本文基於Reissner 混合變分理論(Reissner’s mixed variational theorem, RMVT)，發展有限層板法(finite layer methods, FLMs)，並應用於具簡支承功能性梯度(functionally graded, FG)奈米碳管加勁複合(carbon nanotube-reinforced composite, CNTRC)材料核心層結合壓電材料面層之積層板，承受單軸（或雙軸）壓力作用下，該板之挫屈行為。文中根據三維線性挫屈理論，假設臨界挫屈發生前，存在一組面內正向應力，此正向應力將被視為初始應力，引用於文中之挫屈分析。文中依奈米碳管加勁物沿厚度方向之函數分佈區分為：均勻分佈(uniformly distributed-type, UD-type)、功能性梯度菱型分佈(FG rhombus-type, FG R-type)、功能性梯度X型分佈(FG X-type, FG X-type)，而FG CNTRC之有效材料參數將依兩相混合法則求得，積層板之上、下表面條件考慮為開放（或封閉）迴路。理論推衍過程中，將平板分割成數個矩形板，其面內和面外主變數沿x-y 平面和厚度方向之變化，分別考慮為三角函數和Lagrange多項式分佈。文中亦將探討有限層板法之主要變數所選用形狀函數次數對於本解收斂性與精確性之比較。

Based on Reissner’s mixed variational theorem (RMVT), we develop a unified formulation of finite layer methods (FLMs) for the three-dimensional (3D) buckling analysis of simply-supported, functionally graded (FG) carbon nanotube-reinforced composite (CNTRC) plates with surface-bonded piezoelectric actuator and sensor layers and under bi-axial compressive loads. In this work, a set of membrane stresses is assumed to exist just before instability occurs, and determined using the predefined 3D deformations for the prebuckling state. The carbon nanotubes (CNTs) are considered to be uniformly distributed (UD), FG rhombus- , and X-type variations through the thickness coordinate, and the effective material properties of the FG CNTRC layer are evaluated using the rule of mixtures with two different surface conditions, open- and closed-circuit, are considered. In the formulation, the plate is divided into a number of finite rectangular layers, in which the trigonometric functions and Lagrange polynomials are used to interpolate the in- and out-of-plane variations for the field variables of each individual layer, respectively. The accuracy and convergence of the FLMs with various orders used for the expansion of each field variable in the thickness are assessed by comparing their solutions with the exact 3D ones available in the literature.

[1] Thostenson ET, Ren Z, Chou TW. Advances in the science and technology of carbon nanotubes and their composites: A review. Compos Sci Technol 2001;61:1899-912.
[2] 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.
[3] Esawi AMK, Farag MM. Carbon nanotube reinforced composites: Potential and current challenges. Mater Des 2007;28:2394-401.
[4] Chou TW, Gao L, Thostenson ET, Zhang Z, Byun JH. An assessment of the science and technology of carbon nanotube-based fibers and composites. Compos Sci Technol 2010;70:1-19.
[5] Davis DC, Wilkerson JW, Zhu J, Hadjiev VG. A strategy for improving mechanical properties of a fiber reinforced epoxy composite using functionalized carbon nanotubes. Compos Sci Technol 2011;71:1089-97.
[6] Li C, Thostenson ET, Chou TW. Sensors and actuators based on carbon nanotubes and their composites: A review. Compos Sci Technol 2008;68:1227-49.
[7] Ramaratnam A, Jalili N. Reinforcement of piezoelectric polymers with carbon nanotubes: Pathway to next-generation sensors. J Intell Mater Sys Struct 2006;17:199-208.
[8] Chee CYK, Tong L, Steven GP. Review on the modeling of piezoelectric sensors and actuators incorporated in intelligent structures. J Intell Mater Sys Struct 1998;9:3-19.
[9] Saravanos DA, Heyliger PR. Mechanics and computational models for laminated piezoelectric beams, plates, and shells. Appl Mech Rev 1999;52:305-20.
[10] Wu CP, Chiu KH, Wang YM. A review on the three-dimensional analytical approaches of multilayered and functionally graded piezoelectric plates and shells. CMC: Comput Mater Continua 2008;8: 93-132.
[11] Mackerle J. Smart materials and structures-A finite-element approach: A bibliography (1986-1997). Model Simul Mater Sci Eng 1998;6:293-334.
[12] Mackerle J. Smart materials and structures-A finite-element approach-An addendum: A bibliography (1997-2002). Model Simul Mater Sci Eng 2003;11:707-44.
[13] Mackerle J. Smart materials and structures-FEM and BEM simulations: A bibliography (1997-1999). Finite Elem Anal Des 2001;37:71-83.
[14] Mackerle J. Stability problems analyzed by finite element and boundary element techniques-A bibliography (1994-1996). Finite Elem Anal Des 1997;26:337-53.
[15] Shen HS. Buckling and postbuckling of anisotropic laminated cylindrical shells with piezoelectric fiber reinforced composite actuators. Mech Adv Mater Struct 2010;17:268-79.
[16] Jadhav PA, Bajoria KM. Buckling of piezoelectric functionally graded plate subjected to electro-mechanical loading. Smart Mater Struct 2012;21:105005.
[17] Shen HS, Zhu ZH. Compressive and thermal postbuckling behaviors of laminated plates with piezoelectric fiber reinforced composite actuators. Appl Math Model 2011;35:1829-45.
[18] Shen HS. A comparison of buckling and postbuckling behavior of FGM plates with piezoelectric fiber reinforced composite actuators. Compos Struct 2009;91:375-84.
[19] Kapuria S, Achary GGS. Nonlinear coupled zigzag theory for buckling of hybrid piezoelectric plates. Compos Struct 2006;74:253-64.
[20] Kapuria S, Achary GGS. Nonlinear zigzag theory for electrothermomechanical buckling of piezoelectric composite and sandwich plates. Acta Mech 2006;184:61-76.
[21] Dumir PC, Kumari P, Kapuria S. Assessment of third order smeared and zigzag theories for buckling and vibration of flat angle-ply hybrid piezoelectric panels. Compos Struct 2009;90:346-62.
[22] Kapuria S, Achary GGS. Exact 3D piezoelasticity solution for buckling of hybrid cross-ply plates using transfer matrices. Acta Mech 2004;170:25-45.
[23] Santos H, Mota Soares CM, Mota Soares CA, Reddy JN. A finite element model for the analysis of 3D axisymmetric laminated shells with piezoelectric sensors and actuators. Compos Struct 2006;75: 170-8.
[24] Santos H, Mota Soares CM, Mota Soares CA, Reddy JN. A finite element model for the analysis of 3D axisymmetric laminated shells with piezoelectric sensors and actuators. Compos Struct 2006;75: 170-8.
[25] Reissner E. On a certain mixed variational theory and a proposed application. Int J Numer Methods Eng 1984;20:1366-8.
[26] Reissner E. On a mixed variational theorem and on a shear deformable plate theory. Int J Numer Methods Eng 1986;23:193-8.
[27] Reissner E. On a certain mixed variational theorem and on laminated elastic shell theory. Proc Euromech-Colloquium 1986;219:17-27.
[28] Carrera E. Theories and finite elements for multilayered plates and shells: A unified compact formulation with numerical assessment and benchmarks. Arch Comput Methods Eng 2003;10:215-96.
[29] Carrera E, Ciuffreda A. A unified formulation to assess theories of multilayered plates for various bending problems. Compos Struct 2005;69:271-93.
[30] Nali P, Carrera E, Lecca S. Assessments of refined theories for buckling analysis of laminated plates. Compos Struct 2011;93:456-64.
[31] Ballhause D, D’Ottavio M, B, Carrera E. A unified formulation to assess multilayered theories for piezoelectric plates. Comput Struct 2005;83:1217-35.
[32] Brischetto S, Carrera E. Refined 2D models for the analysis of functionally graded piezoelectric plates. J Intell Mater Sys Struct 2009;20:1783-97.
[33] Robaldo A, Carrera E, Benjeddou A. A unified formulation for finite element analysis of piezoelectric adaptive plates. Comput Struct 2006;84:1494-505.
[34] Carrera E, Boscolo M. Classical and mixed finite elements for static and dynamic analysis of piezoelectric plates. Int J Numer Methods Eng 2007;70:1135-81.
[35] Carrera E, A, P. Nali P. Mixed elements for the analysis of anisotropic multilayered piezoelectric plates. J Intell Mater Sys Struct 2010;21:701-17.
[36] Kim J, Reddy JN. Analytical solutions for bending, vibration, and buckling of FGM plates using a couple stress-based third-order theory. Compos Struct 2013;103:86-98.
[37] Reddy JN, Kim J. A nonlinear modified couple stress-based third-order theory of functionally graded plates. Compos Struct 2011;94:1128-43.
[38] Na KS, Kim JH. Thermal postbuckling investigations of functionally graded plates using 3D finite element method. Finite Elem Anal Des 2006;42:749-56.
[39] Na KS, Kim JH. Volume fraction optimization of functionally graded composite panels for stress reduction and critical temperature. Finite Elem Anal Des 2009;45:845-51.
[40] Shen HS. Nonlinear bending of functionally graded carbon nanotubed-reinforced composite plates in thermal environments. Compos Struct 2009;91:9-19.
[41] Shen HS, Xiang Y. Nonlinear vibration of nanotube-reinforced composite cylindrical shells in thermal environments. Comput Methods Appl Mech Engrg 2012;213:196-205.
[42] Mori T, Tanaka K. Average stress in matrix and average elastic energy of materials with misfitting inclusions. Acta Metall 1973;21:571-4.
[43] Hill R. A self-consistent mechanics of composite materials. J Mech Phys Solids 1965;13:213-22.
[44] Wang ZX, Shen HS. Nonlinear vibration and bending of sandwich plates with nanotube-reinforced composite face sheets. Compos Part B: Eng 2012; 43:411-21.
[45] Shen HS. Postbuckling of nanotube-reinforced composite cylindrical shells in thermal environments, Part I; Axially-loaded shells. Compos Struct 2011;93:2096-108.
[46] Shen HS. Postbuckling of nanotube-reinforced composite cylindrical shells in thermal environments, Part II: Pressure-loaded shells. Compos Struct 2011;93:2496-503.
[47] Shen HS, Xiang Y. Postbuckling of nanotube-reinforced composite cylindrical shells under combined axial and radial mechanical loads in thermal environment. Compos Part B: Eng 2013;52:311-22.
[48] Lei ZX, Liew KM, Yu JL. Buckling analysis of functionally graded carbon nanotube-reinforced composite plates using the element-free kp-Ritz method. Compos Struct 2013;98:160-8.
[49] Zhu P, Lei ZX, Liew KM. Static and free vibration analyses of carbon nanotube-reinforced composite plates using finite element method with first order shear deformation plate theory. Compos Struct 2012;94:1450-60.
[50] Alibeigloo A. Static analysis of functionally graded carbon nanotube-reinforced composite plate embedded in piezoelectric layers by using theory of elasticity. Compos Struct 2013;95:612-22.
[51] Yas MH, Pourasghar A, Kamarian S, Heshmati M. Three-dimensional free vibration analysis of functionally graded nanocomposite cylindrical panels reinforced by carbon nanotube. Mater Des 2013;49:583-90.
[52] Brischetto S, Carrera E. Classical and refined shell models for the analysis of nano-reinforced structures. Int J Mech Sci 2012;55:104-17.
[53] Brischetto S, Carrera E. Analysis of nano-reinforced layered plates via classical and refined two-dimensional theories. Multidisc Model Mater 2012;8:4-31.
[54] Cheung YK, Chakrabarti S. Free vibration of thick, layered rectangular plates by finite layer method. J Sound Vib 1072;21:277-84.
[55] Zhou D, Cheung YK, Kong J. Free vibration of thick, layered rectangular plates with point supports by finite layer method. Int J Solids Struct 2000;37:1483-99.
[56] Cheung YK, Jiang CP. Finite layer method in analysis of piezoelectric composite laminates. Comput. Methods Appl Mech Engrg 2001;191:879-901.
[57] Akhras G, Li WC. Three-dimensional thermal buckling analysis of piezoelectric composite plates using the finite layer method. Smart Mater Struct 2008;17:055004.
[58] Akhras G, Li WC. Three-dimensional stability analysis of piezoelectric antisymmetric angle-ply laminates using finite layer method. J Intell Mater Sys Struct 2010;21:719-27.
[59] Akhras G, Li WC. Three-dimensional thermal buckling analysis of piezoelectric antisymmetric angle-ply laminates using finite layer method. Compos Struct 2010;92:31-8.
[60] Akhras G, Li WC. Three-dimensional static, vibration and stability analysis of piezoelectric composite plates using a finite layer method. Smart Mater Struct 2007;16:561-9.
[61] Carrera E. Developments, ideas, and evaluations based upon Reissner’s Mixed Variational Theorem in the modeling of multilayered plates and shells. Appl Mech Rev 2001;54:301-29.
[62] Carrera E, Brischetto S, Cinefra M, Soave M. Refined and advanced models for multilayered plates and shells embedding functionally graded material layers. Mech Adv Mater Struct 2010;17:603-21.
[63] D’Ottavio M, Carrera E. Variable-kinematics approach for linearized buckling analysis of laminated plates and shells. AIAA J 2010;48:1987-96.
[64] Wu CP, Li HY. The RMVT- and PVD-based finite layer methods for the three-dimensional analysis of multilayered composite and FGM plates. Compos Struct 2010;92:2476-96.
[65] Wu CP, Chang RY. A unified formulation of RMVT-based finite cylindrical layer methods for sandwich circular hollow cylinders with an embedded FGM layer. Compos Part B: Eng 2012;43:3318-33.
[66] Wu CP, Li HY. An RMVT-based finite rectangular prism rectangular prism method for the 3D analysis of sandwich FGM plates with various boundary conditions. CMC: Comput Mater Continua 2013;34:27-62.
[67] Wu CP, Li HY. RMVT-based finite cylindrical prism methods for multilayered functionally graded circular hollow cylinders with various boundary conditions. Compos Struct 2013;100:592-608.
[68] Wu CP, Lu YC. A modified Pagano method for the 3D dynamic responses of functionally graded magneto-electro-elastic plates. Compos Struct 2009;90:363-73.