本文發展混合等參有限元素法(Iso-parametric finite element method , IFEM)，據以進行功能性(Functionally graded , FG)石墨烯片加勁複合 (Graphene platelets-reinforced composite , GPLRC) 材料同心圓環殼之三維自然振動分析。文中考慮五種沿厚度方向不同的石墨烯片(Graphene platelets , GPLs)分布，其中有效楊氏係數使用Halpin-Tsai模型來評估，而有效柏松比與質量密度都是使用二相混合物法則來計算，為了處理此問題，我們依據Hamilton定理和Reissner應變能，發展出一套混合等參有限元素法，此方法在計算功能性石墨烯片加勁複合材料同心圓環殼的自然振動頻率時，可以快速收斂，且其在均質等向性(homogeneous isotropic)材料和功能性橫向等向性(transversely isotropic)材料之同心圓環殼的收斂解與文獻中的精確解極為吻合。在數值範例中，石墨烯片分佈、石墨烯片的重量比和半徑與厚度比對功能性石墨烯片加勁複合材料同心圓環殼的基本自然振動頻率的影響進行探討。

This paper is concerned with that we develop a mixed iso-parametric finite element method (IFEM) based on the weak form of three-dimensional (3D) elasticity theory to analyze the free vibration behavior of freely supported, functionally graded (FG) graphene platelets-reinforced composite (GPLRC) toroidal shells. Five different distribution patterns of graphene platelets (GPLs) through the thickness direction are considered, the Halpin-Tsai model, the rule of mixtures are used to estimate the effective Young’s modulus, the effective Poisson’s ratio and the effective mass density, respectively. By using Hamilton’s principle and Reissner’s strain energy function, we derive the system equations of the mixed IFEM. This method can quickly converge when calculating the 3D solutions for the natural frequency parameters of the FG-GPLRC toroidal shells, and the convergent solutions in homogeneous isotropic toroidal shells and FG transversely isotropic toroidal shells are in strong agreement with the accurate solutions in the literature. In the numerical examples, the effects of GPL distribution, the weight fraction of the GPL, and the radius-to-thickness ratio on the lowest natural frequencies of the FG-GPLRC toroidal shells are discussed.

[1] Reddy JN. Mechanics of laminated composite plates and shells: theory and analysis. Boca Raton: CRC Press; 2003.
[2] Sodel W. Vibrations of shells and plates. New York: Marcel Dekker Inc; 1993.
[3] Gould, PL. Analysis of shells and plates. New Jersey: Prentice-Hall, Inc; 1999.
[4] Love AEH. A treatise on the mathematical theory of elasticity. 4th ed. New York: Dover Publications; 1927.
[5] Senjanovic I, Alujevic N, Catipovic I, Cakmak D, Vladimir N. Vibration analysis of rotating shell by the Rayleigh-Ritz method and Fourier series. Eng Struct 2018:173;870-91.
[6] Reddy JN. Energy and variational methods in applied mechanics. New York: John Wiley & Sons, Inc; 1984.
[7] Senjanovic I, Catipovic I, Alujevic N, Cakmak D, Vladimir N. A finite strip for the vibration analysis of rotating toroidal shell under internal pressure. J Vib Acoust 2019:141;021013.
[8] Senjanovic I, Cakmak D, Alujevic N, Catipovic I, Vladimir N, Cho DS. Pressure and rotation induced tensional forces of toroidal shell and their influence on natural vibrations. Mech Res Commun 2019:96;1-6.
[9] Sanders Jr JL. An improved first-approximation theory for thin shells. NASA Tech Rep 1959:R-24;1-11.
[10] Budiansky B. Notes on nonlinear shell theory. J Appl Mech 1968:40;393-401.
[11] Wang XH, Xu B, Redekop D. Theoretical natural frequencies and mode shapes for thin and thick curved pipes and toroidal shells. J Sound Vib 2006:292;424-34.
[12] Wang XH, Redekop D. Natural frequencies and mode shapes of an orthotropic thin shell of revolution. Thin-Walled Struct 2005:43;735-50.
[13] Bert CW, Malik M. Differential quadrature method in computational mechanics: a review. Appl Mech Rev 1996:49;1-28.
[14] Du H, Lim MK, Lin RM. 1994. Application of generalized differential quadrature method to structural problems. Int J Numer Methods Eng 1994:37;1881-96.
[15] Wu CP, Lee CY. Differential quadrature solution for the free vibration analysis of laminated conical shells with variable stiffness. Int J Mech Sci 2001:43;1853-69.
[16] Buchanan GR, Liu YJ. An analysis of the free vibration of thick-walled isotropic toroidal shells. Int J Mech Sci 2005:47;277-92.
[17] Liu YJ, Buchanan GR. Free vibration of transversely isotropic solid and thick-walled toroidal shells. Int J Struct Stab Dyn 2006:6;359-75.
[18] Washizu K. Variational methods in elasticity & plasticity. 3rd ed. New York: Pergamon Press; 1982.
[19] Jones RM. Mechanics of composite materials. New York: McGraw-Hill Book Comp; 1975.
[20] Noor AK, Peters JM. Stress and vibration analyses of anisotropic shells of revolution. Int J Numer Methods Eng 1988:26;1145-67.
[21] Noor AK, Peters JM. Vibration analysis of laminated anisotropic shells of revolution. Comput Methods Appl Mech Eng 1987:61;277-301.
[22] Tornabene F. 2-D GDQ solution for free vibrations of anisotropic doubly-curved shells and panels of revolution. Compos Struct 2011:93;1854-76.
[23] Tornabene F., Liverani A, Caligiana G. General anisotropic doubly-curved shell theory: A differential quadrature solution for free vibrations of shells and panels of revolution with a free-form meridian. J Sound Vib 2012:331;4848-69.
[24] Qatu MS. Accurate theory for laminated composite deep thick shells. Int J Solids Struct 1999:36;2917-41.
[25] Toorani MH, Lakis AA. General equations of anisotropic plates and shells including transverse shear deformations, rotary inertia and initial curvature effects. J Sound Vib 2000:237;561-615.
[26] Reissner E. The effect of transverse shear deformation on the bending of elastic plates. J Appl Mech 1945:12;66-77.
[27] Kumar A, Chakrabarti A, Bhargava P. Vibration of laminated composites and sandwich shells based on higher order zigzag theory. Eng Struct 2013:56;880-8.
[28] Nayfeh AH. Introduction to perturbation techniques. New York: John Wiley & Sons, Inc; 1993.
[29] Sadd MH. Elasticity: theory, applications, and numerics. Burlinton: Elsevier Inc; 2005.
[30] Wu CP, Tarn JQ, Chi SM. An asymptotic theory for dynamic response of doubly curved laminated shells. Int J Solid Struct 1996:33;3813-41.
[31] Wu CP, Lo JY. An asymptotic theory for dynamic response of laminated piezoelectric shells. Acta Mech 2006:183;177-208.
[32] Novoselov KS, Geim AK, Morozov SV, Jiang D, Zhang Y, Dubonos SV et al. Electric field effect in atomically thin carbon films. Sci 2004:306;666-9.
[33] Stankovich S, Dikin DA, Dommett GH, Kohlhaas KM, Zimney EJ, Stach EA et al. Graphene-based composite materials. Nature 2006:442;282-6.
[34] Huang X, Qi X, Boey F, Zhang H. Graphene-based composites. Chem Soc Rev 2012:41;666-86.
[35] Geim AK, Novoselov KS. The rise of graphene. Nat Mater 2007:6;183-91.
[36] Yang J, Wu H, Kitipornchai S. Buckling and postbuckling of functionally graded multilayer graphene platelet-reinforced composite beams. Compos Struct 2017:161;111-8.
[37] Feng C, Kitipornchai S, Yang J. Nonlinear bending of polymer nanocomposite beams reinforced with non-uniformly distributed graphene platelets (GPLs). Compos Part B 2017:110;132-40.
[38] Gao K, Gao W, Chen D, Yang J. Nonlinear free vibration of functionally graded graphene platelets reinforced porous nanocomposite plates resting on elastic foundation. Compos Struct 2018:204;831-46.
[39] Song M, Kitipornchai S, Yang J. Free and forced vibrations of functionally graded polymer composite plates reinforced with graphene nanoplatelets. Compos Struct 2017:159;579-88.
[40] Bidzard A, Malekzadeh P, Mohebpour SR. Vibration of multilayer FG-GPLRC toroidal panels with elastically restrained against rotation edges. Thin-Walled Struct 2019:143;106209.
[41] Bidzard A, Malekzadeh P, Mohebpour S. Influences of pressure and thermal environment on nonlinear vibration characteristics of multilayer FG-GPLRC toroidal panels on nonlinear elastic foundation. Compos Struct 2021:259;113503.
[42] Heydapour Y, Malekzadeh P, Dimitri R, Tornabene F. Thermoelastic analysis of rotating multilayer FG-GPLRC truncated conical shells based on a coupled TDQM-NURBS scheme. Compos Struct 2020:235;111707.
[43] Wang Y, Feng C, Yang J, Zhou D, Liu W. Static response of functionally graded graphene platelet-reinforced composite plate with dielectric property. J Intell Mater Syst Struct 2020:31;2211-2228.
[44] Wang Y, Zhou Y, Feng C, Yang J, Zhou D, Wang S. Numerical analysis on stability of functionally graded platelets (GPLs) reinforced dielectric composite plate. Appl Math Modell 2022:101;239-258.
[45] Wang Y, Feng C, Yang J, Zhou D, Wang S. Nonlinear vibration of FG-GPLRC dielectric plate with active tuning using differential quadrature method. Comput Meth Appl Mech Eng 2021:379;113761.
[46] Liu D, Kitipornchai S, Chen W, Yang J. Three-dimensional buckling and free vibration analyses of initially stressed functionally graded graphene reinforced composite cylindrical shell. Compos Struct 2018:189;560-9.
[47] Liu D, Zhou Y, Zhu J. On the free vibration and bending analysis of functionally graded nanocomposite spherical shells reinforced with graphene nanoplatelets: Three-dimensional elasticity solutions. Eng Struct 2021:226;111376.
[48] Alibeigloo A. Three-dimensional thermoelasticity analysis of graphene platelets reinforced cylindrical panel. Eur J Mech A/Solids 2020:81;103941.
[49] Rahimi A, Alibeigloo A, Safarpour M. Three-diemnsional static and free vibration analysis of graphene platelet-reinforced porous composite cylindrical shell. J Vib Contr 2020:26;1627-45.
[50] Qin B, Wang Q, Zhong R, Zhao X, Shuai C. A three-dimensional solution for free vibration of FGP-GPLRC cylindrical shells resting on elastic foundations: a comparative and parametric study. Int J Mech Sci 2020:187;105896.
[51] Safarpour M, Rahimi AR, Alibeigloo A. Static and free vibration analysis of graphene platelets reinforced composite truncated conical shell, cylindrical shell, and annular plate using theory of elasticity and DQM. Mech Based Des Struct Mach. 202:48;496-524.
[52] Zhao S, Zhao Z, Yang Z, Ke L, Kitipornchai S, Yang J. Functionally graded graphene reinforced composite structures: A review. Eng Struct 2020:210;110339.
[53] Liew KM, Lei ZX, Zhang LW. Mechanical analysis of functionally graded carbon nanotube reinforced composites: A review. Compos Struct 2015:120;90-7.
[54] Thai HT, Kim SE. A review of theories for the modeling and analysis of functionally graded plates and shells. Compos Struct 2015:128;70-86.
[55] Jha DK, Kant T, Singh RK. A critical review of recent research on functionally graded plates. Compos Struct 2013:96;833-49.
[56] Wu CP, Liu YC. A review of semi-analytical numerical methods for laminated composite and multilayered functionally graded elastic/piezoelectric plates and shells. Compos Struct 2016:147;1-15.
[57] Reissner E. On a certain mixed variational theorem and a proposed application. Int J Numer Methods Eng 1984:20;1366-1368.
[58] Reissner E. On a mixed variational theorem and on shear deformable plate theory. Int J Numer Methods Eng 1986:23;193-198.
[59] Carrera E. A Reissner’s mixed variational theorem applied to vibration analysis of multilayered shell. J Appl Mech 1999:66;69-78.
[60] 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;301-329.
[61] Wu CP, Li E. A semi-analytical FE method for the 3D bending analysis of nonhomogeneous orthotropic toroidal shells. Steel Compos Struct 2021:39;291-306.
[62] Saada AS. Elasticity: theory and applications. New York: Pergamon Press; 1974.
[63] Piggott, M. Load bearing fibre composites. New York: Kluwer Academic Publishers; 2002.
[64] Redekop D. Three-dimensional free vibration analysis of inhomogeneous thick orthotropic shells of revolution using differential quadrature. J Sound Vib 2006:291;1029-40.
[65] Yang J, Chen D, Kitipornchai S. Buckling and free vibration analyses of functionally graded graphene reinforced porous nanocomposite plates based on Chebyshev-Ritz method. Compos Struct 2018:193;281-94.