复合材料及压电材料层合板壳的动力学研究
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摘要
以弹性力学的Hamilton体系为基础,将一般弹性材料的Hellinger-Reissner (H-R)变分原理引入到考虑粘滞阻尼力的一般复合结构、压电材料结构、热弹性结构的动力学分析中,扩展了Hamilton正则方程理论的应用范畴。本文主要包含以下几个方面:
1.建立了一般复合材料的包含粘滞阻尼力修正后的H-R变分原理,推导了对应的状态向量方程。结合精细积分法和米勒法分析了四边简支的开口层合壳的简谐振动问题,研究了粘滞阻尼对壳振动的影响作用。依据线性阻尼振动理论,简要地给出了开口层合壳的欠阻尼、临界阻尼和过阻尼等三种自由运动的通解公式。
2.扩展了一种Laplace数值逆变换的数值方法,通过实例分析验证了该方法的准确性。然后,根据压电体的混合变分原理导出了压电材料动力学问题的状态向量方程,并给出了四边简支压电材料矩形板动力学问题的精确解。对于混合层合板,我们利用传递矩阵的方法进行求解。该方法不仅考虑了弹性层和压电层界面间广义位移和广义应力的连续性,还考虑到了结构的转动惯量和剪切变形。作为Laplace数值逆变换在本文中的应用,具体研究了简谐振动和瞬态响应的问题,获得高精度的数值结果。
3.根据广义的Hamilton变分原理分别推导出了热弹性复合材料和压电热弹性材料非齐次的Hamilton正则方程。然后分析了温度载荷和力载荷作用下压电热弹性材料层合板壳的响应问题;在研究热弹性复合材料层合板的响应问题时,同时考虑了温度和粘滞阻尼力对结构振动的影响。
Based upon the Hamilton systematic methodology, the Hellinger-Reissner (H-R) mixed variational principle for general elastic materials was introduced into analyzing the dynamic problems of general composite structures with viscous damping force, piezoelectric and thermoelastic structures. The application area of the Hamilton canonical equation theory was extended. The main work could be outlined as follows:
1. A modified mixed variational principle for general composite materials with viscous damping force was established and the corresponding state-vector equation was reduced. Combining the precise integration method and Muller method, a new solution for the harmonic vibration of simply supported open cylindrical laminates was proposed. The general solutions for the free vibration of underdamping, critical damping, overdamping of cylindrical laminates was simply given in terms of the linear damp vibration theory. The effect of viscous damp on the vibration of open cylindrical laminates was investigated.
2. A numerical method for the inversion of Laplace transform was developed and its accuracy was shown through a lot of examples. Then, a state-vector equation for the dynamic problems of piezoelectric plates was deduced from a modified mixed variational principle for piezoelectric bodies and its exact solution for the dynamic problems of simply supported rectangle piezoelectric plate was given. For multilayered hybrid plates, the solution in terms of the propagator matrices was derived. The techniques not only accounts for the compatibility of generalized displacements and generalized stresses on the interface both the elastic layers and piezoelectric layers, but also the rotary inertia of laminate and the transverse shear deformation are considered in the general equation of structure. As an application of the numerical inversion of Laplace transform presented in this paper, the problems of harmonic vibration and transient response were proposed and discussed, and we obtained the highly accurate numerical results.
3. According to the generalized Hamilton variation principle, the non-homogeneous Hamilton canonical equation for piezothermoelastic bodies was derived. Then, the numerical responses of the laminated piezothermoelastic plates and shells under thermal and dynamic load were studied; the effects of temperature and viscous damp on the vibration of laminated plate were also investigated when we studied the response problems of the thermoelastic composite laminated plates.
1.建立了一般复合材料的包含粘滞阻尼力修正后的H-R变分原理,推导了对应的状态向量方程。结合精细积分法和米勒法分析了四边简支的开口层合壳的简谐振动问题,研究了粘滞阻尼对壳振动的影响作用。依据线性阻尼振动理论,简要地给出了开口层合壳的欠阻尼、临界阻尼和过阻尼等三种自由运动的通解公式。
2.扩展了一种Laplace数值逆变换的数值方法,通过实例分析验证了该方法的准确性。然后,根据压电体的混合变分原理导出了压电材料动力学问题的状态向量方程,并给出了四边简支压电材料矩形板动力学问题的精确解。对于混合层合板,我们利用传递矩阵的方法进行求解。该方法不仅考虑了弹性层和压电层界面间广义位移和广义应力的连续性,还考虑到了结构的转动惯量和剪切变形。作为Laplace数值逆变换在本文中的应用,具体研究了简谐振动和瞬态响应的问题,获得高精度的数值结果。
3.根据广义的Hamilton变分原理分别推导出了热弹性复合材料和压电热弹性材料非齐次的Hamilton正则方程。然后分析了温度载荷和力载荷作用下压电热弹性材料层合板壳的响应问题;在研究热弹性复合材料层合板的响应问题时,同时考虑了温度和粘滞阻尼力对结构振动的影响。
Based upon the Hamilton systematic methodology, the Hellinger-Reissner (H-R) mixed variational principle for general elastic materials was introduced into analyzing the dynamic problems of general composite structures with viscous damping force, piezoelectric and thermoelastic structures. The application area of the Hamilton canonical equation theory was extended. The main work could be outlined as follows:
1. A modified mixed variational principle for general composite materials with viscous damping force was established and the corresponding state-vector equation was reduced. Combining the precise integration method and Muller method, a new solution for the harmonic vibration of simply supported open cylindrical laminates was proposed. The general solutions for the free vibration of underdamping, critical damping, overdamping of cylindrical laminates was simply given in terms of the linear damp vibration theory. The effect of viscous damp on the vibration of open cylindrical laminates was investigated.
2. A numerical method for the inversion of Laplace transform was developed and its accuracy was shown through a lot of examples. Then, a state-vector equation for the dynamic problems of piezoelectric plates was deduced from a modified mixed variational principle for piezoelectric bodies and its exact solution for the dynamic problems of simply supported rectangle piezoelectric plate was given. For multilayered hybrid plates, the solution in terms of the propagator matrices was derived. The techniques not only accounts for the compatibility of generalized displacements and generalized stresses on the interface both the elastic layers and piezoelectric layers, but also the rotary inertia of laminate and the transverse shear deformation are considered in the general equation of structure. As an application of the numerical inversion of Laplace transform presented in this paper, the problems of harmonic vibration and transient response were proposed and discussed, and we obtained the highly accurate numerical results.
3. According to the generalized Hamilton variation principle, the non-homogeneous Hamilton canonical equation for piezothermoelastic bodies was derived. Then, the numerical responses of the laminated piezothermoelastic plates and shells under thermal and dynamic load were studied; the effects of temperature and viscous damp on the vibration of laminated plate were also investigated when we studied the response problems of the thermoelastic composite laminated plates.
引文
[1] Pagano, N J. Exact solutions for rectangular bidirectional composites and sandwich plates [J]. Journal of Composite materials 1970.4: 20-34.
[2] Ray M C, Rao K M and Samanta B. Exact analysis of coupled electroelastic behavior of a piezoelectric plate under cylindrical bending [J]. Computers & Structures 1993, 47(6):1031-1042.
[3] Batra R C and Liang X Q. The vibration of a rectangular laminated elastic plate with embedded piezoelectric sensors and actuators [J]. Computer & Structure 1997, 63(2): 213-216.
[4] Ray M C, Bhattacharya R and Samanta B. Exact solutions for dynamic analysis of composite plate with distributed piezoelectric layers [J]. Compute & Structures 1998, 66(6): 737-743.
[5] Vel S S, Batra R C. Generalized plane strain thermopiezoelectric analysis of multilayered plates [J]. Journal of Thermal Stresses, 2003, 26: 353-377.
[6]王敏中,赵颖涛.二维各向异性弹性力学的Stroh公式应用推广[J].力学与实践, 2001, 23(6): 7-15
[7] Qing T C T. Anisotropic elasticity: theory and application [J], Oxford University Press. 1996.
[8] Heyliger P R, Brooks P. Exact solution for laminated piezoelectric plates in cylindrical bending [J]. Journal of Applied Mechanics, 1996. 63(4): 903-910.
[9] Xu K, Noor A K and Tang Y Y. Three-dimensional solutions for coupled thermoelectroelastic response of multilayered plates [J], Computer Methods in Applied Mechanics and Engineering, 1995 126(3-4): 355-371.
[10] Ootao Y, Tanigawa Y. Control of transient thermoelastic displacement of a two-layered composite constructed of isotropic elastic and piexoelectric layers due to nonuniform heating [J]. Archive of Applied Mechanics, 2001 71(4-5): 207-230.
[11] Jiann Q T. A state space formalism for piezotheroelasticity [J]. International Journal of Solids and Structures, 200239(20): 5173-5184.
[12] Zhang C, Cheung Y K and Di S. The exact solution of coupled thermoelectroelastic behavior of piezoelectric laminates [J]. Computers & Structures, 2002, 80(13): 1202-1212.
[13] Mindlin R D. Equations of high frequency vibrations of thermopiezoelectric crystal plates [J]. International Journal Solids and Structures, 1974, l0(6): 625-637.
[14] Zhang C, Cheung Y K, Di S and Zhang N. The exact solution of coupledthermoelectroelastic behavior of piezoelectric laminates [J]. Computers & Structures, 2002, 80(13): 1201-1202.
[15] Xu K M, Noor A K and Tang Y. Three-dimensional solutions for coupled thermoelectroelastic response of multilayered plates [J]. Computer Method Applied Mechanics and Engineering 1995, 126(3-4): 355-371
[16] Vel S S, Batra R C. Generalized plane strain thermopiezoelectric analysis of multilayered plates [J]. Journal of Thermal Stresses, 2003, 26(4): 353-377.
[17] Ootao Y, Tanigawa Y. Control of transient thermoelastic displacement of a two-layered composite plate constructed of isotropic elastic and piezoelectric layers due to nonuniform heating [J]. Archive of Applied Mechanics, 2001, 71(4-5): 207-230.
[18] Kapuria S, Dumir P C and Sengupta S. Three-dimensional solution for shape control of a simply supported rectangular hybrid plate [J]. Journal of Thermal Stresses, 1999,22:159-176
[19] Steele C R and Kim Y Y. Modified mixed variational principle and the state-vector equation for elastic bodies and shells of revolution [J]. Journal of Applied Mechanics, 1992, 59(3): 587-598.
[20]邹贵平.层合板壳问题哈密顿体系与哈密顿型广义变分原理[J],上海大学学报(自然科学版),1996,2(2):119-128.
[21]卿光辉,邱家俊,塔娜.压电体的变分原理及叠层板的自由振动分析[J],振动工程学报,2004,17(3):285-290
[22] Qing G H, Qiu J J, Liu Y H. A semi-analytical solution for dynamic analysis of plate with piezoelectric patches [J]. International Journal of Solids and Structures, 2006,43(6): 1388～1403.
[23] Qing G H, Qiu J J, Liu Y H. Free Vibration Analysis of Stiffened Laminated Plates [J]. International Journal of Solids and Structures, 2006,43(6): 1357～1371.
[24]钟万勰著.应用力学对偶体系[M],北京:科学出版社,2003.
[25]钟万勰著.应用力学的辛数值方法[M],北京:科学出版社,2006.
[26] Qing T C T. Anisotropic elasticity: theory and application [J], Oxford University Press. New York, 1996.
[27]钟万勰著.弹性力学求解新体系[M],大连:大连理工大学出版社, 1995.
[28]范家让,张巨勇.层合开口圆柱壳的静、动态和稳定问题的精确解[J].中国科学: E辑, 1992, 6: 623~632.
[29]唐立民等.混合状态Hamiltonian元的半解析解和层合板的计算[J].计算结构力学及其应用, 1992, 9(4):347~360.
[30]范家让著.强厚度层合板壳的精确理论,北京:科学出版社,1996.
[31] Srinivas S and Rao, A K. Bending Vibration and Buckling of Simply Supported ThickOrthotropic Rectangular Plates and Laminates [J], International Journal of Solids and Structures, 1970,6(11): 1463-1481.
[32] Durbin F. Numerical inversion of Laplace transforms: an efficient improvement to Duber and Abate’s method [J]. Comput J, 1974, 17(4): 371-376.
[33] Zhao X. An efficient approach for the numerical inversion of Laplace transform and its application in dynamic fracture analysis of a piezoelectric laminate [J]. International Journal of Solids and Structures, 2004; 41(13): 3653-3674.
[34] Nowacki W. Some General Theorems of Thermopizeoelectricity [J]. Journal of Thermal Stresses, 1978, 1(2): 171~182.
[35] Iesan D. On Some Theorems of Thermopizeoelectricity [J]. Journal of Thermal Stresses, 1989, 12: 209~223.
[36] Tauchert T R. Pizeothermoelastic Behavior of a Laminated Plate [J]. Journal of Thermal Stresses, 1992, 15: 25~37.
[37] Jonnalagadda K D, Blandford G E, and Tauchert T R. Piezothermoelastic Composite Plate Analysis Using First-Oder Shear-Deformation Theory [J]. Computers and Structures, 1994, 51(1): 79~89.
[38] Tauchert T R, Ashida F, Noda N. Recent Developments in Piezothermoelasticity: Inverse Problems Relevant to Smart Structures [J]. Japan Society of Mechanical Engineering International Journal, 1999, 42(4): 452~458.
[39]盛宏玉,张伟林,高荣誉.一般边界条件下压电层合厚板的精确解,安徽建筑工业学院学报[J],2002,10(4): 1~6.
[40] Tarn J Q. A state space formalism for piezothermoelasticity [J]. International Journal of Solids and Structures, 2002, 39(20): 5173~5184.
[41] Ting T C T. Anisotropic Elasticity Theory and Applications [J]. New York, Oxford University Press, 1996, 506-507.
[2] Ray M C, Rao K M and Samanta B. Exact analysis of coupled electroelastic behavior of a piezoelectric plate under cylindrical bending [J]. Computers & Structures 1993, 47(6):1031-1042.
[3] Batra R C and Liang X Q. The vibration of a rectangular laminated elastic plate with embedded piezoelectric sensors and actuators [J]. Computer & Structure 1997, 63(2): 213-216.
[4] Ray M C, Bhattacharya R and Samanta B. Exact solutions for dynamic analysis of composite plate with distributed piezoelectric layers [J]. Compute & Structures 1998, 66(6): 737-743.
[5] Vel S S, Batra R C. Generalized plane strain thermopiezoelectric analysis of multilayered plates [J]. Journal of Thermal Stresses, 2003, 26: 353-377.
[6]王敏中,赵颖涛.二维各向异性弹性力学的Stroh公式应用推广[J].力学与实践, 2001, 23(6): 7-15
[7] Qing T C T. Anisotropic elasticity: theory and application [J], Oxford University Press. 1996.
[8] Heyliger P R, Brooks P. Exact solution for laminated piezoelectric plates in cylindrical bending [J]. Journal of Applied Mechanics, 1996. 63(4): 903-910.
[9] Xu K, Noor A K and Tang Y Y. Three-dimensional solutions for coupled thermoelectroelastic response of multilayered plates [J], Computer Methods in Applied Mechanics and Engineering, 1995 126(3-4): 355-371.
[10] Ootao Y, Tanigawa Y. Control of transient thermoelastic displacement of a two-layered composite constructed of isotropic elastic and piexoelectric layers due to nonuniform heating [J]. Archive of Applied Mechanics, 2001 71(4-5): 207-230.
[11] Jiann Q T. A state space formalism for piezotheroelasticity [J]. International Journal of Solids and Structures, 200239(20): 5173-5184.
[12] Zhang C, Cheung Y K and Di S. The exact solution of coupled thermoelectroelastic behavior of piezoelectric laminates [J]. Computers & Structures, 2002, 80(13): 1202-1212.
[13] Mindlin R D. Equations of high frequency vibrations of thermopiezoelectric crystal plates [J]. International Journal Solids and Structures, 1974, l0(6): 625-637.
[14] Zhang C, Cheung Y K, Di S and Zhang N. The exact solution of coupledthermoelectroelastic behavior of piezoelectric laminates [J]. Computers & Structures, 2002, 80(13): 1201-1202.
[15] Xu K M, Noor A K and Tang Y. Three-dimensional solutions for coupled thermoelectroelastic response of multilayered plates [J]. Computer Method Applied Mechanics and Engineering 1995, 126(3-4): 355-371
[16] Vel S S, Batra R C. Generalized plane strain thermopiezoelectric analysis of multilayered plates [J]. Journal of Thermal Stresses, 2003, 26(4): 353-377.
[17] Ootao Y, Tanigawa Y. Control of transient thermoelastic displacement of a two-layered composite plate constructed of isotropic elastic and piezoelectric layers due to nonuniform heating [J]. Archive of Applied Mechanics, 2001, 71(4-5): 207-230.
[18] Kapuria S, Dumir P C and Sengupta S. Three-dimensional solution for shape control of a simply supported rectangular hybrid plate [J]. Journal of Thermal Stresses, 1999,22:159-176
[19] Steele C R and Kim Y Y. Modified mixed variational principle and the state-vector equation for elastic bodies and shells of revolution [J]. Journal of Applied Mechanics, 1992, 59(3): 587-598.
[20]邹贵平.层合板壳问题哈密顿体系与哈密顿型广义变分原理[J],上海大学学报(自然科学版),1996,2(2):119-128.
[21]卿光辉,邱家俊,塔娜.压电体的变分原理及叠层板的自由振动分析[J],振动工程学报,2004,17(3):285-290
[22] Qing G H, Qiu J J, Liu Y H. A semi-analytical solution for dynamic analysis of plate with piezoelectric patches [J]. International Journal of Solids and Structures, 2006,43(6): 1388～1403.
[23] Qing G H, Qiu J J, Liu Y H. Free Vibration Analysis of Stiffened Laminated Plates [J]. International Journal of Solids and Structures, 2006,43(6): 1357～1371.
[24]钟万勰著.应用力学对偶体系[M],北京:科学出版社,2003.
[25]钟万勰著.应用力学的辛数值方法[M],北京:科学出版社,2006.
[26] Qing T C T. Anisotropic elasticity: theory and application [J], Oxford University Press. New York, 1996.
[27]钟万勰著.弹性力学求解新体系[M],大连:大连理工大学出版社, 1995.
[28]范家让,张巨勇.层合开口圆柱壳的静、动态和稳定问题的精确解[J].中国科学: E辑, 1992, 6: 623~632.
[29]唐立民等.混合状态Hamiltonian元的半解析解和层合板的计算[J].计算结构力学及其应用, 1992, 9(4):347~360.
[30]范家让著.强厚度层合板壳的精确理论,北京:科学出版社,1996.
[31] Srinivas S and Rao, A K. Bending Vibration and Buckling of Simply Supported ThickOrthotropic Rectangular Plates and Laminates [J], International Journal of Solids and Structures, 1970,6(11): 1463-1481.
[32] Durbin F. Numerical inversion of Laplace transforms: an efficient improvement to Duber and Abate’s method [J]. Comput J, 1974, 17(4): 371-376.
[33] Zhao X. An efficient approach for the numerical inversion of Laplace transform and its application in dynamic fracture analysis of a piezoelectric laminate [J]. International Journal of Solids and Structures, 2004; 41(13): 3653-3674.
[34] Nowacki W. Some General Theorems of Thermopizeoelectricity [J]. Journal of Thermal Stresses, 1978, 1(2): 171~182.
[35] Iesan D. On Some Theorems of Thermopizeoelectricity [J]. Journal of Thermal Stresses, 1989, 12: 209~223.
[36] Tauchert T R. Pizeothermoelastic Behavior of a Laminated Plate [J]. Journal of Thermal Stresses, 1992, 15: 25~37.
[37] Jonnalagadda K D, Blandford G E, and Tauchert T R. Piezothermoelastic Composite Plate Analysis Using First-Oder Shear-Deformation Theory [J]. Computers and Structures, 1994, 51(1): 79~89.
[38] Tauchert T R, Ashida F, Noda N. Recent Developments in Piezothermoelasticity: Inverse Problems Relevant to Smart Structures [J]. Japan Society of Mechanical Engineering International Journal, 1999, 42(4): 452~458.
[39]盛宏玉,张伟林,高荣誉.一般边界条件下压电层合厚板的精确解,安徽建筑工业学院学报[J],2002,10(4): 1~6.
[40] Tarn J Q. A state space formalism for piezothermoelasticity [J]. International Journal of Solids and Structures, 2002, 39(20): 5173~5184.
[41] Ting T C T. Anisotropic Elasticity Theory and Applications [J]. New York, Oxford University Press, 1996, 506-507.