Document Type : Research Paper


1 Department of Mechanical Engineering

2 Department of Mechanical engineering


This paper presents closed-form formulations of higher order shear deformation theory (HSDT) to analyse the functionally graded plates (FGPs) acted upon a thermo-mechanical load for simply supported (SS) conditions. This theory assumes nullity conditions for transverse stress on bottom and top face of the FGPs. Moreover, considers the influence of both stresses and strains in the axial and transversal direction. In these improvements an accurate parabolic variation is assumed in the thickness direction for transverse shear strains. Therefore, this theory omits the use of correction factor for accurately estimating the shear stress. The physical properties of the FGPs are considered to change along the thickness using a power law. The equilibrium relations and constriants on all edges are attained by considering the virtual work. Numerical evaluations are attained based on Navier’s approach. The exactness and consistency of the developed theory are ascertained with numerical results for deflections and stresses of SS FGPs and it is deemed that numerical solutions for thermo-mechanical load will utilize as a reference in the future.

Graphical Abstract

Thermomechanical behaviour of Functionally Graded Plates with HSDT


Main Subjects

[1]   E.Carrera, S. Brischetto, M. Cinefra, and M, “Soave Effects of thickness stretching in functionally graded plates and shells”, Compos Part B: Eng, Vol. 42, pp.123-133, (2011).


[2]   Mohammad Talha and B N Singh, “Thermo-mechanical deformation behavior of functionally graded rectangular plates subjected to various boundary conditions and loadings”, International Journal of Aerospace and Mechanical Engineering, Vol. 6, No.1, pp. 14-25,(2012).


[3]   J.L. Mantari,  and C. Guedes Soares, “A novel higher-order shear deformation theory with stretching effect for functionally graded plates”, Composites: Part B, Vol. 45, pp. 268–281 (2013).


[4]   B. Sidda Redddy, J. Suresh Kumar, C. Eswara Reddy and K. Vijaya Kumar Reddy, Free Vibration Behaviour of Functionally Graded Plates Using Higher-Order Shear Deformation Theory, Journal of Applied Science and Engineering, Vol. 17, No. 3, pp. 231-241,(2014) DOI: 10.6180/jase.2014.17.3.03


[5]   J.L.Mantari and E.V.Granados, “A refined FSDT for the static analysis of functionally graded sandwich plates”, Thin-Walled Structures, Vol. 90, pp. 150-158, (2015).


[6]   Zhan Zhao,Chuang Feng, YuWang, and JieYang, “Bending and vibration analysis of functionally graded trapezoidal nano-composite plates reinforced with grapheme nano-platelets (GPLs)”, Composite Structures, Vol. 180, No.15, pp. 799-808, (2017).


[7]   J.H.Tian and K.Jiang, “Heat conduction investigation of the functionally graded materials plates with variable gradient parameters under exponential heat source load”, International Journal of Heat and Mass Transfer, Vol.122, pp. 22-30, (2018).


[8]   JoséS.Moita, Victor Franco, CorreiaCristóvão, M.Mota Soares and JoséHerskovits, “Higher-order finite element models for the static linear and nonlinear behaviour of functionally graded material plate-shell structures”, Composite Structures,Vol. 212, No.15, pp.465-475, (2019).


[9]   M.Mohammadi, E.Mohseni and  M.Moeinfar, “Bending, buckling and free vibration analysis of incompressible functionally graded plates using higher order shear and normal deformable plate theory”, Applied Mathematical Modelling, Vol. 69, pp. 47-62 (2019).


[10]  M. E. Golmakani, and M. Kadkhodayan,  “Nonlinear bending analysis of annular FGM plates using higher order shear deformation plate theories”, Composite struct, Vol. 93, pp. 973-982, (2011).


[11]  Hiroyuki Matsunaga, “Free vibration and stability of functionally graded plates according to a 2-D higher-order deformation theory”, Composite struct,  Vol. 82, pp. 499-512, (2008).


[12]  Hiroyuki Matsunaga, “Stress analysis of functionally graded plates subjected to thermal and mechanical loadings,  Composite struct, Vol.87, pp. 344-357, (2009).


[13]  Song Xiang and Gui-wen Kang, “A nth-order shear deformation theory for the bending analysis on the functionally graded plates”,  European Journal of Mechanics A/Solids, Vol. 37, pp. 336-343, (2013).


[14]  Jyothula Suresh Kumar, Bathini Sidda Reddy, C. Eswara Reddy, and Kondakkagari Vijaya Kumar Reddy, “Nonlinear Thermal Analysis of Functionally Graded Plates Using Higher Order Theory”, Innovative Systems Design and Engineering, Vol. 2 No.5, pp. 1-13, (2011).


[15]  B. Sidda Reddy, J. Suresh Kumar, C. Eswara Reddy, and K. Vijaya Kumar Reddy,  “Buckling Analysis of Functionally Graded Material Plates Using Higher Order Shear Deformation Theory”, Journal of Composites,  Vol. 2013, Article ID 808764, pp.12 pages, (2013).


[16]  J. Suresh Kumar, B. Sidda Reddy, C. Eswara Reddy. “Nonlinear bending analysis of functionally graded plates using higher order theory”, International Journal of Engineering Science and Technology (IJEST), 3(4), 3010-3022, (2011).


[17]  J.R Cho and J.T. Oden, “Functionally gradient material: a parametric study on thermal stress characteristics using Galerkin scheme”, Computer methods in applied mechanics and engineering, Vol. 188, pp. 103-115, (2000).


[18]  Dan Sun and Song-Nan Luo “Wave propagation and transient response of a functionally graded material plate under a point impact load in thermal environments”, Applied Mathematical Modelling,  Vol. 36, No.1, pp. 444–462, (2012).


[19]  S. Jafari Mehrabadi, and  B. Sobhani Aragh, “On the thermal analysis of 2-D temperature-dependent functionally graded open cylindrical shell”, Composite Structures,Vol. 96, pp. 773–785, (2013).


[20]  A.Wagih, M.A. Attia, A.A.Abdel Rahman, K. Bendine, and T.A.Sebaey, “On the indentation of elastoplastic functionally graded materials”, Mechanics of Materials, Vol. 129, pp. 169-188, (2019).


[21]  Slimane Merdaci, and HakimaBelghoul, “High-order shear theory for static analysis of functionally graded plates with porosities, Comptes Rendus Mécanique, Vol. 347, No. 3, pp. 207-217, (2019).


[22]  Amina Attia, Abdelmoumen Anis Bousahla, Abdelouahed Tounsi, S.R. Mahmoud and Afaf S. Alwabl, “A refined four variable plate theory for thermoelastic analysis of FGM plates resting on variable elastic foundations”, Structural Engineering and Mechanics,   Vol. 65, No. 4,pp.453-464,(2018).


[23]  Asmaa Fahsi, Abdelouahed Tounsi, Habib Hebali, Abdelbaki Chikh, E.A. Adda Bedia and S.R. Mahmoud, “A four variable refined nth-order shear deformation theory for mechanical and thermal buckling analysis of functionally graded plates”, Geomechanics and Engineering,   Vol. 13, No. 3, pp. 385-410, (2017).

[24]  Abdelbaki Chikh, Abdelouahed Tounsi, Habib Hebali and S. R. Mahmoud, “Thermal buckling analysis of cross-ply laminated plates using a simplified HSDT”, Smart Structures and Systems ,  Vol. 19, No. 3, pp. 289-297, (2017).

[25]  Fouzia El-Haina, Ahmed Bakora, Abdelmoumen Anis Bousahla, Abdelouahed Tounsi and S.R. Mahmoud, “A simple analytical approach for thermal buckling of thick functionally graded sandwich plates”, Structural Engineering and Mechanics,   Vol. 63, No. 5, pp. 585-595, (2017).


[26]   Menasria, Abderrahmane, Bouhadra, Abdelhakim, Tounsi, Abdelouahed, Bousahla, Abdelmoumen Anis, Mahmoud, S.R, “A new and simple HSDT for thermal stability analysis of FG sandwich plates”, Vo. 25, No. 2, pp.157-175 (2017).


[27]  Youcef Beldjelili, Abdelouahed Tounsi and S.R. Mahmoud, “Hygro-thermo-mechanical bending of S-FGM plates resting on variable elastic foundations using a four-variable trigonometric plate theory”, Smart Structures and Systems,   Vol. 18, No. 4, pp. 755-786, (2016).

[28]  Sabrina Boutaleb, Kouider Halim Benrahou, Ahmed Bakora, Ali Algarni, Abdelmoumen Anis Bousahla, Abdelouahed Tounsi, Abdeldjebbar Tounsi and S.R. Mahmoud, “Dynamic analysis of nanosize FG rectangular plates based on simple nonlocal quasi 3D HSDT”, Advances in Nano Research,   Vol. 7, No. 3, pp.189-206, (2019). 


[29]   Zoulikha Boukhlif, Mohammed Bouremana, Fouad Bourada, Abdelmoumen Anis Bousahla, Mohamed Bourada, Abdelouahed Tounsi and Mohammed A. Al-Osta, “A simple quasi-3D HSDT for the dynamics analysis of FG thick plate on elastic foundation”, Steel and Composite Structures,   Vol. 31, No. 5, pp.503-516 (2019).


[30]  Soumia Bouanati, Kouider Halim Benrahou, Hassen Ait Atmane, Sihame Ait Yahia, Fabrice Bernard, Abdelouahed Tounsi and E.A. Adda Bedia, “Investigation of wave propagation in anisotropic plates via quasi 3D HSDT”, Geomechanics and Engineering,   Vol. 18, No. 1, pp. 85-96 (2019). 


[31]  Hassen, Ait Atmane & Tounsi, Abdelouahed & Bernard, Fabrice, “Effect of thickness stretching and porosity on mechanical response of a functionally graded beams resting on elastic foundations”. International Journal of Mechanics and Materials in Design. Vol. 13, No. 1, pp. 71-84, ((2017). 


[32]  Abdelkarim Benahmed, Mohammed Sid Ahmed Houari, Samir Benyoucef, Khalil Belakhdar and Abdelouahed Tounsi, “A novel quasi-3D hyperbolic shear deformation theory for functionally graded thick rectangular plates on elastic foundation”, Geomechanics and Engineering,   Vol. 12, No. 1, pp. 9-34, (2017).

[33]  Behrouz Karami, Maziar Janghorban, Davood Shahsavari and Abdelouahed Tounsi, “A size-dependent quasi-3D model for wave dispersion analysis of FG nanoplates”, Steel and Composite Structures,   Vol. 28, No. 1, pp. 99-110, (2018).

[34]  Fatima Zohra, Zaoui & Ouinas, Djamel & Tounsi, Abdelouahed, “New 2D and quasi-3D shear deformation theories for free vibration of functionally graded plates on elastic foundations”, Composites Part B Engineering. Vol.159, pp. 231-247 ((2019). 


[35]  Abdelhakim Bouhadra, Abdelouahed Tounsi, Abdelmoumen Anis Bousahla, Samir Benyoucef and S.R. Mahmoud, “Improved HSDT accounting for effect of thickness stretching in advanced composite plates”, Structural Engineering and Mechanics,   Vol. 66, No. 1, pp. 61-73 (2018), 


[36]  Abderahman Younsi, Abdelouahed Tounsi, Fatima Zohra Zaoui, Abdelmoumen Anis Bousahla and S.R. Mahmoud, “Novel quasi-3D and 2D shear deformation theories for bending and free vibration analysis of FGM plates”, Geomechanics and Engineering,   Vol. 14, No. 6, pp. 519-532, (2018).


[37]  M. Abualnour, Houari, M.S.A., Tounsi, A., Bedia, E.A.A., Mahmoud, S.R., “A novel quasi-3D trigonometric plate theory for free vibration analysis of advanced composite plates”, Composite Structures (2017), DOI: