Multifield vibration analysis of a porosity-tailored bidirectional functionally graded piezoelectric nano-plate on variable elastic foundations under hygro-thermoelectric effects

Abstract

This study investigates the multifield vibration behavior of a porosity-dependent bidirectional functionally graded piezoelectric nano-plate (FGPN) subjected to hygrothermal and thermoelectric loading. The material composition is defined by sigmoid and power-law distributions along both transverse and axial directions, accommodating even, uneven, and symmetrically centered porosity patterns. The model incorporates nonclassical elasticity theory and von Kármán nonlinear strains, with the governing equations formulated using a modified first-order shear deformation theory and derived through the energy principle. A higher-order finite element formulation, coupled with a modified Newton–Raphson procedure, ensures robust computational accuracy, validated through convergence tests. The analysis delves into the influence of porosity distribution, bidirectional material variations, non-uniform thickness, thickness ratios, variable elastic foundations, and boundary conditions on vibrational behavior. Additionally, the study explores the interplay of hygrothermal and electrical loading conditions in diverse configurations. The findings highlight the pivotal role of bidirectional material gradation in shaping the vibrational response of porous FGPN structures, offering valuable insights for the design of nano-plates in hygrothermal and thermoelectric applications.

Graphical abstract

Keywords

Porosity / Bidirectional exponent / nano-plate / functionally graded piezoelectric plate / FEM / Elastic foundations

References

[1] Mason W P . Piezoelectricity, its history and applications. Journal of the Acoustical Society of America, 1981, 70(6): 1561–1566

[2] Komijani M , Kiani Y , Esfahani S E , Eslami M R . Vibration of thermo-electrically post-buckled rectangular functionally graded piezoelectric beams. Composite Structures, 2013, 98: 143–152

[3] He X Q , Ng T Y , Sivashanker S , Liew K M . Active control of FGM plates with integrated piezoelectric sensors and actuators. International Journal of Solids and Structures, 2001, 38(9): 1641–1655

[4] Yang J , Xiang H J . Thermo-electro-mechanical characteristics of functionally graded piezoelectric actuators. Smart Materials and Structures, 2007, 16(3): 784–797

[5] Ferreira A J M , Batra R C , Roque C M C , Qian L F , Jorge R M N . Natural frequencies of functionally graded plates by a meshless method. Composite Structures, 2006, 75(1–4): 593–600

[6] Vel S S , Batra R C . Three-dimensional analysis of transient thermal stresses in functionally graded plates. International Journal of Solids and Structures, 2003, 40(25): 7181–7196

[7] Xiang T , Natarajan S , Man H , Song C , Gao W . Free vibration and mechanical buckling of plates with in-plane material inhomogeneity––A three-dimensional consistent approach. Composite Structures, 2014, 118: 634–642

[8] Yin S , Yu T , Bui T Q , Zheng X , Tanaka S . In-plane material inhomogeneity of functionally graded plates: A higher-order shear deformation plate isogeometric analysis. Composites. Part B, Engineering, 2016, 106: 273–284

[9] Ebrahimi F , Barati M R . Vibration analysis of piezoelectrically actuated curved nanosize FG beams via a nonlocal strain–electric field gradient theory. Mechanics of Advanced Materials and Structures, 2018, 25(4): 350–359

[10] KumarPAimmaneeS. Thermoelectrical vibration and bending analysis of multidirectional functionally graded circular piezoelectric porous sigmoid plate resting on variable elastic foundations. International Journal of Mechanics and Materials in Design, 2025: 9779

[11] Asemi S R , Farajpour A . Thermo-electro-mechanical vibration of coupled piezoelectric-nanoplate systems under non-uniform voltage distribution embedded in Pasternak elastic medium. Current Applied Physics, 2014, 14(5): 814–832

[12] Pradhan S C , Phadikar J K . Nonlocal elasticity theory for vibration of nanoplates. Journal of Sound and Vibration, 2009, 325(1–2): 206–223

[13] Eltaher M A , Emam S A , Mahmoud F F . Free vibration analysis of functionally graded size-dependent nanobeams. Applied Mathematics and Computation, 2012, 218(14): 7406–7420

[14] Natarajan S , Chakraborty S , Thangavel M , Bordas S , Rabczuk T . Size-dependent free flexural vibration behavior of functionally graded nanoplates. Computational Materials Science, 2012, 65: 74–80

[15] Liu C , Ke L L , Wang Y S , Yang J , Kitipornchai S . Thermo-electro-mechanical vibration of piezoelectric nanoplates based on the nonlocal theory. Composite Structures, 2013, 106: 167–174

[16] Jandaghian A A , Rahmani O . Vibration analysis of functionally graded piezoelectric nanoscale plates by nonlocal elasticity theory: An analytical solution. Superlattices and Microstructures, 2016, 100: 57–75

[17] Li S , Zheng S , Chen D . Porosity-dependent isogeometric analysis of bi-directional functionally graded plates. Thin-walled Structures, 2020, 156: 106999

[18] Kumar P , Harsha S P . Dynamic analysis of porosity dependent functionally graded sigmoid piezoelectric (FGSP) plate. Structures, 2022, 46: 1737–1752

[19] Kumar P , Harsha S P . Static analysis of porous core functionally graded piezoelectric (PCFGP) sandwich plate resting on the Winkler/Pasternak/Kerr foundation under thermo-electric effect. Materials Today. Communications, 2022, 32: 103929

[20] Zenkour A M , Aljadani M H . Porosity effect on thermal buckling behavior of actuated functionally graded piezoelectric nanoplates. European Journal of Mechanics. A, Solids, 2019, 78: 103835

[21] Kumar P , Harsha S P . Vibration response analysis of sigmoidal functionally graded piezoelectric (FGP) porous plate under thermo-electric environment. Mechanics Based Design of Structures and Machines, 2021, 51(8): 4604–4634

[22] Kumar P , Harsha A . Vibration response analysis of the bi-directional porous functionally graded piezoelectric (BD-FGP) plate. Mechanics Based Design of Structures and Machines, 2022, 52(1): 126–151

[23] Lieu Q X , Lee S , Kang J , Lee J . Bending and free vibration analyzes of in-plane bi-directional functionally graded plates with variable thickness using isogeometric analysis. Composite Structures, 2018, 192: 434–451

[24] Tang Y , Ding Q . Nonlinear vibration analysis of a bi-directional functionally graded beam under hygro-thermal loads. Composite Structures, 2019, 225: 111076

[25] Harsha A , Kumar P . Thermoelectric elastic analysis of bi-directional three-layer functionally graded porous piezoelectric (FGPP) plate resting on elastic foundation. Forces in Mechanics, 2022, 8: 100112

[26] Brischetto S , Cesare D . Three-dimensional vibration analysis of multilayered composite and functionally graded piezoelectric plates and shells. Composite Structures, 2024, 346: 118413

[27] Alzahrani E O , Zenkour A M , Sobhy M . Small scale effect on hygro-thermo-mechanical bending of nanoplates embedded in an elastic medium. Composite Structures, 2013, 105: 163–172

[28] Zenkour A M , Alghanmi R A . Hygro-thermo-electro-mechanical bending analysis of sandwich plates with FG core and piezoelectric faces. Mechanics of Advanced Materials and Structures, 2021, 28(3): 282–294

[29] Li S , Xu C , Zhang W , Zhang C , Yao W , Chen W . On thermo-mechanical buckling of porous bi-directional functionally graded plates using isogeometric analysis. Aerospace Science and Technology, 2024, 155: 109520

[30] Kumar P , Harsha S P . Static and vibration response analysis of sigmoid function-based functionally graded piezoelectric non-uniform porous plate. Journal of Intelligent Material Systems and Structures, 2022, 33(17): 2197–2227

[31] Tornabene F , Fantuzzi N , Viola E , Reddy J N . Winkler–Pasternak foundation effect on the static and dynamic analyses of laminated doubly-curved and degenerate shells and panels. Composites. Part B, Engineering, 2014, 57: 269–296

[32] Sobhy M . Thermoelastic response of FGM plates with temperature-dependent properties resting on variable elastic foundations. International Journal of Applied Mechanics, 2015, 7(6): 1550082

[33] Mudhaffar I M , Tounsi A , Chikh A , Al-Osta M A , Al-Zahrani M M , Al-Dulaijan S U . Hygro-thermo-mechanical bending behavior of advanced functionally graded ceramic metal plate resting on a viscoelastic foundation. Structures, 2021, 33: 2177–2189

[34] Ansari R , Shahabodini A , Shojaei M F . Nonlocal three-dimensional theory of elasticity with application to free vibration of functionally graded nanoplates on elastic foundations. Physica E, Low-Dimensional Systems and Nanostructures, 2016, 76: 70–81

[35] Ke L L , Liu C , Wang Y S . Free vibration of nonlocal piezoelectric nanoplates under various boundary conditions. Physica E, Low-Dimensional Systems and Nanostructures, 2015, 66: 93–106

[36] Xu T F , Xing Y F . Closed-form solutions for free vibration of rectangular FGM thin plates resting on elastic foundation. Acta Mechanica Sinica, 2016, 32(6): 1088–1103

[37] Dehsaraji M L , Arefi M , Loghman A . Size dependent free vibration analysis of functionally graded piezoelectric micro/nano shell based on modified couple stress theory with considering thickness stretching effect. Defense Technology, 2021, 17(1): 119–134

[38] Kumar P , Harsha S P . Vibration response analysis of exponential functionally graded piezoelectric (EFGP) plate subjected to thermo-electro-mechanical load. Composite Structures, 2021, 267: 113901

[39] Pham Q H , Malekzadeh P , Tran V K , Nguyen-Thoi T . Free vibration analysis of functionally graded porous curved nanobeams on elastic foundation in hygro-thermo-magnetic environment. Frontiers of Structural and Civil Engineering, 2023, 17(4): 584–605

[40] Duong K D , Mai D N , Minh P V , Ke T V . An isogeometric approach to free vibration analysis of bi-directional functionally graded porous doubly-curved shallow microshells with variable length-scale parameters. Frontiers of Structural and Civil Engineering, 2023, 17(12): 1871–1894

[41] Nguyen V C , Tran T T , Nguyen-Thoi T , Pham Q H . A novel finite element formulation for static bending analysis of functionally graded porous sandwich plates. Frontiers of Structural and Civil Engineering, 2022, 16(12): 1599–1620

[42] Kumar P , Harsha S P . Vibration response analysis of PZT-4/PZT-5H based functionally graded tapered plate subjected to electromechanical loading. Mechanics Research Communications, 2021, 116: 103765

[43] Kumar P , Harsha S P . Static, buckling and vibration response analysis of three-layered functionally graded piezoelectric plate under thermo-electric mechanical environment. Journal of Vibration Engineering & Technologies, 2022, 10(4): 1561–1598

[44] Kumar P , Harsha S P . Response analysis of hybrid functionally graded material plate subjected to thermo-electro-mechanical loading. In: Proceedings of the Institution of Mechanical Engineers, Part L: Journal of Materials: Design and Applications. Thousand Oaks, CA: SAGE Publications, 2021, 235(4): 813–827

[45] Ebrahimi F , Salari E . Effect of non-uniform temperature distributions on nonlocal vibration and buckling of inhomogeneous size-dependent beams. Advances in Nano Research, 2018, 6(4): 377

[46] Ebrahimi F , Salari E . Thermal loading effects on electro-mechanical vibration behavior of piezoelectrically actuated inhomogeneous size-dependent Timoshenko nanobeams. Advances in Nano Research, 2016, 4(3): 197–228

[47] Ehyaei J , Ebrahimi F , Salari E . Nonlocal vibration analysis of FG nano beams with different boundary conditions. Advances in Nano Research, 2016, 4(2): 85–111

[48] Kumar P , Harsha S P . Modal analysis of functionally graded piezoelectric material plates. Materials Today: Proceedings, 2020, 28: 1481–1486

[49] Salari E , Ashoori A , Sadough Vanini S A . Porosity-dependent asymmetric thermal buckling of inhomogeneous annular nanoplates resting on elastic substrate. Advances in Nano Research, 2019, 7(1): 25–38

[50] Salari E , Ashoori A R , Sadough Vanini S A , Akbarzadeh A H . Nonlinear dynamic buckling and vibration of thermally post-buckled temperature-dependent FG porous nanobeams based on the nonlocal theory. Physica Scripta, 2022, 97(8): 085216

[51] Kumar P , Harsha S P . Electroelastic static and vibration response analysis of sigmoid PZT-5A/Pt-based smart functionally graded plate. International Journal of Structural Stability and Dynamics, 2022, 22(14): 2250155

[52] Harsha A , Kumar P . Impact of the porosity and elastic foundation on frequency and buckling response of bidirectional functionally graded piezoelectric porous plate. International Journal of Structural Stability and Dynamics, 2024, 24(7): 2450077

[53] Kumar P , Harsha S P . Thermoelectric nonlinear vibration and buckling analysis of the smart porous core sandwich plate (SPCSP) resting on the elastic foundation. Journal of Intelligent Material Systems and Structures, 2023, 34(14): 1587–1616

[54] Ebrahimi F , Salari E . Semi-analytical vibration analysis of functionally graded size-dependent nanobeams with various boundary conditions. Smart Structures and Systems, 2017, 19(3): 243–257

[55] Ebrahimi F , Salari E . Analytical modeling of dynamic behavior of piezo-thermo-electrically affected sigmoid and power-law graded nanoscale beams. Applied Physics. A, Materials Science & Processing, 2016, 122(9): 793

[56] Ashoori A R , Sadough Vanini S A , Salari E . Size-dependent axisymmetric vibration of functionally graded circular plates in bifurcation/limit point instability. Applied Physics. A, Materials Science & Processing, 2017, 123(4): 226

[57] Salari E , Sadough Vanini S A , Ashoori A . Nonlinear thermal stability and snap-through buckling of temperature-dependent geometrically imperfect graded nanobeams on nonlinear elastic foundation. Materials Research Express, 2020, 6(12): 1250j6

[58] Salari E , Sadough Vanini S A . Small/large amplitude vibration, snap-through and nonlinear thermo-mechanical instability of temperature-dependent FG porous circular nanoplates. Engineering with Computers, 2023, 39(3): 2295–2326

[59] Salari E , Sadough Vanini S A . Nonlocal nonlinear static/dynamic snap-through buckling and vibration of thermally post-buckled imperfect functionally graded circular nanoplates. Waves in Random and Complex Media, 2025, 35(2): 3805–3851

[60] Ghadiri M , Ebrahimi F , Salari E , Hosseini S A H , Shaghaghi G R . Electro-thermo-mechanical vibration analysis of embedded single-walled boron nitride nanotubes based on nonlocal third-order beam theory. International Journal for Multiscale Computational Engineering, 2015, 13(5): 443–461

[61] Sharma N , Nishad M , Maiti D K , Sunny M R , Singh B N . Uncertainty quantification in buckling strength of variable stiffness laminated composite plate under thermal loading. Composite Structures, 2021, 275: 114486

[62] Sharma N , Swain P K , Maiti D K . Aeroelastic control of delaminated variable angle tow laminated composite plate using piezoelectric patches. Journal of Composite Materials, 2022, 56(29): 4375–4408

[63] Ebrahimi F , Shaghaghi G R , Salari E . Vibration analysis of size-dependent nano beams basedon nonlocal Timoshenko beam theory. Journal of Mechanical Engineering and Technology, 2014, 6(2): 340

[64] Nasri M R , Salari E , Salari A , Sadough Vanini S A . Nonlinear bending and buckling analysis of 3D-printed meta-sandwich curved beam with auxetic honeycomb core. Aerospace Science and Technology, 2024, 152: 109339

[65] Kumar P , Harsha S P . Response analysis of functionally graded piezoelectric plate resting on elastic foundation under thermo-electro environment. Journal of Composite Materials, 2022, 56(24): 3749–3767

[66] Prakash A , Kumar P , Saran V H , Harsha S P . NURBS based thermoelastic behaviour of thin functionally graded sigmoidal (TFGS) porous plate resting on variable Winkler’s foundation. International Journal of Mechanics and Materials in Design, 2023, 19(4): 831–860

[67] Kumar P , Harsha S P . Hygrothermal static bending and deflection responses of porous multidirectional nanofunctionally graded piezoelectric (NFGP) plates with variable thickness on elastic foundations. International Journal of Mechanical System Dynamics, 2025, 5(1): 40–66

[68] KumarPHarshaS P. Static and vibration response analysis of PZT-5A/PT based smart functionally graded (SFG) plate subjected to electromechanical loading. In: International Conference on Vibration Engineering and Technology of Machinery. Singapore: Springer Singapore, 2021, 553–575

[69] Ghasemi F , Salari E , Rastgoo A , Li D , Deng J . Nonlinear vibration analysis of pre/post-buckled 3D-printed tubular metastructures. Engineering Analysis with Boundary Elements, 2024, 165: 105777

[70] GhasemiFSalariEZamanianA HRastgooA. Experimental mechanical properties, nonlinear bending and instability analysis of 3D-printed auxetic tubular metastructures using multiscale finite element and Ritz methods. Mechanics of Advanced Materials and Structures, 2024: 1–24

[71] Sharma N , Swain P K , Maiti D K , Singh B N . Stochastic frequency analysis of laminated composite plate with curvilinear fiber. Mechanics of Advanced Materials and Structures, 2022, 29(6): 933–948

[72] Sharma N , Kumar Swain P , Kumar Maiti D , Nath Singh B . Vibration and uncertainty analysis of functionally graded sandwich plate using layerwise theory. AIAA Journal, 2022, 60(6): 3402–3423

[73] Ashoori A R , Salari E , Sadough Vanini S A . A thermo-electro-mechanical vibration analysis of size-dependent functionally graded piezoelectric nanobeams. Advances in High Temperature Ceramic Matrix Composites and Materials for Sustainable Development: Ceramic Transactions, 2017, 263: 547–558

[74] KumarPSharmaNSuYHarshaS P. Thermoelectric buckling response of nonuniform sigmoid functionally graded piezoelectric (FGPS) plate: Influence of elastic foundation and porosity. International Journal of Computational Materials Science and Engineering, 2025: 2550014

[75] SharmaNChandrakarPTiwariPMaitiD K. Numerical analysis on dynamic response of the variable stiffness composite structures including the damage effect. Mechanics Based Design of Structures and Machines, 2025: 1–25

[76] Sharma N , Chandrakar P , Maiti D K . Influence of delamination on uncertain dynamic characteristics of variable angle tow laminates using polynomial neural network. Acta Mechanica, 2024, 235(9): 5789–5823

[77] Nishad M , Sharma N , Sunny M R , Singh B N , Maiti D K . Stochastic critical buckling speed analysis of rim-driven rotating composite plate using NURBS-based isogeometric approach and HSDT. Mechanics Based Design of Structures and Machines, 2024, 52(10): 7402–7429

[78] Sharma N , Swain P K , Maiti D K , Singh B N . Static and free vibration analyses and dynamic control of smart variable stiffness laminated composite plate with delamination. Composite Structures, 2022, 280: 114793

[79] GhasemiFSalariESalariARastgooALiDDengJ. Integrating analytical and machine learning methods for investigating nonlinear bending and post-buckling behavior of 3D-printed auxetic tubes. Engineering with Computers, 2024: 1–38

[80] Ezzati H , Ebrahimi F , Salari E . Exploring graphene origami-enabled metamaterials: A review. Journal of Computational Applied Mechanics, 2025, 56(1): 249–263

[81] Sharma N , Swain P K , Maiti D K . Static and dynamic control of smart damaged variable stiffness laminated composite plate with piezoelectric layers. Mechanics Based Design of Structures and Machines, 2024, 52(6): 3527–3551

[82] KumarPAimmaneeSHarshaS P. Effect of orthotropic variable foundations and unconventional support conditions on nonlinear hygrothermoelectric vibration of porous multidirectional piezoelectric functionally graded nonuniform plate. International Journal of Mechanical System Dynamics, 2025: 70027

[83] Singh B , Mulik R S , Harsha S P . Dynamic response analysis of functionally graded gears. In: Proceedings of the Institution of Mechanical Engineers, Part L: Journal of Materials: Design and Applications. Thousand Oaks, CA: SAGE Publications, 2023, 237(1): 52–69

[84] Kumar V , Singh S J , Saran V H , Harsha S P . Vibration characteristics of porous FGM plate with variable thickness resting on Pasternak’s foundation. European Journal of Mechanics. A, Solids, 2021, 85: 104124

[85] van Minh C , Do Van T , van Minh P , Nguyen C T , Doan T L , Nguyen H H . Investigation of mechanical responses of flexo-magnetic variable thickness nanoplates resting on elastic foundations, taking into account geometrical imperfections. Frontiers of Structural and Civil Engineering, 2024, 18(12): 1951–1970

[86] Pham Q H , Tran V K , Nguyen P C . An isogeometric approach to static and transient analysis of fluid-infiltrated porous metal foam piezoelectric nanoplates with flexoelectric effects and variable nonlocal parameters. Frontiers of Structural and Civil Engineering, 2024, 18(3): 461–489

[87] Thuy T T T . Static and dynamic analysis of functionally graded fluid-infiltrated porous skew and elliptical nanoplates using an isogeometric approach. Frontiers of Structural and Civil Engineering, 2023, 17(3): 477–502

[88] KumarPSharmaNSuYHarshaS P. Effect of Winkler’s-Pasternak foundations and porosity on the electromechanical Buckling responses of PZT-4/PZT-5H Smart graded plate subjected to thermal loading. International Journal of Computational Materials Science and Engineering, 2025: 2550019

[89] Singh B , Mulik R S , Harsha S P . Static and vibration analysis of functionally graded gears. Mechanics Based Design of Structures and Machines, 2023, 51(12): 6928–6946

[90] ReddyJ N. Theory and Analysis of Elastic Plates and Shells. Boca Raton, FL: CRC Press, 1999

[91] YangJ. An Introduction to the Theory of Piezoelectricity, Vol. 9. New York, NY: Springer, 2005

[92] SihG CMichopoulosJ GChouS C. Hygrothermoelasticity. Berlin: Springer Science & Business Media, 1986

[93] Bergan P G , Clough R W . Convergence criteria for iterative processes. AIAA Journal, 1972, 10(8): 1107–1108

[94] BatheK J. Finite Element Procedures. Upper Saddle River, NJ: Prentice Hall, 2006

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