Static Analysis of Functionally Graded Piezoelectric Plates under Electro-thermo-mechanical Loading Using a Meshfree Method Based on RPIM

Document Type: Original Article


Mechanical Engineering Faculty, Sahand University of Technology, Tabriz, Iran.



In this paper, the static response of functionally graded piezoelectric plates under mechanical, electrical, and thermal loads is studied using a meshless method. The Radial Point Interpolation Method (RPIM) is used to create the shape function to approximate field variables. Given that RPIM shape functions pass Kronecker delta condition, boundary conditions can be applied directly. The First-order Shear Deformation Plate Theory (FSDT) is used to model the behavior of the plate. Power law distribution through the thickness is considered for all of mechanical, thermal, and piezoelectric properties. Effective parameters on deflection and stresses of Functionally Graded Piezoelectric
Material (FGPM), including different electrical and mechanical loads, thermal loads, thickness, and different boundary conditions are studied. In this paper, the effect of power law index on the deflection and stresses of the functionally graded piezoelectric plate under external loads is investigated and different results are obtained in each case of mechanical, electrical, and thermal loading. By analyzing the results of this paper, the effective structure design and sensor/actuator behavior of the plate subjected to thermal and electrical loading could be obtained.


[1] J. Belinha, A.L. Araújo, A.J.M. Ferreira, L.M.J.S. Dinis, R.M. Natal Jorge, The analysis of laminated
plates using distinct advanced discretization meshless techniques, Compos. Struct., 143 (2016) 165-179.
[2] P. Zhu, K.M. Liew, Free vibration analysis of moderately thick functionally graded plates by local
Kriging meshless method, Compos. Struct., 93(11) (2011) 2925-2944.
[3] P.J.C. Branco, J.A. Dente, On the electromechanics of a piezoelectric transducer using a bimorph cantilever undergoing asymmetric sensing and actuation, Smart Mater. Struct., 13(4) (2004) 631-642.
[4] H. Gu, Y. Moslehy, D. Sanders, G. Song, Y.L. Mo, Multi-functional smart aggregate-based structural
health monitoring of circular reinforced concrete columns subjected to seismic excitations, Smart
Mater. Struct., 19(6) (2010) 065026.
[5] K.M. Liew, X.Q. He, T.Y. Ng, S. Sivashanker, Active control of FGM plates subjected to a temperature gradient: Modelling via finite element method based on FSDT, Int. J. Numer. Methods Eng., 52(11) (2001) 1253-1271.
[6] X. Guo, D. Fang, A.K. Soh, H.C. KIM, J.J. Lee, Analysis of piezoelectric ceramic multilayer actuators based on an electro-mechanical coupled meshless method, Acta Mech. Sin., 22(1) (2006) 34-39.
[7] X.H.Wu, Y.P. Shen, X.G. Tian, A high order theory for functionally graded piezoelectric shells, Int. J. Numer. Methods Eng., 39(20) (2002) 5325-5344.
[8] L. Qirong, L. Zhengxing, J. Zhanli, A close-form solution to simply supported piezoelectric beams under uniform exterior pressure, Appl. Math. Mech., 21(6) (2000) 681-690.
[9] L. Qi-rong, L. Zheng-xing, W. Zong-li, Analysis of beams with piezoelectric actuators, Appl. Math.
Mech., 22(9) (2001) 1074-1081.
[10] Z. Lin-nan, S. Zhi-fei, Analytical solution of a simply supported piezoelectric beam subjected to a
uniformly distributed loading, Appl. Math. Mech., 24(10) (2003) 1215-1223.
[11] A.J.M. Ferreira, R.C. Batra, C.M.C. Roque, L.F. Qian, P.A.L.S. Martins, Static analysis of functionally graded plates using third-order shear deformation theory and a meshless method, Compos. Struct., 69(4) (2005) 449-457.
[12] J. Yang, H.J. Xiang, Thermo-electro-mechanical characteristics of functionally graded piezoelectric
actuators, Smart Mater. Struct., 16(3) (2007) 784-797.
[13] X.L. Chen, Z.Y. Zhao, K.M. Liew, Stability of piezoelectric FGM rectangular plates subjected to non-uniformly distributed load, heat and voltage, Adv. Eng. Software, 39(2) (2008) 121-131.
[14] Z. Yan, M. Zaman, L. Jiang, Thermo-electromechanical analysis of a curved functionally graded
piezoelectric actuator with sandwich structure, Materials, 4(12) (2011) 2151-2170.
[15] A. Komeili, A.H. Akbarzadeh, A. Doroushi, M.R. Eslami, Static analysis of functionally graded piezoelectric beams under thermo-electro-mechanical loads, Adv. Eng. Software, 3 (2011) 153731.
[16] J. Singh, K.K. Shukla, Nonlinear flexural analysis of functionally graded plates under different loadings using RBF based meshless method, Eng. Anal. Boundary Elem., 36(12) (2012) 1819-1827.
[17] P. Staňák, V. Sládek, J. Sládek, S. Krahuleca, L. Sátor, Application of patch test in meshless analysis of continuously non-homogeneous piezoelectric circular plate, Appl. Comput. Anal., 7(1) (2013) 65-76.
[18] J. Sládek, V. Sládek„ P. Stanak, C. Zhang, M. Wünsche, Analysis of the bending of circular piezoelectric plates with functionally graded material properties by a MLPG method, Eng. Struct., 47 (2013) 81-89.
[19] P. Zhu, L.W. Zhang, K.M. Liew, Geometrically nonlinear thermomechanical analysis of moderately thick functionally graded plates using a local Petrov–Galerkin approach with moving Kriging interpolation, Compos. Struct., 107 (2014) 298-314.
[20] L. Sator, V. Sladek, J. Sladek, Coupling effects in elastic analysis of FGM composite plates by meshfree methods, Compos. Struct., 115 (2014) 100-110.
[21] L. Sator, J. Sládek, V. Sládek, D.L. Young, Elastodynamics of FGM plates by mesh-free method, Compos. Struct., 140 (2016) 309-322.
[22] P. Staňák, J. Sládek, V. Sládek, S. Krahulec, Numerical MLPG Analysis of Piezoelectric Sensor in
Structures, Slovak J. Civ. Eng., 22(2) (2014) 15-20.
[23] P. Stanak, J. Sládek, V. Sládek„ A. Tadeu, Three-Dimensional Meshless Modelling of Functionally Graded Piezoelectric Sensor, In: Březina T., Jabloński R. (eds) Mechatronics 2013, Cham: Springer International Publishing, (2014).
[24] S. Li, L. Yao, S. Yi, W. Wang, A meshless radial basis function based on partition of unity method
for piezoelectric structures, Math. Prob. Eng., 2016 (2016) 7632176.
[25] S. Mikaeeli, B. Behjat, Three-dimensional analysis of thick functionally graded piezoelectric plate using EFG method, Compos. Struct., 154 (2016) 591-599.
[26] M.R. Barati, A.M. Zenkour, Electro-thermoelastic vibration of plates made of porous functionally
graded piezoelectric materials under various boundary conditions, J. Vib. Control, 24(10) (2016) 1910-1926.
[27] M.R. Barati, H. Shahverdi, A.M. Zenkour, Electro-mechanical vibration of smart piezoelectric FG plates with porosities according to a refined four-variable theory, Mech. Adv. Mater. Struct., 24(12) (2017) 987-998.
[28] P. Phung Van, T. Cuong Le, H. Nguyen-Xuan, M. Abdel Wahab, Nonlinear transient isogeometric
analysis of FG-CNTRC nanoplates in thermal environments, Compos. Struct., 201 (2018) 882-892.
[29] A.M. Zenkour, M.H. Aljadani, Thermo-electrical buckling response of actuated functionally graded
piezoelectric nanoscale plates, Results Phys., 13 (2019) 102192.
[30] S.Q. Zhang, Y.S. Gao, G.Z. Zhao, Y.J. Yu, M.Chen, X.F. Wang, Geometrically nonlinear analysis of CNT-reinforced functionally graded composite plates integrated with piezoelectric layers, Compos. Struct., 234 (2020) 111694.
[31] H.H. Phan-Dao, C.H. Thai, J. Lee, H. NguyenXuan, Analysis of laminated composite and sandwich plate structures using generalized layerwise HSDT and improved meshfree radial point interpolation method, Aerosp. Sci. Technol., 58 (2016) 641-660.
[32] G.R. Liu, Mesh Free Methods: Moving Beyond The Finite Element Method, CRC Press, New York, (2009).
[33] G.R. Liu, Y.T. Gu, An Introduction to Meshfree Methods and Their Programming, Springer, (2005).
[34] M.A. Golberg, C.S. Chen, H. Bowman, Some recent results and proposals for the use of radial basis
functions in the BEM, Eng. Anal. Boundary Elem., 23(4) (1999) 285-296.
[35] B. Yildirim, S. Dag, F. Erdogan, Three dimensional fracture analysis of FGM coatings under thermomechanical loading, Int. J. Fract., 132(4) (2005) 371-397.
[36] J. Reddy, Analysis of functionally graded plates, Int. J. Numer. Methods Eng., 47(1-3) (2000) 663-684.
[37] T.Y. Ng, K.Y. Lam, K.M. Liew, J.N. Reddy, Dynamic stability analysis of functionally graded cylindrical shells under periodic axial loading, Int. J. Numer. Methods Eng., 38(8) (2001) 1295-1309.
[38] J. Yang, H.J. Xiang, Thermo-electro-mechanical characteristics of functionally graded piezoelectric
actuators, Smart Mater. Struct., 16(3) (2007) 784-797.
[39] J.N. Reddy, On laminated composite plates withintegrated sensors and actuators, Eng. Struct., 21(7) (1999) 568-593.
[40] M.H. Babaei, G. Akhras, Graded piezoelectric cylinders subjected to high electric fields and comparison of their frequency response with piezoelectric plates, Meccanica, 49(6) (2014) 1527-1538.
[41] A. Armin, I. Shafieenejad, N. Moallemi, A.B. Novinzadeh, Comparison between hpm and finite
fourier solution in static analysis of fgpm beam under thermal load, J. Theor. Appl. Mech., 48(1) (2010) 173-189.
[42] K. Takagi, J.F. Li, S. Yokoyama, R. Watanabe, A. Almajid, M.Taya, Design and fabrication of functionally graded PZT/Pt piezoelectric bimorph actuator, Sci. Technol. Adv. Mater., 3(2) (2002) 217-224.
[43] M. Taya, A.A. Almajid, M. Dunn, H. Takahashi, Design of bimorph piezo-composite actuators with
functionally graded microstructure, Sens. Actuators A, 107(3) (2003) 248-260.
[44] D. Varelis, D.A. Saravanos, Non-linear coupled multi-field mechanics and finite element for active
multi-stable thermal piezoelectric shells, Int. J. Numer. Methods Eng., 76(1) (2008) 84-107.
[45] S. Cen, A.K. Soh, Y.Q. Long, Z.H.Yao, A new 4-node quadrilateral FE model with variable electrical degrees of freedom for the analysis of piezoelectric laminated composite plates, Compos. Struct., 58(4) (2002) 583-599.