Skip to main content
SpringerLink
Log in

A review of mechanical analyses of rectangular nanobeams and single-, double-, and multi-walled carbon nanotubes using Eringen’s nonlocal elasticity theory

  • Published:
Archive of Applied Mechanics Aims and scope Submit manuscript

Abstract

This article is intended to present an overview of various mechanical analyses of rectangular nanobeams and single-, double-, and multi-walled (SW-, DW-, and MW-) carbon nanotubes (CNTs) with combinations of simply supported, free, and clamped edge conditions embedded or non-embedded in an elastic medium, including bending, free vibration, buckling, coupled thermo-elastic and hygro-thermo-elastic, dynamic instability, wave propagation, geometric nonlinear bending, and large amplitude vibration analyses. This review introduces the development of various nonlocal beam and shell theories incorporating Eringen’s nonlocal elasticity theory and the application of strong- and weak-form-based formulations to the current issue. Based on the principle of virtual displacements and Reissner’s mixed variational theorem, the corresponding strong- and weak-form formulations of the local Timoshenko beam theory are reformulated for the free vibration analysis of rectangular nanobeams and SW-, DW-, and MW-CNTs, and presented for illustrative purposes. A comparative study of the results obtained using assorted nonlocal beam and shell theories in combination with the analytical and numerical methods is carried out.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3

Similar content being viewed by others

References

  1. Iijima, S.: Helica microtubes of graphitic carbon. Nature 354, 56–58 (1991)

    Google Scholar 

  2. Li, C., Thostenson, E.T., Chou, T.W.: Sensors and actuators based on carbon nanotubes and their composites: a review. Compos. Sci. Technol. 68, 1227–1249 (2008)

    Google Scholar 

  3. Gao, C., Guo, Z., Liu, J.H., Huang, X.J.: The new age of carbon nanotubes: an updated review of functionalized carbon nanotubes in electrochemical sensors. Nanoscale 4, 1948–1963 (2012)

    Google Scholar 

  4. Wang, Q., Arash, B.: A review on applications of carbon nanotubes and graphenes as nano-resonator sensors. Comput. Mater. Sci. 82, 350–360 (2014)

    Google Scholar 

  5. Chen, W.X., Tu, J.P., Wang, L.Y., Gan, H.Y., Xu, Z.D., Zhang, X.B.: Tribological application of carbon nanotubes in a metal-based composite coating and composites. Carbon 41, 215–222 (2003)

    Google Scholar 

  6. Mittal, G., Dhand, V., Rhee, K.Y., Park, S.J., Lee, W.R.: A review on carbon nanotubes and graphenes as fillers in reinforced polymer nanocomposites. J. Ind. Eng. Chem. 21, 11–25 (2015)

    Google Scholar 

  7. Janas, D., Koziol, K.K.: A review of production methods of carbon nanotube and graphene thin films for electrothermal applications. Nanoscale 6, 3037–3045 (2014)

    Google Scholar 

  8. Ma, L., Dong, X., Chen, M., Zhu, L., Wang, C., Yang, F., Dong, Y.: Fabrication and water treatment applications of carbon nanotubes (CNTs)-based composite membranes: a review. Membranes 7, 16 (2017)

    Google Scholar 

  9. Ren, X., Chen, C., Nagatsu, M., Wang, X.: Carbon nanotubes as adsorbents in environmental pollution management: a review. Chem. Eng. J. 170, 395–410 (2011)

    Google Scholar 

  10. Gohardani, O., Elola, M.C., Elizetxea, C.: Potential and prospective implementation of carbon nanotubes on next generation aircraft and space vehicles: a review of current and expected applications in aerospace sciences. Prog. Aerosp. Sci. 70, 42–68 (2014)

    Google Scholar 

  11. Khare, R., Bose, S.: Carbon nanotube based composites—a review. J. Miner. Mater. Charact. Eng. 4, 31–46 (2005)

    Google Scholar 

  12. Ng, K.W., Lam, W.H., Pichiah, S.: A review on potential applications of carbon nanotubes in marine current turbines. Renew. Sustain. Energy Rev. 28, 331–339 (2013)

    Google Scholar 

  13. Tan, J.M., Arulselvan, P., Fakurazi, S., Ithnin, H., Hussein, M.Z.: A review on characterization and biocompatibility of functionalized carbon nanotubes in drug delivery design. J. Nanomater. 2014, 917024 (2014)

    Google Scholar 

  14. Alothman, Z.A., Wabaidur, S.M.: Application of carbon nanotubes in extraction and chromatographic analysis: a review. Arabian J. Chem. (2018) https://doi.org/10.1016/j.arabjc.2018.05.012

  15. Bianco, A., Kostarelos, K., Prato, M.: Applications of carbon nanotubes in drug delivery. Curr. Opin. Chem. Biol. 9, 674–679 (2005)

    Google Scholar 

  16. Casas, C.D.L., Li, W.: A review of application of carbon nanotubes for lithium ion battery anode material. J. Power Sour. 208, 74–85 (2012)

    Google Scholar 

  17. Datsyuk, V., Kalyva, M., Papagelis, K., Parthenios, J., Tasis, D., Siokou, A., Kallitsis, I., Galiotis, C.: Chemical oxidation of multiwalled carbon nanotubes. Carbon 46, 833–840 (2008)

    Google Scholar 

  18. Jakubus, A., Paszkiewicz, M., Stepnowski, P.: Carbon nanotubes application in the extraction techniques of pesticides: a review. Crit. Rev. Anal. Chem. 47, 76–91 (2017)

    Google Scholar 

  19. Luo, C., Xie, L., Wang, Q., Luo, G., Liu, C.: A review of the application and performance of carbon nanotubes in fuel cells. J. Nanomater. 2015, 560392 (2015)

    Google Scholar 

  20. Ong, Y.T., Ahmad, A.L., Zein, S.H.S., Tan, S.H.: A review on carbon nanotubes in an environmental protection and green engineering perspective. Braz. J. Chem. Eng. 27, 227–242 (2010)

    Google Scholar 

  21. Harris, P.J.F.: Carbon Nanotube Science: Synthesis Properties and Applications. Cambridge University Press, Cambridge (2009)

    Google Scholar 

  22. Thostenson, E.T., Ren, Z., Chou, T.W.: Advances in the science and technology of carbon nanotubes and their composites: a review. Compos. Sci. Technol. 61, 1899–1912 (2001)

    Google Scholar 

  23. De Volder, M.F.L., Tawfick, S.H., Baughman, R.H., Hart, A.J.: Carbon nanotubes: present and future commercial applications. Science 339, 535–539 (2013)

    Google Scholar 

  24. Eringen, A.C.: On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves. J. Appl. Phys. 54, 4703–4710 (1983)

    Google Scholar 

  25. Eringen, A.C.: Nonlocal Continuum Field Theories. Springer-Verlag, New York (2002)

    MATH  Google Scholar 

  26. Eringen, A.C., Edelen, D.G.B.: On nonlocal elasticity. Int. J. Eng. Sci. 10, 233–248 (1972)

    MathSciNet  MATH  Google Scholar 

  27. Reissner, E.: On a certain mixed variational theorem and a proposed application. Int. J. Numer. Methods Eng. 20, 1366–1368 (1984)

    MATH  Google Scholar 

  28. Reissner, E.: On a mixed variational theorem and on a shear deformable plate theory. Int. J. Numer. Methods Eng. 23, 193–198 (1986)

    MATH  Google Scholar 

  29. Wang, Q., Wang, C.: The constitutive relation and small scale parameter of nonlocal continuum mechanics for modelling carbon nanotubes. Nanotechnology 18, 075702 (2007)

    Google Scholar 

  30. He, A.Q., Kitipornchai, S., Wang, C.M., Liew, K.M.: Modeling of van der Waals force for infinitesimal deformation of multi-walled carbon nanotubes treated as cylindrical shells. Int. J. Solids Struct. 42, 6032–6047 (2005)

    MATH  Google Scholar 

  31. He, X.Q., Kitipornchai, S., Liew, K.M.: Buckling analysis of multi-walled carbon nanotubes: a continuum model accounting for van der Waals interaction. J. Mech. Phys. Solids 53, 303–326 (2005)

    MATH  Google Scholar 

  32. Ru, C.Q.: Effect of van der Waals forces on axial buckling of a double-walled carbon nanotube. J. Appl. Phys. 87, 7227–7231 (2000)

    Google Scholar 

  33. Ru, C.Q.: Elastic models for carbon nanotubes. Encycl. Nanosci. Nanotech. 2, 731–744 (2004)

    Google Scholar 

  34. Wu, C.P., Hong, Z.L., Wang, Y.M.: Geometrically nonlinear static analysis of an embedded multi-walled carbon nanotube and the van der Waals interaction. J. Nanomech. Micromech. 7, 04017012 (2017)

    Google Scholar 

  35. Wu, C.P., Lin, C.H., Wang, Y.M.: Nonlinear finite element analysis of a multi-walled carbon nanotube resting on a Pasternak foundation. Mech. Adv. Mater. Struct. (2018). https://doi.org/10.1080/15376494.2018.1444222

    Google Scholar 

  36. Wu, C.P., Chen, Y.H., Hong, Z.L., Lin, C.H.: Nonlinear vibration analysis of an embedded multi-walled carbon nanotube. Adv. Nano Res. 6, 163–182 (2018)

    Google Scholar 

  37. Strozzi, M., Pellicano, F.: Linear vibrations of triple-walled carbon nanotubes. Math. Mech. Solids 23, 1456–1481 (2018)

    MathSciNet  Google Scholar 

  38. Natsuki, T., Ni, Q.Q., Endo, M.: Analysis of the vibration characteristics of double-walled carbon nanotubes. Carbon 46, 1570–1573 (2008)

    Google Scholar 

  39. Kumar, B.R.: Investigation on mechanical vibration of double-walled carbon nanotubes with inter-tube van der Walls forces. Adv. Nano Res. 6, 135–145 (2018)

    Google Scholar 

  40. Pentaras, D., Elishakoff, I.: Effective approximations for natural frequencies of double-walled carbon nanotubes based on Donnell shell theory. J. Nanotechnol. Eng. Med. 2, 021023 (2011)

    Google Scholar 

  41. Reddy, J.N.: Nonlocal theories for bending, buckling and vibration of beams. Int. J. Eng. Sci. 45, 288–307 (2007)

    MATH  Google Scholar 

  42. Reddy, J.N., Pang, S.D.: Nonlocal continuum theories of beams for the analysis of carbon nanotubes. J. Appl. Phys. 103, 023511 (2008)

    Google Scholar 

  43. Thai, H.T.: A nonlocal beam theory for bending, buckling, and vibration of nanobeams. Int. J. Eng. Sci. 52, 56–64 (2012)

    MathSciNet  MATH  Google Scholar 

  44. Thai, H.T., Vo, T.P.: A nonlocal sinusoidal shear deformation beam theory with application to bending, buckling, and vibration of nanobeams. Int. J. Eng. Sci. 54, 58–66 (2012)

    MathSciNet  MATH  Google Scholar 

  45. Aissani, K., Bouiadjra, M.B., Ahouel, M., Tounsi, A.: A new nonlocal hyperbolic shear deformation theory for nanobeams embedded in an elastic medium. Struct. Eng. Mech. 55, 743–763 (2015)

    Google Scholar 

  46. Aydogdu, M.: A general nonlocal beam theory: its application to nanobeam bending, buckling and vibration. Physica E 41, 1651–1655 (2009)

    Google Scholar 

  47. Zhang, Y.Y., Wang, C.M., Challamel, N.: Bending, buckling, and vibration of micro/nanobeams by hybrid nonlocal beam model. J. Eng. Mech. 136, 562–574 (2010)

    Google Scholar 

  48. Ansari, R., Sahmani, S.: Small scale effect on vibrational response of single-walled carbon nanotubes with different boundary conditions based on nonlocal beam models. Commun. Nonlinear Sci. Numer. Simult. 17, 1965–1979 (2012)

    MathSciNet  Google Scholar 

  49. Kiani, K., Mehri, B.: Assessment of nanotube structures under a moving nanoparticle using nonlocal beam theories. J. Sound Vib. 329, 2241–2264 (2010)

    Google Scholar 

  50. Mehdipour, I., Soltani, P., Ganji, D.D.: Nonlinear vibration and bending instability of a single-walled carbon nanotube using nonlocal elastic beam theory. Int. J. Nanosci. 10, 447–453 (2011)

    Google Scholar 

  51. Setoodeh, A.R., Khosrownejad, M., Malekzadeh, P.: Exact nonlocal solution for postbuckling of single-walled carbon nanotubes. Physica E 43, 1730–1737 (2011)

    Google Scholar 

  52. Shen, H.S., Zhang, C.L.: Nonlocal beam model for nonlinear analysis of carbon nanotubes on elastomeric substrates. Comput. Mater. Sci. 50, 1022–1029 (2011)

    Google Scholar 

  53. Simsek, M.: 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 (2010)

    Google Scholar 

  54. Ansari, R., Gholami, R., Darabi, M.A.: Thermal buckling analysis of embedded single-walled carbon nanotubes with arbitrary boundary conditions using the nonlocal Timoshenko beam theory. J. Therm. Stress. 4, 1271–1281 (2011)

    Google Scholar 

  55. Benzair, A., Tounsi, A., Besseghier, A., Heireche, H., Moulay, N., Boumia, L.: The thermal effect on vibration of single-walled carbon nanotubes using nonlocal Timoshenko beam theory. J. Phys. D: Appl. Phys. 41, 225404 (2008)

    Google Scholar 

  56. Boumia, L., Zidour, M., Benzair, A., Tounsi, A.: A Timoshenko beam model for vibration analysis of chiral single-walled carbon nanotubes. Physica E 59, 186–191 (2014)

    Google Scholar 

  57. Hosseini-Hashemi, S., Nazemnezhad, R., Bedroud, M.: Surface effects on nonlinear free vibration of functionally graded nanobeams using nonlocal elasticity. Appl. Math. Modell. 38, 3538–3553 (2014)

    MathSciNet  MATH  Google Scholar 

  58. Lee, H.L., Chang, W.J.: Surface effects on frequency analysis of nanotubes using nonlocal Timoshenko beam theory. J. Appl. Phys. 108, 093503 (2010)

    Google Scholar 

  59. Murmu, T., Pradhan, S.C.: 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, 1232–1239 (2009)

    Google Scholar 

  60. Wang, C.M., Zhang, Y.Y., Ramesh, S.S., Kitipornchai, S.: Buckling analysis of micro- and nano-rods/tubes based on nonlocal Timoshenko beam theory. J. Phys. D: Appl. Phys. 39, 3904–3909 (2006)

    Google Scholar 

  61. Wu, C.P., Lai, W.W.: Free vibration of an embedded single-walled carbon nanotube with various boundary conditions using the RMVT-based nonlocal Timoshenko beam theory and DQ method. Physica E 68, 8–21 (2015)

    Google Scholar 

  62. Wu, C.P., Lai, W.W.: Reissner’s mixed variational theorem-based nonlocal Timoshenko beam theory for a single-walled carbon nanotube embedded in an elastic medium and with various boundary conditions. Compos. Struct. 122, 390–404 (2015)

    Google Scholar 

  63. Wu, C.P., Liou, J.Y.: RMVT-based nonlocal Timoshenko beam theory for stability analysis of embedded single-walled carbon nanotubes with various boundary conditions. Int. J. Struct. Stab. Dyn. 16, 1550068 (2016)

    MathSciNet  MATH  Google Scholar 

  64. Yang, Y., Lim, C.W.: A variational principle approach for buckling of carbon nanotubes based on nonlocal Timoshenko beam models. Nano: Brief Rep. Rev. 6, 363–377 (2011)

    Google Scholar 

  65. Yang, J., Ke, L.L., Kitipornchai, S.: Nonlinear free vibration of single-walled carbon nanotubes using nonlocal Timoshenko beam theory. Physica E 42, 1727–1735 (2010)

    Google Scholar 

  66. Zidour, M., Benrahou, K.H., Semmah, A., Naceri, M., Belhadj, H.A., Bakhti, K., Tounsi, A.: The thermal effect on vibration of zigzag single walled carbon nanotubes using nonlocal Timoshenko beam theory. Comput. Mater. Sci. 51, 252–260 (2012)

    Google Scholar 

  67. Arash, B., Wang, Q.: A review on the application of nonlocal elastic models in modeling of carbon nanotubes and graphenes. Comput. Mater. Sci. 51, 303–313 (2012)

    Google Scholar 

  68. Behera, L., Chakraverty, S.: Recent researches on nonlocal elasticity theory in the vibration of carbon nanotubes using beam models: a review. Arch. Comput. Methods Eng. 24, 481–494 (2017)

    MathSciNet  MATH  Google Scholar 

  69. Ebrahimi, F., Salari, E.: Thermo-mechanical vibration analysis of a single-walled carbon nanotube embedded in an elastic medium based on higher-order shear deformation beam theory. J. Mech. Sci. Technol. 29, 3797–3803 (2015)

    Google Scholar 

  70. Maachou, M., Zidour, M., Baghdadi, H., Ziane, N., Tounsi, A.: A nonlocal Levinson beam model for free vibration analysis of zigzag single-walled carbon nanotubes including thermal effects. Solid State Commun. 151, 1467–1471 (2011)

    Google Scholar 

  71. Rafiee, R., Moghadam, R.M.: On the modeling of carbon nanotubes: a critical review. Compos. Part B 56, 435–449 (2014)

    Google Scholar 

  72. Adali, S.: Variational principles for transversely vibrating multiwalled carbon nanotubes based on nonlocal Euler–Bernoulli beam model. Nano Lett. 9, 1737–1741 (2009)

    Google Scholar 

  73. Ansari, R., Ramezannezhad, H., Gholami, R.: Nonlocal beam theory for nonlinear vibrations of embedded multiwalled carbon nanotubes in thermal environment. Nonlinear Dyn. 67, 2241–2254 (2012)

    MathSciNet  MATH  Google Scholar 

  74. Arani, A.G., Rabbani, H., Amir, S., Maraghi, Z.K., Mohammadimehr, M., Haghparast, E.: Analysis of nonlinear vibrations for multi-walled carbon nanotubes embedded in an elastic medium. J. Solid Mech. 3, 258–270 (2011)

    Google Scholar 

  75. Chen, X., Fang, C.Q., Wang, X.: The influence of surface effect on vibration behaviors of carbon nanotubes under initial stress. Physica E 85, 47–55 (2017)

    Google Scholar 

  76. Civalek, O., Demir, C.: Bending analysis of microtubules using nonlocal Euler–Bernoulli beam theory. Appl. Math. Modell. 35, 2053–2067 (2011)

    MathSciNet  MATH  Google Scholar 

  77. Ehteshami, H., Hajabasi, M.A.: Analytical approaches for vibration analysis of multi-walled carbon nanotubes modeled as multiple nonlocal Euler beams. Physica E 44, 270–285 (2011)

    Google Scholar 

  78. Fang, B., Zhen, Y.X., Zhang, C.P., Tang, Y.: Nonlinear vibration analysis of double-walled carbon nanotubes based on nonlocal elasticity theory. Appl. Math. Modell. 37, 1096–1107 (2013)

    MathSciNet  MATH  Google Scholar 

  79. Khosrozadeh, A., Hajabasi, M.A.: Free vibration of embedded double-walled carbon nanotubes considering nonlinear interlayer van der Waals forces. Appl. Math. Modell. 36, 997–1007 (2012)

    MathSciNet  MATH  Google Scholar 

  80. Lu, P., Lee, H.P., Lu, C., Zhang, P.Q.: Application of nonlocal beam models for carbon nanotubes. Int. J. Solids Struct. 44, 5289–5300 (2007)

    MATH  Google Scholar 

  81. Mohammadimehr, M., Saidi, A.R., Arani, A.G., Arefmanesh, A., Han, Q.: Torsional buckling of a DWCNT embedded on Winkler and Pasternak foundations using nonlocal theory. J. Mech. Sci. Technol. 24, 1289–1299 (2010)

    Google Scholar 

  82. Shakouri, A., Lin, R.M., Ng, T.Y.: Free flexural vibration studies of double-walled carbon nanotubes with different boundary conditions and modeled as nonlocal Euler beams via the Galerkin method. J. Appl. Phys. 106, 094307 (2009)

    Google Scholar 

  83. Wang, B., Deng, Z., Zhang, K., Zhou, J.: Dynamic analysis of embedded curved double-walled carbon nanotubes based on nonlocal Euler–Bernoulli beam theory. Multidisc. Model Mater. Struct. 8, 432–453 (2012)

    Google Scholar 

  84. Yan, Y., Wang, W., Zhang, L.: Applied multiscale method to analysis of nonlinear vibration for double-walled carbon nanotubes. Appl. Math. Modell. 35, 2279–2289 (2011)

    MathSciNet  MATH  Google Scholar 

  85. Yoon, J., Ru, C.Q., Mioduchowski, A.: Noncoaxial resonance of an isolated multiwall carbon nanotube. Phys. Rev. B 66, 233402 (2002)

    Google Scholar 

  86. Yoon, J., Ru, C.Q., Mioduchowski, A.: Vibration of an embedded multiwalled carbon nanotube. Compos. Sci. Technol. 63, 1533–1542 (2003)

    Google Scholar 

  87. Ansari, R., Gholami, R., Darabi, M.A.: Nonlinear free vibration of embedded double-walled carbon nanotubes with layerwise boundary conditions. Acta Mech. 223, 2523–2536 (2012)

    MathSciNet  MATH  Google Scholar 

  88. Ansari, R., Gholami, R., Sahmani, S., Norouzzadeh, A., Bazdid-Vahdati, M.: Dynamic stability analysis of embedded multi-walled carbon nanotubes in thermal environment. Acta Mech. Solida Sinica 28, 659–667 (2015)

    Google Scholar 

  89. Ansari, R., Ramezannezhad, H.: Nonlocal Timoshenko beam model for the large-amplitude vibrations of embedded multiwalled carbon nanotubes including thermal effects. Physica E 43, 1171–1178 (2011)

    Google Scholar 

  90. Benguediab, S., Tounsi, A., Zidour, M., Semmah, A.: Chirality and scale effects on mechanical buckling properties of zigzag double-walled carbon nanotubes. Compos. Part B 57, 21–24 (2014)

    Google Scholar 

  91. Ece, M.C., Aydogdu, M.: Nonlocal elasticity effect on vibration of in-plane loaded double-walled carbon nano-tubes. Acta Mech. 190, 185–195 (2007)

    MATH  Google Scholar 

  92. Ke, L.L., Xiang, Y., Yang, J., Kitipornchai, S.: Nonlinear free vibration of embedded double-walled carbon nanotubes based on nonlocal Timoshenko beam theory. Comput. Mater. Sci. 47, 409–417 (2009)

    Google Scholar 

  93. Kucuk, I., Sadek, I.S., Adali, S.: Variational principles for multiwalled carbon nanotubes undergoing vibrations based on nonlocal Timoshenko beam theory. J. Nanomater. 2010, 461252 (2010)

    Google Scholar 

  94. Soltani, P., Bahar, P., Farshidianfar, A.: An efficient GDQ model for vibration analysis of a multiwall carbon nanotube on Pasternak foundation with general boundary conditions. Proc. IMechE Part C: J. Mech. Eng. Sci. 225, 1730–1741 (2011)

    Google Scholar 

  95. Wang, C.M., Tan, V.B.C., Zhang, Y.Y.: Timoshenko beam model for vibration analysis of multi-walled carbon nanotubes. J. Sound Vib. 294, 1060–1072 (2006)

    Google Scholar 

  96. Aydogdu, M.: Effects of shear deformation on vibration of doublewalled carbon nanotubes embedded in an elastic medium. Arch. Appl. Mech. 78, 711–723 (2008)

    MATH  Google Scholar 

  97. Barretta, R., Sciarra, F.M.: Analogies between nonlocal and local Bernoulli–Euler nanobeams. Arch. Appl. Mech. 85, 89–99 (2015)

    MATH  Google Scholar 

  98. Barretta, R., Canadija, M., Sciarra, F.M.: A higher-order Eringen model for Bernoulli–Euler nanobeams. Arch. Appl. Mech. 86, 483–495 (2016)

    MATH  Google Scholar 

  99. Behera, L., Chakraverty, S.: Application of differential quadrature method in free vibration analysis of nanobeams based on various nonlocal theories. Comput. Math. Appl. 69, 1444–1462 (2015)

    MathSciNet  Google Scholar 

  100. Eltaher, M.A., Khairy, A., Sadoun, A.M., Omer, E.I.: Coupling effects of nonlocal and surface energy on vibration analysis of nanobeams. Appl. Math. Comput. 224, 760–774 (2013)

    MathSciNet  Google Scholar 

  101. Eltaher, M.A., Khater, M.E., Emam, S.A.: A review on nonlocal elastic models for bending, buckling, vibrations, and wave propagation of nanoscale beams. Appl. Math. Modell. 40, 4109–4128 (2016)

    MathSciNet  Google Scholar 

  102. Gheshlaghi, B., Hasheminejad, S.M.: Surface effects on nonlinear free vibration of nanobeams. Compos. Part B 42, 934–937 (2011)

    Google Scholar 

  103. Janghorban, M.: Two different types of differential quadrature methods for static analysis of microbeams based on nonlocal thermal elasticity theory in thermal environment. Arch. Appl. Mech. 82, 669–675 (2012)

    MATH  Google Scholar 

  104. Reddy, J.N.: Nonlocal nonlinear formulations for bending of classical and shear deformation theories of beams and plates. Int. J. Eng. Sci. 48, 1507–1518 (2010)

    MathSciNet  MATH  Google Scholar 

  105. Sahmani, S., Ansari, R.: Nonlocal beam models for buckling of nanobeams using state-space method regarding different boundary conditions. J. Mech. Sci. Technol. 25, 2365–2375 (2011)

    Google Scholar 

  106. Togun, N., Bagdatli, S.M.: Nonlinear vibration of a nanobeam on a Pasternak elastic foundation based on non-local Euler–Bernoulli beam theory. Math. Comput. Appl. 21, 3 (2016)

    MathSciNet  Google Scholar 

  107. Wang, Y.Z., Li, F.M.: Nonlinear primary resonance of nano beam with axial initial load by nonlocal continuum theory. Int. J. Non-Linear Mech. 61, 74–79 (2014)

    Google Scholar 

  108. Ebrahimi, F., Nasirzadeh, P.: A nonlocal Timoshenko beam theory for vibration analysis of thick nanobeams using differential transform method. J. Theoret. Appl. Mech. 53, 1041–1052 (2015)

    Google Scholar 

  109. Wang, C.M., Kitipornchai, S., Lim, C.W., Eisenberger, M.: Beam bending solutions based on nonlocal Timoshenko beam theory. J. Eng. Mech. 134, 475–481 (2008)

    Google Scholar 

  110. Yang, Q., Lim, C.W., Xiang, Y.: Nonlinear thermal bending for shear deformable nanobeams based on nonlocal elasticity theory. Int. J. Aerosp. Lightweight Struct. 1, 89–107 (2011)

    Google Scholar 

  111. Karlicic, D., Kozic, P., Pavlovic, R.: Nonlocal vibration and stability of a multiple-nanobeam system coupled by the Winkler elastic medium. Appl. Math. Modell. 40, 1599–1614 (2016)

    MathSciNet  Google Scholar 

  112. Khaniki, H.B.: On vibrations of nanobeam systems. Int. J. Eng. Sci. 124, 85–103 (2018)

    MathSciNet  MATH  Google Scholar 

  113. Murmu, T., Adhikari, S.: Axial instability of double-nanobeam-systems. Phys. Lett. A 375, 601–608 (2011)

    Google Scholar 

  114. Murmu, T., Adhikari, S.: Nonlocal elasticity based vibration of initially pre-stressed coupled nanobeam systems. Eur. J. Mech. A/Solids 34, 52–62 (2012)

    MathSciNet  MATH  Google Scholar 

  115. Zhou, Z., Li, Y., Fan, J., Rong, D., Sui, G., Xu, G.: Exact vibration analysis of a double-nanobeam-systems embedded in an elastic medium by a Hamiltonian-based method. Physica E 99, 220–235 (2018)

    Google Scholar 

  116. Lei, X.W., Natsuki, T., Shi, J.X., Ni, Q.Q.: Surface effects on the vibrational frequency of double-walled carbon nanotubes using the nonlocal Timoshenko beam model. Compos. Part B 43, 64–69 (2012)

    Google Scholar 

  117. Murmu, T., Adhikari, S.: Nonlocal transverse vibration of double-nanobeam-systems. J. Appl. Phys. 108, 083514 (2010)

    Google Scholar 

  118. Ebrahimi, F., Salari, E.: Nonlocal thermos-mechanical vibration analysis of functionally graded nanobeams in thermal environment. Acta Astronaut. 113, 29–50 (2015)

    Google Scholar 

  119. Ebrahimi, F., Barati, M.R.: Small-scale effects on hygro-thermo-mechanical vibration of temperature-dependent nonhomogeneous nanoscale beams. Mech. Adv. Mater. Struct. 24, 924–936 (2017)

    Google Scholar 

  120. Ebrahimi, F., Ghadiri, M., Salari, E., Hosein-Hoseini, S.A., Shaghaghi, G.R.: Application of the differential transformation method for nonlocal vibration analysis of functionally graded nanobeams. J. Mech. Sci. Technol. 29, 1207–1215 (2015)

    Google Scholar 

  121. Ebrahimi, F., Salari, E.: Thermal buckling and free vibration analysis of size dependent Timoshenko FG nanobeams in thermal environments. Compos. Struct. 128, 363–380 (2015)

    Google Scholar 

  122. Ebrahimi, F., Salari, E.: Size-dependent free flexural vibrational behavior of functionally graded nanobeams using semi-analytical differential transform method. Compos. Part B 79, 156–169 (2015)

    Google Scholar 

  123. Ebrahimi, F., Salari, E.: Effect of various thermal loadings on buckling and vibrational characteristics of nonlocal temperature-dependent functionally graded nanobeams. Mech. Adv. Mater. Struct. 23, 1379–1397 (2016)

    Google Scholar 

  124. Ebrahimi, F., Salari, E., Hosseini, S.A.H.: Thermomechanical vibration behavior of FG nanobeams subjected to linear and nonlinear temperature distributions. J. Therm. Stress. 38, 1360–1386 (2015)

    Google Scholar 

  125. El-Borgi, S., Fernandes, R., Reddy, J.N.: Non-local free and forced vibrations of graded nanobeams resting on a non-linear elastic foundation. Int. J. Non-linear Mech. 77, 348–363 (2015)

    Google Scholar 

  126. Eptaimeros, K.G., Koutsoumaris, C.C., Tsamasphyros, G.J.: Nonlocal integral approach to the dynamical response of nanobeams. Int. J. Mech. Sci. 115–116, 68–80 (2016)

    Google Scholar 

  127. Li, L., Hu, Y.: Nonlinear bending and free vibration analyses of nonlocal strain gradient beams made of functionally graded material. Int. J. Eng. Sci. 107, 77–97 (2016)

    MathSciNet  MATH  Google Scholar 

  128. Niknam, H., Aghdam, M.M.: A semi analytical approach for large amplitude free vibration and buckling of nonlocal FG beams resting on elastic foundation. Compos. Struct. 119, 452–462 (2015)

    Google Scholar 

  129. Ebrahimi, F., Barati, M.R.: A nonlocal higher-order shear deformation beam theory for vibration analysis of size-dependent functionally graded nanobeams. Arab. J. Sci. Eng. 41, 1679–1690 (2016)

    MathSciNet  MATH  Google Scholar 

  130. Ebrahimi, F., Barati, M.R.: Buckling analysis of smart size-dependent higher order magneto-electro-thermo-elastic functionally graded nanosize beams. J. Mech. 33, 23–33 (2017)

    Google Scholar 

  131. Ebrahimi, F., Salari, E.: Thermo-mechanical vibration analysis of nonlocal temperature-dependent FG nanobeams with various boundary conditions. Compos. Part B-Eng. 78, 272–290 (2015)

    Google Scholar 

  132. Ebrahimi, F., Shafiei, N.: Application of Eringen’s nonlocal elasticity theory for vibration analysis of rotating functionally graded nanobeams. Smart Struct. Syst. 17, 837–857 (2016)

    Google Scholar 

  133. Eltaher, M.A., Khairy, A., Sadoun, A.M., Omer, F.A.: Static and buckling analysis of functionally graded Timoshenko nanobeams. Appl. Math. Comput. 229, 283–295 (2014)

    MathSciNet  MATH  Google Scholar 

  134. Rahmani, O., Pedram, O.: Analysis and modeling the size effect on vibration of functionally graded nanobeams based on nonlocal Timoshenko beam theory. Int. J. Eng. Sci. 77, 55–70 (2014)

    MathSciNet  MATH  Google Scholar 

  135. Li, L., Li, X., Hu, Y.: Free vibration analysis of nonlocal strain gradient beams made of functionally graded material. Int. J. Eng. Sci. 102, 77–92 (2016)

    MATH  Google Scholar 

  136. Ebrahimi, F., Barati, M.R.: Vibration analysis of embedded size dependent FG nanobeams based on third-order shear deformation beam theory. Struct. Eng. Mech. 61, 721–736 (2017)

    Google Scholar 

  137. Eltaher, M.A., Emam, S.A., Mahmoud, F.F.: Static and stability analysis of nonlocal functionally graded nanobeams. Compos. Struct. 96, 82–88 (2013)

    Google Scholar 

  138. Eltaher, M.A., Alshorbagy, A.E., Mahmoud, F.F.: Vibration analysis of Euler–Bernoulli nanobeams by using finite element method. Appl. Math. Modell. 37, 4787–4797 (2013)

    MathSciNet  MATH  Google Scholar 

  139. Nguyen, N.T., Kim, N.I., Lee, J.: Mixed finite element analysis of nonlocal Euler–Bernoulli nanobeams. Finites Elem. Anal. Des. 106, 65–72 (2015)

    Google Scholar 

  140. Pradhan, S.C.: Nonlocal finite element analysis and small scale effects of CNTs with Timoshenko beam theory. Finites Elem. Anal. Des. 50, 8–20 (2012)

    Google Scholar 

  141. Pradhan, S.C., Mandal, U.: Finite element analysis of CNT’s based on nonlocal elasticity and Timoshenko beam theory including thermal effect. Physica E 53, 223–232 (2013)

    Google Scholar 

  142. Merzouki, T., Ganapathi, M., Polit, O.: A nonlocal higher-order curved beam finite model including thickness stretching effect for bending analysis of curved nanobeams. Mech. Adv. Mater. Struct. (2017). https://doi.org/10.1080/15376494.2017.1410903

    Google Scholar 

  143. Polit, O., Merzouki, T., Ganapathi, M.: Elastic stability of curved nanobeam based on higher-order shear deformation theory and nonlocal analysis by finite element approach. Finites Elem. Anal. Des. 146, 1–15 (2018)

    MathSciNet  Google Scholar 

  144. Reddy, J.N., El-Borgi, S.: Eringen’s nonlocal theories of beams accounting for moderate rotations. Int. J. Eng. Sci. 82, 159–177 (2014)

    MathSciNet  MATH  Google Scholar 

  145. Reddy, J.N., El-Borgi, S., Romanoff, J.: Nonlinear analysis of functionally graded microbeams using Eringen’s nonlocal differential model. Int. J. Nonlinear Mech. 67, 308–318 (2014)

    Google Scholar 

  146. Kuo, Y.L.: Nonlinear finite element analysis of nonlocal elastic nanobeams with large-amplitude vibrations. J. Comput. Theor. Nanosci. 10, 488–495 (2013)

    Google Scholar 

  147. Nguyen, N.T., Kim, N.I., Lee, J.: Static behavior of nonlocal Euler–Bernoulli beam model embedded in an elastic medium using mixed finite element formulation. Struct. Eng. Mech. 63, 137–146 (2017)

    Google Scholar 

  148. Ribeiro, P., Thomas, O.: Nonlinear modes of vibration and internal resonances in nonlocal beams. J. Comput. Nonlinear Dyn. 12, 031017 (2017)

    Google Scholar 

  149. Ansari, R., Rajabiehfard, R., Arash, B.: Thermal buckling of multiwalled carbon nanotubes using a semi-analytical finite element approach. J. Therm. Stress. 34, 817–834 (2011)

    Google Scholar 

  150. Alshorbagy, A.E., Eltaher, M.A., Mahmoud, F.F.: Static analysis of nanobeams using nonlocal FEM. J. Mech. Sci. Technol. 27, 2035–2041 (2013)

    Google Scholar 

  151. Demir, C., Civalek, O.: A new nonlocal FEM via Hermitian cubic shape functions for thermal vibration of nano beams surrounded by an elastic matrix. Compos. Struct. 168, 872–884 (2017)

    Google Scholar 

  152. Eltaher, M.A., Hamed, M.A., Sadoun, A.M., Mansour, A.: Mechanical analysis of higher order gradient nanobeams. Appl. Math. Comput. 229, 260–272 (2014)

    MathSciNet  MATH  Google Scholar 

  153. Lignola, G.P., Spena, F.R., Prota, A., Manfredi, G.: Exact stiffness-matrix of two nodes Timoshenko beam on elastic medium: an analogy with Eringen model of nonlocal Euler–Bernoulli nanobeams. Comput. Struct. (2017). https://doi.org/10.1016/j.compstruc.2016.12.003

    Google Scholar 

  154. Mahmoud, F.F., Eltaher, M.A., Alshorbagy, A.E., Meletis, E.I.: Static analysis of nanobeams including surface effects by nonlocal finite element. J. Mech. Sci. Technol. 26, 3555–3563 (2012)

    Google Scholar 

  155. Phadikar, J.K., Pradhan, S.C.: Variational formulation and finite element analysis for nonlocal elastic nanobeams and nanoplates. Comput. Mater. Sci. 49, 492–499 (2010)

    Google Scholar 

  156. Sciarra, F.M.: Finite element modelling of nonlocal beams. Physica E 59, 144–149 (2014)

    Google Scholar 

  157. Alotta, G., Failla, G., Zingales, M.: Finite element method for a nonlocal Timoshenko beam model. Fin. Elem. Anal. Des. 89, 77–92 (2014)

    Google Scholar 

  158. Preethi, K., Rajagopal, A., Reddy, J.N.: Surface and nonlocal effects for nonlinear analysis of Timoshenko beams. Int. J. Nonlinear Mech. 76, 100–111 (2015)

    Google Scholar 

  159. Gholami, R., Ansari, R.: Nonlinear resonance responses of geometrically imperfect shear deformable including surface stress effects. Int. J. Nonlinear Mech. 97, 115–125 (2017)

    Google Scholar 

  160. Eltaher, M.A., Emam, S.A., Mahmoud, F.F.: Free vibration analysis of functionally graded size-dependent nanobeams. Appl. Math. Comput. 218, 7406–7420 (2012)

    MathSciNet  MATH  Google Scholar 

  161. Hamed, M.A., Eltaher, M.A., Sadoun, A.M., Almitani, K.H.: Free vibration of symmetric and sigmoid functionally graded nanobeams. Appl. Phys. A 122, 829–839 (2016)

    Google Scholar 

  162. Eltaher, M.A., Mahmoud, F.F., Assie, A.E., Meletis, F.A.: Static and buckling analysis of functionally graded Timoshenko nanobeams. Appl. Math. Comput. 229, 283–295 (2014)

    MathSciNet  Google Scholar 

  163. Carrera, E.: An assessment of mixed and classical theories on global and local response of multilayered orthotropic plates. Compos. Struct. 50, 183–198 (2000)

    Google Scholar 

  164. Carrera, E.: An assessment of mixed and classical theories for the thermal stress analysis of orthotropic multilayered plates. J. Therm. Stress. 23, 797–831 (2000)

    Google Scholar 

  165. Carrera, E.: Historical review of zig-zag theories for multilayered plates and shells. Appl. Mech. Rev. 56, 287–308 (2003)

    Google Scholar 

  166. Carrera, E.: Assessment of theories for free vibration analysis of homogeneous and multilayered plates. Shock Vib. 11, 261–270 (2004)

    Google Scholar 

  167. Wu, C.P., Li, H.Y.: The RMVT- and PVD-based finite layer methods for the three-dimensional analysis of multilayered composite and FGM plates. Compos. Struct. 92, 2476–2496 (2010)

    Google Scholar 

  168. Wu, C.P., Li, H.Y.: RMVT- and PVD-based finite layer methods for the quasi-3D free vibration analysis of multilayered composite and FGM plates. CMC Comput. Mater. Contin. 19, 155–198 (2010)

    Google Scholar 

  169. Wu, C.P., Chiu, K.H., Wang, Y.M.: A review on the three-dimensional analytical approaches of multilayered and functionally graded piezoelectric plates and shells. CMC Comput. Mater. Contin. 8, 93–132 (2008)

    Google Scholar 

  170. Brischetto, S., Carrera, E.: Refined 2D models for the analysis of functionally graded piezoelectricity plates. J. Intell. Mater. Syst. Struct. 20, 1783–1797 (2009)

    Google Scholar 

  171. Brischetto, S., Carrera, E.: Advanced mixed theories for bending analysis of functionally graded plates. Comput. Struct. 88, 1474–1483 (2010)

    Google Scholar 

  172. Brischetto, S., Carrera, E.: Coupled thermos-electro-mechanical analysis of smart plates embedding composite and piezoelectric layers. J. Therm. Stress. 35, 766–804 (2012)

    Google Scholar 

  173. Wu, C.P., Li, H.Y.: RMVT-based finite cylindrical prism methods for multilayered functionally graded circular hollow cylinders with various boundary conditions. Compos. Struct. 100, 592–608 (2013)

    Google Scholar 

  174. Wu, C.P., Liu, Y.C.: A review of semi-analytical numerical methods for laminated composite and multilayered functionally graded elastic/piezoelectric plates and shells. Compos. Struct. 147, 1–15 (2016)

    Google Scholar 

  175. Wu, C.P., Peng, S.T., Chen, Y.C.: RMVT- and PVD-based finite cylindrical layer methods for the three-dimensional buckling analysis of multilayered FGM cylinders under axial compression. Appl. Math. Modell. 38, 233–252 (2014)

    MathSciNet  MATH  Google Scholar 

  176. Wang, Y.M., Chen, S.M., Wu, C.P.: A meshless collocation method based on the differential reproducing kernel interpolation. Comput. Mech. 45, 585–606 (2010)

    MathSciNet  MATH  Google Scholar 

  177. Bert, C.W., Malik, M.: Differential quadrature method in computational mechanics: a review. Appl. Mech. Rev. 49, 1–27 (1996)

    Google Scholar 

  178. Du, H., Lim, M.K., Lin, R.M.: Application of generalized differential quadrature method to structural problems. Int. J. Numer. Methods Eng. 37, 1881–1896 (1994)

    MathSciNet  MATH  Google Scholar 

  179. Wu, C.P., Lee, C.Y.: Differential quadrature solution for the free vibration analysis of laminated conical shells with variable stiffness. Int. J. Mech. Sci. 43, 1853–1869 (2001)

    MATH  Google Scholar 

  180. Cowper, G.R.: The shear coefficient in Timoshenko’s beam theory. J. Appl. Mech. 33, 335–340 (1966)

    MATH  Google Scholar 

  181. Chen, S.M., Wu, C.P., Wang, Y.M.: Hermite DRK interpolation-based collocation method for the analysis of Bernoulli–Euler beams and Kirchhoff–Love plates. Comput. Mech. 47, 425–453 (2011)

    MathSciNet  MATH  Google Scholar 

  182. Reddy, J.N.: An Introduction to the Finite Element Method. McGraw-Hill, New York (1984)

    MATH  Google Scholar 

  183. Strozzi, M., Smirnov, V.V., Manevitch, L.I., Pellicano, F.: Nonlinear vibrations and energy exchange of single-walled carbon nanotubes. Radial breathing modes. Compos. Struct. 184, 613–632 (2018)

    MATH  Google Scholar 

  184. Strozzi, M., Smirnov, V.V., Manevitch, L.I., Milani, M., Pellicano, F.: Nonlinear vibrations and energy exchange of single-walled carbon nanotubes. Circumferential flexural modes. J. Sound Vib. 381, 156–178 (2016)

    MATH  Google Scholar 

  185. Strozzi, M., Manevitch, L.I., Pellicano, F., Smirnov, V.V., Shepelev, D.S.: Low-frequency linear vibrations of single-walled carbon nanotubes: analytical and numerical models. J. Sound Vib. 333, 2936–2957 (2014)

    Google Scholar 

  186. Hu, Y.G., Liew, K.M., Wang, Q., He, X.Q., Yakobson, B.I.: Nonlocal shell model for elastic wave propagation in single- and double-walled carbon nanotubes. J. Mech. Phys. Solids 56, 3475–3485 (2008)

    MATH  Google Scholar 

  187. Shen, H.S., Zhang, C.L.: Torsional buckling and postbuckling of double-walled carbon nanotubes by nonlocal shear deformable shell model. Compos. Struct. 92, 1073–1084 (2010)

    Google Scholar 

  188. Shen, H.S., Zhang, C.L., Xiang, Y.: Nonlocal shear deformable shell model for thermal postbuckling of axially compressed double-walled carbon nanotubes. Philos. Mag. 90, 3189–3214 (2010)

    Google Scholar 

  189. Shen, H.S., Zhang, C.L.: Nonlocal shear deformable shell model for post-buckling of axially compressed double-walled carbon nanotubes embedded in an elastic matrix. J. Appl. Mech. 77, 041006 (2010)

    Google Scholar 

  190. Ansari, R., Rouhi, H., Sahmani, S.: Free vibration analysis of single- and double-walled carbon nanotubes based on nonlocal elastic shell models. J. Vib. Control 20, 670–678 (2014)

    MathSciNet  MATH  Google Scholar 

  191. Ansari, R., Rouhi, H.: Analytical treatment of the free vibration of single-walled carbon nanotubes based on the nonlocal Flugge shell theory. J. Eng. Mater. Technol. 134, 011008 (2012)

    Google Scholar 

  192. Arani, A.G., Barzoki, A.A.M., Kolahchi, R., Loghman, A.: Pasternak foundation effect on the axial and torsional waves propagation in embedded DWCNTs using nonlocal elasticity cylindrical shell theory. J. Mech. Sci. Tech. 25, 2385–2391 (2011)

    Google Scholar 

  193. Ansari, R., Sahmani, A., Rouhi, H.: Axial buckling analysis of single-walled carbon nanotubes in thermal environments via the Rayleigh–Ritz technique. Comput. Mater. Sci. 50, 3050–3055 (2011)

    Google Scholar 

  194. Ansari, R., Shahabodini, A., Rouhi, H.: A thickness-independent nonlocal shell model for describing the stability behavior of carbon nanotubes under compression. Compos. Struct. 100, 323–331 (2013)

    Google Scholar 

  195. Fazelzadeh, S.A., Ghavanloo, E.: Nonlocal anisotropic elastic shell model for vibrations of single-walled carbon nanotubes with arbitrary chirality. Compos. Struct. 94, 1016–1022 (2012)

    Google Scholar 

  196. Ghavanloo, E., Fazelzadeh, S.A.: Vibration characteristics of single-walled carbon nanotubes based on an anisotropic elastic shell model including chirality effect. Appl. Math. Modell. 36, 4988–5000 (2012)

    Google Scholar 

  197. Ghorbanpour Arani, A., Mohammadimehr, M., Arefmanesh, A., Ghasemi, A.: Transverse vibration of short carbon nanotubes using cylindrical shell and beam models. Proc. Inst. Mech. Eng. Part C J. Mech. Eng. Sci. 224, 745–756 (2010)

    Google Scholar 

  198. Hosseini-Ara, R., Mirdamadi, H.R., Khademyzadeh, H., Salimi, H.: Thermal effect on dynamic stability of single-walled carbon nanotubes in low and high temperatures based on nonlocal shell theory. Adv. Mater. Res. 622, 959–964 (2013)

    Google Scholar 

  199. Rahmanian, M., Torkaman-Asadi, M.A., Firouz-Abadi, R.D., Kouchakzadeh, M.A.: Free vibration analysis of carbon nanotubes resting on Winkler foundations based on nonlocal models. Physica B: Condens. Matter. 484, 83–94 (2016)

    Google Scholar 

  200. Rouhi, H.: Free vibration analysis of single- and double-walled carbon nanotubes based on nonlocal elastic shell models. J. Vib. Control 20, 670–678 (2014)

    MathSciNet  MATH  Google Scholar 

  201. Soltani, P., Saberian, J., Bahramian, R.: Nonlinear vibration analysis of single-walled carbon nanotube with shell model based on the nonlocal elasticity theory. J. Comput. Nonlinear Dyn. 11, 011002 (2016)

    Google Scholar 

  202. Wang, Q., Varadan, V.K.: Application of nonlocal elastic shell theory in wave propagation analysis of carbon nanotubes. Smart Mater. Struct. 16, 178–190 (2007)

    Google Scholar 

  203. Zhang, Y.Y., Wang, C.M., Duan, W.H., Zong, Z.: Assessment of continuum mechanica models in predicting buckling strains of single-walled carbon nanotubes. Nanotechnology 20, 395707 (2009)

    Google Scholar 

  204. Ansari, R., Torabi, J., Faghih Shojaei, M.: An efficient numerical method for analyzing the thermal effects on the vibration embedded single-walled carbon nanotubes based on the nonlocal shell model. Mech. Adv. Mater. Struct. 25, 500–511 (2018)

    Google Scholar 

  205. Brischetto, S.: A continuum elastic three-dimensional model for natural frequencies of single-walled carbon nanotubes. Compos. Part B 61, 222–228 (2014)

    Google Scholar 

  206. Gafour, Y., Zidour, M., Tounsi, A., Heireche, H., Semmah, A.: Sound wave propagation in zigzag double-walled carbon nanotubes embedded in an elastic medium using nonlocal elasticity theory. Physica E 48, 118–123 (2013)

    Google Scholar 

  207. Rouhi, H., Ansari, R.: Nonlocal analytical Fugge shell model for axial buckling of double-walled carbon nanotubes with different end conditions. Nano 7, 1250018 (2012)

    Google Scholar 

  208. Ru, C.Q.: Axially compressed buckling of a doublewalled carbon nanotube embedded in an elastic medium. J. Mech. Phys. Solids 49, 1265–1279 (2001)

    MATH  Google Scholar 

  209. Ansari, R., Rouhi, H., Sahmani, S.: Thermal effect on axial buckling behavior of multi-walled carbon nanotubes based on nonlocal shell model. Physica E 44, 373–378 (2011)

    Google Scholar 

  210. Ansari, R., Shahabodini, A., Rouhi, H., Alipour, A.: Thermal buckling analysis of multi-walled carbon nanotubes through a nonlocal shell theory incorporating interatomic potentials. J. Therm. Stress. 36, 56–70 (2013)

    Google Scholar 

  211. He, X.Q., Eisenberger, M., Liew, K.M.: The effect of van der Waals interaction modeling on the vibration characteristics of multiwalled carbon nanotubes. J. Appl. Phys. 100, 124317 (2006)

    Google Scholar 

  212. He, X.Q., Kitipornchai, S., Wang, C.M., Xiang, Y., Zhou, Q.: A nonlinear van der Waals force model for multiwalled carbon nanotubes modeled by a nested system of cylindrical shells. J. Appl. Mech. 77, 061006 (2010)

    Google Scholar 

  213. Ru, C.Q.: Column buckling of multiwalled carbon nanotubes with interlayer radial displacements. Phys. Rev. B 62, 16962 (2000)

    Google Scholar 

  214. Wang, C.Y., Ru, C.Q., Mioduchowski, A.: Free vibration of multiwall carbon nanotubes. J. Appl. Phys. 97, 114323 (2005)

    Google Scholar 

  215. Yan, Y., Wang, W.Q., Zhang, L.X.: Nonlocal effect on axially compressed buckling of triple-walled carbon nanotubes under temperature field. Appl. Math. Modell. 34, 3422–3429 (2010)

    MathSciNet  MATH  Google Scholar 

  216. Zhang, Y.Y., Wang, C.M., Duan, W.H., Xiang, Y., Zong, Z.: Assessment of continuum mechanics models in predicting buckling strains of single-walled carbon nanotubes. Nanotechnology 20, 395707 (2009)

    Google Scholar 

  217. Love, A.E.H.: A Treatise on the Mathematical Theory of Elasticity. Dover, New York (1944)

    MATH  Google Scholar 

  218. Flügge, W.: Stresses in Shells. Springer-Verlag, New York (1973)

    MATH  Google Scholar 

  219. Donnell, L.H.: Beams, Plates, and Shells. McGraw-Hill, New York (1976)

    MATH  Google Scholar 

  220. Sanders, J.L.: An Improved First Approximation Theory for Thin Shells. NASA-TR-R24 (1959)

  221. Soedel, W.: Vibrations of Shells and Plates. Marcel Dekker Inc, New York (1993)

    MATH  Google Scholar 

  222. Leissa, A.W.: Vibration of Shells. NASA Report No. SP-288 (1973)

  223. Soldatos, K.P.: A comparison of some shell theories used for the dynamic analysis of cross-ply laminated circular cylindrical panels. J. Sound Vib. 97, 305–319 (1984)

    Google Scholar 

  224. Chandrashekhara, K., Kumar, D.V.T.G.P.: Assessment of shell theories for the static analysis of cross-ply laminated circular cylindrical shells. Thin-Walled Struct. 22, 291–318 (1995)

    Google Scholar 

  225. Silvestre, N.: On the accuracy of shell models for torsional buckling of carbon nanotubes. Eur. J. Mech. A Solids 32, 103–108 (2012)

    MATH  Google Scholar 

  226. Silvestre, N., Wang, C.M., Zhang, Y.Y., Xiang, Y.: Sanders shell model for buckling of single-walled carbon nanotubes with small aspect ratio. Compos. Struct. 93, 1683–1692 (2011)

    Google Scholar 

  227. Wang, C.M., Tay, Z.Y., Chowdhuary, A.N.R., Duan, W.H., Zhang, Y.Y., Silvestre, N.: Examination of cylindrical shell theories for buckling of carbon nanotubes. Int. J. Struct. Stab. Dyn. 11, 1025–1058 (2011)

    Google Scholar 

  228. Duan, W.H., Wang, C.M., Zhang, Y.Y.: Calibration of nonlocal scaling effect parameter for free vibration of carbon nanotubes by molecular dynamics. J. Appl. Phys. 101, 024305 (2007)

    Google Scholar 

Download references

Acknowledgements

This work was supported by the Ministry of Science and Technology of the Republic of China through Grant MOST 106-2221-E-006-036-MY3.

Author information

Authors and Affiliations

  1. Department of Civil Engineering, National Cheng Kung University, Tainan, Taiwan

    Chih-Ping Wu & Jung-Jen Yu

Authors
  1. Chih-Ping Wu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jung-Jen Yu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Chih-Ping Wu.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Appendix

Appendix

The differential operators \( L_{ij}\) (i, j=1-3) in Eqs. (60)–(62) are given as

$$\begin{aligned}&L_{11} =\left( {A_{11} +\delta _{iF} \Delta {\bar{A}}_{11} } \right) \partial _{xx} + \left\{ {\left[ {\left( {A_{66} +\delta _{iF} \Delta {\tilde{A}}_{66} } \right) /R_k^2 } \right] -\delta _{iS} \left( {B_{66} /R_k^3 } \right) +\delta _{iS} \left( {D_{66} /4R_k^4 } \right) } \right\} \partial _{\theta \theta } , \\&L_{12} =\left\{ \left[ {\left( {A_{12} +A_{66} +\delta _{iF} \Delta {\tilde{A}}_{66} } \right) /R_k } \right] +\left( {\delta _{iS} +\delta _{iF} } \right) \left[ {\left( {B_{12} +B_{66} +\delta _{iF} B_{66} +\delta _{iF} \Delta {\tilde{B}}_{66} } \right) /R_k^2 } \right] \right. \\&\qquad \left. -\delta _{iS} \left( {3D_{66} /4R_k^3 } \right) \right\} \partial _{x\theta } , \\&L_{13} =-\left( {B_{11} +\delta _{iF} \Delta {\bar{B}}_{11} } \right) \partial _{xxx} -\left[ {\left( {B_{12} +2B_{66} +\delta _{iF} \Delta {\tilde{B}}_{66} } \right) /R_k^2 -\delta _{iS} \left( {D_{66} /R_k^3 } \right) } \right] \partial _{x\theta \theta } +\left( {A_{12} /R_k } \right) \partial _x , \\&L_{21} =\left[ {\left( {A_{12} +A_{66} } \right) /R_k +\left( {\delta _{iS} +\delta _{iF} } \right) \left( {B_{12} +B_{66} } \right) /R_k^2 -\delta _{iS} \left( {3D_{66} /4R_k^3 } \right) } \right] \partial _{x\theta } , \\&L_{22} =\left\{ {A_{66} +\left( {3\delta _{iS} +3\delta _{iF} } \right) \left( {B_{66} +\delta _{iF} \Delta {\bar{B}}_{66} /3} \right) /R_k } \right. +\left[ {\left( {9/4} \right) \delta _{iS} +2\delta _{iF} } \right] \left( {D_{66} +0.5\delta _{iF} \Delta {\bar{D}}_{66} } \right) /\left. {R_k^2 } \right\} \partial _{xx} \\&\qquad +\left\{ \left[ {\left( {A_{22} +\delta _{iF} \Delta {\tilde{A}}_{22} } \right) /R_k^2 } \right] +\left( {2\delta _{iS} +2\delta _{iF} } \right) \left( {B_{22} +\delta _{iF} \Delta {\tilde{B}}_{22} } \right) /R_k^3\right. \\&\qquad \left. +\left( {\delta _{iS} +\delta _{iF} } \right) \left( {D_{22} +\delta _{iF} \Delta {\tilde{D}}_{22} } \right) /R_k^4 \right\} \partial _{\theta \theta } , \\&L_{23} =-\left[ \left( {B_{12} +2B_{66} +\delta _{iF} \Delta {\bar{B}}_{66} } \right) /R_k +\left( {\delta _{iS} +\delta _{iF} } \right) \left( {D_{12} /R_k^2 } \right) \right. \\&\qquad \qquad \! \left. +\left( {3\delta _{iS} +2\delta _{iF} } \right) \left( {D_{66} +0.5\delta _{iF} \Delta {\bar{D}}_{66} } \right) /R_k^2 \right] \partial _{xx\theta } \\&\qquad \qquad \!-\left[ {\left( {B_{22} +\delta _{iF} \Delta {\tilde{B}}_{22} } \right) /R_k^3 +\left( {\delta _{iS} +\delta _{iF} } \right) \left( {D_{22} +\delta _{iF} \Delta {\tilde{D}}_{22} } \right) /R_k^4 } \right] \partial _{\theta \theta \theta } +\left[ {\left( {A_{22} +\delta _{iF} \Delta {\tilde{A}}_{22} } \right) /R_k^2 } \right. \left. \right. \\&\qquad \qquad \! \left. +\left( {\delta _{iS} +\delta _{iF} } \right) \left( {B_{22} +\delta _{iF} \Delta {\tilde{B}}_{22} } \right) /R_k^3 \right] \partial _\theta , \\&L_{31} =\left( {B_{11} +\delta _{iF} \Delta {\bar{B}}_{11} } \right) \partial _{xxx} +\left[ {\left( {B_{12} +2B_{66} +\delta _{iF} \Delta {\tilde{B}}_{66} } \right) /R_k^2 -\delta _{iS} \left( {D_{66} /R_k^3 } \right) } \right] \partial _{x\theta \theta } -\left( {A_{12} /R_k } \right) \partial _x , \\&L_{32} =\left[ \left( {B_{12} +2B_{66} +\delta _{iF} \Delta {\tilde{B}}_{66} } \right) /R_k +\left( {3\delta _{iS} +4\delta _{iF} } \right) \left( {D_{66} +0.25\delta _{iF} \Delta {\bar{D}}_{66} +0.25\delta _{iF} \Delta {\tilde{D}}_{66} } \right) /R_k^2 \right. \left. \right. \\&\qquad \qquad \! \left. +\left( {\delta _{iS} +\delta _{iF} } \right) \left( {D_{12} /R_k^2 } \right) \right] \partial _{xx\theta } +\left[ {\left( {B_{22} +\delta _{iF} \Delta {\tilde{B}}_{22} } \right) /R_k^3 +\left( {\delta _{iS} +\delta _{iF} } \right) \left( {D_{22} +\delta _{iF} \Delta {\tilde{D}}_{22} } \right) /R_k^4 } \right] \partial _{\theta \theta \theta }\\&\qquad \qquad \! -\left[ {\left( {A_{22} +\delta _{iF} \Delta {\tilde{A}}_{22} } \right) /R_k^2 +\left( {\delta _{iS} +\delta _{iF} } \right) \left( {B_{22} +\delta _{iF} \Delta {\tilde{B}}_{22} } \right) /R_k^3 } \right] \partial _\theta , \\&L_{33} =-\left( {D_{11} +\delta _{iF} \Delta {\bar{D}}_{11} } \right) \partial _{xxxx} -\left[ {\left( {2D_{12} +4D_{66} +\delta _{iF} \Delta {\bar{D}}_{66} +\delta _{iF} \Delta {\tilde{D}}_{66} } \right) /R_k^2 } \right] \partial _{xx\theta \theta }\\&\qquad \qquad \!-\left[ {\left( {D_{22} +\delta _{iF} \Delta {\tilde{D}}_{22} } \right) /R_k^4 } \right] \partial _{\theta \theta \theta \theta } +\left( {2B_{12} /R_k } \right) \partial _{xx} +\left[ {2\left( {B_{22} +\delta _{iF} \Delta {\tilde{B}}_{22} } \right) /R_k^3 } \right] \partial _{\theta \theta }\\&\qquad \qquad \!-\left( {A_{22} +\delta _{iF} \Delta {\tilde{A}}_{22} } \right) /R_k^2 , \\&\left( {A_{ij} \quad B_{ij} \quad D_{ij} } \right) =\int _{-h/2}^{h/2} {\;Q_{ij} \left( {1\quad z\quad z^{2}} \right) \;\mathrm{d}z\quad \quad (i,\;j=1,\;2,\;6),} \\&\left( {\Delta {\bar{A}}_{ij} \quad \Delta {\bar{B}}_{ij} \quad \Delta {\bar{D}}_{ij} } \right) =\int _{-h/2}^{h/2} {\;Q_{ij} \frac{z}{R}\left( {1\quad z\quad z^{2}} \right) \;\mathrm{d}z\quad \quad (i,\;j=1,\;2,\;6)} , \\&\left( {\Delta {\tilde{A}}_{ij} \quad \Delta {\tilde{B}}_{ij} \quad \Delta {\tilde{D}}_{ij} } \right) =-\int _{-h/2}^{h/2} {\;Q_{ij} \frac{z}{\left( {R+z} \right) }\left( {1\quad z\quad z^{2}} \right) \;\mathrm{d}z } . \end{aligned}$$

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Wu, CP., Yu, JJ. A review of mechanical analyses of rectangular nanobeams and single-, double-, and multi-walled carbon nanotubes using Eringen’s nonlocal elasticity theory. Arch Appl Mech 89, 1761–1792 (2019). https://doi.org/10.1007/s00419-019-01542-z

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00419-019-01542-z

Keywords

Search

Navigation