各向异性功能梯度平面梁的弹性力学解
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摘要
功能梯度材料是一种特殊的非均匀材料,具有材料组分沿某个方向连续分布的特点,结构不存在内部界面。与传统层合复合材料相比,具有无可比拟的优势。功能梯度梁在结构工程和智能结构中将有广泛的应用。本文系统地研究了各向异性功能梯度材料的有限长弹性梁、压电梁和磁电梁在多项式分布载荷和任意分布载荷作用时,各种端部条件下的弹性力学解。
首先从功能梯度各向异性弹性材料、压电材料或磁电材料的基本方程出发,导出平面问题的应力函数,电位移函数或磁感应强度函数所满足的偏微分方程。方程中的弹性柔度系数、压电电压系数、自由介电隔离率、磁电系数或磁通系数是梁厚度坐标的函数。当梁的上下表面受多项式载荷作用时,在假定梁的材料参数沿梁厚度方向是任意连续函数的情况下,都可以得到问题的解析解。当梁的上下表面受任意载荷作用,在求取解析解时,梁的材料参数假定为沿梁的厚度方向是指数函数或某种幂函数形式;在求取半解析解时,则可指定任意连续函数形式。其次,当梁的上下表面受多项式分布载荷作用时,设定应力函数、电位移函数或磁感应强度函数分别为长度方向的多项式形式,其中包含厚度方向的待定函数;当梁的上下表面受任意形式的分布载荷作用时,则设定应力函数、电位移函数或磁感应强度函数分别由两部分叠加而成。一是长度方向的三角函数与高度方向为待定函数的乘积,另一为长度方向的一次多项式函数形式。接着利用应变协调方程、电场协调方程和磁场协调方程等,求解出含待定常数的应力函数、电位移函数和磁感应强度函数表达式。然后得到应力、电位移、磁感应强度、轴力、剪力、弯矩、平均电位移、平均磁感应强度、位移和电势的表达式。再利用边界条件完全确定应力函数、电位移函数和磁感应强度函数。最后利用应力函数、电位移函数和磁感应强度函数得到应力、电位移、磁感应强度和位移、电势、磁势等物理量,从而得到了各向异性功能梯度梁的各种弹性力学解。
本文还给出若干解析解的数值算例,并与其他研究工作的数值算例和有限元算例进行了比较,发现并讨论了其中相同和不同之处。
Functionally graded materials(FGMs) are a kind of special non-homogenous materials,whose properties varying continuously along certain directions.There are no internal interfaces in FGMs.Compared with traditional composites,FGMs have a great deal of superiorities.FGMs will be widely applied in structure engineering and intelligent structures.Elasticity solutions are studied in this work for anisotropic functionally graded elastic,piezoelectric and magneto-electro-elastic finite-length beams with different kinds of conditions at their two ends,and subjected to polynomial and arbitrary loads on their upper and lower surfaces.
Firstly,based on elementary equations of anisotropic elastic materials,piezoelectric materials and magneto-electro-elastic materials,the partial differential equations are derided for the stress functions,electric displacement function and magnetic induction function of the plane problems.The coefficients of elastic compliance,piezoelectricity, dielectric impermeability,piezo-magnetism,magneto-electricity,and magnetic permeability are functions of the thickness coordinate.The material coefficients are assumed to be arbitrary continuous functions of the thickness coordinate so as to obtain the analytical solutions of the beams subjected to polynomial loads.When the beams are subjected to arbitrary loads,these coefficients are assumed to be exponential or power functions for obtaining analytical solutions,and to be any arbitrary functions for seeking semi-analytical solutions.Secondly,for beams subjected to polynomial loads,the stress function,electric displacement function and magnetic induction function are assumed in forms of polynomials in the longitudinal coordinate,with undetermined functions of the thickness coordinate.For beams subjected to arbitrary loads,the stress function,electric displacement function and magnetic induction function are assumed to consist of two parts,respectively.One is a product of a trigonometric function of the longitudinal coordinate and an undetermined function of the thickness coordinate,and the other a linear polynomial of longitudinal coordinate with unknown coefficients depending on thickness coordinate.Thirdly,the expressions of stress function,electric displacement function and magnetic induction function with undetermined constants are acquired by virtue of the compatibility equations of strain,electric field and magnetic field.The analytical expressions of stresses,electric displacements,magnetic inductions,axial force, bending moment,shear force,average electric displacement,average magnetic induction, displacements,electric potential and magnetic potential are then deduced,with integral constants determinable from the boundary conditions.Finally,the expressions are educed of the stresses,electric displacements,magnetic inductions,displacements,electric potential and magnetic potential.Thus the elasticity solutions are obtained for the anisotropic functionally graded beams.
Numerical examples of these elasticity solutions are presented.Comparisons are made among numerical examples of other literatures,by finite element method and by this work.Similarities and differences are found and discussed.
首先从功能梯度各向异性弹性材料、压电材料或磁电材料的基本方程出发,导出平面问题的应力函数,电位移函数或磁感应强度函数所满足的偏微分方程。方程中的弹性柔度系数、压电电压系数、自由介电隔离率、磁电系数或磁通系数是梁厚度坐标的函数。当梁的上下表面受多项式载荷作用时,在假定梁的材料参数沿梁厚度方向是任意连续函数的情况下,都可以得到问题的解析解。当梁的上下表面受任意载荷作用,在求取解析解时,梁的材料参数假定为沿梁的厚度方向是指数函数或某种幂函数形式;在求取半解析解时,则可指定任意连续函数形式。其次,当梁的上下表面受多项式分布载荷作用时,设定应力函数、电位移函数或磁感应强度函数分别为长度方向的多项式形式,其中包含厚度方向的待定函数;当梁的上下表面受任意形式的分布载荷作用时,则设定应力函数、电位移函数或磁感应强度函数分别由两部分叠加而成。一是长度方向的三角函数与高度方向为待定函数的乘积,另一为长度方向的一次多项式函数形式。接着利用应变协调方程、电场协调方程和磁场协调方程等,求解出含待定常数的应力函数、电位移函数和磁感应强度函数表达式。然后得到应力、电位移、磁感应强度、轴力、剪力、弯矩、平均电位移、平均磁感应强度、位移和电势的表达式。再利用边界条件完全确定应力函数、电位移函数和磁感应强度函数。最后利用应力函数、电位移函数和磁感应强度函数得到应力、电位移、磁感应强度和位移、电势、磁势等物理量,从而得到了各向异性功能梯度梁的各种弹性力学解。
本文还给出若干解析解的数值算例,并与其他研究工作的数值算例和有限元算例进行了比较,发现并讨论了其中相同和不同之处。
Functionally graded materials(FGMs) are a kind of special non-homogenous materials,whose properties varying continuously along certain directions.There are no internal interfaces in FGMs.Compared with traditional composites,FGMs have a great deal of superiorities.FGMs will be widely applied in structure engineering and intelligent structures.Elasticity solutions are studied in this work for anisotropic functionally graded elastic,piezoelectric and magneto-electro-elastic finite-length beams with different kinds of conditions at their two ends,and subjected to polynomial and arbitrary loads on their upper and lower surfaces.
Firstly,based on elementary equations of anisotropic elastic materials,piezoelectric materials and magneto-electro-elastic materials,the partial differential equations are derided for the stress functions,electric displacement function and magnetic induction function of the plane problems.The coefficients of elastic compliance,piezoelectricity, dielectric impermeability,piezo-magnetism,magneto-electricity,and magnetic permeability are functions of the thickness coordinate.The material coefficients are assumed to be arbitrary continuous functions of the thickness coordinate so as to obtain the analytical solutions of the beams subjected to polynomial loads.When the beams are subjected to arbitrary loads,these coefficients are assumed to be exponential or power functions for obtaining analytical solutions,and to be any arbitrary functions for seeking semi-analytical solutions.Secondly,for beams subjected to polynomial loads,the stress function,electric displacement function and magnetic induction function are assumed in forms of polynomials in the longitudinal coordinate,with undetermined functions of the thickness coordinate.For beams subjected to arbitrary loads,the stress function,electric displacement function and magnetic induction function are assumed to consist of two parts,respectively.One is a product of a trigonometric function of the longitudinal coordinate and an undetermined function of the thickness coordinate,and the other a linear polynomial of longitudinal coordinate with unknown coefficients depending on thickness coordinate.Thirdly,the expressions of stress function,electric displacement function and magnetic induction function with undetermined constants are acquired by virtue of the compatibility equations of strain,electric field and magnetic field.The analytical expressions of stresses,electric displacements,magnetic inductions,axial force, bending moment,shear force,average electric displacement,average magnetic induction, displacements,electric potential and magnetic potential are then deduced,with integral constants determinable from the boundary conditions.Finally,the expressions are educed of the stresses,electric displacements,magnetic inductions,displacements,electric potential and magnetic potential.Thus the elasticity solutions are obtained for the anisotropic functionally graded beams.
Numerical examples of these elasticity solutions are presented.Comparisons are made among numerical examples of other literatures,by finite element method and by this work.Similarities and differences are found and discussed.
引文
[1]王保林,韩杰才,张幸红,2003.非均匀材料力学.科学出版社,北京.
[2]Yamanouchi,M.,Koizumi,M.,Hirai,T.,Shiota,I.,1990.Proceedings of the First International Symposium Functionally Gradient Materials.Japan,1-15.
[3]Koizumi,M.,1993.Concept of FGM.Ceramic Transaction:Functionally gradient materials,34:3-10.
[4]Koizumi,M.,1997.FGM activities in Japan.Composites B,28:1-4.
[5]Rabin,B.H.,Shiota,I.,1995.Functionally gradient materials.Materials Research Bulletin,20:14-18.
[6]余茂黎,魏明坤,1992.功能梯度材料的研究动态.功能材料,23:184-191.
[7]林启荣,刘振兴,2000.噪声控制压电智能系统研究的现状和展望.力学季刊,21(1):27-32.
[8]Rao,S.S.,Sunar,M.,1994.Piezoelectricity and its use in disturbance sensing and control of flexible structures.Applied Mechanics Reviews,47:113-123.
[9]Sosa,H.A.,Castro,M.A.,1993.Electroelastic analysis of Piezoelectric laminated structures.Applied Mechanics Reviews,46(11):521-528.
[10]Cady,W.G.,1964.Piezoelectricity,revised edition,Dover Publications,Inc.,Newyork.
[11]Jafe,B.,Cook,W.R.,Jafe,H.,1971.Piezoelectricity Ceramics,Academic Press,New York.
[12]Uchino,K.,1997.Piezoelectric actuators and ultrasonic motors.Kiuwer Academic Publishers Boston/Dordrecht/London.
[13]Wu,C.C.M.,Kahn,M.,Moy,W.,1996.Piezoelectric ceramics with functional gradients:a new application in material design.Journal of the American Ceramic Society,79(3):809-812
[14]Zhu,X.H.,Wang,Q.,Meng,Z.Y.,1995.A functionally graded piezoelectric actuator prepared by powder metallurgical process in PNN-PZ-PT system.Journal of Materials Science Letters,14516-14518.
[15]Hirai,T.,Chen,L.,1999.Recent and prospective development of functionally graded materials in Japan.Materials Science Forum,308-311,509-514.
[16]Miyamoto,Y.,Niino,M.,Koizumi,M.,1997.FGM research program in Japan-from structural to functional uses.Functionally Graded Materials,Ed.By Shiota and Miyamoto,Pub.By Elsevier Science B,V:1-8.
[17]李永,宋健,张志民,2003.梯度功能力学.清华大学出版社,北京.
[18]Functionally Graded Materials Ⅶ,Materials Science Forum:Vol.423425(2003),by Trans Tech Publications Inc.Publishers in Science and Engineering,Proceeding of the Seventh International Symposium on Functionally Graded Materials,October 15-18,2002,Beijing,China.
[19]李碧容,雍歧龙,张国亮,1999.功能梯度材料的发展、制备方法及应用前景.云南工业大学报,15(4):56-58.
[20]金灯仁,2003.PZT、PZT/ZnO系梯度功能压电陶瓷及其执行器的研制与表征.上海交通大学博士论文.
[21]Luo,Y.,Li,S.,Chen,J.,2003.Preparation and characterization of A1203-Ti3SiC2 composites and its functionally graded materials.Materials Research Bulletin,38(1):69-78.
[22]Luo,Y.,Wei,P.,Li,S.,Wang,R.G.,Li,J.Q.,2003.A novel functionally graded material in the Ti-Si-C system.Materials Science and Engineering(A),345(1-2):99-105.
[23]Ling,Y.H.,Li,J.T,Ge,C.C.,Bai,X.D.,2002.Fabrication and evaluation of SiC/Cu functionally graded material used for plasma facing components in a fusion reactor.Journal of Nuclear Materials,303(2-3):188-195.
[24]Zeng,Y.P.,Jiang,D.L.,2000.Fabrication and properties of tape-cast laminated and functionally gradient alumina-tianium carbide materials.Journal of the American Ceramic Society,83(12):2999-3003.
[25]Xiong,H.P.,Shen,Q.,Li,J.G.,Zhang,L.M.,Yuan,R.Z.,2000.Design and microstructures of Ti/TiAl/Al system functionally graded material.Journal of Materials Science Letters,19:989-993.
[26]Lekhnitskii,S.G,1968.Anisotropic Plate.Gordon and Breach,New York.
[27]Noda,N.,Tsuji,T.,1991.Steady thermal stresses in a plate of functionally gradient material.Journal of Japanese Society for Mechanical Engineering(A),57(535):625-631.
[28]Obata,Y.,Noda,N.,1994.Steady thermal stresses in a hollow circular cylinder and a hollow sphere of a functionally graded materials.Journal of Thermal Stresses,17:471-178.
[29]Aboudi,J.,Pindera,M.J.,Arnold,S.M.,1995.A coupled higher-order theory for functionally graded composites with partial homogenization.Composites Engineering,5:771-792.
[30]Aboudi,J.,Pindera,M.J.,Arnold,S.M.,1999.Higher-order theory for functionally graded materials.Composites B,30(8):777-832.
[31]Mian,A.M.,Spencer,A.J.M.,1998.Exact solutions for functionally graded and laminated elastic materials.Journal oft he Mechanics and Physics of Solids,42(12):2283-2295.
[32]赵军,艾兴,张建华,1998.基于高承载能力的梯度功能材料设计.陶瓷学报,19:125-129.
[33]李春雨,邹振祝,1998.受内压的功能梯度材料圆筒应力分析.工程力学增刊,153-156.
[34]Reddy,J.N.,Chin,C.D.,1998.Thermomachanical analysis of functionally graded cylinders and plates.Journal of Thermal stresses,21:593-626.
[35]Reddy,J.N.,Wang,C.,Kitipomchai,M.S.,1999.Axisymmetric bending of functionally graded circular and annular plates.European Journal of Mechanics-A/Solids,18:185-199.
[36]Reddy,J.N.,1984.A Simple higher-order theory for laminated composite plates.Journal of Applied Mechanics,51:745-752.
[37]Reddy,J.N.,2000.Analysis of functionally graded plates.International Journal for Numerical Methods in Engineering,47:663-684.
[38]Cheng,Z.Q.,Batra R.C.,2000.Three-dimensional thermoelastic deformations of a functionally graded elliptic plate.Composites B,31:97-106.
[39]Reddy,J.N.,Cheng,Z.Q.,2001.Three-dimensional thermomechanical deformations of functionally graded rectangular plates.European Journal of Mechanics-A/Solids,20:841-855.
[40]Cheng,Z.Q.,2001.Nonlinear bending of inhomogeneous plates.Engineering Structures,23:1359-1363.
[41]Vel,S.S.,Batra,R.C.,2002.Exact solution for thermoelastic deformation of functionally graded thick rectangular plates.AIAA Journal,40:1421-1433.
[42]Vel,S.S.,Barea,R.C.,2003.Three-dimensional analysis of transient thermal stresses in functionally graded rectangular plates.International Journal of Solids and Structures,40(25):7181-7196.
[43]Vel,S.S.,Barea,R.C.,2004.Three-dimensional Exact solution for the vibration of functionally graded rectangular plates.Journal of Sound and Vibration,272:703-730.
[44]Woo,J.,Meguid,S.A.,2001.Nonlinear analysis of functionally graded plates and shallow shells.International Journal of Solids and Structures,38:7409-7421.
[45]Rooney,F.M.,2001.Tension,bending,and flexure of functionally graded cylinders.International Journal of Solids and Structures,38:413-421.
[46]Sankar,B.V.,2001.An elasticity solution for functionally graded beams.Composites Science and Technology,61:689-696.
[47]Sankar,B.V,Tzeng,J.T.,2002.Thermal stresses in functionaly graded beams.AIAA Journal,40(6):1228-1232.
[48]Zhu,H.,Sankar,B.V.,2004.A combined Fourier Series-Galerkin method for the analysis of functionally graded beams.Journal of Applied Mechanics,71(3):421-424.
[49]Shen,H.S.,2002.Postbuckling analysis of axially loaded functionally graded cylindrical panels in thermal environments.International Journal of Solids and Structures,39:5991-6010.
[50]Shen,H.S.,2002.Nonlinear bending response of functionally graded plates subjected to transverse loads and in thermal environments.International Journal of Mechanical Sciences,44:561-584.
[51]Huang,X.L.,Shen,H.S.,2004.Nonlinear vibration and dynamic response of functionally graded plates in thermal environments.International Journal of Solids and Structures,41:2403-2427.
[52]Shen,H.S.,2002.Postbuckling analysis of axially-loaded functionally graded cylindrical shells in thermal environments.Composites Science and Technology,62:977-987.
[53]Shen,H.S.,2004.Thermal postbuckling behavior of functionally graded cylindrical shells with temperature-dependent properties.International Journal of Solids and Structures,41:1961-1974.
[54]Yang,Y.Y.,2000.Time-dependent stress analysis in functionally graded materials International Journal of Solids and Structures,37:7593-7608.
[55]杨杰,沈惠申,2002.面内预应力功能梯度材料矩形板的横向弯曲.力学季刊,23:342-346.
[56]杨杰,沈惠申,2003.热/机械载荷下功能梯度材料矩形板的弯曲行为.固体力学学报,24(1):119-124.
[57]Yang,J.,Shen,H.S.,2003.Nonlinear bending analysis of shear deformable functionally graded plates subjected to thermo-mechanical loads under various boundary conditions.Composites B,34(3):103-115.
[58]Yang,J.,Shen,H.S.,2003.Free vibration and parametric resonance of shear deformable functionally graded cylindrical panels.Journal of Sound and Vibration,261:871-893.
[59]Yang,J.,Liew,K.M.,Wu,Y.E,Kitipornchai,S.,2006.Thermo-mechanical post-buckling of FGM cylindrical panels with temperature-dependent properties.International Journal of Solids and Structures,43(2):307-324.
[60]Javaheri,R.,2002.Thermal bulking of functionally graded plates.AIAA Journal,40:162-168.
[61]李水,宋健,张志民,2002.功能梯度材料悬臂梁受复杂载荷作用的分层剪切理论.宇航学报,23:62-67.
[62]李永,张志民,马淑雅,2002.梯度功能材料层梁受机械/热载作用的结构特性分析.强度与环境29:20-26.
[63]Saizonou,C.,Njiwa,R.K.,Stebut,J.V.,2002.Surface engineering with functionally graded coatings:a numerical study based on the boundary element method.Surface and Coatings Technology,153:290-297.
[64]陈伟球,边祖光,丁皓江,2002.功能梯度矩形厚板的三维热弹性分析.力学季刊,23(4):443-449.
[65]Chakrabortya,A.,Gopalakrishnana,S.,Reddy J.N.,2003.A new beam finite element for the analysis of functionally graded materials.International Journal of Mechanical Sciences,45:519-539.
[66]Ma,L.S.,Wang,T.J.,2003.Nonlinear bending and post-buckling of a functionally graded circular plate under mechanical and thermal loadings.International Journal of Solids and Structures,40:3311-3330.
[67]Ma,L.S.,Wang,T.J.,2004.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,41:85-101.
[68]刘进,武兰河,张晓炜,2003.功能梯度材料板的弯曲问题.石家庄铁道学院学报,16:1-5.
[69]Soldatos,K.P.,2004.Complex potential formalisms for bending of inhomogeneous monoclinic plates including transverse shear deformation.Journal of the Mechanics and Physics of Solids,52:341-357.
[70]杨正光,仲政,戴瑛,2004.功能梯度矩形板的三维弹性分析.力学季刊,25(1):15-20.
[71]Kashtalyan,M.,2004.Three-dimensional elastic solutions for bending of functionally graded rectangular plates.European Journal of Mechanics -A/Solids,23:853-864.
[72]马连生,赵永刚,杨静宁,2004.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30(6):139-142.
[73]王铁军,马连生,石朝锋,2004.功能梯度中厚圆/环板轴对称弯曲问题的解析解.力学学报,36(3):348-353.
[74]Qian,L.F.,Batra,R.C.,Chen,L.M.,2004.Static and dynamic deformations of thick functionally graded elastic plate by using higher-order shear and normal deformable plate theory and meshless local Petrov-Galerkin method.Composites B,35:85-697.
[75]Ramirez,F.,Heyliger,P.R.,Pan,E.,2004.Static analysis of functionally graded elastic anisotropic plates using a discrete layer approach.Composites B,37:10-20.
[76]舒小平,2003.梯度功能材料板热弹性分析模型.复合材料学报,20:51-54.
[77]校金友,张铎,白宏伟,2005.用应力函数法求功能梯度梁的弹性理论解.强度与环境,32(4):17-21
[78]仲政,于涛,2006.功能梯度悬臂梁弯曲问题的解析解.同济大学学报(自然科学版),34(4):443-447.
[79]于涛,仲政,2006.均布荷载作用下功能梯度悬臂梁弯曲问题的解析解.固体力学学报,27(1):15-20
[80]Zhong,Z.,Yu,T.,2007.Analytical solution of a cantilever functionally graded beam.Composites Science and Technology,67:481-488.
[81]吕朝锋,陈伟球,仲政,2006.功能梯度厚梁的二维热弹性力学解.中国科学(G辑)36(4):384-392.
[82]Chi,S.H.,Chung,Y.L.,2006.Mechanical behavior of functionally graded material plates under transverse load,Part Ⅰ:Analysis.International Journal of Solids and Structures,43:3657-3674.
[83]Chi,S.H.,Chung,Y.L.,2006.Mechanical behavior of functionally graded material plates under transverse load,Part Ⅱ:Numerical results.International Journal of Solids and Structures,43:3675-3691.
[84]Ding,H.J.,Huang,D.J.,Chen W.Q.,2007.Elasticity solutions for plane anisotropic functionally graded beams.International Journal of Solids and Structures,44:176-196.
[85]Huang,D.J.,Ding,H.J.,Chen,W.Q.,2007.Analytical solution for functionally graded anisotropic cantilever beam subjected to linearly distributed load.Applied Mathematics and Mechanics(English Edition),28(7):855-860.
[86]Huang,D.J.,Ding,H.J.,Chen,W.Q.,2007.Analytical solution for functionally graded anisotropic cantilever beam under thermal and uniformly distributed load.Journal of Zhejiang University (Science),8 A(9):1351-1355.
[87]Wu,Z.,Chen,W.J.,2006.A higher-order theory and refined three-node triangular element for functionally graded plates.European Journal of Mechanics-A/Solids,25:447-463.
[88]Ashida,F.,1999.Reduction of applied electric potential controlling thermoelastic displacement in a piezoelectric actuators.Archive of Applied Mechchanics 69(7):443-454.
[89]Clark,W.W.,2000.Vibration control with state-switched piezoelectric material.Journal of Intelligent Material Systems and Stractures,11(4):263-271.
[90]Corr,L.R.,Clark,W.W.,2001.Energy dissipation analysis of piezoceramic semi-active vibration control.Journal of Intelligent Material Systems and Structures,12(11):729-736.
[91]Va'zquez,C.A.,Kenji,U.,2001,Novel piezoelectric-based power supply for driving piezoelectric actuators designed for active vibration damping applications.Journal of Electroceramics 7(3):197-21.
[92]Alexander,P.W.,Brei,D.,2003.Piezoceramic telescopic actuator quasi-static experimental characterization.Journal of Intelligent Materials Systems and Structures,14(10):643-665.
[93]Sirohi,J.,Chopra,I.,2000.Fundamental behavior of piezoceramic sheet actuators.Journal of Intelligent Material Systems and Structures,11(1):47-61.
[94]Vendlinski,J.,Brei,D.,2003.Dynamic behavior of telescopic actuators.Journal of Intelligent Materials Systems and Structures,14(9):577-586.
[95]Zhong,Z.,Shang E.T.,2003.Three-dimensional exact analysis of a simply supported functionally gradient piezoelectric plate.International Journal of Solids and Structures,40(20):5335-5352.
[96]Saravanos,D.A.,Heyliger,P.R.,1999.Mechanics and computational models for laminated piezoelectric beams,plates,and shells.Applied Mechanics Review,52(10):305-320.
[97]Tauchert,T.R.,Ashida,F.,Noda,N.,Adali,S.Verijenko,V.,2000.Developments in thermopiezoelasticity with relevance to smart composite structures.Composite Structures,48:31-38.
[98]Chen,W.Q.,Ding H.J.,2000.Bending of functionally graded piezoelectric rectangular plates.Acta Mechanica Solids Sinica,13(4):312-319.
[99]Ootao,Y.,Tanigawa,Y.,2000.Three-dimensional transient piezothermoelasticity in functionally graded rectangular plate bonded to a piezoelectric plate.International Journal of Solids and Structures,37:4377-4401.
[100]Reddy,J.N.,Cheng,Z.Q.,2001.Three-dimensional solutions ofsmart functionally graded plate.Journal of Applied Mechanics,68:234-241.
[101]Wang,B.L.,Noda,N.,2001.Design of a smart functionally graded thermopiezoelectric composite structure.Smart Materials and Structures,35(2):93-109.
[102]Almajid,A.,Taya,M.,Hudnut,S.,2001.Analysis of out-of-plane displacement and stress field in a piezocomposite plate with functionally graded microstructure.International Journal of Solids and Structures,38(19):3377-3391.
[103]吴瑞安,仲政,金波,2002.功能梯度压电材料矩形板的三维分析.固体力学学报,23(1):43-49.
[104]吴瑞安,2003.功能梯度压电材料矩形板自由振动方程的精确解.工程物理研究院科技年报,11(2):414-415.
[105]仲政,尚尔涛,2003.功能梯度热释电材料矩形板的三维精确分析.力学学报,35(5):542-552.
[106]Liew,K.M.,He,X.Q.,Kitipomchai,S.,2004.Finite element method for the feedback control of FGM shells in frequency domain via piezoelectric sensors and actuators.Computer Methods in Applied Mechanics and Engeering,193:257-273.
[107]朱吴文,李尧臣,杨吕锦,2005.功能梯度压电材料板的有限元解.力学季刊,26(4):567-571.
[108]Lu,P.,Lee,H.E.,Lu,C.,2006.Exact solutions for simply supported functionally graded piezoelectric laminates by Stroch-like formalism.Composite Structures,72:352-363.
[109]Marcus,M.A.,1984.Performance characteristics of piezoelectric polymer flexure mode devices,Ferroelectrics,57:203-220.
[110]DeVoe,D.L.,Pisano,A.P.,1997.Modeling and optimal design of piezoelectric cantilever microactuators.Journal of Microelectromechanical Systems,6(3):266-270.
[111]Hauke,T,Kouvatov,A.,Steinhausen,R.,Seifert,W.,Beige,H.,Langhammer,H.T.,Abicht,H.P.,2000.Bending behavior of functionally gradient materials.Ferroelectrics,238:195-202.
[112]Kruusing,A.,2000.Analysis and optimization of loaded cantilever beam microactuators.Smart Materials and Structures,9:186-196.
[113]Almajid,A.,Taya,M.,Hudnut,S.,2001.Analysis of out-of-plane displacement and stress field in a piezocomposite plate with functionally graded microstructure.International Journal of Solids and Structures,38:3377-3391.
[114]Shi,Z.F.,2002.General solution of a density functionally gradient piezoelectric cantilever and its applications.Smart Materials and Structures,11(1):122-129.
[115]Shi,Z.F.,Chen,Y.,2004.Functionally graded piezoelectric cantilever beam under load.Archive of Applied Mechanics,74(3-4):237-247.
[116]Liu,T.T.,Shi,Z.E,2004.Bending behavior of functionally gradient piezoelectric cantilever.Ferroelectrics,308:43-51.
[117]Chen,Y.,Shi,Z.F.,2005.Exact solutions of functionally gradient piezothermoelastic cantilevers and parameter identification.Journal of Intelligent Material Systems and Structures,16:531-539.
[118]Lee,H.J.,2005.Laywise Laminate analysis of functionally graded piezoelectric bimorph beams.Journal of Intelligent Material Systems and Structures,16(4):365-371.
[119]Yu,T.,Zhong,Z.,2007.Bending analysis of a functionally graded piezoelectric cantilever beam.Science in China(Series G),50(1):97-108.
[120]Zhong,Z.,Yu,T.,2008.Electroelastic analysis of functionally graded piezoelectric material beams.Journal of Intelligent Material Systems and Structures,19(6):707-713.
[121]Huang,D.J.,Ding,H.J.,Chen,W.Q.,2007.Piezoelasticity solutions for functionally graded piezoelectric beams.Smart Materials and Structures,16(3):687-695.
[122]Huang,D.J.,Ding,H.J.,Chen,W.Q.,2008.Analysis of functionally graded and laminated piezoelectric cantilever actuators subjected to constant voltage.Smart Materials and Structures 17(6),art.no.065002.
[123]丁皓江,国风林,2000.横观各向同性热电材料简支矩形板的自由振动.力学学报,7(2):45-50.
[124]Chen,W.Q.,Ding,H.J.,2002.On free vibration of a functionally graded piezoelectric rectangular.Acta Mechanics,15(3):207-216.
[125]陈伟球,徐荣桥,丁皓江,王国庆,2004.压电复合材料矩形板自由振动的三维理论的分析.力学学报,18(1):150-159.
[126]曹宗杰,闻邦椿,陈塑寰,2001.含压电材料智能结构动态特性的研究.计算力学学报,8(3):53-69.
[127]Liu,G.R.,Dai,K.Y.,Han,X.,Ohyoshi,T.,2003.Dispersion of waves and characteristic wave surfaces in functionally graded piezoelectric plates.Journal of Sound and vibration,268:131-147.
[128]伍晓红,沈亚鹏,2003.压电功能梯度板自由振动的三维解.固体力学学报,24(1):75-82.
[129]黄小林,沈惠申,2004.带压电层功能梯度复合材料混合层合板的自由振动和动力响应.强度与环境,31(2):13-19.
[130]黄小林,沈惠中,2005.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,22(3):456-460.
[131]Zhong,Z.,Yu,T.,2006.Vibration of a simply supported functionally graded piezoelectric rectangular plate.Smart Materials and Structures,15(5):1404-1412.
[132]Wu,T.L.,Huang,J.H.,2000.Closed-form solutions for the coupling coeffients in fibrous composites with piezoelectric and piezomagnetic phases.International Journal of Solids and Structures,37(21): 2981-3009.
[133] Van Suchtelen, J., 1972. Product properties: a new application of composite materials. Philips Research Reports, 27: 28-37.
[134] Sih, G.C., Song Z.F., 2003. Magnetic and electric poling effects associated with crack growth in BaTiO3-CoFe2O4 composite. Theoretical and Applied Fracture Mechanics, 39 (3): 209-227.
[135] Song, Z.F., Sih, GC, 2003. Crack initiation behavior in magnetoelectroelastic composite under in-plane deformation. Theoretical and Applied Fracture Mechanics, 39 (3): 189-207.
[136] Harshe, G, Dougherty, J.P., Newnham, R.E., 1993. Theoretical modelling of 3-0/0-3 magnetoelectric composites. International journal of applied electromagnetics in materials, 4(2): 161-171.
[137] Avellaneda, M., Harshe, G, 1994. Magnetoelectric effect in piezoelectric/magneto-strictive multilayer (2-2) composites. Journal of Intelligent Material Systems and Structures, 5(4): 501-513.
[138] Nan, C.W., 1994. Magnetoelectric effect in composites of piezoelectric and piezomagnetic phases.Physical Review B, 50(9): 6082-6088.
[139] Benveniste, Y., 1995. Magnetoelectric effect in fibrous composites with piezoelectric and piezomagnetic phases. Physical Review B, 51(22): 16424-16427.
[140] Hou, P.F., Ding, H.J, Chen, J.Y., 2005. Green's functions for transversely isotropic magnetoelectroelastic media. International Journal of Engineering Science, 43(10): 826-858.
[141] Liu, J.X., Liu, X.L. Zhao,Y.B., 2001. Green's functions for an-isotropic magneto-electro-elastic solids with an elliptical cavity or a crack. International Journal of Engineering Science, 39: 1405-1418.
[142] Pan, E., 2002. Three-dimensional Green's function in an-isotropic magnetoelectro-elastic bi-materials.Zeitschrift f(?)r Angewandte Mathematik und Physik, 53(5): 815-838.
[143] Hou, P.F., Leung, Y., Ding, H.J., 2003. The elliptical Hertzian contact of transversely isotropic magneto-electro-elastic bodies. International Journal of Solids and Structures, 40(11): 2833-2850.
[144] Wang, X., Zhong, Z., 2003. The general solution of spherical magneto-electro-elastic media and applications. European Journal of Mechanics, A/Solids, 22(6): 953-969.
[145] Chen, W.Q., Lee, K.Y., Ding, H.J., 2004. General solution for transversely isotropic magneto-electro-thermo-elasticity and the potential theory method. International Journal of Engineering Science, 42(13-14): 1361-1379.
[146] Wang, X., Shen, Y.P., 2002. The general solution of three-dimensional problems in magneto-electro-elastic media. International Journal of Engineering Science, 40(10): 1069-1080.
[147] Ding, H.J., Jiang, A.M., 2003. Fundamental solutions for transversely isotropic magneto-electro-elastic media and boundary integral formulation, Science in China (Series E), 46(6): 607-619.
[148] Ding, H.J., Jiang, A.M., 2004. A boundary integral formulation and solution for 2D problems in magneto-electro-elastic media. Computers & Structures, 82(20-21): 1599-1607.
[149] Jiang, A.M., Ding, H.J., 2004. Green's functions and boundary element methodfor an infinite magneto-electro-elastic half-plane, Computational Mechanics ( WCCM VI in conjunction with APCOM'04, Beijing, China).
[150] Jiang, A.M., Ding, H.J., 2004. Analytical solutions to magneto-electro-elastic beams. Structural Engineering & Mechanics, 18(2): 195-209.
[151] Li, J.Y., 2000. Magnetoelectroelastic multi-inclusion and inhomogeneity problems and their applications in composite materials. International Journal of Engineering Science, 38(18): 1993-2011.
[152] Aboudi, J., 2001. Micromechanical analysis of fully coupled electro-magneto-thermo-elastic multi- phase composites. Smart Materials and Structures, 10: 867-877.
[153] Pan, E., 2001. Exact solution for simply supported and multilayered magneto-electro-elastic plates. Journal of Applied Mechanics, 68: 608-618.
[154] Pan, E., Heyliger, P.R., 2002. Free vibration of simply supported and mutilayered magneto-electro- elastic plates. Journal of Sound and Vibration, 252(3): 429-442.
[155] Pan, E., Heyliger, P., 2003. Exact solutions for magneto-electro-elastic laminates in cylindrical bending. International Journal of Solids and Structures, 40: 6859-6876.
[156] Chen, J., Chen, H., Pan, E., Heyliger, P.R., 2007. Modal analysis of magneto-electro-elastic plates using the state-vector approach. Journal of Sound and vibration, 304: 722-734.
[157] Pan, E., Han, F., 2005. Exact solution for functionally graded and layered magneto-electro-elastic plates. International Journal of Engineering Science, 43: 321-339.
[158] Wang, X., Zhong, Z., 2003. A finitely long circular cylindrical shell of piezoelectric/piezomagnetic composite under pressuring and temperature change. International Journal of Engineering Science,41:2429-2445.
[159] Wang, J., Chen, L., Fang, S., 2003. State vector approach to analysis of multilayered magneto- electro- elastic plates. International Journal of Solids and Structures, 40: 1669-1680.
[160] Chen, W.Q., Lee, K.Y., 2003. Alternative state space formulations for magnetoelectric thermo-elasticity with transverse isotropy and the application to bending analysis of non-homogeneous plates,International Journal of Solids and Structures, 40: 5689-5705.
[161] Chen, W.Q., Lee, K.Y., Ding, H.J., 2005. On free vibration of non-homogeneous transversely isotropic magneto-electro-elastic plates. Journal of Sound and vibration, 279: 237-251.
[162] Heyliger, P.R., Ramirez, F., Pan, E., 2004. Two-dimensional state fields in magneto-electro-elastic laminates. Journal of Intelligent Material Systems and Structures, 15,689-709.
[163] Buchanan, G.R., 2003. Free vibration of an infinite magneto-electro-elastic cylinder. Journal of Sound and Vibration, 268:413-426.
[164] Buchanan, G.R., 2004. Layered verses multiphase magneto-electro-elastic composites. Composites B,3: 413-420.
[165] Lage, G.R., Soares, C.M., Soares, C.A., Reddy, J.N., 2004. Layer wise partial mixed finite element analysis of magneto-electro-elastic plates. Computers and Structures, 76: 299-317.
[166] Bhangale, R.K., Ganesan, N., 2005. Free vibration studies of simply supported nonhomogeneous Functionally graded magneto-electroelastic finite cylindrical shells. Journal of Sound and Vibration 288(1-2): 412-422.
[167] Bhangale, R.K., Ganesan, N., 2006, Free vibration of simply supported functionally graded and layered magneto-electroelastic plates by finite element method. Journal of Sound and Vibration 294(4): 1016- 1038.
[168] Bhangale, R.K., Ganesan, N., 2006. Free vibration of functionally graded non-homogeneous magneto-electro-elastic cylindrical shell. International Journal of Computational Methods in Engineering Science and Mechanics, 7(3): 191-200.
[169] Bhangale, R.K., Ganesan, N., 2006. Static analysis of simply supported functionally graded and layered magneto-electroelastic plates. International Journal of Solids and Structures 43(10): 3230-3253.
[170] Ramirez, F., Heyliger, P.R., Pan, E., 2006. Discrete layer solution to free vibrations of functionally graded magneto-electro-elastic plates. Mechanics of Advanced Materials and Structures, 13: 249-266.
[171]Ramirez,F.,Heyliger,P.R.,Pan,E.,2006.Static analysis of functionally graded elastic anisotropic plates using a discrete layer approach.Composites Part B,37:10-20.
[172]陈江义,熊滨生,陈柳,2006.用状态空间法求功能梯度电磁弹性多层板的场变量.郑州大学学报(理学版),38(1):55-59.
[173]Chen,J.Y.,Chen,H.L.,Pan,E.N.,2006.Free vibration of functionally graded,magneto-electroelastic,and multilayered plates.Acta Mechanica Solida Sinica,19(2):160-166.
[174]Wang,H.M.,Ding,H.J.,2006.Spherically symmetric transient responses of functionally graded magnetoelectro-elastic hollow sphere.Structural Engineering and Mechanics,23(5):525-542.
[175]Wu,C.P.,Tsai,Y.H.,2007.Static behavior of functionally graded magneto-electro-elastic shells under electric displacement and magnetic flux.International Journal of Engineering Science,45:744-769.
[176]Tsai,Y.H.,Wu,C.P.,2008.Three-dimensional analysis of doubly curved functionally graded magneto-electro-elastic shells.European Journal of Mechanics,A/Solids,27:75-105.
[177]Tsai,Y.H.,Wu,C.P.,2008.Dynamic responses of functionally graded magneto-electroelastic shells with open-circuit surface conditions.International Journal of Engineering Science,46:843-57.
[178]Tsai,Y.H.,Wu,C.P.,Syu,Y.S.,2008.Three-dimensional analysis of doubly curved functionally graded magneto-electroelastic shells.European Journal of Mechanics,A/Solids,27(1):79-105.
[179]Huang,D.J.,Ding,H.J.,Chen,W.Q.,2007.Analytical solution for functionally graded magnetoelectro -elastic plane beams.International Journal of Engineering Science,45(2-8):467-485.
[180]Timoshenko,S.P.,Goodier,J.N.,1970.Theory of Elasticity(3rd ED),New York:McGraw Hill.
[181]Silverman,I.K.,1964.Orthotropic beams under polynomial loads.Proceeding ASCE,Journal of the Engineering Mechanics Division,90(EM 5):293-319
[182]Hashin,Z.,1967.Plane anisotropic beams.Journal of Applied Mechanics,34(2):257-263.
[183]Ahmed,S.R.,Idris,B.M.,Uddin,M.W.,1996.Numerical Solution of Both Ends Fixed Deep Beams.Computer & Structures,61(1):21-29.
[184]Ahmed,S.R.,Khan,M.R.,Islam,K.M.S.,Uddin,M.W.,1998.Investigation of stress at the fixed end of deep cantilever beams.Computers & Structures,69(3):329-338.
[185]Ding,H.J.,Huang,D.J.,Wang,H.M.,2005.Analytical solution for fixed-end beam subjected to uniform load.Journal of Zhejiang University(SCIENCE),6A(8):779-783.
[186]Jiang,A.M,Ding,H.J.,2005.The analytical solutions for orthotropic cantilever beams(Ⅱ):Solutions for density functionally graded beams.Journal of Zhejiang University(SCIENCE),6A(3):155-158.
[187]Gere,J.M.,Timoshenko,S.,1984.Mechanics of Materials(2nd ED).Boston,PWS-KENT Publishing Company.
[188]Timoshenko,S.P.,1925.Analysis of Bi-metal thermostats.Journal of the Optical Society of America,11:233-255.
[189]Suhir,E.,1986.Stress in bimetal thermostats.Journal of applied mechanics,33:657-660.
[190]Suhir,E.,1989.Interfacial stress in bimetal thermostats.Journal of applied mechanics,56:595-600.
[191]蔡乾煌,1994.双材料板条的界面应力——两端面承受一般载荷的情况.工程力学,11(3):121-128.
[192]程军,蔡红查,2004.热/均布载荷下双材料悬臂梁的解析解.力学季刊,25(4):583-588.
[193]L(u|¨),C.F.,Chen,W.Q.,Zhong,Z.,2006.Two-dimensional thermoelasticity solution for functionally graded thick beams Science in China(Series G),49(4):451-460.
[194]Sosa,H.A.,Castro,M.A.,1994.On concentrated loads at the boundary of a piezoelectric half-plane.Journal of the mechanics and Physics of Solid,42:1105-1122.
[195]Ding,H.J.,Wang,GQ.,Chen,W.Q.,1997.Green's functions for a two-phase infinite piezoelectric plane.Proceeding of the Royal Society of London,Series A,453:2241-2257.
[196]Ding,H.J.,Wang,G.Q.,Chen,W.Q.,1998.Green's functions for a piezoelectric half plane,Science in China(Series E),41:70-75.
[197]Ding,H.J.,Wang,G.Q.,Chen,W.Q.,1998.A boundary integral formulation and 2D fundamental solutions for piezoelectric media.Computer Methods in Applied Mechanics and Engineering 158:65-80.
[198]Ding,H.J.,Jiang,A.M.,2005.Polynomial solutions to piezoelectric beams(Ⅰ):several exact solutions.Applied Mathematics and Mechanics 26(9):1107-1114.
[199]Ding,H.J.,Jiang,A.M.,2005.Polynomial solutions to piezoelectric beams(Ⅱ):analytical solutions to typical problems.Applied Mathematics and Mechanics,26(9):1115-1120.
[200]Uchino,K.,Takahashi,S.,1996.Multilayer ceramic actuators.Current opinion in solid state and materias science,1(5):698-705.
[201]Kouvatov,A.,Steinhausen,R.,Seifert,W.,Hauke,T.,Langhammer,H.T.,Beige,H.,Abicht,H.,1999.Comparison between bimorphic and polymorphic bending devices.Journal of the European Ceramic Society,19:1153-1156.
[202]王荣津,1983.水声材料手册.科学出版社,北京.
[2]Yamanouchi,M.,Koizumi,M.,Hirai,T.,Shiota,I.,1990.Proceedings of the First International Symposium Functionally Gradient Materials.Japan,1-15.
[3]Koizumi,M.,1993.Concept of FGM.Ceramic Transaction:Functionally gradient materials,34:3-10.
[4]Koizumi,M.,1997.FGM activities in Japan.Composites B,28:1-4.
[5]Rabin,B.H.,Shiota,I.,1995.Functionally gradient materials.Materials Research Bulletin,20:14-18.
[6]余茂黎,魏明坤,1992.功能梯度材料的研究动态.功能材料,23:184-191.
[7]林启荣,刘振兴,2000.噪声控制压电智能系统研究的现状和展望.力学季刊,21(1):27-32.
[8]Rao,S.S.,Sunar,M.,1994.Piezoelectricity and its use in disturbance sensing and control of flexible structures.Applied Mechanics Reviews,47:113-123.
[9]Sosa,H.A.,Castro,M.A.,1993.Electroelastic analysis of Piezoelectric laminated structures.Applied Mechanics Reviews,46(11):521-528.
[10]Cady,W.G.,1964.Piezoelectricity,revised edition,Dover Publications,Inc.,Newyork.
[11]Jafe,B.,Cook,W.R.,Jafe,H.,1971.Piezoelectricity Ceramics,Academic Press,New York.
[12]Uchino,K.,1997.Piezoelectric actuators and ultrasonic motors.Kiuwer Academic Publishers Boston/Dordrecht/London.
[13]Wu,C.C.M.,Kahn,M.,Moy,W.,1996.Piezoelectric ceramics with functional gradients:a new application in material design.Journal of the American Ceramic Society,79(3):809-812
[14]Zhu,X.H.,Wang,Q.,Meng,Z.Y.,1995.A functionally graded piezoelectric actuator prepared by powder metallurgical process in PNN-PZ-PT system.Journal of Materials Science Letters,14516-14518.
[15]Hirai,T.,Chen,L.,1999.Recent and prospective development of functionally graded materials in Japan.Materials Science Forum,308-311,509-514.
[16]Miyamoto,Y.,Niino,M.,Koizumi,M.,1997.FGM research program in Japan-from structural to functional uses.Functionally Graded Materials,Ed.By Shiota and Miyamoto,Pub.By Elsevier Science B,V:1-8.
[17]李永,宋健,张志民,2003.梯度功能力学.清华大学出版社,北京.
[18]Functionally Graded Materials Ⅶ,Materials Science Forum:Vol.423425(2003),by Trans Tech Publications Inc.Publishers in Science and Engineering,Proceeding of the Seventh International Symposium on Functionally Graded Materials,October 15-18,2002,Beijing,China.
[19]李碧容,雍歧龙,张国亮,1999.功能梯度材料的发展、制备方法及应用前景.云南工业大学报,15(4):56-58.
[20]金灯仁,2003.PZT、PZT/ZnO系梯度功能压电陶瓷及其执行器的研制与表征.上海交通大学博士论文.
[21]Luo,Y.,Li,S.,Chen,J.,2003.Preparation and characterization of A1203-Ti3SiC2 composites and its functionally graded materials.Materials Research Bulletin,38(1):69-78.
[22]Luo,Y.,Wei,P.,Li,S.,Wang,R.G.,Li,J.Q.,2003.A novel functionally graded material in the Ti-Si-C system.Materials Science and Engineering(A),345(1-2):99-105.
[23]Ling,Y.H.,Li,J.T,Ge,C.C.,Bai,X.D.,2002.Fabrication and evaluation of SiC/Cu functionally graded material used for plasma facing components in a fusion reactor.Journal of Nuclear Materials,303(2-3):188-195.
[24]Zeng,Y.P.,Jiang,D.L.,2000.Fabrication and properties of tape-cast laminated and functionally gradient alumina-tianium carbide materials.Journal of the American Ceramic Society,83(12):2999-3003.
[25]Xiong,H.P.,Shen,Q.,Li,J.G.,Zhang,L.M.,Yuan,R.Z.,2000.Design and microstructures of Ti/TiAl/Al system functionally graded material.Journal of Materials Science Letters,19:989-993.
[26]Lekhnitskii,S.G,1968.Anisotropic Plate.Gordon and Breach,New York.
[27]Noda,N.,Tsuji,T.,1991.Steady thermal stresses in a plate of functionally gradient material.Journal of Japanese Society for Mechanical Engineering(A),57(535):625-631.
[28]Obata,Y.,Noda,N.,1994.Steady thermal stresses in a hollow circular cylinder and a hollow sphere of a functionally graded materials.Journal of Thermal Stresses,17:471-178.
[29]Aboudi,J.,Pindera,M.J.,Arnold,S.M.,1995.A coupled higher-order theory for functionally graded composites with partial homogenization.Composites Engineering,5:771-792.
[30]Aboudi,J.,Pindera,M.J.,Arnold,S.M.,1999.Higher-order theory for functionally graded materials.Composites B,30(8):777-832.
[31]Mian,A.M.,Spencer,A.J.M.,1998.Exact solutions for functionally graded and laminated elastic materials.Journal oft he Mechanics and Physics of Solids,42(12):2283-2295.
[32]赵军,艾兴,张建华,1998.基于高承载能力的梯度功能材料设计.陶瓷学报,19:125-129.
[33]李春雨,邹振祝,1998.受内压的功能梯度材料圆筒应力分析.工程力学增刊,153-156.
[34]Reddy,J.N.,Chin,C.D.,1998.Thermomachanical analysis of functionally graded cylinders and plates.Journal of Thermal stresses,21:593-626.
[35]Reddy,J.N.,Wang,C.,Kitipomchai,M.S.,1999.Axisymmetric bending of functionally graded circular and annular plates.European Journal of Mechanics-A/Solids,18:185-199.
[36]Reddy,J.N.,1984.A Simple higher-order theory for laminated composite plates.Journal of Applied Mechanics,51:745-752.
[37]Reddy,J.N.,2000.Analysis of functionally graded plates.International Journal for Numerical Methods in Engineering,47:663-684.
[38]Cheng,Z.Q.,Batra R.C.,2000.Three-dimensional thermoelastic deformations of a functionally graded elliptic plate.Composites B,31:97-106.
[39]Reddy,J.N.,Cheng,Z.Q.,2001.Three-dimensional thermomechanical deformations of functionally graded rectangular plates.European Journal of Mechanics-A/Solids,20:841-855.
[40]Cheng,Z.Q.,2001.Nonlinear bending of inhomogeneous plates.Engineering Structures,23:1359-1363.
[41]Vel,S.S.,Batra,R.C.,2002.Exact solution for thermoelastic deformation of functionally graded thick rectangular plates.AIAA Journal,40:1421-1433.
[42]Vel,S.S.,Barea,R.C.,2003.Three-dimensional analysis of transient thermal stresses in functionally graded rectangular plates.International Journal of Solids and Structures,40(25):7181-7196.
[43]Vel,S.S.,Barea,R.C.,2004.Three-dimensional Exact solution for the vibration of functionally graded rectangular plates.Journal of Sound and Vibration,272:703-730.
[44]Woo,J.,Meguid,S.A.,2001.Nonlinear analysis of functionally graded plates and shallow shells.International Journal of Solids and Structures,38:7409-7421.
[45]Rooney,F.M.,2001.Tension,bending,and flexure of functionally graded cylinders.International Journal of Solids and Structures,38:413-421.
[46]Sankar,B.V.,2001.An elasticity solution for functionally graded beams.Composites Science and Technology,61:689-696.
[47]Sankar,B.V,Tzeng,J.T.,2002.Thermal stresses in functionaly graded beams.AIAA Journal,40(6):1228-1232.
[48]Zhu,H.,Sankar,B.V.,2004.A combined Fourier Series-Galerkin method for the analysis of functionally graded beams.Journal of Applied Mechanics,71(3):421-424.
[49]Shen,H.S.,2002.Postbuckling analysis of axially loaded functionally graded cylindrical panels in thermal environments.International Journal of Solids and Structures,39:5991-6010.
[50]Shen,H.S.,2002.Nonlinear bending response of functionally graded plates subjected to transverse loads and in thermal environments.International Journal of Mechanical Sciences,44:561-584.
[51]Huang,X.L.,Shen,H.S.,2004.Nonlinear vibration and dynamic response of functionally graded plates in thermal environments.International Journal of Solids and Structures,41:2403-2427.
[52]Shen,H.S.,2002.Postbuckling analysis of axially-loaded functionally graded cylindrical shells in thermal environments.Composites Science and Technology,62:977-987.
[53]Shen,H.S.,2004.Thermal postbuckling behavior of functionally graded cylindrical shells with temperature-dependent properties.International Journal of Solids and Structures,41:1961-1974.
[54]Yang,Y.Y.,2000.Time-dependent stress analysis in functionally graded materials International Journal of Solids and Structures,37:7593-7608.
[55]杨杰,沈惠申,2002.面内预应力功能梯度材料矩形板的横向弯曲.力学季刊,23:342-346.
[56]杨杰,沈惠申,2003.热/机械载荷下功能梯度材料矩形板的弯曲行为.固体力学学报,24(1):119-124.
[57]Yang,J.,Shen,H.S.,2003.Nonlinear bending analysis of shear deformable functionally graded plates subjected to thermo-mechanical loads under various boundary conditions.Composites B,34(3):103-115.
[58]Yang,J.,Shen,H.S.,2003.Free vibration and parametric resonance of shear deformable functionally graded cylindrical panels.Journal of Sound and Vibration,261:871-893.
[59]Yang,J.,Liew,K.M.,Wu,Y.E,Kitipornchai,S.,2006.Thermo-mechanical post-buckling of FGM cylindrical panels with temperature-dependent properties.International Journal of Solids and Structures,43(2):307-324.
[60]Javaheri,R.,2002.Thermal bulking of functionally graded plates.AIAA Journal,40:162-168.
[61]李水,宋健,张志民,2002.功能梯度材料悬臂梁受复杂载荷作用的分层剪切理论.宇航学报,23:62-67.
[62]李永,张志民,马淑雅,2002.梯度功能材料层梁受机械/热载作用的结构特性分析.强度与环境29:20-26.
[63]Saizonou,C.,Njiwa,R.K.,Stebut,J.V.,2002.Surface engineering with functionally graded coatings:a numerical study based on the boundary element method.Surface and Coatings Technology,153:290-297.
[64]陈伟球,边祖光,丁皓江,2002.功能梯度矩形厚板的三维热弹性分析.力学季刊,23(4):443-449.
[65]Chakrabortya,A.,Gopalakrishnana,S.,Reddy J.N.,2003.A new beam finite element for the analysis of functionally graded materials.International Journal of Mechanical Sciences,45:519-539.
[66]Ma,L.S.,Wang,T.J.,2003.Nonlinear bending and post-buckling of a functionally graded circular plate under mechanical and thermal loadings.International Journal of Solids and Structures,40:3311-3330.
[67]Ma,L.S.,Wang,T.J.,2004.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,41:85-101.
[68]刘进,武兰河,张晓炜,2003.功能梯度材料板的弯曲问题.石家庄铁道学院学报,16:1-5.
[69]Soldatos,K.P.,2004.Complex potential formalisms for bending of inhomogeneous monoclinic plates including transverse shear deformation.Journal of the Mechanics and Physics of Solids,52:341-357.
[70]杨正光,仲政,戴瑛,2004.功能梯度矩形板的三维弹性分析.力学季刊,25(1):15-20.
[71]Kashtalyan,M.,2004.Three-dimensional elastic solutions for bending of functionally graded rectangular plates.European Journal of Mechanics -A/Solids,23:853-864.
[72]马连生,赵永刚,杨静宁,2004.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30(6):139-142.
[73]王铁军,马连生,石朝锋,2004.功能梯度中厚圆/环板轴对称弯曲问题的解析解.力学学报,36(3):348-353.
[74]Qian,L.F.,Batra,R.C.,Chen,L.M.,2004.Static and dynamic deformations of thick functionally graded elastic plate by using higher-order shear and normal deformable plate theory and meshless local Petrov-Galerkin method.Composites B,35:85-697.
[75]Ramirez,F.,Heyliger,P.R.,Pan,E.,2004.Static analysis of functionally graded elastic anisotropic plates using a discrete layer approach.Composites B,37:10-20.
[76]舒小平,2003.梯度功能材料板热弹性分析模型.复合材料学报,20:51-54.
[77]校金友,张铎,白宏伟,2005.用应力函数法求功能梯度梁的弹性理论解.强度与环境,32(4):17-21
[78]仲政,于涛,2006.功能梯度悬臂梁弯曲问题的解析解.同济大学学报(自然科学版),34(4):443-447.
[79]于涛,仲政,2006.均布荷载作用下功能梯度悬臂梁弯曲问题的解析解.固体力学学报,27(1):15-20
[80]Zhong,Z.,Yu,T.,2007.Analytical solution of a cantilever functionally graded beam.Composites Science and Technology,67:481-488.
[81]吕朝锋,陈伟球,仲政,2006.功能梯度厚梁的二维热弹性力学解.中国科学(G辑)36(4):384-392.
[82]Chi,S.H.,Chung,Y.L.,2006.Mechanical behavior of functionally graded material plates under transverse load,Part Ⅰ:Analysis.International Journal of Solids and Structures,43:3657-3674.
[83]Chi,S.H.,Chung,Y.L.,2006.Mechanical behavior of functionally graded material plates under transverse load,Part Ⅱ:Numerical results.International Journal of Solids and Structures,43:3675-3691.
[84]Ding,H.J.,Huang,D.J.,Chen W.Q.,2007.Elasticity solutions for plane anisotropic functionally graded beams.International Journal of Solids and Structures,44:176-196.
[85]Huang,D.J.,Ding,H.J.,Chen,W.Q.,2007.Analytical solution for functionally graded anisotropic cantilever beam subjected to linearly distributed load.Applied Mathematics and Mechanics(English Edition),28(7):855-860.
[86]Huang,D.J.,Ding,H.J.,Chen,W.Q.,2007.Analytical solution for functionally graded anisotropic cantilever beam under thermal and uniformly distributed load.Journal of Zhejiang University (Science),8 A(9):1351-1355.
[87]Wu,Z.,Chen,W.J.,2006.A higher-order theory and refined three-node triangular element for functionally graded plates.European Journal of Mechanics-A/Solids,25:447-463.
[88]Ashida,F.,1999.Reduction of applied electric potential controlling thermoelastic displacement in a piezoelectric actuators.Archive of Applied Mechchanics 69(7):443-454.
[89]Clark,W.W.,2000.Vibration control with state-switched piezoelectric material.Journal of Intelligent Material Systems and Stractures,11(4):263-271.
[90]Corr,L.R.,Clark,W.W.,2001.Energy dissipation analysis of piezoceramic semi-active vibration control.Journal of Intelligent Material Systems and Structures,12(11):729-736.
[91]Va'zquez,C.A.,Kenji,U.,2001,Novel piezoelectric-based power supply for driving piezoelectric actuators designed for active vibration damping applications.Journal of Electroceramics 7(3):197-21.
[92]Alexander,P.W.,Brei,D.,2003.Piezoceramic telescopic actuator quasi-static experimental characterization.Journal of Intelligent Materials Systems and Structures,14(10):643-665.
[93]Sirohi,J.,Chopra,I.,2000.Fundamental behavior of piezoceramic sheet actuators.Journal of Intelligent Material Systems and Structures,11(1):47-61.
[94]Vendlinski,J.,Brei,D.,2003.Dynamic behavior of telescopic actuators.Journal of Intelligent Materials Systems and Structures,14(9):577-586.
[95]Zhong,Z.,Shang E.T.,2003.Three-dimensional exact analysis of a simply supported functionally gradient piezoelectric plate.International Journal of Solids and Structures,40(20):5335-5352.
[96]Saravanos,D.A.,Heyliger,P.R.,1999.Mechanics and computational models for laminated piezoelectric beams,plates,and shells.Applied Mechanics Review,52(10):305-320.
[97]Tauchert,T.R.,Ashida,F.,Noda,N.,Adali,S.Verijenko,V.,2000.Developments in thermopiezoelasticity with relevance to smart composite structures.Composite Structures,48:31-38.
[98]Chen,W.Q.,Ding H.J.,2000.Bending of functionally graded piezoelectric rectangular plates.Acta Mechanica Solids Sinica,13(4):312-319.
[99]Ootao,Y.,Tanigawa,Y.,2000.Three-dimensional transient piezothermoelasticity in functionally graded rectangular plate bonded to a piezoelectric plate.International Journal of Solids and Structures,37:4377-4401.
[100]Reddy,J.N.,Cheng,Z.Q.,2001.Three-dimensional solutions ofsmart functionally graded plate.Journal of Applied Mechanics,68:234-241.
[101]Wang,B.L.,Noda,N.,2001.Design of a smart functionally graded thermopiezoelectric composite structure.Smart Materials and Structures,35(2):93-109.
[102]Almajid,A.,Taya,M.,Hudnut,S.,2001.Analysis of out-of-plane displacement and stress field in a piezocomposite plate with functionally graded microstructure.International Journal of Solids and Structures,38(19):3377-3391.
[103]吴瑞安,仲政,金波,2002.功能梯度压电材料矩形板的三维分析.固体力学学报,23(1):43-49.
[104]吴瑞安,2003.功能梯度压电材料矩形板自由振动方程的精确解.工程物理研究院科技年报,11(2):414-415.
[105]仲政,尚尔涛,2003.功能梯度热释电材料矩形板的三维精确分析.力学学报,35(5):542-552.
[106]Liew,K.M.,He,X.Q.,Kitipomchai,S.,2004.Finite element method for the feedback control of FGM shells in frequency domain via piezoelectric sensors and actuators.Computer Methods in Applied Mechanics and Engeering,193:257-273.
[107]朱吴文,李尧臣,杨吕锦,2005.功能梯度压电材料板的有限元解.力学季刊,26(4):567-571.
[108]Lu,P.,Lee,H.E.,Lu,C.,2006.Exact solutions for simply supported functionally graded piezoelectric laminates by Stroch-like formalism.Composite Structures,72:352-363.
[109]Marcus,M.A.,1984.Performance characteristics of piezoelectric polymer flexure mode devices,Ferroelectrics,57:203-220.
[110]DeVoe,D.L.,Pisano,A.P.,1997.Modeling and optimal design of piezoelectric cantilever microactuators.Journal of Microelectromechanical Systems,6(3):266-270.
[111]Hauke,T,Kouvatov,A.,Steinhausen,R.,Seifert,W.,Beige,H.,Langhammer,H.T.,Abicht,H.P.,2000.Bending behavior of functionally gradient materials.Ferroelectrics,238:195-202.
[112]Kruusing,A.,2000.Analysis and optimization of loaded cantilever beam microactuators.Smart Materials and Structures,9:186-196.
[113]Almajid,A.,Taya,M.,Hudnut,S.,2001.Analysis of out-of-plane displacement and stress field in a piezocomposite plate with functionally graded microstructure.International Journal of Solids and Structures,38:3377-3391.
[114]Shi,Z.F.,2002.General solution of a density functionally gradient piezoelectric cantilever and its applications.Smart Materials and Structures,11(1):122-129.
[115]Shi,Z.F.,Chen,Y.,2004.Functionally graded piezoelectric cantilever beam under load.Archive of Applied Mechanics,74(3-4):237-247.
[116]Liu,T.T.,Shi,Z.E,2004.Bending behavior of functionally gradient piezoelectric cantilever.Ferroelectrics,308:43-51.
[117]Chen,Y.,Shi,Z.F.,2005.Exact solutions of functionally gradient piezothermoelastic cantilevers and parameter identification.Journal of Intelligent Material Systems and Structures,16:531-539.
[118]Lee,H.J.,2005.Laywise Laminate analysis of functionally graded piezoelectric bimorph beams.Journal of Intelligent Material Systems and Structures,16(4):365-371.
[119]Yu,T.,Zhong,Z.,2007.Bending analysis of a functionally graded piezoelectric cantilever beam.Science in China(Series G),50(1):97-108.
[120]Zhong,Z.,Yu,T.,2008.Electroelastic analysis of functionally graded piezoelectric material beams.Journal of Intelligent Material Systems and Structures,19(6):707-713.
[121]Huang,D.J.,Ding,H.J.,Chen,W.Q.,2007.Piezoelasticity solutions for functionally graded piezoelectric beams.Smart Materials and Structures,16(3):687-695.
[122]Huang,D.J.,Ding,H.J.,Chen,W.Q.,2008.Analysis of functionally graded and laminated piezoelectric cantilever actuators subjected to constant voltage.Smart Materials and Structures 17(6),art.no.065002.
[123]丁皓江,国风林,2000.横观各向同性热电材料简支矩形板的自由振动.力学学报,7(2):45-50.
[124]Chen,W.Q.,Ding,H.J.,2002.On free vibration of a functionally graded piezoelectric rectangular.Acta Mechanics,15(3):207-216.
[125]陈伟球,徐荣桥,丁皓江,王国庆,2004.压电复合材料矩形板自由振动的三维理论的分析.力学学报,18(1):150-159.
[126]曹宗杰,闻邦椿,陈塑寰,2001.含压电材料智能结构动态特性的研究.计算力学学报,8(3):53-69.
[127]Liu,G.R.,Dai,K.Y.,Han,X.,Ohyoshi,T.,2003.Dispersion of waves and characteristic wave surfaces in functionally graded piezoelectric plates.Journal of Sound and vibration,268:131-147.
[128]伍晓红,沈亚鹏,2003.压电功能梯度板自由振动的三维解.固体力学学报,24(1):75-82.
[129]黄小林,沈惠申,2004.带压电层功能梯度复合材料混合层合板的自由振动和动力响应.强度与环境,31(2):13-19.
[130]黄小林,沈惠中,2005.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,22(3):456-460.
[131]Zhong,Z.,Yu,T.,2006.Vibration of a simply supported functionally graded piezoelectric rectangular plate.Smart Materials and Structures,15(5):1404-1412.
[132]Wu,T.L.,Huang,J.H.,2000.Closed-form solutions for the coupling coeffients in fibrous composites with piezoelectric and piezomagnetic phases.International Journal of Solids and Structures,37(21): 2981-3009.
[133] Van Suchtelen, J., 1972. Product properties: a new application of composite materials. Philips Research Reports, 27: 28-37.
[134] Sih, G.C., Song Z.F., 2003. Magnetic and electric poling effects associated with crack growth in BaTiO3-CoFe2O4 composite. Theoretical and Applied Fracture Mechanics, 39 (3): 209-227.
[135] Song, Z.F., Sih, GC, 2003. Crack initiation behavior in magnetoelectroelastic composite under in-plane deformation. Theoretical and Applied Fracture Mechanics, 39 (3): 189-207.
[136] Harshe, G, Dougherty, J.P., Newnham, R.E., 1993. Theoretical modelling of 3-0/0-3 magnetoelectric composites. International journal of applied electromagnetics in materials, 4(2): 161-171.
[137] Avellaneda, M., Harshe, G, 1994. Magnetoelectric effect in piezoelectric/magneto-strictive multilayer (2-2) composites. Journal of Intelligent Material Systems and Structures, 5(4): 501-513.
[138] Nan, C.W., 1994. Magnetoelectric effect in composites of piezoelectric and piezomagnetic phases.Physical Review B, 50(9): 6082-6088.
[139] Benveniste, Y., 1995. Magnetoelectric effect in fibrous composites with piezoelectric and piezomagnetic phases. Physical Review B, 51(22): 16424-16427.
[140] Hou, P.F., Ding, H.J, Chen, J.Y., 2005. Green's functions for transversely isotropic magnetoelectroelastic media. International Journal of Engineering Science, 43(10): 826-858.
[141] Liu, J.X., Liu, X.L. Zhao,Y.B., 2001. Green's functions for an-isotropic magneto-electro-elastic solids with an elliptical cavity or a crack. International Journal of Engineering Science, 39: 1405-1418.
[142] Pan, E., 2002. Three-dimensional Green's function in an-isotropic magnetoelectro-elastic bi-materials.Zeitschrift f(?)r Angewandte Mathematik und Physik, 53(5): 815-838.
[143] Hou, P.F., Leung, Y., Ding, H.J., 2003. The elliptical Hertzian contact of transversely isotropic magneto-electro-elastic bodies. International Journal of Solids and Structures, 40(11): 2833-2850.
[144] Wang, X., Zhong, Z., 2003. The general solution of spherical magneto-electro-elastic media and applications. European Journal of Mechanics, A/Solids, 22(6): 953-969.
[145] Chen, W.Q., Lee, K.Y., Ding, H.J., 2004. General solution for transversely isotropic magneto-electro-thermo-elasticity and the potential theory method. International Journal of Engineering Science, 42(13-14): 1361-1379.
[146] Wang, X., Shen, Y.P., 2002. The general solution of three-dimensional problems in magneto-electro-elastic media. International Journal of Engineering Science, 40(10): 1069-1080.
[147] Ding, H.J., Jiang, A.M., 2003. Fundamental solutions for transversely isotropic magneto-electro-elastic media and boundary integral formulation, Science in China (Series E), 46(6): 607-619.
[148] Ding, H.J., Jiang, A.M., 2004. A boundary integral formulation and solution for 2D problems in magneto-electro-elastic media. Computers & Structures, 82(20-21): 1599-1607.
[149] Jiang, A.M., Ding, H.J., 2004. Green's functions and boundary element methodfor an infinite magneto-electro-elastic half-plane, Computational Mechanics ( WCCM VI in conjunction with APCOM'04, Beijing, China).
[150] Jiang, A.M., Ding, H.J., 2004. Analytical solutions to magneto-electro-elastic beams. Structural Engineering & Mechanics, 18(2): 195-209.
[151] Li, J.Y., 2000. Magnetoelectroelastic multi-inclusion and inhomogeneity problems and their applications in composite materials. International Journal of Engineering Science, 38(18): 1993-2011.
[152] Aboudi, J., 2001. Micromechanical analysis of fully coupled electro-magneto-thermo-elastic multi- phase composites. Smart Materials and Structures, 10: 867-877.
[153] Pan, E., 2001. Exact solution for simply supported and multilayered magneto-electro-elastic plates. Journal of Applied Mechanics, 68: 608-618.
[154] Pan, E., Heyliger, P.R., 2002. Free vibration of simply supported and mutilayered magneto-electro- elastic plates. Journal of Sound and Vibration, 252(3): 429-442.
[155] Pan, E., Heyliger, P., 2003. Exact solutions for magneto-electro-elastic laminates in cylindrical bending. International Journal of Solids and Structures, 40: 6859-6876.
[156] Chen, J., Chen, H., Pan, E., Heyliger, P.R., 2007. Modal analysis of magneto-electro-elastic plates using the state-vector approach. Journal of Sound and vibration, 304: 722-734.
[157] Pan, E., Han, F., 2005. Exact solution for functionally graded and layered magneto-electro-elastic plates. International Journal of Engineering Science, 43: 321-339.
[158] Wang, X., Zhong, Z., 2003. A finitely long circular cylindrical shell of piezoelectric/piezomagnetic composite under pressuring and temperature change. International Journal of Engineering Science,41:2429-2445.
[159] Wang, J., Chen, L., Fang, S., 2003. State vector approach to analysis of multilayered magneto- electro- elastic plates. International Journal of Solids and Structures, 40: 1669-1680.
[160] Chen, W.Q., Lee, K.Y., 2003. Alternative state space formulations for magnetoelectric thermo-elasticity with transverse isotropy and the application to bending analysis of non-homogeneous plates,International Journal of Solids and Structures, 40: 5689-5705.
[161] Chen, W.Q., Lee, K.Y., Ding, H.J., 2005. On free vibration of non-homogeneous transversely isotropic magneto-electro-elastic plates. Journal of Sound and vibration, 279: 237-251.
[162] Heyliger, P.R., Ramirez, F., Pan, E., 2004. Two-dimensional state fields in magneto-electro-elastic laminates. Journal of Intelligent Material Systems and Structures, 15,689-709.
[163] Buchanan, G.R., 2003. Free vibration of an infinite magneto-electro-elastic cylinder. Journal of Sound and Vibration, 268:413-426.
[164] Buchanan, G.R., 2004. Layered verses multiphase magneto-electro-elastic composites. Composites B,3: 413-420.
[165] Lage, G.R., Soares, C.M., Soares, C.A., Reddy, J.N., 2004. Layer wise partial mixed finite element analysis of magneto-electro-elastic plates. Computers and Structures, 76: 299-317.
[166] Bhangale, R.K., Ganesan, N., 2005. Free vibration studies of simply supported nonhomogeneous Functionally graded magneto-electroelastic finite cylindrical shells. Journal of Sound and Vibration 288(1-2): 412-422.
[167] Bhangale, R.K., Ganesan, N., 2006, Free vibration of simply supported functionally graded and layered magneto-electroelastic plates by finite element method. Journal of Sound and Vibration 294(4): 1016- 1038.
[168] Bhangale, R.K., Ganesan, N., 2006. Free vibration of functionally graded non-homogeneous magneto-electro-elastic cylindrical shell. International Journal of Computational Methods in Engineering Science and Mechanics, 7(3): 191-200.
[169] Bhangale, R.K., Ganesan, N., 2006. Static analysis of simply supported functionally graded and layered magneto-electroelastic plates. International Journal of Solids and Structures 43(10): 3230-3253.
[170] Ramirez, F., Heyliger, P.R., Pan, E., 2006. Discrete layer solution to free vibrations of functionally graded magneto-electro-elastic plates. Mechanics of Advanced Materials and Structures, 13: 249-266.
[171]Ramirez,F.,Heyliger,P.R.,Pan,E.,2006.Static analysis of functionally graded elastic anisotropic plates using a discrete layer approach.Composites Part B,37:10-20.
[172]陈江义,熊滨生,陈柳,2006.用状态空间法求功能梯度电磁弹性多层板的场变量.郑州大学学报(理学版),38(1):55-59.
[173]Chen,J.Y.,Chen,H.L.,Pan,E.N.,2006.Free vibration of functionally graded,magneto-electroelastic,and multilayered plates.Acta Mechanica Solida Sinica,19(2):160-166.
[174]Wang,H.M.,Ding,H.J.,2006.Spherically symmetric transient responses of functionally graded magnetoelectro-elastic hollow sphere.Structural Engineering and Mechanics,23(5):525-542.
[175]Wu,C.P.,Tsai,Y.H.,2007.Static behavior of functionally graded magneto-electro-elastic shells under electric displacement and magnetic flux.International Journal of Engineering Science,45:744-769.
[176]Tsai,Y.H.,Wu,C.P.,2008.Three-dimensional analysis of doubly curved functionally graded magneto-electro-elastic shells.European Journal of Mechanics,A/Solids,27:75-105.
[177]Tsai,Y.H.,Wu,C.P.,2008.Dynamic responses of functionally graded magneto-electroelastic shells with open-circuit surface conditions.International Journal of Engineering Science,46:843-57.
[178]Tsai,Y.H.,Wu,C.P.,Syu,Y.S.,2008.Three-dimensional analysis of doubly curved functionally graded magneto-electroelastic shells.European Journal of Mechanics,A/Solids,27(1):79-105.
[179]Huang,D.J.,Ding,H.J.,Chen,W.Q.,2007.Analytical solution for functionally graded magnetoelectro -elastic plane beams.International Journal of Engineering Science,45(2-8):467-485.
[180]Timoshenko,S.P.,Goodier,J.N.,1970.Theory of Elasticity(3rd ED),New York:McGraw Hill.
[181]Silverman,I.K.,1964.Orthotropic beams under polynomial loads.Proceeding ASCE,Journal of the Engineering Mechanics Division,90(EM 5):293-319
[182]Hashin,Z.,1967.Plane anisotropic beams.Journal of Applied Mechanics,34(2):257-263.
[183]Ahmed,S.R.,Idris,B.M.,Uddin,M.W.,1996.Numerical Solution of Both Ends Fixed Deep Beams.Computer & Structures,61(1):21-29.
[184]Ahmed,S.R.,Khan,M.R.,Islam,K.M.S.,Uddin,M.W.,1998.Investigation of stress at the fixed end of deep cantilever beams.Computers & Structures,69(3):329-338.
[185]Ding,H.J.,Huang,D.J.,Wang,H.M.,2005.Analytical solution for fixed-end beam subjected to uniform load.Journal of Zhejiang University(SCIENCE),6A(8):779-783.
[186]Jiang,A.M,Ding,H.J.,2005.The analytical solutions for orthotropic cantilever beams(Ⅱ):Solutions for density functionally graded beams.Journal of Zhejiang University(SCIENCE),6A(3):155-158.
[187]Gere,J.M.,Timoshenko,S.,1984.Mechanics of Materials(2nd ED).Boston,PWS-KENT Publishing Company.
[188]Timoshenko,S.P.,1925.Analysis of Bi-metal thermostats.Journal of the Optical Society of America,11:233-255.
[189]Suhir,E.,1986.Stress in bimetal thermostats.Journal of applied mechanics,33:657-660.
[190]Suhir,E.,1989.Interfacial stress in bimetal thermostats.Journal of applied mechanics,56:595-600.
[191]蔡乾煌,1994.双材料板条的界面应力——两端面承受一般载荷的情况.工程力学,11(3):121-128.
[192]程军,蔡红查,2004.热/均布载荷下双材料悬臂梁的解析解.力学季刊,25(4):583-588.
[193]L(u|¨),C.F.,Chen,W.Q.,Zhong,Z.,2006.Two-dimensional thermoelasticity solution for functionally graded thick beams Science in China(Series G),49(4):451-460.
[194]Sosa,H.A.,Castro,M.A.,1994.On concentrated loads at the boundary of a piezoelectric half-plane.Journal of the mechanics and Physics of Solid,42:1105-1122.
[195]Ding,H.J.,Wang,GQ.,Chen,W.Q.,1997.Green's functions for a two-phase infinite piezoelectric plane.Proceeding of the Royal Society of London,Series A,453:2241-2257.
[196]Ding,H.J.,Wang,G.Q.,Chen,W.Q.,1998.Green's functions for a piezoelectric half plane,Science in China(Series E),41:70-75.
[197]Ding,H.J.,Wang,G.Q.,Chen,W.Q.,1998.A boundary integral formulation and 2D fundamental solutions for piezoelectric media.Computer Methods in Applied Mechanics and Engineering 158:65-80.
[198]Ding,H.J.,Jiang,A.M.,2005.Polynomial solutions to piezoelectric beams(Ⅰ):several exact solutions.Applied Mathematics and Mechanics 26(9):1107-1114.
[199]Ding,H.J.,Jiang,A.M.,2005.Polynomial solutions to piezoelectric beams(Ⅱ):analytical solutions to typical problems.Applied Mathematics and Mechanics,26(9):1115-1120.
[200]Uchino,K.,Takahashi,S.,1996.Multilayer ceramic actuators.Current opinion in solid state and materias science,1(5):698-705.
[201]Kouvatov,A.,Steinhausen,R.,Seifert,W.,Hauke,T.,Langhammer,H.T.,Beige,H.,Abicht,H.,1999.Comparison between bimorphic and polymorphic bending devices.Journal of the European Ceramic Society,19:1153-1156.
[202]王荣津,1983.水声材料手册.科学出版社,北京.