Abstract
Present study considers the longitudinal dynamic analysis of carbon nanotubes which has been modeled as an axially functionally graded Rayleigh-Bishop rod with the help of nonlocal stress gradient elasticity theory. Governing equation of motion and boundary conditions for the axial dynamics of nanorod have been obtained with variational formulation. Ritz method has been used in the solution of the problem. Effects of lateral inertia and transverse deformation of carbon nanotubes, material composition properties and grading power-law index on the axial dynamics response have been investigated. Mode shapes of functionally graded Rayleigh-Bishop nanorod have been depicted in various cases. Boundary conditions and power-law index have important effect on dynamics of axially graded Rayleigh-Bishop nanorod. Present results could be useful at designing of nano-electromechanical devices.
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References
Adali S (2008) Variational principles for multi-walled carbon nanotubes undergoing buckling based on nonlocal elasticity theory. Phys Lett A 372:5701–5705. https://doi.org/10.1016/j.physleta.2008.07.003
Adali S (2009) Variational principles for transversely vibrating multiwalled carbon nanotubes based on nonlocal Euler–Bernoulli beam model. Nano Lett 9:1737–1741. https://doi.org/10.1021/nl8027087
Adali S (2015) Variational principles for vibrating carbon nanotubes conveying fluid, based on the nonlocal beam model. East Asian J Appl Math 5:209–221. https://doi.org/10.4208/eajam.130814.250515a
Akbaş ŞD (2015) Post-buckling analysis of axially functionally graded three-dimensional beams. Int J Appl Mech 07:1550047. https://doi.org/10.1142/s1758825115500477
Akgöz B, Civalek Ö (2013a) Longitudinal vibration analysis of strain gradient bars made of functionally graded materials (FGM). Compos Part B Eng 55:263–268. https://doi.org/10.1016/j.compositesb.2013.06.035
Akgöz B, Civalek Ö (2013b) Free vibration analysis of axially functionally graded tapered Bernoulli-Euler microbeams based on the modified couple stress theory. Compos Struct 98:314–322. https://doi.org/10.1016/j.compstruct.2012.11.020
Akgöz B, Civalek Ö (2014) Longitudinal vibration analysis for microbars based on strain gradient elasticity theory. J Vib Control 20:606–616. https://doi.org/10.1177/1077546312463752
Akgöz B, Civalek Ö (2016) Bending analysis of embedded carbon nanotubes resting on an elastic foundation using strain gradient theory. Acta Astronaut 119:1–12. https://doi.org/10.1016/j.actaastro.2015.10.021
Alterman Z, Karal FC (1970) Propagation of elastic waves in a semi-infinite cylindrical rod using finite difference methods. J Sound Vib 13:115-IN1. https://doi.org/10.1016/S0022-460X(70)81169-5
Anderson SP (2006) Higher-order rod approximations for the propagation of longitudinal stress waves in elastic bars. J Sound Vib 290:290–308. https://doi.org/10.1016/j.jsv.2005.03.031
Aydogdu M (2008) Semi-inverse method for vibration and buckling of axially functionally graded beams. J Reinf Plast Compos 27:683–691. https://doi.org/10.1177/0731684407081369
Aydogdu M (2012) Longitudinal wave propagation in nanorods using a general nonlocal unimodal rod theory and calibration of nonlocal parameter with lattice dynamics. Int J Eng Sci 56:17–28. https://doi.org/10.1016/j.ijengsci.2012.02.004
Aydogdu M, Arda M (2016) Forced vibration of nanorods using nonlocal elasticity. Adv Nano Res 4:265–279. https://doi.org/10.12989/anr.2016.4.4.265
Aydogdu M, Arda M, Filiz S (2018) Vibration of axially functionally graded nano rods and beams with a variable nonlocal parameter. Adv Nano Res 6:257–278. https://doi.org/10.12989/anr.2018.6.3.257
Babaei A, Yang CX (2019) Vibration analysis of rotating rods based on the nonlocal elasticity theory and coupled displacement field. Microsyst Technol 25:1077–1085. https://doi.org/10.1007/s00542-018-4047-3
Barretta R, Faghidian SA, Marotti de Sciarra F (2019) Aifantis versus Lam strain gradient models of Bishop elastic rods. Acta Mech. https://doi.org/10.1007/s00707-019-02431-w
Bishop RED (1952) Longitudinal waves in beams. Aeronaut Q 3:280–293
Calim FF (2016) Free and forced vibration analysis of axially functionally graded Timoshenko beams on two-parameter viscoelastic foundation. Compos Part B Eng 103:98–112. https://doi.org/10.1016/j.compositesb.2016.08.008
Civalek Ö, Uzun B, Yaylı M, Akgöz B (2020) Size-dependent transverse and longitudinal vibrations of embedded carbon and silica carbide nanotubes by nonlocal finite element method. Eur Phys J Plus 135:381. https://doi.org/10.1140/epjp/s13360-020-00385-w
Ebrahimi F, Dabbagh A (2018) NSGT-based acoustical wave dispersion characteristics of thermo-magnetically actuated double-nanobeam systems. Struct Eng Mech 68:701–711. https://doi.org/10.12989/sem.2018.68.6.701
Ebrahimi F, Dehghan M, Seyfi A (2019) Eringen’s nonlocal elasticity theory for wave propagation analysis of magneto-electro-elastic nanotubes. Adv Nano Res 7:1–11. https://doi.org/10.12989/ANR.2019.7.1.001
Ecsedi I, Baksa A (2017) Free axial vibration of nanorods with elastic medium interaction based on nonlocal elasticity and Rayleigh model. Mech Res Commun 86:1–4. https://doi.org/10.1016/j.mechrescom.2017.10.003
Eltaher MA, Abdraboh AM, Almitani KH (2018) Resonance frequencies of size dependent perforated nonlocal nanobeam. Microsyst Technol 24:3925–3937. https://doi.org/10.1007/s00542-018-3910-6
Eringen AC (1965) Linear theory of micropolar elasticity. DTIC Document
Eringen AC (1972a) Nonlocal polar elastic continua. Int J Eng Sci. https://doi.org/10.1016/0020-7225(72)90070-5
Eringen AC (1972b) Linear theory of nonlocal elasticity and dispersion of plane waves. Int J Eng Sci 10:425–435. https://doi.org/10.1016/0020-7225(72)90050-X
Eringen AC (1983) On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves. J Appl Phys 54:4703–4710. https://doi.org/10.1063/1.332803
Eringen AC (2004) Nonlocal Continuum Field Theories. Springer New York, New York
Eringen AC, Suhubi ES (1964) Nonlinear theory of simple micro-elastic solids—I. Int J Eng Sci. https://doi.org/10.1016/0020-7225(64)90004-7
Eringen AC, Edelen DGB (1972) On nonlocal elasticity. Int J Eng Sci 10:233–248. https://doi.org/10.1016/0020-7225(72)90039-0
Fedotov IA, Polyanin AD, Shatalov MY (2007) Theory of free and forced vibrations of a rigid rod based on the Rayleigh model. Dokl Phys 52:607–612. https://doi.org/10.1134/S1028335807110080
Fedotov IA, Polyanin AD, Shatalov MY, Tenkam HM (2010) Longitudinal vibrations of a Rayleigh-Bishop rod. Dokl Phys 55:609–614. https://doi.org/10.1134/S1028335810120062
Feynman RP (2011) There’s plenty of room at the bottom. Resonance 16:890–905. https://doi.org/10.1007/s12045-011-0109-x
Ghayesh MH (2018) Nonlinear vibration analysis of axially functionally graded shear-deformable tapered beams. Appl Math Model 59:583–596. https://doi.org/10.1016/j.apm.2018.02.017
Ghayesh MH, Farokhi H, Gholipour A, Tavallaeinejad M (2017) Nonlinear bending and forced vibrations of axially functionally graded tapered microbeams. Int J Eng Sci 120:51–62. https://doi.org/10.1016/j.ijengsci.2017.03.010
Golmakani ME, Vahabi H (2017) Nonlocal buckling analysis of functionally graded annular nanoplates in an elastic medium with various boundary conditions. Microsyst Technol 23:3613–3628. https://doi.org/10.1007/s00542-016-3210-y
Gopalakrishnan S, Narendar S (2013) Wave propagation in nanostructures-nonlocal continuum mechanics formulations. Springer International Publishing, Cham
Güven U (2014) Love-Bishop rod solution based on strain gradient elasticity theory. Comptes Rendus - Mec 342:8–16. https://doi.org/10.1016/j.crme.2013.10.011
Han JB, Hong SY, Song JH, Kwon HW (2014) Vibrational energy flow models for the Rayleigh-Love and Rayleigh-Bishop rods. J Sound Vib 333:520–540. https://doi.org/10.1016/j.jsv.2013.08.027
Hosseini M, Mofidi MR, Jamalpoor A, Safi Jahanshahi M (2018) Nanoscale mass nanosensor based on the vibration analysis of embedded magneto-electro-elastic nanoplate made of FGMs via nonlocal Mindlin plate theory. Microsyst Technol 24:2295–2316. https://doi.org/10.1007/s00542-017-3654-8
Hu Y, Liew K, Wang Q et al (2008) Nonlocal shell model for elastic wave propagation in single- and double-walled carbon nanotubes. J Mech Phys Solids 56:3475–3485. https://doi.org/10.1016/j.jmps.2008.08.010
Huang Y, Li XF (2010) A new approach for free vibration of axially functionally graded beams with non-uniform cross-section. J Sound Vib 329:2291–2303. https://doi.org/10.1016/j.jsv.2009.12.029
Huang Y, Yang LE, Luo QZ (2013) Free vibration of axially functionally graded Timoshenko beams with non-uniform cross-section. Compos Part B Eng 45:1493–1498. https://doi.org/10.1016/j.compositesb.2012.09.015
Iijima S (1991) Helical microtubules of graphitic carbon. Nature 354:56–58
Karličić DZ, Ayed S, Flaieh E (2018) Nonlocal axial vibration of the multiple Bishop nanorod system. Math Mech Solids. https://doi.org/10.1177/1081286518766577
Kiani K (2016a) Free dynamic analysis of functionally graded tapered nanorods via a newly developed nonlocal surface energy-based integro-differential model. Compos Struct 139:151–166. https://doi.org/10.1016/j.compstruct.2015.11.059
Kiani K (2016b) Thermo-elasto-dynamic analysis of axially functionally graded non-uniform nanobeams with surface energy. Int J Eng Sci 106:57–76. https://doi.org/10.1016/j.ijengsci.2016.05.004
Krawczuk M, Grabowska J, Palacz M (2006) Longitudinal wave propagation. Part I-Comparison of rod theories. J Sound Vib 295:461–478. https://doi.org/10.1016/j.jsv.2005.12.048
Kucuk I, Sadek IS, Adali S (2010) Variational principles for multiwalled carbon nanotubes undergoing vibrations based on nonlocal timoshenko beam theory. J Nanomater https://doi.org/10.1155/2010/461252
Li C (2017) Nonlocal Thermo-Electro-Mechanical coupling vibrations of axially moving piezoelectric nanobeams. Mech Based Des Struct Mach 45:463–478. https://doi.org/10.1080/15397734.2016.1242079
Li L, Guo Y (2016) Analysis of longitudinal waves in rod-type piezoelectric phononic crystals. Crystals 6:45. https://doi.org/10.3390/cryst6040045
Li C, Li S, Yao L, Zhu Z (2015) Nonlocal theoretical approaches and atomistic simulations for longitudinal free vibration of nanorods/nanotubes and verification of different nonlocal models. Appl Math Model 39:4570–4585. https://doi.org/10.1016/j.apm.2015.01.013
Li X, Li L, Hu Y et al (2017a) Bending, buckling and vibration of axially functionally graded beams based on nonlocal strain gradient theory. Compos Struct 165:250–265. https://doi.org/10.1016/j.compstruct.2017.01.032
Li XF, Shen Z, Bin, Lee KY (2017b) Axial wave propagation and vibration of nonlocal nanorods with radial deformation and inertia. ZAMM Zeitschrift fur Angew Math Mech 97:602–616. https://doi.org/10.1002/zamm.201500186
Li C, Guo H, Tian X (2019) Nonlocal second-order strain gradient elasticity model and its application in wave propagating in carbon nanotubes. Microsyst Technol 25:2215–2227. https://doi.org/10.1007/s00542-018-4085-x
Loghmani M, Hairi Yazdi MR, Nikkhah Bahrami M (2018) Longitudinal vibration analysis of nanorods with multiple discontinuities based on nonlocal elasticity theory using wave approach. Microsyst Technol 24:2445–2461. https://doi.org/10.1007/s00542-017-3619-y
Love AEH (1893) A treatise on the mathematical theory of elasticity. University Press, Cambridge
Marais J, Fedotov I, Shatalov M (2015) Longitudinal vibrations of a cylindrical rod based on the Rayleigh-Bishop theory. Afrika Mat 26:1549–1560. https://doi.org/10.1007/s13370-014-0286-3
Nguyen DK (2013) Large displacement response of tapered cantilever beams made of axially functionally graded material. Compos Part B Eng 55:298–305. https://doi.org/10.1016/j.compositesb.2013.06.024
Numanoğlu HM, Akgöz B, Civalek Ö (2018) On dynamic analysis of nanorods. Int J Eng Sci 130:33–50. https://doi.org/10.1016/j.ijengsci.2018.05.001
Peddieson J, Buchanan GR, McNitt RP (2003) Application of nonlocal continuum models to nanotechnology. Int J Eng Sci 41:305–312. https://doi.org/10.1016/S0020-7225(02)00210-0
Pichugin AV, Askes H, Tyas A (2008) Asymptotic equivalence of homogenisation procedures and fine-tuning of continuum theories. J Sound Vib 313:858–874. https://doi.org/10.1016/j.jsv.2007.12.005
Pochhammer VHL (1876) Ueber die Fortpflanzungsgeschwindigkeiten kleiner Schwingungen in einem unbegrenzten isotropen Kreiscylinder. J für die reine und Angew Math (Crelle’s Journal) https://doi.org/10.1515/crll.1876.81.324
Rahmani O, Hosseini SAH, Parhizkari M (2017) Buckling of double functionally-graded nanobeam system under axial load based on nonlocal theory: an analytical approach. Microsyst Technol 23:2739–2751. https://doi.org/10.1007/s00542-016-3127-5
Rayleigh JWSB (1877) The theory of sound. Macmillan, New York
Reddy JN (2002) Energy principles and variational methods in applied mechanics. Wiley, Hoboken
Rosenfeld G, Keller JB (1974) Wave propagation in elastic rods of arbitrary cross section. J Acoust Soc Am 55:555–561. https://doi.org/10.1121/1.1914563
Seemann W (1996) Transmission and reflection coefficients for longitudinal waves obtained by a combination of refined rod theory and FEM. J Sound Vib 198:571–587. https://doi.org/10.1006/jsvi.1996.0589
Shafiei N, Kazemi M, Ghadiri M (2016) Comparison of modeling of the rotating tapered axially functionally graded Timoshenko and Euler–Bernoulli microbeams. Phys E Low-dimensional Syst Nanostr 83:74–87. https://doi.org/10.1016/j.physe.2016.04.011
Shafiei N, Kazemi M, Ghadiri M (2016b) On size-dependent vibration of rotary axially functionally graded microbeam. Int J Eng Sci 101:29–44. https://doi.org/10.1016/j.ijengsci.2015.12.008
Shafiei N, Kazemi M, Ghadiri M (2016c) Nonlinear vibration of axially functionally graded tapered microbeams. Int J Eng Sci 102:12–26. https://doi.org/10.1016/j.ijengsci.2016.02.007
Shafiei N, Kazemi M, Safi M, Ghadiri M (2016d) Nonlinear vibration of axially functionally graded non-uniform nanobeams. Int J Eng Sci 106:77–94. https://doi.org/10.1016/j.ijengsci.2016.05.009
Shahba A, Rajasekaran S (2012) Free vibration and stability of tapered Euler-Bernoulli beams made of axially functionally graded materials. Appl Math Model 36:3094–3111. https://doi.org/10.1016/j.apm.2011.09.073
Shahba A, Attarnejad R, Hajilar S (2011a) Free vibration and stability of axially functionally graded tapered Euler-Bernoulli beams. Shock Vib 18:683–696. https://doi.org/10.3233/SAV-2010-0589
Shahba A, Attarnejad R, Marvi MT, Hajilar S (2011b) Free vibration and stability analysis of axially functionally graded tapered Timoshenko beams with classical and non-classical boundary conditions. Compos Part B Eng 42:801–808. https://doi.org/10.1016/j.compositesb.2011.01.017
Shahba A, Attarnejad R, Hajilar S (2013) A mechanical-based solution for axially functionally graded tapered euler-bernoulli beams. Mech Adv Mater Struct 20:696–707. https://doi.org/10.1080/15376494.2011.640971
Shatalov M, Fedotov I, Tenkam H (2009) Comparison of classical and modern theories of longitudinal wave propagation in elastic rods. In: Proceedings of the 16th International Congress on Sound and Vibration. Kraków, pp 5–9
Shatalov M, Marais J, Fedotov I, Djouosseu M (2011) Longitudinal Vibration of Isotropic Solid Rods: From Classical to Modern Theories. Adv Comput Sci Eng 1877:408–409. https://doi.org/10.5772/15662
Shen JP, Li C, Fan XL, Jung CM (2017) Dynamics of silicon nanobeams with axial motion subjected to transverse and longitudinal loads considering nonlocal and surface effects. Smart Struct Syst 19:105–113. https://doi.org/10.12989/sss.2017.19.1.105
Shen JP, Wang PY, Gan WT, Li C (2020) Stability of Vibrating Functionally Graded Nanoplates with Axial Motion Based on the Nonlocal Strain Gradient Theory. Int J Struct Stab Dyn S0219455420500881. https://doi.org/10.1142/S0219455420500881
Şimşek M (2012) Nonlocal effects in the free longitudinal vibration of axially functionally graded tapered nanorods. Comput Mater Sci 61:257–265. https://doi.org/10.1016/j.commatsci.2012.04.001
Şimşek M (2015) Size dependent nonlinear free vibration of an axially functionally graded (AFG) microbeam using He’s variational method. Compos Struct 131:207–214. https://doi.org/10.1016/j.compstruct.2015.05.004
Steigmann DJ, Faulkner MG (1993) Variational theory for spatial rods. J Elast 33:1–26. https://doi.org/10.1007/BF00042633
Sudak LJ (2003) Column buckling of multiwalled carbon nanotubes using nonlocal continuum mechanics. J Appl Phys 94:7281–7287. https://doi.org/10.1063/1.1625437
Uzun B, Kafkas U, Yaylı M (2020) Stability analysis of restrained nanotubes placed in electromagnetic field. Microsyst Technol. https://doi.org/10.1007/s00542-020-04847-0
Wang Q (2005) Wave propagation in carbon nanotubes via nonlocal continuum mechanics. J Appl Phys 98:124301. https://doi.org/10.1063/1.2141648
Wu L, Wang QS, Elishakoff I (2005) Semi-inverse method for axially functionally graded beams with an anti-symmetric vibration mode. J Sound Vib 284:1190–1202. https://doi.org/10.1016/j.jsv.2004.08.038
Yao LQ, Ji CJ, Shen JP, Li C (2020) Free vibration and wave propagation of axially moving functionally graded Timoshenko microbeams. J Brazilian Soc Mech Sci Eng 42:137. https://doi.org/10.1007/s40430-020-2206-9
Yayli M (2020) Axial vibration analysis of a Rayleigh nanorod with deformable boundaries. Microsyst Technol. https://doi.org/10.1007/s00542-020-04808-7
Zak A, Krawczuk M (2010) Assessment of rod behaviour theories used in spectral finite element modelling. J Sound Vib 329:2099–2113. https://doi.org/10.1016/j.jsv.2009.12.019
Zarrinzadeh H, Attarnejad R, Shahba A (2012) Free vibration of rotating axially functionally graded tapered beams. Proc Inst Mech Eng Part G J Aerosp Eng 226:363–379. https://doi.org/10.1177/0954410011413531
Zeighampour H, Tadi Beni Y (2015) Free vibration analysis of axially functionally graded nanobeam with radius varies along the length based on strain gradient theory. Appl Math Model 39:5354–5369. https://doi.org/10.1016/j.apm.2015.01.015
Zeighampour H, Tadi Beni Y, Botshekanan Dehkordi M (2018) Wave propagation in viscoelastic thin cylindrical nanoshell resting on a visco-Pasternak foundation based on nonlocal strain gradient theory. Thin-Walled Struct 122:378–386. https://doi.org/10.1016/j.tws.2017.10.037
Zenkour AM (2018) Nonlocal elasticity and shear deformation effects on thermal buckling of a CNT embedded in a viscoelastic medium. Eur Phys J Plus 133:196. https://doi.org/10.1140/epjp/i2018-12014-2
Zhu X, Li L (2017) On longitudinal dynamics of nanorods. Int J Eng Sci 120:129–145. https://doi.org/10.1016/j.ijengsci.2017.08.003
Zhuang X, Ning CZ, Pan A (2012) Composition and bandgap-graded semiconductor alloy nanowires. Adv Mater 24:13–33. https://doi.org/10.1002/adma.201103191
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Arda, M. Axial dynamics of functionally graded Rayleigh-Bishop nanorods. Microsyst Technol 27, 269–282 (2021). https://doi.org/10.1007/s00542-020-04950-2
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DOI: https://doi.org/10.1007/s00542-020-04950-2