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http://dx.doi.org/10.12989/anr.2020.8.4.307

Theoretical impact of Kelvin's theory for vibration of double walled carbon nanotubes  

Hussain, Muzamal (Department of Mathematics, Government College University Faisalabad)
Naeem, Muhammad N. (Department of Mathematics, Government College University Faisalabad)
Asghar, Sehar (Department of Mathematics, Government College University Faisalabad)
Tounsi, Abdelouahed (Materials and Hydrology Laboratory, University of Sidi Bel Abbes, Algeria Faculty of Technology Civil Engineering Department)
Publication Information
Advances in nano research / v.8, no.4, 2020 , pp. 307-322 More about this Journal
Abstract
In this article, free vibration of double-walled carbon nanotubes (DWNT) based on nonlocal Kelvin's model have been investigated. For this purpose, a nonlocal Kelvin's model is established to observe the small scale effect. The wave propagation is employed to frame the governing equations as eigenvalue system. The influence of nonlocal parameter subjected to different end supports has been overtly examined. The new set of inner and outer tubes radii investigated in detail against aspect ratio. The influence of boundary conditions via nonlocal parameter is shown graphically. Due to small scale effect fundamental frequency ratio decreases as length to diameter ratio increases. Small scale effect becomes negligible on all end supports for the higher values of aspect ratio. With the smaller inner tube radius double-walled CNT behaves more sensitive towards nonlocal parameter. The results generated furnish the evidence regarding applicability of nonlocal model and also verified by earlier published literature.
Keywords
free vibration; nonlocal material; double-walled CNTs; Kelvin's model; WPA;
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1 Reddy, J.N. (2007), "Nonlocal theories for bending, buckling and vibration of beams", Int. J. Eng. Sci., 45, 288-307. https://doi.org/10.1016/j.ijengsci.2007.04.004   DOI
2 Rouhi, H., Ansari, R. and Arash, B. (2013), "Vibrational analysis of double-walled carbon nanotubes based on the nonlocal Donnell shell theory via a new numerical approach", Int J. Mech. Sei., 37, 91-105.
3 Rouhi, H., BazdidVahdati, M. and Ansari, R. (2015), "Rayleigh-Rits vibrational analysis of multi-walled carbon nanotubes based on the non-local Flugge shell theory", J. Compos., 750392. https://doi.org/10.1155/2015/750392
4 Safa, A., Hadji, L., Bourada, M. and Zouatnia, N. (2019), "Thermal vibration analysis of FGM beams using an efficient shear deformation beam theory", Earthq. Struct., Int. J., 17(3), 329-336. https://doi.org/10.12989/eas.2019.17.3.329
5 Ansari, R. and Rouhi, H. (2013), "Nonlocal analytical Flugge shell model forr the vibrations of double-walled carbon nanotubes with different end conditions", Int. J. Appl. Mech., 80, 021006-1. https://doi.org/10.1142/S179329201250018X   DOI
6 Ansari, R., Sahmani, S. and Arash, B. (2010), "Nonlocal plate model for free vibrations of single-layered graphene sheets", Phy. Letters A., 375(1), 53-62. https://doi.org/10.1016/j.physleta.2010.10.028   DOI
7 Ansari, R., Hemmatnezhad, M. and Rezapour, J. (2011), "The thermal effect on nonlinear oscillations of carbon nanotubes with arbitrary boundary conditions", Current Appl. Phys., 11(3), 692-697. https://doi.org/10.1016/j.cap.2010.11.034   DOI
8 Arani, A.J. and Kolahchi, R. (2016), "Buckling analysis of embedded concrete columns armed with carbon nanotubes", Comput. Concrete, Int. J., 17(5), 567-578. https://doi.org/10.12989/cac.2016.17.5.567   DOI
9 Mehar, K. and Panda, S.K. (2016b), "Free vibration and bending behaviour of CNT reinforced composite plate using different shear deformation theory", Proceedings of IOP Conference Series: Materials Science and Engineering, 115(1), 012014. https://doi.org/10.1088/1757-899X/115/1/012014   DOI
10 Mehar, K. and Panda, S.K. (2018a), "Dynamic response of functionally graded carbon nanotube reinforced sandwich plate", Proceedings of IOP Conference Series: Materials Science and Engineering, 338(1), p. 012017. https://doi.org/10.1088/1757-899X/338/1/012017   DOI
11 Mehar, K. and Panda, S.K. (2018b), "Thermal free vibration behavior of FG-CNT reinforced sandwich curved panel using finite element method", Polym. Compos., 39(8), 2751-2764. https://doi.org/10.1002/pc.24266   DOI
12 Mehar, K. and Panda, S.K. (2019), "Multiscale modeling approach for thermal buckling analysis of nanocomposite curved structure", Adv. Nano Res., Int. J., 7(3), 181-190. https://doi.org/10.12989/anr.2019.7.3.181   DOI
13 Mehar, K., Panda, S.K., Dehengia, A. and Kar, V.R. (2016), "Vibration analysis of functionally graded carbon nanotube reinforced composite plate in thermal environment", J. Sandw. Struct. Mater., 18(2), 151-173. https://doi.org/10.1177/1099636215613324   DOI
14 Mehar, K., Panda, S.K. and Mahapatra, T.R. (2017a), "Thermoelastic nonlinear frequency analysis of CNT reinforced functionally graded sandwich structure", Eur. J. Mech.- A/Solids, 65, 384-396. https://doi.org/10.1016/j.euromechsol.2017.05.005   DOI
15 Mehar, K., Panda, S.K., Bui, T.Q. and Mahapatra, T.R. (2017b), "Nonlinear thermoelastic frequency analysis of functionally graded CNT-reinforced single/doubly curved shallow shell panels by FEM", J. Thermal Stress., 40(7), 899-916. https://doi.org/10.1080/01495739.2017.1318689   DOI
16 Wang, Q., Varadan, V.K. and Quek, S.T. (2006a), "Small scale effect on elastic buckling of carbon nanotubes with nonlocal continuum models", Phys. Lett. A, 357(2), 130-135. https://doi.org/10.1016/j.physleta.2006.04.026   DOI
17 Mehar, K., Panda, S.K. and Mahapatra, T.R. (2017c), "Theoretical and experimental investigation of vibration characteristic of carbon nanotube reinforced polymer composite structure", Int. J. Mech. Sci., 133, 319-329. https://doi.org/10.1016/j.ijmecsci.2017.08.057   DOI
18 Vodenitcharova, T. and Zhang, L.C. (2003), "Effective wall thickness of single walled carbon nanotubes", Phy. Rev. B, 68, 165401. https://doi.org/10.1103/PhysRevB.68.165401   DOI
19 Wang, J. and Gao, Y. (2016), "Nonlocal orthotropic shell model applied on wave propagation in microtubules", Appl. Mathe. Model., 40(11-12), 5731-5744. https://doi.org/10.1016/j.apm.2016.01.013   DOI
20 Wang, C.Y. and Zhang, L.C. (2007), "Modeling the free vibration of single-walled carbon nanotubes", Proceedings of the 5th Australasian Congress on Applied Mechanics, ACAM, Brisbane, Australia, pp. 252-257.
21 Wang, Q., Zhou, G.Y. and Lin, K.C. (2006b), "Scale effect on wave propagation of double-walled carbon nanotubes", Int. J. Solids Struct., 43, 6071-6084. https://doi.org/10.1016/j.ijsolstr.2005.11.005   DOI
22 Xiaobin, L., Shuangxi, X., Weiguo, W. and Jun, L. (2014), "An exact dynamic stiffness matrix for axially loaded double-beam systems", Sadhana, 39(3), 607-623. https://doi.org/10.1007/s12046-013-0214-5   DOI
23 Xu, K.U., Aifantis, E.C. and Yan, Y.H. (2008), "Vibrations of double-walled carbon nanotubes with different boundary conditions between inner and outer tubes", J. Appl. Mech., 75(2), 021013-1. https://doi.org/10.1115/1.2793133   DOI
24 Fatahi-Vajari. A., Azimzadeh, Z. and Hussain. M. (2019), "Nonlinear coupled axial-torsional vibration of single-walled carbon nanotubes using Galerkin and Homotopy perturbation method", Micro Nano Lett., 14(14), 1366-1371. https://doi.org/10.1049/mnl.2019.0203   DOI
25 Flugge, W. (1962), Statik und Dynamik der Scahlen, Springer, Berlin, Germany.
26 Jassas, M.R., Bidgoli, M.R. and Kolahchi, R. (2019), "Forced vibration analysis of concrete slabs reinforced by agglomerated SiO2 nanoparticles based on numerical methods", Constr. Build. Mater., 211, 796-806. https://doi.org/10.1016/j.conbuildmat.2019.03.263   DOI
27 Hussain, M., Naeem, M.N., Sehar, A. and Tounsi, A. (2020e), "Eringen's nonlocal model sandwich with Kelvin's theory for vibration of DWCNT", Comput. Concrete, Int. J., 25(4), 343-354. https://doi.org/10.12989/cac.2020.25.4.343
28 Hussain, M., Naeem, M.N., Khan, M.S. and Tounsi, A. (year), "Computer-aided approach for modelling of FG cylindrical shell sandwich with ring supports", Comput. Concrete, Int. J., 25(5), 411-425. https://doi.org/10.12989/cac.2020.25.5.411
29 Iijima, S. (1991), "Helical microtubules of graphitic carbon", Nature, 354(7), 56-58. https://doi.org/10.1038/354056a0   DOI
30 Karami, B., Janghorban, M. and Tounsi, A. (2018), "Variational approach for wave dispersion in anisotropic doubly-curved nanoshells based on a new nonlocal strain gradient higher order shell theory", Thin-Wall. Struct., 129, 251-264. https://doi.org/10.1016/j.tws.2018.02.025   DOI
31 Karami, B., Shahsavari, D., Janghorban, M. and Li, L. (2019), "Elastic guided waves in fully-clamped functionally graded carbon nanotube-reinforced composite plates", Mater. Res. Express, 6(9), 0950a9. https://doi.org/10.1088/2053-1591/ab3474   DOI
32 Ke, L.L., Xiang, Y., Yang, J. and Kitipornchai, S. (2009), "Nonlinear free vibration of embedded double-walled carbon nanotubes based on nonlocal Timoshenko beam theory", Computat. Mater. Sci., 47(2), 409-417. https://doi.org/10.1016/j.commatsci.2009.09.002   DOI
33 Khosrazadeh, A. and Hajabasi, M.A. (2012), "Free vibrations of embedded doube-walled carbon nanotubes considering nonlinear interlayer van der Waals forces", Appl. Mathe. Model., 36(3), 997-1007 https://doi.org/10.1016/j.apm.2011.07.063   DOI
34 Mehar, K., Mahapatra, T.R., Panda, S.K., Katariya, P.V. and Tompe, U.K. (2018c), "Finite-element solution to nonlocal elasticity and scale effect on frequency behavior of shear deformable nanoplate structure", J. Eng. Mech., 144(9), 04018094. https://doi.org/10.1061/(ASCE)EM.1943-7889.0001519   DOI
35 Mehar, K., Panda, S.K. and Patle, B.K. (2017d), "Thermoelastic vibration and flexural behavior of FG-CNT reinforced composite curved panel", Int. J. Appl. Mech., 9(4), 1750046. https://doi.org/10.1142/S1758825117500466   DOI
36 Mehar, K., Panda, S.K. and Patle, B.K. (2018a), "Stress, deflection, and frequency analysis of CNT reinforced graded sandwich plate under uniform and linear thermal environment: A finite element approach", Polym. Compos., 39(10), 3792-3809. https://doi.org/10.1002/pc.24409   DOI
37 Mehar, K., Panda, S.K. and Mahapatra, T.R. (2018b), "Nonlinear frequency responses of functionally graded carbon nanotubereinforced sandwich curved panel under uniform temperature field", Int. J. Appl. Mech., 10(3), 1850028. https://doi.org/10.1142/S175882511850028X   DOI
38 Mehar, K., Panda, S.K., Devarajan, Y. and Choubey, G. (2019), "Numerical buckling analysis of graded CNT-reinforced composite sandwich shell structure under thermal loading", Compos. Struct., 216, 406-414. https://doi.org/10.1016/j.compstruct.2019.03.002   DOI
39 Mercan, K. and Civalek, O. (2016), "DSC method for buckling analysis of boron nitride nanotube (BNNT) surrounded by an elastic matrix", Compos. Struct., 143, 300-309. https://doi.org/10.1016/j.compstruct.2016.02.040   DOI
40 Mohsen, M. and Eyvazian, A. (2020), "Post-buckling analysis of Mindlin Cut out-plate reinforced by FG-CNTs", Steel Compos. Struct., Int. J., 34(2), 289-297. https://doi.org/10.12989/scs.2020.34.2.289
41 Moradi-Dastjerdi, R. and Payganeh, G. (2017), "Transient heat transfer analysis of functionally graded CNT reinforced cylinders with various boundary conditions", Steel Compos. Struct., Int. J., 24(3), 359-367. https://doi.org/10.12989/scs.2017.24.3.359
42 Motezaker, M., Jamali, M. and Kolahchi, R. (2020), "Application of differential cubature method for nonlocal vibration, buckling and bending response of annular nanoplates integrated by piezoelectric layers based on surface-higher order nonlocalpiezoelasticity theory", J. Computat. Appl. Mathe., 369, 112625. https://doi.org/10.1016/j.cam.2019.112625   DOI
43 Motezaker, M. and Eyvazian, A. (2020), "Buckling load optimization of beam reinforced by nanoparticles", Struct. Eng. Mech., Int. J., 73(5), 481-486. https://doi.org/10.12989/sem.2020.73.5.481
44 Motezaker, M. and Kolahchi, R. (2017a), "Seismic response of concrete columns with nanofiber reinforced polymer layer", Comput. Concrete, Int. J., 20(3), 361-368. https://doi.org/10.12989/cac.2017.20.3.361
45 Motezaker, M. and Kolahchi, R. (2017b), "Seismic response of $SiO_2$ nanoparticles-reinforced concrete pipes based on DQ and newmark methods", Comput. Concrete, Int. J., 19(6), 745-753. https://doi.org/10.12989/cac.2017.19.6.745
46 Narwariya, M., Choudhury, A. and Sharma, A.K. (2018), "Harmonic analysis of moderately thick symmetric cross-ply laminated composite plate using FEM", Adv. Computat. Des., Int. J., 3(2), 113-132. https://doi.org/10.12989/acd.2018.3.2.113
47 Natsuki, T., Endo, M. and Tsuda, H. (2006), "Vibration analysis of embedded carbon nanotubes using wave propagation approach", J. Appl. Phys., 99(3), 034311. https://doi.org/10.1063/1.2170418   DOI
48 Kumar, B.R. (2018), "Investigation on mechanical vibration of double-walled carbon nanotubes with inter-tube Van der waals forces", Adv. Nano Res., Int. J., 6(2), 135. https://doi.org/10.12989/anr.2018.6.2.135
49 Kolahchi, R., Keshtegar, B. and Fakhar, M.H. (2020), "Optimization of dynamic buckling for sandwich nanocomposite plates with sensor and actuator layer based on sinusoidal-viscopiezoelasticity theories using Grey Wolf algorithm", J. Sandw. Struct. Mater., 22(1), 3-27. https://doi.org/10.1177/1099636217731071   DOI
50 Kroner, E. (1967), "Elasticity theory of materials with long range cohesive forces", Int. J. Solids Struct., 3(5),731-742. https://doi.org/10.1016/0020-7683(67)90049-2   DOI
51 Lal, A., Jagtap, K.R. and Singh, B.N. (2017), "Thermomechanically induced finite element based nonlinear static response of elastically supported functionally graded plate with random system properties", Adv. Computat. Des., Int. J., 2(3), 165-194. https://doi.org/10.12989/acd.2017.2.3.165
52 Lei, Z. and Zhang, Y. (2018), "Characterizing buckling behavior of matrix-cracked hybrid plates containing CNTR-FG layers", Steel Compos. Struct., Int. J., 28(4), 495-508. https://doi.org/10.12989/scs.2018.28.4.495
53 Li, C. and Chou, T.W. (2003), "A structural mechanics approach for the analysis of carbon nanotubes", Int. J. Solids Struct., 40(10), 2487-2499. https://doi.org/10.1016/S0020-7683(03)00056-8   DOI
54 Madani, H., Hosseini, H. and Shokravi, M. (2016), "Differential cubature method for vibration analysis of embedded FG-CNTreinforced piezoelectric cylindrical shells subjected to uniform and non-uniform temperature distributions", Steel Compos. Struct., Int. J., 22(4), 889-913. https://doi.org/10.12989/scs.2016.22.4.889   DOI
55 Mehar, K. and Panda, S.K. (2016a), "Geometrical nonlinear free vibration analysis of FG-CNT reinforced composite flat panel under uniform thermal field", Compos. Struct., 143, 336-346. https://doi.org/10.1016/j.compstruct.2016.02.038   DOI
56 Akgoz, B. and Civalek, O. (2011), "Buckling analysis of cantilever carbon nanotubes using the strain gradient elasticity and modified couple stress theories", J. Computat. Theor. Nanosci., 8, 1821-1827. https://doi.org/10.1166/jctn.2011.1888   DOI
57 Natsuki, T., Ni, Q.Q. and Endo, M. (2007), "Wave propagation in single-walled and double-walled carbon nanotubes filled with fluids", J. Appl Phys., 101(3), 034319-034319-5. https://doi.org/10.1063/1.2432025   DOI
58 Zou, R.D. and Foster, C.G. (1995), "Simple solution for buckling of orthotropic circular cylindrical shells", Thin-Wall. Struct., 22(3), 143-158. https://doi.org/10.1016/0263-8231(94)00026-V   DOI
59 Adela, I. (2018), Computational Fluid Dynamics, Romania.
60 Akgoz, B. and Civalek, O. (2015), "A microstructure-dependent sinusoidal plate model based on the strain gradient elasticity theory", Acta Mechanica, 226(7), 2277-2294. https://doi.org/10.1007/s00707-015-1308-4   DOI
61 Amara, K., Tounsi, A., Mechab, I. and Adda-Bedia, E.A. (2010), "Nonlocal elasticity effect on column buckling of multiwalled carbon nanotubes under temperature field", Appl. Mathe. Model., 34(12), 3933-3942. https://doi.org/10.1016/j.apm.2010.03.029   DOI
62 Paliwal, D.N., Kanagasabapathy, H. and Gupta, K.M. (1995), "The large deflection of an orthotropic cylindrical shell on a Pasternak foundation", Compos. Struct., 31(1), 31-37. https://doi.org/10.1016/0263-8223(94)00068-9   DOI
63 Amnieh, H.B., Zamzam, M.S. and Kolahchi, R. (2018), "Dynamic analysis of non-homogeneous concrete blocks mixed by $SiO_2$ nanoparticles subjected to blast load experimentally and theoretically", Constr. Build. Mater., 174, 633-644. https://doi.org/10.1016/j.conbuildmat.2018.04.140   DOI
64 Ansari, R. and Arash, B. (2013), "Nonlocal Flugge shell model for vibrations of double-walled carbon nanotubes with different boundary conditions", J. Appl. Mech., 80(2), 021006. https://doi.org/10.1115/1.4007432   DOI
65 Ansari, R. and Rouhi, H. (2012), "Nonlocal analytical Flugge shell model for the axial buckling of double-walled carbon nanotubes with different end conditions", Int. J. Nano, 7, 1250081. https://doi.org/10.1142/S179329201250018X
66 Peddieson, J., Buchanan, G.R. and McNitt, R.P. (2003), "Application of Nonlocal Continuum Models to Nanotechnology", Int. J. Eng. Sei., 41, 305-312. https://doi.org/10.1016/S0020-7225(02)00210-0   DOI
67 Pradhan, S.C. and Phadikar, J.K. (2009), "Small scale effect on vibration of embedded multilayered graphene sheets based on nonlocal continuum models", Phy. Lett. A, 373(11), 1062-1069. https://doi.org/10.1016/j.physleta.2009.01.030   DOI
68 Qian, D., Wagner, G.J., Liu, W.K., Yu, M.F. and Ruoff, R.S. (2002), "Mechanics of carbon nanotubes", Appl. Mech. Rev., 55(6), 495-533. https://doi.org/10.1115/1.1490129   DOI
69 Rakrak, K., Zidour, M., Heireche, H., Bousahla, A.A. and Chemi, A. (2016), "Free vibration analysis of chiral double-walled carbon nanotube using non-local elasticity theory", Adv. Nano Res., Int. J., 4(1), 31-44. https://doi.org/10.12989/anr.2016.4.1.031   DOI
70 Arefi, M., Mohammadi, M., Tabatabaeian, A., Dimitri, R. and Tornabene, F. (2018), "Two-dimensional thermo-elastic analysis of FG-CNTRC cylindrical pressure vessels", Steel Compos. Struct., Int. J., 27(4), 525-536. https://doi.org/10.12989/scs.2018.27.4.525
71 Asghar, S., Hussain, M. and Naeem, M. (2019), "Non-local effect on the vibration analysis of double walled carbon nanotubes based on Donnell shell theory", Physica E: Low-dimens. Syst. Nanostruct., 116, 113726. https://doi.org/10.1016/j.physe.2019.113726   DOI
72 Avcar, M. (2015), "Effects of rotary inertia shear deformation and non-homogeneity on frequencies of beam", Struct. Eng. Mech., Int. J., 55(4), 871-884. https://doi.org/10.12989/sem.2015.55.4.871   DOI
73 Avcar, M. (2019), "Free vibration of imperfect sigmoid and power law functionally graded beams", Steel Compos. Struct., Int. J., 30(6), 603-615. https://doi.org/10.12989/scs.2019.30.6.603
74 Aydogdu, M.A. (2009), "A general nonlocal beam theory: Its application to nanobeam bending, buckling and vibration", Physica E, 41, 1651-1655. https://doi.org/10.1016/j.physe.2009.05.014   DOI
75 Sahouane, A., Hadji, L. and Bourada, M. (2019), "Numerical analysis for free vibration of functionally graded beams using an original HSDBT", Earthq. Struct., Int. J., 17(1), 31-37. https://doi.org/10.12989/eas.2019.17.1.031
76 Salah, F., Boucham, B., Bourada, F., Benzair, A., Bousahla, A.A. and Tounsi, A. (2019), "Investigation of thermal buckling properties of ceramic-metal FGM sandwich plates using 2D integral plate model", Steel Compos. Struct., Int. J., 33(6), 805-822. https://doi.org/10.12989/scs.2019.33.6.805
77 Sanchez-Portal, D., Artacho, E., Soler, J.M., Rubio, A. and Ordejon, P. (1999), "Ab-initio structural, elastic, and Vibrational Properties of Carbon Nanotubes", Phys. Rev. B, 59, 12678-2688. http://dx.doi.org/10.1103/PhysRevB.59.12678   DOI
78 Gafour, Y., Hamidi, A., Benahmed, A., Zidour, M. and Bensattalah, T. (2020), "Porosity-dependent free vibration analysis of FG nanobeam using non-local shear deformation and energy principle", Adv. Nano Res., Int. J., 8(1), 37-47. https://doi.org/10.12989/anr.2020.8.1.037   DOI
79 Flugge, S. (1973), Stresses in Shells, Springer, 2nd Edition, Berlin, Germany.
80 Fu, Y.M., Hong, J.W. and Wang, X.Q. (2006), "Analysis of nonlinear vibration for embedded carbon nanotubes", J. Sound Vib., 296(4-5), 746-756. https://doi.org/10.1016/j.jsv.2006.02.024   DOI
81 Gao, Y. and An, L. (2010), "A nonlocal elastic anisotropic shell model for microtubule buckling behaviors in cytoplasm", Physica E: Low-dimens. Syst. Nanostruct., 42(9), 2406-2415. https://doi.org/10.1016/j.physe.2010.05r.022   DOI
82 Ghodrati, B., Yaghootian, A., Ghanbar Zadeh, A. and Mohammad-Sedighi, H. (2018), "Lamb wave extraction of dispersion curves in micro/nano-plates using couple stress theories", Waves Random Complex Media, 28(1), 15-34. https://doi.org/10.1080/17455030.2017.1308582   DOI
83 Hadji, L., Zouatnia, N. and Bernard, F. (2019), "An analytical solution for bending and free vibration responses of functionally graded beams with porosities: Effect of the micromechanical models", Struct. Eng. Mech., Int. J., 69(2), 231-241. https://doi.org/10.12989/sem.2019.69.2.231
84 Hajmohammad, M.H., Farrokhian, A. and Kolahchi, R. (2018a), "Smart control and vibration of viscoelastic actuator-multiphase nanocomposite conical shells-sensor considering hygrothermal load based on layerwise theory", Aerosp. Sci. Technol., 78, 260-270. https://doi.org/10.1016/j.ast.2018.04.030   DOI
85 Yakobson, B.I., Brabec, C.J. and Bernholc, J. (1996), "Nanomechanics of carbon tubes: instabilities beyond linear response", Phy. Rev. Lett, 76, 2511-2514. https://doi.org/10.1103/PhysRevLett.76.2511   DOI
86 Kolahchi, R. and Cheraghbak, A. (2017), "Agglomeration effects on the dynamic buckling of viscoelastic microplates reinforced with SWCNTs using Bolotin method", Nonlinear Dyn., 90(1), 479-492. https://doi.org/10.1007/s11071-017-3676-x   DOI
87 Kolahchi, R. (2017), "A comparative study on the bending, vibration and buckling of viscoelastic sandwich nano-plates based on different nonlocal theories using DC, HDQ and DQ methods", Aerosp. Sci. Technol., 66, 235-248. https://doi.org/10.1016/j.ast.2017.03.016   DOI
88 Kolahchi, R. and Bidgoli, A.M. (2016), "Size-dependent sinusoidal beam model for dynamic instability of single-walled carbon nanotubes", Appl. Mathe. Mech., 37(2), 265-274. https://doi.org/10.1007/s10483-016-2030-8   DOI
89 Iijima, S., Brabec, C., Maiti, A. and Bernholc, J. (1996), "Structural flexibility of carbon nanotubes", J. Chem. Phys., 104(5), 2089. https://doi.org/10.1063/1.470966   DOI
90 Kolahchi, R., Hosseini, H. and Esmailpour, M. (2016a), "Differential cubature and quadrature-Bolotin methods for dynamic stability of embedded piezoelectric nanoplates based on visco-nonlocal-piezoelasticity theories", Compos. Struct., 157, 174-186. https://doi.org/10.1016/j.compstruct.2016.08.032   DOI
91 Kolahchi, R., Safari, M. and Esmailpour, M. (2016b), "Dynamic stability analysis of temperature-dependent functionally graded CNT-reinforced visco-plates resting on orthotropic elastomeric medium", Compos. Struct., 150, 255-265. https://doi.org/10.1016/j.compstruct.2016.05.023   DOI
92 Kolahchi, R., Zarei, M.S., Hajmohammad, M.H. and Nouri, A. (2017a), "Wave propagation of embedded viscoelastic FG-CNTreinforced sandwich plates integrated with sensor and actuator based on refined zigzag theory", Int. J. Mech. Sci., 130, 534-545. https://doi.org/10.1016/j.ijmecsci.2017.06.039   DOI
93 Sharma, P., Singh, R. and Hussain, M. (2019), "On modal analysis of axially functionally graded material beam under hygrothermal effect", Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science. https://doi.org/10.1177/0954406219888234
94 Sedighi, H.M. and Sheikhanzadeh, A. (2017), "Static and dynamic pull-in instability of nano-beams resting on elastic foundation based on the nonlocal elasticity theory", Chin. J. Mech. Eng., 30, 385-397. https://doi.org/10.1007/s10033-017-0079-3   DOI
95 Sehar, A., Hussain, M., Naeem, M.N. and Tounsi, A. (2020), "Prediction and assessment of nonlocal natural frequencies of DWCNTs: Vibration analysis", Comput. Concrete, Int. J., 25(2), 133-144. https://doi.org/10.12989/cac.2020.25.2.133
96 Shafiei, H. and Setoodeh, A.R. (2017), "Nonlinear free vibration and post-buckling of FG-CNTRC beams on nonlinear foundation", Steel Compos. Struct., Int. J., 24(1), 65-77. https://doi.org/10.12989/scs.2017.24.1.065   DOI
97 She, G.L., Ren, Y.R. and Yuan, F.G. (2019), "Hygro-thermal wave propagation in functionally graded double-layered nanotubes systems", Steel Compos. Struct., Int. J., 31(6), 641-653. https://doi.org/10.12989/scs.2019.31.6.641
98 Batou, B., Nebab, M., Bennai, R., Atmane, H.A., Tounsi, A. and Bouremana, M. (2019), "Wave dispersion properties in imperfect sigmoid plates using various HSDTs", Steel Compos. Struct., Int. J., 33(5), 699-716. https://doi.org/10.12989/scs.2019.33.5.699
99 Behera, S. and Kumari, P. (2018), "Free vibration of Levy-type rectangular laminated plates using efficient zig-zag theory", Adv. Computat. Des., Int. J., 3(3), 213-232. https://doi.org/10.12989/acd.2018.3.3.213
100 Benguediab, S., Tounsi, A., Zidour, M. and Semmah, A. (2014), "Chirality and scale effects on mechanical and buckling properties of zigzag double-walled carbon nanotubes", Composites Part B, 57, 21-24. https://doi.org/10.1016/j.compositesb.2013.08.020   DOI
101 Zamanian, M., Kolahchi, R. and Bidgoli, M.R. (2017), "Agglomeration effects on the buckling behaviour of embedded concrete columns reinforced with $SiO_2$ nano-particles", Wind Struct., Int. J., 24(1), 43-57. https://doi.org/10.12989/was.2017.24.1.043   DOI
102 Yakobson, B.I., Campbell, M.P., Brabec, C.J. and Bemholc, J. (1997), "High strain rate fracture and C-chain unravelling in carbon nanotubes", Comput. Mater. Sei., 8(4), 341-348. https://doi.org/10.1016/S0927-0256(97)00047-5   DOI
103 Yoon, J., Ru, C.Q. and Mioduchowski, A. (2003), "Vibration of an embedded multiwall carbon nanotube", Compos. Sei. Tech., 63(11), 1533-1542. https://doi.org/10.1016/S0266-3538(03)00058-7   DOI
104 Youcef, D.O., Kaci, A., Benzair, A., Bousahla, A.A. and Tounsi, A. (2018), "Dynamic analysis of nanoscale beams surface stress effects", Smart Struct. Syst., Int. J., 21(1), 65-74. https://doi.org/10.12989/sss.2018.21.1.065
105 Zarei, M.S., Kolahchi, R., Hajmohammad, M.H. and Maleki, M. (2017), "Seismic response of underwater fluid-conveying concrete pipes reinforced with $SiO_2$ nanoparticles and fiber reinforced polymer (FRP) layer", Soil Dyn. Earthq. Eng., 103, 76-85. https://doi.org/10.1016/j.soildyn.2017.09.009   DOI
106 Zemri, A., Houari, M.S.A., Bousahla, A.A. and Tounsi, A. (2015), "A mechanical response of functionally graded nanoscale beam: an assessment of a refined nonlocal shear deformation theory beam theory", Struct. Eng. Mech., Int. J., 54(4), 693-710. http://dx.doi.org/10.12989/sem.2015.54.4.693   DOI
107 Zidour, M., Daouadji, T.H., Benrahou, K.H., Tounsi, A., Bedia, E.A.A. and Hadji, L. (2014), "Buckling analysis of chiral singlewalled carbon nanotubes by using the nonlocal Timoshenko beam theory", Mech. Compos. Mater., 50(1), 95-104. https://doi.org/10.1007/s11029-014-9396-0   DOI
108 Bensattalah, T., Bouakkaz, K., Zidour, M. and Daouadji, T.H. (2018), "Critical buckling loads of carbon nanotube embedded in Kerr's medium", Adv. Nano Res., Int. J., 6(4), 339-356. https://doi.org/10.12989/anr.2018.6.4.339
109 Kolahchi, R., Zarei, M.S., Hajmohammad, M.H. and Oskouei, A.N. (2017b), "Visco-nonlocal-refined Zigzag theories for dynamic buckling of laminated nanoplates using differential cubature-Bolotin methods", Thin-Wall. Struct., 113, 162-169. https://doi.org/10.1016/j.tws.2017.01.016   DOI
110 Kolahchi, R., Hosseini, H., Fakhar, M.H., Taherifar, R. and Mahmoudi, M. (2019), "A numerical method for magneto-hygrothermal postbuckling analysis of defective quadrilateral graphene sheets using higher order nonlocal strain gradient theory with different movable boundary conditions", Comput. Mathe. Applicat., 78(6), 2018-2034. https://doi.org/10.1016/j.camwa.2019.03.042   DOI
111 Bilouei, B.S., Kolahchi, R. and Bidgoli, M.R. (2016), "Buckling of concrete columns retrofitted with Nano-Fiber Reinforced Polymer (NFRP)", Comput. Concrete, Int. J., 18(5), 1053-1063. https://doi.org/10.12989/cac.2016.18.5.1053   DOI
112 Bouadi, A., Bousahla, A.A., Houari, M.S.A., Heireche, H. and Tounsi, A. (2018), "A new nonlocal HSDT for analysis of stability of single layer graphene sheet", Adv. Nano Res., Int. J., 6(2), 147-162. https://doi.org/10.12989/anr.2018.6.2.147
113 Brischotto, S. (2015), "A continuum shell model including van der Waals interaction for free vibrations of double-walled carbon nanotubes", CMES, 104, 305-327.
114 Chemi, A., Zidour, M., Heireche, H., Rakrak, K. and Bousahla, A.A. (2018), "Critical Buckling Load of Chiral Double-Walled Carbon Nanotubes Embedded in an Elastic Medium", Mech. Compos. Mater., 53(6), 827-836. https://doi.org/10.1007/s11029-018-9708-x   DOI
115 Shen, H.S. and Zhang, C.L. (2010), "Torsional buckling and post buckling of double-walled carbon nanotubes by nonlocal shear deformable shell model", Compos. Struct., 92(5), 1073-1084. https://doi.org/10.1016/j.compstruct.2009.10.002   DOI
116 Soldano, C. (2015), "Hybrid metal-based carbon nanotubes", "Novel platform for multifunctional applications", Progress in Mater. Sci., 69, 183-212. https://doi.org/10.1016/j.pmatsci.2014.11.001   DOI
117 Sosa, E.D., Darlington, T.K., Hanos, B.A. and O'Rourke, M.J.E. (2014), "Multifunctional thermally remendable nanocomposites", J. Compos., 12 p. http://dx.doi.org/10.1155/2014/705687
118 Hajmohammad, M.H., Kolahchi, R., Zarei, M.S. and Nouri, A.H. (2019), "Dynamic response of auxetic honeycomb plates integrated with agglomerated CNT-reinforced face sheets subjected to blast load based on visco-sinusoidal theory", Int. J. Mech. Sci., 153, 391-401. https://doi.org/10.1016/j.ijmecsci.2019.02.008   DOI
119 Hajmohammad, M.H., Maleki, M. and Kolahchi, R. (2018b), "Seismic response of underwater concrete pipes conveying fluid covered with nano-fiber reinforced polymer layer", Soil Dyn. Earthq. Eng., 110, 18-27. https://doi.org/10.1016/j.soildyn.2018.04.002   DOI
120 Hajmohammad, M.H., Kolahchi, R., Zarei, M.S. and Maleki, M. (2018c), "Earthquake induced dynamic deflection of submerged viscoelastic cylindrical shell reinforced by agglomerated CNTs considering thermal and moisture effects", Compos. Struct., 187, 498-508. https://doi.org/10.1016/j.compstruct.2017.12.004   DOI
121 Hajnayeb, A. and Khadem, S.E. (2015), "An analytical study on the nonlinear vibration of a double walled carbon nanotube", Struct. Eng. Mech., Int. J., 54(5), 987-998. https://doi.org/10.12989/sem.2015.54.5.987   DOI
122 Hao, M.J., Guo, X.M. and Wang, Q. (2010), "Small-scale effect on torsional buckling of multi-walled carbon nanotubes", Eur. J. Mech. A/Solids, 29(1), 49-55. https://doi.org/10.1016/j.euromechsol.2009.05.008   DOI
123 Hernandez, E., Goze, C., Bemier, P. and Rubio, A. (1998), "Elastic properties of C and BxCyNz composite nanotubes", Phys. Rev. Lett., 80, 4502-4505. https://doi.org/10.1103/PhysRevLett.80.4502   DOI
124 Heydarpour, Y., Aghdam, M.M. and Malekzadeh, P. (2014), "Free vibration analysis of rotating functionally graded carbon nanotube-reinforced composite truncated conical shells", Compos. Struct., 117, 187-200. https://doi.org/10.1016/j.compstruct.2014.06.023   DOI
125 Taj, M., Safeer, M., Hussain, M., Naeem, M.N., Majeed, A., Ahmad, M., Khan, H.U. and Tounsi, A. (2020a), "Non-local orthotropic elastic shell model for vibration analysis of protein microtubules", Comput. Concrete, Int. J., 25(3), 245-253. https://doi.org/10.12989/cac.2020.25.3.245
126 Sudak, L.J. (2003), "Column buckling of multi-walled carbon nanotubes using nonlocal continuum mechanics", J. Appl. Phys., 94, 7281-7287. https://doi.org/10.1063/1.1625437   DOI
127 Sun, C.T. and Zhang, H. (2002), "Size-dependent elastic moduli of plate like nanomaterials", J. Appl. Phys., 93, 212-1218. https://doi.org/10.1063/1.1530365
128 Tahouneh, V. (2017), "Effects of CNTs waviness and aspect ratio on vibrational response of FG-sector plate", Steel Compos. Struct., Int. J., 25(6), 649-661. https://doi.org/10.12989/scs.2017.25.6.649
129 Taj, M., Safeer, M., Hussain, M., Naeem, M.N., Ahmad, M., Abbas, K., Khan, A.Q. and Tounsi, A. (2020b), "Effect of external force on buckling of cytoskeleton intermediate filaments within viscoelastic media", Comput. Concrete, Int. J., 25(3), 205-214. https://doi.org/10.12989/cac.2020.25.3.205
130 Taj, M., Hussain, M., Naeem, M.N. and Tounsi, A. (2020c), "Effects of elastic medium on buckling of microtubules due to bending and torsion", Adv. Concrete Constr., Int. J., 9(5), 491-501. https://doi.org/10.12989/acc.2020.9.5.491
131 Das, B., Mandal, M., Upadhyay, A., Chattopadhyay, P. and Karak, N. (2013), "Bio-based hyperbranched polyurethane/Fe3O4 nanocomposites: smart antibacterial biomaterials for biomedical devices and implants", Biomed. Mater., 8(3), 035003. https://doi.org/10.1088/1748-6041/8/3/035003   DOI
132 Do, Q.C., Pham, D.N., Vu, D.Q., Vu, T.T.A. and Nguyen, D.D. (2019), "Nonlinear buckling and post-buckling of functionally graded CNTs reinforced composite truncated conical shells subjected to axial load", Steel Compos. Struct., Int. J., 31(3), 243-259. https://doi.org/10.12989/scs.2019.31.3.243
133 Hu, Y.G., Liew, K.M. and Wang, Q. (2012), "Modeling of vibrations of carbon nanotubes", Procedia Eng., 31, 343-347. https://doi.org/10.1016/j.proeng.2012.01.1034   DOI
134 Hosseini, H. and Kolahchi, R. (2018), "Seismic response of functionally graded-carbon nanotubes-reinforced submerged viscoelastic cylindrical shell in hygrothermal environment", Physica E: Low-dimens. Syst. Nanostruct., 102, 101-109. https://doi.org/10.1016/j.physe.2018.04.037   DOI
135 Zouatnia, N. and Hadji, L. (2019), "Effect of the micromechanical models on the bending of FGM beam using a new hyperbolic shear deformation theory", Earthq. Struct., Int. J., 16(2), 177-183. https://doi.org/10.12989/eas.2019.16.2.177
136 Hu, Y.G., Liew, K.M., Wang, Q., He, X.Q. and Yakobson, B.I. (2008), "Nonlocal shell model for elastic wave propagation in single- and double-walled carbon nanotubes", J. Mech. Phy. Solids, 56, 3475-3485. https://doi.org/10.1016/j.jmps.2008.08.010   DOI
137 Hussain, M. and Naeem, M.N. (2017), "Vibration analysis of single-walled carbon nanotubes using wave propagation approach", Mech. Sci., 8(1), 155-164. https://doi.org/10.5194/ms-8-155-2017   DOI
138 Hussain, M. and Naeem, M. (2018a), "Vibration of single-walled carbon nanotubes based on Donnell shell theory using wave propagation approach", Chapter, Intechopen, In: Novel Nanomaterials - Synthesis and Applications. ISBN 978-953-51-5896-7 https://doi.org/10.5772 /intechopen.73503
139 Hussain, M. and Naeem, M.N. (2018b), "Effect of various edge conditions on free vibration characteristics of rectangular plates", Chapter, Intechopen, In: Advance Testing and Engineering. ISBN 978-953-51-6706-8
140 Hussain, M. and Naeem, M.N. (2019a), "Rotating response on the vibrations of functionally graded zigzag and chiral single walled carbon nanotubes". Appl. Math. Modeling, 75, 506-520. https://doi.org/10.1016/j.apm.2019.05.039   DOI
141 Eringen, A.C. (1972), "Linear theory of nonlocal elasticity and dispersion of plane waves", Int. J. Eng. Sci., 10(5), 425-435. https://doi.org/10.1016/0020-7225(72)90050-X   DOI
142 Ebrahimi, F. and Mahmoodi, F. (2018), "Vibration analysis of carbon nanotubes with multiple cracks in thermal environment", Adv. Nano Res., Int. J., 6(1), 57-80. https://doi.org/10.12989/anr.2018.6.1.057
143 Ehyaei, J. and Daman, M. (2017), "Free vibration analysis of double walled carbon nanotubes embedded in an elastic medium with initial imperfection", Adv. Nano Res., Int. J., 5(2), 179-192. https://doi.org/10.12989/anr.2017.5.2.179
144 Eltaher, M., Emam, S.A. and Mahmoud, F. (2013), "Static and stability analysis of nonlocal functionally graded nanobeams", Compos. Struct., 96, 82-88. https://doi.org/10.1016/j.compstruct.2012.09.030   DOI
145 Eringen, A.C. (1983), "On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves", J. Appl. Phys., 54, 4703-4710. https://doi.org/10.1063/1.33280   DOI
146 Eringen, A.C. (2002), Nonlocal Continuum Field Theories, Springer Science & Business Media.
147 Fakhrabadi, M.M.S., Rastgoo, A. and Ahmadian, M.T. (2015), "Application of electrostatically actuated carbon nanotubes in nanofluidic and bio-nanofluidic sensors and actuators", Measurement, 73, 127-136. https://doi.org/10.1016/j.measurement.2015.05.009   DOI
148 Tounsi, A., Benguediab, S., Semmah, A. and Zidour, M. (2013), "Nonlocal effects on thermal buckling properties of doublewalled carbon nanotubes", Adv. Nano Res., Int. J., 1(1), 1-11. https://doi.org/10.12989/anr.2013.1.1.001   DOI
149 Usuki, T. and Yogo, K. (2009), "Beam equations for multi-walled carbon nanotubes derived from Flugge shell theory", Proceedings of Royal Society A, 465(2104). https://doi.org/10.1098/rspa.2008.0394
150 Hussain, M. and Naeem, M.N. (2019b), "Effects of ring supports on vibration of armchair and zigzag FGM rotating carbon nanotubes using Galerkin's method", Compos. Part B: Eng., 163, 548-561. https://doi.org/10.1016/j.compositesb.2018.12.144   DOI
151 Hussain, M. and Naeem, M.N. (2019c), "Vibration characteristics of zigzag and chiral functionally graded material rotating carbon nanotubes sandwich with ring supports", J. Mech. Eng. Sci., Part C, 233(16), 5763-5780. https://doi.org/10.1177/0954406219855095   DOI
152 Hussain, M. and Naeem, M. (2019d), "Rotating response on the vibrations of functionally graded zigzag and chiral single walled carbon nanotubes", Appl. Mathe. Model., 75, 506-520. https://doi.org/10.1016/j.apm.2019.05.039   DOI
153 Hussain, M. and Naeem, M. (2019e), "Vibration characteristics of single-walled carbon nanotubes based on non-local elasticity theory using wave propagation approach (WPA) including chirality", Chapter, Intechopen, In: Perspective of Carbon Nanotubes. https://doi.org/10.5772/intechopen.85948
154 Hussain, M. and Naeem, M.N. (2020a), "Mass density effect on vibration of zigzag and chiral SWCNTs", J. Sandw. Struct. Mater. https://doi.org/10.1177/1099636220906257
155 Hussain, M. and Naeem, M.N. (2020b), "Vibration characteristics of zigzag FGM single-walled carbon nanotubes based on Ritz method with ring-stiffeners", Indian J. Phys. [In Press]
156 Hussain, M., Naeem, M.N., Shahzad, A. and He, M. (2017), "Vibrational behavior of single-walled carbon nanotubes based on cylindrical shell model using wave propagation approach", AIP Advances, 7(4), 045114. https://doi.org/10.1063/1.4979112   DOI
157 Hussain, M., Naeem, M., Shahzad, A. and He, M. (2018a), "Vibration characteristics of fluid-filled functionally graded cylindrical material with ring supports", Chapter, Intechopen, Computational Fluid Dynamics. ISBN 978-953-51-5706-9 https://doi.org/10.5772 /intechopen.72172
158 Hussain, M., Naeem, M.N., Tounsi, A. and Taj, M. (2019a), "Nonlocal effect on the vibration of armchair and zigzag SWCNTs with bending rigidity", Adv. Nano Res., Int. J., 7(6), 431-442. https://doi.org/10.12989/anr.2019.7.6.431   DOI
159 Hussain, M., Naeem, M.N., Shahzad, A., He, M. and Habib, S. (2018b), "Vibrations of rotating cylindrical shells with functionally graded material using wave propagation approach", IMechE Part C: J. Mech. Eng. Sci., 232(23), 4342-4356. https://doi.org/10.1177/0954406218802320   DOI
160 Hussain, M., Naeem, M.N. and Isvandzibaei, M. (2018c), "Effect of Winkler and Pasternak elastic foundation on the vibration of rotating functionally graded material cylindrical sheel", Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science, 232(24), 4564-4577. https://doi.org/10.1177/0954406217753459   DOI
161 Hussain, M., Naeem, M.N. and Taj, M. (2019b), "Effect of length and thickness variations on the vibration of SWCNTs based on Flugge's shell model", Micro Nano Lett., 15(1), 1-6. https://doi.org/10.1049/mnl.2019.0309   DOI
162 Hussain, M., Naeem, M.N. and Tounsi, A. (2020a), "Simulating vibration of single-walled carbon nanotube using Rayleigh-Ritz's method", 8(3), 215-228. https://doi.org/10.12989/anr.2020.8.3.215   DOI
163 Hussain, M., Naeem, M.N. and Tounsi, A. (2020b), "On mixing the Rayleigh-Ritz formulation with Hankel's function for vibration of fluid-filled Fluid-filled cylindrical shell", Adv. Computat. Des., Int. J. [In Press]
164 Hussain, M., Naeem, M.N. and Tounsi, A. (2020c), "Numerical Study for nonlocal vibration of orthotropic SWCNTs based on Kelvin's model", Adv. Concrete Constr., Int. J., 9(3), 301-312. https://doi.org/10.12989/acc.2020.9.3.301
165 Hussain, M., Naeem, M.N. and Tounsi, A. (2020d), "Response of orthotropic Kelvin modeling for single-walled carbon nanotubes: Frequency analysis", Adv. Nano Res., Int. J., 8(3), 229-244. https://doi.org/10.12989/anr.2020.8.3.229