Free vibration analysis of nonlocal nanobeams: a comparison of the one-dimensional nonlocal integral Timoshenko beam theory with the two-dimensional nonlocal integral elasticity theory
Restricted accessResearch articleFirst published online April, 2022
Free vibration analysis of nonlocal nanobeams: a comparison of the one-dimensional nonlocal integral Timoshenko beam theory with the two-dimensional nonlocal integral elasticity theory
Beam theories such as the Timoshenko beam theory are in agreement with the elasticity theory. However, due to the different nonlocal averaging processes, they are expected to yield different results in their nonlocal forms. In the present work, the free vibration behavior of nonlocal nanobeams is studied using the nonlocal integral Timoshenko beam theory (NITBT) and two-dimensional nonlocal integral elasticity theory (2D-NIET) with different kernels and their results are compared. A new kernel, termed the compensated two-phase (CTP) kernel, is introduced, which entirely compensates for the boundary effects and does not suffer from the ill-posedness of previous kernels. Using the finite element method, the free vibration analysis is performed for different boundary conditions based on the first three natural frequencies. For both the NITBT and 2D-NIET with both the two-phase (TP) and CTP kernels, the nonlocal parameter has a softening effect on the natural frequencies for all the boundary conditions, without observing the paradoxical behaviors of the nonlocal differential theory. For both theories, the softening effect of the nonlocal parameter is more pronounced for the TP kernel compared to the CTP kernel. The sensitivity of the 2D-NIET to the nonlocal parameter is found to be higher than that of the NITBT. Also, the softening effects for different vibration modes are compared to each other for both theories and both kernels. The obtained results can be extended for various important beam problems with nonlocal effects and help obtain a better understanding of applicable nonlocal theories.
MindlinRDEshelNN.On first strain-gradient theories in linear elasticity. Int J Solids Struct1968; 4: 109–124.
2.
LamDCCYangFChongACM, et al. Experiments and theory in strain gradient elasticity. J Mech Phys Solids2003; 51: 1477–1508.
3.
AskesHAifantisEC.Gradient elasticity in statics and dynamics: an overview of formulations, length scale identification procedures, finite element implementations and new results. Int J Solids Struct2011; 48: 1962–1990. https://doi.org/10.1016/j.ijsolstr.2011.03.006.
4.
NiiranenJBalobanovVKiendlJ, et al. Variational formulations, model comparisons and numerical methods for Euler–Bernoulli micro- and nano-beam models. Math Mech Solids2017; 24: 312–335.
5.
dell’IsolaFSteigmannD.A Two-dimensional gradient-elasticity theory for woven fabrics. J Elast2015; 118: 113–125.
6.
GiorgioIGrygorukRdell’IsolaF, et al. Pattern formation in the three-dimensional deformations of fibered sheets. Mech Res Commun2015; 69: 164–171.
7.
AndreausUdell’IsolaFGiorgioI, et al. Numerical simulations of classical problems in two-dimensional (non) linear second gradient elasticity. Int J Eng Sci2016; 108: 34–50.
8.
ToupinRA.Elastic materials with couple-stresses. Arch Ration Mech Anal1962; 11: 385–414.
9.
ToupinRA.Theories of elasticity with couple-stress. Arch Ration Mech Anal1964; 17: 85–112.
10.
AlaviSESadighiMPazhoohMD, et al. Development of size-dependent consistent couple stress theory of Timoshenko beams. Appl Math Model2020; 79: 685–712.
11.
MisraAPoorsolhjouyP.Granular micromechanics based micromorphic model predicts frequency band gaps. Contin Mech Thermodyn2016; 28: 215–234.
12.
GrekovaEFPorubovAVdell’IsolaF.Reduced linear constrained elastic and viscoelastic homogeneous cosserat media as acoustic metamaterials. Symmetry (Basel)2020; 12: 521.
13.
EringenACEdelenDGB. On nonlocal elasticity. Int J Eng Sci1972; 10: 233–248.
14.
EringenAC.Linear theory of nonlocal elasticity and dispersion of plane waves. Int J Eng Sci1972; 10: 425–435.
15.
dell’IsolaFAndreausUPlacidiL.At the origins and in the vanguard of peridynamics, non-local and higher-gradient continuum mechanics: an underestimated and still topical contribution of Gabrio Piola. Math Mech Solids2015; 20: 887-928.
16.
dell’IsolaFDella CorteAEspositoR, et al. Some cases of unrecognized transmission of scientific knowledge: from antiquity to Gabrio Piola’s peridynamics and generalized continuum theories. In: AltenbachHForestS (eds) Generalized continua as models for classical and advanced materials. Cham: Springer, 2016, 42, 77–128.
JavanbakhtMLevitasVI.Interaction between phase transformations and dislocations at the nanoscale. Part 2: phase field simulation examples. J Mech Phys Solids2015; 82: 164–185.
19.
JavanbakhtMLevitasVI.Phase field simulations of plastic strain-induced phase transformations under high pressure and large shear. Phys Rev B2016; 94: 214104.
20.
JavanbakhtMAdaeiM.Formation of stress- and thermal-induced martensitic nanostructures in a single crystal with phase-dependent elastic properties. J Mater Sci2020; 5: 2544–2563.
21.
JavanbakhtMRahbarHAshourianM. Explicit nonlinear finite element approach to the Lagrangian-based coupled phase field and elasticity equations for nanoscale thermal- and stress-induced martensitic transformations. Continuum Mech Thermodyn2020.
22.
LevitasVIJafarzadehHFarrahiGH, et al. Thermodynamically consistent and scale-dependent phase field approach for crack propagation allowing for surface stresses. Int J Plast2018; 111: 1–35.
23.
JafarzadehHLevitasVIFarrahiGH, et al. Phase field approach for nanoscale interactions between crack propagation and phase transformation. Nanoscale2019; 11: 22243–22247.
24.
JavanbakhtMGhaediMS.Phase field approach for void dynamics with interface stresses at the nanoscale. Int J Eng Sci2020; 154: 103279.
25.
JavanbakhtMGhaediMSBarchiesiE, et al. The effect of a pre-existing nanovoid on martensite formation and interface propagation: a phase field study. Math Mech Solids2020: 1–20.
26.
JavanbakhtMGhaediMS.Nanovoid induced multivariant martensitic growth under negative pressure: effect of misfit strain and temperature on PT threshold stress and phase evolution. Mech Mater2020; 1–16.
27.
JavanbakhtMGhaediMS.Nanovoid induced martensitic growth under uniaxial stress: effect of misfit strain, temperature and nanovoid size on PT threshold stress and nanostructure in NiAl. Comput Mater Sci2020; 184: 103627.
28.
JavanbakhtMSadegh GhaediM.Thermal induced nanovoid evolution in the vicinity of an immobile austenite-martensite interface. Comput Mater Sci2020; 1–12.
29.
IslandF. et al. Of a controversial principle of Lagrange’s analytical mechanics and its multiple applications. In: dell’IsolaF.MaierG.PeregoU.AndreausU.EspositoR.ForestS. (eds) The complete works of Gabrio Piola: Volume I. Advanced Structured Materials. 2014, 38. Springer, Cham.
30.
KrönerE.Elasticity theory of materials with long range cohesive forces. Int J Solids Struct1967; 3: 731–742.
31.
KuninIA.On foundations of the theory of elastic media with microstructure. Int J Eng Sci1984; 22: 969–978.
32.
KrumhanslJA. Some considerations of the relation between solid state physics and generalized continuum mechanics. In: KrönerE (ed.) Mechanics of generalized continua. Berlin, Heidelberg: Springer, 1968, 298–311.
33.
EdelenDGBLawsN.On the thermodynamics of systems with nonlocality. Arch Ration Mech Anal1971; 43: 24–35.
34.
EringenACKimBS.Stress concentration at the tip of crack. Mech Res Commun1974; 1: 233–237.
35.
EringenACSpezialeCGKimBS.Crack-tip problem in non-local elasticity. J Mech Phys Solids1977; 25: 339–355.
36.
EringenAC.Line crack subject to shear. Int J Fract1978; 14: 367–379.
37.
EringenAC.Line crack subject to antiplane shear. Eng Fract Mech1979; 12: 211-219.
38.
EringenAC.Theory of nonlocal elasticity and some applications. Princeton, NJ: Princeton Univ Dept of Civil Engineering, 1984.
39.
AltanSB.Uniqueness of initial-boundary value problems in nonlocal elasticity. Int J Solids Struct1989; 25: 1271-1278.
40.
AltanSB.Existence in nonlocal elasticity. Arch Mech1989; 41: 25–36.
41.
EvgrafovABellidoJC.From non-local Eringen’s model to fractional elasticity. Math Mech Solids2019; 24: 1935–1953.
42.
EringenAC.On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves. J Appl Phys1983; 54: 4703.
43.
ReddyJN.Nonlocal theories for bending, buckling and vibration of beams. Int J Eng Sci2007; 45: 288–307.
44.
AydogduM.A general nonlocal beam theory: its application to nanobeam bending, buckling and vibration. Phys E Low Dim Syst Nanostruct2009; 41: 1651–1655.
45.
MalikanM.On the buckling response of axially pressurized nanotubes based on a novel nonlocal beam theory. J Appl Comput Mech2019; 5: 103–112.
46.
MalikanMTornabeneFDimitriR.Nonlocal three-dimensional theory of elasticity for buckling behavior of functionally graded porous nanoplates using volume integrals. Mater Res Express2018; 5: 95006.
47.
NguyenVBMalikanM.A novel one-variable first-order shear deformation theory for biaxial buckling of a size-dependent plate based on Eringen’s nonlocal differential law. World J Eng2018; 15: 633–645.
48.
LuPLeeHPLuC, et al. Dynamic properties of flexural beams using a nonlocal elasticity model. J Appl Phys2006; 99: 73510.
49.
WangCMZhangYYHeXQ.Vibration of nonlocal Timoshenko beams. Nanotechnology2007; 18: 105401.
50.
PeddiesonJBuchananGRMcNittRP.Application of nonlocal continuum models to nanotechnology. Int J Eng Sci2003; 41: 305–312.
51.
KhodabakhshiPReddyJN.A unified integro-differential nonlocal model. Int J Eng Sci2015; 95: 60–75.
52.
Fernández-SáezJZaeraRLoyaJA, et al. Bending of Euler-Bernoulli beams using Eringen’s integral formulation: a paradox resolved. Int J Eng Sci2016; 99: 107–116.
53.
TunaMKircaM.Exact solution of Eringen’s nonlocal integral model for bending of Euler-Bernoulli and Timoshenko beams. Int J Eng Sci2016; 105: 80–92.
54.
BažantZPJirásekM.Nonlocal integral formulations of plasticity and damage: survey of progress. J Eng Mech2002; 128: 1119–1149.
PolizzottoC. Remarks on some aspects of nonlocal theories in solid mechanics. In: Proceedings of the 6th Congress of Italian Society for Applied and Industrial Mathematics (SIMAI), 2002.
DaneshH.JavanbakhtM.Mohammadi AghdamM. A comparative study of 1D nonlocal integral Timoshenko beam and 2D nonlocal integral elasticity theories for bending of nanoscale beams. Continuum Mech. Thermodyn2021.
59.
EptaimerosKGKoutsoumarisCCKaryofyllisIG.Eigenfrequencies of microtubules embedded in the cytoplasm by means of the nonlocal integral elasticity. Acta Mech2020; 231: 1669–1684.
60.
FuschiPPisanoAAPolizzottoC.Size effects of small-scale beams in bending addressed with a strain-difference based nonlocal elasticity theory. Int J Mech Sci2019; 151: 661–671.
61.
RomanoGBarrettaR.Stress-driven versus strain-driven nonlocal integral model for elastic nano-beams. Compos Part B Eng2017; 114: 184–188.
62.
BarrettaRCaporaleAFaghidianSA, et al. A stress-driven local-nonlocal mixture model for Timoshenko nano-beams. Compos Part B Eng2019; 164: 590–598.
63.
PisanoAAFuschiP.Closed form solution for a nonlocal elastic bar in tension. Int J Solids Struct2003; 40: 13–23.
64.
MikhasevGAvdeichikEPrikazchikovD.Free vibrations of nonlocally elastic rods. Math Mech Solids2018; 24: 1279–1293.
65.
YanJWTongLHLiC, et al. Exact solutions of bending deflections for nano-beams and nano-plates based on nonlocal elasticity theory. Compos Struct2015; 125: 304–313.
66.
WangYHuangKZhuX, et al. Exact solutions for the bending of Timoshenko beams using Eringen’s two-phase nonlocal model. Math Mech Solids2018; 24: 559–572.
67.
PolizzottoC.Nonlocal elasticity and related variational principles. Int J Solids Struct2001; 38: 7359–7380.
68.
PisanoAASofiAFuschiP.Nonlocal integral elasticity: 2D finite element based solutions. Int J Solids Struct2009; 46: 3836–3849.
69.
PhadikarJKPradhanSC.Variational formulation and finite element analysis for nonlocal elastic nanobeams and nanoplates. Comput Mater Sci2010; 49: 492–499.
70.
AnsariRRajabiehfardRArashB.Nonlocal finite element model for vibrations of embedded multi-layered graphene sheets. Comput Mater Sci2010; 49: 831–838.
71.
PradhanSCMandalU.Finite element analysis of CNTs based on nonlocal elasticity and Timoshenko beam theory including thermal effect. Phys E Low Dim Syst Nanostruct2013; 53: 223–232.
72.
AriaAIFriswellMI.A nonlocal finite element model for buckling and vibration of functionally graded nanobeams. Compos Part B Eng2019; 166: 233–246.
73.
NorouzzadehAAnsariR.Finite element analysis of nano-scale Timoshenko beams using the integral model of nonlocal elasticity. Phys E Low Dim Syst Nanostruct2017; 88: 194–200.
74.
TunaMKircaM.Bending, buckling and free vibration analysis of Euler-Bernoulli nanobeams using Eringen’s nonlocal integral model via finite element method. Compos Struct2017; 179: 269–284.
75.
EptaimerosKGKoutsoumarisCCTsamasphyrosGJ.Nonlocal integral approach to the dynamical response of nanobeams. Int J Mech Sci2016; 115–116: 68–80.
76.
NaghinejadMOvesyHR.Free vibration characteristics of nanoscaled beams based on nonlocal integral elasticity theory. J Vib Control2017; 24: 3974–3988.
77.
AnsariRTorabiJNorouzzadehA.Bending analysis of embedded nanoplates based on the integral formulation of Eringen’s nonlocal theory using the finite element method. Phys B Condens Matter2018; 534: 90–97.
78.
FakherMRahmanianSHosseini-HashemiS.On the carbon nanotube mass nanosensor by integral form of nonlocal elasticity. Int J Mech Sci2019; 150: 445–457.
79.
FakherM.Hosseini-HashemiS. Vibration of two-phase local/nonlocal Timoshenko nanobeams with an efficient shear-locking-free finite-element model and exact solution. Engineering with Computers2020.
80.
MikhasevG. Free high-frequency vibrations of nonlocally elastic beam with varying cross-section area. Continuum Mech. Thermodyn. (2021). https://doi.org/10.1007/s00161-021-00977-6
81.
MalikanMUglovNSEremeyevVA.On instabilities and post-buckling of piezomagnetic and flexomagnetic nanostructures. Int J Eng Sci2020; 157: 103395.
82.
MalikanMEremeyevVA. On the geometrically nonlinear vibration of a piezo-flexomagnetic nanotube. Math Methods Appl Sci2020. https://doi.org/10.1002/mma.6758
83.
MalikanMEremeyevVAŻurKK.Effect of axial porosities on flexomagnetic response of in-plane compressed piezomagnetic nanobeams. Symmetry2020, 12, 1935; doi:10.3390/sym12121935, 1–16.
84.
MalikanMEremeyevVA.On nonlinear bending study of a piezo-flexomagnetic nanobeam based on an analytical-numerical solution. Nanomater2020, 10(9): 1–22.
85.
MalikanMEremeyevVA.On the Dynamics of a visco–piezo–flexoelectric nanobeam. Symmetry2020; 12(4): 1–21.
86.
DaviesRMTaylorGI.A critical study of the Hopkinson pressure bar. Philos Trans R Soc London Ser A Math Phys Sci1948; 240: 375–457.
87.
LabuschagneAvan RensburgNFJvan der MerweAJ.Comparison of linear beam theories. Math Comput Model2009; 49: 20–30.
88.
SayyadAS.Comparison of various refined beam theories for the bending and free vibration analysis of thick beams. Appl Comput Mech2011; 5: 217–230.
89.
CazzaniAStochinoFTurcoE.On the whole spectrum of Timoshenko beams. Part I: a theoretical revisitation. Zeitschrift Für Angew Math Und Phys2016; 67: 24.
90.
CazzaniAStochinoFTurcoE.On the whole spectrum of Timoshenko beams. Part II: further applications. Zeitschrift Für Angew Math Und Phys2016; 67: 25.
91.
AbdollahiRBoroomandB.On using mesh-based and mesh-free methods in problems defined by Eringen’s non-local integral model: issues and remedies. Meccanica2019; 54: 1801–1822.
92.
RomanoGBarrettaRDiacoM, et al. Constitutive boundary conditions and paradoxes in nonlocal elastic nanobeams. Int J Mech Sci2017; 121: 151–156.
93.
Fernández-SáezJZaeraR.Vibrations of Bernoulli-Euler beams using the two-phase nonlocal elasticity theory. Int J Eng Sci2017; 119: 232–248.
94.
MikhasevGNobiliA.On the solution of the purely nonlocal theory of beam elasticity as a limiting case of the two-phase theory. Int J Solids Struct2020; 190: 47–57.
95.
MikhasevGIEremeyevVAWildeK, et al. Assessment of dynamic characteristics of thin cylindrical sandwich panels with magnetorheological core. J Intell Mater Syst Struct2019; 30: 2748–2769.
EugsterSRHeschCBetschP, et al. Director-based beam finite elements relying on the geometrically exact beam theory formulated in skew coordinates. Int J Numer Methods Eng2014; 97: 111–129.
98.
GrecoLCuomoM.Consistent tangent operator for an exact Kirchhoff rod model. Contin Mech Thermodyn2015; 27: 861–877.
99.
SpagnuoloMAndreausU.A targeted review on large deformations of planar elastic beams: extensibility, distributed loads, buckling and post-buckling. Math Mech Solids2018; 24: 258–280.
100.
TurcoEBarchiesiEGiorgioI, et al. A Lagrangian Hencky-type non-linear model suitable for metamaterials design of shearable and extensible slender deformable bodies alternative to Timoshenko theory. Int J Non Linear Mech2020; 123: 103481.
