Abstract
Discontinuities such as cracks and steps on the length, and the mass of attached buckyball on the tip, of nanoresonators have large effects on nanoresonators natural frequencies. In this paper, free longitudinal vibration of multiple cracked, stepped nanorods with a buckyball at tip is studied using the Eringen nonlocal elasticity theory. From wave viewpoint, vibrations can be considered as travelling waves along structures. Waves propagate in a waveguide and transmit and reflect at discontinuities. The propagation, transmission and reflection functions are derived for nanorods. Cracks and steps as discontinuities, and buckyball at the end, clamped end and free end for boundary conditions, are considered. Cracks are modelled by an infinitesimal length, massless linear spring. Steps are formed by connecting two nanorods with different cross section areas and mechanical properties and buckyballs are considered as lumped masses. These propagation, transmission and reflection functions are combined to provide a short comprehensive systematic analytical approach to analyze the free longitudinal vibration of nanorods. Also, explicit expressions for natural frequencies are derived. The effects of crack intensity, sectional change as step, mass of buckyball, location of crack and step and also small-scale effect on natural frequencies are discussed. The approach is explained using several examples and the closed-form solutions are derived for some cases. The results are compared with the existing methods and can be used as benchmark for other future works.
Similar content being viewed by others
References
Achenbach JD (1973) Wave propagation in elastic solids. North-Holland Publishing Company, Amsterdam
Adhikari S, Chowdhury R (2010) The calibration of carbon nanotube based bionanosensors. J Appl Phys 107(12):124322. https://doi.org/10.1063/1.3435316
Adhikari S, Murmu T, McCarthy MA (2013) Dynamic finite element analysis of axially vibrating nonlocal rods. Finite Elem Anal Des 63:42–50. https://doi.org/10.1016/j.finel.2012.08.001
Ansari R, Gholami R (2016) Size-dependent modeling of the free vibration characteristics of postbuckled third-order shear deformable rectangular nanoplates based on the surface stress elasticity theory. Compos Part B Eng 95:301–316. https://doi.org/10.1016/j.compositesb.2016.04.002
Arash B, Jiang JW, Rabczuk T (2015) A review on nanomechanical resonators and their applications in sensors and molecular transportation. Appl Phys Rev 2:021301. https://doi.org/10.1063/1.4916728
Arlett JL, Myers EB, Roukes ML (2011) Comparative advantages of mechanical biosensors. Nat Nanotechnol 6:203–215. https://doi.org/10.1038/nnano.2011.44
Aydogdu M (2009) Axial vibration of the nanorods with the nonlocal continuum rod model. Physica E 41(5):861–864. https://doi.org/10.1016/j.physe.2009.01.007
Aydogdu M (2012) Axial vibration analysis of nanorods (carbon nanotubes) embedded in an elastic medium using nonlocal elasticity. Mech Res Commun 43:34–40. https://doi.org/10.1016/j.mechrescom.2012.02.001
Aydogdu M (2014) Longitudinal wave propagation in multiwalled carbon nano-tubes. Compos Struct 107:578–584. https://doi.org/10.1016/j.compstruct.2013.08.031
Aydogdu M, Elishakoff I (2014) On the vibration of nanorods restrained by a linear spring in-span. Mech Res Commun 57:90–96. https://doi.org/10.1016/j.mechrescom.2014.03.003
Aydogdu M, Filiz S (2011) Modeling carbon nanotube-based mass sensors using axial vibration and nonlocal elasticity. Physica E 42:1229–1234. https://doi.org/10.1016/j.physe.2011.02.006
Babaei H, Shahidi AR (2011) Small-scale effects on the buckling of quadrilateral nanoplates based on nonlocal elasticity theory using the Galerkin method. Arch Appl Mech 81(8):1051–1062. https://doi.org/10.1007/s00419-010-0469-9
Chang TP (2013) Axial vibration of non-uniform and non-homogeneous nanorods based on nonlocal elasticity theory. Appl Math Comput 219:4933–4941. https://doi.org/10.1016/j.amc.2012.11.059
Chowdhury R, Adhikari S, Mitchell J (2009) Vibrating carbon nanotube based bio-sensors. Physica E 42(2):104–109. https://doi.org/10.1016/j.physe.2009.09.007
Ciekot A (2012) Free axial vibration of a nanorod using the WKB method. Sci Res Inst Math Comput Sci 11:29–34
Danesh M, Farajpour A, Mohammadi M (2012) Axial vibration analysis of a tapered nanorod based on nonlocal elasticity theory and differential quadrature method. Mech Res Commun 39:23–27. https://doi.org/10.1016/j.mechrescom.2011.09.004
Dimarogonas AD (1996) Vibration of cracked structures: a state of the art review. Eng Fract Mech 55(5):831–857. https://doi.org/10.1109/JMEMS.2015.2434390
Ebrahimi F, Barati MR (2016) Vibration analysis of smart piezoelectrically actuated nanobeams subjected to magneto-electrical field in thermal environment. J Vib Control. https://doi.org/10.1177/1077546316646239
Ebrahimi F, Barati MR (2017) Small-scale effects on hygro-thermomechanical vibration of temperature-dependent, nonhomogeneous nanoscale beams. Mech Adv Mater Struc 24(11):924–936. https://doi.org/10.1080/15376494.2016.1196795
Ebrahimi F, Hosseini SHS (2016) Thermal effects on nonlinear vibration behavior of viscoelastic nanosize plates. J Thermal Stresses 39(5):606–625. https://doi.org/10.1080/01495739.2016.1160684
Ebrahimi F, Jafari A (2016) Higher-order thermomechanical vibration analysis of temperature-dependent FGM beams with porosities. J Engrg Math 2016:9561504. https://doi.org/10.1155/2016/9561504
Eom K, Park HS, Yoon DS, Kwon T (2011) Nanomechanical resonators and their applications in biological/chemical detection: nanomechanics principles. Phys Rep 503:115–163. https://doi.org/10.1016/j.physrep.2011.03.002
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, Edelen DGB (1972) On nonlocal elasticity. Int J Eng Sci 10:233–248. https://doi.org/10.1016/0020-7225(72)90039-0
Farajpour A, Rastgoo A (2017) Size-dependent static stability of magneto-electro-elastic CNT/MT-based composite nanoshells under external electric and magnetic fields. Microsyst Technol. https://doi.org/10.1007/s00542-017-3440-7
Farajpour A, Shahidi AR, Mohammadi M, Mahzoon M (2012) Buckling of orthotropic micro/nanoscale plates under linearly varying in-plane load via nonlocal continuum mechanics. Compos Struct 94(5):1605–1615. https://doi.org/10.1016/j.compstruct.2011.12.032
Farajpour A, Rastgoo A, Mohammadi M (2017) Vibration, buckling and smart control of microtubules using piezoelectric nanoshells under electric voltage in thermal environment. Physica B 509:100–114. https://doi.org/10.1016/j.physb.2017.01.006
Filiz S, Aydogdu M (2010) Axial vibration of carbon nanotube heterojunctions using nonlocal elasticity. Comput Mater Sci 49:619–627. https://doi.org/10.1016/j.commatsci.2010.06.003
Firouz-Abadi RD, Fotouhi MM, Haddadpour H (2011) Free vibration analysis of nanocones using a nonlocal continuum model. Phys Lett A 375(41):3593–3598. https://doi.org/10.1016/j.physleta.2011.08.035
Gil-Santos E, Ramos D, Martinez J, Fernindez-Regulez M, Garcia R, San Paulo A, Calleja M, Tamayo J (2010) Nanomechanical mass sensing and stiffness spectrometry based on two-dimensional vibrations of resonant nanowires. Nat Nanotechnol 5:641–645. https://doi.org/10.1038/nnano.2010.151
Guo SQ, Yang SP (2012) Axial vibration analysis of nanocones elasticity theory. Acta Mech Sin 28(3):801–807. https://doi.org/10.1007/s10409-012-0109-4
Harland NR, Mace BR, Jones RW (2001) Wave propagation, reflection and transmission in tunable fluid-filled beams. J Sound Vib 241(5):735–754. https://doi.org/10.1006/jsvi.2000.3316
Hermanson GT (2008) Buckyballs, fullerenes, and carbon nanotubes. Bioconj Tech. https://doi.org/10.1016/B978-0-12-370501-3.00015-1
Hsu JC, Lee HL, Chang WJ (2011) Longitudinal vibration of cracked nanobeams using nonlocal elasticity theory. Curr Appl Phys 11(6):1384–1388. https://doi.org/10.1016/j.cap.2011.04.026
Huang Z (2012) Nonlocal effects of longitudinal vibration in nanorod with internal long-range interactions. Int J Solids Struct 46:2150–2154. https://doi.org/10.1016/j.ijsolstr.2012.04.020
Iijima S (1991) Helical microtubules of graphitic carbon. Nature 354:56–58
Joshi AY, Harsha SP, Sharma SC (2010a) Vibration signature analysis of single walled carbon nanotube based nanomechanical sensors. Physica E 42(8):2115–2123. https://doi.org/10.1016/j.physe.2010.03.033
Joshi AY, Sharma SC, Harsha SP (2010b) Analysis of crack propagation in fixed-free single-walled carbon nanotube under tensile loading using XFEM. J Nanotechnol Eng Med 1(4):041008–041017. https://doi.org/10.1115/1.4002417
Kiani K (2010) Free longitudinal vibration of tapered nanowires in the context of nonlocal continuum theory of Eringen via a perturbation technique. Physica E 43(1):387–397. https://doi.org/10.1016/j.physe.2010.08.022
Kirkham M, Wang ZL, Snyder RL (2008) In situ growth kinetics of ZnO nanobelts. Nanotechnology 19(44):5708–5714. https://doi.org/10.1088/0957-4484/19/44/445708
Lee SK, Mace BR, Brennan MJ (2007) Wave propagation, reflection and transmission in non-uniform one-dimensional waveguides. J Sound Vib 304:31–49. https://doi.org/10.1016/j.jsv.2007.01.039
Loya J, López-Puente J, Zaera R, Fernández-Sáez J (2009) Free transverse vibrations of cracked nanobeams using a nonlocal elasticity model. J Appl Phys 105(4):044309. https://doi.org/10.1063/1.3068370
Mace BR (1984) Wave reflection and transmission in beams. J Sound Vib 97:237–246. https://doi.org/10.1016/0022-460X(84)90320-1
Mei C (2001) Free vibration studies of classical beams/rods with lumped masses at boundaries using an approach based on wave vibration. Int J Mech Eng Educ 39(3):256–268. https://doi.org/10.7227/IJMEE.39.3.7
Mei C (2012) Wave analysis of in-plane vibrations of L-shaped and portal planar frame structures. J Vib Acoust 134(2):021011–021012. https://doi.org/10.1115/1.4005014
Mei C (2013) Comparison of the four rod theories of longitudinally vibrating rods. J Vib Control 21(8):1639–1656. https://doi.org/10.1177/1077546313494216
Mei C, Mace BR (2005) Wave reflection and transmission in Timoshenko beams and wave analysis of Timoshenko beam structures. J Vib Acoust 127(4):382–394. https://doi.org/10.1115/1.1924647
Mei C, Sha H (2015) Analytical and experimental study of vibrations in simple spatial structures. J Vib Control 22(17):3711–3735. https://doi.org/10.1177/1077546314565807C
Mei C, Karpenko Y, Moody S, Allen D (2006) Analytical approach to free and forced vibrations of axially loaded cracked Timoshenko beams. J Sound Vib 291:1041–1060. https://doi.org/10.1016/j.jsv.2005.07.017
Mohammadi M, Ghayour M, Farajpour A (2013) Free transverse vibration analysis of circular and annular graphene sheets with various boundary conditions using the nonlocal continuum plate model. Compos B Eng 45(1):32–42. https://doi.org/10.1016/j.compositesb.2012.09.011
Mohammadi M, Farajpour A, Moradi A, Ghayour M (2014) Shear buckling of orthotropic rectangular graphene sheet embedded in an elastic medium in thermal environment. Compos B Eng 56:629–637. https://doi.org/10.1016/j.compositesb.2013.08.060
Mohammadi M, Safarabadi M, Rastgoo A, Farajpour A (2016) Hygro-mechanical vibration analysis of a rotating viscoelastic nanobeam embedded in a visco-Pasternak elastic medium and in a nonlinear thermal environment. Acta Mech 227(8):2207–2232. https://doi.org/10.1007/s00707-016-1623-4
Murmu T, Adhikari S (2010) Nonlocal effects in the longitudinal vibration of double-nanorod systems. Physica E 43(1):415–422. https://doi.org/10.1016/j.physe.2010.08.023
Murmu T, Adhikari S (2011) Nonlocal vibration of carbon nanotubes with attached buckyballs at tip. Mech Res Commun 38(1):62–67. https://doi.org/10.1016/j.mechrescom.2010.11.004
Narendar S (2011) Terahertz wave propagation in uniform nanorods: a nonlocal continuum mechanics formulation including the effect of lateral inertia. Physica E 43:1015–1020. https://doi.org/10.1016/j.physe.2010.12.004
Narendar S, Gopalakrishnan S (2010) Nonlocal scale effects on ultrasonic wave characteristics of nanorods. Physica E 42(5):1601–1604. https://doi.org/10.1016/j.physe.2010.01.002
Narendar S, Gopalakrishnan S (2011) Axial wave propagation in coupled nanorod system with nonlocal small scale effects. Compos B Eng 42(7):2013–2023. https://doi.org/10.1016/j.compositesb.2011.05.021
Phadikar JK, Pradhan SC (2010) Variational formulation and finite element analysis for nonlocal elastic nanobeams and nanoplates. Comp Mater Sci 49(3):492–499. https://doi.org/10.1016/j.commatsci.2010.05.040
Pisano AA, Sofi A, Fuschi P (2009) Nonlocal integral elasticity: 2D finite element based solutions. Int J Solids Struct 46(21):3836–3849. https://doi.org/10.1016/j.ijsolstr.2009.07.009
Polizzotto C (2001) Nonlocal elasticity and related variational principles. Int J Solids Struct 38(42–43):7359–7380. https://doi.org/10.1016/S0020-7683(01)00039-7
Pradhan SC (2009) Buckling of single layer graphene sheet based on nonlocal elasticity and higher order shear deformation theory. Phys Lett A 373(45):4182–4188. https://doi.org/10.1016/j.physleta.2009.09.021
Pradhan SC, Kumar A (2011) Vibration analysis of orthotropic graphene sheets using nonlocal elasticity theory and differential quadrature method. Compos Struct 93(2):774–779. https://doi.org/10.1016/j.compstruct.2010.08.004
Pugno N (2006) Nanocomplex oscillations as forewarning of fatigue collapse of NEMS. Curr Top Acoust Res 4:11–15
Pugno N (2007) Damage assessment of nanostructures. Key Eng Mat 347:199–204. https://doi.org/10.4028/www.scientific.net/KEM.347.199
Rao CNR, Seshadri R, Govindaraj A, Sen R (1995) Fullerenes, nanotubes, onions and related carbon structures. Mater Sci Eng, R 15(6):209–262. https://doi.org/10.1016/S0927-796X(95)00181-6
Roostai H, Haghpanahi M (2014) Vibration of nanobeams of different boundary conditions with multiple cracks based on nonlocal elasticity theory. Appl Math Model 38(3):1159–1169. https://doi.org/10.1016/j.apm.2013.08.011
Şimşek M (2012) Nonlocal effects in the free longitudinal vibration of axially functionally graded tapered nanorods. Mech Res Commun 61:257–265. https://doi.org/10.1016/j.commatsci.2012.04.001
Sun Y, Gao J, Zhu R, Xu J, Chen L, Zhang J, Zhao Q, Yu D (2010) In situ observation of ZnO nanowire growth on zinc film in environmental scanning electron microscope. J Chem Phys 132(12):124705–124714. https://doi.org/10.1063/1.3370339
Tan CA, Kang B (1998) Wave reflection and transmission in an axially strained, rotating Timoshenko shaft. J Sound Vib 213(3):483–510. https://doi.org/10.1006/jsvi.1998.1517
Wang Q (2005) Wave propagation in carbon nanotubes via nonlocal continuum mechanics. J Appl Phys 98(12):124301–124306. https://doi.org/10.1063/1.2141648
Wang Q, Arash B (2014) A review on applications of carbon nanotubes and graphenes as nano-resonator sensors. Comput Mater Sci 82:350–360. https://doi.org/10.1016/j.commatsci.2013.10.010
Wang Q, Varadan VK (2006) Wave characteristics of carbon nanotubes. Int J Solids Struct 43:254–265. https://doi.org/10.1016/j.ijsolstr.2005.02.047
Wang CM, Tan VBC, Zhang YY (2006) Timoshenko beam model for vibration analysis of multi-walled carbon nanotubes. J Sound Vib 294:1060–1072. https://doi.org/10.1016/j.jsv.2006.01.005
Yayli MÖ (2016) An efficient solution method for the longitudinal vibration of nanorods with arbitrary boundary conditions via a hardening nonlocal approach. J Vib Control Des 15:1–17. https://doi.org/10.1177/1077546316684042
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Loghmani, M., Hairi Yazdi, M.R. & Nikkhah Bahrami, M. Longitudinal vibration analysis of nanorods with multiple discontinuities based on nonlocal elasticity theory using wave approach. Microsyst Technol 24, 2445–2461 (2018). https://doi.org/10.1007/s00542-017-3619-y
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00542-017-3619-y