Analysis of Stability and Vibrations of Porous Power and Sigmoid Functionally Graded Sandwich Plates by the R-Functions Method

Authors

Lidiya V. Kurpa, National Technical University “Kharkiv Polytechnic Institute” (2, Kyrpychova str., Kharkiv, 61002, Ukraine), e-mail: kurpalidia@gmail.com, ORCID:0000-0002-4459-8249

Tetyana V. Shmatko, National Technical University “Kharkiv Polytechnic Institute” (2, Kyrpychova str., Kharkiv, 61002, Ukraine), e-mail: ktv_ua@yahoo.com, ORCID: 0000-0003-3386-8343

Anna B. Linnik, National Technical University “Kharkiv Polytechnic Institute” (2, Kyrpychova str., Kharkiv, 61002, Ukraine), e-mail: linnik2105@gmail.com, ORCID: 000-0003-4227-3210

Abstract

In this paper, the R-functions method is used for the first time to study the stability and vibrations of porous functionally graded (FG) sandwich plates with a complex geometric shape. It is assumed that the face layers of the plate are made of functionally graded materials, and the middle layer is isotropic, namely ceramic. Differential equations of motion were obtained using the first-order shear deformation theory with a given shear coefficient (FSDT). Two models of porosity distribution according to the power (P-law) and sigmoid (S-law) laws were studied. Analytical expressions for calculating the effective mechanical characteristics of functionally graded materials with even and uneven porosity distribution were obtained. Proposed approach takes into account the fact that the subcritical state of the plate can be heterogeneous, and therefore, first of all, the stresses in the middle plane of the plate are determined, and then the eigenvalue problem is solved in order to find the critical load. To determine the critical load and plate frequencies, the Ritz method combined with the R-functions theory was used. Developed algorithms and software are tested on case studies and compared with known results obtained by another methods. A number of problems of stability and vibrations of the porous functionally graded sandwich plates with a complex geometric shape for various layer arrangement schemes, various boundary conditions and laws of porosity distribution have been solved.

Keywords: stability, vibrations, sandwich plates, porosity, functionally graded material, R-functions method, Ritz method.

Full text: Download in PDF

References

1. Thai, H.-T. & Kim, S.-E. (2015). A review of theories for the modeling and analysis of functionally graded plates and shells. Composite Structures, vol. 128, pp. 70–86. https://doi.org/10.1016/j.compstruct.2015.03.010.
2. Swaminathan, K., Naveenkumar, D. T., Zenkour, A. M, & Carrera, E. (2015). Stress, vibration and buckling analyses of FGV plates – A state-of-the-art review. Composite Structures, vol. 120, pp. 10–31. https://doi.org/10.1016/j.compstruct.2014.09.070.
3. Kumar, Y. (2017). The Rayleigh–Ritz method for linear dynamic, static and buckling behavior of beams, shells and plates: A literature review. Journal of Vibration and Control, vol. 24, iss. 7, pp. 1205–1227. https://doi.org/10.1177/1077546317694724.
4. Elmeiche, N., Tounsi, A., Ziane, N., Mechab, I., & El Abbes, A. B. (2011). A new hyperbolic shear deformation theory for buckling and vibration of functionally graded sandwich plate. International Journal of Mechanical Sciences, vol. 53, iss. 4, pp. 237–247. https://doi.org/10.1016/j.ijmecsci.2011.01.004.
5. Neves, A. M. A., Ferreira, A. J. M., Carrera, E., Cinefra, M., Jorge, R. M. N., & Soares, C. M. M. (2012). Buckling analysis of sandwich plates with functionally graded skins using a new quasi-3D hyperbolic sine shear deformation theory and collocation with radial basis functions. Journal of Applied Mathematics and Mechanics, vol. 92, iss. 9, pp. 749–766. https://doi.org/10.1002/zamm.201100186.
6. Yaghoobi, H. & Yaghoobi, P. (2013). Buckling analysis of sandwich plates with FGM face sheets resting on elastic foundation with various boundary conditions: An analytical approach. Meccanica, vol. 48, pp. 2019–2035. https://doi.org/10.1007/s11012-013-9720-0.
7. Singh, S. J. & Harsha, S. P. (2019). Exact solution for free vibration and buckling of sandwich S-FGM plates on Pasternak elastic foundation with various boundary conditions. International Journal of Structural Stability and Dynamics, vol. 19, no. 3, paper 1950028. https://doi.org/10.1142/S0219455419500287.
8. Li, D., Zhu, H., & Gong, X. (2021). Buckling analysis of functionally graded sandwich plates under both mechanical and thermal loads. Materials, vol. 14, iss. 23, paper 7194. https://doi.org/10.3390/ma14237194.
9. Zencour, A. M. (2005). A comprehensive analysis of functionally graded sandwich plates: Part 2 – Buckling and free vibration. International Journal of Solids and Structures, vol. 42, iss. 18–19, pp. 5243–5258. https://doi.org/10.1016/j.ijsolstr.2005.02.016.
10. Daikh, A. A. & Zenkour, A. M. (2019). Free vibration and buckling of porous power-law and sigmoid functionally graded sandwich plates using a simple higher-order shear deformation theory. Materials Research Express, vol. 6, no. 11, paper 115707. https://doi.org/10.1088/2053-1591/ab48a9.
11. Le, C. I, Tran, Q. D., Pham, V. N., & Nguyen, D. K. (2021). Free vibration and buckling of bidirectional functionally graded sandwich plates using an efficient Q9 element. Vietnam Journal of Mechanics, vol. 43, no. 3, pp. 277–295. https://doi.org/10.15625/0866-7136/15981.
12. Kurpa, L., Mazur, O., & Tkachenko, V. (2013). Dynamical stability and parametrical vibrations of the laminated plates with complex shape. Latin American Journal of Solids and Structures, vol. 10, iss. 1, pp. 175–188. https://doi.org/10.1590/S1679-78252013000100017.
13. Awrejcewicz, J., Kurpa, L., & Mazur, O. (2016). Dynamical instability of laminated plates with external cutout. International Journal of Non-Linear Mechanics, vol. 81, pp. 103–114. https://doi.org/10.1016/j.ijnonlinmec.2016.01.002.
14. Awrejcewicz, J., Kurpa, L., & Shmatko, T. (2017). Analysis of geometrically nonlinear vibrations of functionally graded shallow shells of a complex shape. Latin American Journal of Solids and Structures, vol. 14, iss. 9, pp. 1648–1668. https://doi.org/10.1590/1679-78253817.
15. Kurpa, L. V. & Shmatko, T. V. (2020). Investigation of free vibrations and stability of functionally graded three-layer plates by using the R-functions theory and variational methods. Journal of Mathematical Sciences, vol. 249, no. 3, pp. 496–520. https://doi.org/10.1007/s10958-020-04955-2.
16. Kurpa, L. V. & Shmatko, T. V. (2020). Buckling and free vibration analysis of functionally graded sandwich plates and shallow shells by the Ritz method and the R-functions theory. Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science, vol. 235, iss. 20, pp. 4582–-4593. https://doi.org/10.1177/0954406220936304.
17. Shen, H.-S. (2011). Functionally graded materials. Nonlinear analysis of plates and shells. USA, Boca Raton: CRC Press, 280 p. https://doi.org/10.1201/9781420092578.
18. Rvachev, V. L. (1982). Teoriya R-funktsiy i nekotoryye yeye prilozheniya [The R-functions theory and some of its applications]. Kyiv: Naukova dumka, 552 p. (in Russian).
19. Lekhnitskiy, S. G. (1957). Anizotropnyye plastinki [Anisotropic plates]. Moscow: Gostekhizdat, 464 p. (in Russian).

Received 08 August 2023

Published 30 December 2023