热弹性材料层合板壳静力学研究
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摘要
基于弹性力学和传热学理论,结合状态空间法和对偶理论,研究了热弹性复合材料和压电材料层合板及壳问题。
首先,分别推导了机-热耦合的热弹性复合材料和机-电-热耦合的热弹性压电材料的非齐次状态方程;再由对偶理论,将热传导方程并入到复合材料和压电材料的本构关系中,分别得到热弹性复合材料和压电材料的增维本构关系,由此增维本构关系,通过状态空间法,便可直接推导出热弹性复合材料机-热耦合和热弹性压电材料机-电-热耦合问题的齐次状态方程。
齐次状态方程的导出,避免了非齐次状态方程求解时,必须先求解由热传导方程和热平衡方程导出的,关于温度的二阶微分方程,可大大简化求解过程,提高计算精度,并可直接进行有限元离散,为工程实践中,求解复杂边界条件下层合板及壳的半解析解提供了一种新方法。
文中除了详细推导了热弹性复合材料层合板、开口圆柱壳和正交双曲壳,以及热弹性压电材料层合板和开口圆柱壳的非齐次状态方程和齐次状态方程外,以非齐次状态方程和齐次状态方程为求解对象,分析了简支边界条件下多种板壳结构的精确解。两种方法的求解结果完全吻合,从而验证了齐次状态方程的正确性。
Based upon the theories of elastic mechanics and heat transfer, uniting the symplectic theory and state space method, the static mechanics problems of the laminated plates and laminated shells of thermoelastic composites and piezothermoelastic composites were studied.
First, deduced the non-homogenous state equations of thermoelastic composites with mechanic-heat coupling effect and piezothermoelastic composites with mechanic-electric-heat coupling effect respectively. Secondly, according to the symplectic theory, a corresponding augment dimensions constitutive relationships was established by combining the heat exchange equations with the constitutive relationships of thermoelastic composite materials and piezothermoelastic composite materials. The homogenous state equations of thermoelastic composites with mechanic-heat coupling effect and piezothermoelastic composites with mechanic-electric-heat coupling effect, wer directly derived by the augment dimensions constitutive relationships and the state space method.
The treatment avoided the solution of the second order differential equation on the thermal equilibrium equations and thermal gradient relationships, which was necessary performed to solving non-homogeneous state equations, greatly simplified the solution procedure of laminated structures, and improved numerical precision. Notably the homogenous state equations could be straightly discreted in finite element format, presented a new method for the semi-analytical solution of laminated plates and shells with complicated boundary condition in practice.
In this paper, non-homogenous state equations and homogenous state equations were detailedly derived of thermoelastic composites laminated plates, cylindrical open shells, orthogonal generic shells, and piezothermoelastic composites laminated plates, cylindrical open shells, and respectively analyzed their exact solution with four simply supported, the results validated the correctness of the homogenous state equations.
首先,分别推导了机-热耦合的热弹性复合材料和机-电-热耦合的热弹性压电材料的非齐次状态方程;再由对偶理论,将热传导方程并入到复合材料和压电材料的本构关系中,分别得到热弹性复合材料和压电材料的增维本构关系,由此增维本构关系,通过状态空间法,便可直接推导出热弹性复合材料机-热耦合和热弹性压电材料机-电-热耦合问题的齐次状态方程。
齐次状态方程的导出,避免了非齐次状态方程求解时,必须先求解由热传导方程和热平衡方程导出的,关于温度的二阶微分方程,可大大简化求解过程,提高计算精度,并可直接进行有限元离散,为工程实践中,求解复杂边界条件下层合板及壳的半解析解提供了一种新方法。
文中除了详细推导了热弹性复合材料层合板、开口圆柱壳和正交双曲壳,以及热弹性压电材料层合板和开口圆柱壳的非齐次状态方程和齐次状态方程外,以非齐次状态方程和齐次状态方程为求解对象,分析了简支边界条件下多种板壳结构的精确解。两种方法的求解结果完全吻合,从而验证了齐次状态方程的正确性。
Based upon the theories of elastic mechanics and heat transfer, uniting the symplectic theory and state space method, the static mechanics problems of the laminated plates and laminated shells of thermoelastic composites and piezothermoelastic composites were studied.
First, deduced the non-homogenous state equations of thermoelastic composites with mechanic-heat coupling effect and piezothermoelastic composites with mechanic-electric-heat coupling effect respectively. Secondly, according to the symplectic theory, a corresponding augment dimensions constitutive relationships was established by combining the heat exchange equations with the constitutive relationships of thermoelastic composite materials and piezothermoelastic composite materials. The homogenous state equations of thermoelastic composites with mechanic-heat coupling effect and piezothermoelastic composites with mechanic-electric-heat coupling effect, wer directly derived by the augment dimensions constitutive relationships and the state space method.
The treatment avoided the solution of the second order differential equation on the thermal equilibrium equations and thermal gradient relationships, which was necessary performed to solving non-homogeneous state equations, greatly simplified the solution procedure of laminated structures, and improved numerical precision. Notably the homogenous state equations could be straightly discreted in finite element format, presented a new method for the semi-analytical solution of laminated plates and shells with complicated boundary condition in practice.
In this paper, non-homogenous state equations and homogenous state equations were detailedly derived of thermoelastic composites laminated plates, cylindrical open shells, orthogonal generic shells, and piezothermoelastic composites laminated plates, cylindrical open shells, and respectively analyzed their exact solution with four simply supported, the results validated the correctness of the homogenous state equations.
引文
[1]张丽华,范玉清.复合材料在飞机上的应用综述[J].航空制造技术, 2006, 39(3):64-66
[2]田海民,缑新科.压电材料与智能结构在振动控制中的研究与前景展望[J].仪表技术与传感器, 2007, 8:7-9
[3]陈亚莉.智能材料在飞机结构上的新近应用研究[J].航空科学技术, 1997, 50(1):32-33,46
[4]范家让.强厚度叠层板壳的精确理论[M] .北京:科学出版社,1996. 1-100,130-376
[5]朱谷君.工程传热传质学[M].北京:航空工业出版社,1989. 2-20
[6] Cleve M, Charles V L. Nineteen dubious ways to compute the exponential of a matrix[J]. Society for Industrial and Applied Mathematics, 1978, 20(4): 801-836
[7]钟万勰.结构动力方程的精细时程积分法[J].大连理工大学学报, 1994, 34(2): 131-136
[8] Yang P C, Norris C H, Stavsky Y. Elastic wave propagation in heterogeneous plates[J]. International Journal of Solids and Structures, 1966, 2(4): 665-684
[9] Whitney J W. The effect of transverse shear deformation on the bending of laminated plates[J]. Journal of Compute Materials, 1969,3(3): 534-547
[10] Srinivas S, Rao A K. Bending, flexure of simple supported thick homogenous and laminated Plates [J]. Mathematica Mechanics , 1969,49(4): 449-458
[11] Srinivas S, Rao A K. Bending, vibration and buckling of simply supported thick orthotropic rectangular plates[J]. International Journal of Solids and Structures, 1970,6(11): 1463-1481
[12] Chandrashekara S, Santhosh U. Natural frequencies of cross-ply laminates by state space approach[J]. Journal of Sound and Vibration, 1990,136(3): 413-424
[13] Ting T C T. Anisotropic elasticity theory and applications[M]. New York, Oxford University Press, 1996. 506-507
[14]邹贵平.层合板热应力分析的辛正交解析法[J].强度与环境, 1997,3(1): 1-8
[15]丁皓江,国凤林,侯鹏飞.耦合热弹性问题的一般解[J].应用数学和力学, 2000, 21(6): 573-577
[16]钟万勰.应用力学对偶体系[M] .北京:科学出版社,2002. 289-574
[17]钟万勰.应用力学的辛数学方法[M] .北京:高等教育出版社,2006. 1-214
[18]丁克伟,唐立明.层合闭口厚柱壳的温度应力[J].上海力学, 1998,19(1): 29-37
[19]丁克伟,唐立明.任意厚度开口柱壳的温度应力[J].应用力学学报, 1999,16(2): 68-74
[20]丁克伟,唐立明.层合开口柱壳热应力问题的解析解[J].工程力学, 1999,16(3): 1-6
[21]丁克伟,唐立明.复合材料连续柱壳轴对称问题的热应力研究[J].大连理工大学学报, 1999,39(5): 607-611
[22]高荣誉,盛宏玉.变温场中正交异性叠成闭口厚柱壳的解析解[J].中国科学技术大学学报, 2000,30(2): 228-234
[23]盛宏玉,范家让.具有固支边的叠层开口柱壳的解析解[J].复合材料学报, 2000,17(4): 100-104
[24]邹贵平,唐立明.层合圆柱厚壳热应力分析的状态方程及其半解析法[J].力学学报, 1995,27(3): 336-342
[25]邹贵平,唐立明.层合双曲率厚壳的状态方程及其半解析法[J].航空学报, 1994,15(2): 1424-1432
[26] Sheng H Y, Ye J Q. A three-dimensional state space finite element solution for laminated composite cylindrical shells[J]. Computer Methods in Applied Mechanics and Engineering, 2003,192(22-24): 2441-2459
[27] Sosa H A, Castro M A. On the elastic and electric analyses of piezoelectric solids[J]. American Society of Mechanical Engineers, Aerospace Division (Publication) AD, Adaptive Structures and Material Systems, 1993, 35(2): 209-215
[28] Lee J S, Jiang L Z. Exact electroelastic analysis of piezoelectric laminae via state space approach[J]. International Journal Solids and Structures, 1996,33(7): 977-990
[29]陈伟球,梁剑,丁浩江.压电复合材料矩形厚板弯曲的三维分析[J].复合材料学报, 1997,14(1): 108-115
[30]丁皓江,梁剑.横观各向同性压电半无限体的Green函数[J].固体力学学报, 1998,19(2): 180-183
[31] Ding H J, Xu R Q, Chi Y W, eta. Free axisymmetric vibration of transversely isotropic piezoelectric circular plates[J]. International Journal Solids and Structures, 1999,36(30): 4629-4652
[32] Xu R Q, Ding H J, Chen W Q. Three-dimensional static analysis of multi-layered piezoelectric hollow spheres via the state space method[J]. International Journal of Solids and Structures, 2001, 38(28-29): 4921-4936
[33]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[J].安徽建筑工业学院学报(自然科学版), 2002,10(4): 1-6
[34]盛宏玉,董朝文.任意厚度压电层合闭口柱壳的精确解[J].合肥工业大学学报(自然科学版), 2003, 26(5): 980-985
[35] Mindlin R D. Equations of high frequency vibrations of thermopiezoelectric crystal plates[J]. International Journal of Solids and Structures, 1974, 10(6): 625-632
[36] Nowacki W. Some general theorems of thermopizeoelectricity[J]. Journal of Thermal Stresses, 1978, 1(1): 171-182
[37] Iesan D. On some theorems of thermopizeoelectricity[J]. Journal of Thermal Stresses, 1989,12(2): 209-223
[38] Tauchert T R. Pizeothermoelastic behavior of a laminated plate[J]. Journal of Thermal Stresses, 1992,15(1): 25-37
[39] Jonnalagadda K D, Blandford G E, Tauchert T R. Piezothermoelastic composite plate analysis using first-oder shear-deformation theory[J]. Computers and Structures, 1994,51(1): 79-89
[40] 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
[41] Xu K M, Noor A K, 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
[42] Kapuria S, Dumir P C, Sengupta S. Three-dimensional solution for shape control of a simply supported rectangular hybrid plate[J]. Journal of Thermal Stresses, 1999,22(1): 159-176
[43] Senthil S Vel, Batra R C. Generalized plane strain thermopiezoelectric analysis of multilayered plates[J], Journal of Thermal Stresses, 2003,26(4): 353-378
[44] 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): 207-220
[45] Tarn J Q. A state space formalism for piezothermoelasticity[J]. International Journal of Solids and Structures, 2002, 39(20): 5173-5184
[46] Qing G H, Liu Y H, Qiu J J, et al. A semi-analytical method for the free vibration analysis of thick double-shell systems[J]. Finite Elements in Analysis and Design, 2006, 42(10): 837-845
[47] 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
[48] 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
[2]田海民,缑新科.压电材料与智能结构在振动控制中的研究与前景展望[J].仪表技术与传感器, 2007, 8:7-9
[3]陈亚莉.智能材料在飞机结构上的新近应用研究[J].航空科学技术, 1997, 50(1):32-33,46
[4]范家让.强厚度叠层板壳的精确理论[M] .北京:科学出版社,1996. 1-100,130-376
[5]朱谷君.工程传热传质学[M].北京:航空工业出版社,1989. 2-20
[6] Cleve M, Charles V L. Nineteen dubious ways to compute the exponential of a matrix[J]. Society for Industrial and Applied Mathematics, 1978, 20(4): 801-836
[7]钟万勰.结构动力方程的精细时程积分法[J].大连理工大学学报, 1994, 34(2): 131-136
[8] Yang P C, Norris C H, Stavsky Y. Elastic wave propagation in heterogeneous plates[J]. International Journal of Solids and Structures, 1966, 2(4): 665-684
[9] Whitney J W. The effect of transverse shear deformation on the bending of laminated plates[J]. Journal of Compute Materials, 1969,3(3): 534-547
[10] Srinivas S, Rao A K. Bending, flexure of simple supported thick homogenous and laminated Plates [J]. Mathematica Mechanics , 1969,49(4): 449-458
[11] Srinivas S, Rao A K. Bending, vibration and buckling of simply supported thick orthotropic rectangular plates[J]. International Journal of Solids and Structures, 1970,6(11): 1463-1481
[12] Chandrashekara S, Santhosh U. Natural frequencies of cross-ply laminates by state space approach[J]. Journal of Sound and Vibration, 1990,136(3): 413-424
[13] Ting T C T. Anisotropic elasticity theory and applications[M]. New York, Oxford University Press, 1996. 506-507
[14]邹贵平.层合板热应力分析的辛正交解析法[J].强度与环境, 1997,3(1): 1-8
[15]丁皓江,国凤林,侯鹏飞.耦合热弹性问题的一般解[J].应用数学和力学, 2000, 21(6): 573-577
[16]钟万勰.应用力学对偶体系[M] .北京:科学出版社,2002. 289-574
[17]钟万勰.应用力学的辛数学方法[M] .北京:高等教育出版社,2006. 1-214
[18]丁克伟,唐立明.层合闭口厚柱壳的温度应力[J].上海力学, 1998,19(1): 29-37
[19]丁克伟,唐立明.任意厚度开口柱壳的温度应力[J].应用力学学报, 1999,16(2): 68-74
[20]丁克伟,唐立明.层合开口柱壳热应力问题的解析解[J].工程力学, 1999,16(3): 1-6
[21]丁克伟,唐立明.复合材料连续柱壳轴对称问题的热应力研究[J].大连理工大学学报, 1999,39(5): 607-611
[22]高荣誉,盛宏玉.变温场中正交异性叠成闭口厚柱壳的解析解[J].中国科学技术大学学报, 2000,30(2): 228-234
[23]盛宏玉,范家让.具有固支边的叠层开口柱壳的解析解[J].复合材料学报, 2000,17(4): 100-104
[24]邹贵平,唐立明.层合圆柱厚壳热应力分析的状态方程及其半解析法[J].力学学报, 1995,27(3): 336-342
[25]邹贵平,唐立明.层合双曲率厚壳的状态方程及其半解析法[J].航空学报, 1994,15(2): 1424-1432
[26] Sheng H Y, Ye J Q. A three-dimensional state space finite element solution for laminated composite cylindrical shells[J]. Computer Methods in Applied Mechanics and Engineering, 2003,192(22-24): 2441-2459
[27] Sosa H A, Castro M A. On the elastic and electric analyses of piezoelectric solids[J]. American Society of Mechanical Engineers, Aerospace Division (Publication) AD, Adaptive Structures and Material Systems, 1993, 35(2): 209-215
[28] Lee J S, Jiang L Z. Exact electroelastic analysis of piezoelectric laminae via state space approach[J]. International Journal Solids and Structures, 1996,33(7): 977-990
[29]陈伟球,梁剑,丁浩江.压电复合材料矩形厚板弯曲的三维分析[J].复合材料学报, 1997,14(1): 108-115
[30]丁皓江,梁剑.横观各向同性压电半无限体的Green函数[J].固体力学学报, 1998,19(2): 180-183
[31] Ding H J, Xu R Q, Chi Y W, eta. Free axisymmetric vibration of transversely isotropic piezoelectric circular plates[J]. International Journal Solids and Structures, 1999,36(30): 4629-4652
[32] Xu R Q, Ding H J, Chen W Q. Three-dimensional static analysis of multi-layered piezoelectric hollow spheres via the state space method[J]. International Journal of Solids and Structures, 2001, 38(28-29): 4921-4936
[33]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[J].安徽建筑工业学院学报(自然科学版), 2002,10(4): 1-6
[34]盛宏玉,董朝文.任意厚度压电层合闭口柱壳的精确解[J].合肥工业大学学报(自然科学版), 2003, 26(5): 980-985
[35] Mindlin R D. Equations of high frequency vibrations of thermopiezoelectric crystal plates[J]. International Journal of Solids and Structures, 1974, 10(6): 625-632
[36] Nowacki W. Some general theorems of thermopizeoelectricity[J]. Journal of Thermal Stresses, 1978, 1(1): 171-182
[37] Iesan D. On some theorems of thermopizeoelectricity[J]. Journal of Thermal Stresses, 1989,12(2): 209-223
[38] Tauchert T R. Pizeothermoelastic behavior of a laminated plate[J]. Journal of Thermal Stresses, 1992,15(1): 25-37
[39] Jonnalagadda K D, Blandford G E, Tauchert T R. Piezothermoelastic composite plate analysis using first-oder shear-deformation theory[J]. Computers and Structures, 1994,51(1): 79-89
[40] 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
[41] Xu K M, Noor A K, 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
[42] Kapuria S, Dumir P C, Sengupta S. Three-dimensional solution for shape control of a simply supported rectangular hybrid plate[J]. Journal of Thermal Stresses, 1999,22(1): 159-176
[43] Senthil S Vel, Batra R C. Generalized plane strain thermopiezoelectric analysis of multilayered plates[J], Journal of Thermal Stresses, 2003,26(4): 353-378
[44] 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): 207-220
[45] Tarn J Q. A state space formalism for piezothermoelasticity[J]. International Journal of Solids and Structures, 2002, 39(20): 5173-5184
[46] Qing G H, Liu Y H, Qiu J J, et al. A semi-analytical method for the free vibration analysis of thick double-shell systems[J]. Finite Elements in Analysis and Design, 2006, 42(10): 837-845
[47] 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
[48] 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