热磁弹性耦合作用下扁锥薄壳非线性动态和准静态响应分析
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摘要
本文对处于随时间变化磁场中扁锥薄壳的热磁弹性行为进行了动态和准静态研究。分析了在机械载荷，电磁场和温度场耦合作用下扁锥薄壳的位移和应力动态和准静态响应行为。
第一章对热磁弹性的研究进行了比较详尽的综述，对热磁弹性的定义、应用及其力学研究现状和进展进行了概述，同时介绍了这一领域的研究历史及其发展。最后，介绍本文的研究内容和研究方法。
第二章研究了处于交变电磁场中扁锥壳的热磁弹性问题的动态行为。基于麦克斯韦方程和欧姆定律，导出了电磁场控制方程。基于焦耳热效应、热传导方程以及热平衡方程，导出温度场的控制方程。基于经典板壳非线性运动方程且考虑洛伦磁力和温度应力的耦合作用，导出扁锥薄壳的非线性热磁弹性控制方程。然后采用分离变量法对电磁场和温度场控制方程进行求解，采用Galerkin方法和四阶龙格库塔法对非线性弹性场控制方程进行求解。得到了在外加磁场和表面均布定常机械载荷耦合作用下的温度、磁场、应力和位移随时间变化的规律；讨论了边界条件、几何参数以及外加磁场的变化频率对扁锥壳的应力和位移波形的影响。
第三章研究了处于交变电磁场中扁锥薄壳的热磁弹性问题的准静态行为。建立准静态的磁场、温度场和弹性场耦合作用的非线性控制方程，然后采用Galerkin方法对非线性弹性场控制方程进行求解。得到了应力和位移随时间的变化规律；讨论边界条件、几何参数以及外加磁场的变化频率对扁锥壳的准静态应力和位移分布的影响。
第四章对全文进行了总结，得出了一些有用的结论，并对今后进一步的研究方向进行了展望。
The problems of nonlinear dynamic and quasi-static response for shallow conical shellsunder a time-varying magnetic field are investigated. The dynamic and quasi-static responsesof displacement and stress for shallow conical shells are analyzed due tomagneto-thermo-elastic interaction.
In Chapter Ⅰ, The researches of the history and development inmagneto-thermo-elastic field are introduced. And the definition, application and mechanicalresearch status and progress of the magneto-thermo-elastic theory are summarized. Finally,the content and investigating methods in this dissertation are introduced.
In Chapter Ⅱ, the problem of nonlinear dynamic response for shallow conical shellsunder a time-varying magnetic field when temperature field, electromagnetic field and elasticfield are coupled is analyzed. The nonlinear governing equations of shallow conical shells arededuced under magneto-thermo-mechanical field.The electromagnetic governing equationsand temperature governing equations are solved by the separation of variables.The elasticfield governing equations are solved by Galerkin method and the4th Runge-Kutta to achievethe loads-displacement relation. The regularities of distribution of temperature, magneticfield strength stress and strain for shallow conical shells under a time-varying magneticfield are derived.
In Chapter Ⅲ, the problem of nonlinear quasi-static response for Shallow ConicalShells under a time-varying magnetic field when temperature field, electromagnetic field andelastic field are coupled is analyzed. The nonlinear governing equation of shallow conicalshells is deduced under magneto-thermo-mechanical field. The basic governing equationsare solved by Galerkin method to achieve the loads-displacement relation. The regularities ofdistribution of stress and strain for Shallow Conical Shells under a time-varying magneticfield are derived.
In Chapter Ⅳ, the whole paper is summarized and some useful conclusions areobtained. Then, farther investigations about this problem are prospected.
第一章对热磁弹性的研究进行了比较详尽的综述，对热磁弹性的定义、应用及其力学研究现状和进展进行了概述，同时介绍了这一领域的研究历史及其发展。最后，介绍本文的研究内容和研究方法。
第二章研究了处于交变电磁场中扁锥壳的热磁弹性问题的动态行为。基于麦克斯韦方程和欧姆定律，导出了电磁场控制方程。基于焦耳热效应、热传导方程以及热平衡方程，导出温度场的控制方程。基于经典板壳非线性运动方程且考虑洛伦磁力和温度应力的耦合作用，导出扁锥薄壳的非线性热磁弹性控制方程。然后采用分离变量法对电磁场和温度场控制方程进行求解，采用Galerkin方法和四阶龙格库塔法对非线性弹性场控制方程进行求解。得到了在外加磁场和表面均布定常机械载荷耦合作用下的温度、磁场、应力和位移随时间变化的规律；讨论了边界条件、几何参数以及外加磁场的变化频率对扁锥壳的应力和位移波形的影响。
第三章研究了处于交变电磁场中扁锥薄壳的热磁弹性问题的准静态行为。建立准静态的磁场、温度场和弹性场耦合作用的非线性控制方程，然后采用Galerkin方法对非线性弹性场控制方程进行求解。得到了应力和位移随时间的变化规律；讨论边界条件、几何参数以及外加磁场的变化频率对扁锥壳的准静态应力和位移分布的影响。
第四章对全文进行了总结，得出了一些有用的结论，并对今后进一步的研究方向进行了展望。
The problems of nonlinear dynamic and quasi-static response for shallow conical shellsunder a time-varying magnetic field are investigated. The dynamic and quasi-static responsesof displacement and stress for shallow conical shells are analyzed due tomagneto-thermo-elastic interaction.
In Chapter Ⅰ, The researches of the history and development inmagneto-thermo-elastic field are introduced. And the definition, application and mechanicalresearch status and progress of the magneto-thermo-elastic theory are summarized. Finally,the content and investigating methods in this dissertation are introduced.
In Chapter Ⅱ, the problem of nonlinear dynamic response for shallow conical shellsunder a time-varying magnetic field when temperature field, electromagnetic field and elasticfield are coupled is analyzed. The nonlinear governing equations of shallow conical shells arededuced under magneto-thermo-mechanical field.The electromagnetic governing equationsand temperature governing equations are solved by the separation of variables.The elasticfield governing equations are solved by Galerkin method and the4th Runge-Kutta to achievethe loads-displacement relation. The regularities of distribution of temperature, magneticfield strength stress and strain for shallow conical shells under a time-varying magneticfield are derived.
In Chapter Ⅲ, the problem of nonlinear quasi-static response for Shallow ConicalShells under a time-varying magnetic field when temperature field, electromagnetic field andelastic field are coupled is analyzed. The nonlinear governing equation of shallow conicalshells is deduced under magneto-thermo-mechanical field. The basic governing equationsare solved by Galerkin method to achieve the loads-displacement relation. The regularities ofdistribution of stress and strain for Shallow Conical Shells under a time-varying magneticfield are derived.
In Chapter Ⅳ, the whole paper is summarized and some useful conclusions areobtained. Then, farther investigations about this problem are prospected.
引文
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[2]侯鹏飞，骆伟，郭丽娟.耦合均载作用下的电磁热弹性简支圆板.工程力学.2007,11:47-62.
[3]陈兴华，王亮.热磁电弹性材料层合板的半解析法.装备制造技术2011，4：24-28.
[4]朱为国，白象忠.四边固支热磁弹性矩形薄板的分岔与混沌.振动与冲击.2009,5:59-62
[5]田振国，杨阳，白象忠.载流球台薄壳的热磁弹性分析.工程力学，2010，10：224-229.
[6]陈燕,杨万里,郑小玲.轴对称热弹性问题有限元基本方程求解.机械设计与制造工程,2002,4：14-17.
[7]白象忠.磁弹性、热磁弹性理论及其应用.力学进展,1996,3(26):389-406.
[8] G.Para, Magneto-elasticity and magneto-thermo-elasticity. Adv. Appl.Mech,1967，10:73-112.
[9].H.S.Paul,N.Muthigala. Free vibrations of an infinite isotropic magneto-thermo-elasticplate. Acta Mech,1972,14:147-156.
[10] M.Higuchi,R,Kawamura,Y.Tanigawa,H.Fujieda. Dynamic and quasi-static behaviors ofmagneto-thermo-elastic stresses in a conducting infinite plate subjected to arbitrary variationof magnetic field. Acta Mech,2007,191:135-154.
[11] F.c.Moon,P.H.Pao. Vibration and dynamic instability of a beam-plate in a transversemagnetic field[J]. Appl.Mech.,Trans.ASME,1969,36:92-100.
[12]侯鹏飞，骆伟，郭丽娟.耦合均载作用下的电磁热弹性简支圆板.工程力学.2007,11:47-62.
[13]陈兴华，王亮.热磁电弹性材料层合板的半解析法.装备制造技术2011，4：24-28.
[14]朱为国，白象忠.四边固支热磁弹性矩形薄板的分岔与混沌.振动与冲击.2009,5:59-62
[15]田振国，杨阳，白象忠.载流球台薄壳的热磁弹性分析.工程力学，2010，10：224-229.
[16]陈燕,杨万里,郑小玲.轴对称热弹性问题有限元基本方程求解.机械设计与制造工程,2002,4：14-17.
[17]白象忠.磁弹性、热磁弹性理论及其应用.力学进展,1996,3(26):389-406.
[18] P.Pratar. Plane waves in thermoelasticity-magnetothermoelasticity. int.J. Eng.Sci,1972,10(5):467-477.
[19] A.Nagfeh,S.Nemat-Nasser. Electromagneto-thermoealstic palne waves in solids withthermal relaxation[J]. Appl.Mech.Ser,1972,39:108-113.
[20] F.C. Moon, S. Chattopadhyay, Magnetically induced stress waves in a conductingsolid-theory and experiment[J]. Appl. Mech., Trans. ASME,1974,41:641–646.
[21] C.T. Chian, F.C. Moon. Magnetically induced cylindrical stress waves in a thermoelasticconductor, Int.[J]. Solids Struct,1981,17:1021–1035.
[22] Hany, H.Sherie and Magdy, A.Ezzat. A Thermal-shock problem for a half-space inmagneto-thermo-elasticity with thermal relaxation. Int.J.solidsstructures,1996,33(30):4449-4457.
[23] S. Banerjee, S.K.Roychoudhuri. Magneto-thermo-elastic interactions in an infiniteisotropic elastic cylinder subjected to a periodic loading. Int. J. Engng. Sci,1997,35(4):437–444.
[24] M.Rakshit,B.Mukhopadhyay. An electro-magneto-thermo-Visco-elastic problem in aninfinite medium with a cylindrical hole. Int.J,Eng.Sci,2005,43:925-936.
[25] A.Baksi,R.K.Bera. Eigenfunction expansion method for the solution ofmagneto-thermoelastic problems with relaxation and heat source in three dimensions.Math.Comput.Model,2005,42533-552.
[26]侯鹏飞，骆伟，郭丽娟.耦合均载作用下的电磁热弹性简支圆板.工程力学.2007,11:47-62.
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