ZHAO Li, CHEN Wei-qiu, Lü Chao-feng. Two-Dimensional Complete Rational Analysis of Functionally Graded Beams Within the Symplectic Framework[J]. Applied Mathematics and Mechanics, 2012, 33(10): 1143-1155. doi: 10.3879/j.issn.1000-0887.2012.10.001
Citation: ZHAO Li, CHEN Wei-qiu, Lü Chao-feng. Two-Dimensional Complete Rational Analysis of Functionally Graded Beams Within the Symplectic Framework[J]. Applied Mathematics and Mechanics, 2012, 33(10): 1143-1155. doi: 10.3879/j.issn.1000-0887.2012.10.001

Two-Dimensional Complete Rational Analysis of Functionally Graded Beams Within the Symplectic Framework

doi: 10.3879/j.issn.1000-0887.2012.10.001
  • Received Date: 2012-01-18
  • Rev Recd Date: 2012-03-21
  • Publish Date: 2012-10-15
  • Exact solutions for generally supported functionally graded plane beams were given within the framework of symplectic elasticity. The Young’s modulus was assumed to vary exponentially along the longitudinal direction while Poisson’s ratio remained constant. The state equation with a shiftHamiltonian operator matrix had been established in our previous work, but limited to the SaintVenant solution. Here it was presented that a complete rational analysis of the displacement and stress distributions in the beam by exploring the eigensolutions which were usually covered up by the SaintVenant principle. These solutions played a significant role on local behavior of materials that was usually ignored by the conventional elasticity methods but may be crucial to the failure of the materials/structures. The analysis made full use of symplectic orthogonality of the eigensolutions. Two illustrative examples were presented to compare the displacement and stress results with those for homogenous materials, and to demonstrate the effect of material inhomogeneity.
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  • [1]
    Timoshenko S P, Goodier J N. Theory of Elasticity[M]. New York: McGraw-Hill, 1970.
    [2]
    DING Hao-jiang, HUANG De-jin, CHEN Wei-qiu. Elasticity solutions for plane anisotropic functionally graded beams[J]. International Journal of Solids and Structures, 2007, 44(1): 176-196.
    [3]
    黄德进, 丁皓江, 陈伟球. 线性分布载荷作用下功能梯度各向异性悬臂梁的解析解[J]. 应用数学和力学, 2007, 28(7): 763-768.(HUANG De-jin, DING Hao-jiang, CHEN Wei-qiu. Analytical solution for functionally graded anisotropic cantilever beam subjected to linearly distributed load[J]. Applied Mathematics and Mechanics(English Edition), 2007, 28(7): 855-860.)
    [4]
    HUANG De-jin, DING Hao-jiang, CHEN Wei-qiu. Analytical solution and semi-analytical solution for anisotropic functionally graded beam subject to arbitrary loading[J]. Science in China (Series G: Physics Mechanics and Astronomy), 2009, 52(8): 1244-1256.
    [5]
    钟万勰. 弹性力学求解新体系[M]. 大连:大连理工大学出版社, 1995. (ZHONG Wan-xie. A New Systematic Methodology for Theory of Elasticity[M]. Dalian: Publishing House of Dalian University of Technology, 1995. (in Chinese))
    [6]
    姚伟岸, 钟万勰. 辛弹性力学[M]. 北京: 高等教育出版社, 2002. (YAO Wei-an, ZHONG Wan-xie. Symplectic Elasticity[M]. Beijing: Higher Education Press, 2002. (in Chinese))
    [7]
    ZHANG Hong-wu, ZHONG Wan-xie, LI Yun-peng. Stress singularity analysis at crack tip on bi-material interfaces based on Hamiltonian principle[J]. Acta Mechanics Solida Sinica, 1996, 9(2): 124-138.
    [8]
    XU Xin-sheng, ZHONG Wan-xie, ZHANG Hong-wu. The Saint-Venant problem and principle in elasticity[J]. International Journal of Solids and Structures, 1997, 34(22): 2815-2827.
    [9]
    梁以德,郑建军. 二维弹性平面问题中任意边界条件下应力分布的封闭解[J]. 应用数学和力学, 2007, 28(12): 1455-1467. (LEUNG Andrew Y T, ZHENG Jian-jun. Closed form stress distribution in 2D elasticity for all boundary conditions[J]. Applied Mathematics and Mechanics (English Edition), 2007, 28(12): 1629-1642.)
    [10]
    YAO Wei-an, YANG Hai-tian. Hamiltonian system based Saint Venant solutions for multi-layered composite plane anisotropic plates[J]. International Journal of Solids and Structures, 2001, 38(32/33): 5807-5817.
    [11]
    YAO Wei-an, XU C. A restudy of paradox on an elastic wedge based on the Hamiltonian system[J]. Journal of Applied Mechanics, 2001, 68(4): 678-681.
    [12]
    LIM Chee-wah, CUI S, YAO Wei-an. On new symplectic elasticity approach for exact bending solutions of rectangular thin plates with two opposite sides simply supported[J]. International Journal of Solids and Structures, 2007, 44(16): 5396-5411.
    [13]
    TARN Jiann-quo, TSENG Wei-der, CHANG Hsi-hung. A circular elastic cylinder under its own weight[J]. International Journal of Solids and Structures, 2009, 46(14/15): 2886-2896.
    [14]
    ZHONG Yang, LI Rui. Exact bending analysis of fully clamped rectangular thin plates subjected to arbitrary loads by new symplectic approach[J]. Mechanics Research Communications, 2009, 36(6): 707-714.
    [15]
    ZHONG Yang, LI Rui, LIU Yue-mei, TIAN Bin. On new symplectic approach for exact bending solutions of moderately thick rectangular plates with two opposite edges simply supported[J]. International Journal of Solids and Structures, 2009, 46(11/12): 2506-2513.
    [16]
    XU Xin-sheng, LEUNG Andrew Y T, GU Qian, YANG Hao, ZHENG Jian-jun. 3D symplectic expansion for piezoelectric media[J]. International Journal for Numerical Method in Engineering, 2008, 74(12): 1848-1871.
    [17]
    姚伟岸, 李晓川. 平面电磁弹性固体的辛对偶体系[J]. 应用数学和力学, 2006, 27(2): 177-185.(YAO Wei-an, LI Xiao-chuan. Symplectic duality system on plane magnetoelectroelastic solids[J]. Applied Mathematics and Mechanics (English Edition), 2006, 27(2): 195-205.)
    [18]
    陈伟球,赵莉. 功能梯度材料平面问题的新弹性力学解法[J].力学学报, 2009, 41(4): 589-594. (CHEN Wei-qiu, ZHAO Li. The symplectic method for plane elasticity problem of functionally graded materials[J]. Acta Mech Sinica, 2009, 41(4): 588-594. (in Chinese))
    [19]
    ZHAO Li, CHEN Wei-qiu. Symplectic analysis of plane problems of functionally graded piezoelectric materials[J]. Mechanics of Materials, 2009, 41(12): 1330-1339.
    [20]
    ZHAO Li, CHEN Wei-qiu. Plane analysis for functionally graded magneto-electro-elastic materials via the symplectic framework[J]. Composite Structures, 2010, 92(7): 1753-1761.
    [21]
    ZHAO Li, CHEN Wei-qiu. On the numerical calculation in symplectic approach for elasticity problems[J]. Journal of Zhejiang University(Science A), 2008, 9(5): 583-588.
    [22]
    谢贻权, 林钟祥, 丁皓江. 弹性力学[M]. 杭州: 浙江大学出版社, 1988. (XIE Yi-quan, LIN Zhong-xiang, DING Hao-jiang. Elasticity[M]. Hangzhou: Zhejiang University Publishers, 1988. (in Chinese))
    [23]
    CHEN Wei-qiu, DING Hao-jiang. Bending of functionally graded piezoelectric rectangular plates[J]. Acta Mechanica Solida Sinica, 2000, 13(4): 312-319.
    [24]
    Ruddock G J, Spencer A J M. A new approach to stress analysis of anisotropic laminated  ̄elastic cylinders[J]. Proceedings of the Royal Society of London, Series A, 1997, 453(1960): 1067-1082.
    [25]
    SHENG Hong-yu, YE Jian-qiao. A state space finite element for laminated composite plates[J]. Computer Methods in Applied Mechanics and Engineering, 2002, 191(37/38): 4259-4276.
    [26]
    CHEN Wei-qiu, L Chao-feng, BIAN Zu-guang. Elasticity solution for free vibration of laminated beams[J]. Composite Structures, 2003, 62(1): 75-82.
    [27]
    吕朝锋. 基于状态空间架构的微分求积法及其应用[D]. 博士学位论文. 杭州: 浙江大学, 2006.(L Chao-feng. State-space-based differential quadrature method and its applicatications[D]. Ph D Dissertions. Hangzhou: Zhejiang University, 2006. (in Chinese))
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