Geometrically Nonlinear Analysis of Functionally Graded Beams Under Thermomechanical Loading
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摘要: 基于经典梁理论,运用虚功原理和变分法推导了均匀变温场与横向均布荷载联合作用的功能梯度梁的几何非线性控制方程.考虑端部不可移夹紧边界条件,运用打靶法求解了该两点边值问题.当横向均布荷载为0时,考察了功能梯度梁的热屈曲临界升温和屈曲平衡路径.当均匀变温与横向均布荷载都不为0时,考察了功能梯度梁的荷载挠度曲线.数值结果表明:随材料体积分数指数增加,梁的有量纲热屈曲临界升温显著减小,后屈曲变形显著增加;变温对功能梯度梁的荷载挠度曲线影响非常显著.发现了功能梯度梁的双稳态构形及其转换现象,梁的最终平衡位形不但与变温及荷载参数有关,还与加载历程有关.Abstract: Based on the classical beam theory, the geometric nonlinear governing equations for FGM beams under uniform temperature field and uniform transverse loading were derived according to the principle of virtual work and the variational method. In view of the immovably clamped boundary conditions, the 2-point boundary value problem was solved with the shooting method. For the zero uniform transverse loading, the thermal buckling critical temperature and equilibrium path of the FGM beam were investigated. The load-deflection curves of the FGM beam were given for the nonzero uniform temperature and the nonzero transverse uniform loading. The numerical results show that, the dimensional thermal buckling critical temperature of the beam decreases significantly and the post-buckling deformation increases significantly with the material volume fraction index increases, and the temperature variation has a heavy influence on the load-deflection curves. The bistable configurations and the switch of the FGM beam were found. The final equilibrium shape of the beam is not only related to the variable temperature and loading parameters, but also to the loading process.
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[1] KOIZUMI M. FGM activities in Japan[J]. Composites Part B: Engineering,1997,28(1/2): 1-4. [2] 李永, 宋健, 张志民. 梯度功能力学[M]. 北京: 清华大学出版社, 2003.(LI Yong, SONG Jian, ZHANG Zhimin. Gradient Functional Mechanic s[M]. Beijing: Tsinghua University Press, 2003.(in Chinese)) [3] MA L S, WANG T J. Relationships between axisymmetric bending and buckling solutions of FGM circular plates based on third-order plate theory and classical plate theory[J]. International Journal of Solids and Structure,2004,41(1): 85-101. [4] REDDY J N, WANG C M, KITIPORNCHAI S. Axisymmetric bending of functionally graded circular and annular plates[J]. European Journal of Mechanics: A/Solids,1999,18(2): 185-199. [5] MOITA J S, ARAJO A L, MOTA SOARES C M, et al. Material and geometric nonlinear analysis of functionally graded plate-shell type structures[J]. Applied Composite Materials,2016,23(4): 537-554. [6] 胡超, 郑日恒, 孙旭峰, 等. 梯度材料平板弯拉耦合力学的精确化支配方程[J]. 应用数学和力学, 2016,37(7): 756-765.(HU Chao, ZHENG Riheng, SUN Xufeng, et al. Refined equations for functionally graded material plates under bending-tension coupling[J]. Applied Mathematics and Mechanics,2016,37(7): 756-765.(in Chinese)) [7] 张莹, 梅靖, 陈鼎, 等. 功能梯度圆板和环板受周边力作用的弹性力学解[J]. 应用数学和力学, 2018,39(5): 538-547.(ZHANG Ying, MEI Jing, CHEN Ding, et al. Elasticity solutions for functionally graded circular and annular plates subjected to boundary forces and moments[J]. Applied Mathematics and Mechanics,2018,〖STHZ〗 39(5): 538-547.(in Chinese)) [8] SHEN H S. Nonlinear bending response of functionally graded plates subjected to transverse loads and in thermal environments[J]. International Journal of Mechanical Sciences,2002,44(3): 561-584. [9] SANKAR B V, TZENG J T. Thermal stresses in functionally graded beams[J]. AIAA Journal,2002,40(6): 1228-1232. [10] 仲政, 于涛. 功能梯度悬臂梁弯曲问题的解析解[J]. 同济大学学报(自然科学版), 2006,34(3): 443-447.(ZHONG Zheng, YU Tao. Analytical bending solution of functionally graded cantilever-beam[J]. Journal of Tongji University(Natural Science),2006,34(3): 443-447.(in Chinese)) [11] KADOLI R, AKHTAR K, GANESAN N. Static analysis of functionally graded beams using higher order shear deformation theory[J]. Applied Mathematical Modelling,2008,32(12): 2509-2525. [12] 李世荣, 张靖华, 赵永刚. 功能梯度材料Timoshenko梁的热过屈曲分析[J]. 应用数学和力学, 2006,27(6): 709-715.(LI Shirong, ZHANG Jinghua, ZHAO Yonggang. Thermal post-buckling of functionally graded material Timoshenko beams[J]. Applied Mathematics and Mechanics,2006,27(6): 709-715.(in Chinese)) [13] LIBRESCU L, OH S Y, SONG O. Thin-walled beams made of functionally graded materials and operating in a high temperature environment: vibration and stability[J]. Journal of Thermal Stresses,2005,28(6/7): 649-712. [14] 钟万勰. 应用力学对偶体系[M]. 北京: 科学出版社, 2002.(ZHONG Wan-xie. Duality System of Applied Mechanics [M]. Beijing: Science Press, 2002.(in Chinese)) [15] 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)) [16] LIEW K M, YANG J, KITIPORNCHAI S. Postbuckling of piezoelectric FGM plates subject to thermo-electro-mechanical loading[J]. International Journal of Solids and Structures,2003,40(15): 3869-3892. [17] 沈惠申. 功能梯度复合材料板壳结构的弯曲、屈曲和振动[J]. 力学进展, 2004,34(1): 53-60.(SHEN Huishen. Bending, buckling and vibration of functionally graded plates[J]. Advances in Mechanics,2004,34(1): 53-60.(in Chinese)) [18] 赵伟东, 高士武, 马宏伟. 扁球壳在热-机械荷载作用下的稳定性分析[J]. 应用数学和力学, 2017,38(10): 1146-1154.(ZHAO Weidong, GAO Shiwu, MA Hongwei. Thermomechanical stability analysis of shallow spherical shells[J]. Applied Mathematics and Mechanics,2017,38(10): 1146-1154.(in Chinese)) [19] BICH D H, VAN TUNG H. Non-linear axisymmetric response of functionally graded shallow spherical shells under uniform external pressure including temperature effects[J]. International Journal of Non-Linear Mechanics,2011,46(9): 1195-1204. [20] 朱媛媛, 胡育佳, 程昌钧. Euler型梁-柱结构的非线性稳定性和后屈曲分析[J]. 应用数学和力学, 2011,32(6): 674-682.(ZHU Yuanyuan, HU Yujia, CHENG Changjun. Analysis of nonlinear stability and post-buckling for Euler-type beam-column structure[J]. Applied Mathematics and Mechanics,2011,32(6): 674-682.(in Chinese)) [21] 周承倜. 弹性稳定理论[M]. 成都: 四川人民出版社, 1981.(ZHOU Chengti. Elastic Stability Theory [M]. Chengdu: Sichuan People’s Publishing House, 1981.(in Chinese))
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