Dynamic stiffness formulation for the vibration response of functionally graded nanoplates considering the neutral surface position with microstructural defects
Restricted accessResearch articleFirst published online May, 2025
Dynamic stiffness formulation for the vibration response of functionally graded nanoplates considering the neutral surface position with microstructural defects
In the present paper, the dynamic stiffness matrix of a functionally graded nanoplate (FG-nP) has been developed and used to carry out a vibration analysis. The nonlocal elasticity theory in conjunction with classical plate theory (CPT) has been used to drive the governing equations of motion using Hamilton’s principle. The nonlocal elasticity theory incorporates the length scale parameter, which can withhold the small-scale effect that is necessary for model FG-nP. The pioneer Wittrick–Williams algorithm is employed to evaluate the dynamic stiffness matrix. The property of the FG-nP has been calculated using the power law. The authenticity and efficacy of the present formulation have been ascertained using various validation studies. The accurate prediction of the effect of porosity inclusions on different volume fraction indexes, boundary conditions, and higher vibration modes has been reported. The influence of the nonlocal parameter, boundary conditions, various gradation techniques, and geometric parameters on vibration behavior has been extensively explored. The results show the versatility of the proposed approach in addressing small scaled complex engineering structures. This research significantly contributes to the understanding and analysis of functionally graded nanoplates, providing valuable insights for applications in aerospace, structural systems, sensors, actuators, and energy harvesting devices.
KangTJFangSKozlovME, et al. Electrical power from nanotube and graphene electrochemical thermal energy harvesters. Adv Funct Mater2012; 22: 477–489.
2.
EbrahimyFHosseiniSHS. Nonlinear electroelastic vibration analysis of NEMS consisting of double-viscoelastic nanoplates. Appl Phys A Mater Sci Process2016; 122.
3.
LiuXMHuangZDOhSW, et al. Carbon nanotube (CNT)-based composites as electrode material for rechargeable Li-ion batteries: a review. Compos Sci Technol2012; 72: 121–144.
4.
GhayeshMHFarokhiHGholipourA, et al. Dynamic characterisation of functionally graded imperfect Kirchhoff microplates. Compos Struct2017; 179: 720–731.
5.
ChenLChinLCAshbyPD, et al. Single-walled carbon nanotube AFM probes: optimal imaging resolution of nanoclusters and biomolecules in ambient and fluid environments. Nano Lett2004; 4: 1725–1731.
6.
EbrahimiFDabbaghA.Magnetic field effects on thermally affected propagation of acoustical waves in rotary double-nanobeam systems. Waves Random Complex Media2021; 31: 25–45.
7.
EbrahimiFSeyfiADabbaghA.Dispersion of waves in FG porous nanoscale plates based on NSGT in thermal environment. Adv Nano Res2019; 7: 325–335.
8.
EbrahimiFDabbaghA.Wave dispersion characteristics of heterogeneous nanoscale beams via a novel porosity-based homogenization scheme. Eur Phys J Plus2019; 134: 157.
9.
EbrahimiFNopourRDabbaghA.Smart laminates with an auxetic ply rested on visco-Pasternak medium: active control of the system’s oscillation. Eng Comput2023; 39: 221–231.
10.
EbrahimiFDabbaghA.Wave propagation analysis of magnetostrictive sandwich composite nanoplates via nonlocal strain gradient theory. Proc IMechE, Part C: J Mechanical Engineering Science2018; 232: 4180–4192.
EbrahimiFDabbaghATornabeneF, et al. Hygro-thermal effects on wave dispersion responses of magnetostrictive sandwich nanoplates. Adv Nano Res7: 157–167.
13.
EringenAC.On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves. J Appl Phys1983; 54: 4703–4710.
14.
EringenACEdelenDGB. On nonlocal elasticity. Int J Eng Sci1972; 10: 233–248.
15.
LimCWZhangGReddyJN.A higher-order nonlocal elasticity and strain gradient theory and its applications in wave propagation. J Mech Phys Solids2015; 78: 298–313.
16.
ZenkourAM.Modified couple stress theory for micro-machined beam resonators with linearly varying thickness and various boundary conditions. Arch Mech Eng2018; 65: 43–64.
17.
ZareMNazemnezhadRHosseini-HashemiS.Natural frequency analysis of functionally graded rectangular nanoplates with different boundary conditions via an analytical method. Meccanica2015; 50: 2391–2408.
18.
NatarajanSChakrabortySThangavelM, et al. Size-dependent free flexural vibration behavior of functionally graded nanoplates. Comput Mater Sci2012; 65: 74–80.
19.
Hosseini-HashemiSBedroudMNazemnezhadR.An exact analytical solution for free vibration of functionally graded circular/annular Mindlin nanoplates via nonlocal elasticity. Compos Struct2013; 103: 108–118.
20.
AghababaeiRReddyJN.Nonlocal third-order shear deformation plate theory with application to bending and vibration of plates. J Sound Vib2009; 326: 277–289.
21.
ThaiHTVoTP.A size-dependent functionally graded sinusoidal plate model based on a modified couple stress theory. Compos Struct2013; 96: 376–383.
22.
EbrahimiFKhosraviKDabbaghA.A novel spatial–temporal nonlocal strain gradient theorem for wave dispersion characteristics of FGM nanoplates. Waves Random Complex Media2024; 34: 3490–3509.
23.
EbrahimiFKhosraviKDabbaghA.Wave dispersion in viscoelastic FG nanobeams via a novel spatial–temporal nonlocal strain gradient framework. Waves Random Complex Media2024; 34: 2962–2984.
24.
EbrahimiFGhazaliMDabbaghA. Hygro-thermo-viscoelastic wave propagation analysis of FGM nanoshells via nonlocal strain gradient fractional time–space theory. Waves Random Complex Media2022: 1–20.
25.
UzunBCivalekÖYaylıMÖ. Vibration of FG nano-sized beams embedded in Winkler elastic foundation and with various boundary conditions. Mech Based Des Struct Mach2020: 1–20.
26.
TalebizadehsardariPSalehipourHShahgholian-GhahfarokhiD, et al. Free vibration analysis of the macro-micro-nano plates and shells made of a material with functionally graded porosity: a closed-form solution. Mech Based Des Struct Mach2022; 50: 1054–1080.
27.
FattahiAMSahmaniSAhmedNA.Nonlocal strain gradient beam model for nonlinear secondary resonance analysis of functionally graded porous micro/nano-beams under periodic hard excitations. Mech Based Des Struct Mach2020; 48: 403–432.
28.
WangHChenC-SHsuC-Y, et al. Vibration and stability of initially stressed functionally graded carbon nanotube-reinforced hybrid composite plates in thermal environments. Mech Based Des Struct Mach2022; 50: 1298–1313.
29.
DabbaghAGolpaygani SaniSEbrahimiF.Piezoelectrically controlled wave propagation in laminates with auxetic core: transient analysis incorporated with electrical stability monitoring. Eur Phys J Plus2023; 138: 801.
30.
GarousiMHKarimiMCasoliP, et al. Vibration analysis of a centrifugal pump with healthy and defective impellers and fault detection using multi-layer perceptron. Eng2024; 5: 2511–2530.
31.
ZhuJLaiZYinZ, et al. Fabrication of ZrO2–NiCr functionally graded material by powder metallurgy. Mater Chem Phys2001; 68: 130–135.
32.
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.
33.
WattanasakulpongNUngbhakornV.Linear and nonlinear vibration analysis of elastically restrained ends FGM beams with porosities. Aerosp Sci Technol2014; 32: 111–120.
34.
AllahkaramiFTohidiHDimitriR, et al. Dynamic stability of bi-directional functionally graded porous cylindrical shells embedded in an elastic foundation. Appl Sci2020; 10: 1345.
35.
FanFXuYSahmaniS, et al. Modified couple stress-based geometrically nonlinear oscillations of porous functionally graded microplates using NURBS-based isogeometric approach. Comput Methods Appl Mech Eng2020; 372: 113400.
36.
ShahverdiHBaratiMR.Vibration analysis of porous functionally graded nanoplates. Int J Eng Sci2017; 120: 82–99.
37.
BaratiMRShahverdiH.Forced vibration of porous functionally graded nanoplates under uniform dynamic load using general nonlocal stress–strain gradient theory. J Vib Control2018; 24: 4700–4715.
38.
MarjanovićMKolarevićNNefovska-DanilovićM, et al. Free vibration study of sandwich plates using a family of novel shear deformable dynamic stiffness elements: limitations and comparison with the finite element solutions. Thin-Walled Struct2016; 107: 678–694.
39.
BanerjeeJR. Dynamic stiffness formulation for structural elements: a general approach. Comput Struct1997; 63: 101–103.
40.
ShamsSTorabiARNarabMF, et al. Free vibration analysis of a laminated beam using dynamic stiffness matrix method considering delamination. Thin-Walled Struct2021; 166: 107952.
41.
BanerjeeJR. Review of the dynamic stiffness method for free-vibration analysis of beams. Transp Saf Environ2019; 1: 106–116.
42.
LiuXBanerjeeJR. An exact spectral-dynamic stiffness method for free flexural vibration analysis of orthotropic composite plate assemblies – part I: theory. Compos Struct2015; 132: 1274–1287.
43.
KolarevicNMarjanovićMNefovska-DanilovicM, et al. Free vibration analysis of plate assemblies using the dynamic stiffness method based on the higher order shear deformation theory. J Sound Vib2016; 364: 110–132.
44.
ZhangCJinGYeT, et al. Harmonic response analysis of coupled plate structures using the dynamic stiffness method. Thin-Walled Struct2018; 127: 402–415.
45.
AliMIAzamMSRanjanV, et al. Free vibration of sigmoid functionally graded plates using the dynamic stiffness method and the Wittrick-Williams algorithm. Comput Struct2021; 244: 106424.
46.
KumarSRanjanVJanaP.Free vibration analysis of thin functionally graded rectangular plates using the dynamic stiffness method. Compos Struct2018; 197: 39–53.
47.
KumarSJanaP.Application of dynamic stiffness method for accurate free vibration analysis of sigmoid and exponential functionally graded rectangular plates. Int J Mech Sci2019; 163: 105105.
48.
EringenAWegnerJ.Nonlocal continuum field theories. Appl Mech Rev2003; 56: B20–B22.
49.
LiQIuVPKouKP.Three-dimensional vibration analysis of functionally graded material sandwich plates. J Sound Vib2008; 311: 498–515.
50.
HarrisonHRNettletonT.Hamilton’s principle. In: HarrisonHRNettletonT (eds.) Advanced engineering dynamics. New York, NY: Elsevier, 1997, pp.46–54.
51.
ThaiHTKimSE.Levy-type solution for free vibration analysis of orthotropic plates based on two variable refined plate theory. Appl Math Model2012; 36: 3870–3882.
52.
BoscoloMBanerjeeJR. Dynamic stiffness elements and their applications for plates using first order shear deformation theory. Comput Struct2011; 89: 395–410.
53.
GuptaATalhaM.Recent development in modeling and analysis of functionally graded materials and structures. Prog Aerosp Sci2015; 79: 1–14.
54.
ToudehdehghanALimJWFooKE, et al. A brief review of functionally graded materials. MATEC Web Conf2017; 131: 1–6.
55.
El HartiKRahmouneMSanbiM, et al. Finite element model of vibration control for an exponential functionally graded Timoshenko beam with distributed piezoelectric sensor/actuator. Actuators2019; 8: 19.
56.
BelarbiMLiLSidM, et al. Nonlocal vibration of functionally graded nanoplates using a layerwise theory. Math Mech Solids2022; 27: 2634–2661.
