Advances in nano research

Techno-Press (테크노프레스)
권호 표지
DOI QR코드

DOI QR Code

Longitudinal vibration of a nanorod embedded in viscoelastic medium considering nonlocal strain gradient theory

  • Balci, Mehmet N. (Department of Mechanical Engineering, Hacettepe University)
  • Received : 2021.01.26
  • Accepted : 2022.04.23
  • Published : 2022.08.25
⟨ Previous Next ⟩

Abstract

This article investigates the longitudinal vibration of a nanorod embedded in viscoelastic medium according to the nonlocal strain gradient theory. Viscoelastic medium is considered based on Kelvin-Voigt model. Governing partial differential equation is derived based on longitudinal equilibrium and analytical solution is obtained by adopting harmonic motion solution for the nanorod. Modal frequencies and corresponding damping ratios are presented to demonstrate the influences of nonlocal parameter, material length scale, elastic and damping parameters of the viscoelastic medium. It is observed that material length scale parameter is very influential on modal frequencies especially at lower values of nonlocal parameter whereas increase in length scale parameter has less effect at higher values of nonlocal parameter when the medium is purely elastic. Elastic stiffness and damping coefficient of the medium have considerable impacts on modal frequencies and damping ratios, and the highest impact of these parameters on frequency and damping ratio is seen in the first mode. Results calculated based on strain gradient theory are quite different from those calculated based on classical elasticity theory. Hence, nonlocal strain gradient theory including length scale parameter can be used to get more accurate estimations of frequency response of nanorods embedded in viscoelastic medium.

Keywords

References

  1. 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.
  2. Aifantis, E.C. (1999), "Strain gradient interpretation of size effects", Int. J. Fract., 95, 299-314. https://doi.org/10.1023/A:1018625006804.
  3. Akbas, S.D. (2020), "Modal analysis of viscoelastic nanorods under an axially harmonic load", Adv. Nano Res., 8(4), 277-282. https://doi.org/10.12989/anr.2020.8.4.277.
  4. Al-Furjan, M.S.H., Dehini, R., Khorami, M., Habibi, M. and Jung, D.W. (2020), "On the dynamics of the ultra-fast rotating cantilever orthotropic piezoelectric nanodisk based on nonlocal strain gradient theory", Compos. Struct., 255, 112990. https://doi.org/10.1016/j.compstruct.2020.112990.
  5. Alizadeh-Hamidi, B., Hassannejad, R. and Omidi, Y. (2021), "Size-dependent thermos-mechanical vibration of lipid supramolecular nano-tubules via nonlocal strain gradient Timoshenko beam theory", Comput. Biol. Med., 134, 104475. https://doi: 10.1016/j.compbiomed.2021.104475.
  6. Allen, M.P. (2004), "Introduction to molecular dynamics simulation", Comput. Soft Matter, 23, 1-28.
  7. Andrews, R. and Weisenberger, M.C. (2004), "Carbon nanotube polymer composites", Curr. Opin. Solid State Mater. Sci., 8(1), 31-37. https://doi.org/10.1016/j.cossms.2003.10.006.
  8. Ansari, R., Sahmani, S. and Rouhi, H. (2011), "Axial buckling analysis of single-walled carbon nanotubes in thermal environments via the Rayleigh-Ritz technique", Comput. Mater. Sci., 50, 3050-3055. https://doi.org/10.1016/j.commatsci.2011.05.027.
  9. Arda, M. and Aydogdu, M. (2015), "Analysis of free torsional vibration in carbon nanotubes embedded in a viscoelastic medium", Adv. Sci. Technol. Res. J., 9(26), 28-33, https://doi.org/10.12913/22998624/2361.
  10. Arda, M. and Aydogdu, M. (2019), "Torsional dynamics of coaxial nanotubes with different lengths in viscoelastic medium", Microsyst. Technol., 25, 3943-3957. https://doi.org/10.1007/s00542-019-04446-8.
  11. Aydogdu, M. (2009), "Axial vibration of nanorods with the nonlocal continuum rod model", Physica E, 41, 861-864. https://doi.org/10.1016/j.physe.2009.01.007.
  12. Aydogdu, M. (2012), "Axial vibration analysis of nanorods (carbon nanotubes) embedded in elastic medium using nonlocal elasticity", Mech. Res. Commun., 43, 34-40. https://doi.org/10.1016/j.mechrescom.2012.02.001.
  13. Bilal Tahir, M., Riaz, K.N. and Asiri, A.M. (2019), "Boosting the performance of visible light-driven WO3/g-C3N4 anchored with BiVO4 nanoparticles for photocatalytic hydrogen evolution", Int. J. Energy Res., 43, 5747-5758. https://doi.org/10.1002/er.4673.
  14. Bensaid, I. Bekhadda, A. and Kerboua, B. (2018), "Dynamic analysis of higher order shear-deformable nanobeams resting on elastic foundation based on nonlocal strain gradient theory", Adv. Nano Res., 6(3), 279-298. https://doi.org/10.12989/anr.2018.6.3.279.
  15. Challamel, N., Rakotomanana, L. and Marrec, L. (2009), "A dispersive wave equation using nonlocal elasticity", Comptes Rendus Mecanique, 337(8), 591-595. https://doi.org/10.1016/j.crme.2009.06.028.
  16. Challamel, N. (2013), "Variational formulation of gradient or/and nonlocal higher-order shear elasticity beams", Compos. Struct., 105, 351-368. https://doi.org/10.1016/j.compstruct.2013.05.026.
  17. Cruz, F.J.A.L., de Pablo, J.J. and Mota, J.P.B. (2014), "Endohedral confinement of a DNA dodecamer onto pristine carbon nanotubes and the stability of the canonical B form", J. Chem. Phys., 140(22), 225103. https://doi.org/10.1063/1.4881422.
  18. Cruz, F.J.A.L. and Mota, J.P.B., (2016), "Conformational thermodynamics of DNA strands in hydrophilic nanopores", J. Phys. Chem. C, 120(36), 20357-20367. https://doi.org/10.1021/acs.jpcc.6b06234.
  19. Dalton, A.B., Collins, S., Munoz, E., Razal, J.M., Ebron, V.H., Ferraris, J.P., Coleman, J.N., Kim, B.G. and Baughman, R.H. (2003), "Super-tough carbon-nanotube fibres", Nature, 423, 703. https://doi.org/10.1038/423703a.
  20. Ebrahimi, F., Dehghan, M. and Seyfi, A. (2019a), "Eringen's nonlocal elasticity theory for wave propagation analysis of magneto-electro-elastic nanotubes", Adv. Nano Res., 7(1), 1-11. https://doi.org/10.12989/anr.2019.7.1.001.
  21. Ebrahimi, F., Karimiasl, M. and Mahesh, V. (2019b), "Vibration analysis of magneto-flexo-electrically actuated porous rotary nanobeams considering thermal effects via nonlocal strain gradient elasticity theory", Adv. Nano Res., 7(4), 223-231. https://doi.org/10.12989/anr.2019.7.4.223.
  22. Ebrahimi, F., Daman, M. and Mahesh, V. (2019c), "Thermomechanical vibration analysis of curved imperfect nano-beams based on nonlocal strain gradient theory", Adv. Nano Res., 7(4), 249-263. https://doi.org/10.12989/anr.2019.7.4.249.
  23. Eltaher, M.A., Emam, S.A. and Mahmoud, F.F. (2012), "Free vibration analysis of functionally graded size-dependent nanobeams", Appl. Math. Comput., 218, 7406-7420. https://doi.org/10.1016/j.amc.2011.12.090.
  24. Eringen, A.C. (1967), "Theory of micropolar plates", Zeitschrift fur Angewandte Mathematik und Physik, 18, 12-30. https://doi.org/10.1007/BF01593891
  25. Eringen, A.C. (1972), "Nonlocal Polar Elastic Continua", Int. J. Eng. Sci., 10, 1-16. https://doi.org/10.1016/0020-7225(72)90070-5.
  26. 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/10.1063/1.332803.
  27. Eringen, A.C. (2002), Nonlocal Continuum Field Theories, Springer, U.S.A.
  28. Esen, I. (2020), "Response of a micro-capillary system exposed to a moving mass in magnetic field using nonlocal strain gradient theory", Int. J. Mech. Sci., 188, 105937. https://doi.org/10.1016/j.ijmecsci.2020.105937.
  29. Fu, Y., Li, L. and Hu, Y. (2018), "Enlarging quality factor in microbeam resonators by topology optimization", J. Therm. Stresses, 42(3), 341-360. https://doi.org/10.1080/01495739.2018.1489744.
  30. Fu, G., Zhou, S. and Qi, L., (2020), "On the strain gradient elasticity theory for isotropic materials", Int. J. Eng. Sci., 154, 103348. https://doi.org/10.1016/j.ijengsci.2020.103348.
  31. Ghavanloo, E., Rafiei, M. and Daneshmand, F. (2011), "In-plane vibration analysis of curved carbon nanotubes conveying fluid embedded in viscoelastic medium", Phys. Lett. A, 375, 1994-1999. https://doi.org/10.1016/j.physleta.2011.03.025.
  32. Guz, L., Fama, L., Candal, R. and Goyanes, S. (2017), "Size effect of ZnO nanorods on physicochemical properties of plasticized starch composites", Carbohydrate Polymers, 157, 1611-1619. https://doi.org/10.1016/j.carbpol.2016.11.041.
  33. Hashemi, H. and Khaniki, H.B. (2017), "Dynamic behavior of multi-layered viscoelastic nanobeam system embedded in a viscoelastic medium with a moving nanoparticle", J. Mech., 33(5), 559-575. https://doi.org/10.1017/jmech.2016.91.
  34. Hsu, T.W., Yang, C.C., Chu, C.Y., Tung, Y.H., Kao, C.W., Wu, W.C. and Lin, K.S. (2019), "Size effect on the structure and magnetic properties of SmMn2O5 nanorods", Chin. J. Phys., 62, 368-373. https://doi.org/10.1016/j.cjph.2019.10.012.
  35. Imboden, M. and Mohanty, P. (2014), "Dissipation in nanoelectromechanical systems", Phys. Rep. 534, 89-146. http://doi.org/10.1016/j.physrep.2013.09.003.
  36. Karlicic, D., Kozic, P. and Pavlovic, R. (2014), "Free transverse vibration of nonlocal viscoelastic orthotropic multi-nanoplate system (MNPS) embedded in a viscoelastic medium", Compos. Struct., 115, 89-99. https://doi.org/10.1016/j.compstruct.2014.04.002.
  37. Karlicic, D., Cajic, M., Murmu, T. and Adhikari, S. (2015), "Nonlocal longitudinal vibration of viscoelastic coupled doublenanorod systems", Eur. J. Mech. A-Solid, 49, 183-196. https://doi.org/10.1016/j.euromechsol.2014.07.005.
  38. Karlicic, D., Kozic, P., Pavlovic, R. and Nesic, N. (2017), "Dynamic stability of single-walled carbon nanotube embedded in a viscoelastic medium under the influence of the axially harmonic load", Compos. Struct., 162, 227-243. https://doi.org/10.1016/j.compstruct.2016.12.003.
  39. Kazemi-Lari, M.A., Ghavanloo, E. and Fazelzadeh, S.A. (2013), "Structural instability of carbon nanotubes embedded in viscoelastic medium and subjected to distributed tangential load", J. Mech. Sci. Technol., 27(7), 2085-2091. https://doi.org/10.1007/s12206-013-0522-z.
  40. Khosravi, F. and Hosseini, S.A. (2020), "On the viscoelastic carbon nanotube mass nanosensor using torsional forced vibration and Eringen's nonlocal model", Mech. Based Des. Struct. Mach., 50(3), 1030-1053. https://doi.org/ 10.1080/15397734.2020.1744001.
  41. Kung, S.W. and Singh, R. (1998a), "Vibration analysis of beams with multiple constrained layer damping patches", J. Sound Vib., 212(5), 1-28. https://doi.org/10.1006/jsvi.1997.1409.
  42. Kung, SW. and Singh, R. (1998b), "Complex eigensolutions of rectangular plates with damping patches", J. Sound Vib., 216(1), 1-28. https://doi.org/10.1006/jsvi.1998.1644.
  43. Li, L. and Hu, Y. (2015), "Buckling analysis of size-dependent nonlinear beams based on a nonlocal strain gradient theory", Int. J. Eng. Sci., 97, 84-94. https://doi.org/10.1016/j.ijengsci.2015.08.013.
  44. Li, L. and Hu, Y. (2016), "Wave propagation in fluid-conveying viscoelastic carbon nanotubes based on nonlocal strain gradient theory", Comput. Mater. Sci., 112, 282-288. https://doi.org/10.1016/j.commatsci.2015.10.044.
  45. Li, L., Hu, Y. and Li, X. (2016a), "Longitudinal vibration of sizedependent rods via nonlocal strain gradient theory", Int. J. Mech. Sci., 115-116, 135-144. https://doi.org/10.1016/j.ijmecsci.2016.06.011.
  46. Li, L., Hu, Y. and Ling, L. (2016b), "Wave propagation in viscoelastic single-walled carbon nanotubes with surface effect under magnetic field based on nonlocal strain gradient theory", Physica E, 75, 118-124, https://doi.org/10.1016/j.physe.2015.09.028.
  47. Li, X.F, Shen, Z.B. and Lee, K.Y. (2017), "Axial wave propagation and vibration of nonlocal nanorods with radial deformation and inertia", Z. Angew. Math. Mech., 97(5), 602-616. https://doi.org/10.1002/zamm.201500186.
  48. Lim, C.W., Zhang, G. and Reddy, J.N. (2015), "A higher-order nonlocal elasticity and strain gradient theory and its applications in wave propagation", J. Mech. Phys. Solids, 78, 298-313. https://doi.org/10.1016/j.jmps.2015.02.001.
  49. Lu, L., Guo, X. and Zhao, J. (2017), "A unified nonlocal strain gradient model for nanobeams and the importance of higher order terms", Int. J. Eng. Sci., 119, 265-277. https://doi.org/10.1016/j.ijengsci.2017.06.024.
  50. Ma, H.M., Gao, X.L. and Reddy, J.N. (2008), "A microstructuredependent Timoshenko beam model based on a modified couple stress theory", J. Mech. Phys. Solids, 56, 3379-3391. https://doi.org/10.1016/j.jmps.2008.09.007.
  51. Malikan, M. and Nguyen, V.B. (2018), "Buckling analysis of piezo-magnetic nanoplates in hygrothermal environment based on a novel one variable plate theory combining with higherorder nonlocal strain gradient theory", Physica E, 102, 8-28. https://doi.org/10.1016/j.physe.2018.04.018.
  52. Mindlin, R. (1964), "Micro-structure in linear elasticity", Arch. Ration. Mech. Anal., 16, 52-78. https://doi.org/10.1007/BF00248490.
  53. Mindlin, R. (1965), "Second gradient of strain and surface-tension in linear elasticity", Int. J. Solids Struct., 1, 414-438. https://doi.org/10.1007/BF00248490.
  54. Mirjavadi, S.S., Forsat, M., Nia, A.F., Badnava, S. and Hamouda, A.M.S. (2020), "Nonlocal strain gradient effects on forced vibrations of porous FG cylindrical nanoshells", Adv. Nano Res., 8(2), 149-156. https://doi.org/10.12989/anr.2020.8.2.149.
  55. Mohammadian, M., Hosseini, S.M. and Abolbashari, M.H. (2019), "Lateral vibrations of embedded hetero-junction carbon nanotubes based on the nonlocal strain gradient theory: Analytical and differential quadrature element (DQE) methods", Physica E, 105, 68-82. https://doi.org/10.1016/j.physe.2018.08.022.
  56. Mohammadimehr, M., Monajemi, A.A. and Moradi, M. (2015), "Vibration analysis of viscoelastic tapered micro-rod based on strain gradient theory resting on visco-pasternak foundation using DQM", J. Mech. Sci. Technol., 29(6), 2297-2305. https://doi.org/10.1007/s12206-015-0522-2.
  57. 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 using DQM", Physica E, 41, 1232-1239. https://doi.org/10.1016/j.physe.2009.02.004.
  58. Murmu, T. and Adhikari, S. (2011), "Axial instability of doublenanobeam- systems", Phys. Lett. A, 375, 601-608. https://doi.org/10.1016/j.physleta.2010.11.007.
  59. Namazu, T., Isono, Y. and Tanaka, T. (2000), "Evaluation of size effect on mechanical properties of single crystal silicon by nanoscale bending test using AFM", J. Microelectromech Syst., 9(4), 450-459. https://doi.org/10.1109/84.896765.
  60. Park, S.K. and Gao, X.L. (2006), "Bernoulli-Euler beam model based on a modified couple stress theory", J. Micromech. Microeng., 16, 2355-2359. https://doi.org/10.1088/0960-1317/16/11/015.
  61. Peddieson, J., Buchanan, G.R. and McNitt, R.P. (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.
  62. Pradhan, S.C. and Reddy, G.K. (2011), "Buckling analysis of single walled carbon nanotube on Winkler foundation using nonlocal elasticity theory and DTM", Comput. Mater. Sci., 50, 1052-1056. https://doi.org/10.1016/j.commatsci.2010.11.001.
  63. Rafique, M., Bilal Tahir, M., Rafique, M.S., Safdar, N. and Tahir, R. (2020), "Chapter 2- nanostructure materials and their classification by dimensionality, nanotechnology and photocatalysis for environmental applications", Micro Nanotechnol., 27-44. https://doi.org/10.1016/B978-0-12-821192-2.00002-4.
  64. Rao, S.S. (2007), Vibration of Continuous systems, John Wiley & Sons, Inc. Hoboken, New Jersey., U.S.A.
  65. 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.
  66. Reddy, J.N. (2011), "Microstructure-dependent couple stress theories of functionally graded beams", J. Mech. Phys. Solids, 59, 2382-2399, https://doi.org/10.1016/j.jmps.2011.06.008.
  67. Roudbari, M.A., Jorshari, T.D., Lu, C., Ansari, R., Kouzani, A.Z. and Amabili, M. (2022), "A review of size-dependent continuum mechanics models for micro- and nano-structures", Thin Wall. Struct, 170, 108562. https://doi.org/10.1016/j.tws.2021.108562.
  68. Ruan, S.L., Gao, P., Yang, X.G. and Yu, T.X. (2003), "Toughening high performance ultrahigh molecular weight polyethylene using multiwalled carbon nanotubes", Polymer, 44(19), 5643-5654. https://doi.org/10.1016/S0032-3861(03)00628-1.
  69. Safeer, M., Taj, M. and Abbas, S.S. (2019), "Effect of viscoelastic medium on wave propagation along protein microtubules", AIP Adv., 9, 045108. https://doi.org/10.1063/1.5086216.
  70. Sahmani, S., Aghdam, M.M. and Rabzcuk, T. (2018), "Nonlinear bending of functionally graded porous micro/nano-beams reinforced with graphene platelets based upon nonlocal strain gradient theory", Compos. Struct., 186, 68-78. https://doi.org/10.1016/j.compstruct.2017.11.082.
  71. She, G.L., Yuan, F.G., Ren, Y.R., Liu, H.B. and Xiao, W.S. (2018), "Nonlinear bending and vibration analysis of functionally graded porous tubes via a nonlocal strain gradient theory", Compos. Struct., 203, 614-623. https://doi.org/10.1016/j.compstruct.2018.07.063.
  72. 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, 182-191. https://doi.org/10.1016/j.physe.2010.07.003.
  73. Simsek, M. (2011a), "Forced vibration of an Embedded Single-Walled Carbon Nanotube Traversed by a Moving Load Using Nonlocal Timoshenko Beam Theory", Steel Compos. Struct., 11(1), 59-76. https://doi.org/10.12989/scs.2011.11.1.059.
  74. Simsek, M. (2012), "Nonlocal effects in then 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.
  75. 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, 182-191. https://doi.org/10.1016/j.physe.2010.07.003
  76. Simsek, M. (2011), "Nonlocal effects in the forced vibration of an elastically connected double-carbon nanotube system under a moving nanoparticle", Comput. Mater. Sci., 50, 2112-2123. https://doi.org/10.1016/j.commatsci.2011.02.017.
  77. Simsek, M. and Reddy, J.N. (2013a), "A unified higher order beam theory for buckling of a functionally graded microbeam embedded in elastic medium using modified couple stress theory", Compos. Struct., 101, 47-58. https://doi.org/10.1016/j.compstruct.2013.01.017.
  78. Simsek, M. and Reddy, J.N. (2013b), "Bending and vibration of functionally graded microbeams using a new higher order beam theory and the modified couple stress theory", Int. J. Eng. Sci., 64, 37-53. https://doi.org/10.1016/j.ijengsci.2012.12.002.
  79. Simsek, M. (2014), "Nonlinear static and free vibration analysis of microbeams based on the nonlinear elastic foundation using modified couple stress theory and He's variational method", Compos. Struct., 112, 264-272. https://doi.org/10.1016/j.compstruct.2014.02.010.
  80. Simsek, M. (2016a), "Nonlinear free vibration of a functionally graded nanobeam using nonlocal strain gradient theory and a novel Hamiltonian approach", Int. J. Eng. Sci., 105, 12-27. https://doi.org/10.1016/j.ijengsci.2016.04.013.
  81. Simsek, M. (2016b), "Axial vibration analysis of a nonorod embedded in elastic medium using nonlocal strain gradient theory", C ukurova University Journal of the Faculty of Engineering and Architecture, 31(1), 213-221. https://doi.org/10.21923/jesd.553328.
  82. Soltani, P., Taherian, M.M and Farshidianfar, A. (2010), "Vibration and instability of a viscous-fluid-conveying single-walled carbon nanotube embedded in viscoelastic medium", Phys. Lett. A, 43, 401-425.
  83. Tang, Y., Liu, Y. and Zhao, D. (2017), "Wave dispersion in viscoelastic single walled carbon nanotubes based on the nonlocal strain gradient Timoshenko beam model", Physica E, 87, 301-307. https://doi.org/10.1016/j.physe.2016.10.046.
  84. Tang, H., Li, L., Hu, Y., Meng, W. and Duan, K. (2019), "Vibration of nonlocal strain gradient beams incorporating Poisson's ratio and thickness effects", Thin Walled Struct., 137, 377-391. https://doi.org/10.1016/j.tws.2019.01.027.
  85. Thai, H.T. (2012), "A nonlocal beam theory for bending, buckling and vibration of nanobeams", Int. J. Eng. Sci., 52, 56-64. https://doi.org/10.1016/j.ijengsci.2011.11.011.
  86. Thang, P.T., Nguyen-Thoi, T. and Lee, J. (2021a), "Modelling and analysis of bi-directional functionally graded nanobeams based on nonlocal strain gradient theory", Appl. Math. Comput., 407, 126303. https://doi.org/10.1016/j.amc.2021.126303.
  87. Thang, P.T., Tran, P. and Nguyen-Thoi, T. (2021b), "Applying nonlocal strain gradient theory to size-dependent analysis of functionally graded carbon nanotube-reinforced composite nanoplates", Appl. Math. Model., 93, 775-791. https://doi.org/10.1016/j.apm.2021.01.001.
  88. Thang, P.T., Do, D.T.T., Lee, J. and Nguyen-Thoi, T. (2021c), "Size-dependent analysis of functionally graded carbon nanotube-reinforced composite nanoshells with double curvature based on nonlocal strain gradient theory", Eng. Comput., 1-20. https://doi.org/10.1007/s00366-021-01517-1.
  89. Tsepoura, K.G., Papargyri-Beskou, S., Polyzos, D., Beskos, D.E. (2002), "Static and dynamic analysis of a gradient-elastic bar in tension", Arch. Appl. Mech., 72, 483-497. https://doi.org/10.1007/s00419-002-0231-z.
  90. Tung, Y.H., Chen, Y.J., Yang, C.C., Weng, C.Y., Huang, Y.K., Chen, Y.Y. and Wu, M.K. (2021), "Size effect on multiferroicity of GdMn2O5 nanorods", Chin. J. Phys., 70, 336-342. https://doi.org/10.1016/j.cjph.2021.01.011.
  91. Wang, J., Shen, H., Zhang, B., Liu, J. and Zhang, Y. (2018), "Complex modal analysis of transverse free vibrations for axially moving nanobeams based on the nonlocal strain gradient theory", Physica E, 101, 85-93. https://doi.org/10.1016/j.physe.2018.03.017.
  92. Wang, X.Q. and Lee, J.D. (2010), "Micromorphic theory: A gateway to nano world", Int. J. Smart Nano Mater., 1(2), 115- 135. https://doi.org/10.1080/19475411.2010.484207.
  93. Wu, C.P. and Li, W.C. (2017), "Asymptotic nonlocal elasticity theory for the buckling analysis of embedded single-layered nanoplates/graphene sheets under biaxial compression", Physica E, 89, 160-169. https://doi.org/10.1016/j.physe.2017.01.027.
  94. Yan, Y., Li, J.X., Ma, X.F. and Wang, W.Q. (2021), "Application and dynamical behavior of CNTs as fluidic nanosensors based on the nonlocal strain gradient theory", Sens. Actuator A Phys., 330, 112836. https://doi.org/10.1016/j.sna.2021.112836.
  95. Yang, F., Chong, A.C.M., Lam, D.C.C. and Tong, P. (2002), "Couple stress based strain gradient theory for elasticity", Int. J. Solids Struct., 39, 2731-2743. https://doi.org/10.1016/S0020-7683(02)00152-X.