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PROPER GENERALIZED DECOMPOSITION METHOD FOR STRESS ANALYSIS OF FUNCTIONALLY GRADED MATERIALS

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Abstract

Proper generalized decomposition (PGD), an advanced numerical simulation method, has been successfully used as a model reduction method in many fields. In this study, functionally graded materials are analyzed using the Airy stress function method, and a compatible equation representing the stress function is obtained. PGD is used to solve a compatible equation, which is a complex high-order partial differential equation, by decomposing it into two one-dimensional problems in space. The solution of the complex high-order partial differential equation is then obtained by solving the two one-dimensional problems using Chebyshev’s method. The accuracy of the proposed method is verified by comparing the PGD solution with the analytical solution. Numerical examples show that highly accurate results can be obtained by using several iterative modes. Therefore, the PGD method is an effective numerical technique for the stress analysis of functionally graded materials. This paper provides a theoretical basis for future research on the stress concentrations of irregular shape and rough surface.

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REFERENCES

  1. S. Abrate, “Functionally graded plates behave like homogeneous plates,” Compos. Part B: Eng. 39 (1), 151–158 (2008).

    Article  Google Scholar 

  2. A. Gupta and M. Talha, “Recent development in modeling and analysis of functionally graded materials and structures,” Prog. Aerosp. Sci. 79, 1–14 (2015).

    Article  Google Scholar 

  3. D. K. Jha, T. Kant and R. K. Singh, “A critical review of recent research on functionally graded plates,” Compos. Struct. 96, 833–849 (2013).

    Article  Google Scholar 

  4. B. Victor and L. W. Byrd, “Modeling and analysis of functionally graded materials and structures,” Appl. Mech. Rev. 60, 195–216 (2007).

    Article  ADS  Google Scholar 

  5. M. Kashtalyan, “Three-dimensional elasticity solution for bending of functionally graded rectangular plates,” Eur. J. Mech. A Solid 23 (5), 853–864 (2004).

  6. H. A. Atmane, E. A. Bedia, M. Bouazza, et al., “On the thermal buckling of simply supported rectangular plates made of a sigmoid functionally graded Al/Al2O3 based material,” Mech. Solids 51, 177–187 (2016).

    Article  ADS  Google Scholar 

  7. S. Suresh and A. Mortensen, Fundamentals of Functionally Graded Materials, 2nd ed. (Cambridge University, London, 1998).

    Google Scholar 

  8. B. V. Sankar, “An elasticity solution for functionally graded beams,” Compos. Sci. Technol. 61 (5), 689–696 (2001).

    Article  Google Scholar 

  9. B. V. Sankar and J. T. Tzeng, “Thermal stresses in functionally graded beams,” AIAA J. 40 (6), 1228–1232 (2002). https://doi.org/10.2514/2.1775

    Article  ADS  Google Scholar 

  10. L. Della Croce and P. Venini, “Finite elements for functionally graded Reissner–Mindlin plates,” Comput. Methods Appl. Mech. Eng. 193 (9-11), 705–725 (2003).

    Article  MathSciNet  Google Scholar 

  11. S. H. Chi and Y. L. Chung, “Mechanical behavior of functionally graded material plates under transverse load-part I: analysis,” Int. J. Solids Struct. 43 (13), 3657–3674 (2006).

    Article  Google Scholar 

  12. S. Brischetto, “Exact three-dimensional static analysis of single- and multi-layered plates and shells,” Compos. Part B: Eng. 119 (10), 230–252 (2017).

    Article  Google Scholar 

  13. Y. Y. Lu, J. T. Shi, G. J. Nie, et al., “An elasticity solution for transversely isotropic, functionally graded circular plates,” Mech. Adv. Mater. Struc. 23 (4), 451–457 (2016).

    Article  Google Scholar 

  14. G. N. Praveen and J. N. Reddy, “Nonlinear transient thermoelastic analysis of functionally graded ceramic-metal plates,” Int. J. Solids Struct. 35 (33),4457–4476 (1998).

    Article  Google Scholar 

  15. S. S. Hosseini, H. Bayesteh, and S. Mohammadi, “Thermo-mechanical XFEM crack propagation analysis of functionally graded materials,” A-Structural Mat. 561, 285–302 (2013).

    Google Scholar 

  16. S. Bhattacharya and K. Sharma, “Fatigue crack growth simulations of FGM plate under cyclic thermal load by XFEM,” Proc. Eng. 86, 727–731 (2014).

    Article  Google Scholar 

  17. K. M. Liew, X. Zhao, and A. J. Ferreira, “A review of meshless methods for laminated and functionally graded plates and shells,” Compos. Struct. 93 (8), 2031–2041 (2011).

    Article  Google Scholar 

  18. A. J. Ferreira, R. C. Batra, C. M. Roque, et al., “Natural frequencies of functionally graded plates by a meshless method,” Compos. Struct. 75 (1), 593–600 (2006).

    Article  Google Scholar 

  19. G. Kerschen, J. C. Golinval, A. F. Vakakis, et al., “The method of proper orthogonal decomposition for dynamical characterization and order reduction of mechanical systems: an overview,” Nonlin. Dyn. 41 (1-3), 147–169 (2005).

    Article  MathSciNet  Google Scholar 

  20. B. J. Zheng, Y. Liang, X. W. Gao, et al., “Analysis for dynamic response of functionally graded materials using pod based reduced order model,” J. Theor. App. Mech. Pol. 50 (4), 787–797 (2018).

    Google Scholar 

  21. F. Chinesta, A. Ammar, and E. Cueto, “Recent advances and new challenges in the use of the proper generalized decomposition for solving multidimensional models,” Arch. Comput. Method Eng. 17 (4), 327–350 (2010).

    Article  MathSciNet  Google Scholar 

  22. F. Chinesta, R. Keunings, and A. Leygue, The Proper Generalized Decomposition for Advanced Numerical Simulations: A Primer (Springer, New York, 2013).

    MATH  Google Scholar 

  23. B. Bognet, F. Bordeu, F. Chinesta, et al., “Advanced simulation of models defined in plate geometries: 3D solutions with 2D computational complexity,” Comput. Method Appl. M. 201, 1–12 (2011).

    MATH  Google Scholar 

  24. E. Giner, B. Bognet, J. Rodenas, et al., “The proper generalized decomposition (PGD) as a numerical procedure to solve 3D cracked plates in linear elastic fracture mechanics,” Int. J. Solids Struct. 50 (10),1710–1720 (2013).

    Article  Google Scholar 

  25. F. Chinesta, P. Ladeveze and E. Cueto, “A short review on model order reduction based on proper generalized decomposition,” Arch. Comput. Method Eng. 18 (4), 395–404 (2011).

    Article  Google Scholar 

  26. C. Quesada, G. T. Xu, D. González, et al., “Un método de descomposición propia generalizada para operadores diferenciales de alto orden,” Rev. Int. Metod Numer. Calc. Diseno Ing. 31 (3), 188–197 (2015). https://doi.org/10.1016/j.rimni.2014.09.001

    Article  Google Scholar 

  27. R. Ibañez,D. Borzacchiello, J. V. Aguado, et al., “Data-driven non-linear elasticity: constitutive manifold construction and problem discretization,” Comput. Mech. 60 (5), 813–826 (2017).

    Article  MathSciNet  Google Scholar 

  28. F. Delale and F. Erdogan, “The crack problem for a non-homogeneous plane,” ASME J. Appl. Mech. 50 (3), 609–614 (1983).

    Article  ADS  Google Scholar 

  29. P. Chu, X. F. Li, J. X. Wu, et al., “Two-dimensional elasticity solution of elastic strips and beams made of functionally graded materials under tension and bending,” Acta. Mech. 226 (7), 2235–2253 (2015).

    Article  MathSciNet  Google Scholar 

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Funding

The work was supported by the National Natural Science Foundation of China (Nos. 11702252 and U1804254) and the Key Teachers Program for the University of Henan Provence (2019GGJS005).

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Authors and Affiliations

  1. School of Mechanical and Power Engineering, Zhengzhou University, 450001, Zhengzhou, Henan, China

    Guang Tao Xu & Shao Kang Wu

  2. Henan Key Engineering Laboratory for Anti-fatigue Manufacturing Technology, Zhengzhou University, 450001, Zhengzhou, Henan, China

    Guang Tao Xu & Shao Kang Wu

Authors
  1. Guang Tao Xu
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  2. Shao Kang Wu
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Correspondence to Guang Tao Xu.

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Xu, G.T., Wu, S.K. PROPER GENERALIZED DECOMPOSITION METHOD FOR STRESS ANALYSIS OF FUNCTIONALLY GRADED MATERIALS. Mech. Solids 56, 430–442 (2021). https://doi.org/10.3103/S0025654421030146

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  • DOI: https://doi.org/10.3103/S0025654421030146

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