摘要
压电传感器等部件在工作中其内部常常存在着不均匀的温度场,研究压电材料等类似材料的温度场—电场—变形等的耦合问题对于工程应用有很大的理论意义。本文基于弹性体的本构关系、热平衡方程和导热方程,根据弹性体修正后的H-R变分原理建立了热弹性材料、压电热弹性材料及电磁热弹性材料在温度梯度下的广义H-R变分原理,并推导了相应的非齐次广义Hamilton正则方程。然后根据对偶变量理论,将热流量及温度变量类比为应力与位移的对偶关系,导出了增维的齐次向量方程。齐次向量方程的特点是:方程中的变量是偶数,它们可分为广义的平面外应力与广义位移两大类,而且这两类变量是完全对偶的;另一方面,齐次向量方程具有控制方程特性,即可以独立求解稳态的三维板壳问题。采用各类材料相应的层合板、壳实例验证了所得出的齐次向量方程的正确性。
本论文工作的主要意义是将复杂问题简单化,这主要表现在:
(1)考虑温度梯度关系,建立了热弹性体、压电热弹性体和电磁热弹性体修正后的广义H-R变分原理,并推导了相应的非齐次Hamilton正则方程。广义H-R变分原理的主要应用价值是:一方面,基于变分原理可简化Hamilton正则方程的推导过程;另一方面,基于广义H-R变分原理,为有关复杂工程结构问题的数值分析(如有限元法,渐近法等)的推导提供了必要的理论基础。
(2)根据热平衡和导热方程中变量的对偶关系,将非齐次的Hamilton正则方程化为能独立求解稳态的三维问题的齐次向量方程。齐次向量方程不但可大大简化分析耦合问题时通常要求联立非齐次Hamilton正则方程及由热平衡方程和导热方程导出的二阶微分方程的繁琐方法,而且也避免了求解非齐次方程时的矩阵求逆或卷积运算,在提高了运算效率的同时保证了数值结果的稳定性。
总之,本论文的工作为相关工程问题分析提供了一种简便、可靠的方法。
In order to investigate the generalized Hamilton canonical equation theory of thermoelasticity, piezothermoelasticity and electromagnetothermoelasticity, based on the constitutive relationships of thermoelastical solids, the modified generaliezed H-R (Hellinger-Reissner) variation principles of thermoelasticity, piezothermoelasticity and electromagnetothermoelasticity were established in terms of the modified H-R variation formulation of elastical solid. Meantime, the relevant nonhomogeneous generalized Hamilton canonical equations were derived from these principles in this paper. And then, based upon the symplectic variable theory, the nonhomogeneous generalized Hamilton canonical equations were transform into the homogeneous vector equations with the dimension expanding according to the symplectic relationship of the heat dynsity in the heat equilibrium equations and the temperature variable in the heat conduction equation were analogied as the symplectic variables. Moreover, many numerical examples (laminated plate/shell) with the simply supported boundary condition were used to verify the correctness of these homogeneous vector equations. There exit several charactisitics in the homogeneous vector equations: the numbers of variables in these equatinos is even, and these variables, which are symplectic variables, can be classified into two groups, namely, the generalized outplane stress and the generalized displacement. The homogeneous vector equations is governing equations, that is to say, these equations can be used to solve independly the stable three dimensional plate/shell problems.
Main purpose of the paper is to simply the complex problems, main works are listed as follows:
(1) Consider the gradient relationship of temperature field, the modified generalized H-R variation formulations of the thermoelastictity, piezothermoelasticity and electromagnetothermoelasticity was established and the nonhomogeneous generalized Hamilton canonical equations of the above three materials were derived in detail. Several nonhomogeneous generalized Hamilton canonical equations were derived in detail. The main values of application of these variation principles can be marked as follows: On the one hand, the deriving approach of Hamilton canonical equation is simpled greatly based on these variation theorems. On the other hand, these variation principles provides a fundation for the derivation of complex engeneering problems of nnumerical method (eg. the finite element methods).
(2) On the basis of the dual relationships of variables in the heat equilibrium equations and the heat conduction equation, the nonhomogeneous generalized Hamilton canonical equations were simplied to the homogeneous vector equations. And they can be used independly to analyze the stable three dimensional problems. The main advantages of the homogeneous vector equations simplifies greatly the solution programs which are often performed to combining nonhomogeneous Hamilton canonical equation and second order differential equation on the thermal equilibrium and the gradent relationship. Meanwhile, this algorithm avoids the inverse matrix calculations or the convolution operation of nonhomogeneous equations, improves the computing efficiency and ensures the stablitiness of numerical resultions.
In a word, a simply and reliable method for relevant engeneering problems is provided in this paper.
引文
[1]杨大智.智能材料与智能系统[M].天津:天津大学出版社, 2000, 1-4.
[2]李卓球,宋显辉.智能复合材料结构体系[M].武汉:武汉理工大学出版社, 2005, 1 -7.
[3] Nowinski J L. Theory of thermoelasticity with application [M]. The Netherlands: Sijthoff and Noordhoff International Publishers, 1978.
[4] Rao S S, Sunar M. Piezoelectricity and its use in disturbance sensing and control of flexible structures:A survey[J]. Applied Mechanics Review. 1994, 47(4): 113-123.
[5]周又和,郑晓静.电磁固体结构力学[M].北京:科学出版社, 1999.
[6] Wang X Z, Zhou Y H, Zheng X J. A generalized variational model of magneto-thermo-elasticity for nonlinear magnetized ferromagnetic bodies[J]. Internatioanal Journal of Engneering Science. 2002, 40(18): 1957-1973.
[7] Ting T C T. Anisotropic elasticity theory and applications[M]. New York: Oxford University Press, 1996.
[8]郑哲敏,周恒,张涵信等. 21世纪初的力学发展趋势[J].力学进展. 1995, 25(4): 433-441.
[9] Lord H W, Shulman Y. A Generalized Dynamical Theory of Thermoelasticity[J]. J Mech Phys Solids. 1967, 15(5): 299-309.
[10] Green A E, Lindsay K A. Thermoelasticity[J]. Journal of Elasticity. 1972, 2(1): 1-7.
[11] Mindlin R D. Equations of high frequency bibrations of thermopiezoelectric crystal plates[J]. Int. J. Solids Struct. 1974, 10: 625-632.
[12] Nowacki W. Some general theorems of thermopiezoelectricity[J]. Journal of Thermal Stresses. 1978, 1(2): 171-182.
[13] Iean D. On Some theorems in thermopiezoelectricity[J]. Journal of Thermal Stresses. 1989, 12(2): 209-223.
[14] Huang Y N, Batra R C. A theory of thermoviscoelastic dielectrics[J]. Journal of Thermal Stresses. 1996, 19(5): 419-430.
[15] Yang J S. Nonlinear equations of thermoviscoelectricity[J]. Math. Mech. Solids. 2009, 2: 113-124.
[16] Tauchert, Source T R. Piezothermoelastic behavior of a laminated plate[J]. Journal of Thermal Stresses. 1992, 15(1): 25-37.
[17] Jonnalagadda K D, Blandford G E, Tauchert T R. Piezothermoelastic composite plate analysis using first-order shear deformation theory[J]. Computers and Structures. 1994, 51(1): 79-83.
[18] Rao S S, Sunar M. Analysis of distributed thermopiezoelectric sensors and actuators in advanced intelligent structures [J]. AIAA Journal. 1993, 31(7): 1280-1286.
[19] Tzou H S, Ye R. Piezothermoelasticity and precision control of piezoelectric systems: Theory and finite element analysis [J]. Journal of Vibration and Acoustics, Transactions of the ASME. 1994, 116(4): 489-495.
[20] Tang Y Y, Noor A K, Xu K. Assessment of computational models for thermoelectroelastic multilayered plates [J]. Computers and Structures. 1996, 61(5): 915-933.
[21] Lee H J, Saravanos D A. Coupled layerwise analysis of thermopiezoelectric composite beams [J]. AIAA Journal. 1996, 34(6): 1231-1237.
[22] Ishihara M, Noda N. Piezothermoelastic analysis of a cross-ply laminate considering the effects of transverse shear and coupling [J]. Journal of Thermal Stresses. 2000, 23(5): 441-461.
[23] Dube G P, Kapuria S, Dumir. Exact piezothermoelastic solution of simply-supported orthotropic flat panel in cylindrical bending[J]. International Journal of Mechanical Sciences. 1996, 38(11): 1161-1177.
[24] Dube G P, Upadhyay M M, Dumir P C, et al. Piezothermoelastic solution for angle-ply laminated plate in cylindrical bending[J]. Structural Engineering and Mechanics. 1998, 6(5).
[25] Kapuria S, Dube G P, Dumir P C. Exact piezothermoelastic solution for simply supported laminated flat panel in cylindrical bending[J]. Zeitschrift fuer Angewandte Mathematik und Mechanik, ZAMM, Applied Mathematics and Mechanics. 1997, 77(4): 281.
[26] Shang F, Wang Z, Li Z. Analysis of thermally induced cylindrical flexure of laminated plates with piezoelectric layers[J]. Composites Part B: Engineering. 1997, 28 B(3).
[27] Xu K, 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.
[28] Tang Y Y, Xu K. Dynamic analysis of a piezothermoelastic laminated plate[J]. Journal of Thermal Stresses. 1995, 18(1): 87-104.
[29] Ootao Y, Tanigawa Y. Three-dimensional transient piezothermoelasticity for a rectangular composite plate composed of cross-ply and piezoelectric laminate[J]. International Journal of Engineering Science. 2000, 38(1): 47-71.
[30] Cheng Z, Batra R C. Three-dimensional asymptotic scheme for piezothermoelastic laminates[J]. Journal of Thermal Stresses. 2000, 23(2): 95-110.
[31] Ashida F, Tauchert T R. A general plane-stress solution in cylindrical coordinates for a piezothermoelastic plate[J]. International Journal of Solids and Structures. 2001, 38(28-29): 4969-4985.
[32] Yang J S, Batra R C. Free vibrations of a linear thermopiezoelectric body[J]. Journal of Thermal Stresses. 1995, 18(2): 247-262.
[33] Tauchert T R, Ashida F, Noda N, et al. Developments in thermopiezoelectricity with relevance to smart composite structures[J]. Composite Structure. 2000, 48: 31-38.
[34] Eshelby J D, Read W T, Shockley W. Anisotropic elasticity with applications to dislocation theory[J]. Acta Metal. 1953, 1: 251-259.
[35] Stroh A N. Dislocations and cracks in anisotropic elasticity[J]. Philos. Mag. 1958, 3: 625-646.
[36] Vel S S, Batra R C. Cylindrical bending of laminated plates with distributed and segmented piezoelectric actuators/sensors[J]. AIAA journal. 2000, 38(5): 857-867.
[37] Vel S S, Batra, R.c. Source:, V 38, N 8, P 1395-1414 F. Generalized plane strain thermoelastic deformation of laminated anisotropic thick plates[J]. International Journal of Solids and Structures. 2001, 38(8): 1395-1414.
[38] Rakshit M, Mukhopadhyay B. An electro-magneto-thermo-visco-elastic problem in an infinite medium with a cylindrical hole[J]. International Journal of Engineering Science. 2005, 43: 925-936.
[39] Ezzat M A, Youssef H M. Generalized magneto-thermoelasticity in a perfectly conducting medium[J]. International Journal of Solids and Structures. 2005, 42: 6319-6334.
[40] Othman M I A, Song Y. Effect of rotation on plane waves of generalized electromagneto-thermoviscoelasticity with two relaxation times[J]. Applied Mathematical Modelling. 2008, 32(5): 811-825.
[41] Zheng X J, Liu X E. A nonlinear constitutive model for Terfenol-D rods[J]. Journal of Applied Physics. 2005, 97(5): 1-8.
[42] Zheng X J, Sun L. A nonlinear constitutive model of magneto-thermo-mechanical coupling for giant magnetostrictive materials[J]. Journal of Applied Physics. 2006, 100(6).
[43] Zheng X, Wang X. A magnetoelastic theoretical model for soft ferromagnetic shell in magnetic field[J]. International Journal of Solids and Structures. 2003, 40(24): 6897-6912.
[44] Zheng X J, Zhu L, Zhou Y, et al. Impact of grain sizes on phonon thermal conductivity of bulk thermoelectric materials[J]. Applied Physics Letters. 2005, 87(24): 1-3.
[45] Zheng X, Gou X, Zhou Y. Influence of flux creep on dynamic behavior of magnetic levitation systems with a high-Tc superconductor[J]. IEEE Transactions on Applied Superconductivity. 2005, 15(3): 3856-3863.
[46]钟万勰.应用力学对偶体系.北京:科学出版社, 2002.
[47] Stroh A N. Steady state problems in anisotropic elasticity[J]. Mathematic and Physics. 1962, 41(1): 77-103.
[48] Cohen G A. Computer analysis of asymmetrical deformation of orthotropic shells of revolution[J]. AIAA Journal. 1964, 2(5): 932-934.
[49] Ingebrigsten K A, Tonnig A. Elastic surface waves in crystals[J]. physics Review. 1969(184): 942-951.
[50] Cohen G A. Numerical integration of shell equation using the field method[J]. ASME Journal of applied mechanics. 1974, 41(2): 261-266.
[51] Cohen G A. FASOR-A second generation shell of revolution code[J]. Computers and Structures. 1979(10): 301-309.
[52] Chandrashekara S, Santhosh U. Natural frequencies of cross-ply laminated by state space approach[J]. Journal of Sound and Vibration. 1990, 136(3): 413-424.
[53] Fang J R, Ye J Q. An exact solution for the static and dynamics of laminated thick plate with orthotropic[J]. International Journal of Solids and Structures. 1990, 26(5): 655-662.
[54] Fang J R, Ye J Q. A series solution of the exact equation for thick orthotropic plate[J]. International Journal of Solids and Structures. 1990, 26(7): 773-778.
[55]范家让.强厚度叠层板壳的精确解法[M].北京:科学出版社, 1996: 130-351.
[56]唐立民.弹性力学的混合方程和Hamilton正则方程[J].计算结构力学及其应用. 1991, 8(4): 343-349.
[57]钟万勰.条形域平面弹性问题和哈密尔顿体系[J].大连理工大学学报. 1991, 31(4): 373-384.
[58]钟万勰.弹性力学求解新体系[M].大连:大连理工出版社, 1995: 1-250.
[59]唐立民.混合状态Hamilton元的半解析解和叠层板的计算[J].计算结构力学及其应用. 1992, 9(4): 347-360.
[60]欧阳华江,钟万勰.一类基于Hamilton体系的半解析法[J].计算结构力学及其应用. 1993, 10(2): 129-136.
[61]赵家祥.民用航空和先进复合材料[J].高科技纤维与应用. 2007, 32(2): 6-10.
[62]邹贵平,唐立民.极坐标系中弹性力学平面问题的Hamilton正则方程及状态空间有限元法[J].工程力学. 1993, 10(1): 19-29.
[63]卿光辉,邱家俊,刘艳红.电磁弹性体修正后的H-R变分原理及其正则方程[J].天津大学学报. 2006, 39(1): 78-82.
[64] Zou G P, Tang L M. A semi-analytical solution for thermal stress analysis of laminated composite plate in the Hamiltonian system[J]. Computer & Structure. 1995, 55(1): 113-118.
[65]邹贵平,唐立民.厚壁板动力学分析的混合状态Hamiltonian等参元[J].振动工程学报. 1994, 7(2): 23-31.
[66]邹贵平,唐立民.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.
[67] Zou G P, Tang L M. A semi-analytical solution for the dynamic response of thick composite in Hamilton systems[J]. Finite Elements in Analysis and Design. 1996, 21: 201-212.
[68]邹贵平,唐立民.反对称铺设层合板动力问题的Hamilton体系及辛几何解法[J].固体力学学报. 1996, 17(4): 312-319.
[69]王治国,朱德懋.复合材料叠层圆柱曲板振动分析的半解析解[J].振动工程学报. 1995, 8(2): 92-97.
[70]唐立民,齐朝晖,丁克伟.弹性力学弱形式广义基本方程的建立和应用[J].大连理工大学学报. 2001, 41(1): 1-8.
[71] Ding H J, Chen W Q, Xu R Q. New state space formulations for transversely isotropic piezoelectricity with application[J]. Mechanics Research Communications. 2000, 27(3): 319-326.
[72] Cheng Z Q, Batra R C. Three-Dimensional Asymptotic Analysis of Multiple-Electroded Piezoelectric Laminates[J]. AIAA Journal. 2000, 38(2): 317-325.
[73] 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.
[74]郭广峰.基于哈密顿体系辛算法喇叭天线口径场的研究[D].大连:大连理工大学, 2006.
[75]陈杰夫.电磁波导的辛分析与对偶有限元计算[D].大连:大连理工大学, 2006.
[76]朱谷君.工程传热介质学[M].北京:航空工业出版社, 1989.
[77]陈浩然,杨正林,唐立民.复合材料层合板固化过程的数值模拟[J].应用力学学报. 1998, 15(3): 30-36.
[78]邹贵平.层合板壳问题的哈密顿体系与哈密顿型广义变分原理[J].上海大学学报. 1996, 2(2): 119-128.
[79]邹贵平,唐立民.圆柱厚壳混合状态等参元的半解析解[J].强度与环境. 1993, 93(4): 61-67.
[80]卿光辉,王喆,刘艳红.壁厚不连续不对称圆柱壳和开口壳的自由振动分析[J].工程力学. 2006, 23(Sup.1): 25-29.
[81]卿光辉,邱家俊,刘艳红.压电弹性体修正后的H-R混合变分原理及其层合板的精确解法[J].工程力学. 2005, 22(5): 43-47.
[82] Qing G H, Qiu J J, Liu Y H. A simi-analytical solution for dynamic analysis of plate with piexoelectric patches[J]. International Journal of Solids and Structures. 2006, 43(6): 1388-1403.
[83] Lee J S, Jiang L Z. Exact electroelastic analysis of piezoelectric laminate via state space approach[J]. International Journal Solids and Structures. 1996, 33(7): 977-990.
[84]刘艳红.压电弹性体壳的广义变分原理和状态向量方程[J].中国民航学院学报. 2004, 22(5): 38-42.
[85] Qing G H, Liu Y H, Guo Q, et al. Dynamic analysis for three-dimensional laminated plates and panels with damping[J]. International Journal of Mechanical Sciences. 2008, 50(1): 83-91.
[86]钟万勰.结构动力方程的精细时程积分[J].大连理工大学学报. 1994, 34(4): 131-136.
[87]顾元宪,陈飚松,张洪武.结构动力方程的增维精细积分法[J].力学学报. 2000, 32(4): 447-456.
[88]林家浩.计算结构力学[M].北京:高等教育出版社, 1989.
[89]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.
[90]丁皓江,国凤林,侯鹏飞.耦合热弹性问题的一般解[J].应用数学与力学. 200, 21(6): 573-577.
[91]邹贵平.层合板壳问题的哈密顿体系与哈密顿广义变分原理[J].上海大学学报. 1996, 2(2): 119-128.
[92] Pagano N J. Exact solutions for rectangular bidirectional composites and sandwich plates[J]. Journal of Composite materials. 1970, 4(1): 20-34.
[93] Ray M C, Rao K M, Samanta B. Exact analysis of coupled electroelastic behavior of a piezoelectric plate under cylindrical bending[J]. Computers & Structures. 1992(45): 667-677.
[94] Batra R C, Liang X Q. The vibration of a rectangular laminated elastic plate with embedded piezoelectric sensors and actuators[J]. Computer & Structure. 1997, 63(2): 203-216.
[95] Ray M C, Bhattacharya R, Samanta B. Exact solutions for dynamic analysis of composite plate with distributed piezoelectric layers[J]. Computer & Structures. 1998, 66(6): 737-743.
[96] Vel S S, Batrs R C. Generalized plane strain thermopiezoelectric analysis of multilayered plates[J]. Journal of Thermal Stresses. 2003, 26(4): 353-377.
[97] Heyliger P R, Brooks P. Exact solution for laminated piezoelectric plates in cylindrical bending[J]. Journal of Applied Mechanics. 1996, 63(6): 903-910.
[98] Xu K, 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.
[99] 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.
[100] Quo T J. A state space formalism for piezothermoelasticity[J]. International Journal of Solids and Structures. 2002, 39(20): 5173-5184.
[101] Kapuria S, dumir P C. Three-dimensional solution for shape control of a simply supported rectangular hybrid plate[J]. Journal of Thermal Stresses. 2003, 22(2): 159-176.
[102] Zhang C, Cheung Y K, Di S, et al. The exact solution of coupled thermoelectroelastic behavior of piezoelectric laminates[J]. Computers & Structures. 2002, 80(13): 1201-1212.
[103] Ding H J, Wang H M, Ling D S. Analytical solution of a pyroelectric hollow cylinder for piezothermoelastic axisymmetric dynamic problems[J]. Journal of Thermal Stresses. 2003, 26(3): 261-276.
[104]卿光辉,邱家俊,刘艳红.电磁弹性体修正后的H-R混合变分原理和状态向量方程[J].应用数学和力学. 2005, 26(6): 665-670.
[105]范家让,盛宏玉.具有固支边的强厚度叠层板的精确解[J].力学学报. 1992, 24(5): 574-583.
[106]范家让,丁克伟.周边固支强厚度叠层开口圆柱壳的精确解[J].应用数学和力学. 1994, 15(4): 343-354.
[107] Chen W Q, Kang Y L. Alternative state space formulations for magnetoelectric thermoelasticity with transverse isotropy and the application to bending analysis of nonhomogeneous plates[J]. International Journal of Solids and Structures. 2003, 40(21): 5689-5705.
[108] Yoshihiro O, Yoshinobu T. Transient analysis of multilayered magneto-electro-thermoelastic strip due to nonuniform heat supply[J]. Composite Structures. 2005, 68(4): 471-480.
[109]曾丹苓.工程非平衡态动力学[M].北京:科学出版社,1997
[110] Y.L.姚著,鲜于玉琼,张经坤译[M].不可逆过程热力学.北京:科学出版社,1981
