Two-dimensional complete rational analysis of functionally graded beams within symplectic framework

Li ZHAO;Wei-qiu CHEN;Chao-feng L&#0;

doi:10.1007/s10483-012-1617-8

Applied Mathematics and Mechanics >

2012 , Vol. 33 >Issue 10: 1225 - 1238

DOI: https://doi.org/10.1007/s10483-012-1617-8

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Two-dimensional complete rational analysis of functionally graded beams within symplectic framework

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1. Department of Civil Engineering, Ningbo University of Technology, Ningbo 315016, Zhejiang Province, P. R. China;
2. Department of Engineering Mechanics, Zhejiang University, Hangzhou 310027, P. R. China;
3. Department of Civil Engineering, Zhejiang University, Hangzhou 310058, P. R. China

Received date: 2012-01-18

Revised date: 2012-03-21

Online published: 2012-10-10

Supported by

Project supported by the National Natural Science Foundation of China (Nos. 11090333 and 10972193) and the Natural Science Foundation of Ningbo City of China (No. 2011A610077)

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Abstract

Exact solutions for generally supported functionally graded plane beams are given within the framework of symplectic elasticity. The Young’s modulus is assumed to exponentially vary along the longitudinal direction while Poisson’s ratio remains constant. The state equation with a shift-Hamiltonian operator matrix has been established in the previous work, which are limited to the Saint-Venant solution. Here, a complete rational analysis of the displacement and stress distributions in the beam is presented by exploring the eigensolutions that are usually covered up by the Saint-Venant principle. These solutions play a significant role in the local behavior of materials that is usually ignored in the conventional elasticity methods but possibly crucial to the materials/structures failures. The analysis makes full use of the symplectic orthogonality of the eigensolutions. Two illustrative examples are presented to compare the displacement and stress results with those for homogenous materials, demonstrating the effects of material inhomogeneity.

Key words： functionally graded material (FGM); exact solution; expansion of eigenfunction; symplectic elasticity; semi-infinite orthotropic strip; stress singularity at clamped end; generalized Cauchy type singular integral equation

Cite this article

Li ZHAO;Wei-qiu CHEN;Chao-feng L� . Two-dimensional complete rational analysis of functionally graded beams within symplectic framework[J]. Applied Mathematics and Mechanics, 2012 , 33(10) : 1225 -1238 . DOI: 10.1007/s10483-012-1617-8

References

[1] Timoshenko, S. P. and Goodier, J. N. Theory of Elasticity, McGraw-Hill, New York (1970)
[2] Ding, H. J., Huang, D. J., and Chen, W. Q. Elasticity solutions for plane anisotropic functionallygraded beams. International Journal of Solids and Structures, 44, 176-196 (2007)
[3] Huang, D. J., Ding, H. J., and Chen, W. Q. Analytical solution for functionally graded anisotropiccantilever beam subjected to linearly distributed load. Applied Mathematics and Mechanics (En-glish Edition), 28(7), 855-860 (2007) DOI 10.1007/s10483-007-0702-1
[4] Huang, D. J., Ding, H. J., and Chen, W. Q. Analytical solution and semi-analytical solution foranisotropic functionally graded beam subject to arbitrary loading. Science in China Series G:Physics, Mechanics and Astronomy, 52(8), 1244-1256 (2009)
[5] Zhong, W. X. A New Systematic Methodology for Theory of Elasticity (in Chinese), PublishingHouse of Dalian University of Technology, Dalian (1995)
[6] Yao, W. A. and Zhong, W. X. Symplectic Elasticity (in Chinese), Higher Education Press, Beijing(2002)
[7] Zhang, H. W., Zhong, W. X., and Li, Y. P. Stress singularity analysis at crack tip on bi-materialinterfaces based on Hamiltonian principle. ACTA Mechanics Solida Sinica, 9, 124-138 (1996)
[8] Xu, X. S., Zhong, W. X., and Zhang, H. W. The Saint-Venant problem and principle in elasticity.International Journal of Solids and Structures, 34, 2815-2827 (1997)
[9] Leung, A. Y. T. and Zheng, J. J. Closed form stress distribution in 2D elasticity for all boundaryconditions. Applied Mathematics and Mechanics (English Edition), 28(12), 1629-1642 (2007) DOI10.1007/s10483-007-1210-z
[10] Yao, W. A. and Zhong, W. X. Hamiltonian system based Saint Venant solutions for multi-layeredcomposite plane anisotropic plates. International Journal of Solids and Structures, 38, 5807-5817(2001)
[11] Yao, W. A. and Xu, C. A restudy of paradox on an elastic wedge based on the Hamiltoniansystem. Journal of Applied Mechanics, 68, 678-681 (2001)
[12] Lim, C. W., Cui, S., and Yao, W. A. On new symplectic elasticity approach for exact bending solutionsof rectangular thin plates with two opposite sides simply supported. International Journalof Solids and Structures, 44, 5396-5411 (2007)
[13] Tarn, J. Q., Tseng, W. D., and Chang, H. H. A circular elastic cylinder under its own weight.International Journal of Solids and Structures, 46, 2886-2896 (2009)
[14] Zhong, Y. and Li, R. Exact bending analysis of fully clamped rectangular thin plates subjected toarbitrary loads by new symplectic approach. Mechanics Research Communications, 36(6), 707-714(2009)
[15] Zhong, Y., Li, R., and Tian, B. On new symplectic approach for exact bending solutions of moderatelythick rectangular plates with two opposite edges simply supported. International Journalof Solids and Structures, 46, 2506-2513 (2009)
[16] Xu, X. S., Leung, A. Y. T., Gu, Q., Yang, H., and Zheng, J. J. 3D symplectic expansion forpiezoelectric media. International Journal for Numerical Method in Engineering, 74, 1848-1871(2008)
[17] Yao, W. A. and Li, X. C. Symplectic duality system on plane magnetoelectroelastic solids. AppliedMathematics and Mechanics (English Edition), 27(2), 195-205 (2006) DOI 10.1007/s10483-006-0207-z
[18] Chen, W. Q. and Zhao, L. The symplectic method for plane elasticity problem of functionallygraded materials (in Chinese). Acta Mech Sinica, 41(4), 588-594 (2009)
[19] Zhao, L. and Chen, W. Q. Symplectic analysis of plane problems of functionally graded piezoelectricmaterials. Mechanics of Materials, 41(12), 1330-1339 (2009)
[20] Zhao, L. and Chen, W. Q. Plane analysis for functionally graded magneto-electro-elastic materialsvia the symplectic framework. Composite Structures, 92(7), 1753-1761 (2010)
[21] Zhao, L. and Chen, W. Q. On the numerical calculation in symplectic approach for elasticityproblems. Journal of Zhejiang University (SCIENCE A), 9(5), 583-588 (2008)
[22] Xie, Y. Q., Lin, Z. X., and Ding, H. J. Mechanics of Elasticity (in Chinese), Zhejiang UniversityPress, Hangzhou (1988)
[23] Chen, W. Q. and Ding, H. J. Bending of functionally graded piezoelectric rectangular plates. ActaMechanica Solida Sinica, 13, 312-319 (2000)
[24] Ruddock, G. J. and Spencer, A. J. M. A new approach to stress analysis of anisotropic laminatedelastic cylinders. Proceedings of the Royal Society of London, Series A, 453(1960), 1067-1082(1997)
[25] Sheng, H. Y. and Ye, J. Q. A state space finite element for laminated composite plates. ComputerMethods in Applied Mechanics and Engineering, 191(37-38), 4259-4276 (2002)
[26] Chen, W. Q., L?, C. F., and Bian, Z. G. Elasticity solution for free vibration of laminated beams.Composite Structures, 62(1), 75-82 (2003)
[27] L?, C. F. State-Space-Based Differential Quadrature Method and Its Applications (in Chinese),Ph.D. dissertation, Zhejiang University (2006)

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