Vibrations of the Euler–Bernoulli Beam Under a Moving Force based on Various Versions of Gradient Nonlocal Elasticity Theory: Application in Nanomechanics

[1]A. C. Eringen, Nonlocal polar elastic continua, International Journal of Engineering Science 1972; 10: 1–16.EringenA. C.Nonlocal polar elastic continuaInternational Journal of Engineering Science19721011610.1016/0020-7225(72)90070-5Search in Google Scholar

[2]A. C. Eringen, Linear theory of nonlocal elasticity and dispersion of plane waves, International Journal of Engineering Science 1972; 5: 425–435.EringenA. C.Linear theory of nonlocal elasticity and dispersion of plane wavesInternational Journal of Engineering Science1972542543510.1016/0020-7225(72)90050-XSearch in Google Scholar

[3]A. C. Eringen, D. G. B. Edelen, On nonlocal elasticity, International Journal of Engineering Science 1972; 3: 233–248.EringenA. C.EdelenD. G. B.On nonlocal elasticityInternational Journal of Engineering Science1972323324810.1016/0020-7225(72)90039-0Search in Google Scholar

[4]A. C. Eringen, On differential-equations of nonlocal elasticity and solutions of screw dislocation and surface waves, Journal of Applied Physics 1983; 9: 4703–4710.EringenA. C.On differential-equations of nonlocal elasticity and solutions of screw dislocation and surface wavesJournal of Applied Physics198394703471010.1063/1.332803Search in Google Scholar

[5]M. Aydogdu, Axial vibration of the nanorods with the nonlocal continuum rod model, Physica E Low-dimensional Systems Nanostructures 2009; 41(5): 861–864.AydogduM.Axial vibration of the nanorods with the nonlocal continuum rod modelPhysica E Low-dimensional Systems Nanostructures200941586186410.1016/j.physe.2009.01.007Search in Google Scholar

[6]M. Danesh, A. Farajpour, M. Mohammadi, Axial vibration analysis of a tapered nanorod based on nonlocal elasticity theory and differential quadrature method, Mechanics Research Communications 2012; 39 (1): 23–27.DaneshM.FarajpourA.MohammadiM.Axial vibration analysis of a tapered nanorod based on nonlocal elasticity theory and differential quadrature methodMechanics Research Communications2012391232710.1016/j.mechrescom.2011.09.004Search in Google Scholar

[7]Z. X. Huang, Nonlocal effects of longitudinal vibration in nanorod with internal long range interactions, International Journal of Solid and Structures 2012; 49: 2150–2154.HuangZ. X.Nonlocal effects of longitudinal vibration in nanorod with internal long range interactionsInternational Journal of Solid and Structures2012492150215410.1016/j.ijsolstr.2012.04.020Search in Google Scholar

[8]M. Aydogdu, M. Arda, Force vibration of nanorods using nonlocal elasticity, Advances in Nano Research 2016; 4: (4): 265–279.AydogduM.ArdaM.Force vibration of nanorods using nonlocal elasticityAdvances in Nano Research20164426527910.12989/anr.2016.4.4.265Search in Google Scholar

[9]J. Peddieson, G. R. Buchanan, R. P. McNitt, Application of nonlocal continuum models to nanotechnology, International Journal of Engineering Science 2003; 41: 305–312.PeddiesonJ.BuchananG. R.McNittR. P.Application of nonlocal continuum models to nanotechnologyInternational Journal of Engineering Science20034130531210.1016/S0020-7225(02)00210-0Search in Google Scholar

[10]J. N. Reddy, Nonlocal theories for bending, buckling and vibration of beam, International Journal of Engineering Science 2007; 45: 288–307.ReddyJ. N.Nonlocal theories for bending, buckling and vibration of beamInternational Journal of Engineering Science20074528830710.1016/j.ijengsci.2007.04.004Search in Google Scholar

[11]R. Ansari, R. Gholami, H. Rouchi, Vibration analysis of single-walled carbon nanotubes using gradient elasticity theories, Composites: Part B 2012; 43: 2985–2989.AnsariR.GholamiR.RouchiH.Vibration analysis of single-walled carbon nanotubes using gradient elasticity theoriesComposites: Part B2012432985298910.1016/j.compositesb.2012.05.049Search in Google Scholar

[12]M. Aydogdu, A general nonlocal beam theory: Its application to nanobeam bending, buckling and vibration, Physica E 2009; 41: 1651–1655.AydogduM.A general nonlocal beam theory: Its application to nanobeam bendingbuckling and vibrationPhysica E2009411651165510.1016/j.physe.2009.05.014Search in Google Scholar

[13]D. Karličić, T. Murmu, S. Adhikari, M. McCarthy, Non-local Structural Mechanics 2016; WILEY.KarličićD.MurmuT.AdhikariS.McCarthyM.Non-local Structural Mechanics2016WILEY10.1002/9781118572030Search in Google Scholar

[14]H. Askes, E. C. Aifantis, Gradient elasticity and flexural wave dispersion in carbon nanotubes, Physical Review B: Condensed Matter and Materials Physics 2009; 80: 1955412.AskesH.AifantisE. C.Gradient elasticity and flexural wave dispersion in carbon nanotubesPhysical Review B: Condensed Matter and Materials Physics200980195541210.1103/PhysRevB.80.195412Search in Google Scholar

[15]D. Karličić, P. Kozić, R. Pavlović, Flexural vibration and buckling analysis of single-walled carbon nanotubes using different gradient elasticity theories based on Reddy and Huu-Tai formulations, Journal of Theoretical and Applied Mechanics, 2015; 53 (1): 217–233.KarličićD.KozićP.PavlovićR.Flexural vibration and buckling analysis of single-walled carbon nanotubes using different gradient elasticity theories based on Reddy and Huu-Tai formulationsJournal of Theoretical and Applied Mechanics201553121723310.15632/jtam-pl.53.1.217Search in Google Scholar

[16]R. Rafie, R. M. Moghadam, On the modeling of carbon nanotubes: A critical review, Composites: Part B 2014; 56: 435–449.RafieR.MoghadamR. M.On the modeling of carbon nanotubes: A critical reviewComposites: Part B20145643544910.1016/j.compositesb.2013.08.037Search in Google Scholar

[17]S. I. Yengejeh, S. A. Kazami, A. Öchsner, Advances in mechanical analysis of structurally and atomically modified carbon nanotubes and degenerated nanostructures: A review, Composites Part B 2016; 86: 95–107.YengejehS. I.KazamiS. A. A.Öchsner, Advances in mechanical analysis of structurally and atomically modified carbon nanotubes and degenerated nanostructures: A reviewComposites Part B2016869510710.1016/j.compositesb.2015.10.006Search in Google Scholar

[18]S. Gopalakrishnan, S. Narendar, Wave Propagation in Nanostructures 2013; Springer.GopalakrishnanS.NarendarS.Wave Propagation in Nanostructures2013Springer10.1007/978-3-319-01032-8Search in Google Scholar

[19]Elishakoff I., Carbon Nanotubes and Nanosensors: Vibration, Buckling and Balistic Impact, ISTE, London and John Wiley & Sons, New York, 2012.ElishakoffI.Carbon Nanotubes and Nanosensors: Vibration, Buckling and Balistic ImpactISTE, London and John Wiley & SonsNew York201210.1002/9781118562000Search in Google Scholar

[20]K. Kiani, B. Mehri, Assessment of nanotube structures under a moving nanoparticle using nonlocal beam theories, Journal of Sound and Vibration 2010;329 (11): 2241–2264.KianiK.MehriB.Assessment of nanotube structures under a moving nanoparticle using nonlocal beam theoriesJournal of Sound and Vibration2010329112241226410.1016/j.jsv.2009.12.017Search in Google Scholar

[21]K. Kiani, Application of nonlocal beam models to double walled carbon nanotubes under a moving nanoparticle. Part I: theoretical formulations, Acta Mechanica 2011; 216: 165–195.KianiK.Application of nonlocal beam models to double walled carbon nanotubes under a moving nanoparticle. Part I: theoretical formulationsActa Mechanica201121616519510.1007/s00707-010-0362-1Search in Google Scholar

[22]K. Kiani, Application of nonlocal beam models to double-walled carbon nanotubes under a moving nanoparticle. Part II: parametric study, Acta Mechanica 2011; 216: 197–206.KianiK.Application of nonlocal beam models to double-walled carbon nanotubes under a moving nanoparticle. Part II: parametric studyActa Mechanica201121619720610.1007/s00707-010-0363-0Search in Google Scholar

[23]K. Kiani, Longitudinal and transverse vibration of a single-walled carbon nanotube subjected to a moving nanoparticle accounting for both nonlocal and inertial effects, Physica E: Low-dimensional Systems and Nanostructures 2010; 42 (9): 2391–2401.KianiK.Longitudinal and transverse vibration of a single-walled carbon nanotube subjected to a moving nanoparticle accounting for both nonlocal and inertial effectsPhysica E: Low-dimensional Systems and Nanostructures20104292391240110.1016/j.physe.2010.05.021Search in Google Scholar

[24]K. Kiani, A. Nikkhoo, B. Mehri, Prediction capabilities of classical and shear deformable beam theories excited by a moving mass, Journal of Sound and Vibration 2009; 320: 632–648.KianiK.NikkhooA.MehriB.Prediction capabilities of classical and shear deformable beam theories excited by a moving massJournal of Sound and Vibration200932063264810.1016/j.jsv.2008.08.010Search in Google Scholar

[25]K. Kiani, Small-scale effect on the vibration of thin nanoplates subjected to a moving nanoparticle via nonlocal continuum theory, Journal of Sound and Vibration 2011; 330; 4896–4914.KianiK.Small-scale effect on the vibration of thin nanoplates subjected to a moving nanoparticle via nonlocal continuum theoryJournal of Sound and Vibration20113304896491410.1016/j.jsv.2011.03.033Search in Google Scholar

[26]M. Şimşek, Vibration analysis of a single-walled carbon nanotube under action of a moving harmonic load based on nonlocal elasticity theory, Physica E 2010; 43: 182–191.ŞimşekM.Vibration analysis of a single-walled carbon nanotube under action of a moving harmonic load based on nonlocal elasticity theoryPhysica E20104318219110.1016/j.physe.2010.07.003Search in Google Scholar

[27]S.A.H. Hosseini, O. Rahmani, Exact solution for axial and transverse dynamic response of functionally graded nanobeam under moving constant load based on nonlocal elasticity theory, Meccanica 2017, 52: 1441–1457.HosseiniS.A.H.RahmaniO.Exact solution for axial and transverse dynamic response of functionally graded nanobeam under moving constant load based on nonlocal elasticity theoryMeccanica2017521441145710.1007/s11012-016-0491-2Search in Google Scholar

[28]M. Pourseifi, O. Rahmami, S.A.H. Hoseini, Active vibration control of nanotube structures under a moving nanoparticle based on the nonlocal continuum theories, Meccanica 2015; 50 (5):1351–1369.PourseifiM.RahmamiO.HoseiniS.A.H.Active vibration control of nanotube structures under a moving nanoparticle based on the nonlocal continuum theoriesMeccanica20155051351136910.1007/s11012-014-0096-6Search in Google Scholar

[29]B. Mehri, A. Davar, O. Rahmani, Dynamic Green function solution of beams under a moving load with different boundary conditions, Scientia Iranica 2009; 16 (3): 273–279.MehriB.DavarA.RahmaniO.Dynamic Green function solution of beams under a moving load with different boundary conditionsScientia Iranica2009163273279Search in Google Scholar

[30]G. Szefer, D. Jasińska, Continuum molecular modelling of nanostructured materials, 2010, 189–201, in Alberts (eds.) Continuous Media with Microstructure, Springer, Berlin, Heidelberg.SzeferG.JasińskaD.Continuum molecular modelling of nanostructured materials2010189201inAlberts(eds.)Continuous Media with Microstructure, SpringerBerlin, Heidelberg10.1007/978-3-642-11445-8_16Search in Google Scholar

[31]G. Szefer, Molecular modeling of stresses and deformations in nanostructured materials, International Journal of Applied Mathematics and Computer Science, 2004; 14 (4): 541–548.SzeferG.Molecular modeling of stresses and deformations in nanostructured materialsInternational Journal of Applied Mathematics and Computer Science2004144541548Search in Google Scholar

[32]Y. Shirai, J.F. Morin, T. Sasaki, J.M. Guerrero, J.M. Recent progress on nanovehicles, Chemical Society Reviews 2006; 35 (11): 1043–1055.ShiraiY.MorinJ.F.SasakiT.GuerreroJ.M.J.M. Recent progress on nanovehiclesChemical Society Reviews200635111043105510.1039/b514700j17057834Search in Google Scholar

[33]R. Lipowsky, S. Klumpp, “Life is motion”: multiscale motility of molecular motors, Physica A- Statistical Mechanics and its Applications 2005; 352 (1): 53–112.LipowskyR.KlumppS.“Life is motion”: multiscale motility of molecular motorsPhysica A- Statistical Mechanics and its Applications200535215311210.1016/j.physa.2004.12.034Search in Google Scholar

[34]L. Fryba, Vibration of Solids and Structures under Moving Loads, Telford, London, 1999.FrybaL.Vibration of Solids and Structures under Moving LoadsTelford, London199910.1680/vosasuml.35393Search in Google Scholar

[35]G. Romano, R. Barretta, Stress-driven versus strain-driven nonlocal integral model for elastic nano-beams, Composite Part B: Engineering 2017; 114: 184–188.RomanoG.BarrettaR.Stress-driven versus strain-driven nonlocal integral model for elastic nano-beamsComposite Part B: Engineering201711418418810.1016/j.compositesb.2017.01.008Search in Google Scholar

[36]E.C. Aifantis, On the role of gradients in the localization of deformation and fracture, International Journal of Engineering Science, 1992, 30, 1279–1299.AifantisE.C.On the role of gradients in the localization of deformation and fractureInternational Journal of Engineering Science1992301279129910.1016/0020-7225(92)90141-3Search in Google Scholar

[37]E.C. Aifantis, Gradient deformation models at nano, micro, and macro scales, Journal of Engineering Materials and Technology, ASCE, 1999, 121, April, 189–202.AifantisE.C.Gradient deformation models at nano, micro, and macro scalesJournal of Engineering Materials and TechnologyASCE1999121April18920210.1115/1.2812366Search in Google Scholar

[38]E.C. Aifantis, On the gradient approach- Relation to Eringen’s nonlocal theory, International Journal of Engineering Science 2011; 49: 1367–1377.AifantisE.C.On the gradient approach- Relation to Eringen’s nonlocal theoryInternational Journal of Engineering Science2011491367137710.1016/j.ijengsci.2011.03.016Search in Google Scholar

[39]C.W. Lim, G. Zhang, J.N. Reddy, A higher-order nonlocal elasticity and strain gradient theory and its applications in wave propagation, Journal of the Mechanics and Physics of Solids, 2015, 78, 298–313.LimC.W.ZhangG.ReddyJ.N.A higher-order nonlocal elasticity and strain gradient theory and its applications in wave propagationJournal of the Mechanics and Physics of Solids20157829831310.1016/j.jmps.2015.02.001Search in Google Scholar

[40]J. Fernandez-Saez, R. Zaera, J.A. Loya, J.N. Reddy, Bending of Euler-Bernoulli beams using Eringen’s integral formulation: A paradox resolved, International Journal of Engineering Science 2016; 99: 107–116.Fernandez-SaezJ.ZaeraR.LoyaJ.A.ReddyJ.N.Bending of Euler-Bernoulli beams using Eringen’s integral formulation: A paradox resolvedInternational Journal of Engineering Science20169910711610.1016/j.ijengsci.2015.10.013Search in Google Scholar

[41]P. Śniady, Dynamic response of a Timoshenko beam to a moving force, Journal of Applied Mechanics, ASME 2008; 75: 024503–1-024503-4.ŚniadyP.Dynamic response of a Timoshenko beam to a moving forceJournal of Applied MechanicsASME200875024503-1024503-410.1115/1.2775500Search in Google Scholar

[42]J. Rusin, P. Śniady, P. Śniady, Vibrations of double-string complex system under moving force. Closed solutions, Journal of Sound and Vibration 2011; 330: 404–415.RusinJ.ŚniadyP.ŚniadyP.Vibrations of double-string complex system under moving forceClosed solutions, Journal of Sound and Vibration201133040441510.1016/j.jsv.2010.08.021Search in Google Scholar

[43]K. Misiurek, P. Śniady, Vibrations of sandwich beam due to a moving force, Composite Structures 2013; 104: 85–93.MisiurekK.ŚniadyP.Vibrations of sandwich beam due to a moving forceComposite Structures2013104859310.1016/j.compstruct.2013.04.007Search in Google Scholar

[44]P. Śniady, M. Podwórna, R. Idzikowski, Stochastic vibrations of the Euler-Bernoulli beam based on various versions of the gradient nonlocal elasticity theory, Probabilistic Engineering Mechanics 2019; 56: 27–34.ŚniadyP.PodwórnaM.IdzikowskiR.Stochastic vibrations of the Euler-Bernoulli beam based on various versions of the gradient nonlocal elasticity theoryProbabilistic Engineering Mechanics201956273410.1016/j.probengmech.2019.03.001Search in Google Scholar