考虑热对材料参数影响的FGM梁热后屈曲特性研究
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摘要
功能梯度梁热后屈曲特性研究对于推进功能梯度材料在航天器热防护设计中的应用有着重要意义。基于经典梁的几何非线性理论和物理中面的概念,建立了热载荷作用下功能梯度梁的运动微分方程,通过化简得到一个仅关于挠度的四阶微分-积分方程,并与固支边界条件构成特征值问题,分析研究了功能梯度梁的热后屈曲及在此基础上的振动问题。首先证明了通过哈密顿原理推导的运动方程是轴线可伸长理论的近似形式。接着考虑热对材料物性参数的影响,并对梁的长细比、功能梯度指数和温度比作了详细分析,研究了这些参数对热后屈曲路径和后屈曲振动的影响规律。结果表明:只有在长细比较大时才可以不考虑温度对材料物性参数的影响,否则误差较大;长细比、功能梯度指数和温度比的增大会增大无量纲热屈曲载荷,同时使屈曲路径和频率-载荷曲线向热载荷增大的方向移动。
Thermal post-buckling analysis of functionally graded beams is of great significance for the practical implementation of functionally graded materials for the thermal insulation of spacecrafts. The governing equations of the functionally graded beam in thermal environment are obtained based on the geometrical nonlinear theory and the introduction of the physical neutral surface. By simplification, a fourth-order integral-differential equation with respect to transverse deformation is obtained, and the eigenvalue problem is solved with the clamped boundary condition. The thermal post-buckling and the vibration on the basis of the buckled configuration are investigated. Firstly, the governing equations obtained by the Hamilton's principle are proved to be approximate equations obtained by the axial extension theory. Then, considering the heat effect on the properties of the material, the influence of the slenderness ratio, functionally gradient index and temperature ratio on post-buckling vibration is analyzed in detail. The heat effect on the properties of the material can be neglected when the slenderness is large enough. Increasing the slenderness ratio, functionally gradient index or temperature ratio will increase the non-dimensional thermal buckling load.
Thermal post-buckling analysis of functionally graded beams is of great significance for the practical implementation of functionally graded materials for the thermal insulation of spacecrafts. The governing equations of the functionally graded beam in thermal environment are obtained based on the geometrical nonlinear theory and the introduction of the physical neutral surface. By simplification, a fourth-order integral-differential equation with respect to transverse deformation is obtained, and the eigenvalue problem is solved with the clamped boundary condition. The thermal post-buckling and the vibration on the basis of the buckled configuration are investigated. Firstly, the governing equations obtained by the Hamilton's principle are proved to be approximate equations obtained by the axial extension theory. Then, considering the heat effect on the properties of the material, the influence of the slenderness ratio, functionally gradient index and temperature ratio on post-buckling vibration is analyzed in detail. The heat effect on the properties of the material can be neglected when the slenderness is large enough. Increasing the slenderness ratio, functionally gradient index or temperature ratio will increase the non-dimensional thermal buckling load.
引文
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[21]Sun Y,Li S R,Batra R C.Thermal buckling and post-buckling of FGM Timoshenko beams on nonlinear elastic foundation[J].Journal of Thermal Stresses,2016,39(1):11―26.
[22]Nayfeh A H,Emam S A.Exact solution and stability of postbuckling configurations of beams[J].Nonlinear Dynamics,2008,54(4):395―408.
[23]马连生,顾春龙.剪切可变形梁热过屈曲解析解[J].工程力学,2012,29(2):172―176.Ma Liansheng,Gu Chunlong.Exact solutions for thermal post-buckling of shear deformable beams[J].Engineering Mechanics,2012,29(2):172―176.(in Chinese)
[24]Ma L S,Lee D W.A further discussion of nonlinear mechanical behavior for FGM beams under in-plane thermal loading[J].Composite Structures,2011,93(2):831―842.
[25]Levyakov S V.Elastica solution for thermal bending of a functionally graded beam[J].Acta Mechanica,2013,224(8):1731―1740.
[26]崔德福.含粘滞-滑移-约束边界的梁和板的热屈曲及热振动研究[D].南京:南京航空航天大学,2015.Cui Defu.Thermal buckling and thermal vibration of beams and plates with stick-slip-stop boundaries[D].Nanjing:Nanjing University of Aeronautics and Astronautics,2015.(in Chinese)
[27]毛丽娟,马连生.非均匀热载荷作用下功能梯度梁的非线性静态响应[J].工程力学,2017,34(6):1―8.Mao Lijuan,Ma Liansheng.Nonlinear static responses of FGM beams under non-uniform thermal loading[J].Engineering Mechanics,2017,34(6):1―8.(in Chinese)
[28]Ebrahimi F,Jafari A.A higher-order thermomechanical vibration analysis of temperature-dependent fgm beams with porosities[J].Journal of Engineering,2016,2016(2):1―20.
[29]Ebrahimi F,Jafari A.A four-variable refined shear-deformation beam theory for thermo-mechanical vibration analysis of temperature-dependent FGM beams with porosities[J].Mechanics of Advanced Materials&Structures,2018,25(3):212―224.
[30]钮鹏,李旭,李世荣,等.弹性约束下Timoshenko夹层梁的热屈曲行为研究[J].工程力学,2018,35(增刊1):13―16,39.Niu Peng,Li Xu,Li Shirong,et al.The thermal buckling behavior of Timoshenko sandwich beam under elastic constraint[J].Engineering Mechanics,2018,35(Suppl1):13―16,39.(in Chinese)
[31]Taati E.On buckling and post-buckling behavior of functionally graded micro-beams in thermal environment[J].International Journal of Engineering Science,2018,128:63―78.
[32]Bagheri H,Kiani Y,Eslami M R.Asymmetric thermal buckling of temperature dependent annular FGM plates on a partial elastic foundation[J].Computers and Mathematics with Applications,2018,75(5):1566―1581.
[33]Do Q C,Dao V D,Le K H.Thermal buckling analysis of stiffened FGM truncated conical shells resting on elastic foundations using FSDT[J].Acta Mechanica,2018,229(5):2221―2249.
[34]赵伟东,高士武,马宏伟.功能梯度扁球壳在热-机械荷载作用下的屈曲分析[J].工程力学,2018,35(12):220―228.Zhao Weidong,Gao Shiwu,Ma Hongwei.Thermomechanical buckling analysis of functionally graded shallow spherical shells[J].Engineering Mechanics,2018,35(12):220―228.(in Chinese)
[35]Shen H S.Functionally graded materials nonlinear analysis of plates and shells[M].USA:CRC Press,2009:3―7.
[36]Kim Y W.Temperature dependent vibration analysis of functionally graded rectangular plates[J].Journal of Sound&Vibration,2005,284(3):531―549.
[37]Reddy J N,Chin C D.Thermomechanical analysis of funuctionally graded cylinders and plates[J].Journal of Thermal Stresses,1998,21(6):593―626.
[38]Yang J,Shen H S.Vibration characteristics and transient response of shear-deformable functionally graded plates in thermal environments[J].Journal of Sound&Vibration,2002,255(3):579―602.
[39]马连生,张璐.面内热载荷作用下功能梯度梁热过屈曲精确解[J].兰州理工大学学报,2015,41(1):164―167.Ma Liansheng,Zhang Lu.Exact solutions for thermo-post-buckling of functionally graded material beams under in-plane thermal loading[J].Journal of Lanzhou University of Technology,2015,41(1):164―167.(in Chinese)
[40]Shahrjerdi A,Bayat M,Majid D L A.Free vibration analysis of solar functionally graded plates with temperature-dependent material properties using second order shear deformation theory[J].Journal of Mechanical Science&Technology,2011,25(9):2195―2209.
[41]牛牧华,马连生.基于物理中面FGM梁的非线性力学行为[J].工程力学,2011,28(6):219―225.Niu Muhua,Ma Liansheng.Nonlinear mechanical behaviors of FGM beams based on the physical neutral surface[J].Engineering Mechanics,2011,28(6):219―225.(in Chinese)
[2]Duc N D,et al.Nonlinear stability of eccentrically stiffened S-FGM elliptical cylindrical shells in thermal environment[J].Thin-Walled Structures,2016,108:280―290.
[3]Huang H W,Zhang Y Q,Han Q.Stability of hydrostatic-pressured FGM thick rings with material nonlinearity[J].Applied Mathematical Modelling,2017,45:55―64.
[4]李清禄,栾玮荻,李世荣.功能梯度材料圆板在随从力作用下的稳定性[J].玻璃钢/复合材料,2016(10):5―10.Li Qinglu,Luan Weidi,Li Shirong.The stability of FGMcircular plates subjected to follower force[J].Fiber Reinforced Plastics/Composites,2016(10):5―10.(in Chinese)
[5]Bouderba B,Houari M S A,Tounsi A,et al.Thermal stability of functionally graded sandwich plates using a simple shear deformation theory[J].Structural Engineering&Mechanics,2016,58(3):397―422.
[6]Majumadar A,Das D.A study on thermal buckling load of clamped functionally graded beams under linear and nonlinear thermal gradient across thickness[J].Proceedings of the Institution of Mechanical Engineers,Part L:Journal of Materials:Design and Applications,2018,232(9):769―784.
[7]Eroglu U.In-plane free vibrations of circular beams made of functionally graded material in thermal environment:Beam theory approach[J].Composite Structures,2015,122:217―228.
[8]Ebrahimi F,Barati M R.Vibration analysis of nonlocal beams made of functionally graded material in thermal environment[J].European Physical Journal Plus,2016,131(279):1―22.
[9]蒋坤.功能梯度材料薄板热模态分析[D].河北:河北工程大学,2017.Jiang Kun.Thermal modal analysis of functionally graded plate[D].Hebei:Hebei University of Engineering,2017.(in Chinese)
[10]刘文光,舒斌,郭隆清,等.热环境对FGM壳模态频率的影响[J].振动与冲击,2017,36(4):127―131.Liu Wenguang,Shu Bin,Guo Longqing,et al.Impacts of thermal environment on modal frequency of FGM shells[J].Journal of Vibration and Shock,2017,36(4):127―131.(in Chinese)
[11]Bhangale R K,Ganesan N.Thermoelastic buckling and vibration behavior of a functionally graded sandwich beam with constrained viscoelastic core[J].Journal of Sound&Vibration,2006,295(1/2):294―316.
[12]Kapuria S,Bhattacharyya M,Kumar A N.Bending and free vibration response of layered functionally graded beams:A theoretical model and its experimental validation[J].Composite Structures,2008,82(3):390―402.
[13]Kocaturk T,Akbas S D.Post-buckling analysis of Timoshenko beams made of functionally graded material under thermal loading[J].Structural Engineering&Mechanics,2012,41(6):775―789.
[14]Lee Y Y,Zhao X,Reddy J N.Postbuckling analysis of functionally graded plates subject to compressive and thermal loads[J].Computer Methods in Applied Mechanics&Engineering,2010,199(25/26/27/28):1645―1653.
[15]Wattanasakulpong N,Prusty B G,Kelly D W.Thermal buckling and elastic vibration of third-order shear deformable functionally graded beams[J].International Journal of Mechanical Sciences,2011,53(9):734―743.
[16]Shen H S.Postbuckling of shear deformable FGMcylindrical shells surrounded by an elastic medium[J].International Journal of Mechanical Sciences,2009,51(5):372―383.
[17]She G L,Yuan F G,Ren Y R.Thermal buckling and post-buckling analysis of functionally graded beams based on a general higher-order shear deformation theory[J].Applied Mathematical Modelling,2017,47:340―357.
[18]Li S R,Batra R C.Thermal buckling and postbuckling of euler-bernoulli beams supported on nonlinear elastic foundations[J].Aiaa Journal,2007,3(45):712―720.
[19]李清禄,李世荣.功能梯度材料梁在后屈曲构形附近的自由振动[J].振动与冲击,2011,30(9):76―78.Li Qinglu,Li Shirong.Free vibration of FGM Euler beam with post-buckling configuration subjected to axial force[J].Journal of Vibration and Shock,2011,30(9):76―78.(in Chinese)
[20]钮鹏,李旭,李世荣,等.弹性地基上复合材料夹层梁的热过屈曲[J].工程力学,2017,34(增刊1):26―30.Niu Peng,Li Xu,Li Shirong,et al.The thermal buckling of composite sandwich beams on elastic foundations[J].Engineering Mechanics,2017,34(suppl 1):26―30.(in Chinese)
[21]Sun Y,Li S R,Batra R C.Thermal buckling and post-buckling of FGM Timoshenko beams on nonlinear elastic foundation[J].Journal of Thermal Stresses,2016,39(1):11―26.
[22]Nayfeh A H,Emam S A.Exact solution and stability of postbuckling configurations of beams[J].Nonlinear Dynamics,2008,54(4):395―408.
[23]马连生,顾春龙.剪切可变形梁热过屈曲解析解[J].工程力学,2012,29(2):172―176.Ma Liansheng,Gu Chunlong.Exact solutions for thermal post-buckling of shear deformable beams[J].Engineering Mechanics,2012,29(2):172―176.(in Chinese)
[24]Ma L S,Lee D W.A further discussion of nonlinear mechanical behavior for FGM beams under in-plane thermal loading[J].Composite Structures,2011,93(2):831―842.
[25]Levyakov S V.Elastica solution for thermal bending of a functionally graded beam[J].Acta Mechanica,2013,224(8):1731―1740.
[26]崔德福.含粘滞-滑移-约束边界的梁和板的热屈曲及热振动研究[D].南京:南京航空航天大学,2015.Cui Defu.Thermal buckling and thermal vibration of beams and plates with stick-slip-stop boundaries[D].Nanjing:Nanjing University of Aeronautics and Astronautics,2015.(in Chinese)
[27]毛丽娟,马连生.非均匀热载荷作用下功能梯度梁的非线性静态响应[J].工程力学,2017,34(6):1―8.Mao Lijuan,Ma Liansheng.Nonlinear static responses of FGM beams under non-uniform thermal loading[J].Engineering Mechanics,2017,34(6):1―8.(in Chinese)
[28]Ebrahimi F,Jafari A.A higher-order thermomechanical vibration analysis of temperature-dependent fgm beams with porosities[J].Journal of Engineering,2016,2016(2):1―20.
[29]Ebrahimi F,Jafari A.A four-variable refined shear-deformation beam theory for thermo-mechanical vibration analysis of temperature-dependent FGM beams with porosities[J].Mechanics of Advanced Materials&Structures,2018,25(3):212―224.
[30]钮鹏,李旭,李世荣,等.弹性约束下Timoshenko夹层梁的热屈曲行为研究[J].工程力学,2018,35(增刊1):13―16,39.Niu Peng,Li Xu,Li Shirong,et al.The thermal buckling behavior of Timoshenko sandwich beam under elastic constraint[J].Engineering Mechanics,2018,35(Suppl1):13―16,39.(in Chinese)
[31]Taati E.On buckling and post-buckling behavior of functionally graded micro-beams in thermal environment[J].International Journal of Engineering Science,2018,128:63―78.
[32]Bagheri H,Kiani Y,Eslami M R.Asymmetric thermal buckling of temperature dependent annular FGM plates on a partial elastic foundation[J].Computers and Mathematics with Applications,2018,75(5):1566―1581.
[33]Do Q C,Dao V D,Le K H.Thermal buckling analysis of stiffened FGM truncated conical shells resting on elastic foundations using FSDT[J].Acta Mechanica,2018,229(5):2221―2249.
[34]赵伟东,高士武,马宏伟.功能梯度扁球壳在热-机械荷载作用下的屈曲分析[J].工程力学,2018,35(12):220―228.Zhao Weidong,Gao Shiwu,Ma Hongwei.Thermomechanical buckling analysis of functionally graded shallow spherical shells[J].Engineering Mechanics,2018,35(12):220―228.(in Chinese)
[35]Shen H S.Functionally graded materials nonlinear analysis of plates and shells[M].USA:CRC Press,2009:3―7.
[36]Kim Y W.Temperature dependent vibration analysis of functionally graded rectangular plates[J].Journal of Sound&Vibration,2005,284(3):531―549.
[37]Reddy J N,Chin C D.Thermomechanical analysis of funuctionally graded cylinders and plates[J].Journal of Thermal Stresses,1998,21(6):593―626.
[38]Yang J,Shen H S.Vibration characteristics and transient response of shear-deformable functionally graded plates in thermal environments[J].Journal of Sound&Vibration,2002,255(3):579―602.
[39]马连生,张璐.面内热载荷作用下功能梯度梁热过屈曲精确解[J].兰州理工大学学报,2015,41(1):164―167.Ma Liansheng,Zhang Lu.Exact solutions for thermo-post-buckling of functionally graded material beams under in-plane thermal loading[J].Journal of Lanzhou University of Technology,2015,41(1):164―167.(in Chinese)
[40]Shahrjerdi A,Bayat M,Majid D L A.Free vibration analysis of solar functionally graded plates with temperature-dependent material properties using second order shear deformation theory[J].Journal of Mechanical Science&Technology,2011,25(9):2195―2209.
[41]牛牧华,马连生.基于物理中面FGM梁的非线性力学行为[J].工程力学,2011,28(6):219―225.Niu Muhua,Ma Liansheng.Nonlinear mechanical behaviors of FGM beams based on the physical neutral surface[J].Engineering Mechanics,2011,28(6):219―225.(in Chinese)