Closed-Form Formulation for Bending Analysis of Functionally Graded Thick Plates

Subject Areas : Structural Mechanics

Mahdi Shaban ¹ , Mohammad Javad Khoshgoftar ²

1 - Mechanical Engineering Department, Bu-Ali Sina University, Hamadan, Iran
2 - Department of Mechanical Engineering, Faculty of Engineering, Arak University, Arak, Iran

Received: 2023-02-17 Accepted : 2023-04-05 Published : 2023-06-01

Keywords: Elasticity approach, Exact solution, Functionally graded material, Thick plate, Bending Analysis,

Abstract :

Due to their continuous material variation and eliminating the mismatch stress field in the thickness direction, Functionally Graded Materials (FGMs) have found wide applications in aerospace and mechanical engineering. This article presents closed-form solution for thick functionally graded plate based on three-dimensional elasticity theory. To this end, first, the characteristic equation of FG plate is derived and general closed-form is obtained analytically. Both positive and negative discriminant of characteristic equation is considered and solved. The presented method is validated with finite element results by considering isotropic thick plate. Several parametric studies are carried out to investigate the effect of geometric and material parameters. The aim of this research is to present analytical solution form for thick FG plate and work out the problem of inconsistency for corresponding displacements field. The presented solution can be used to examine accuracy of various plate theories such as first-order, third order shear deformation theories and other equivalent plate theories.

References:

1. Ghorbanpour-Arani A. H., Rastgoo A., Sharafi M. M., Kolahchi R., Ghorbanpour Arani A., 2016, Nonlocal viscoelasticity based vibration of double viscoelastic piezoelectric nanobeam systems, Meccanica 51(1): 25-40.


2. Ghorbanpour-Arani A. H., Rastgoo A., Hafizi Bidgoli A., Kolahchi R., Ghorbanpour Arani A., 2017, Wave propagation of coupled double-DWBNNTs conveying fluid-systems using different nonlocal surface piezoelasticity theories, Mechanics of Advanced Materials and Structures 24(14): 1159-1179.


3. Ghorbanpour Arani A., Jamali M., Ghorbanpour-Arani A. H., Kolahchi R., Mosayyebi M., 2017, Electro-magneto wave propagation analysis of viscoelastic sandwich nanoplates considering surface effects, Proceedings of the Institution of Mechanical Engineers Part C, Journal of Mechanical Engineering Science 231(2): 387-403.


4. Ghorbanpour-Arani A. H., Abdollahian M., Ghorbanpour Arani A., 2020, Nonlinear dynamic analysis of temperature-dependent functionally graded magnetostrictive sandwich nanobeams using different beam theories, Journal of the Brazilian Society of Mechanical Sciences and Engineering42(6): 314.


5. Shaban M., Alipour M. M., 2011, Semi-analytical solution for free vibration of thick functionally graded plates rested on elastic foundation with elastically restrained edge, Acta Mechanica Solida Sinica 24(4): 340-354.


6. Alipour M. M., Shariyat M., Shaban M., 2010, A semi-analytical solution for free vibration of variable thickness two-directional-functionally graded plates on elastic foundations, International Journal of Mechanics and Materials in Design 6(4): 293-304.


7. Shaban M., Mazaheri H., 2021, Bending analysis of five-layer curved functionally graded sandwich panel in magnetic field: closed-form solution, Applied Mathematics and Mechanics 42(2): 251-274.


8. Alipour M. M., Shariyat M., Shaban M., 2020, Free vibration analysis of bidirectional functionally graded conical/cylindrical shells and annular plates on nonlinear elastic foundations, based on a unified differential transform analytical formulation, Journal of Solid Mechanics 12(2): 385-400.


9. Swaminathan K., Naveenkumar D. T., Zenkour A. M., Carrera E., 2015, Stress, vibration and buckling analyses of FGM plates-A state-of-the-art review, Composite Structures 120: 10-31.


10. Pagano N. J., 1970, Exact solutions for rectangular bidirectional composites and sandwich plates, Journal of Composite Materials 4(1): 20-34.


11. Pan E., 2003, Exact solution for functionally graded anisotropic elastic composite laminates, Journal of Composite Materials 37(21): 1903-1920.


12. Li X. Y., Ding H. J., Chen W. Q., 2006, Pure bending of simply supported circular plate of transversely isotropic functionally graded material, Journalof Zheijang University Science 7(8): 1324-1328.


13. Li X. Y., Ding H. J., Chen W. Q., 2008, Elasticity solutions for a transversely isotropic functionally graded circular plate subject to an axisymmetric transverse load qrk, International Journal of Solids and Structures 45(1): 191-210.


14. Zenkour A. M., 2007, Benchmark trigonometric and 3-D elasticity solutions for an exponentially graded thick rectangular plate, Archive of Applied Mechanics 77(4): 197-214.


15. Liu N. W., Sun Y. L., Chen W. Q., Yang B., Zhu J., 2019, 3D elasticity solutions for stress field analysis of FGM circular plates subject to concentrated edge forces and couples, Acta Mechanica 230(8): 2655-2668.


16. Khoshgoftar M. J., Mirzaali M. J., Rahimi G. H., 2015, Thermoelastic analysis of non-uniform pressurized functionally graded cylinder with variable thickness using first order shear deformation theory (FSDT) and perturbation method, Chinese Journal of Mechanical Engineering 28(6): 1149-1156.


17. Khoshgoftar M. J., 2018, Second order shear deformation theory for functionally graded axisymmetric thick shell with variable thickness under non-uniform pressure, Thin-walled Structures Comprise 144: 106286.


18. Omidi M., Loghman A., Arefi M., 2019,The effect of grading index on two-dimensional stress and strain distribution of FG rotating cylinder resting on a friction bed under thermomechanical loading, Journal of Stress Analysis 3(2): 75-82.


19. Kumari P., Singh A., 2019, Recent Advances in Structural Engineering, Springer Singapore.


20. Lu C. F., Lim C. W., Chen W. Q., 2009, Semi-analytical analysis for multi-directional functionally graded plates: 3-D elasticity solutions, International Journal for Numerical Methods in Engineering 79: 25-44.


21. Habibi N., Asadi S., Moradikhah R., 2017, Evaluation of SIF in FGM thick-walled cylindrical vessel, Journal of Stress Analysis 2: 2588-2597.


22. Kardomateas G. A., Phan C. N., 2011, Three-dimensional elasticity solution for sandwich beams/wide plates with orthotropic phases: The negative discriminant case, Journal of Sandwich Structures and Materials 13(6): 641-661.


23. Jones R. M., 1975, Mechanics of Composite Materials,Washington, Scripta Book Co.


24. Alibeigloo A., Alizadeh M., 2015, Static and free vibration analyses of functionally graded sandwich plates using state space differential quadrature method, European Journal of Mechanics, A/Solids 54: 252-266.


25. Filippi M., Carrera E., Zenkour A. M., 2014, Static analyses of FGM beams by various theories and finite elements, Composites Part B: Engineering 72: 1-9.