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Three-dimensional elasticity solutions for bending of generally supported thick functionally graded plates

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Abstract

Three-dimensional elasticity solutions for static bending of thick functionally graded plates are presented using a hybrid semi-analytical approach-the state-space based differential quadrature method (SSDQM). The plate is generally supported at four edges for which the two-way differential quadrature method is used to solve the in-plane variations of the stress and displacement fields numerically. An approximate laminate model (ALM) is exploited to reduce the inhomogeneous plate into a multi-layered laminate, thus applying the state space method to solve analytically in the thickness direction. Both the convergence properties of SSDQM and ALM are examined. The SSDQM is validated by comparing the numerical results with the exact solutions reported in the literature. As an example, the Mori-Tanaka model is used to predict the effective bulk and shear moduli. Effects of gradient index and aspect ratios on the bending behavior of functionally graded thick plates are investigated.

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References

  1. Koizumi, M. FGM activities in Japan. Composites Part B: Engineering, 28(1–2), 1–4 (1997)

    Article  Google Scholar 

  2. Fereidoon, A., Asghardokht, S. M., and Mohyeddin, A. Bending analysis of thin functionally graded plates using generalized differential quadrature method. Archive of Applied Mechanics, 81, 1523–1539 (2011)

    Article  MATH  Google Scholar 

  3. Zenkour, A. M. A comprehensive analysis of functionally graded sandwich plates, part 1: deflection and stresses. International Journal of Solids and Structures, 42(18–19), 5224–5242 (2005)

    Article  MATH  Google Scholar 

  4. Nguyen, T. K., Sab, K., and Bonnet, G. First-order shear deformation plate models for functionally graded materials. Composite Structures, 83(1), 25–36 (2008)

    Article  Google Scholar 

  5. Yang, J. and Sheng, H. S. Nonlinear bending analysis of shear deformable functionally graded plates subjected to thermo-mechanical loads under various boundary conditions. Composites Part B: Engineering, 34(1), 103–115 (2003)

    Article  Google Scholar 

  6. Reddy, J. N. Analysis of functionally graded plates. International Journal for Numerical Methods in Engineering, 47(1–3), 663–684 (2000)

    Article  MATH  Google Scholar 

  7. Wu, C. P. and Li, H. Y. An RMVT-based third-order shear deformation theory of multilayered functionally graded material plates. Composite Structures, 92(10), 2591–2605 (2010)

    Article  Google Scholar 

  8. Gilhooley, D. F., Batra, R. C., Xiao, J. R., McCarthy, M. A., and Gillespie, J. W. Analysis of thick functionally graded plates by using higher-order shear and normal deformable plate theory and MLPG method with radial basis functions. Composite Structures, 80(4), 539–552 (2007)

    Article  MATH  Google Scholar 

  9. Matsunaga, H. Free vibration and stability of functionally graded plates according to a 2-D higherorder deformation theory. Composite Structures, 82(4), 499–512 (2008)

    Article  Google Scholar 

  10. Sahraee, S. and Saidi, A. R. Axisymmetric bending analysis of thick functionally graded circular plates using fourth-order shear deformation theory. European Journal of Mechanics A—Solids, 28(5), 974–984 (2009)

    Article  MATH  Google Scholar 

  11. Batra, R. C. and Vel, S. S. Exact solution for thermoelastic deformations of functionally graded thick rectangular plates. AIAA Journal, 40(7), 1421–1433 (2001)

    Google Scholar 

  12. Reddy, J. N. and Cheng, Z. Q. Three-dimensional thermomechanical deformations of functionally graded rectangular plates. European Journal of Mechanics A—Solids, 20(5), 841–855 (2001)

    Article  MATH  Google Scholar 

  13. Wen, P. H., Sladek, J., and Sladek, V. Three-dimensional analysis of functionally graded plates. International Journal for Numerical Methods in Engineering, 87(10), 923–942 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  14. Kashtalyan, M. Three-dimensional elasticity solution for bending of functionally graded rectangular plates. European Journal of Mechanics A—Solids, 23(5), 853–864 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  15. Huang, Z. Y., Lü, C. F., and Chen, W. Q. Benchmark solutions for functionally graded thick plates resting on Winkler-Pasternak elastic foundations. Composite Structures, 85(1), 95–104 (2008)

    Article  Google Scholar 

  16. Xu, Y. P. and Zhou, D. Three-dimensional elasticity solution of functionally graded rectangular plates with variable thickness. Composite Structures, 91(1), 56–65 (2009)

    Article  Google Scholar 

  17. Alibeigloo, A. Three-dimensional exact solution for functionally graded rectangular plate with integrated surface piezoelectric layers resting on elastic foundation. Mechanics of Advanced Materials and Structures, 17, 183–195 (2010)

    Article  Google Scholar 

  18. Wu, C. P., Chiu, K. H., and Wang, Y. M. RMVT-based meshless collocation and element-free Galerkin methods for the quasi-3D analysis of multilayered composite and FGM plates. Composite Structures, 93, 923–943 (2011)

    Article  Google Scholar 

  19. Chen, W. Q., Bian, Z. G., and Ding, H. J. Three-dimensional analysis of a thick FGM rectangular plate in thermal environment. Journal of Zhejiang University Science A, 4(1), 1–7 (2003)

    Article  Google Scholar 

  20. Vaghefi, R., Baradaran, G. H., and Koohkan, H. Three-dimensional static analysis of thick functionally graded plates by using meshless local Petrov-Galerkin (MLPG) method. Engineering Analysis with Boundary Elements, 34, 564–573 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  21. Chen, W. Q., Lü, C. F., and Bian, Z. G. Elasticity solution for free vibration of laminated beams. Composite Structures, 62(1), 75–82 (2003)

    Article  Google Scholar 

  22. Lü, C. F. State-Space-Based Differential Quadrature Method and Its Applications, Ph. D. dissertation, Zhejiang University (2006)

    Google Scholar 

  23. Lü, C. F., Zhang, Z. C., and Chen, W. Q. Free vibration of generally supported rectangular Kirchhoff plates: state-space-based differential quadrature method. International Journal for Numerical Methods in Engineering, 70(12), 1430–1450 (2007)

    Article  MATH  Google Scholar 

  24. Lü, C. F., Chen, W. Q., Xu, R. Q., and Lim, C. W. Semi-analytical elasticity solutions for bi-directional functionally graded beams. International Journal of Solids and Structures, 45(1), 258–275 (2008)

    Article  MATH  Google Scholar 

  25. Lü, C. F., Chen, W. Q., and Shao, J. W. Semi-analytical three-dimensional elasticity solutions for generally laminated composite plates. European Journal of Mechanics A—Solids, 27(5), 899–917 (2008)

    Article  MATH  Google Scholar 

  26. Mori, T. and Tanaka, K. Average stress in matrix and average elastic energy of materials with misfitting inclusions. Acta Metallurgica, 21(5), 571–574 (1973)

    Article  Google Scholar 

  27. Chen, W. Q. and Ding, H. J. Bending of functionally graded piezoelectric rectangular plates. Acta Mechanica Solida Sinica, 13(4), 312–319 (2000)

    MathSciNet  Google Scholar 

  28. Sherbourne, A. N. and Pandey, M. D. Differential quadrature method in the buckling analysis of beams and composite plates. Computers and Structures, 40(4), 903–913 (1991)

    Article  MATH  Google Scholar 

  29. Hill, R. A self-consistent mechanics of composite materials. Journal of the Mechanics and Physics of Solids, 13(2), 213–222 (1965)

    Article  Google Scholar 

  30. Voigt, W. Ueber die beziehung zwischen den beiden elasticitätsconstanten isotroper körper. Annalen der Physik, 274(12), 573–587 (1889)

    Article  MATH  Google Scholar 

  31. Librescu, L., Oh, S. Y., and Song, O. Thin-walled beams made of functionally graded materials and operating in a high temperature environment: vibration and stability. Journal of Thermal Stresses, 28(6–7), 649–712 (2005)

    Article  Google Scholar 

  32. Huang, C. S., McGee, O. G., and Chang, M. J. Vibrations of cracked rectangular FGM thick plates. Composite Structures, 93(7), 1747–1764 (2011)

    Article  Google Scholar 

  33. Shen, H. S. and Wang, Z. X. Assessment of Voigt and Mori-Tanaka models for vibration analysis of functionally graded plates. Composite Structures, 94(7), 2197–2208 (2012)

    Article  Google Scholar 

  34. Shen, H. S. Nonlinear vibration of shear deformable FGM cylindrical shells surrounded by an elastic medium. Composite Structures, 94(3), 1144–1154 (2012)

    Article  Google Scholar 

  35. Shackelford, J. F. and Alexander, W. CRC Materials Science and Engineering Handbook, CRC Press, Boca Raton (2000)

    Google Scholar 

  36. Nakamura, T., Wang, T., and Sampath, S. Determination of properties of graded materials by inverse analysis and instrumented indentation. Acta Materialia, 48(17), 4293–4306 (2000)

    Article  Google Scholar 

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Authors and Affiliations

  1. College of Civil Engineering and Architecture, Zhejiang University, Hangzhou, 310058, P.R. China

    He Zhang  (张 鹤)

  2. School of Civil Engineering, Zhejiang University City College, Hangzhou, 310058, P.R. China

    Ji-qing Jiang  (蒋吉清)

  3. Department of Civil Engineering, Zhejiang University, Hangzhou, 310058, P.R. China

    Zhi-cheng Zhang  (张治成)

Authors
  1. He Zhang  (张 鹤)
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  2. Ji-qing Jiang  (蒋吉清)
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  3. Zhi-cheng Zhang  (张治成)
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Corresponding author

Correspondence to Zhi-cheng Zhang  (张治成).

Additional information

Project supported by the National Natural Science Foundation of China (Nos. 51108412, 11472244, and 11202186), the National Basic Research Program of China (973 Program) (No. 2013CB035901), the Fundamental Research Funds for the Central Universities (No. 2014QNA4017), and the Zhejiang Provincial Natural Science Foundation of China (No. LR13A020001)

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Zhang, H., Jiang, Jq. & Zhang, Zc. Three-dimensional elasticity solutions for bending of generally supported thick functionally graded plates. Appl. Math. Mech.-Engl. Ed. 35, 1467–1478 (2014). https://doi.org/10.1007/s10483-014-1871-7

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  • DOI: https://doi.org/10.1007/s10483-014-1871-7

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2010 Mathematics Subject Classification

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