摘要
功能梯度材料(Functionally Graded Materials,简称FGMs)通常是由陶瓷和金属复合而成的一种新型先进非均匀复合材料,在航空航天、核工业、动力机械等领域内有着极为广阔的应用前景。关于该类材料结构在复杂的机械荷载、热荷载及电荷载作用下屈曲前和屈曲后的振动特性研究具有重要的理论与应用价值。
本文的研究工作主要包括:(1)基于Reddy高阶剪切变形板理论和广义Kármán型方程,给出了几类现有研究成果较少或没有彻底解决的FGM混合板结构的屈曲前和屈曲后振动特性研究的控制方程,成功将摄动-Galerkin混合法应用于求解过程,并给出了各控制方程相对直观且易于计算的解析解形式;(2)单向压电纤维增强复合材料(Piezoelectric Fiber Reinforced Composite,简称PFRC)是一种新型压电材料,本文将其应用到了结构振动控制研究当中,发现通过压电纤维体积含量的选择可以实现混合板结构的优化振动控制;(3)围绕几种有代表性的FGM混合板结构,讨论其屈曲后的振动特性,在分析中同时考虑热传导,材料物性参数的温度相关性,以及机/热/电荷载耦合效应对非线性振动特性的影响,最后给出了大量数值计算结果。着重讨论了FGM材料组分指数、温度变化、长宽比、边厚比、屈曲与振动模态、PFRC压电纤维体积含量、控制电压、芯层与面板厚度比等因素对各类FGM混合板结构屈曲前及屈曲后振动特性的影响。
针对功能梯度材料主要应用于高温环境中的特点,本文考虑了所有参与分析材料的物性参数的温度相关性。又由于金属和陶瓷的储热能力不同,热传导难以避免,故在每章开头都先给出了各类功能梯度材料混合板结构在热场中的温度分布函数。
随后,基于Reddy高阶剪切变形板理论和广义Kármán型方程,给出了各类功能梯度材料混合板结构的振动以及屈曲后的振动的控制方程,并采用改进的摄动-Galerkin混合法进行求解。
接着,重点分析了表面覆盖纤维增强压电材料作动器的功能梯度材料混合板结构的非线性自由振动和受迫振动,具体讨论了材料组分指数、温度变化、压电纤维体积含量及控制电压等因素对表面覆盖纤维增强压电材料作动器的功能梯度材料混合板结构的非线性振动和动力响应的影响。在这之后,详细分析了对称功能梯度材料板结构屈曲后的振动特性,将控制方程拆分为屈曲和振动两个部分并分别用摄动-Galerkin混合法进行求解,具体讨论了材料组分指数、温度变化、边厚比、长宽比、屈曲与振动模态等因素对对称功能梯度材料板结构屈曲后的振动特性的影响,并对由两组不同材料构成的对称功能梯度材料板结构的屈曲后的振动特性进行了比较。还研究了热环境下表面覆盖或埋置纤维增强压电材料作动器的功能梯度材料混合板结构的屈曲后的振动特性,通过参数分析讨论了材料组分指数、温度变化、压电纤维体积含量及控制电压等因素对表面覆盖或埋置纤维增强压电材料作动器的功能梯度材料混合板屈曲后的振动特性的影响。最后,研究了以FGM为面板的夹芯板在热环境下屈曲后的线性自由振动以及非线性振动。详细讨论了材料组分指数、温度变化、芯层与面板厚度比等因素对FGM夹芯板屈曲后的振动特性的影响。
在本文在理论分析结果的基础上,采用FORTRAN语言编制扩充了多个数值分析程序包,分别用于各类功能梯度材料复合板结构的振动以及屈曲后的振动特性计算。
本文所取得的研究成果有助于加深对功能梯度复合材料板结构的振动特性的认识,且对于此类新型复合材料结构的工程实际应用具有一定的积极意义。
Functionally graded materials (FGMs), usually made from a mixture of metals and ceramics, have been regarded as one of the advanced inhomogeneous composite materials with great application potential in many engineering sectors such as aerospace vehicles, nuclear reactors, power generators and so on. Investigations on the vibration characteristics of pre- and post-buckled FGM hybrid plate structures under various combinations of mechanical, thermal and electrical loading conditions have been, therefore, receiving considerably more attention in recent years due to their prime importance in both theoretical and practical aspects.
This paper mainly consists of three parts: (1) Based on the Reddy’s higher-order shear deformation plate theory and general von Kármán-type equations, put forward the governing equations of pre- and post-buckled vibration of several hybrid FGM plate forms which have not been studied adequately, triumphantly applying the perturbation technique combined with Galerkin method into equations solving, and get the comparatively simple result forms with complicated deduction at last; (2) As a new type of materials, piezoelectric fiber reinforced composite (PFRC) actuators were applied to optimal vibration control, interrelated informatons be depicted in this paper; (3) During the investigative cource, some factors not been considered adequently in other researches were discussed fully in the postbuckled vibration analysis, such as the temperature-dependent material coefficients, heat conduction and nonlinear vibration characteristic etc. Bring forward a great deal of numerical results, and discuss the effect of volume fraction index, temperature change, aspect ratio, side-to-thickness ratio, mode, piezoelectric fiber volume fraction, control voltage and substrate-to-face sheet thickness ratio on the pre- and postbuckled vibration characteristics of FGM hybrid plates.
Since FGM are used in high-temperature environments, all the constituents of FGM sheets considered herein possess temperature-dependent properties. On the other hand, ceramics and metal used in FGM do store different amounts of heat, and therefore heat conduction usually occurs, therewith temperature distribution was deduced at the beginning of every chapter.
Subsequently, based on the Reddy’s higher-order shear deformation plate theory and general von Kármán-type equations, the governing equations of nonlinear vibration and postbuckling vibration are advanced for FGM hybrid plates and solved by using an improved perturbation technique combined with Galerkin method.
In succession, the free and forced vibration of FGM hybrid plates bonded with PFRC actuators in thermal environments are analyzed, the effect of volume fraction index, temperature change, piezoelectric fiber volume fraction, and control voltage on the nonlinear vibration and dynamic response of FGM hybrid plates are discussed. After that, the pre- and postbuckled vibration characteristics of symmetrical FGM plates are studied, through dividing the governing equations into two sets for postbuckling and vibration separately, and solving them with an improved perturbation technique combined with Galerkin method, the effect of volume fraction index, temperature change, side-to-thickness ratio, aspect ratio and mode on the postbuckling vibration characteristics of symmetrical FGM plates are discussed amply. Afterwards, comparisons of the pre- and postbuckled vibration characteristics of two type of symmetrical FGM plates with different constituents are given. whereafter, the pre- and postbuckled vibration characteristics of FGM plates bonded or imbedded with PFRC actuators in thermal environments are canvassed, including probe into the effect of volume fraction index, temperature change, piezoelectric fiber volume fraction, and control voltage on the postbuckling vibration characteristics of FGM hybrid plates. At last, the vibration of pre- and post-buckled sandwich plates with FGM face sheets in thermal environment is discussed. The effect of volume fraction index, temperature change, and substrate-to-face sheet thickness ratio on the vibration characteristics are studied.
As the result of extensive theoretical and numerical analysis by using the computer program packages developed with FORTRAN language, this paper provides comprehensive first-ever-known results which are helpful for better understanding the vibration behavior of the FGM plate structures.
引文
[1]陈华辉,邓海金,李明等.现代复合材料.北京:中国物资出版社, 1998.
[2]李进,田西华.功能梯度材料的研究现状及应用.宁夏工程技术, 2007, 6(1): 80-83.
[3]杨杰.功能梯度复合材料板结构的非线性力学行为与动力特性. [博士学位论文],上海,上海交通大学, 2002.
[4] Reddy JN, Chin CD. Thermomechanical analysis of functionally graded cylinders and plates. Journal of Thermal Stresses 1998, 21: 593-629.
[5]黄小林.船舶与海洋工程中复合材料板构件的非线性振动和动力响应. [博士学位论文],上海,上海交通大学, 2005.
[6] Koizumi M. The concept of FGM. Ceramic Transactions, Functionally Gradient Materials, 1993, 34: 3-10.
[7] Touloukian YS. Thermophysical properties of high temperature solid materials, McMillan, New York, 1967.
[8]陈东,杨光义,王国元等.功能梯度材料的进展.青岛建筑工程学院学报, 2001, 22(4): 92-95.
[9]李晓娟,李全禄,谢妙霞等.国内外压电陶瓷的新进展及新应用.硅酸盐通报, 2006, 25(4): 101-107.
[10] Mallik N, Ray MC. Effective coefficients of piezoelectric fiber-reinforced composites. AIAA Journal 2003, 41: 704-710.
[11]沈惠申.功能梯度复合材料板壳结构的弯曲、屈曲和振动.力学进展, 2004, 34(1): 53-60.
[12] Cheng ZQ, Batra RC. Exact correspondence between eigenvalues of membranes and functionally graded simply supported polygonal plates. Journal of Sound and Vibration 2000, 229: 879-895.
[13] Cheng ZQ, Reddy JN. Frequency of functionally graded plates with-dimensional asymptotic approach. Journal of Engineering Mechanics ASCE 2003, 129: 896-900.
[14] Vel SS, Batra RC. Three-dimensional exact solution for the vibration of functionally graded rectangular plates. Journal of Sound and Vibration 2004, 272: 703-730.
[15] Elishakoff I, Gentilini C, Viola E. Forced vibrations of functionally graded plates in the three-dimensional setting. AIAA Journal 2005, 43: 2000-2007.
[16] Abrate S. Free vibration, buckling, and static deflections of functionally graded plates.Composites Science and Technology 2006, 66: 2383-2394.
[17] Efraim E, Eisenberger M. Exact vibration analysis of variable thickness thick annular isotropic and FGM plates. Journal of Sound and Vibration 2007, 299: 720-738.
[18] Han SC, Lomboy GR, Kim KD. Mechanical vibration and buckling analysis of FGM plates and shells using a four-node quasi-conforming shell element. International Journal of Structural Stability and Dynamics 2008, 8: 203-229.
[19] Chen CS, Fung CP, Yu SY. The investigation on the vibration and stability of functionally graded plates. Journal of Reinforced Plastics and Composites 2008, 27: 1435-1447.
[20] Li Q, Iu VP, Kou KP. Three-dimensional vibration analysis of functionally graded material sandwich plates. Journal of Sound and Vibration 2008, 311: 498-515.
[21]黄小林,沈惠申.热环境下功能梯度材料板的自由振动和动力响应.工程力学, 2005, 22(3): 224-227.
[22] Kim KS, Noda N. A Green’s function approach to the deflection of a FGM plate under transient thermal loading. Archive of Applied Mechanics 2002, 72: 127-137.
[23] Kim YW. Temperature dependent vibration analysis of functionally graded rectangular plates. Journal of Sound and Vibration 2005, 284: 531-549.
[24] Prakash T, Ganapathi M. Asymmetric flexural vibration and thermoelastic stability of FGM circular plates using finite element method. Composites Part B 2006, 37: 642-649.
[25] Praveen GN, Reddy JN, Nonlinear transient thermoelastic analysis of functionally graded ceramic-metal plates. International Journal of Solids and Structures 1998, 35: 4457-4476.
[26] Reddy JN. Analysis of functionally graded plates. International Journal for Numerical Methods in Engineering 2000, 47: 663-684.
[27] Kitipornchai S, Yang J, Liew KM. Semi-analytical solution for nonlinear vibration of laminated FGM plates with geometric imperfections. International Journal of Solids and Structures 2004, 41: 2235-2257.
[28] Chen CS. Nonlinear vibration of a shear deformable functionally graded plate. Composite Structures 2005, 68: 295-302.
[29] Chen CS, Chen TJ, Chien RD. Nonlinear vibration of initially stressed functionally graded plates. Thin-Walled Structures 2006, 44: 844-851.
[30] Fung CP, Chen CS. Inperfection sensitivity in the nonlinear vibration of functionally gradedplates. European Journal of Mechanics - A/Solids 2006, 25: 425-436.
[31] Chen CS, Tan AH. Imperfection sensitivity in the nonlinear vibration of initially stresses functionally graded plates. Composite Structures 2007, 78: 529-536.
[32] Allahverdizadeh A, Naei MH, Bahrami MN. Nonlinear free and forced vibration analysis of thin circular functionally graded plates. Journal of Sound and Vibration 2008, 310: 966-984.
[33] Allahverdizadeh A, Naei MH, Bahrami MN. Vibration amplitude and thermal effects on the nonlinear behavior of thin circular functionally graded plates. International Journal of Mechanical Sciences 2008, 50: 445-454.
[34] Allahverdizadeh A, Rastgo A, Naei MH. Nonlinear analysis of a thin circular functionally graded plate and large deflection effects on the forces and moments. Journal of Engineering Materials and Technology-ASME 2008, 130: Article Number: 011009.
[35] Sundararajan N, Prakash T, Ganapathi M. Nonlinear free flexural vibrations of functionally graded rectangular and skew plates under thermal environments. Finite Elements in Analysis and Design 2005, 42: 152-168.
[36] Yang J, Huang XL. Nonlinear transient response of functionally graded plates with general imperfections in thermal environments. Computer Methods in Applied Mechanics and Engineering 2007, 196: 2619-2630.
[37] Reddy JN, Cheng ZQ. Three-dimensional solutions of smart functionally graded plates. Journal of Applied Mechanics ASME 2001, 68: 234-241.
[38] He XQ, Ng TY, Sivashanker S, Liew KM. Active control of FGM plates with integrated piezoelectric sensors and actuators. International Journal of Solids and Structures 2001, 38: 1641-1655.
[39] Liew KM, He XQ, Ng TY, Sivashanker S. Active control of FGM plates subjected to a temperature gradient: modelling via finite element method based on FSDT. International Journal for Numerical Methods in Engineering 2001, 52: 1253-1271.
[40] Liew KM, He XQ, Ng TY, Kitipornchai S. Finite element piezothermoelasticity analysis and the active control of FGM plates with integrated piezoelectric sensors and actuators. Computational Mechanics 2003, 31: 350-358.
[41] Liew KM, Sivashanker S, He XQ, Ng TY. The modelling and design of smart structures using functionally graded materials and piezoelectric sensor/actuator patches. SmartMaterials and Structures 2003, 12: 647-655.
[42] Liew KM, He XQ, Ray T. On the use of computational intelligence in the optimal shape control of functionally graded smart plates. Computational Methods in Applied Mechanics and Engineering 2004, 193: 4475-4492.
[43] Ebrahimi F, Rastgoo A, Kargarnovin MH. Analytical investigation on axisymmetric free vibrations of moderately thick circular functionally graded plate integrated with piezoelectric layers. Journal of Mechanical Science and Technology 2008, 22: 1058-1072.
[44] Ebrahimi F, Rastgoo A. Free vibration analysis of smart annular FGM plates integrated with piezoelectric layers. Smart Materials and Structures 2008, 17: 15-44.
[45] Yang J, Kitipornchai S, Liew KM. Large amplitude vibration of thermo-electric -mechanically stressed FGM laminated plates. Computational Methods in Applied Mechanics and Engineering 2003, 192: 3861-3885.
[46] Dai KY, Liu GR, Han X, Lim KM. Thermomechanical analysis of functionally graded material (FGM) plates using element-free Galerkin method. Computers and Structures 2005, 83: 1487-1502.
[47] Shariyat M. Vibration and dynamic buckling control of imperfect hybrid FGM plates with temperature-dependent material properties subjected to thermo-electro-mechanical loading conditions. Composite Structures 2008, doi:10.1016/j.compstruct.2008.04.003.
[48] Najafizadeh MM, Eslami MR. Buckling analysis of circular plates of functionally graded materials under uniform radial compression. International Journal of Mechanical Sciences 2002, 44: 2479-2493.
[49] Najafizadeh MM, Eslami MR. First-order-theory-based thermoelastic stability of functionally graded material circular plates. AIAA Journal 2002, 40: 1444-1450.
[50] Najafizadeh MM, Heydari HR. Refined theory for thermoelastic stability of functionally graded circular plates. Journal of Thermal Stresses 2004, 27: 857-880.
[51] Najafizadeh MM, Heydari HR. Thermal buckling of functionally graded circular plates based on higher order shear deformation plate theory. European Journal of Mechanics A: Solids 2004, 23: 1085-1100.
[52] Chen XL, Liew KM. Buckling of rectangular functionally graded material plates subjected to nonlinearly distributed in-plane edge loads. Smart Materials & Structures 2004, 13:1430-1437.
[53] Naei MH, Masoumi A, Shamekhi A. Buckling analysis of circular functionally graded material plate having variable thickness under uniform compression by finite-element method. Proceedings of the Institution of Mechanical Engineers Part C-Journal of Mechanical Engineering Science 2007, 221: 1241-1247.
[54] Najafizadeh MM, Heydari HR. An exact solution for buckling of functionally graded circular plates based on higher order shear deformation plate theory under uniform radial compression. International Journal of Mechanical Sciences 2008, 50: 603-612.
[55] Woo J, Meguid SA, Stranart JC, Liew KM. Thermomechanical postbuckling analysis of moderately thick functionally graded plates and shallow shells. International Journal of Mechanical Sciences 2005, 47: 1147-1171.
[56] Yang J, Liew KM, Kitipornchai S. Imperfection sensitivity of the post-buckling behavior of higher-order shear deformable functionally graded plates. International Journal of Solids and Structures 2006, 43: 5247-5266.
[57] Wu TL, Shukla KK, Huang JH. Post-buckling analysis of functionally graded rectangular plates. Composite Structures 2007, 81: 1-10.
[58] Javaheri R, Eslami MR. Buckling of functionally graded plates under in-plane compressive loading. ZAMM 2002, 82: 277-283.
[59] Javaheri R, Eslami MR. Thermal buckling of functionally graded plates. AIAA Journal 2002, 40: 162-169.
[60] Javaheri R, Eslami MR. Thermal buckling of functionally graded plates based on higher order theory. Journal of Thermal Stresses 2002, 25: 603-625.
[61] Na KS, Kim JH. Three-dimensional thermal buckling analysis of functionally graded materials. Composites Part B: Engineering 2004, 35: 429-437.
[62] Na KS, Kim JH. Thermal postbuckling investigations of functionally graded plates using 3-D finite element method. Finite Elements in Analysis and Design 2006, 42: 749-756.
[63] Na KS, Kim JH. Three-dimensional thermomechanical buckling analysis of functionally graded composite plates. Composite Structures 2006, 73: 413-422.
[64] Shariat BAS, Eslami MR. Effect of initial imperfections on thermal buckling of functionally graded plates. Journal of Thermal Stresses 2005, 28: 1183-1198.
[65] Shariat BAS, Eslami MR. Thermal buckling of imperfect functionally graded plates. International Journal of Solids and Structures 2006, 43: 4082-4096.
[66] Ganapathi M, Prakash T, Sundararajan N. Influence of functionally graded material on buckling of skew plates under mechanical loads. Journal of Engineering Mechanics ASCE 2006, 132: 902-905.
[67] Ganapathi M, Prakash T. Thermal buckling of simply supported functionally graded skew plates. Composite Structures 2006, 74: 247-250.
[68] Chen XL, Zhao ZY, Liew KM. Stability of piezoelectric FGM rectangular plates subjected to non-uniformly distributed load, heat and voltage. Advances in Engineering Software 2008, 39: 121-131.
[69] Shukla KK, Kumar KVR, Pandey R, Nath Y. Postbuckling response of functionally graded rectangular plates subjected to thermo-mechanical loading. International Journal of Structural Stability and Dynamics 2007, 7: 519-541.
[70] Liew KM, Yang J, Kitipornchai S. Postbuckling of piezoelectric FGM plates subject to thermo-electro-mechanical loading. International Journal of Solids and Structures 2003, 40: 3869-3892.
[71] Liew KM, Yang J, Kitipornchai S. Thermal post-buckling of laminated plates comprising functionally graded materials with temperature-dependent properties. Journal of Applied Mechanics-ASME 2004, 71: 839-850.
[72] Birman V. Stability of functionally graded hybrid composite plates. Composites Engineering 1995, 5: 913-921.
[73] Feldman E, Aboudi J. Buckling analysis of functionally graded plates subjected to uniaxial loading. Composite Structures 1997, 38: 29-36.
[74] Prakash T, Singha MK, Ganapathi M. Thermal postbuckling analysis of FGM skew plates. Engineering Structures 2008, 30: 22-32.
[75] Bisplinghoff RL, Pian THH. On the vibration of thermally buckled bars and plates. Proceedings of the 9th International Congress for Applied Mechanics 1957, 7: 307-318.
[76] Yang TY, Han AD. Buckled plate vibrations and large amplitude vibrations using high-order triangular elements. AIAA Journal 1983, 21: 758-766.
[77] Zhou RC, Xue DY, Mei C, Gray CC. Vibration of thermally buckled composite plates withinitial deflections using triangular elements. 34th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics and Materials Conference 1993, pt 1: 226-235.
[78] Lee DM, Lee I. Vibration behaviors of thermally postbuckled anisotropic plates using first-order shear deformable plate theory. Computers and Structures 1997, 63: 371-378.
[79] Lee DM, Lee I. Vibration analysis of stiffened laminated plates including thermally postbuckled deflection effect. Journal of Reinforced Plastics and Composites 1997, 16: 1138-1154.
[80] Shiau LC, Wu TY. Free vibration of buckled laminated plates by finite element method. Journal of Vibration and Acoustics ASME 1997, 119: 635-640.
[81] Shiau LC, Kuo SY. Free vibration of thermally buckled composite sandwich plates. Journal of Vibration and Acoustics ASME 2006, 128: 1-7.
[82] Girish J, Ramachandra LS. Thermal postbuckled vibrations of symmetrically laminated composite plates with initial geometric imperfections. Journal of Sound and Vibration 2005, 282: 1137-1153.
[83] Singha MK, Ramachandra LS, Bandyopadhyay JN. Vibration behavior of thermally stressed composite skew plate. Journal of Sound and Vibration 2006, 296: 1093-1102.
[84] Park JS, Kim JH. Thermal postbuckling and vibration analyses of functionally graded plates. Journal of Sound and Vibration 2006, 289: 77-93.
[85] Varelis D, Saravanos DA. Small-amplitude free-vibration analysis of piezoelectric composite plates subject to large deflections and initial stresses. Journal of Vibration and Acoustics ASME 2006, 128: 41-49.
[86] Oh IK, Han JH, Lee I. Postbuckling and vibration characteristics of piezolaminated composite plate subjected to thermo-piezoelectric loads. Journal of Sound and Vibration 2000, 233: 19-40.
[87] Yang J, Shen HS. Nonlinear analysis of functionally graded plates under transverse and in-plane loads. International Journal of Non-linear Mechanics 2003, 38: 467-482.
[88] Yang J, Shen HS. Dynamic response of initially stressed functionally graded rectangular thin plates. Composite Structures 2001, 54: 497-508.
[89] Yang J, Shen HS. Vibration characteristic and transient response of shear-deformable functionally graded plates in thermal environment. Journal of Sound and Vibration 2002,255: 579-602.
[90] Huang XL, Shen HS. Nonlinear and dynamic response of functionally graded plates in thermal environments. International Journal of Solids and Structures 2004, 41: 2403-2427.
[91]黄小林,沈惠申.热环境下贴压电层的功能梯度材料板的自由振动和动力响应.应用力学学报, 2005, 22(3): 456-460.
[92] Huang XL, Shen HS. Vibration and dynamic response of functionally graded plates with piezoelectric actuators in thermal environments. Journal of Sound and Vibration 2006, 289: 25-53.
[93] Shen HS. Nonlinear bending response of functionally graded plates subjected to transverse loads and in thermal environments. International Journal of Mechanical Sciences 2002, 44: 561-584.
[94] Shen HS. Postbuckling of FGM plates with piezoelectric actuators under thermo-electro-mechanical loadings. International Journal of Solids and Structures 2005, 42: 6101-6121.
[95] Shen HS. Thermal postbuckling behavior of shear deformable FGM plates with temperature-dependent properties. International Journal of Mechanical Sciences 2007, 49: 466-478.
[96] Shen HS, Li SR. Postbuckling of sandwich plates with FGM face sheets and temperature-dependent properties. Composites Part B: Engineering 2008, 39: 332-344.
[97]陈伟球,叶贵如,蔡金标,丁皓江.横观各向同性功能梯度材料矩形板的自由振动.振动工程学报, 2001, 33(4): 263-267.
[98] Chen WQ, Bian ZG, Ding HJ. Three-dimensional analysis of a thick FGM rectangular plate in thermal environment. Journal of Zhejiang University: Science 2003, 4: 1-7.
[99] Li XY, Ding HJ, Chen WQ. Pure bending of simply supported circular plate of transversely isotropic functionally graded material. Journal of Zhejiang University: Science 2006, 7: 1324-1328.
[100] Chen JY, Chen WQ. 3D analytical solution for a rotating transversely isotropic annular plate of functionally materials. Journal of Zhejiang University: Science A 2007, 8: 1038-1043.
[101] Li XY, Ding HJ, Chen WQ. Elasticity solutions for a transversely isotropic functionally graded circular plate subject to an axisymmetric transverse load qrk. Journal of Solid andStructures 2008, 45: 191-210.
[102] Li XY, Ding HJ, Chen WQ. Axisymmetric elasticity solutions for a uniformly loaded annular plate of transversely isotropic functionally graded materials. Acta Mechanica 2008, 196: 139-159.
[103] Bian ZG, Ying J, Cheng WQ, Ding HJ. Bending and free vibration analysis of a smart functionally graded plate. Structural Engineering and Mechanics 2006, 23: 97-113.
[104]边祖光,王亚明,陈伟球. Winkler地基上功能梯度材料厚板的自由振动.工程力学, 2001, (增刊): 181-185.
[105] Huang ZY, Lu CF, Chen WQ. Benchmark solutions for functionally graded thick plate resting on Winkler-Pasternak elastic foundations. Composite Structures 2008, 85: 95-104.
[106]曹志远.功能梯度板的非线性动力分析.固体力学学报, 2006, 27(1): 21-25.
[107]程红梅,曹志远.沿板平面变异FGM板件的三维动力特性.振动工程学报, 2007, 20(3): 238-242.
[108]杨正光,仲政,戴瑛.功能梯度矩形板的三维弹性分析.力学季刊, 2004, 25(1): 15-20.
[109]张晓日,仲政.功能梯度压电圆板自由振动问题的三维精确分析.力学季刊, 2005, 26(1): 81-86.
[110] Zhong Z, Shang ET. Closed-form solutions of three-demensional functionally graded plates. Mechanics of Advanced Materials and Structures 2008, 15: 355-363.
[111] Nie GJ, Zhong Z. Vibration analysis of functionally graded annular sectorial plates with simply supported radial edges. Composite Structures 2008, 84: 167-176.
[112] Nie GJ, Zhong Z. Semi-analytical solution for three-dimensional vibration of functionally graded circular plates. Computer Methods in Applied Mechanics and Engineering 2007, 196: 4901-4910.
[113]马连生,赵永刚,杨静宁.功能梯度圆板的轴对称非线性分析——大挠度问题.兰州理工大学学报, 2004, 30(6): 139-142.
[114]马连生,赵永刚.功能梯度材料圆(环)板的屈曲分析.甘肃工业大学学报, 2003, 29(4): 140-143.
[115]马连生,欧志英.剪切变形对FGM圆板轴对称屈曲的影响.应用力学学报, 2004, 21(2): 129-131.
[116] Li SR, Zhang JH, Zhao YG. Nonlinear thermomechanical post-buckling of circular FGMplate with geometric imperfection. Thin-Walled Structures 2007, 45: 528-536.
[117]李世荣,范亮亮.热环境中功能梯度材料圆板的自由振动.振动工程学报, 2007, 20(4): 353-360.
[118]周凤玺,李世荣.功能梯度材料圆板的三维弹性分析.振动与冲击, 2008, 27(1): 115-118, 130.
[119]王铁军,马连生,石朝锋.功能梯度中厚圆/环板轴对称弯曲问题的解析解.力学学报, 2004, 36(3): 348-353.
[120] Ma LS, Wang TJ. Axisymmetric post-buckling of a functionally graded circular plate subjected to uniformly distributed radial compression. Materials Science Forum 2003, 423-425: 719-724.
[121] Ma LS, Wang TJ. Nonlinear bending and post-buckling of a functionally graded circular plate under mechanical and thermal loadings. International Journal of Solids and Structures 2003, 40: 3311-3330.
[122] Ma LS, Wang TJ. Relationships between axisymmetric bending and buckling solutions of FGM circular plates based on third-order plate theory and classical plate theory. International Journal of Solids and Structures 2004, 41: 85–101.
[123]武兰河,王立彬,刘淑红.四边简支功能梯度矩形板的热屈曲分析.工程力学, 2004, 21(2): 152-156, 166.
[124] Wu LH. Thermal buckling of a simply supported moderately thick rectangular FGM plate. Composite Structures 2004, 64: 211-218.
[125] Wu LH, Wang HJ, Wang DB. Dynamic stability analysis of FGM plates by the moving least squares differential quadrature method. Composite Structures 2007, 77: 383-394.
[126]杨志安,贾尚帅.受机械力作用金属/陶瓷功能梯度板的非线性振动模型.功能材料, 2007, (增刊): 3638-3640.
[127]赵凤群.非保守FGM杆和板的非线性动力学特性研究. [博士学位论文],西安,西安理工大学, 2007.
[128]刘玮,闫铂.具有压电元件的功能梯度弹性薄板的振动特性.振动与冲击, 2007, 26(5): 1-3, 7.
[129] Reissner E. The effect of transverse shear deformation on the bending of elastic plates. Journal of Applied Mechanics 1945, 12: A69-A77.
[130] Hencky H. Uber die berucksichtibung der schubverzerrungen in ebenen platten. Ing.-Archiv. 1947, 16: 72-76.
[131] Mindlin RD. Influence of rotatory inertia and shear on flexural motions of isotropic, elastic plates, Journal of Applied Mechanics 1951, 18: 31-38.
[132] Reddy JN. A refined nonlinear theory of plates with transverse shear deformation. International Journal of Solids and Structures 1984, 20: 881-896.
[133]沈惠申.高阶剪切变形板理论Kármán型方程及在热后屈曲分析中的应用.应用数学和力学, 1997, 18 (12): 1059-1073.
[134]陈予恕,唐云等.非线性动力学现代分析方法(第二版).北京:科学出版社, 2000.
[135]朱位秋.随机振动.北京:科学出版社, 1992.
[136] Shen HS. Hygrothermal effects on the nonlinear bending of shear deformable laminated plates. Journal of Engineering Mechanics ASCE 2002, 128: 493-496.
[137] Shen HS. Postbuckling of shear of deformable laminated plates with piezoelectric actuators under complex loading conditions. International Journal Solids and Structures 2001, 38: 7703-7721.
[1] He XQ, Ng TY, Sivashanker S, Liew KM. Active control of FGM plates with integrated piezoelectric sensors and actuators. International Journal of Solids and Structures 2001, 38: 1641-1655.
[2] Liew KM, He XQ, Ng TY, Sivashanker S. Active control of FGM plates subjected to a temperature gradient: modelling via finite element method based on FSDT. International Journal for Numerical Methods in Engineering 2001, 52: 1253-1271.
[3] Liew KM, He XQ, Ng TY, Kitipornchai S. Finite element piezothermoelasticity analysis and the active control of FGM plates with integrated piezoelectric sensors and actuators. Computational Mechanics 2003, 31: 350-358.
[4] Reddy JN, Cheng ZQ. Three-dimensional solutions of smart functionally graded plates. Journal of Applied Mechanics ASME 2001, 68: 234-241.
[5] Aryana F, Bahai H, Mirzaeifar R, Yeilaghi A. Modification of dynamic characteristics of FGM plates with integrated piezoelectric layers using first- and second-order approximations. International Journal for Numerical Methods in Engineering 2007, 70: 1409-1429.
[6] Yang J, Kitipornchai S, Liew KM. Large amplitude vibration of thermo-electric -mechanically stressed FGM laminated plates. Computational Methods in Applied Mechanics and Engineering 2003, 192: 3861-3885.
[7] Huang XL, Shen HS. Vibration and dynamic response of functionally graded plates with piezoelectric actuators in thermal environments. Journal of Sound and Vibration 2006, 289: 25-53.
[8] Mallik N, Ray MC. Effective coefficients of piezoelectric fiber-reinforced composites. AIAA Journal 2003, 41: 704-710.
[9] Reddy BA, Ray MC. Optimal control of smart functionally graded plates using piezoelectric fiber reinforced composites. Journal of Vibration and Control 2007, 13: 795-814.
[10] Panda S, Ray MC. Active constrained layer damping of geometrically nonlinear vibrations of functionally graded plates using piezoelectric fiber-reinforced composites. Smart Materials and Structures 2008, 17: 025012.
[11] Reddy JN. Refined nonlinear theory of plates with transverse shear deformation. International Journal of Solids and Structures 1984, 20: 881-896.
[12] Shen HS. Kármán-type equations for a higher–order shear deformation plate theory and its use in the thermal postbuckling analysis. Applied Mathematics and Mechanics 1997, 18: 1137-1152.
[13] Shen HS. Hygrothermal effects on the postbuckling of shear deformable laminated plate. International Journal of Mechanical Science 2001, 43: 1259-1281.
[14] Tsai SW, Hahn HT. Introduction to Composite Materials, Westport, CT: Technomic Publishing Co., 1980.
[15] Reddy JN, Chin CD. Thermomechanical analysis of functionally graded cylinders and plates. Journal of Thermal Stresses 1998, 21: 593-629.
[16] Shen HS. Nonlinear bending response of functionally graded plates subjected to transverse loads and in thermal environments. International Journal of Mechanical Sciences 2002, 44: 561-584.
[17] Reddy JN. On laminated composite plates with integrated sensors and actuators. Engineering Structures 1999, 21: 568-593.
[18]黄小林.船舶与海洋工程中复合材料板构件的非线性振动和动力响应. [博士学位论文],上海,上海交通大学, 2005.
[19]王海期.非线性振动.高等教育出版社,北京, 1992.
[20] Pearson CE. Numerical Methods in Engineering and Science. Van Nostrand Reinhold Company INC., 1996.
[21] Hussein M, Heyliger P. Three-dimensional vibrations of laminated piezoelectric cylinders. Journal of Engineering Mechanics ASCE 1998, 124:1294-1298.
[22] Touloukian YS. Thermophysical properties of high temperature solid materials, McMillan, New York, 1967.
[23] Aglietti GS, Cunningham PR. Is a simple support really that simple? Journal of Sound and Vibration 2002, 257: 321-335.
[24] Park JS, Kim JH. Thermal postbuckling and vibration analyses of functionally graded plates. Journal of Sound and Vibration 2006, 289: 77-93.
[25] Abrate S. Free vibration, buckling, and static deflections of functionally graded plates.Composites Science and Technology 2006, 66: 2383-2394.
[26] Sundararajan N, Prakash T, Ganapathi M. Nonlinear free flexural vibrations of functionally graded rectangular and skew plates under thermal environments. Finite Elements in Analysis and Design 2005, 42: 152-168.
[27] Praveen GN, Reddy JN, Nonlinear transient thermoelastic analysis of functionally graded ceramic-metal plates. International Journal of Solids and Structures 1998, 35: 4457-4476.
[28] Xia XK, Shen HS. Nonlinear vibration and dynamic response of FGM plates with piezoelectric fiber reinforced composite actuators. (submitted).
[1] Bisplinghoff RL, Pian THH. On the vibration of thermally buckled bars and plates. Proceedings of the 9th International Congress for Applied Mechanics 1957, 7: 307-318.
[2] Yang TY, Han AD. Buckled plate vibrations and large amplitude vibrations using high-order triangular elements. AIAA Journal 1983, 21: 758-766.
[3] Zhou RC, Xue DY, Mei C, Gray CC. Vibration of thermally buckled composite plates with initial deflections using triangular elements. 34th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics and Materials Conference 1993, pt 1: 226-235.
[4] Lee DM, Lee I. Vibration behaviors of thermally postbuckled anisotropic plates using first-order shear deformable plate theory. Computers and Structures 1997, 63: 371-378.
[5] Lee DM, Lee I. Vibration analysis of stiffened laminated plates including thermally postbuckled deflection effect. Journal of Reinforced Plastics and Composites 1997, 16: 1138-1154.
[6] Shiau LC, Wu TY. Free vibration of buckled laminated plates by finite element method. Journal of Vibration and Acoustics ASME 1997, 119: 635-640.
[7] Shiau LC, Kuo SY. Free vibration of thermally buckled composite sandwich plates. Journal of Vibration and Acoustics ASME 2006, 128: 1-7.
[8] Girish J, Ramachandra LS. Thermal postbuckled vibrations of symmetrically laminated composite plates with initial geometric imperfections. Journal of Sound and Vibration 2005, 282: 1137-1153.
[9] Singha MK, Ramachandra LS, Bandyopadhyay JN. Vibration behavior of thermally stressed composite skew plate. Journal of Sound and Vibration 2006, 296: 1093-1102.
[10] Park JS, Kim JH. Thermal postbuckling and vibration analyses of functionally graded plates. Journal of Sound and Vibration 2006, 289: 77-93.
[11] Varelis D, Saravanos DA. Small-amplitude free-vibration analysis of piezoelectric composite plates subject to large deflections and initial stresses. Journal of Vibration and Acoustics ASME 2006, 128: 41-49.
[12] Oh IK, Han JH, Lee I. Postbuckling and vibration characteristics of piezolaminatedcomposite plate subjected to thermo-piezoelectric loads. Journal of Sound and Vibration 2000, 233: 19-40.
[13] Shen HS. Nonlinear bending response of functionally graded plates subjected to transverse loads and in thermal environments. International Journal of Mechanical Sciences 2002, 44: 561-584.
[14] Shen HS. Thermal postbuckling behavior of shear deformable FGM plates with temperature-dependent properties. International Journal of Mechanical Sciences 2007, 49: 466-478.
[15] Reddy JN. A refined nonlinear theory of plates with transverse shear deformation. International Journal of Solids and Structures 1984, 20: 881-896.
[16]沈惠申,板壳后屈曲行为.上海:上海科学技术出版社, 2002.
[17] Shen HS. Nonlinear thermal bending response of FGM plates due to heat conduction. Composites Part B: Engineering 2007, 38: 201-215.
[18]王海期.非线性振动.高等教育出版社,北京, 1992.
[19] Tang S. Natural vibration of isotropic plates with temperature-dependent properties. AIAA Journal 1969, 7: 725-727.
[20] Ilanko S, Dickinson SM. The vibration and post-buckling of geometrically imperfect, simply supported, rectangular plate under uni-axial loading, Part I: Theoretical approach. Journal of Sound and vibration 1987, 118: 313-336.
[21] Eisley JG. Nonlinear vibrations of beams and rectangular plates. ZAMP 1964, 15: 167-175.
[22] Reddy JN, Chin CD. Thermomechanical analysis of functionally graded cylinders and plates. Journal of Thermal Stresses 1998, 21: 593-629.
[23]夏贤坤,沈惠申.功能梯度材料剪切板屈曲后的自由振动.固体力学学报, 2008, 29(2): 129-133.
[24]夏贤坤,沈惠申.功能梯度材料剪切板热屈曲后的非线性振动.振动工程学报, 2008, 21(2): 120-125.
[1] Oh IK, Han JH, Lee I. Postbuckling and vibration characteristics of piezolaminated composite plate subjected to thermo-piezoelectric loads. Journal of Sound and Vibration 2000, 233: 19-40.
[2] Varelis D, Saravanos DA. Small-amplitude free-vibration analysis of piezoelectric composite plates subject to large deflections and initial stresses. Journal of Vibration and Acoustics ASME 2006, 128: 41-49.
[3] Mallik N, Ray MC. Effective coefficients of piezoelectric fiber-reinforced composites. AIAA Journal 2003, 41: 704-710.
[4] Reddy JN. A refined nonlinear theory of plates with transverse shear deformation. International Journal of Solids and Structures 1984, 20: 881-896.
[5] Shen H-S. Kármán-type equations for a higher–order shear deformation plate theory and its use in the thermal postbuckling analysis. Applied Mathematics and Mechanics 1997, 18: 1137-1152.
[6] Girish J, Ramachandra LS. Thermal postbuckled vibrations of symmetrically laminated composite plates with initial geometric imperfections. Journal of Sound and Vibration 2005, 282: 1137-1153.
[7] Hussein M, Heyliger P. Three-dimensional vibrations of laminated piezoelectric cylinders. Journal of Engineering Mechanics ASCE 1998, 124:1294-1298.
[8] Reddy JN, Chin CD. Thermomechanical analysis of functionally graded cylinders and plates. Journal of Thermal Stresses 1998, 21: 593-629.
[9] Xia XK, Shen HS. Vibration of postbuckled FGM hybrid laminated plates in thermal environment. Engineering Structures 2008, 30: 2420-2435.
[10] Xia XK, Shen HS. A comparison of vibration characteristics for postbuckled FGM plates with piezoelectric fiber reinforced composite actuators. International Journal of Structural Stability and Dynamics (accepted).
[1] Yang J, Shen HS, Zhang L. Nonlinear local response of foam-filled sandwich plates with laminated faces under combined transverse and in-plane loads. Composite Structures 2001, 52: 137-148.
[2] Nayak AK, Moy SSJ, Shenoi RA. A higher order finite element theory for buckling and vibration analysis of initially stressed composite sandwich plates. Journal of Sound and Vibration 2005, 286: 763-780.
[3] Kokhanenko, Yu V. Numerical solution to the buckling problem for a sandweich plate under uniaxial compression. International Applied Mechanics 2006, 42:1045-1051.
[4] Wang TG, Rajaram S, Nutt SR. Free vibration analysis of sandwich plate with consistent higer-order approach. 43rd AIAA Aerospace Sciences Meeting and Exhibit - Meeting Papers 2005, pt: 7023-7030.
[5] Chakrabarti A, Abdul H. Buckling of laminated sandwich plates using an efficient plate model. International Shipbuilding Progress 2007, 54: 63-81.
[6] Shen HS, Li SR. Postbuckling of sandwich plates with FGM face sheets and temperature-dependent properties. Composites Part B: Engineering 2008, 39: 332-344.
[7] Zhao J, Li YZ, AI X. Analysis of transient thermal stress in sandwich plate with functionally graded coatings. Thin Solid Films 2008, 516: 7581-7587.
[8] Li Q, Lu VP, Kou KP. Three-dimensional vibration analysis of functionally graded material sandwich plates. Journal of Sound and Vibration 2008, 311: 498-515.
[9] Shiau LC, Kuo SY. Free vibration of thermally buckled composite sandwich plates. Journal of Vibration and Acoustics ASME 2006, 128: 1-7.
[10] Reddy JN. A refined nonlinear theory of plates with transverse shear deformation. International Journal of Solids and Structures 1984, 20: 881-896.
[11] Shen H-S. Kármán-type equations for a higher–order shear deformation plate theory and its use in the thermal postbuckling analysis. Applied Mathematics and Mechanics 1997, 18: 1137-1152.
[12]夏贤坤.功能梯度材料夹芯板的后屈曲.强度与环境, 2007, 34(3): 46-52.
[13] Xia XK, Shen HS. Vibration of post-buckled sandwich plates with FGM face sheets in thermal environment. Journal of Sound and Vibration 2008, 314: 254-274.