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Forced vibration of an embedded single-walled carbon nanotube traversed by a moving load using nonlocal Timoshenko beam theory

  • Simsek, Mesut (Yildiz Technical University, Department of Civil Engineering, Davutpasa Campus)
  • Received : 2010.03.21
  • Accepted : 2010.12.09
  • Published : 2011.01.25
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Abstract

Dynamic analysis of an embedded single-walled carbon nanotube (SWCNT) traversed by a moving nanoparticle, which is modeled as a moving load, is investigated in this study based on the nonlocal Timoshenko beam theory, including transverse shear deformation and rotary inertia. The governing equations and boundary conditions are derived by using the principle of virtual displacement. The Galerkin method and the direct integration method of Newmark are employed to find the dynamic response of the SWCNT. A detailed parametric study is conducted to study the influences of the nonlocal parameter, aspect ratio of the SWCNT, elastic medium constant and the moving load velocity on the dynamic responses of SWCNT. For comparison purpose, free vibration frequencies of the SWCNT are obtained and compared with a previously published study. Good agreement is observed. The results show that the above mentioned effects play an important role on the dynamic behaviour of the SWCNT.

Keywords

References

  1. Abu-Hilal, M. and Mohsen, M. (2000), "Vibration of beams with general boundary conditions due to moving harmonic load", J. Sound Vib., 232(4), 703-717. https://doi.org/10.1006/jsvi.1999.2771
  2. Adali, S. (2008), "Variational principles for multi-walled carbon nanotubes undergoing buckling based on nonlocal elasticity theory", Phys. Lett. A, 372(35), 5701-5705. https://doi.org/10.1016/j.physleta.2008.07.003
  3. Aydogdu, M. and Ece, M.C. (2007), "Vibration and buckling of in-plane loaded double-walled carbon nanotubes", Turkish J. Eng. Env. Sci., 31, 305-310.
  4. Aydogdu, M. (2008), "Vibration of multi-walled carbon nanotubes by generalized shear deformation theory", Int. J. Mech. Sci., 50(4), 837-844. https://doi.org/10.1016/j.ijmecsci.2007.10.003
  5. 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
  6. Aydogdu, M. (2009), "A general nonlocal beam theory: Its application to nanobeam bending, buckling and vibration", Physica E, 41(9), 1651-1655. https://doi.org/10.1016/j.physe.2009.05.014
  7. Aydogdu, M. (2009), "A new shear deformation theory for laminated composite plates", Compos. Struct., 89(1), 94-101. https://doi.org/10.1016/j.compstruct.2008.07.008
  8. Chopra, A.K. (2001), Dynamics of structures, Prentice Hall, Inc, New Jersey.
  9. Civalek, O., and Akgoz, B. (2009), "Static analysis of single walled carbon nanotubes (SWCNT) based on Eringen's elasticity theory", Int. J. Eng. Appl. Sci., 2, 47-56.
  10. Civalek, O., Demir, C. and Akgoz, B. (2010), "Free vibration and bending analyses of cantilever microtubules based on nonlocal continuum model", Math. Comput. Appl., 15, 289-298.
  11. Civalek, O., Akgoz, B. (2010), "Free vibration analysis of microtubules as cytoskeleton components: Nonlocal Euler-Bernoulli beam modeling", Scientica Iranica, Transaction B- Mech. Eng., 17, 367-375.
  12. Cowper, G.R. (1966), "The shear coefficient in Timoshenko's beam theory", ASME J. Appl. Mech., 33, 335-340. https://doi.org/10.1115/1.3625046
  13. Dedkov, G.V. and Kyasov, A.A. (2007), "Fluctuational-electromagnetic interaction of a moving nanoparticle with walls of a flat dielectric gap", Tech. Phys. Lett., 33(1), 51-53. https://doi.org/10.1134/S1063785007010142
  14. Demir, C., Civalek, O. and Akgoz, B. (2010), "Free vibration analysis of carbon nanotubes Based on shear deformable beam theory by discrete singular convolution technique", Math. Comput. Appl., 15, 57-65.
  15. Ece, M.C. and Aydogdu, M. (2007), "Nonlocal elasticity effect on vibration of in-plane loaded double-walled carbon nano-tubes", Acta. Mech., 190(1-4), 185-195. https://doi.org/10.1007/s00707-006-0417-5
  16. Eringen, A.C. (1972), "Nonlocal polar elastic continua", Int. J. Eng. Sci., 10(1), 1-16. https://doi.org/10.1016/0020-7225(72)90070-5
  17. Eringen, A.C. and Edelen, D.G.B. (1972), "On nonlocal elasticity", Int. J. Eng. Sci., 10(3), 233-248. https://doi.org/10.1016/0020-7225(72)90039-0
  18. Eringen, A.C. (1983), "On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves", J. Appl. Phys., 54(9), 4703-4710. https://doi.org/10.1063/1.332803
  19. Fryba, L. (1972), Vibration of solids and structures under moving loads, the netherlands: Noordhoff International, Groningen.
  20. Garinei, A. (2006), "Vibrations of simple beam-like modeled bridge under harmonic moving loads", Int. J. Eng. Sci., 44(11-12), 778-787. https://doi.org/10.1016/j.ijengsci.2006.04.013
  21. Hu, Y.G., Liew, K.M. and Wang, Q. (2009), "Nonlocal elastic beam models for flexural wave propagation in double-walled carbon nanotubes", J. Appl. Phys., 106(4), 044301. https://doi.org/10.1063/1.3197857
  22. Heireche, H., Tounsi, A., Benzair, A., Maachou, M. and Adda, B.E.A. (2008), "Sound wave propagation in single-walled carbon nanotubes using nonlocal elasticity", Physica E, 40(8), 2791-2799. https://doi.org/10.1016/j.physe.2007.12.021
  23. Hummer, G., Rasaiah. J.C. and Noworyta, J.P. (2001), "Water conduction through the hydrophobic channel of a carbon nanotube", Nature, 414, 188-190. https://doi.org/10.1038/35102535
  24. Iijima, S. (1991), "Helical microtubes of graphitic carbon", Nature, 354, 56-58. https://doi.org/10.1038/354056a0
  25. 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", Comput. Mater. Sci., 47(2), 409-417. https://doi.org/10.1016/j.commatsci.2009.09.002
  26. Kiani K. and Mehri, B., (2010), "Assessment of nanotube structures under a moving nanoparticle using nonlocal beam theories", J. Sound Vib. 329(11), 2241-2264. https://doi.org/10.1016/j.jsv.2009.12.017
  27. Klasztorny, M. (2001), "Vertical vibrations of a multi-span beam steel bridge induced by a superfast passenger train" Struct. Eng. Mech., 12(3), 267-281. https://doi.org/10.12989/sem.2001.12.3.267
  28. Kocaturk, T. and im ek, M. (2006a), "Vibration of viscoelastic beams subjected to an eccentric compressive force and a concentrated moving harmonic force", J. Sound Vib., 291(1-2), 302-322. https://doi.org/10.1016/j.jsv.2005.06.024
  29. Kocaturk, T. and Simsek, M. (2006b), "Dynamic analysis of eccentrically prestressed viscoelastic Timoshenko beams under a moving harmonic load", Comput. Struct., 84(31-32), 2113-2127. https://doi.org/10.1016/j.compstruc.2006.08.062
  30. Kumar, D., Heinrich, C. and Waas, A.M. (2008), "Buckling analysis of carbon nanotubes modeled using nonlocal continuum theories", J. Appl. Phys., 103(7), 073521. https://doi.org/10.1063/1.2901201
  31. Lee, H.P. (1994), "Dynamic response of a beam with intermediate point constraints subject to a moving load", J. Sound Vib., 171(3), 361-368. https://doi.org/10.1006/jsvi.1994.1126
  32. Lee, H.W. and Chang, W.J. (2009), "Vibration analysis of a viscous-fluid-conveying single-walled carbon nanotube embedded in an elastic medium", Physica E, 41(4), 529-532. https://doi.org/10.1016/j.physe.2008.10.002
  33. Li, X.F. and Wang, B.L. (2009), "Vibrational modes of Timoshenko beams at small scales", Appl. Phys. Lett., 94(10), 101903. https://doi.org/10.1063/1.3094130
  34. Lim, C.W. and Wang, C.M. (2007), "Exact variational nonlocal stress modeling with asymptotic higher-order strain gradients for nanobeams", J. Appl. Phys., 101(5), 054312. https://doi.org/10.1063/1.2435878
  35. Lim, C.W., Li, C. and Yu, J.L. (2009), "The effects of stiffness strengthening nonlocal stress and axial tension on free vibration of cantilever nanobeams", Interaction and Multiscale Mechanics, 2, 223-233. https://doi.org/10.12989/imm.2009.2.3.223
  36. Lu, P., Lee, H.P., Lu, C. and Zhang, P.Q. (2006), "Dynamics properties of flexural beams using a nonlocal elasticity model", J. Appl. Phys., 99(7), 073510. https://doi.org/10.1063/1.2189213
  37. Lu, P., Lee, H.P., Lu, C. and Zhang, P.Q. (2007), "Application of nonlocal beam models for carbon nanotubes", Int. J. Solids. Struct., 44(16), 5289-5300. https://doi.org/10.1016/j.ijsolstr.2006.12.034
  38. Lu, P. (2007), "Dynamic analysis of axially prestressed micro/nanobeam structures based on nonlocal beam theory", J. Appl. Phys., 101(7), 073504. https://doi.org/10.1063/1.2717140
  39. Mir, M., Hosseini, A., Majzoobi, G.H. (2008), "A numerical study of vibrational properties of single-walled carbon nanotubes", Comput. Mater. Sci., 43(3), 540-548. https://doi.org/10.1016/j.commatsci.2007.12.024
  40. Murmu, T. and Pradhan, S.C. (2009), "Buckling analysis of a single-walled carbon nanotube embedded in an elastic medium based on nonlocal elasticity and Timoshenko beam theory and using DQM", Physica E, 41(7), 1232-1239. https://doi.org/10.1016/j.physe.2009.02.004
  41. Murmu, T. and Pradhan, S.C. (2009), "Small-scale effect on the vibration of nonuniform nanocantilever based on nonlocal elasticity theory", Physica E, 41(8), 1451-1456. https://doi.org/10.1016/j.physe.2009.04.015
  42. Murmu, T. and Pradhan, S.C. (2009), "Thermo-mechanical vibration of a single-walled carbon nanotube embedded in an elastic medium based on nonlocal elasticity theory", Comput. Mater. Sci., 46(4), 854-859. https://doi.org/10.1016/j.commatsci.2009.04.019
  43. Murmu, T. and Pradhan, S.C. (2010), "Thermal effects on the stability of embedded carbon nanotubes", Comput. Mater. Sci., 47(3), 721-726. https://doi.org/10.1016/j.commatsci.2009.10.015
  44. Narendar, S. and Gopalakrishnan, S. (2009), "Nonlocal scale effects on wave propagation in multi-walled carbon nanotubes", Comput. Mater. Sci., 47(2), 526-538. https://doi.org/10.1016/j.commatsci.2009.09.021
  45. Newmark, N.M. (1959), "A method of computation for structural dynamics", ASCE Eng. Mech. Div., 85, 67-94.
  46. Peddieson, J., Buchanan, G.R. and McNitt, R.P. (2003), "Application of nonlocal continuum models to nanotechnology", Int. J. Eng. Sci., 41(3-5), 305-312. https://doi.org/10.1016/S0020-7225(02)00210-0
  47. Pradhan, S.C., Sarkar, A. (2009), "Analyses of tapered fgm beams with nonlocal theory", Struct. Eng. Mech., 32(6), 811-833. https://doi.org/10.12989/sem.2009.32.6.811
  48. Pradhan, S.C. and Phadikar, J.K. (2009), "Bending, buckling and vibration analyses of nonhomogeneous nanotubes using GDQ and nonlocal elasticity theory", Struct. Eng. Mech., 33, 193-213. https://doi.org/10.12989/sem.2009.33.2.193
  49. Reddy, J.N. (2007), "Nonlocal theories for bending, buckling and vibration of beams", Int. J. Eng. Sci., 45(2-8), 288-307. https://doi.org/10.1016/j.ijengsci.2007.04.004
  50. Reddy, J.N. and Pang, S.D. (2008), "Nonlocal continuum theories of beams for the analysis of carbon nanotubes", J. Appl. Phys., 103(2), 023511. https://doi.org/10.1063/1.2833431
  51. Sato, M. and Shima, H. (2009), "Buckling characteristics of multiwalled carbon nanotubes under external pressure", Interaction and Multiscale Mechanics, 2, 209-222. https://doi.org/10.12989/imm.2009.2.2.209
  52. Sears, A. and Batra, R.C. (2004), "Macroscopic properties of carbon nanotubes from molecular-mechanics simulations", Phys. Rev. B, 69(23), 235406. https://doi.org/10.1103/PhysRevB.69.235406
  53. Sniady, P. (2008), "Dynamic response of a Timoshenko beam to a moving force", ASME J. Appl. Mech., 75(2), 0245031-0245034
  54. Sudak, L.J. (2003), "Column buckling of multiwalled carbon nanotubes using nonlocal continuum mechanics", J. Appl. Phys., 94(11), 7281. https://doi.org/10.1063/1.1625437
  55. Simsek, M. and Kocaturk, T. (2007), "Dynamic analysis of an eccentrically prestressed damped beam under a moving harmonic force using higher order shear deformation theory", ASCE J. Struct. Eng., 133(12), 1733- 1741. https://doi.org/10.1061/(ASCE)0733-9445(2007)133:12(1733)
  56. Simsek, M. and Kocaturk, T. (2009a), "Nonlinear dynamic analysis of an eccentrically prestressed damped beam under a concentrated moving harmonic load", J. Sound Vib., 320(1-2), 235-253. https://doi.org/10.1016/j.jsv.2008.07.012
  57. Simsek, M. and Kocaturk, T. (2009b), "Free and forced vibration of a functionally graded beam subjected to a concentrated moving harmonic load", Compos. Struct., 90(4), 465-473. https://doi.org/10.1016/j.compstruct.2009.04.024
  58. Simsek. (2010), "Vibration analysis of a functionally graded beam under a moving mass by using different beam theories", Compos. Struct., 92(4), 904-917. https://doi.org/10.1016/j.compstruct.2009.09.030
  59. Simsek, M. (2010), "Non-linear vibration analysis of a functionally graded Timoshenko beam under action of a moving harmonic load", Compos. Struct., 92(10), 2532-2546. https://doi.org/10.1016/j.compstruct.2010.02.008
  60. Simsek, M. (2010), "Vibration analysis of a single-walled carbon nanotube under action of a moving harmonic load based on nonlocal elasticity theory", Physica E, 43(1), 182-191. https://doi.org/10.1016/j.physe.2010.07.003
  61. Timoshenko, S. and Young, D.H. (1955), Vibration Problems in Engineering, Van Nostrand Company, New York.
  62. Tounsi, A., Heireche, H., Berrabah, H.M., Benzair, A. and Boumia, L. (2008), "Effect of small size on wave propagation in double-walled carbon nanotubes under temperature field", J. Appl. Phys., 104(10), 104301. https://doi.org/10.1063/1.3018330
  63. Wang, R.T. (1997), "Vibration of multi-span Timoshenko beams to a moving force", J. Sound Vib., 207(5), 731-742. https://doi.org/10.1006/jsvi.1997.1188
  64. Wang, R.T. and Lin, J.S. (1998), "Vibration of T-type Timoshenko frames subjected to moving loads", Struct. Eng. Mech., 6, 229-243. https://doi.org/10.12989/sem.1998.6.2.229
  65. Wang, R.T. and Sang, Y.L. (1999), "Out-of-plane vibration of multi-span curved beam due to moving loads", Struct. Eng. Mech., 7(4), 361-375. https://doi.org/10.12989/sem.1999.7.4.361
  66. Wang, C.M., Tan, V.B.C. and Zhang, Y.Y. (2006), "Timoshenko beam model for vibration analysis of multiwalled carbon nanotubes", J. Sound. Vib., 294(4-5), 1060-1072. https://doi.org/10.1016/j.jsv.2006.01.005
  67. Wang, Q. (2005), "Wave propagation in carbon nanotubes via nonlocal continuum mechanics", J. Appl. Phys., 98(12), 124301. https://doi.org/10.1063/1.2141648
  68. Wang, L. and Hu, H. (2005), "Flexural wave propagation in single-walled carbon nanotubes", Phys. Rev. B, 71(19), 195412. https://doi.org/10.1103/PhysRevB.71.195412
  69. Wang, Q., Varadan, V.K. and Quek, S.T. (2006), "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
  70. Wang, Q. and Liew, K.M. (2007), "Application of nonlocal continuum mechanics to static analysis of micro- and nano-structures", Phys. Lett. A, 363(3), 236-242. https://doi.org/10.1016/j.physleta.2006.10.093
  71. Wang, L. (2009), "Vibration and instability analysis of tubular nano- and micro-beams conveying fluid using nonlocal elastic theory", Physica E, 41(10), 1835-1840. https://doi.org/10.1016/j.physe.2009.07.011
  72. Wang, L. (2009), "Dynamical behaviors of double-walled carbon nanotubes conveying fluid accounting for the role of small length scale", Comput. Mater. Sci., 45(2), 584-588. https://doi.org/10.1016/j.commatsci.2008.12.006
  73. XiaoDong, Y. and Lim, C.W. (2009), "Nonlinear vibrations of nano-beams accounting for nonlocal effect using a multiple scale method", Science in China Series E: Technological Sciences, 52(3), 617-621. https://doi.org/10.1007/s11431-009-0046-z
  74. Yoon, J., Ru, C.Q. and Mioduchowski, A. (2003), "Vibration of an embedded multiwall carbon nanotube", Compos. Sci. Techn., 63(11), 1533-1542. https://doi.org/10.1016/S0266-3538(03)00058-7
  75. Yoon, J., Ru, C.Q. and Mioduchowski, A. (2004), "Timoshenko-beam effects on transverse wave propagation in carbon nanotubes", Compos. Part B: Eng., 35(2), 87-93. https://doi.org/10.1016/j.compositesb.2003.09.002
  76. Yu, L., Chan, T.H.T. and Zhu, J.H. (2008), "A MOM-based algorithm for moving force identification: Part II. Experiment and comparative studies", Struct. Eng. Mech., 29(2), 155-169. https://doi.org/10.12989/sem.2008.29.2.155
  77. Zhang, Y., Liu, G. and Han, X. (2005), "Transverse vibrations of double-walled carbon nanotubes under compressive axial load", Phys. Lett. A, 340(1-4), 258-266. https://doi.org/10.1016/j.physleta.2005.03.064
  78. Zhang, Y.Q., Liu, G.R and Wang, J.S. (2004), "Small-scale effects on buckling of multiwalled carbon nanotubes under axial compression", Phys. Rev. B, 70(20), 205430. https://doi.org/10.1103/PhysRevB.70.205430
  79. Zhang, Y.Q., Liu, G.R. and Han, X. (2006), "Effect of small length scale on elastic buckling of multi-walled carbon nanotubes under radial pressure", Phys. Lett. A, 349(5), 370-376. https://doi.org/10.1016/j.physleta.2005.09.036
  80. Zhang, Y.Q., Liu, G.R. and Xie, X.Y. (2005), "Free transverse vibrations of double-walled carbon nanotubes using a theory of nonlocal elasticity", Phys. Rev. B, 71(19), 195404. https://doi.org/10.1103/PhysRevB.71.195404
  81. Zheng, D.Y., Cheung, Y.K., Au, F.T.K. and Cheng, Y.S. (1998), "Vibration of multi-span non-uniform beams under moving loads by using modified beam vibration functions" J. Sound Vib., 212(3), 455-467. https://doi.org/10.1006/jsvi.1997.1435
  82. Zhu, X.Q. and Law, S.S. (1999), "Moving force identification on multi-span continuous bridge", J. Sound Vib., 228(2), 377-396. https://doi.org/10.1006/jsvi.1999.2416

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  38. Dynamic characteristics of viscoelastic nanoplates under moving load embedded within visco-Pasternak substrate and hygrothermal environment vol.4, pp.8, 2017, https://doi.org/10.1088/2053-1591/aa7d89
  39. Dynamical properties of nanotubes with nonlocal continuum theory: A review vol.55, pp.7, 2012, https://doi.org/10.1007/s11433-012-4781-y
  40. Physical insight into Timoshenko beam theory and its modification with extension vol.48, pp.4, 2013, https://doi.org/10.12989/sem.2013.48.4.519
  41. Finite element buckling analysis of multi-layered graphene sheets on elastic substrate based on nonlocal elasticity theory vol.38, pp.24, 2014, https://doi.org/10.1016/j.apm.2014.03.036
  42. Dynamic response of nanobeams subjected to moving nanoparticles and hygro-thermal environments based on nonlocal strain gradient theory pp.1537-6532, 2018, https://doi.org/10.1080/15376494.2018.1444234
  43. The computation of bending eigenfrequencies of single-walled carbon nanotubes based on the nonlocal theory vol.9, pp.2, 2018, https://doi.org/10.5194/ms-9-349-2018
  44. Vibration of initially stressed carbon nanotubes under magneto-thermal environment for nanoparticle delivery via higher-order nonlocal strain gradient theory vol.133, pp.6, 2018, https://doi.org/10.1140/epjp/i2018-12039-5
  45. Nonlocal effects on thermal buckling properties of double-walled carbon nanotubes vol.1, pp.1, 2013, https://doi.org/10.12989/anr.2013.1.1.001
  46. Assessment of various nonlocal higher order theories for the bending and buckling behavior of functionally graded nanobeams vol.23, pp.3, 2011, https://doi.org/10.12989/scs.2017.23.3.339
  47. Damping and vibration analysis of viscoelastic curved microbeam reinforced with FG-CNTs resting on viscoelastic medium using strain gradient theory and DQM vol.25, pp.2, 2011, https://doi.org/10.12989/scs.2017.25.2.141
  48. Size dependent bending analysis of micro/nano sandwich structures based on a nonlocal high order theory vol.27, pp.3, 2018, https://doi.org/10.12989/scs.2018.27.3.371
  49. Buckling analysis of new quasi-3D FG nanobeams based on nonlocal strain gradient elasticity theory and variable length scale parameter vol.28, pp.1, 2011, https://doi.org/10.12989/scs.2018.28.1.013
  50. Nonlinear free and forced vibration analysis of microbeams resting on the nonlinear orthotropic visco-Pasternak foundation with different boundary conditions vol.28, pp.2, 2018, https://doi.org/10.12989/scs.2018.28.2.149
  51. Free axial vibration analysis of axially functionally graded thick nanorods using nonlocal Bishop's theory vol.28, pp.6, 2018, https://doi.org/10.12989/scs.2018.28.6.749
  52. Size-dependent forced vibration response of embedded micro cylindrical shells reinforced with agglomerated CNTs using strain gradient theory vol.22, pp.5, 2011, https://doi.org/10.12989/sss.2018.22.5.527
  53. Wave propagation of functionally graded anisotropic nanoplates resting on Winkler-Pasternak foundation vol.70, pp.1, 2019, https://doi.org/10.12989/sem.2019.70.1.055
  54. Theoretical analysis of chirality and scale effects on critical buckling load of zigzag triple walled carbon nanotubes under axial compression embedded in polymeric matrix vol.70, pp.3, 2019, https://doi.org/10.12989/sem.2019.70.3.269
  55. Some closed-form solutions for static, buckling, free and forced vibration of functionally graded (FG) nanobeams using nonlocal strain gradient theory vol.224, pp.None, 2019, https://doi.org/10.1016/j.compstruct.2019.111041
  56. Effect of nonlocal parameter on nonlocal thermoelastic solid due to inclined load vol.33, pp.1, 2011, https://doi.org/10.12989/scs.2019.33.1.123
  57. Static stability analysis of axially functionally graded tapered micro columns with different boundary conditions vol.33, pp.1, 2019, https://doi.org/10.12989/scs.2019.33.1.133
  58. Nonlocal effect on the vibration of armchair and zigzag SWCNTs with bending rigidity vol.7, pp.6, 2011, https://doi.org/10.12989/anr.2019.7.6.431
  59. Effects of nonlocality and two temperature in a nonlocal thermoelastic solid due to ramp type heat source vol.27, pp.1, 2020, https://doi.org/10.1080/25765299.2020.1825157
  60. An inclined FGM beam under a moving mass considering Coriolis and centrifugal accelerations vol.35, pp.1, 2020, https://doi.org/10.12989/scs.2020.35.1.061
  61. Time harmonic interactions in non local thermoelastic solid with two temperatures vol.74, pp.3, 2011, https://doi.org/10.12989/sem.2020.74.3.341
  62. Effect of Pasternak foundation: Structural modal identification for vibration of FG shell vol.9, pp.6, 2020, https://doi.org/10.12989/acc.2020.9.6.569
  63. Runge-Kutta method for flow of dusty fluid along exponentially stretching cylinder vol.36, pp.5, 2011, https://doi.org/10.12989/scs.2020.36.5.603
  64. Flow of casson nanofluid along permeable exponentially stretching cylinder: Variation of mass concentration profile vol.38, pp.1, 2011, https://doi.org/10.12989/scs.2021.38.1.033
  65. Parametric vibration analysis of single-walled carbon nanotubes based on Sanders shell theory vol.10, pp.2, 2021, https://doi.org/10.12989/anr.2021.10.2.165
  66. On the mechanics of nanocomposites reinforced by wavy/defected/aggregated nanotubes vol.38, pp.5, 2011, https://doi.org/10.12989/scs.2021.38.5.533
  67. Thermal stress effects on microtubules based on orthotropic model: Vibrational analysis vol.11, pp.3, 2011, https://doi.org/10.12989/acc.2021.11.3.255
  68. Effect of suction on flow of dusty fluid along exponentially stretching cylinder vol.10, pp.3, 2011, https://doi.org/10.12989/anr.2021.10.3.263
  69. Aerodynamic Analysis of Temperature-Dependent FG-WCNTRC Nanoplates under a Moving Nanoparticle using Meshfree Finite Volume Method vol.134, pp.None, 2011, https://doi.org/10.1016/j.enganabound.2021.10.021