摘要

基於Reissner混合變分原理(Reissner’s mixed variational theorem，RMVT)，我們推衍有限圓柱層殼法(finite cylindrical layer methods，FCLMs) 對具開放與封閉迴路表面條件及簡支承邊界條件之雙層功能性壓電材料(functionally graded piezoelectric material
，FGPM)的薄膜基層中空圓柱殼進行擬三維(quasi-three-dimensional，3D) 壓電力學分析。本研究之FGPM薄膜基層圓柱殼是由厚且軟的FGPM基層及薄且硬的均質壓電材料(homogeneous piezoelectric material，HPM) 薄膜層組合而成。FGPM層的材料性質則是參照壓電陶瓷材料PZT-4且會依據自然對數的次方規律分佈於厚度方向。FCLM的數值解之收斂率及精確度是透過與文獻中的3D精確解比較而得的。本文將展示FCLMs基本頻率及對應的模態電場與彈性場變數的收斂解。本文也探討各種影響該殼動態反應之因素，如電彈耦合效應，材料性質梯度指數，表面條件及相關尺寸比例造成的影響。

Coupled electro-mechanical effects and the dynamic responses of functionally graded piezoelectric film-substrate circular hollow cylinders
Author: Xian-Foong Lim
Advisor: Prof.Chih-Ping Wu
Department of Civil Engineering, National Cheng Kung University

SUMMARY
Based on Reissner’s mixed variational theorem (RMVT), we developed finite cylindrical layer methods (FCLMs) to investigate the quasi-three-dimensional (3D) dynamic responses of simply-supported, two-layered functionally graded piezoelectric material (FGPM) film-substrate circular hollow cylinders with open- and closed-circuit surface conditions. The FGPM film-substrate cylinder considered in this work consists of a thick and soft FGPM substrate with a surface-bonded thin and stiff homogeneous piezoelectric material (HPM) film. The material properties of the FGPM layer are assumed to obey an exponent-law exponentially varying with the thickness coordinate, and the piezoelectric ceramic material PZT-4 is taken to be the reference material. The accuracy and convergence rate of FCLMs with different orders are assessed by comparing their solutions with the exact 3D ones available in the literature. The convergent solutions of FCLMs for the lowest frequency parameters and their corresponding modal electric and elastic variables of the FGPM film-substrate cylinder are presented. The influences of various factors with regard to some of the key dynamic responses of the cylinder are examined, such as the coupled electro-mechanical characteristics, material-property gradient index, surface boundary conditions, aspect ratio, and film-substrate thickness ratio.
Key words: Finite layer methods, vibration, functionally graded piezoelectric material, film, substrate, cylinders.

INTRODUCTION
Functionally graded piezoelectric materials (FGPMs) are a new class of advanced materials, and these have been used to form beam-, plate- and shell-like smart (or intelligent) structures for sensing, actuating and control purposes in micro- and nano-electro-mechanical systems, due to their electro-mechanical coupling characteristics [1-4]. For example, sandwich elastic structures have been embedded with an FGEM core in order to reduce the stress concentration occurring at the face sheet-core interfaces, as well as some failure phenomena, such as matrix cracking, due to the discontinuous in-surface stresses [8-12].
Wu and Li [66,68,69] and Wu and Chang [67] have been shown that these finite layer methods are accurate and have a fast convergence rate, that the solutions obtained by using the same orders for the primary variables are more computationally efficient than those obtained by using the orders of primary variables different from one another.
We thus extend the unified formulation for these to that for FGPM ones in this article, accounting for the coupled electro-mechanical effects. The newly developed formulation will be applied to the dynamic responses of simply-supported, two-layered FGPM film-substrate circular hollow cylinders with open- and closed-circuit surface conditions, in which the orders used for expansions of the elastic variables through the thickness coordinate remain the same values, while the orders are variable for the electric variables. The influences with regard to some crucial effects on the lowest frequency parameters and their corresponding through-thickness distributions of modal variables are examined, such as the effects with regard to the electro-mechanical coupling characteristics, material-property gradient index, surface conditions, aspect ratio and film-substrate thickness ratio.
FINITE CYLINDRICAL LAYER METHODS
Kinematic and kinetic assumptions
We consider a simply supported, two-layered FGPM film-substrate circular hollow cylinder, as shown in Fig. 1. In the implementation of the developed FCLMs, the cylinder will be artificially divided into cylindrical layers, the thicknesses of which are . A set of local thickness coordinates, , is located at the mid-surface of each individual cylindrical layer, as shown in Fig. 2. The relationship among the global and local thickness coordinates and the radial one in the mth-layer is and , in which , and are the global thickness coordinates measured from the mid-surface of the cylinder to the top and bottom surfaces of the mth-layer, respectively.
For the mth-layer of the cylinder, the linear constitutive equations valid for the orthotropic materials are given, as Eqs.(9)-(10).
Hamilton’s principle
For this analysis, the applied electric and mechanical loads on the lateral surfaces and boundary edges are absent.Using the kinematic and kinetic assumptions, which are given in Eqs. (1)-(4) and (5)-(8), respectively, we express the first-order variation of this energy functional as Eq.(23).
The system motion equations of RMVT-based FCLMs
Three different surface conditions are considered in the article, as Eqs. (26)-(28).The edge boundary conditions of each individual layer are considered as fully simple supports, suitably grounded, and are given as Eq.(29).
By means of the separation of variables, the primary field variables of each individual layer are expanded as the following forms of a double Fourier series in the surface and a harmonic function in the time domain, so that the boundary conditions of the simply supported edges are exactly satisfied. They are given as Eqs. (30)-(32).
Introducing Eqs. (30)-(32) in Eq. (23) and applying the Hamilton principle, which is , we obtain the system of motion equations of the cylinder, as Eq.(33).
ILLUSTRATIVE EXAMPLES
In the following examples, is defined to represent various RMVT-based FCLMs, in which the in- and out-of-surface elastic displacement components and the electric potential one are expanded as the -, - and -order Lagrange polynomials, respectively; and the transverse shear and normal stress components and the normal electric displacement one are expanded as the -, - and -order Lagrange polynomials in the thickness coordinate of each layer. In addition, because an h-refinement process is adopted for this work, the values of are taken as 1, 2 and 3 in the following examples.
FGPM sandwich cylinders
The cylinder considered consists of two HPM face-sheets and an FGPM core. The material properties ( ( )) are assumed to be symmetric with respect to the mid-surface of the sandwich cylinder and obey an exponent-law distribution through the thickness coordinate, as Eq(42).Table 2 presents the various FCLM solutions of the lowest frequency parameters for simply-supported, FGPM sandwich cylinders with different surface boundary conditions (Cases 1-3).
FGPM film-substrate cylinders
In this section, we investigate the free vibration characteristics of simply-supported, FGPM film-substrate cylinders with different surface conditions (Cases 1-3), in which the film and substrate layers are located on the outer and inner layers, respectively. The film layer is considered as an HPM layer, while the substrate layer is an FGPM one, the material properties of which are given as Eq(43).
Tables 3 and 4 show the lowest frequency parameters of L/R=R/h=5 and 20 FGPM film-substrate cylinders, respectively, with different surface conditions (Cases 1-3), vibration modes, material-property gradient indices and film-substrate thickness ratios.
In order to have a more clear picture with regard to how the electric and mechanical variables change along the thickness coordinate in the FGPM FSC, which may be used to assess the basic kinematic and kinetic assumptions of the existing 2D piezoelectric shell theories, we show the through-thickness distributions of these modal variables in Figs. 3-5 for the cylinders with different surface conditions (Cases 1-3).
Figures 6 and 7 show the variations of the lowest frequency parameters with the wave numbers in the circumferential coordinate ( ) by holding a certain wave number in the length coordinate ( ), in which 2.5 and 5, hf: : hs=0.2h : 0.8h, the surface boundary conditions are closed-closed circuit ones, and L/R=R/h=5 and 20 in Figs. 6 and 7, respectively.

CONCLUDING REMARKS
In the implementation of these FCLMs, we found that selecting the same orders for the elastic and electric variables will lead to more efficient performance than other selections based on the considerations of accuracy and time-consuming, and the and FCLMs are thus recommended. It is shown in the illustrative examples that the open surface conditions will make the cylinder slightly stiffer than the cylinder with closed surface conditions, even though the effects of surface boundary conditions on the lowest frequency parameters is minor. The coupled electric-elastic effect increases the gross stiffness of the FGPM sandwich cylinder. The effects of the material-property gradient index and aspect ratio on the lowest frequency parameters of FGPM sandwich and film-substrate cylinders are significant. The fundamental vibration modes occur at =(1, 2) and =(1, 1) for the cases of the thick cylinder (L/R=R/h=5) and thin one (L/R=R/h=20), and these will not be affected by changing the values of the material-property gradient index or the film-substrate thickness ratio. The through-thickness distributions of elastic and electric variables in the FGPM layer appear to be higher-order polynomial function variations, which totally differ from the basic kinematical and kinetic assumptions of the existing 2D equivalent single-layered shell theories, which might not be suitable for the analysis of FGPM cylinders, and these can thus provide a reference for the development of a 2D advanced FGPM shell theory in the future.

參考文獻
[1] Coleman JN, Khan U, Blau WJ, Gun’ko YK. Small but strong: a review of the mechanical properties of carbon nanotube-polymer composites. Carbon 2006;44:1624-52.
[2] Esawi AMK, Farag MM. Carbon nanotube reinforced composites: potential and current challenges. Materials & Design 2007;28:2394-401.
[3] Li C, Thostenson ET, Chou TW. Sensors and actuators based on carbon nanotubes and their composites: a review. Composites Science and Technology 2008;68:1227-49.
[4] Ramaratnam A, Jalili N. Reinforcement of piezoelectric polymers with carbon nanotubes: pathway to next-generation sensors. Journal of Intelligent Material Systems and Structures 2006;17:199-208.
[5] Barzoki AAM, Arani AG, Kolahchi R, Mozdianfard MR. Electro-thermo-mechanical torsional buckling of a piezoelectric polymetric cylindrical shell reinforced by DWBNNTs with an elastic core. Applied Mathematical Modelling 2012;36:2983-95.
[6] Chou TW, Gao L, Thostenson ET, Zhang Z, Byun JH. An assessment of the science and technology of carbon nanotube-based fibers and composites. Composites Science and Technology 2010;70:1-19.
[7] Thostenson ET, Ren Z, Chou TW. Advances in the science and technology of carbon nanotubes and their composites: a review. Composites Science and Technology 2001;61:1899-912.
[8] Kashtalyan M. Three-dimensional elasticity solution for bending of functionally graded rectangular plates. European Journal of Mechanics A-Solids 2004;23:853-864.
[9] Kashtalyan M, Menshykova M. Three-dimensional elasticity solution for sandwich panels with a functionally graded core. Composite Structures 2009; 87:36-43.
[10] Kashtalyan M, Menshykova M. Effect of a functionally graded interlayer on three-dimensional elastic deformation of coated plates subjected to transverse loading. Composite Structures 2009;89:167-76.
[11] Wu CP, Tsai, TC. Exact solutions of functionally graded piezoelectric material sandwich cylinders by a modified Pagano method. Applied Mathematical Modeling 2012;36:1910-30.
[12] Wu CP, Jiang RY. A state space differential reproducing kernel method for the 3D analysis of FGM sandwich circular hollow cylinders with combinations of simply-supported and clamped edges. Composite Structures 2012;94:3401-20.
[13] Noor AK, Burton WS. Assessment of computational models for multilayered anisotropic plates. Composite Structures 1990;14:233-65.
[14] Noor AK, Burton WS. Assessment of computational models for multilayered composite shells. Applied Mechanics Reviews 1990;43:67-97.
[15] Noor AK, Burton WS, Peters, JM. Assessment of computational models for multilayered composite cylinders. International Journal of Solids and Structures 1991;27:1269-86.
[16] Carrera E. An assessment of mixed and classical theories on global and local response of multilayered orthotropic plates. Composite Structures 2000;50:183-98.
[17] Carrera E. Developments, ideas, and evaluations based upon Reissner’s Mixed Variational Theorem in the modeling of multilayered plates and shells. Applied Mechanics Reviews 2001;54:301-29.
[18] Carrera E. Historical review of zig-zag theories for multilayered plates and shells. Applied Mechanics Reviews 2003;56:287-308.
[19] Carrera E. Theories and finite elements for multilayered plates and shells: a unified compact formulation with numerical assessment and benchmarks. Archives of Computational Methods in Engineering 2003;10:215-96.
[20] Neves AMA, Ferreira AJM, Carrera E, Cinefra M, Roque CMC, Jorge RMN, Soares CMM. A quasi-3D hyperbolic shear deformation theory for the static and free vibration analysis of functionally graded plates. Composite Structures 2012;94:1814-25.
[21] Neves AMA, Ferreira AJM, Carrera E, Roque CMC, Cinefra M, Jorge RMN, Soares CMM. A quasi-3D sinusoidal shear deformation theory for the static and free vibration analysis of functionally graded plates. Composites Part B: Engineering 2012;43:711-25.
[22] Neves AMA, Ferreira AJM, Carrera E, Cinefra M, Roque CMC, Jorge RMN, Soares CMM. Free vibration analysis of functionally graded shells by a higher-order shear deformation theory and radial basis functions collocation, accounting for through-the-thickness deformations. European Journal of Mechanics A/Solids 2013;37:24-34.
[23] Neves AMA, Ferreira AJM, Carrera E, Cinefra M, Roque CMC, Jorge RMN, Soares CMM. Static, free vibration and buckling analysis of isotropic and sandwich functionally graded plates using a quasi-3D higher-order shear deformation theory and a meshless technique. Composites Part B: Engineering 2013;44:657-74.
[24] Zenkour AM. On vibration of functionally graded plates according to a refined trigonometric plate theory. International Journal of Structural Stability and Dynamics 2005;5:279-97.
[25] Fazzolari FA, Carrera E. Refined hierarchical kinematics quasi-3D Ritz models for free vibration analysis of doubly curved FGM shells and sandwich shells with FGM core. Journal of Sound and Vibration 2014;333:1485-508.
[26] Reissner E. On a certain mixed variational theory and a proposed application. International Journal for Numerical Methods in Engineering 1984;20:1366-8.
[27] Reissner E. On a mixed variational theorem and on a shear deformable plate theory. International Journal for Numerical Methods in Engineering 1986;23:193-8.
[28] Reissner E. On a certain mixed variational theorem and on laminated elastic shell theory. Proceeding of Euromech-Colloquium 1986;219:17-27.
[29] Cinefra M, Belouettar S, Soave M, Carrera E. Variable kinematic models applied to free-vibration analysis of functionally graded material shells. European Journal of Mechanics A/Solids 2010;29:1078-87.
[30] Baferani AH, Saidi AR, Jomehzadeh E. Exact analytical solution for free vibration of functionally graded thin annular sector plates resting on elastic foundation. Journal of Vibration and Control 2012;18:246-67.
[31] Zhang Y, Wang S, Loughlan J. Free vibration analysis of rectangular composite laminates using a layerwise cubic B-spline finite strip method. Thin-Walled Structures 2006;44:601-22.
[32] Najafov A, Sofiyev A, Ozyigit P, Yucel KT. Vibration and stability of axially compressed truncated conical shells with functionally graded middle layer surrounded by elastic medium. Journal of Vibration and Control 2014;20:303-20.
[33] Tajeddini V, Ohadi A. Three-dimensional vibration analysis of functionally graded thick, annular plates with variable thickness via polynomial-Ritz method. Journal of Vibration and Control 2012;18:1698-707.
[34] Brischetto S. Exact elasticity solution for natural frequencies of functionally graded simply-supported structures. CMES-Computer Modeling in Engineering & Sciences 2013;95:391-430.
[35] Brischetto S, Carrera E. Classical and refined shell models for the analysis of nano-reinforced structures. International Journal of Mechanical Sciences 2012;55:104-17.
[36] Jha DK, Kant T, Singh RK. A critical review of recent research on functionally graded plates. Composite Structures 2013;96:833-49.
[37] Liew KM, Zhao X, Ferreira AJM. A review of meshless methods for laminated and functionally graded plates and shells. Composite Structures 2011;93:2031-41.
[38] Liew KM, Xiang P, Zhang LW. Mechanical properties and characteristics of microtubules: a review. Composite Structures 2015;123:98-108.
[39] Swaminathan K, Naveenkumar DT, Zenkour AM, Carrera E. Stress, vibration and buckling analyses of FGM plates-a state-of-the-art review. Composite Structures 2015;120:10-31.
[40] Thai HT, Kim SE. A review of theories for the modeling and analysis of functionally graded plates and shells. Composite Structures 2015;128:70-86.
[41] Zenkour AM. A comprehensive analysis of functionally graded sandwich plates: part 1-deflection and stresses. International Journal of Solids and Structures 2005;42:5224-42.
[42] Zenkour AM. A comprehensive analysis of functionally graded sandwich plates: part 2-buckling and free vibration. International Journal of Solids and Structures 2005;42:5243-58.
[43] Chen WQ, Ding HJ. On free vibration of a functionally graded piezoelectric rectangular plate. Acta Mechanica 2002;153:207-16.
[44] Chen WQ, Bian ZG, Lv CF, Ding HJ. 3D free vibration analysis of a functionally graded piezoelectric hollow cylinder filled with compressible fluid. International Journal of Solids and Structures 2004;41:947-964.
[45] Alibeigloo A, Kani AM, Pashaei MH. Elasticity solution for the free vibration analysis of functionally graded cylindrical shell bonded to thin piezoelectric layers. International Journal of Pressure Vessels and Piping 2012;89:98-111.
[46] Messina A, Carrera E. Three-dimensional free vibration of multi-layered piezoelectric plates through approximate and exact analyses. Journal of Intelligent Material Systems and Structures 2015;26:489-504.
[47] Wu CP, Syu YS. Asymptotic solutions for multilayered piezoelectric cylinders under electro-mechanical loads. CMC-Computers, Materials & Continua 2006;4:87-108.
[48] Wu CP, Syu YS. Exact solutions of functionally graded piezoelectric shells under cylindrical bending. International Journal of Solids and Structures 2007;44:6450-72.
[49] Wu CP, Tsai TC. Cylindrical bending vibration of functionally graded piezoelectric shells using the method of perturbation. Journal of Engineering Mathematics 2009;63:95-119.
[50] Wu CP, Syu YS, Lo JY. Three-dimensional solutions for multilayered piezoelectric hollow cylinders by an asymptotic approach. International Journal of Mechanical Sciences 2007;49:669-89.
[51] Li Y, Shi Z. Free vibration of a functionally graded piezoelectric beam via state space based differential quadrature. Composite Structures 2009;87:257-64.
[52] Zhang Z, Feng C, Liew KM. Three-dimensional vibration analysis of multilayered piezoelectric composite plates. International Journal of Engineering Science 2006;44:397-408.
[53] Loja MAR, Soares CMM, Barbosa JI. Analysis of functionally graded sandwich plate structures with piezoelectric skins, using B-spline finite strip method. Composite Structures 2013;96:606-15.
[54] Jafari AA, Khalili SMR, Tavakolian M. Nonlinear vibration of functionally graded cylindrical shells embedded with a piezoelectric layer. Thin-Walled Structures 2014;79:8-15.
[55] Zenkour AM, Abbas IA. Magneto-thermo-elastic response of an infinite functionally graded cylinder using the finite element method. Journal of Vibration and Control 2014;20:1907-19.
[56] Beheshti-Aval SB, Shahvaghar-Asl S, Lezgy-Nazargah M, Noori M. A finite element model based on coupled refined high-order global-local theory for static analysis of electromechanical embedded shear-mode piezoelectric sandwich composite beams with various widths. Thin-Walled Structures 2013;72:139-63.
[57] D’Ottavio M, Ballhause D, Kroplin B, Carrera E. Closed-form solutions for the free vibration problem of multilayered piezoelectric shells. Computers & Structures 2006;84:1506-18.
[58] Ebrahimi F, Rastgo A. An analytical study on the free vibration of smart circular thin FGM plate based on classical plate theory. Thin-Walled Structures 2008;46:1402-8.
[59] Wu XH, Chen C, Shen YP, Tian XG. A high order theory for functionally graded piezoelectric shells. International Journal of Solids and Structures 2002;39:5325-44.
[60] Torres DAF, Mendonca PDTR. HSDT-layerwise analytical solution for rectangular piezoelectric laminated plates. Composite Structures 2010;92:1763-74.
[61] Rouzegar J, Abad F. Free vibration analysis of FG plate with piezoelectric layers using four-variable refined plate theory. Thin-Walled Structures 2015;89:76-83.
[62] Saravanos DA, Heyliger PR. Mechanics and computational models for laminated piezoelectric beams, plates, and shells. Applied Mechanics Reviews 1999;52:305-20.
[63] Wu CP, Chiu KH, Wand YM. A review on the three-dimensional analytical approaches of multilayered and functionally graded piezoelectric plates and shells. CMC-Computers, Materials, & Continua 2008;18:93-132.
[64] Carrera E, Brischetto S, Cinefra M, Soave M. Refined and advanced models for multilayered plates and shells embedding functionally graded material layers. Mechanics of Advanced Materials and Structures 2010;17:603-21.
[65] Brischetto S, Carrera E. Refined 2D models for the analysis of functionally graded piezoelectric plates. Journal of Intelligent Material Systems and Structures 2009;20:1783-98.
[66] Wu CP, Li HY. The RMVT- and PVD-based finite layer methods for the three-dimensional analysis of multilayered composite and FGM plates. Composite Structures 2010;92:2476-96.
[67] Wu CP, Chang YT. A unified formulation of RMVT-based finite cylindrical layer methods for sandwich circular hollow cylinders with an embedded FGM layer. Composites Part B: Engineering 2012;43:3318-33.
[68] Wu CP, Li HY. An RMVT-based finite rectangular prism method for the 3D analysis of sandwich FGM plates with various boundary conditions. CMC-Computers, Materials & Continua 2013;34:27-62.
[69] Wu CP, Li HY. RMVT-based finite cylindrical prism methods for multilayered functionally graded circular hollow cylinders with various boundary conditions. Composite Structures 2013;100:592-608.
[70] Wu CP, Chen H. Exact solutions of free vibration of simply-supported functionally graded piezoelectric sandwich cylinders using a modified Pagano method. Journal of Sandwich Structures and Materials 2013;15:229-257.