Isogeometric analysis approach for vibration of tridirectional functionally graded porous plate in high-temperature environment taking into account initial geometrical imperfections
Restricted accessResearch articleFirst published online January, 2026
Isogeometric analysis approach for vibration of tridirectional functionally graded porous plate in high-temperature environment taking into account initial geometrical imperfections
This research investigates the application of the isogeometric analysis (IGA) method, grounded in higher-order shear plate theory, to examine the free vibrations of a tridirectional functionally graded porous (TFGP) plate. The analysis incorporates initial geometrical imperfections and considers the effects of a high-temperature environment while the plate is supported by an elastic foundation. The distribution of symmetric and asymmetric materials adheres to a power law across the three dimensions of the Cartesian coordinate system under consideration. Transcendental functions are employed to articulate two categories of geometric imperfections: cosine-type global/localized imperfections, which are modeled using hyperbolic and cosine functions, and sine-type imperfections, characterized by global eccentricity/symmetry, represented through sine and exponential functions. The amalgamation of various materials imparts numerous essential mechanical properties to the structure, thereby facilitating its application across diverse scientific disciplines. The precision of the method is validated through numerical comparisons with other credible analytical publications or established numerical techniques. A thorough numerical investigation is presented to examine the mechanisms by which material distribution patterns, geometrical imperfection modes, and various boundary conditions affect the natural oscillations of TFGP plates, while also considering initial geometrical imperfections in high-temperature environments. In practical applications involving structures with defects, this study serves a crucial function in the referencing, examination, and computation of such structures.
KoizumiMNiinoM.Overview of FGM research in Japan. MRS Bull1995; 20(1): 19–21.
2.
KumarPSharmaSKSinghRKR. Recent trends and future outlooks in manufacturing methods and applications of FGM: a comprehensive review. Mater Manuf Process2023; 38(9): 1033–1067.
3.
TangHNguyenNVLeeJ.Simultaneous optimal tri-directional distribution of material and porosity in functionally graded plates under free vibration. Thin-Walled Struct2024; 196: 111496.
4.
TangHNguyenNVLeeJ.Accelerating tri-directional material distribution optimization in functionally graded plates with an adaptive design control point variable selection. Comput Methods Appl Mech Eng2024; 418: 116474.
5.
KiranRSinghDSinghSK, et al. On the isogeometric analysis of vibration and buckling of bioinspired composite plates using inverse hyperbolic shear deformation theory. Mech Based Des Struct Mach2025; 53: 1067–1104.
6.
TangYLiCLYangT.Application of the generalized differential quadrature method to study vibration and dynamic stability of tri-directional functionally graded beam under magneto-electro-elastic fields. Eng Anal Bound Elem2023; 146: 808–823.
7.
TaoCDaiTChenY.Thermal postbuckling and thermally induced postbuckled flutter of tri-directional functionally graded plates in yawed supersonic flow. Aerosp Sci Technol2024; 154: 109491.
8.
ShahmohammadiMAAbdollahiPSalehipourH.Geometrically nonlinear analysis of doubly curved imperfect shallow shells made of functionally graded carbon nanotube reinforced composite (FG_CNTRC). Mech Based Des Struct Mach2022; 50(11): 3796–3820.
9.
Van MinhCDo VanTVan MinhP, et al. Investigation of mechanical responses of flexo-magnetic variable thickness nanoplates resting on elastic foundations, taking into account geometrical imperfections. Front Struct Civil Eng2024; 18: 1951–1970.
10.
ThaiS.Optimization of multi-directional functionally graded plates in thermal environment based on 3D isogeometric analysis and adaptive-hybrid evolutionary firefly algorithm. Thin-Walled Struct2023; 190: 111000.
11.
LahdiriAKadriM.Free vibration behaviour of multi-directional functionally graded imperfect plates using 3D isogeometric approach. Earthq Struct2022; 22(5): 527–538.
12.
Hosur ShivaramaiahNKKattimaniSShariatiM, et al. Geometrically nonlinear behavior of two-directional functionally graded porous plates with four different materials. Proc IMechE, Part C: J Mechanical Engineering Science2022; 236(22): 11008–11023.
13.
HanSYeQMahmoudHA, et al. Nonlinear phase velocities in tri-directional functionally graded nanoplates coupled with NEMS patch using multi-physics simulation. Aerosp Sci Technol2025; 156: 109714.
14.
TangYWangGYangT, et al. Nonlinear dynamics of three-directional functional graded pipes conveying fluid with the integration of piezoelectric attachment and nonlinear energy sink. Nonlinear Dyn2023; 111(3): 2415–2442.
15.
ThaiLMLuatDTVan KeT, et al. Finite-element modeling for static bending analysis of rotating two-layer FGM beams with shear connectors resting on imperfect elastic foundations. J Aerosp Eng2023; 36(3). https://ascelibrary.org/doi/abs/10.1061/JAEEEZ.ASENG-477
16.
PhungVM.Static bending analysis of symmetrical three-layer fgm beam with shear connectors under static load. J Sci Tech2020; 15(3): 68–78.
17.
Nguyen ThaiDMinhPVPhan HoangC, et al. Bending of symmetric sandwich FGM beams with shear connectors. Math Probl Eng2021; 2021: 1–15.
18.
SalehiMGholamiRAnsariR.Analytical solution approach for nonlinear vibration of shear deformable imperfect FG-GPLR porous nanocomposite cylindrical shells. Mech Based Des Struct Mach2023; 51(4): 2177–2199.
19.
Kazemzadeh-ParsiMJChinestaFAmmarA.Proper generalized decomposition for parametric study and material distribution design of multi-directional functionally graded plates based on 3D elasticity solution. Materials2021; 14(21). https://www.mdpi.com/1996-1944/14/21/6660
ChenXLuYWuZ, et al. Free vibration of in-plane bi-directional functionally graded materials rectangular plates with geometric imperfections and general elastic restraints. Aerosp Sci Technol2023; 132: 108045.
24.
KatiyarVGuptaA.Vibration response of a geometrically discontinuous bi-directional functionally graded plate resting on elastic foundations in thermal environment with initial imperfections. Mech Based Des Struct Mach2023; 51(6): 3480–3508.
CuongBMTounsiAVan ThomD, et al. Finite element modelling for the static bending response of rotating FG-GPLRC beams with geometrical imperfections in thermal mediums. Comput Concr2024; 33(1): 91–102.
27.
LyHBLeLMDuongHT, et al. Hybrid artificial intelligence approaches for predicting critical buckling load of structural members under compression considering the influence of initial geometric imperfections. Appl Sci2019; 9(11): 2258.
28.
ShayanSRasmussenKJRZhangH.On the modelling of initial geometric imperfections of steel frames in advanced analysis. J Constr Steel Res2014; 98: 167–177.
29.
ArefiM.Third-order electro-elastic analysis of sandwich doubly curved piezoelectric micro shells. Mech Based Des Struct Mach2021; 49(6): 781–810.
30.
ZhangJZhangWZhangYF.Nonlinear resonant responses of hyperelastic cylindrical shells with initial geometric imperfections. Chaos Solitons Fractals2023; 173: 113709.
31.
ChenGZhangHRasmussenKJR, et al. Modeling geometric imperfections for reticulated shell structures using random field theory. Eng Struct2016; 126: 481–489.
32.
CerviCVirginLN.An experimental parametric study assessing the effect of a certain initial geometric imperfection on cylinder buckling. Aerosp Sci Technol2024; 152: 109355.
33.
Zargar ErshadiMFaraji OskouieMAnsariR. Nonlinear vibration analysis of functionally graded porous circular plates under hygro-thermal loading. Mech Based Des Struct Mach2024; 52(2): 1042–1059.
34.
Thai DungNMinh ThaiLVan KeT, et al. Nonlinear static bending analysis of microplates resting on imperfect two-parameter elastic foundations using modified couple stress theory. Comptes Rendus - Mec2022; 350: 121–141.
35.
LiZMQiaoP.Buckling and postbuckling behavior of shear deformable anisotropic laminated beams with initial geometric imperfections subjected to axial compression. Eng Struct2015; 85: 277–292.
36.
NiZYangJFanY, et al. Multi-factors effects analysis of nonlinear vibration of FG-GNPRC membrane using machine learning. Mech Based Des Struct Mach2024; 52: 8988–9014.
37.
CuiXZLiYGHongHP.Effect of spatially correlated initial geometric imperfection on reliability of spherical latticed shell considering global instability. Struct Saf2020; 82: 101895.
38.
RzeszutKGarsteckiA.Modeling of initial geometrical imperfections in stability analysis of thin-walled structures. J Theor Appl Mech2009; 47(3): 667–684.
39.
SekkalMBouiadjraRBBenyoucefS, et al. Effect of material distribution on bending and buckling response of a bidirectional FG beam exposed to a combined transverses and variable axially loads. Mech Based Des Struct Mach2024; 52(4): 2193–2212.
40.
DuongKDMaiDNMinhPV, et al. An isogeometric approach to free vibration analysis of bi-directional functionally graded porous doubly-curved shallow microshells with variable length-scale parameters. Front Struct Civil Eng2023; 17(12): 1871–1894.
41.
Khuat DucDNguyen TuanLDao NhuM, et al. A novel isogeometric model for dynamic buckling analysis of doubly curved two-directional functionally graded porous shallow microshells in thermal environments via variable length-scale parameters. Mech Based Des Struct Mach2024; 52: 8610–8638.
42.
PhamQHNguyenPCTranVK, et al. Isogeometric analysis for free vibration of bidirectional functionally graded plates in the fluid medium. Def Technol2022; 18(8): 1311–1329.
43.
TuPHVan KeTTraiVK, et al. An isogeometric analysis approach for dynamic response of doubly-curved magneto electro elastic composite shallow shell subjected to blast loading. Def Technol2024; 41: 159–180.
44.
PhamQHNguyenPCTranVK, et al. Modified nonlocal couple stress isogeometric approach for bending and free vibration analysis of functionally graded nanoplates. Eng Comput2023; 39(1): 993–1018.
45.
PhamQHTranVKNguyenPC.Exact solution for thermal vibration of multi-directional functionally graded porous plates submerged in fluid medium. Def Technol2024; 35: 77–99.
46.
LeHHTranVKHoangNT, et al. Nonlinear free oscillation of tri-directional functionally graded porous skew-plates with variable thickness in high-thermal environment. Case Stud Therm Eng2025; 70: 106101.
47.
HuangXLShenHS.Nonlinear vibration and dynamic response of functionally graded plates in thermal environments. Int J Solids Struct2004; 41(9-10): 2403–2427.
48.
KianiYEslamiMR.An exact solution for thermal buckling of annular FGM plates on an elastic medium. Compos B Eng2013; 45(1): 101–110.
49.
ThiTHNTranVKPhungVM, et al. Nonlocal isogeometric analysis for bidirectional functionally graded porous curved microbeams with arbitrary boundary conditions. Acta Mech Sin Xuebao2024; 40(8). https://link.springer.com/article/10.1007/s10409-023-23257-x
50.
ShiPDongCShouH, et al. Bending, vibration and buckling isogeometric analysis of functionally graded porous microplates based on the TSDT incorporating size and surface effects. Thin-Walled Struct2023; 191: 111027.
51.
ReddyJN.Analysis of functionally graded plates. Int J Numer Methods Eng2000; 47(1-3): 663–684.
52.
PhamQHNguyenPCTranVK.Effects of hygro-thermal environment on dynamic responses of variable thickness functionally graded porous microplates. Steel Compos Struct2024; 50(5): 563–581.
53.
TranVKPhungVMTruongTHH, et al. Nonlocal strain gradient free vibration analysis of sandwich functionally graded porous nanoshell integrated with piezoelectric surface layers taking into account flexoelectric effect. Front Struct Civil Eng2025; 19: 623–644.
54.
HoaiLTuPHTranVK, et al. An isogeometric approach for vibration characteristics analysis of functionally graded triply periodic minimal sandwich curved-doubly shell integrated with magneto-electro surface layers subjected to low-velocity impact load. Def Technol2025; 50: 101–125. https://www.sciencedirect.com/science/article/pii/S2214914725001266.
55.
KeTVMinhPVThomDV, et al. Static and dynamic analysis of doubly-curved functionally graded porous nanoshells integrated with piezoelectric surface layers and flexoelectric effect. Comput Struct2025; 312: 107737.
BordenMJScottMAEvansJA, et al. Isogeometric finite element data structures based on Bézier extraction of NURBS. Int J Numer Methods Eng2011; 87(1-5): 15–47.
58.
Nguyen ThiTHTranVKPhamQH.An isogeometric approach for nonlocal bending and free oscillation of magneto-electro-elastic functionally graded nanobeam with elastic constraints. Front Struct Civil Eng2024; 18(9): 1401–1423.
59.
HashemiSJafariAA.Nonlinear Free and forced vibrations of In-plane Bi-directional functionally graded rectangular plate with temperature-dependent properties. Int J Struct Stab Dyn2020; 20(08): 2050097.
60.
LieuQXLeeDKangJ, et al. NURBS-based modeling and analysis for free vibration and buckling problems of in-plane bi-directional functionally graded plates. Mech Adv Mater Struct2019; 26(12): 1064–1080.
61.
SonTHuu-TaiT.Free-vibration analysis of multi-directional functionally graded plates based on 3D isogeometric analysis. J Sci Technol Civ Eng - NUCE2019; 13(2): 1–11.
62.
KaramanliAAydogduMVoTP.A comprehensive study on the size-dependent analysis of strain gradient multi-directional functionally graded microplates via finite element model. Aerosp Sci Technol2021; 111: 106550.
63.
PhamQHTranVKNguyenPC.Linear finite element formulation for free vibration and buckling analyses of multi-directional FGP doubly curved shallow shells in thermal environment. Ships Offshore Struct2025; 20: 77–98.
64.
YeTJinGChenY, et al. Three-dimensional vibration analysis of functionally graded material plates in thermal environment. In: INTER-NOISE 2019 MADRID - 48th Int Congr Exhib Noise Control Eng. 2019, Madrid, Spain.
