板电磁—热—力多场耦合分析
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摘要
电磁-热-弹性理论专门研究电磁场、热场与弹性场之间的耦合,即研究电磁场、热场和弹性场的相互作用。电磁场对变形场的作用是由运动方程中的洛仑兹力引起。弹性场会影响磁场的强度、磁弹性波和电磁波的传播速度与相位,具体表现在欧姆定律中多了电流密度增长项,而且该项取决于变形物体在磁场中的速度。变化的电磁场在导体中产生涡流,激发热场作用于介质,产生热应力。热作用于介质又影响变形,弹性场对热场的影响主要表现在热传导方程中多出了应变率项。
本文致力于研究任意时变电磁场作用下的环状柱体的电磁-热-力耦合问题,其力学模型为圆板,分别研究了强耦合和弱耦合两种情况,但不考虑弹性场对温度场的影响。给出了电磁场、热场和弹性场的基本方程,采用高精度和高效的微分求积方法求解了在任意时变磁场作用下的环状圆柱的热磁弹性应力。计算时采用微分求积方法在时间和空间域离散基本微分方程和时变边界条件,在整个时间、空间域求解同时满足微分方程组、边界条件及初始条件的未知量。微分求积方法的效率很高,在时间和空间域只需很少数目的结点就可得到很高精度的结果。以图示的形式给出了磁场、温度变化和应力与位移,还给出了两种耦合情况结果的比较。
Magneto-thermo-elasticity is a special subject, in which we investigate the interaction among the strain, electromagnetic fields and temperature fields. The influence of the electromagnetic field on the strain occurs by means of Lorentz forces appearing in the equations of motion. The Ohm law contains a term describing the increment of density of the electric field developing on the velocity of the material particles moving in the magnetic field. The arbitrary variation of magnetic field can generate eddy current in the media. The eddy current heats the body, which generates thermo stress. The influence of the strain on the temperature field occurs by means of a term containing strain rate appearing in the heat conduction function.
The article aims at the analysis of the coupling problem in a conducting hollow circular cylinder subjected to an arbitrary variation of magnetic field. The model is assumed as a circular plate, and the coupling problem will be discussed in two ways, with and without the coupling term which shows the interact between strain and electromagnetic field, but the influence strain rate to thermal field is not considered. The fundamental equation of electromagnetic field, temperature field and elastic field are formulated. An accurate and efficient differential quadrature method is employed for numerical simulation of magneto-thermo-elastic stresses in a conducting hollow circular cylinder subjected to an arbitrary variation of magnetic field. The fundamental equations and both boundary conditions and initial conditions are discretized in spatial and temporal domain by differential quadrature rules. The unknown variables satisfy the governing equations, the boundary conditions and the initial conditions simultaneously, and they are computed in entire domain by means of DQM. Accurate solutions can be obtained by using less grid points in both spatial and time domain. Solutions of magnetic field, temperature field and stresses and deformation are illustrated graphically. The comparison between two kinds of calculated results is also illustrated graphically and analyzed theoretically.
本文致力于研究任意时变电磁场作用下的环状柱体的电磁-热-力耦合问题,其力学模型为圆板,分别研究了强耦合和弱耦合两种情况,但不考虑弹性场对温度场的影响。给出了电磁场、热场和弹性场的基本方程,采用高精度和高效的微分求积方法求解了在任意时变磁场作用下的环状圆柱的热磁弹性应力。计算时采用微分求积方法在时间和空间域离散基本微分方程和时变边界条件,在整个时间、空间域求解同时满足微分方程组、边界条件及初始条件的未知量。微分求积方法的效率很高,在时间和空间域只需很少数目的结点就可得到很高精度的结果。以图示的形式给出了磁场、温度变化和应力与位移,还给出了两种耦合情况结果的比较。
Magneto-thermo-elasticity is a special subject, in which we investigate the interaction among the strain, electromagnetic fields and temperature fields. The influence of the electromagnetic field on the strain occurs by means of Lorentz forces appearing in the equations of motion. The Ohm law contains a term describing the increment of density of the electric field developing on the velocity of the material particles moving in the magnetic field. The arbitrary variation of magnetic field can generate eddy current in the media. The eddy current heats the body, which generates thermo stress. The influence of the strain on the temperature field occurs by means of a term containing strain rate appearing in the heat conduction function.
The article aims at the analysis of the coupling problem in a conducting hollow circular cylinder subjected to an arbitrary variation of magnetic field. The model is assumed as a circular plate, and the coupling problem will be discussed in two ways, with and without the coupling term which shows the interact between strain and electromagnetic field, but the influence strain rate to thermal field is not considered. The fundamental equation of electromagnetic field, temperature field and elastic field are formulated. An accurate and efficient differential quadrature method is employed for numerical simulation of magneto-thermo-elastic stresses in a conducting hollow circular cylinder subjected to an arbitrary variation of magnetic field. The fundamental equations and both boundary conditions and initial conditions are discretized in spatial and temporal domain by differential quadrature rules. The unknown variables satisfy the governing equations, the boundary conditions and the initial conditions simultaneously, and they are computed in entire domain by means of DQM. Accurate solutions can be obtained by using less grid points in both spatial and time domain. Solutions of magnetic field, temperature field and stresses and deformation are illustrated graphically. The comparison between two kinds of calculated results is also illustrated graphically and analyzed theoretically.
引文
[1]关明智.广义磁热弹介质的动态响应分析[D].兰州:兰州理工大学,2004
[2]白象忠.磁弹性、热磁弹性理论及应用[J].力学进展,1996,V26(3):389-403
[3] Maxwell J.C.. On the dynamic theory of gases[J]. Phil Trans Roy Soc London, 1867, V157:49-88
[4]白象忠.磁弹性、热磁弹性理论及应用[J].力学进展,1996,V26(3):389-403
[5] Tisza L.. Surla supraconductibilite Thermique de l’HelliumⅡliquide et la statistique de Bos-Einstein[J]. C R Acad Sci Paris Ser1 Math,1938,V207(22):1035-1041
[6] Landau L.D.. The theory of superfluidity of heliumⅡ. J Phys, USSR,1941,V5:71-90
[7] Peshkov V.. Second sound in heliumⅡ[J].JPhys,1944,V8:381-386
[8] Cattaneo C.. On the conduction of heat[J]. Atti Sem Mat Fis Univ Modena,1948,V3:3-21
[9] H. Lord, Y. Shulman. A generalized dynamical theory of thermoelasticity[J].Journal of the Mechanics and Physics of Solids,1967,V15 (5):299–309
[10] I. Muller. The coldness, A universal function in thermo-elastic solids[J].Archive for Rational Mechanics and Analysis,1971,V41(5):319-332
[11] A. Green, N. Laws. On the entropy production inequality[J].Archive for Rational Mechanics and Analysis,1972,V 45(1):47-53
[12] A. Green, K. Lindsay, Thermoelasticity[J].Journal of Elasticity,1972,V2(1):1-7
[13] E. Suhubi, Thermoelastic solids[A]. A.C. Eringen. Cont. Phys II[C].New York:Academic Press,1975:ch.2
[14] M. Ezzat. Fundamental solution in thermoelasticity with two relaxation times for cylindrical regions[J]. International Journal of Engineering Science,1995,V33(14): 2011–2020
[15] L. Knopoff. The Interaction between elastic wave motion and a magnetic field in electrical conductors[J].Journal of Geophysical Reserch,1955,V60(4):441–456
[16] P. Chadwick. Elastic wave propagation in a magnetic field[J].Proceedings of the Ninth International. Congress on Applied Mechanics,1957,V7(9):143-158
[17] S. Kaliski, J. Petykiewicz. Equation of motion coupled with the field of temperature in a magnetic field involving mechanical and electrical relaxation for anisotropic bodies[J]. Proc. Vibr. Probl,1959,V4:1-12
[18] Kaliski S, Nowacki W. Combined elastic and electromagnetic waves produced by thermal shock[J].Bulletin de l Academie Polonaise des Sciences-Serie des Sciences Techniques,1962,V10:159–68
[19] Kaliski S, Michalec J. The resonance amplification of a magnetoelastic wave radiated from a cylindrical cavity[J].Proceedings of Vibration Problems,1963,V4:7–15
[20] Purushotham CM. Magneto-elastic coupling effects on the propagation of harmonic waves in an electrically conducting elastic plate[J].Proceedings of the Cambridge Philosophical Society,1967,V63(503):62-63
[21] Moon FC, Pao PH. Magnetoelastic buckling of a thin plate[J].Journal of Applied Mechanics, Transactions of the ASME 1968,V35(1):53–58
[22] Moon FC, Pao PH. Vibration and dynamic instability of a beamplate in a transverse magnetic field[J]. Journal of Applied Mechanics, Transactions of the ASME, 1969, V36: 92–100
[23] M. Higuchi, R. Kawamura, Y. Tanigawa. Magneto-thermo-elastic stresses induced by a transient magnetic field in a conducting solid circular cylinder[J].International Journal of Solids and Structures,2007,V44(16):5316–5335
[24] M. Higuchi, R. Kawamura, Y. Tanigawa. Dynamic and quasi-static behaviors of magneto-thermo-elastic stresses in a conducting hollow circular cylinder subjected to an arbitrary variation of magnetic field[J]. International Journal of Mechanical Sciences, 2008,V50(3):365–379
[25] M. Higuchi, R. Kawamura, Y. Tanigawa, Osaka, H. Fujieda. Dynamic and quasi-static behaviors of magneto-thermo-elastic stresses in a conducting infinite plate subjected to an arbitrary variation of magnetic field[J].Acta Mechanica,2007,V191(3-4):135–154
[26] Moon, F. C., Chattopadhyay, S. Magnetically induced stress waves in a conducting solid– theory and experiment[J].American Society of Mechanical Engineers, U.S. National Congress of Applied Mechanics,1974,V41:641–647
[27] Chian C. T., Moon F. C. Magnetically induced cylindrical stress waves in a thermoelastic conductor[J].International Journal of Solids and Structures,1981,V17(11):1021–1035
[28] A. Nayfeh, S. Nemat-Nasser. Electromagneto-thermoelastic plane waves in solids with thermal relaxation[J].J. Appl. Mech,1972,V39(1):108–113
[29] S. Choudhuri. Electro-magneto-thermo-elastic waves in rotating media with thermal relaxation[J].International Journal of Engineering Science,1984,V22(5):519–530
[30] H. Sherief. Problem in electromagneto thermoelasticity for an infinitely long solid conducting circular cylinder with thermal relaxation[J].International Journal of Engineering Science,1994,V32(7):1137–1149
[31] H. Sherief, M. Ezzat. A Thermal-shock problem in magneto-thermoelasticity with thermalrelaxation[J].International Journal of Solids and Structures,1996,V33(30):4449–4459
[32] H. Sherief, M. Ezzat. A Problem in generalized magneto-thermoelasticity for an infinitely long annular cylinder[J].Journal of Engineering Mathematics,1998,V34(4):387–402
[33] M. Ezzat. Generation of generalized magnetothermoelastic waves by thermal shock in a perfectly conducting halfspace[J],Journal of thermal stresses,1997,V20(6):617–633
[34] M. Ezzat, M. Othman, A. El-Karamany. Electromagneto-thermoelastic plane waves with thermal relaxation in a medium of perfect conductivity[J].Journal of thermal stresses,2000,V38:107-120
[35]朱为国,白象忠.四边固支热磁弹性矩形薄板的分岔与混沌[J].振动与冲击,2009,V28(5):59-62
[36]何天虎,贾维维.旋转半无限大体广义电磁热弹性耦合的二维问题[J].工程力学,2009,V26(2):196-202
[37]田振国,白象忠.载流板壳电磁、热、机械场耦合的弹性效应分析[J].应用基础与工程科学学报,2009,V17(2):318-325
[38]何天虎,田晓耕.半无限大体广义点磁热弹耦合的二维问题[J].应用力学学报,2007,V24(3):343-347
[39]关明智,何天虎,盖超,李洋.半无限长旋转杆件中的广义电磁热弹耦合问题[J].吉林大学学报(理学版),2009,V47(1):92-97
[40]胡宇达,白象忠.磁场中圆柱壳体的轴对称振动[J].工程力学,1999,V16(5):47-64
[41] Y. Xing, B. Liu. A differential quadrature analysis of dynamic and quasi-static magneto-thermo-elastic stresses in a conducting rectangular plate subjected to an arbitrary variation of magnetic field[J].International Journal of Engineering Science. Sci,2010,V48(12):1944-1960
[42] Striz A G, Chen W L, Bert C W. Static analysis of structures by the quadrature element method(QEM)[J].International Journal of Solids and Structures,1994,V31(20):2807-2818
[43] Chen W. Differential quadrature method and its applications to structural engineering[D].上海:上海交通大学,1994
[44]王鑫伟.微分求积法在结构力学中的应用[J].力学进展,1995,V25(2):232-239
[45]王永亮.微分求积法和微分求积单元法—原理与应用[D].南京:南京航空航天大学,2001
[46] Bellman R. E., Cacti J. Differential quadrature and long-term integration[J].J. Math. Anal. Appl.,1971,V34:235-238
[47]王永亮,王鑫伟.多外载联合作用下圆板的非线性弯曲[J].南京航空航天大学学报,1996,V28(1):53-58
[48]王永亮.边缘弹性约束圆薄板的大挠度分析[J].江苏力学,1995,V10:6-12
[49]蔡圣善,朱耘,徐建军.电动力学[M].北京:高等教育出版社,2002:15-26
[50]周又和,郑晓静.电磁固体力学[M].北京:科学出版社,1999:10-20
[51] Moon F. C. Magneto-solid mechanics[J].Physics Today,1985,V38(12):79
[52]郑哲敏,周恒,张涵信.21世纪的力学发展趋势[J].力学进展,1984,V25(4):433-411
[53] G. Paria. Magneto-elasticity and magneto-thermo-elasticity[J].Adv. Appl. Mech.,1967, V10:73-112
[54]徐耀玲,胡宇达,白象忠.薄板薄壳非线性磁弹性问题的基本方程[J].燕山大学学报,1998,V22(4):358-362
[55]马世麟,胡宇达,徐耀玲.广义洛伦兹力的磁弹性效应[J].工程力学,1999,V16(2):79-84
[56]胡宇达,白象忠.磁场环境对导电薄板的磁弹性振动的影响[J].非线性动力学学报,1999,V6(2):134-138
[57]胡宇达,白象忠.倾斜磁场中条形传导薄板的磁弹性振动[J].振动与冲击,2000,V19(2):64-66
[58] Zheng X J, Liu X N. Analysis on dynamic characteristics for ferromagnetic conducting plates in a transverse uniform magnetic field[J].Acta Mechanica Solid Sinica,2000, V13(4):361-368
[59]郑晓静,刘信恩.铁磁导电梁式板在横向均匀磁场中的动力特性分析[J].固体力学学报,2000,V21(3):243-250
[60] M. Higuchi, R. Kawamura, Y. Tanigawa. Dynamic and quasi-static behaviors of magneto-thermo-elastic stresses in a conducting infinite plate subjected to an arbitrary variation of magnetic field[J].Acta Mechanica,2007,V191(3-4):135-154
[61] M. Higuchia, R. Kawamurab, Y. Tanigawa. Dynamic and quasi-static behaviors of magneto-thermo-elastic stresses in a conducting hollow circular culinder subjected to an arbitrary variation of magnetic field[J]. International Journal of Mechanical Sciences, 2008,V50(3): 365-379
[62] R. Quan, C.T. Chang. New insights in solving distributed system equations by the quadrature methods– I. Analysis[J].Computers in Chemical Engineering,1989,V13(7): 779–788
[63] C. Shu, B.E. Richards. Application of generalized differential quadrature to solvetwo-dimensional incompressible Navier-Stokes equations[J].International Journal for Numerical Methods in Fluids,1992,V15(7):791-798
[64] C.W. Bert, M. Malik. Differential quadrature method in computational mechanics: a review[J].Appl. Mech. Rev.,1996,V49(1):1-28
[65] Y.F. Xing, B. Liu. High-Accuracy Differential Quadrature Finite Element Method and Its Application to Free Vibrations of Thin Plate with curvilinear domain[J].International Journal for Numerical methods in Engineering.2010,Vol.2(1):1-20
[66] Y.F. Xing, B. Liu, G. Liu. A differential quadrature finite element method[J].Int. J. of Applied Mechanics.2010,Vol.2(1):207-227
[2]白象忠.磁弹性、热磁弹性理论及应用[J].力学进展,1996,V26(3):389-403
[3] Maxwell J.C.. On the dynamic theory of gases[J]. Phil Trans Roy Soc London, 1867, V157:49-88
[4]白象忠.磁弹性、热磁弹性理论及应用[J].力学进展,1996,V26(3):389-403
[5] Tisza L.. Surla supraconductibilite Thermique de l’HelliumⅡliquide et la statistique de Bos-Einstein[J]. C R Acad Sci Paris Ser1 Math,1938,V207(22):1035-1041
[6] Landau L.D.. The theory of superfluidity of heliumⅡ. J Phys, USSR,1941,V5:71-90
[7] Peshkov V.. Second sound in heliumⅡ[J].JPhys,1944,V8:381-386
[8] Cattaneo C.. On the conduction of heat[J]. Atti Sem Mat Fis Univ Modena,1948,V3:3-21
[9] H. Lord, Y. Shulman. A generalized dynamical theory of thermoelasticity[J].Journal of the Mechanics and Physics of Solids,1967,V15 (5):299–309
[10] I. Muller. The coldness, A universal function in thermo-elastic solids[J].Archive for Rational Mechanics and Analysis,1971,V41(5):319-332
[11] A. Green, N. Laws. On the entropy production inequality[J].Archive for Rational Mechanics and Analysis,1972,V 45(1):47-53
[12] A. Green, K. Lindsay, Thermoelasticity[J].Journal of Elasticity,1972,V2(1):1-7
[13] E. Suhubi, Thermoelastic solids[A]. A.C. Eringen. Cont. Phys II[C].New York:Academic Press,1975:ch.2
[14] M. Ezzat. Fundamental solution in thermoelasticity with two relaxation times for cylindrical regions[J]. International Journal of Engineering Science,1995,V33(14): 2011–2020
[15] L. Knopoff. The Interaction between elastic wave motion and a magnetic field in electrical conductors[J].Journal of Geophysical Reserch,1955,V60(4):441–456
[16] P. Chadwick. Elastic wave propagation in a magnetic field[J].Proceedings of the Ninth International. Congress on Applied Mechanics,1957,V7(9):143-158
[17] S. Kaliski, J. Petykiewicz. Equation of motion coupled with the field of temperature in a magnetic field involving mechanical and electrical relaxation for anisotropic bodies[J]. Proc. Vibr. Probl,1959,V4:1-12
[18] Kaliski S, Nowacki W. Combined elastic and electromagnetic waves produced by thermal shock[J].Bulletin de l Academie Polonaise des Sciences-Serie des Sciences Techniques,1962,V10:159–68
[19] Kaliski S, Michalec J. The resonance amplification of a magnetoelastic wave radiated from a cylindrical cavity[J].Proceedings of Vibration Problems,1963,V4:7–15
[20] Purushotham CM. Magneto-elastic coupling effects on the propagation of harmonic waves in an electrically conducting elastic plate[J].Proceedings of the Cambridge Philosophical Society,1967,V63(503):62-63
[21] Moon FC, Pao PH. Magnetoelastic buckling of a thin plate[J].Journal of Applied Mechanics, Transactions of the ASME 1968,V35(1):53–58
[22] Moon FC, Pao PH. Vibration and dynamic instability of a beamplate in a transverse magnetic field[J]. Journal of Applied Mechanics, Transactions of the ASME, 1969, V36: 92–100
[23] M. Higuchi, R. Kawamura, Y. Tanigawa. Magneto-thermo-elastic stresses induced by a transient magnetic field in a conducting solid circular cylinder[J].International Journal of Solids and Structures,2007,V44(16):5316–5335
[24] M. Higuchi, R. Kawamura, Y. Tanigawa. Dynamic and quasi-static behaviors of magneto-thermo-elastic stresses in a conducting hollow circular cylinder subjected to an arbitrary variation of magnetic field[J]. International Journal of Mechanical Sciences, 2008,V50(3):365–379
[25] M. Higuchi, R. Kawamura, Y. Tanigawa, Osaka, H. Fujieda. Dynamic and quasi-static behaviors of magneto-thermo-elastic stresses in a conducting infinite plate subjected to an arbitrary variation of magnetic field[J].Acta Mechanica,2007,V191(3-4):135–154
[26] Moon, F. C., Chattopadhyay, S. Magnetically induced stress waves in a conducting solid– theory and experiment[J].American Society of Mechanical Engineers, U.S. National Congress of Applied Mechanics,1974,V41:641–647
[27] Chian C. T., Moon F. C. Magnetically induced cylindrical stress waves in a thermoelastic conductor[J].International Journal of Solids and Structures,1981,V17(11):1021–1035
[28] A. Nayfeh, S. Nemat-Nasser. Electromagneto-thermoelastic plane waves in solids with thermal relaxation[J].J. Appl. Mech,1972,V39(1):108–113
[29] S. Choudhuri. Electro-magneto-thermo-elastic waves in rotating media with thermal relaxation[J].International Journal of Engineering Science,1984,V22(5):519–530
[30] H. Sherief. Problem in electromagneto thermoelasticity for an infinitely long solid conducting circular cylinder with thermal relaxation[J].International Journal of Engineering Science,1994,V32(7):1137–1149
[31] H. Sherief, M. Ezzat. A Thermal-shock problem in magneto-thermoelasticity with thermalrelaxation[J].International Journal of Solids and Structures,1996,V33(30):4449–4459
[32] H. Sherief, M. Ezzat. A Problem in generalized magneto-thermoelasticity for an infinitely long annular cylinder[J].Journal of Engineering Mathematics,1998,V34(4):387–402
[33] M. Ezzat. Generation of generalized magnetothermoelastic waves by thermal shock in a perfectly conducting halfspace[J],Journal of thermal stresses,1997,V20(6):617–633
[34] M. Ezzat, M. Othman, A. El-Karamany. Electromagneto-thermoelastic plane waves with thermal relaxation in a medium of perfect conductivity[J].Journal of thermal stresses,2000,V38:107-120
[35]朱为国,白象忠.四边固支热磁弹性矩形薄板的分岔与混沌[J].振动与冲击,2009,V28(5):59-62
[36]何天虎,贾维维.旋转半无限大体广义电磁热弹性耦合的二维问题[J].工程力学,2009,V26(2):196-202
[37]田振国,白象忠.载流板壳电磁、热、机械场耦合的弹性效应分析[J].应用基础与工程科学学报,2009,V17(2):318-325
[38]何天虎,田晓耕.半无限大体广义点磁热弹耦合的二维问题[J].应用力学学报,2007,V24(3):343-347
[39]关明智,何天虎,盖超,李洋.半无限长旋转杆件中的广义电磁热弹耦合问题[J].吉林大学学报(理学版),2009,V47(1):92-97
[40]胡宇达,白象忠.磁场中圆柱壳体的轴对称振动[J].工程力学,1999,V16(5):47-64
[41] Y. Xing, B. Liu. A differential quadrature analysis of dynamic and quasi-static magneto-thermo-elastic stresses in a conducting rectangular plate subjected to an arbitrary variation of magnetic field[J].International Journal of Engineering Science. Sci,2010,V48(12):1944-1960
[42] Striz A G, Chen W L, Bert C W. Static analysis of structures by the quadrature element method(QEM)[J].International Journal of Solids and Structures,1994,V31(20):2807-2818
[43] Chen W. Differential quadrature method and its applications to structural engineering[D].上海:上海交通大学,1994
[44]王鑫伟.微分求积法在结构力学中的应用[J].力学进展,1995,V25(2):232-239
[45]王永亮.微分求积法和微分求积单元法—原理与应用[D].南京:南京航空航天大学,2001
[46] Bellman R. E., Cacti J. Differential quadrature and long-term integration[J].J. Math. Anal. Appl.,1971,V34:235-238
[47]王永亮,王鑫伟.多外载联合作用下圆板的非线性弯曲[J].南京航空航天大学学报,1996,V28(1):53-58
[48]王永亮.边缘弹性约束圆薄板的大挠度分析[J].江苏力学,1995,V10:6-12
[49]蔡圣善,朱耘,徐建军.电动力学[M].北京:高等教育出版社,2002:15-26
[50]周又和,郑晓静.电磁固体力学[M].北京:科学出版社,1999:10-20
[51] Moon F. C. Magneto-solid mechanics[J].Physics Today,1985,V38(12):79
[52]郑哲敏,周恒,张涵信.21世纪的力学发展趋势[J].力学进展,1984,V25(4):433-411
[53] G. Paria. Magneto-elasticity and magneto-thermo-elasticity[J].Adv. Appl. Mech.,1967, V10:73-112
[54]徐耀玲,胡宇达,白象忠.薄板薄壳非线性磁弹性问题的基本方程[J].燕山大学学报,1998,V22(4):358-362
[55]马世麟,胡宇达,徐耀玲.广义洛伦兹力的磁弹性效应[J].工程力学,1999,V16(2):79-84
[56]胡宇达,白象忠.磁场环境对导电薄板的磁弹性振动的影响[J].非线性动力学学报,1999,V6(2):134-138
[57]胡宇达,白象忠.倾斜磁场中条形传导薄板的磁弹性振动[J].振动与冲击,2000,V19(2):64-66
[58] Zheng X J, Liu X N. Analysis on dynamic characteristics for ferromagnetic conducting plates in a transverse uniform magnetic field[J].Acta Mechanica Solid Sinica,2000, V13(4):361-368
[59]郑晓静,刘信恩.铁磁导电梁式板在横向均匀磁场中的动力特性分析[J].固体力学学报,2000,V21(3):243-250
[60] M. Higuchi, R. Kawamura, Y. Tanigawa. Dynamic and quasi-static behaviors of magneto-thermo-elastic stresses in a conducting infinite plate subjected to an arbitrary variation of magnetic field[J].Acta Mechanica,2007,V191(3-4):135-154
[61] M. Higuchia, R. Kawamurab, Y. Tanigawa. Dynamic and quasi-static behaviors of magneto-thermo-elastic stresses in a conducting hollow circular culinder subjected to an arbitrary variation of magnetic field[J]. International Journal of Mechanical Sciences, 2008,V50(3): 365-379
[62] R. Quan, C.T. Chang. New insights in solving distributed system equations by the quadrature methods– I. Analysis[J].Computers in Chemical Engineering,1989,V13(7): 779–788
[63] C. Shu, B.E. Richards. Application of generalized differential quadrature to solvetwo-dimensional incompressible Navier-Stokes equations[J].International Journal for Numerical Methods in Fluids,1992,V15(7):791-798
[64] C.W. Bert, M. Malik. Differential quadrature method in computational mechanics: a review[J].Appl. Mech. Rev.,1996,V49(1):1-28
[65] Y.F. Xing, B. Liu. High-Accuracy Differential Quadrature Finite Element Method and Its Application to Free Vibrations of Thin Plate with curvilinear domain[J].International Journal for Numerical methods in Engineering.2010,Vol.2(1):1-20
[66] Y.F. Xing, B. Liu, G. Liu. A differential quadrature finite element method[J].Int. J. of Applied Mechanics.2010,Vol.2(1):207-227