压电层合板的三维有限元分析
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摘要
本文基于文献调研的基础上，简介了有限单元法的发展历史以及压电复合材料的应用和研究现状。引入了压电材料的空间等参数单元，并且从弹性力学与电学基础上，根据三维有限元法，建立了压电材料的能量泛函。由于最小势能原理，建立了单元刚度矩阵和压电材料的整体刚度方程。并且推导了压电层合板的边界条件。
在此基础上，利用了Matlab语言编制了具有一定工程意义的计算机程序，对于各种边界条件以及各种形状的压电层合板和非压电层合板的位移、应力、电势、电位移以及电场强度等做了计算与分析。其中一部分结果与相应的解析解做了详细的比较，且证明了本文结果的合理性。本文的研究不仅扩大了解决问题的范围，同时也为求解该问题的解析解或半解析解提供了理想的比较结果。
Based on the theory of elasticity and piezoelectricity, formulas for the three-dimensional finite element analysis of laminated piezoelectric plates are presented. According to the fundamental equations of elastic and electric fields, total potential energy for the electro-elastic laminate under general boundary conditions is established. By using 3-D isoparametric element, the global stiffness matrix equation for the piezoelectric laminate is obtained on the bases of minimum principle of total potential energy. The boundary conditions related to electric variables are discussed and this facilitates the solution of global stiffness equation. Numerical examples are given at the end of the paper to calculate laminated plates with various kinds of elastic and electric boundary conditions. Some results are compared with analytical ones.
在此基础上，利用了Matlab语言编制了具有一定工程意义的计算机程序，对于各种边界条件以及各种形状的压电层合板和非压电层合板的位移、应力、电势、电位移以及电场强度等做了计算与分析。其中一部分结果与相应的解析解做了详细的比较，且证明了本文结果的合理性。本文的研究不仅扩大了解决问题的范围，同时也为求解该问题的解析解或半解析解提供了理想的比较结果。
Based on the theory of elasticity and piezoelectricity, formulas for the three-dimensional finite element analysis of laminated piezoelectric plates are presented. According to the fundamental equations of elastic and electric fields, total potential energy for the electro-elastic laminate under general boundary conditions is established. By using 3-D isoparametric element, the global stiffness matrix equation for the piezoelectric laminate is obtained on the bases of minimum principle of total potential energy. The boundary conditions related to electric variables are discussed and this facilitates the solution of global stiffness equation. Numerical examples are given at the end of the paper to calculate laminated plates with various kinds of elastic and electric boundary conditions. Some results are compared with analytical ones.
引文
[1] 王焕定，吴德伦 有限单元法及计算程序，中国建筑工业出版社，1997
[2] 冯康，石钟慈 弹性结构的数学理论，科学出版社，1981
[3] 龙驭球 新型有限元引论，清华大学出版社，1991
[4] 林启荣，刘振兴等，噪声控制压电智能系统研究的现状和展望[J]，力学季刊，2000，21(1)：27-32
[5] Rao S.S., Sunar M., piezoelectricity and its use in disturbance sensing and control of flexible structures [J], App. Mech. Rev., 1994,47:113-123
[6] Allik, H. and Hughes, T.J.R.(1970). Finite element method for piezoelectric vibtration. International Journal of Numerical Methods in Engineering 2,151-157
[7] Bailey, T. and Hubbard, J.E.(1985). Distributed piezoelectric-polymer active vibration control of a cantilever beam. Journal of Guidance,8,605-611
[8] Chandrashekhara, K. and Agarwal, A.N.(1993). Active vibration control of laminated composite plates using piezoelectric devices: a finite element approach. Journal of Intelligent Materials, Systems and Structures 4,496-508
[9] Chandrashekhara, K. and Kolli ,M.(1995). Thermally induced vibration of adaptive doubly curved composite shells with piezoelectric devices. In Proc.36th Structures, Structural Dynamics, and Materials Conference,New Orleans,LA,10-13 April 1995, pp.1628-1636
[10] Crawley, E.F. and de Luis, J. (1987). Use of piezoelectric actuators as elements of intelligent structures. AIAA Journal 25, 1373-1385.
[11] Ha, S.K., Keilers, C. and Chang, F.-K. (1992). Finite element analysis of composite structures containing distributed piezoelectric sensors and actuators. AIAA Journal 30,772-780
[12] Heyliger, RR.,Ramirez,G. and Saravanos, D.A.(1994).Coupled discrete-layer finite elements for laminated piezoelectric plates.Communications in Numerical Methods in Engineering 10, 971-981
[13] Hwang, W.-S. and Park, H. C. (1993). Finite element modeling of piezoelectric sensors and actuators. AIAA Journal 31,930-937
[14] Lammering, R.(1991). The application of a finite shell element for composite containing piezoelectric polymers in vibration control. Computers & Structures 41, 1101-1109
[15] Lee, C.K.(1990). Theory of laminated piezoelectric plates for the design of distributed sensors/actuators. Part 1: governing equations and reciprocal
??relationships. Journal of the Acoustic Society of America 87, 1144-1158
[16] Lee,H.-J. and Saravanos, D.A (1995) On the response of smart piezoelectric composite structures in thermal environments. In Proc.36th Structures, Structural Dynamics, and Materials Conf., New Orleans, LA,10-13 April 1995, pp.2876-2885
[17] Lee, H.-J. and Sravanos, D.A.(1996). Coupled layerwise analysis of thermpiezoelectric composite beams. AIAA Journal 34, 1231-1237
[18] Mindlin, R.D.(1974).Equations of high frequency vibrations of thermopiezoelectric crystal plates. International Journal of Solids and Structures 10,625-632
[19] Rao, S.S. and Sunar,M.(1993). Analysis of distributed thermopiezoelectric crystal plates. International intelligent structures. AIAA Journal 31, 1280-1286
[20] Robbins, D.H..and Reddy, J.N.(1991). Analysis of piezoelectrically actuated beams using a layer-wise displacement theory. Computers & Structures 41,265-279
[21] Saravanos, D.A. and Heyliger, RR.(1995). Coupled layerwise analysis of composite beams with embedded piezoelectric sensors and actuators. Journal of Intelligent Materials, Systems and Structures 6, 350-363
[22] Shieh, R.C.(1994). Governing equations and finite element methods for multiaxial piezoelectric beam sensors/actuators. AIAA Journal 32, 1250-1258
[23] Tauchert, T.R.(1992). Piezothermoelastic behaviour of a laminated plate. Journal of Thermal Stresses 15, 25-37
[24] Tzou, H.S. and Gadre M.(1989). Theoretical analysis of a multi-layered thin shell coupled with piezoelectric shell actuators for distributed vibration controls. Journal of Sound and Vibration 132,433-450
[25] Tzou, H. S. and Tseng, C.I.(1990). Distributed piezoelectric sensor/actuator design for dynamic measurement/control of distributed parameter systems: a piezoelectric finite element approach. Journal of Sound and Vibration 138,17-34
[26] Tzou, H.S. and Ye,R.(1994a). Analysis of laminated piezoelectric shell systerms with C~°piezoelectric triangle finite elements. In Proc. Adaptive Structures and Composite Materials: Analysis and Application. ASME,Vol. AD-45,pp.113-124.
[27] Tzou, H.S. and Ye,R. (1994b).Piezothermoelasticity and precision control of piezoelectric systems: theory and finite element analysis. Journal of Vibration and Acoustics 116, 489-495
[28] Wang, B.-T. and Rogers, C.A.(1991). Laminate plate theory for spatially
??distributed induced strain actuators. Joumal of Computers and Materials 25,433-452.
[29] Reddy, J.N.(1987). A generalization of two-dimensional theories of laminated composite plates. Communications in Numerical Methods in Engineering3,173-180
[30] Reddy, J.N.(1993). An evaluation of equivalent single-layer and layerwise theories of composite laminates. Computers & Structures 25, 21-35
[31] Tzou, H.S. and Howard, R.V(1994). A piezothermoelastic thin shell theory applied to active structures. Journal of Vibration and Acoustics 116, 295-302
[32] Vekovishcheva I A. Plane problem of electroelastic theory for a piezoelectric plates. Sov. Appl. Mech., 1976, 11(2): 180—183
[33] Wang B T, Rogers C A. Laminates plate theory for spatially distributed strain actuators. J composite Materials, 1991, 25:433—452
[34] Mitchell J A, Reddy J N. A refined hybrid plate theory for composite laminates with piezoelectric lamina. Int. J. solids structures, 1995, 32:2345—2367
[35] 曹宗杰，闻邦椿等．含压电材料智能结构动态特性的研究．计算力学学报,200 1，18(3)：0267-06
[36] Srinivas, S. and Rao, A.K. (1970). Bending, vibration and buckling of simply supported thick orthotropic rectangular plates and laminates. Int. J. Solids Structures. 1970, 6: 1463-1481.
[37] Fan J R, Ye J Q. An Exact Solution for the Statics and Dynamics of Laminated Thick Plates with Orthotropic Layers. Int J Solids Structures, 1990, 26(5/6): 655—662.
[38] 盛宏玉，范家让．非平面应变状态下的叠层厚壁筒．应用力学学报，1997，14(2)：64—71
[39] 盛宏玉，范家让．具有固支边的强厚度层合板的一种新解法．计算物理，1999，16(6)：682—687
[40] 高坚新，沈亚鹏，王子昆．有限长压电层合简支板自由振动的三维精确解．力学学报，1998，30(2)，168—177
[41] Lee J S, Jiang L Z. (1996). Exact electroelastic analysis of piezoelectric laminate via state space approach. Int J Solids Structures, 1996, 33:977-990
[42] 龙志飞等 有限元法新论 中国水利水电出版社2001
[43] 丁浩江等 弹性和塑性力学中的有限单元法(第二版)北京：机械工业出版社，1989
[44] H. A. Sosa and M. A. Castro, Electroelastic analysis of Piezoelectric laminated structures[J], Appl. Mech.,Rov., 1993, 46(11):S21-S28
[45] Cady, W. G., Piezoelectricity[M], revised edition, Dover Publications, Inc., New York, 1964
[46] Jaffe, B., Cook, W. R. and Jaffe, H., Piezoelectricity Ceramics[M], Academic Press, New York, 1971
[2] 冯康，石钟慈 弹性结构的数学理论，科学出版社，1981
[3] 龙驭球 新型有限元引论，清华大学出版社，1991
[4] 林启荣，刘振兴等，噪声控制压电智能系统研究的现状和展望[J]，力学季刊，2000，21(1)：27-32
[5] Rao S.S., Sunar M., piezoelectricity and its use in disturbance sensing and control of flexible structures [J], App. Mech. Rev., 1994,47:113-123
[6] Allik, H. and Hughes, T.J.R.(1970). Finite element method for piezoelectric vibtration. International Journal of Numerical Methods in Engineering 2,151-157
[7] Bailey, T. and Hubbard, J.E.(1985). Distributed piezoelectric-polymer active vibration control of a cantilever beam. Journal of Guidance,8,605-611
[8] Chandrashekhara, K. and Agarwal, A.N.(1993). Active vibration control of laminated composite plates using piezoelectric devices: a finite element approach. Journal of Intelligent Materials, Systems and Structures 4,496-508
[9] Chandrashekhara, K. and Kolli ,M.(1995). Thermally induced vibration of adaptive doubly curved composite shells with piezoelectric devices. In Proc.36th Structures, Structural Dynamics, and Materials Conference,New Orleans,LA,10-13 April 1995, pp.1628-1636
[10] Crawley, E.F. and de Luis, J. (1987). Use of piezoelectric actuators as elements of intelligent structures. AIAA Journal 25, 1373-1385.
[11] Ha, S.K., Keilers, C. and Chang, F.-K. (1992). Finite element analysis of composite structures containing distributed piezoelectric sensors and actuators. AIAA Journal 30,772-780
[12] Heyliger, RR.,Ramirez,G. and Saravanos, D.A.(1994).Coupled discrete-layer finite elements for laminated piezoelectric plates.Communications in Numerical Methods in Engineering 10, 971-981
[13] Hwang, W.-S. and Park, H. C. (1993). Finite element modeling of piezoelectric sensors and actuators. AIAA Journal 31,930-937
[14] Lammering, R.(1991). The application of a finite shell element for composite containing piezoelectric polymers in vibration control. Computers & Structures 41, 1101-1109
[15] Lee, C.K.(1990). Theory of laminated piezoelectric plates for the design of distributed sensors/actuators. Part 1: governing equations and reciprocal
??relationships. Journal of the Acoustic Society of America 87, 1144-1158
[16] Lee,H.-J. and Saravanos, D.A (1995) On the response of smart piezoelectric composite structures in thermal environments. In Proc.36th Structures, Structural Dynamics, and Materials Conf., New Orleans, LA,10-13 April 1995, pp.2876-2885
[17] Lee, H.-J. and Sravanos, D.A.(1996). Coupled layerwise analysis of thermpiezoelectric composite beams. AIAA Journal 34, 1231-1237
[18] Mindlin, R.D.(1974).Equations of high frequency vibrations of thermopiezoelectric crystal plates. International Journal of Solids and Structures 10,625-632
[19] Rao, S.S. and Sunar,M.(1993). Analysis of distributed thermopiezoelectric crystal plates. International intelligent structures. AIAA Journal 31, 1280-1286
[20] Robbins, D.H..and Reddy, J.N.(1991). Analysis of piezoelectrically actuated beams using a layer-wise displacement theory. Computers & Structures 41,265-279
[21] Saravanos, D.A. and Heyliger, RR.(1995). Coupled layerwise analysis of composite beams with embedded piezoelectric sensors and actuators. Journal of Intelligent Materials, Systems and Structures 6, 350-363
[22] Shieh, R.C.(1994). Governing equations and finite element methods for multiaxial piezoelectric beam sensors/actuators. AIAA Journal 32, 1250-1258
[23] Tauchert, T.R.(1992). Piezothermoelastic behaviour of a laminated plate. Journal of Thermal Stresses 15, 25-37
[24] Tzou, H.S. and Gadre M.(1989). Theoretical analysis of a multi-layered thin shell coupled with piezoelectric shell actuators for distributed vibration controls. Journal of Sound and Vibration 132,433-450
[25] Tzou, H. S. and Tseng, C.I.(1990). Distributed piezoelectric sensor/actuator design for dynamic measurement/control of distributed parameter systems: a piezoelectric finite element approach. Journal of Sound and Vibration 138,17-34
[26] Tzou, H.S. and Ye,R.(1994a). Analysis of laminated piezoelectric shell systerms with C~°piezoelectric triangle finite elements. In Proc. Adaptive Structures and Composite Materials: Analysis and Application. ASME,Vol. AD-45,pp.113-124.
[27] Tzou, H.S. and Ye,R. (1994b).Piezothermoelasticity and precision control of piezoelectric systems: theory and finite element analysis. Journal of Vibration and Acoustics 116, 489-495
[28] Wang, B.-T. and Rogers, C.A.(1991). Laminate plate theory for spatially
??distributed induced strain actuators. Joumal of Computers and Materials 25,433-452.
[29] Reddy, J.N.(1987). A generalization of two-dimensional theories of laminated composite plates. Communications in Numerical Methods in Engineering3,173-180
[30] Reddy, J.N.(1993). An evaluation of equivalent single-layer and layerwise theories of composite laminates. Computers & Structures 25, 21-35
[31] Tzou, H.S. and Howard, R.V(1994). A piezothermoelastic thin shell theory applied to active structures. Journal of Vibration and Acoustics 116, 295-302
[32] Vekovishcheva I A. Plane problem of electroelastic theory for a piezoelectric plates. Sov. Appl. Mech., 1976, 11(2): 180—183
[33] Wang B T, Rogers C A. Laminates plate theory for spatially distributed strain actuators. J composite Materials, 1991, 25:433—452
[34] Mitchell J A, Reddy J N. A refined hybrid plate theory for composite laminates with piezoelectric lamina. Int. J. solids structures, 1995, 32:2345—2367
[35] 曹宗杰，闻邦椿等．含压电材料智能结构动态特性的研究．计算力学学报,200 1，18(3)：0267-06
[36] Srinivas, S. and Rao, A.K. (1970). Bending, vibration and buckling of simply supported thick orthotropic rectangular plates and laminates. Int. J. Solids Structures. 1970, 6: 1463-1481.
[37] Fan J R, Ye J Q. An Exact Solution for the Statics and Dynamics of Laminated Thick Plates with Orthotropic Layers. Int J Solids Structures, 1990, 26(5/6): 655—662.
[38] 盛宏玉，范家让．非平面应变状态下的叠层厚壁筒．应用力学学报，1997，14(2)：64—71
[39] 盛宏玉，范家让．具有固支边的强厚度层合板的一种新解法．计算物理，1999，16(6)：682—687
[40] 高坚新，沈亚鹏，王子昆．有限长压电层合简支板自由振动的三维精确解．力学学报，1998，30(2)，168—177
[41] Lee J S, Jiang L Z. (1996). Exact electroelastic analysis of piezoelectric laminate via state space approach. Int J Solids Structures, 1996, 33:977-990
[42] 龙志飞等 有限元法新论 中国水利水电出版社2001
[43] 丁浩江等 弹性和塑性力学中的有限单元法(第二版)北京：机械工业出版社，1989
[44] H. A. Sosa and M. A. Castro, Electroelastic analysis of Piezoelectric laminated structures[J], Appl. Mech.,Rov., 1993, 46(11):S21-S28
[45] Cady, W. G., Piezoelectricity[M], revised edition, Dover Publications, Inc., New York, 1964
[46] Jaffe, B., Cook, W. R. and Jaffe, H., Piezoelectricity Ceramics[M], Academic Press, New York, 1971