The investigation conducted in this paper aims to study free vibration and buckling behaviors of size-dependent functionally graded sandwich nanobeams. In order to take into account the small size effects, nonlocal elasticity theory of Eringen's is incorporated. Material properties of the functionally graded sandwich beams are supposed to change continuously through the thickness direction according to two forms of the volume fraction of constituents by power law functionally graded material and sigmoid law functionally graded material. These rules are modified to consider the effect of porosity, which covers four kinds of porosity distributions. Two types of sandwich nanobeams were provided: (a) homogeneous core and functionally graded skins and (b) functionally graded core and homogeneous skins. Third-order shear deformation theory without any shear correction factor in conjunction with Hamilton's principle is used to extract the governing equations of motions of porous functionally graded sandwich nanobeams and then solved analytically for two hinged ends. The effects of nonlocal parameter, length to thickness ratios, material graduation index, amount of porosity, porosity distribution shape, on the nondimensional frequency and critical buckling load of the functionally graded sandwich nanobeams made of porous materials are exhibited by a parametric study.
ThaiH-TVoTP. Bending and free vibration of functionally graded beams using various higher-order shear deformation beam theories. Int J Mech Sci2012; 62: 57–66.
2.
Ould LarbiLKaciAHouariMSA, et al.An efficient shear deformation beam theory based on neutral surface position for bending and free vibration of functionally graded beams. Mech Based Des Struct Mach2013; 41: 421–433.
3.
BensaidICheikhAMangouchiA, et al.Static deflection and dynamic behavior of higher-order hyperbolic shear deformable compositionally graded beams. Adv Mater Res2017; 6: 13–26.
4.
ZenkourAM. A comprehensive analysis of functionally graded sandwich plates: part 2 – buckling and free vibration. J Sol Struct2005; 42: 5243–5258.
5.
AmiraniMCKhaliliSMRNematiN. Free vibration analysis of sandwich beam with FG core using the element free Galerkin method. Compos Struct2009; 90: 373–379.
6.
ZenkourASobhyM. Thermal buckling of various types of FGM sandwich plates. Compos Struct2010; 93: 93–102.
7.
NatarajanSManickamG. Bending and vibration of functionally graded material sandwich plates using an accurate theory. Finite Elem Anal Des2012; 57: 32–42.
8.
YaghoobiHYaghoobiP. Buckling analysis of sandwich plates with FGM face sheets resting on elastic foundation with various boundary conditions: an analytical approach. Meccanica2013; 48: 2019–2035.
9.
BhangaleRKGanesanN. Thermoelastic buckling and vibration behavior of a functionally graded sandwich beam with constrained viscoelastic core. J Sound Vib2006; 295: 294–316.
10.
BuiTQKhosravifardAZhangC, et al.Dynamic analysis of sandwich beams with functionally graded core using a truly meshfree radial point interpolation method. Eng Struct2013; 47: 90–104.
11.
BouradaMTounsiAHouariMSA, et al.A new four-variable refined plate theory for thermal buckling analysis of functionally graded sandwich plates. J Sandw Struct Mater2012; 14: 5–33.
12.
TounsiAHouariMSABenyoucefS, et al.A refined trigonometric shear deformation theory for thermoelastic bending of functionally graded sandwich plates. Aerosp Sci Technol2013; 24: 209–220.
13.
BessaimAHouariMSATounsiA, et al.A new higher-order shear and normal deformation theory for the static and free vibration analysis of sandwich plates with functionally graded isotropic face sheets. J Sandw Struct Mater2013; 15: 671–703.
NguyenT-KNguyenTT-PVoTP, et al.Vibration and buckling analysis of functionally graded sandwich beams by a new higher-order shear deformation theory. Compos B2015; 76: 273–285.
16.
BennaiRAit AtmaneHTounsiA. A new higher order shear and normal deformation theory for functionally graded sandwich beams. Steel Compos Struct2015; 19: 521–546.
17.
OsoferoIVoTPNguyenT-K, et al.Analytical solution for vibration and buckling of functionally graded sandwich beams using various quasi-3D theories. J Sandw Struct Mater2015; 18: 3–29.
18.
NguyenT-KVoTPNguyenB-D, et al.An analytical solution for buckling and vibration analysis of functionally graded sandwich beams using a quasi-3D shear deformation theory. Compos Struct2016; 156: 238–252.
19.
DraouiAZidourMTounsiA, et al.Static and dynamic behavior of nanotubes-reinforced sandwich plates using (FSDT). J Nano Res2019; 57: 117–135.
20.
ZergaDTounsiABousahlaAA, et al.Thermomechanical bending study for functionally graded sandwich plates using a simple quasi-3D shear deformation theory. Steel Compos Struct2015; 32: 389–410.
21.
MahmoudiABenyoucefSTounsiA, et al.A refined quasi-3D shear deformation theory for thermo-mechanical behavior of functionally graded sandwich plates on elastic foundations. J Sandw Struct Mater2019; 21: 1906–1926.
22.
LavrikNVSepaniakMJDatskosPG. Cantilever transducers as a platform for chemical and biological sensors. Rev Sci Instrum2004; 75: 2229–2253.
RahmaniOPedramO. Analysis and modeling the size effect on vibration of functionally graded nanobeams based on nonlocal Timoshenko beam theory. J Eng Sci2014; 77: 55–70.
25.
AlimirzaeiAMohammadimehrMTounsiA. Nonlinear analysis of viscoelastic micro-composite beam with geometrical imperfection using FEM: MSGT electro-magneto-elastic bending, buckling and vibration solutions. Struct Eng Mech2019; 71: 485–502.
26.
Adda BediaWHouariMSABessaimA, et al.A new hyperbolic two-unknown beam model for bending and buckling analysis of a nonlocal strain gradient nanobeams. J Nano Res2019; 57: 175–191.
27.
ShaatMFaroughiS. Influence of long-range interatomic and interlayer interactions on dispersion of acoustic waves by multilayer graphene. Physi E: Low-Dimen Sys and Nanostr2019; 108: 74–82.
28.
AnsariRSahmaniS. Bending behavior and buckling of nanobeams including surface stress effects corresponding to different beam theories. Int J Eng Sci2012; 49: 1244–1255.
29.
EringenACEdelenDGB. On nonlocal elasticity. Int J Eng Sci1972; 10: 233–248.
30.
EringenAC. On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves. J Appl Phys1983; 54: 4703–4710.
31.
ReddyJ. Nonlocal theories for bending, buckling and vibration of beams. Int J Eng Sci2007; 45: 288–307.
32.
ThaiH-T. A nonlocal beam theory for bending, buckling, and vibration of nanobeams. Int J Eng Sci2012; 52: 56–64.
33.
ThaiH-TThucPV. A nonlocal sinusoidal shear deformation beam theory with application to bending, buckling, and vibration of nanobeams. Int J Eng Sci2012; 54: 58–66.
34.
ŞimşekMYurtcuH. Analytical solutions for bending and buckling of functionally graded nanobeams based on the nonlocal Timoshenko beam theory. Compos Struct2013; 97: 378–386.
35.
TounsiABenguediabSHouariMSA, et al.A new nonlocal beam theory with thickness stretching effect for nanobeams. Int J Nanosci2013; 12: 1350025.
36.
BenguediabSSemmahALarbi ChahtF, et al.An investigation on the characteristics of bending, buckling and vibration of nanobeams via nonlocal beam theory. J Compos Meth2014; 11: 13500851–14.
37.
EbrahimiFSalariE. Thermal buckling and free vibration analysis of size dependent Timoshenko FG nanobeams in thermal environments. Compos Struct2015; 128: 363–380.
38.
Larbi ChahtFKaciAHouariMSA, et al.Bending and buckling analyses of functionally graded material (FGM) size-dependent nanoscale beams including the thickness stretching effect. Steel Compos Struct2015; 18: 425–442.
39.
ZemriAHouariMSABousahlaAA, et al.A mechanical response of functionally graded nanoscale beam: an assessment of a refined nonlocal shear deformation theory beam theory. Struct Eng Mech2015; 54: 693–710.
40.
BensaidI. A refined nonlocal hyperbolic shear deformation beam model for bending and dynamic analysis of nanoscale beams. Adv Nano Res Int J2017; 5: 113–126.
41.
EbrahimiFBaratiMR. A unified formulation for dynamic analysis of nonlocal heterogeneous nanobeams in hygro-thermal environment. Appl Phys A2016; 122: 792.
42.
EbrahimiFBaratiMR. A nonlocal strain gradient refined beam model for buckling analysis of size-dependent shear-deformable curved FG nanobeams. Compos Struct2017; 159: 174–182.
43.
LiLLiXHuY. Free vibration analysis of nonlocal strain gradient beams made of functionally graded material. Int J Eng Sci2016; 102: 77–92.
44.
BensaidIBekhaddaA. Thermal stability analysis of temperature dependent inhomogeneous size-dependent nano-scale beams. Adv Mater Res2018; 7: 363–378.
45.
HussainMNaeemMNTounsiA, et al.Nonlocal effect on the vibration of armchair and zigzag SWCNTs with bending rigidity. Adv Nano Res2019; 7: 431–442.
46.
TlidjiYZidourMDraicheK, et al.Vibration analysis of different material distributions of functionally graded microbeam. Struct Eng Mech2019; 69: 637–649.
47.
SemmahAHeirecheHBousahlaAA, et al.Thermal buckling analysis of SWBNNT on Winkler foundation by non local FSDT. Adv Nano Res2019; 7: 89–98.
48.
KaramiBJanghorbanMTounsiA. Galerkin's approach for buckling analysis of functionally graded anisotropic nanoplates/different boundary conditions. Eng Comp2019; 35: 1297–1316.
49.
KaramiBJanghorbanMTounsiA. On pre stressed functionally graded anisotropic nanoshell in magnetic field. J Braz Soc Mech Sci Eng2019; 41: 495.
50.
KaramiBJanghorbanMTounsiA. Wave propagation of functionally graded anisotropic nanoplates resting on Winkler-Pasternak foundation. Stru Eng Mech2019; 7: 55–66.
51.
KaramiBShahsavariDJanghorbanM, et al.Resonance behavior of functionally graded polymer composite nanoplates reinforced with grapheme nanoplatelets. Int J Mech Sci2019; 156: 94–105.
52.
KaramiBJanghorbanMTounsiA. On exact wave propagation analysis of triclinic material using three dimensional bi-Helmholtz gradient plate model. Struct Eng Mech2019; 69: 487–497.
53.
ZhangKGeMHZhaoC. Free vibration of nonlocal Timoshenko beams made of functionally graded materials by Symplectic method. Compos Part B: Eng2019; 156: 174–184.
54.
ArashABabaeiAFaroughiS. Vibration characteristics of functionally graded micro-beam carrying an attached mass. Mech Adv Compos Struct2020; 7: 49–58.
55.
ShaatMFaroughiS. Influence of long-range interatomic and interlayer interactions on dispersion of acoustic waves by multilayer graphene. Eur J of Mech – A/Sol2018; 71: 365–377.
56.
AriaALFriswellMI. A nonlocal finite element model for buckling and vibration of functionally graded nanobeams. Compos Par B: Eng2019; 166: 233–246.
57.
ZhuJLaiZYinZ, et al.Fabrication of ZrO2-NiCr functionally graded material by powder metallurgy. Mater Chem Phys2001; 68: 130–135.
58.
WattanasakulpongNPrustyBGKellyDW, et al.Free vibration analysis of layered functionally graded beams with experimental validation. Mater Des2012; 36: 182–190.
59.
RezaeiASaidiA. Application of Timoshenko unified formulation to study the effect of porosity on natural frequencies of thick porous-cellular plates. Compos Part B: Eng2016; 91: 361–370.
60.
ChenDYangJKitipornchaiS. Elastic buckling and static bending of shear deformable functionally graded porous beam. Compos Struct2015; 133: 54–61.
61.
WattanasakulpongNUngbhakornV. Linear and nonlinear vibration analysis of elastically restrained ends FGM beams with porosities. Aeros Sci Technol2014; 32: 111–120.
62.
WattanasakulpongNChaikittiratanaA. Flexural vibration of imperfect functionally graded beams based on Timoshenko beam theory: Chebyshev collocation method. Meccan2015; 50: 1–12.
63.
YahiaSAAtmaneHAHouariMSA, et al.Wave propagation in functionally graded plates with porosities using various higher-order shear deformation plate theories. Struct Eng Mech2015; 53: 1143–1165.
64.
EbrahimiFGhasemiFSalariE. Investigating thermal effects on vibration behavior of temperature-dependent compositionally graded Euler beams with porosities. Meccan2016; 51: 223–249.
65.
MechabIMechabBBenaissaS, et al.Free vibration analysis of FGM nanoplate with porosities resting on Winkler elastic foundations based on two-variable refined plate theories. J Brazil Soc Mech Sci Eng2016; 38: 1–19.
66.
BensaidIGuenanouA. Bending and stability analysis of size-dependent compositionally graded Timoshenko nanobeams with porosities. Adv Mater Res2017; 1: 45–63.
67.
AddouFYMradjahMBousahlaAA, et al.Influences of porosity on dynamic response of FG plates resting on Winkler/Pasternak/Kerr foundation using quasi 3D HSDT. Comput Concr2019; 24: 347–367.
68.
MedaniMBenahmedAZidourM, et al.Static and dynamic behavior of (FG-CNT) reinforced porous sandwich plate. Steel Comput Struct2019; 32: 595–610.
69.
BerghoutiHAdda BediaEABenkheddaA, et al.Vibration analysis of nonlocal porous nanobeams made of functionally graded material. Adv Nano Res2019; 7: 351–364.
70.
BouradaFBousahlaAABouradaM, et al.Dynamic investigation of porous functionally graded beam using a sinusoidal shear deformation theory. Wind Struct2019; 28: 19–30.
71.
FaroughiSRahmaniAFriswellMI. On wave propagation in two-dimensional functionally graded porous rotating nano-beams using a general nonlocal higher-order beam model. Appl Math Model2020; 80: 169–190.
72.
ThaiHTVoTPNguyenTK, et al.Size-dependent behavior of functionally graded sandwich microbeams based on the modified couple stress theory. Compos Struct2012; 43: 337–349.
73.
TrinhLCVoTPThaiHT, et al.State-space Levy solution for size-dependent static, free vibration and buckling behaviours of functionally graded sandwich plates. Compos B2015; 123: 144–164.
74.
ŞimşekMAl-ShujairiM. Static, free and forced vibration of functionally graded (FG) sandwich beams excited by two successive moving harmonic load. Compos B2017; 108: 18–34.
75.
DraicheKTounsiAMahmoudSR. A refined theory with stretching effect for the flexure analysis of laminated composite plates. Geomech Eng Int J2016; 11: 671–690.
76.
KhilounMBousahlaAAKaciA, et al.Analytical modeling of bending and vibration of thick advanced composite plates using a four variable quasi 3D HSDT. Eng with Comput2019. https://doi.org/10.1007/s00366-019-00732-1.
77.
ChaabaneLABouradaFSekkalM, et al.Analytical study of bending and free vibration responses of functionally graded beams resting on elastic foundation. Struct Eng Mech Int J2019; 71: 185–196.
78.
AbualnourMHouariMSATounsiA, et al.A novel quasi-3D trigonometric plate theory for free vibration analysis of advanced composite plates. Compos Struct2018; 184: 688–697.
79.
ZaouiFZOuinasDTounsiA. New 2D and quasi-3D shear deformation theories for free vibration of functionally graded plates on elastic foundations. Compos Part B2019; 159: 231–247.
80.
MahmoudSRTounsiA. On the stability of isotropic and composite thick plates. Steel Compos Struct2019; 33: 551–568.
81.
BelbachirNDraichKBousahlaAA, et al.Bending analysis of anti-symmetric cross-ply laminated plates under nonlinear thermal and mechanical loadings. Steel Compos Struct2019; 33: 913–924.
82.
WangQ. Wave propagation in carbon nanotubes via nonlocal continuum mechanics. J Appl Phys2005; 98: 124301.
83.
HeirecheHTounsiABenzairA, et al.Sound wave propagation in singlewalled carbon nanotubes using nonlocal elasticity. Phys E2008; 40: 2791–2799.
84.
HeirecheHTounsiABenzairA. Scale effect on wave propagation of double-walled carbon nanotubes with initial axial loading. Nanotechnology2008; 19: 185703.
85.
TounsiASemmahABousahlaAA. Thermal buckling behavior of nanobeams using an efficient higher-order nonlocal beam theory. ASCE J Nanomech Micromech2013; 3: 37–42.
86.
EltaherMAEmamSAMahmoudFF. Free vibration analysis of functionally graded size-dependent nanobeams. Appl Math Comput2012; 218: 7406–7420.
87.
VoTPThaiHTNguyenTK, et al.Finite element model for vibration and buckling of functionally graded sandwich beams based on a refined shear deformation theory. Eng Struct2014; 64: 12–22.
