本文根據三維電彈性力學理論，應用狀態空間法，解析四邊為簡支承雙曲率功能性彈性與壓電材料殼受電場和機械場外載重作用下之結構行為。首先，將三維電彈性力學之22條基本方程式重新組合，消去曲面應力場，並以位移場、橫向應力、電位及電位移等8個場量為主要變數，應用直接消去法化簡成8條常微分方程式。過程中將各場變量無因次化，避免數值計算過程產生病態問題。基於傳遞矩陣近似解法之應用，將殼切割成NL層，透過層與層，由下而上經傳遞矩陣相互連乘之應用，轉換常微分方程組為聯立線性代數方程組。應用上、下表面邊界條件，諸如橫向載重、橫向電位差或橫向電位能已知條件，可進而求得層間之各項場變量。本文解析解將與文獻中其他三維理論解析解綜合比較，藉以評估本解析方法之精確性與收斂速度。

Based on the three-dimensional (3D) piezoelectricity, we present the exact solutions of simply-supported, doubly curved functionally graded (FG) elastic and piezoelectric shells using a state space approach. A set of the dimensionless coordinates and field variables are introduced in the present formulation to prevent from the ill-conditioned problem in the relevant computation. By means of direct elimination, we reduce the twenty-two basic differential equations to a set of eight state variable equations (or state equations) with variable coefficients of the thickness coordinate. By means of the successive approximation method, we artificially divide the shell into a NL-layered shell and the thickness of each layer is small. That leads to a reasonable manipulation to reduce the state equations of a thickness-varying system for each individual layer to those of a thickness-invariant system. Imposition of the boundary conditions on the lateral surfaces of the shell, the state variables through the thickness coordinate can then be determined using the method of propagator matrix. The direct and converse effects on the static behavior of doubly curved, multilayered and FG piezoelectric shells are studied. The accuracy and the rate of convergence of the present state space approach are evaluated.

Bufler, H. (1971): Theory of elasticity of a multilayered medium. J. Elasticity, vol. 1, pp. 125_143.
Chen, W.Q.; Ding, H.J.; Xu, R.Q. (2001): Three-dimensional static analysis of multi-layered piezoelectric hollow spheres via the state space method. Int. J. Solids Struct., vol. 38, pp. 4921_4936.
Dumir, P.C.; Dube, G.P.; Kapuria, S. (1997): Exact piezoelectric solution of simply-supported orthotropic circular cylindrical panel in cylindrical bending. Int. J. Solids Struct., vol. 34, pp. 685_702.
Fan, J.R.; Ye, J.Q. (1990): Exact solutions for axisymmetric vibration of laminated circular plate. J. Engrg. Mech. ASCE, vol. 116, pp. 920_927.
Fan, J.R.; Zhang, J. (1992): Analytical solutions for thick doubly curved laminated shells. J. Engrg. Mech. ASCE, vol. 118, pp. 1338_1356.
Gopinathan, S.V.; Varadan, V.V.; Varadan, V.K. (2000): A review and critique of theories for piezoelectric laminates. Smart Mater. Struct., vol. 9, pp. 24_48.
Heyliger, P. (1997): Exact solutions for simply supported laminated piezoelectric plates. J. Appl. Mech., vol. 64, pp. 299_306.
Heyliger, P.; Brooks, S. (1996): Exact solutions for laminated piezoelectric plates in cylindrical bending. J. Appl. Mech., vol. 63, pp. 903_910.
Kapuria, S.; Achary, G.G.S. (2005): Exact 3D piezoelasticity solution of hybrid cross-ply plates with damping under harmonic electro-mechanical loads. J. Sound Vibr., vol. 282, pp. 617_634.
Lee, J.S.; Jiang, L.Z. (1996): Exact electroelastic analysis of piezoelectric laminae via state space approach. Int. J. Solids Struct., vol. 13, pp. 977_990.
Pan, E. (2003): Exact solution for functionally graded anisotropic elastic composite laminates. J. Compos. Mater., vol. 37, pp. 1903_1920.
Pan, E.; Han, F. (2005): Exact solution for functionally graded and layered magneto-electro-elastic plates. Int. J. Eng. Sci., vol. 43, pp. 321_339.
Ramirez, F.; Heyliger, P.R.; Pan, E. (2006): Static analysis of functionally graded elastic anisotropic plates using a discrete layer approach. Compos. Part B, vol. 37, pp. 10_20.
Ren, J.G. (1989): Analysis of simply supported laminated circular cylindrical shell roofs. Compos. Struct., vol. 11, pp. 277_292.
Rogers, T.G.; Watson, P.; Spencer, A.J.M. (1992): An exact three-dimensional solution for normal loading of inhomogeneous and laminated anisotropic elastic plates of moderate thickness. Proceedings of Royal Society, vol. A437, pp. 199_213.
Rogers, T.G.; Watson, P.; Spencer, A.J.M. (1995): Exact three-dimensional elasticity solutions for bending of thick inhomogeneous and laminated strips under normal pressure. Int. J. Solids Struct., vol. 32, pp. 1659_1673.
Saravanos, D.A.; Heyliger, P.R. (1999): Mechanics and computational models for laminated piezoelectric beams, plates, and shells. Appl. Mech. Rev., vol. 52, pp. 305_319.
Soldatos, K.P.; Hadjigeorgiou, V.P. (1990): Three-dimensional solution of the free vibration problem of homogeneous isotropic cylindrical shells and panels. J. Sound and Vibr., vol. 137, pp. 369_384.
Srinivas, S.; Rao, A.K. (1970): Bending, vibration and buckling of simply supported thick orthotropic rectangular plates and laminates. Int. J. Solids Struct., vol. 6, pp. 1463_1481.
Vlasov, V.Z. (1957): The method of initial functions in problems of theory of thick plates and shells. in Proc. Ninth Int. Congress Appl. Mech., Brussels, pp. 321_330.
Wu, C.P.; Lo, J.Y. (2006): An asymptotic theory for the dynamic response of laminated piezoelectric shells. Acta Mech., vol. 183, pp. 177_208.
Wu, C.P.; Lo, J.Y.; Chao, J.K. (2005): A three-dimensional asymptotic theory of laminated piezoelectric shells. CMC: Comput. Mater. Continua, vol. 2, pp. 119_137.
Wu, C.P.; Syu, Y.S. (2007): Exact solutions of functionally graded piezoelectric shells under cylindrical bending.. Int. J. Solids Struct., doi:10.1016/j.ijsolstr.2007.02.037, in press.
Wu, C.P.; Syu, Y.S.; Lo, J.Y. (2007): Three-dimensional solutions of multilayered piezoelectric hollow cylinders by an asymptotic approach. Int. J. Mech. Sci., vol. 49, pp. 669_689.
Wu, C.P.; Tarn, J.Q.; Chi, S.M. (1996a): Three-dimensional analysis of doubly curved laminated shells. J. Engrg. Mech. ASCE, vol. 122, pp. 391_401.
Wu, C.P.; Tarn, J.Q.; Chi, S.M. (1996b): An asymptotic theory for dynamic response of doubly curved laminated shells. Int. J. Solids Struct., vol. 26, pp. 3813_3841.
Ye, J. (2003): Laminated composite plates and shells-3D modeling. Springer, London, UK.
Zhong, Z.; Shang, E.T. (2003): Three-dimensional exact analysis of a simply supported functionally gradient piezoelectric plate. Int. J. Solids Struct., vol. 40, pp. 5335_5352.