pp. 351-362, 1963.
[15] Prudnikov, A. P., Brychkov, Yu. A. and Marichev, O. I., Integrals and Series. Volume 2: Special Functions, Gordon and Breach Science Publisher, New York, 1986.
[16] Sneddon, I.N., Mixed Boundary Value Problems in Potential Theory, North-Holland, Amsterdam, 1966.
[17] Sneddon, I.N., The use of Integral Transforms, McGraw-Hill, New York, 1972.
[18] Shindo, Y., Narita, F. and Sosa, H., “Electrodelastic analysis of piezoelectric ceramics with surface electrodes”, International Journal of Engineering Scienc., Vol. 36, pp. 1001-1009, 1998.
[19] Wang, B.L., “Circular surface electrode on a piezoelectric layer”, Journal of Applied Physics, Vol. 95, No. 8, pp. 4267-4274, 2004.
[20] Wang, W. M., “A new mechanical algorithm for solving the second kind of Fredholm integral equation”, Applied Mathematics and Computation, Vol. 172, pp. 946-962, 2006.
[21] Yang, F. Q.., “Solution of a dual integral equation for crack and indentation problems”, Theoretical and Applied Fracture Mechanics, Vol. 26, pp. 211-217, 1997.
[22] Yoshida, M., Narita, F. and Shindo, Y., “Electroelastic field concentration by circular electrodes in piezoelectric ceramics”, Smart Materials and Structures, Vol. 12, pp. 972-978, 2003.