The main aim of this paper is to investigate the combined influences of porosity, initial geometric imperfection, nonlinear elastic foundations, tangential elastic constraint of boundary edge and elevated temperature on the nonlinear axisymmetric free vibration of functionally graded material (FGM) shallow spherical caps. To this goal, a semi–analytical method is used. Unlike previous studies, the effects of imperfections in material, geometry and meridional periphery constraint on the nonlinear vibration of shear deformable FGM shallow spherical caps are included. The properties of constituent materials are considered to be temperature–dependent and the effective properties of porous FGM are determined using a modified rule of mixture. Motion and compatibility equations in terms of deflection and stress function are established within the framework of first order shear deformation theory (FSDT) incorporating initial geometric imperfection and interactive pressure from three-parameter elastic foundation. The derived equations are treated employing analytical solutions and Galerkin method for clamped boundary condition to obtain a time–dependent nonlinear ordinary differential equation. This equation is solved by means of fourth-order Runge–Kutta numerical scheme to determine the frequencies of nonlinear free vibration. It is found that curvature ratio, tangential constraint of periphery, initial geometric imperfection and elastic foundations have significant effects on the frequency–amplitude response of porous FGM spherical caps. Additionally, it is revealed that the nonlinear free vibration of porous FGM spherical caps exhibits the softening type when the shell is sufficiently curved and meridional periphery constraint is sufficiently rigorous. The results detect that geometric imperfection has prominent influence on the intensity of softening–type response.
HuiD. Large-amplitude vibrations of geometrically imperfect shallow spherical shells with structural damping. AIAA J1983; 21(12): 1736–1741. https://doi.org/10.2514/3.8317
4.
SinharayGCBanerjeeB. Large amplitude free vibrations of shallow spherical shell and cylindrical shell – a new approach. Int J Non Lin Mech1985; 20(2): 69–78. https://doi.org/10.1016/0020-7462(85)90001-0
5.
DumirPC. Nonlinear axisymmetric response of orthotropic thin spherical caps on elastic foundations. Int J Mech Sci1985; 27(11–12): 751–760. https://doi.org/10.1016/0020-7403(85)90007-4
6.
XuCSChiaCY. Nonlinear vibration of symmetrically laminated spherical caps with flexible supports. Acta Mech1992; 94: 59–72. https://doi.org/10.1007/bf01177005
7.
XuCSChiaCY. Non-linear analysis of unsymmetrically laminated moderately thick shallow spherical shells. Int J Non Lin Mech1994; 29(2): 247–259. https://doi.org/10.1016/0020-7462(94)90043-4
8.
XuC. Multi-mode nonlinear vibration and large deflection of symmetrically laminated imperfect spherical caps on elastic foundations. Int J Mech Sci1992; 34(6): 459–474. https://doi.org/10.1016/0020-7403(92)90012-6
9.
LiD. A time-mode approach to nonlinear vibrations of orthotropic thin shallow spherical shells. Int J Solid Struct1993; 30(22): 3113–3128. https://doi.org/10.1016/0020-7683(93)90142-t
10.
SathyamoorthyM. Vibrations of moderately thick shallow spherical shells at large amplitudes. J Sound Vib1994; 172(1): 63–70. https://doi.org/10.1006/jsvi.1994.1158
TrabelsiSZghalSDammakF. Thermo–elastic buckling and post–buckling analysis of functionally graded thin plate and shell structures. J Braz Soc Mech Sci Eng2020; 42: 233. https://doi.org/10.1007/s40430-020-02314-5
13.
AliAYHasanHM. Nonlinear dynamic stability of an imperfect shear deformable orthotropic functionally graded material toroidal shell segments under the longitudinal constant velocity. Proc IME C J Mech Eng Sci2019; 233(issue 19–20): 6827–6850. https://doi.org/10.1177/0954406219867991
14.
ChenWQWangXDingHJ. Free vibration of a fluid–filled hollow sphere of a functionally graded material with spherical isotropy. J Acoust Soc Am1999; 106(5): 2588–2594. https://doi.org/10.1121/1.428090
GhaheriAAhmadianMTFallahF. Free vibration analysis of a fluid–filled functionally graded spherical shell subjected to internal pressure. Acta Mech2022; 233: 3095–3112. https://doi.org/10.1007/s00707-022-03262-y
17.
HadrichAZghalSKoubaaS, et al.Effects of cracks characteristics on free vibrations of functionally graded plates and shells. Acta Mech2025; 236: 7277–7321. https://doi.org/10.1007/s00707-025-04510-7
PrakashTSundararajanNGanapathiM. On the nonlinear axisymmetric dynamic buckling behavior of clamped functionally graded spherical caps. J Sound Vib2007; 299(1–2): 36–43. https://doi.org/10.1016/j.jsv.2006.06.060
20.
BichDHTungHV. Non-linear axisymmetric response of functionally graded shallow spherical shells under uniform external pressure including temperature effects. Int J Non Lin Mech2011; 46(9): 1195–1204. https://doi.org/10.1016/j.ijnonlinmec.2011.05.015
21.
BichDHDungDVHoaLK. Nonlinear static and dynamic buckling analysis of functionally graded shallow spherical shells including temperature effects. Compos Struct2012; 94(9): 2952–2960. https://doi.org/10.1016/j.compstruct.2012.04.012
BoroujerdyMSEslamiMR. Axisymmetric snap-through behavior of piezo-FGM shallow clamped spherical shells under thermo-electro-mechanical loading. Int J Pres Ves Pip2014; 120–121: 19–26. https://doi.org/10.1016/j.ijpvp.2014.03.008
24.
TungHV. Nonlinear thermomechanical stability of shear deformable FGM shallow spherical shells resting on elastic foundations with temperature dependent properties. Compos Struct2014; 114: 107–116. https://doi.org/10.1016/j.compstruct.2014.04.004
25.
TungHV. Nonlinear axisymmetric response of FGM shallow spherical shells with tangential edge constraints and resting on elastic foundations. Compos Struct2016; 149: 231–238. https://doi.org/10.1016/j.compstruct.2016.04.032
26.
MucAKubisSBratekL, et al.Higher order theories for the buckling and post-buckling studies of shallow spherical shells made of functionally graded materials. Compos Struct2022; 295: 115851. https://doi.org/10.1016/j.compstruct.2022.115851
27.
ZhaoWDGuoDMGongXB, et al.Nonlinear axisymmetric buckling analysis of the FGM sandwich shallow spherical shells under thermomechanical loads. European Journal of Mechanics / A Solids2023; 97: 104841. https://doi.org/10.1016/j.euromechsol.2022.104841
28.
FuYHuSMaoY. Nonlinear transient response of functionally graded shallow spherical shells subjected to mechanical load and unsteady temperature field. Acta Mech Solida Sin2014; 27(5): 496–508. https://doi.org/10.1016/s0894-9166(14)60058-6
29.
BarzegarARFadaeeM. Thermal vibration analysis of functionally graded shallow spherical caps by introducing a decoupling analytical approach. Appl Math Model2018; 58: 473–486. https://doi.org/10.1016/j.apm.2018.02.018
30.
GaoCPangFLiH, et al.An approximate solution for vibrations of uniform and stepped functionally graded spherical cap based on ritz method. Compos Struct2020; 233: 111640. https://doi.org/10.1016/j.compstruct.2019.111640
31.
PhuongNTNamVHDongDT. Nonlinear vibration of functionally graded sandwich shallow spherical shells caps resting on elastic foundations by using first-order shear deformation theory in thermal environment. J Sandw Struct Mater2020; 22(4): 1157–1183.
32.
AliAYHasanHM. Non-linear large amplitude vibration of orthotropic FGM convex and concave toroidal shell segments including the damping effect using the shear deformation theory. Thin-Walled Struct2022; 173: 109035. https://doi.org/10.1016/j.tws.2022.109035
33.
AliAYHasanHMMohammedFM. Nonlinear dynamic buckling of a simply supported imperfect nanocomposite shear deformable plate under the effect of in-plane velocities. Commun Nonlinear Sci Numer Simulat2024; 138: 108232. https://doi.org/10.1016/j.cnsns.2024.108232
34.
HasanHMAliAY. Nonlinear forced vibration of functionally graded graphene-reinforced composite (FG-GRC) laminated cylindrical shells under different boundary conditions with thermal repercussions. Int J Struct Stabil Dynam2024; 24(18): 2450207. https://doi.org/10.1142/s0219455424502079
35.
WattanasakulpongNUngbhakornV. Linear and nonlinear vibration analysis of elastically restrained ends FGM beams with porosities. Aero Sci Technol2014; 32(1): 111–120. https://doi.org/10.1016/j.ast.2013.12.002
36.
AnhNDThinhNVTungHV. Nonlinear vibration analysis of geometrically imperfect FGM Timoshenko beams with porosity and elastically restrained ends in thermal environments. Mech Base Des Struct Mach2024; 52(12): 10148–10169. https://doi.org/10.1080/15397734.2024.2353876
37.
ZghalSAtaouiDDammakF. Static bending analysis of beams made of functionally graded porous materials. Mech Base Des Struct Mach2022; 50(3): 1012–1029. https://doi.org/10.1080/15397734.2020.1748053
38.
ZghalSAtaouiDDammakF. Free vibration analysis of porous beams with gradually varying mechanical properties. Proc IME M J Eng Marit Environ2022; 236(3): 800–812. https://doi.org/10.1177/14750902211047746
39.
RezaeiASSaidiARAbrishamdariM, et al.Natural frequencies of functionally graded plates with porosities via a simple four variable plate theory: an analytical approach. Thin-Walled Struct2017; 120: 366–377. https://doi.org/10.1016/j.tws.2017.08.003
40.
DemirhanPATaskinV. Bending and free vibration analysis of Levy-type porous functionally graded plate using state space approach. Composites Part B2019; 160: 661–676. https://doi.org/10.1016/j.compositesb.2018.12.020
41.
GuptaATalhaM. Influence of porosity on the flexural and free vibration responses of functionally graded plates in thermal environment. Int J Struct Stabil Dynam2018; 18(1): 1850013. https://doi.org/10.1142/s021945541850013x
42.
ThinhNVTungHV. Free vibration and dynamical analyses of FGM plates with porosity and tangential edge constraints. J Vib Eng Technol2024; 12: 5291–5305. https://doi.org/10.1007/s42417-023-01205-y
43.
TrangLTNThinhNVTungHV. Vibration and thermomechanical transient response of doubly curved FGM panels with porosities and elastically restrained edges. Mech Base Des Struct Mach2024; 52(8): 4907–4929. https://doi.org/10.1080/15397734.2023.2242486
44.
ZghalSDammakF. Vibration characteristics of plates and shells with functionally graded pores imperfections using an enhanced finite shell element. Comput Math Appl2021; 99: 52–72. https://doi.org/10.1016/j.camwa.2021.08.001
45.
TrinhMCKimSE. A three variable refined shear deformation theory for porous functionally graded doubly curved shell analysis. Aero Sci Technol2019; 94: 105356. https://doi.org/10.1016/j.ast.2019.105356
46.
LakhdarZChorfiSMBelaliaSA, et al.Free vibration and bending analysis of porous bi-directional FGM sandwich shell using a TSDT p-version finite element method. Acta Mech2024; 235: 3657–3686. https://doi.org/10.1007/s00707-024-03909-y
47.
AbuteirBWBoutagougaD. Free-vibration response of functionally graded porous shell structures in thermal environments with temperature-dependent material properties. Acta Mech2022; 233: 4877–4901. https://doi.org/10.1007/s00707-022-03351-y
48.
ZghalSDammakF. Buckling responses of porous structural components with gradient power-based and sigmoid material variations under different types of compression loads. Compos Struct2021; 273: 114313. https://doi.org/10.1016/j.compstruct.2021.114313
49.
JoueidNZghalSChriguiM, et al.Thermoelastic buckling analysis of plates and shells of temperature and porosity dependent functionally graded materials. Mech Time-Dependent Mater2024; 28: 817–859. https://doi.org/10.1007/s11043-023-09644-6
50.
KalleliSHajlaouiADammakF. Improved-FSDT-based solid-shell element for buckling analysis of plate, spherical cap, and cylindrical shells of FG porous materials. Comput Math Appl2024; 157: 1–14. https://doi.org/10.1016/j.camwa.2023.12.021
51.
TrangLTNTungHV. Influences of porosity and tangential edge constraint on thermo-torsional postbuckling of FGM toroidal shell segments surrounded by an elastic medium. Arch Mech2023; 75(4): 401–429.
52.
TrangLTNTungHV. Thermal nonlinear stability of functionally graded porous material nearly cylindrical shells with surrounding elastic media and tangentially restrained edges. Math Methods Appl Sci2023; 46(6): 7285–7304. https://doi.org/10.1002/mma.8970
53.
LongVTTungHV. Postbuckling responses of porous FGM spherical caps and circular plates including edge constraints and nonlinear three–parameter elastic foundations. Mech Base Des Struct Mach2023; 51(8): 4214–4236. https://doi.org/10.1080/15397734.2021.1956327
54.
JoshiKKKarVR. Geometric and material nonlinear bending analysis of three dimensional functionally graded porous panels. Proc Inst Mech Eng G J Aerosp Eng. 2026; 09544100261419898. https://doi.org/10.1177/09544100261419898, In Press.
55.
ZghalSJoueidNTornabeneF, et al.Time-dependent deflection responses of FG porous structures subjected to different external pulse loads. J Vib Eng Technol2024; 12: 857–876. https://doi.org/10.1007/s42417-023-00880-1
56.
ZghalSDammakF. Functionally graded materials: analysis and applications to FGM, FG-CNTRC and FG porous structures. CRC Press, 2024.
57.
XieKWangYNiuH, et al.Large-amplitude nonlinear free vibrations of functionally graded plates with porous imperfection: a novel approach based on energy balance method. Compos Struct2020; 246: 112367. https://doi.org/10.1016/j.compstruct.2020.112367
58.
AmirMTalhaM. Nonlinear vibration characteristics of shear deformable functionally graded curved panels with porosity including temperature effects. Int J Pres Ves Pip2019; 172: 28–41. https://doi.org/10.1016/j.ijpvp.2019.03.008
59.
AmirMKimSWTalhaM. Comparative study of different porosity models for the nonlinear free vibration analysis of the functionally graded cylindrical panels. Mech Base Des Struct Mach2024; 52(2): 773–799. https://doi.org/10.1080/15397734.2022.2122504
60.
AmirMKimSWLeeSY. Free vibration analysis of the geometrically nonlinear functionally graded porous curved panels in deterministic and stochastic domains considering various boundary conditions. Waves Random Complex Media. 2024; 1–26. https://doi.org/10.1080/17455030.2024.2343100, In Press.
61.
MirjavadiSSForsatMBaratiMR, et al.Geometrically nonlinear vibration analysis of eccentrically stiffened porous functionally graded annular spherical shell segments. Mech Base Des Struct Mach2022; 50(6): 2206–2220. https://doi.org/10.1080/15397734.2020.1771729
62.
RamtekePMKumarVSharmaN, et al.Geometrical nonlinear numerical frequency prediction of porous functionally graded shell panel under thermal environment. Int J Non Lin Mech2022; 143: 104041. https://doi.org/10.1016/j.ijnonlinmec.2022.104041
RamtekePMPandaSKSharmaN. Nonlinear vibration analysis of multidirectional porous functionally graded panel under thermal environment. AIAA J2022; 60(8): 4923–4933. https://doi.org/10.2514/1.j061635
65.
RahmaniAAHosseinnejadFRostamiyanY. Nonlinear vibrations analysis of two-directional functionally graded porous cylindrical shells resting on elastic substrates in a thermal environment based on donnell nonlinear shell theory. Proc Inst Mech Eng G J Aerosp Eng2024; 238(7): 687–710. https://doi.org/10.1177/09544100241234376
66.
AnhNDThinhNVTungHV. Nonlinear thermomechanical vibration of initially stressed functionally graded plates with porosities. Z Angew Math Mech2024; 104(2): e202300528.
67.
TungHVThinhNV. Thermomechanical nonlinear vibration of axially loaded functionally graded material cylindrical panels including porosity. AIAA J2025; 63(3): 1091–1105. https://doi.org/10.2514/1.j064613
68.
TungHVThinhNV. Nonlinear free vibration of geometrically imperfect porous FGM shell panels on nonlinear foundations including elastic edge restraints and elevated temperatures. Acta Mech2025; 236(2): 1091–1115.
69.
ThinhNVTungHV. Nonlinear thermal vibration and postbuckling of shear deformable porous FGM circular plates with geometric imperfection and elastic edge restraint. Mech Base Des Struct Mach2025; 53(10): 6765–6788. https://doi.org/10.1080/15397734.2025.2489068
70.
TrangLTNLongVTThinhNV, et al.Mechanical and thermal nonlinear buckling analysis of FGM shallow spherical caps with porosity and elastically restrained edge. Z Angew Math Mech2025; 105(10): e70264.
71.
ThinhNVTungHV. Nonlinear thermomechanical transient response of FGM circular plates and shallow spherical caps with porosities and tangentially restrained edges. J Aero Eng2026; 39(2): 04025140. https://doi.org/10.1061/jaeeez.aseng-6480
72.
ThinhNVTungHV. Nonlinear axisymmetric free vibration of thin FGM shallow spherical caps with porosity and tangentially restrained edges. International Journal of Computational Materials Science and Engineering. 2025; 2550042-1 to 2550042-26. https://doi.org/10.1142/S2047684125500423, In Press.
73.
HuangXLShenHS. Nonlinear vibration and dynamic response of functionally graded plates in thermal environments. Int J Solid Struct2004; 41: 2403–2427. https://doi.org/10.1016/j.ijsolstr.2003.11.012
74.
TouloukianYS. Thermophysical properties of high temperature solid materials. MacMillan, 1967.
