Skip to main content
SpringerLink
Log in

A review on the application of modified continuum models in modeling and simulation of nanostructures

  • Review Paper
  • Published:
Acta Mechanica Sinica Aims and scope Submit manuscript

Abstract

Analysis of the mechanical behavior of nanostructures has been very challenging. Surface energy and nonlocal elasticity of materials have been incorporated into the traditional continuum analysis to create modified continuum mechanics models. This paper reviews recent advancements in the applications of such modified continuum models in nanostructures such as nanotubes, nanowires, nanobeams, graphenes, and nanoplates. A variety of models for these nanostructures under static and dynamic loadings are mentioned and reviewed. Applications of surface energy and nonlocal elasticity in analysis of piezoelectric nanomaterials are also mentioned. This paper provides a comprehensive introduction of the development of this area and inspires further applications of modified continuum models in modeling nanomaterials and nanostructures.

Graphic abstract

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
Fig. 4
Fig. 5
Fig. 6
Fig. 7

Similar content being viewed by others

References

  1. Li, C., Chou, T.W.: A structural mechanics approach for the analysis of carbon nanotubes. Int. J. Solids Struct. 40, 2487–2499 (2003)

    Article  MATH  Google Scholar 

  2. Chiu, H.Y., Hung, P., Postma, HWCh., et al.: Atom-ic-scale mass sensing using carbon nanotube resonators. Nano Lett. 8, 4342–4346 (2008)

    Article  Google Scholar 

  3. Falvo, M.R., Clary, G.J., Taylor, R.M., et al.: Bending and buckling of carbon nanotubes under large strain. Nature 389, 582–584 (1997)

    Article  Google Scholar 

  4. Iijima, S., Brabec, C., Maiti, A., et al.: Structural flexibility of carbon nanotubes. J. Chem. Phys. 104, 2089–2092 (1996)

    Article  Google Scholar 

  5. Hernandez, E., Goze, C., Bernier, P., et al.: Elastic properties of C and Bx Cy Nz composite nanotubes. Phys. Rev. Lett. 80, 4502–4505 (1998)

    Article  Google Scholar 

  6. Sanchez-Portal, D., Artacho, E., Soler, J.M., et al.: Ab initio structural, elastic, and vibrational properties of carbon nanotubes. Phys. Rev. B 59, 12678–12688 (1999)

    Article  Google Scholar 

  7. Yakobson, B.I., Campbell, M.P., Brabec, C.J., et al.: High strain rate fracture and C-chain unraveling in carbon nanotubes. Comput. Mater. Sci. 8, 341–348 (1997)

    Article  Google Scholar 

  8. Bodily, B.H., Sun, C.T.: Structural and equivalent continuum properties of single-walled carbon nanotubes. Int. J. Mater. Prod. Technol. 18, 381–397 (2003)

    Article  Google Scholar 

  9. Li, C., Chou, T.W.: Single-walled carbon nanotubes as ultrahigh frequency nanomechanical resonators. Phys. Rev. B 68, 073405 (2003)

    Article  Google Scholar 

  10. Yakobson, B.I., Brabec, C.J., Bernholc, J.: Nanomechanics of carbon tubes: instabilities beyond linear response. Phys. Rev. Lett. 76, 2511–2514 (1996)

    Article  Google Scholar 

  11. Krishnan, A., Dujardin, E., Ebbesen, T.W., et al.: Young’s modulus of single-walled nanotubes. Phys. Rev. B 58, 14013–14019 (1998)

  12. Wang, X., Yang, H.K., Dong, K.: Torsional buckling of multi-walled carbon nanotubes. Mater. Sci. Eng. A 404, 314–322 (2005)

  13. Duan, W.H., Wang, Q., Wang, Q., et al.: Modeling the instability of carbon nanotubes: from continuum mechanics to molecular dynamics. J. Nanotechnol. Eng. Med. 1, 011001 (2010)

    Article  Google Scholar 

  14. Kitipornchai, S., He, X.Q., Liew, K.M.: Continuum model for the vibration of multilayered graphene sheets. Phys. Rev. B 72, 075443 (2005)

    Article  Google Scholar 

  15. He, X.Q., Kitipornchai, S., Liew, K.M.: Resonance analysis of multi-layered graphene sheets used as nanoscale resonators. Nanotechnology 16, 2086–2091 (2005)

    Article  Google Scholar 

  16. Maranganti, R., Sharma, P.: A novel atomistic approach to determine strain gradient elasticity constants. J. Mech. Phys. Solids 55, 1832–1852 (2005)

    Google Scholar 

  17. Gurtin, M.E., Murdoch, A.I.: A continuum theory of elastic material surfaces. Arch. Ration. Mech. Anal 57, 291–323 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  18. Gurtin, M.E., Murdoch, A.I.: Surface stress in solids. Int. J. Solids Struct. 14, 431–440 (1978)

    Article  MATH  Google Scholar 

  19. Truesdell, C., Noll, W.: The Nonlinear Field Theories of Mechanics. Springer, Berlin (1992)

    Book  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  Google Scholar 

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

    MATH  Google Scholar 

  23. Miller, R.E., Shenoy, V.B.: Size-dependent elastic properties of nanosized structural elements. Nanotechnology 11, 139 (2000)

    Article  Google Scholar 

  24. Sharma, P., Ganti, S., Bhate, N.: Effect of surfaces on the size-dependent elastic state of nano-inhomogeneities. Appl. Phys. Lett. 82, 535 (2003)

    Article  Google Scholar 

  25. Cammarata, R.C.: Surface and interface stress effects in thin films. Prog. Surf. Sci. 46, 1–38 (1994)

    Article  Google Scholar 

  26. Jing, G.Y., Duan, H.L., Sun, X.M., et al.: Surface effects on elastic properties of silver nanowires: contact atomic-force microscopy. Phys. Rev. B 73, 235409 (2006)

    Article  Google Scholar 

  27. Tabib-Azar, M., Nassirou, M., Wang, R., et al.: Mechanical properties of self-welded silicon nanobridges. Appl. Phys. Lett. 87, 113102 (2005)

    Article  Google Scholar 

  28. Song, J.H., Wang, X.D., Riedo, E., et al.: Elastic property of vertically aligned nanowires. Nano Lett. 5, 1954–1958 (2005)

    Article  Google Scholar 

  29. Cuenot, S., Fretigny, C., Demoustier-Champagne, S., et al.: Surface tension effect on the mechanical properties of nanomaterials measured by atomic force microscopy. Phys. Rev. B 69, 165410 (2004)

    Article  Google Scholar 

  30. He, J., Lilley, C.M.: Surface effect on the elastic behavior of static bending nanowires. Nano Lett. 8, 1798–1802 (2008)

    Article  Google Scholar 

  31. Jiang, L.Y., Yan, Z.: Timoshenko beam model for static bending of nanowires with surface effects. Physica E 42, 2274–2279 (2010)

  32. Wang, G.F.: Effects of surface energy on the mechanical performance of nanosized beams. J. Comput. Theor. Nanosci. 8, 1–5 (2011)

    Article  Google Scholar 

  33. Liu, C., Rajapakse, R.K.N.D., Phani, A.S.: Finite element modeling of beams with surface energy effects. J. Appl. Mech. 78, 031014 (2011)

    Article  Google Scholar 

  34. Ansari, R., Sahmani, S.: Bending behavior and buckling of nanobeams including surface stress effects corresponding to different beam theories. Int. J. Eng. Sci. 49, 1244–1255 (2011)

  35. Wang, K.F., Wang, B.L.: Effects of surface and interface energies on the bending behavior of nanoscale multilayered beams. Physica E 54, 197–201 (2013)

    Article  Google Scholar 

  36. Lu, P., He, L.H., Lee, H.P., et al.: Thin plate theory including surface effects. Int. J. Solids Struct. 43, 4631–4647 (2006)

    Article  MATH  Google Scholar 

  37. Wang, Z., Zhao, Y.: Self-instability and bending behaviors of nano plates, Acta Mech. Solida Sin. 22, 630–643 (2009)

    Article  Google Scholar 

  38. Ansari, R., Shahabodini, A., FaghihShojaei, M., et al.: On the bending and buckling behaviors of Mindlin nanoplates considering surface energies. Physica E 57, 126–137 (2014)

    Article  Google Scholar 

  39. Wang, G.F., Yang, F.: Postbuckling analysis of nanowires with surface effects. J. Appl. Phys. 109, 063535 (2011)

    Article  Google Scholar 

  40. Liu, J.L., Mei, Y., Xia, R., et al.: Large displacement of a static bending nanowire with surface effects. Physica E 44, 2050–2055 (2012)

    Article  Google Scholar 

  41. Sapsathiarn, Y., Rajapakse, R.K.N.D.: A model for large deflections of nanobeams and experimental comparison. IEEE T. Nanotechnol. 11, 247–254 (2012)

    Article  Google Scholar 

  42. Young, S.J., Ji, L.H., Chang, S.J., et al.: Nanoscale mechanical characteristics of vertical ZnO nanowires grown on ZnO:Ga/glass templates. Nanotechnology 18, 225603 (2007)

    Article  Google Scholar 

  43. Ji, L.W., Young, S.J., Fang, T.H., et al.: Buckling characterization of vertical ZnO nanowires using nanoindentation. Appl. Phys. Lett. 90, 033109 (2007)

    Article  Google Scholar 

  44. Wang, G.F., Feng, X.Q.: Surface effects on buckling of nanowires under uniaxial compression. Appl. Phys. Lett. 94, 141913 (2009)

  45. Wang, G.F., Feng, X.Q.: Timoshenko beam model for buckling and vibration of nanowires with surface effects. J. Phys. D 42, 155411 (2009)

    Article  Google Scholar 

  46. Assadi, A., Farshi, B.: Size dependent stability analysis of circular ultrathin films in elastic medium with consideration of surface energies. Physica E 43, 1111–1117 (2011)

    Article  Google Scholar 

  47. Li, B., Li, C.X., Wei, C.L.: Surface effects on the postbuckling of nanowires. Chin. Phys. Lett. 28, 046202 (2011)

    Article  MathSciNet  Google Scholar 

  48. Li, Y.H., Song, J.Z., Fang, B., et al.: Surface effects on the postbuckling of nanowires. J. Phys. D 44, 425304 (2011)

    Article  Google Scholar 

  49. Ansari, R., Mohammadi, V., Faghih Shojaei, M., et al.: Postbuckling characteristics of nanobeams based on the surface elasticity theory. Compos. Part B 55, 240–246 (2013)

    Article  Google Scholar 

  50. Lim, C.W., He, L.H.: Size-dependent nonlinear response of thin elastic films with nano-scale thickness. Int. J. Mech. Sci. 46, 1715–1726 (2004)

    Article  MATH  Google Scholar 

  51. Wang, K.F., Wang, B.L.: Effect of surface energy on the non-linear postbuckling behavior of nanoplates. Int. J. Non-Linear Mech. 55, 19–24 (2013)

    Article  Google Scholar 

  52. Ma, J.B., Jiang, L., Asokanthan, S.F.: Influence of surface effects on the pull-in instability of NEMS electrostatic switches. Nanotechnology 21, 505708 (2010)

    Article  Google Scholar 

  53. Koochi, A., Kazemi, A., Khandani, F., et al.: Influence of surface effects on size-dependent instability of nano-actuators in the presence of quantum vacuum fluctuations. Phys. Scripta 85, 035804 (2012)

    Article  MATH  Google Scholar 

  54. Yang, F., Wang, G.F., Long, J.M., et al.: Influence of surface energy on the pull-in instability of electrostatic nano-switches. J. Comput. Theor. Nanosci. 10, 1273–1277 (2013)

    Article  Google Scholar 

  55. Fu, Y.M., Zhang, J.: Size-dependent pull-in phenomena in electrically actuated nanobeamsincorporating surface energies. Appl. Math. Model. 35, 941–951 (2011)

    Article  MathSciNet  Google Scholar 

  56. Wang, K.F., Wang, B.L.: Influence of surface energy on the non-linear pull-in instability of nano-switches. Int. J. Non-Linear Mech. 59, 69–75 (2014)

    Article  Google Scholar 

  57. Wang, K.F., Wang, B.L.: A general model for nano-cantilever switches with consideration of surface effects and nonlinear curvature. Physica E 66, 197–208 (2015)

    Article  Google Scholar 

  58. Lu, C.F., Wu, D.Z., Chen, W.Q.: Surface effects on the jump-in instability of nanomechanical structures. IEEE Trans. Nanotechnol. 10, 962–967 (2010)

    Article  Google Scholar 

  59. Wang, G.F., Feng, X.Q.: Effects of surface elasticity and residual surface tension on the natural frequency of microbeams. Appl. Phys. Lett. 90, 231904 (2007)

    Article  Google Scholar 

  60. He, J., Lilley, C.M.: Surface stress effect on bending resonance of nanowires with different boundary conditions. Appl. Phys. Lett. 93, 263108 (2008)

    Article  Google Scholar 

  61. He, J., Lilley, C.M.: Resonant frequency analysis of Timoshenko nanowires with surface stress for different boundary conditions. J. Appl. Phys. 112, 074322 (2012)

    Article  Google Scholar 

  62. Farshi, B., Assadi, A., Alinia-ziazi, A.: Frequency analysis of nanotubes with consideration of surface effects. Appl. Phys. Lett. 96, 093105 (2010)

    Article  Google Scholar 

  63. Wang, K.F., Wang, B.L.: Effect of surface energy on the sensing performance of bridged nanotubebased micro-mass sensors. J. Intel. Mater. Syst. Struct. 25, 2177–2186 (2014)

    Article  Google Scholar 

  64. Hasheminejad, B.S.M., Gheshlaghi, B., Mirzaei, Y., et al.: Free transverse vibrations of cracked nanobeams with surface effects. Thin Solid Films 519, 2477–2482 (2011)

    Article  Google Scholar 

  65. Wang, K.F., Wang, B.L.: Timoshenko beam model for the vibration analysis of a cracked nanobeam with surface energy. J. Vib. Control. doi:10.1177/1077546313513054 (2013)

  66. Assadi, A., Farshi, B., Alinia-ziazi, A.: Size dependent dynamic analysis of nanoplates. J. Appl. Phys. 107, 124310 (2010)

    Article  Google Scholar 

  67. Assadi, A., Farshi, B.: Vibration characteristics of circular nanoplates. J. Appl. Phys. 108, 074312 (2010)

    Article  Google Scholar 

  68. Ansari, R., Gholami, R., FaghihShojaei, M., et al.: Surface stress effect on the vibrational response of circular nanoplates with various edge supports. J. Appl. Mech. 80, 021021 (2013)

    Article  Google Scholar 

  69. Ansari, R., Sahmani, S.: Surface stress effects on the free vibration behavior of nanoplates. Int. J. Eng. Sci. 49, 1204–1215 (2011)

    Article  MathSciNet  Google Scholar 

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

    Article  Google Scholar 

  71. Sharabiani, P.A., Yazdi, M.R.H.: Nonlinear free vibrations of functionally graded nanobeams with surface effects. Compos. Part B 45, 581–586 (2013)

    Article  Google Scholar 

  72. Wang, K.F., Wang, B.L.: Effects of residual surface stress and surface elasticity on the nonlinear free vibration of nanoscale plates. J. Appl. Phys. 112, 013520 (2012)

    Article  Google Scholar 

  73. Lu, C.F., Wu, D.Z., Chen, W.Q.: Nonlinear responses of nanoscale FGM films including the effects of surface energies. IEEE Trans. Nanotechnol 10, 1321–1327 (2010)

    Google Scholar 

  74. Huang, G.Y., Yu, S.W.: Effect of surface piezoelectricity on the electromechanical behavior of a piezoelectric ring. Phys. Status Solid B 243, R22–R24 (2006)

    Article  Google Scholar 

  75. Yan, Z., Jiang, L.Y.: Surface effects on the electromechanical coupling and bending behaviours of piezoelectric nanowires. J. Phys. D 44, 075404 (2011)

    Article  Google Scholar 

  76. Xu, X.J., Deng, Z.C., Wang, B.: Closed solutions for the electromechanical bending and vibration of thick piezoelectric nanobeams with surface effects. J. Phys. D 46, 405302 (2013)

  77. Yan, Z., Jiang, L.Y.: Surface effects on the electroelastic responses of a thin piezoelectric plate with nanoscale thickness. J. Phys. D 45, 255401 (2012)

    Article  Google Scholar 

  78. Wang, G.F., Feng, X.Q.: Effect of surface stresses on the vibration and buckling of piezoelectric nanowires. EPL 91, 56007 (2010)

    Article  Google Scholar 

  79. Samaei, A.T., Bakhtiari, M., Wang, G.F.: Timoshenko beam model for buckling of Piezoelectric nanowires with surface effects. Nanoscale Res. Let. 7, 201–206 (2012)

    Article  Google Scholar 

  80. Yan, Z., Jiang, L.Y.: The vibrational and buckling behaviors of piezoelectric nanobeams with surface effects. Nanotechnology 22, 245703 (2011)

    Article  Google Scholar 

  81. Wang, K.F., Wang, B.L.: Surface effects on the buckling of piezoelectric nanobeams. Adv. Mater. Res. 486, 519–523 (2012)

    Article  Google Scholar 

  82. Li, Y.H., Chen, C., Fang, B., et al.: Postbuckling of piezoelectric nanobeams with surface effects. Int. J. Appl. Mech. 04, 1250018 (2012)

    Article  Google Scholar 

  83. Zhang, J., Wang, C.Y., Adhikari, S.: Surface effect on the buckling of piezoelectric nanofilms. J. Phys. D 45, 285301 (2012)

    Article  Google Scholar 

  84. Yan, Z., Jiang, L.Y.: Surface effects on the vibration and buckling of piezoelectric nanoplates. EPL 99, 27007 (2012)

    Article  Google Scholar 

  85. Yan, Z., Jiang, L.Y.: Vibration and buckling analysis of a piezoelectric nanoplate considering surface effects and in-plane constraints. Proc. R. Soc. A 468, 3458–3475 (2012)

    Article  MathSciNet  Google Scholar 

  86. Samaei, A.T., Gheshlaghi, B., Wang, G.F.: Frequency analysis of piezoelectric nanowires with surface effects. Curr. Appl. Phys. 13, 2098–2102 (2013)

    Article  Google Scholar 

  87. Zhang, C.L., Chen, W.Q.: Two-dimensional theory of piezoelectric plate considering surface effect. Eur. J. Mech. A Solids 41, 50–57 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  88. Ibach, H.: Adsorbate-induced surface stress. J. Vac. Sci. Technol. A 12, 2240–2245 (1994)

    Article  Google Scholar 

  89. Seoánez, C., Guines, F.: Surface dissipation in nanoelectromechanical systems: Unified description with the standard tunneling model and effects of metallic electrodes. Phys. Rev. B 77, 125107 (2008)

    Article  Google Scholar 

  90. Zener, C.: Elasticity and Anelasticity of Metals. University of Chicago Press, Chicago (1948)

    MATH  Google Scholar 

  91. Ru, C.Q.: Size effect of dissipative surface stress on quality factor of microbeams Appl. Phys. Lett. 94, 051905 (2009)

    Google Scholar 

  92. Hasheminejad, S.M., Gheshlaghi, B.: Dissipative surface stress effects on free vibrations of nanowires. Appl. Phys. Lett. 97, 253103 (2010)

    Article  Google Scholar 

  93. Gheshlaghi, B., Hasheminejad, S.M.: Size dependent surface dissipation in thick nanowires. Appl. Phys. Lett. 100, 263112 (2012)

  94. Gheshlaghi, B.: Large-amplitude vibrations of nanowires with dissipative surface stress effects. Acta Mech. 224, 1329–1334 (2013)

  95. Hasheminejad, S.M., Gheshlaghi, B.: Eigenfrequencies and quality factors of nanofilm resonators with dissipative surface stress effects. Wave. Motion. 50, 94–100 (2013)

    Article  MathSciNet  Google Scholar 

  96. Gheshlaghi, B., Hasheminejad, S.M.: Size dependent damping in axisymmetric vibrations of circular nanoplates. Thin Solid Films 537, 212–216 (2013)

    Article  Google Scholar 

  97. Wei, G., Yu, S.W., Huang, G.Y.: Finite Element characterization of the size-dependent mechanical behavior in nanosystems. Nanotechnology 17, 1118–1122 (2006)

    Article  Google Scholar 

  98. Tian, L., Rajapakse, R.K.N.D.: Finite element modeling of nanoscale inhomogeneities in an elastic matrix. Comput. Mater. Sci. 41, 44–53 (2007)

    Article  Google Scholar 

  99. Feng, Y.K., Liu, Y.L., Wang, B.: Finite element analysis of resonant properties of silicon nanowires with consideration of surface effects. Acta Mech. 217, 149–155 (2011)

    Article  MATH  Google Scholar 

  100. Wang, K.F., Wang, B.L.: A finite element model for the bending and vibration of nanoscale plates with surface effect. Finite Elem. Anal. Des. 74, 22–29 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  101. Peddieson, J., Buchanan, G.R., McNitt, R.P.: Application of nonlocal continuum models to nanotechnology. Int. J. Eng. Sci. 41, 305–312 (2003)

    Article  Google Scholar 

  102. Li, C., Yao, L.Q., Chen, W.Q., et al.: Comments on nonlocal effects in nano-cantilever beams. Int. J. Eng. Sci. 87, 47–57 (2015)

    Article  Google Scholar 

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

    Article  Google Scholar 

  104. Wang, Q., Liew, K.M.: Application of nonlocal continuum mechanics to static analysis of micro- and nano-structures. Phys. Lett. A 363, 236–242 (2007)

    Article  Google Scholar 

  105. Aydogdu, M.: A general nonlocal beam theory: Its application to nanobeam bending, buckling and vibration. Physica E 41, 651–1655 (2009)

    Google Scholar 

  106. Duan, W.H., Wang, C.M.: Exact solutions for axisymmetric bending of micro/nanoscale circular plates based on nonlocal plate theory. Nanotechnology 18, 385704 (2007)

    Article  Google Scholar 

  107. Wang, Y.Z., Li, F.M.: Static bending behaviors of nanoplate embedded in elastic matrix with small scale effects. Mech. Res. Commun. 41, 44–48 (2012)

    Article  Google Scholar 

  108. 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)

    Article  Google Scholar 

  109. 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)

    Article  MATH  MathSciNet  Google Scholar 

  110. Sudak, L.J.: Column buckling of multiwalled carbon nanotubes using nonlocal continuum mechanics. J. Appl. Phys. 94, 7281–7287 (2003)

    Article  Google Scholar 

  111. Wang, Q., Varadan, V.K., Quek, S.T.: Small scale effect on elastic buckling of carbon nanotubes with nonlocal continuum models. Phys. Lett. A 357, 130–135 (2006)

    Article  Google Scholar 

  112. Zhang, Y.Q., Liu, G.R., Wang, J.S.: Small-scale effects on buckling of multiwalled carbon nanotubes under axial compression. Phys. Rev. B 70, 205430 (2004)

    Article  Google Scholar 

  113. Yang, Y., Lim, C.W.: A variational principle approach for buckling of carbon nanotube on nonlocal Timoshenko beam models. Nano 06, 363 (2011)

    Article  Google Scholar 

  114. Senthilkumar, V., Pradhan, S.C., Prathap, G.: Buckling analysis of carbon nanotube based on nonlocal Timoshenko beam theory using differential transform method. Adv. Sci. Lett. 3, 415–421 (2010)

    Article  Google Scholar 

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

    Article  Google Scholar 

  116. Lim, C.W., Niu, J.C., Yu, Y.M.: Nonlocal stress theory for buckling instability of nanotubes: new predictions on stiffness strengthening effects of nanoscales. J. Comput. Theor. Nanosci. 7, 2104–2111 (2010)

    Article  Google Scholar 

  117. Xie, G.Q., Han, X., Liu, G.R., et al.: Effect of small size-scale on the radial buckling pressure of a simply supported multi-walled carbon nanotube. Smart Mater. Struct. 15, 1143–1149 (2006)

    Article  Google Scholar 

  118. Wang, Y.Z., Li, F.M., Kishimoto, K.: Scale effects on thermal buckling properties of carbon nanotube. Phys. Lett. A 374, 4890–4893 (2010)

    Article  MATH  Google Scholar 

  119. Li, R., Kardomateas, G.A.: Thermal buckling of multi-walled carbon nanotubes by nonlocal elasticity. J. Appl. Mech. 74, 399–405 (2007)

    Article  MATH  Google Scholar 

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

    Article  Google Scholar 

  121. 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. Stresses 34, 1271–1281 (2011)

    Article  Google Scholar 

  122. Arani, A.G., Zarei, M.S., Mohammadimehr, M., et al.: The thermal effect on buckling analysis of a DWCNT embedded on the Pasternak foundation. Physica E 43, 1642–1648 (2011)

    Article  Google Scholar 

  123. Pradhan, S.C., Murmu, T.: Small scale effect on the buckling of single-layered graphene sheets under biaxial compression via nonlocal continuum mechanics. Comput. Mater. Sci. 47, 268–274 (2009)

    Article  Google Scholar 

  124. Pradhan, S.C., Murmu, T.: Small scale effect on the buckling analysis of single-layered graphene sheet embedded in an elastic medium based on nonlocal plate theory. Physica E 42, 1293–1301 (2010)

    Article  Google Scholar 

  125. Malekzadeh, P., Setoodeh, A.R., Alibeygi Beni, A.: Small scale effect on the thermal buckling of orthotropic arbitrary straight-sided quadrilateral nanoplates embedded in an elastic medium. Comput. Struct. 93, 2083–2089 (2011)

    Article  Google Scholar 

  126. Farajpour, A., Mohammadi, M., Shahidi, A.R., et al.: Axisymmetric buckling of the circular graphene sheets with the nonlocal continuum plate model. Physica E 43, 1820–1825 (2011)

    Article  Google Scholar 

  127. Farajpour, A., Shahidi, A.R., Mohammadi, M., et al.: Buckling of orthotropic micro/nanoscale plates under linearly varying in-plane load via nonlocal continuum mechanics. Comput. Struct. 94, 1605–1614 (2012)

    Article  Google Scholar 

  128. Samaeia, A.T., Abbasion, S., Mirsayar, M.M.: Buckling analysis of a single-layer graphene sheet embedded in an elastic medium based on nonlocal Mindlin plate theory. Mech. Res. Commun. 38, 481–485 (2011)

    Article  MATH  Google Scholar 

  129. Wang, C.M., Xiang, Y., Kitipornchai, S.: Postbuckling of nano-rods/tubes based on nonlocal beam theory. Int. J. Appl. Mech. 1, 259–266 (2009)

    Article  Google Scholar 

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

    Article  Google Scholar 

  131. Emam, S.A.: A general nonlocal nonlinear model for buckling of nanobeams. Appl. Math. Model. 37, 6929–6939 (2013)

    Article  MathSciNet  Google Scholar 

  132. Yang, Q., Lim, C.W.: Thermal effects on buckling of shear deformable nanocolumns with von Kármán nonlinearity based on nonlocal stress theory. Nonlinear Anal. Real 13, 905–922 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  133. Shen, H.S.: Nonlocal plate model for nonlinear analysis of thin films on elastic foundations in thermal environments. Comput. Struct. 93, 1143–1152 (2011)

    Article  Google Scholar 

  134. Farajpour, A., Arab Solghar, A.: Postbuckling analysis of multi-layered graphene sheets under non-uniform biaxial compression. Physica E 47, 197–206 (2013)

    Article  Google Scholar 

  135. Yang, J., Jia, X.L., Kitipornchai, S.: Pull-in instability of nano-switches using nonlocal elasticity theory. J. Phys. D 41, 035103 (2008)

    Article  Google Scholar 

  136. Peng, J., Luo, G., Yang, L., et al.: Pull-in instability behaviour of nanoscale actuators using nonlocal elasticity theory. Adv. Mater. Res. 468, 2755–2758 (2012)

    Article  Google Scholar 

  137. Taghavi, N., Nahvi, H.: Pull-in instability of cantilever and fixed-fixed nano-switches. Eur. J Mech. A-Solids 41, 123–133 (2013)

    Article  MathSciNet  Google Scholar 

  138. Mousavi, T., Bornassi, S., Haddadpour, H.: The effect of small scale on the pull-in instability of nano-switches using DQM. Int. J. Solids Struct. 50, 1193–1202 (2013)

    Article  Google Scholar 

  139. Arani, A.G., Ghaffari, M., Jalilvand, A., et al.: Nonlinear nonlocal pull-in instability of boron nitride nanoswitches. Acta Mech. 224, 3005–3019 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  140. Zhang, Y.Q., Liu, G.R., Xie, X.Y.: Free transverse vibrations of double-walled carbon nanotubes using a theory of nonlocal elasticity. Phys. Rev. B 71, 195404 (2005)

    Article  Google Scholar 

  141. Lu, P., Lee, H.P., Lu, C.: Dynamic properties of flexural beams using a nonlocal elasticity model. J. Appl. Phys. 99, 073510 (2006)

    Article  Google Scholar 

  142. Wang, Q., Varadan, V.K.: Vibration of carbon nanotubes studied using nonlocal continuum mechanics. Smart. Mater. Struct. 15, 659–666 (2006)

    Article  Google Scholar 

  143. Wang, C.M., Zhang, Y.Y., He, X.Q.: Vibration of nonlocal Timoshenko beams. Nanotechnology 18, 105401 (2007)

    Article  Google Scholar 

  144. Murmu, T., Pradhan, S.C.: Small-scale effect on the vibration of nonuniform nanocantilever based on nonlocal elasticity theory. Physica E 41, 1451–1456 (2009)

    Article  Google Scholar 

  145. Murmu, T., Pradhan, S.C.: Thermo-mechanical vibration of a single-walled carbon nanotube embedded in an elastic medium based on nonlocal elasticity theory. Comput. Mater. Sci. 46, 854–859 (2009)

    Article  Google Scholar 

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

    Article  Google Scholar 

  147. Murmu, T., Adhikari, S.: Nonlocal vibration of carbon nanotubes with attached buckyballs at tip. Mech. Res. Commun. 38, 62–67 (2011)

    Article  MATH  Google Scholar 

  148. Murmu, T., Adhikari, S., Wang, C.Y.: Torsional vibration of carbon nanotube-buckyball systems based on nonlocal elasticity theory. Physica E 43, 1276–1280 (2011)

    Article  Google Scholar 

  149. Simsek, M.: Nonlocal effects in the forced vibration of an elastically connected double-carbon nanotube system under a moving nanoparticle. Comput. Mater. Sci. 50, 2112–2123 (2011)

    Article  Google Scholar 

  150. Aydogdu, M., Filiz, S.: Modeling carbon nanotube-based mass sensors using axial vibration and nonlocal elasticity. Physica E 43, 1229–1234 (2011)

    Article  Google Scholar 

  151. Lee, H.L., Chang, W.J.: Frequency analysis of carbon-nanotube-based mass sensor using non-local Timoshenko beam theory. Micro Nano Lett. 7, 86–89 (2012)

    Article  Google Scholar 

  152. Wang, K.F., Wang, B.L.: Vibration modeling of carbon nanotube based biosensors incorporating thermal and nonlocal effects. J. Vib. Control. (2014). doi:10.1177/1077546314534718

  153. Adhikari, S., Chowdhury, R.: Zeptogram sensing from gigahertz vibration: Graphene based nanosensor. Physica E 44, 1528–1534 (2012)

    Article  Google Scholar 

  154. Lu, P., Zhang, P.Q., Lee, H.P., et al.: Non-local elastic plate theories. Proc. R. Soc. A 463, 3225–3240 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  155. Pradhan, S.C., Phadikar, J.K.: Nonlocal elasticity theory for vibration of nanoplates. J. Sound Vib. 325, 206–223 (2009)

    Article  MATH  Google Scholar 

  156. Wang, Y.Z., Li, F.M., Kishimoto, K.: Wave propagation characteristics in fluid-conveying double-walled nanotubes with scale effects. Comput. Mater. Sci. 48, 413–418 (2010)

    Article  Google Scholar 

  157. Pradhan, S.C., Kumar, A.: Vibration analysis of orthotropic graphene sheets using nonlocal elasticity theory and differential quadrature method. Compos. Struct. 93, 774–779 (2011)

    Article  Google Scholar 

  158. Murmu, T., Pradhan, S.C.: Vibration analysis of nano-single-layered graphene sheets embedded in elastic medium based on nonlocal elasticity theory. J. Appl. Phys. 105, 064319 (2009)

    Article  Google Scholar 

  159. Ansari, R., Rajabiehfard, R., Arash, B.: Nonlocal finite element model for vibrations of embedded multi-layered graphene sheets. Comput. Mater. Sci. 49, 831–838 (2010)

    Article  Google Scholar 

  160. Yang, X.D., Lim, C.W.: Nonlinear vibrations of nano-beams accounting for nonlocal effect using a multiple scale method. Sci. China Ser. E 52, 617–621 (2009)

    Article  MATH  Google Scholar 

  161. 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)

    Article  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  163. 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)

    Article  Google Scholar 

  164. Shen, L., Shen, H.S., Zhang, C.L.: Nonlocal plate model for nonlinear vibration of single layer graphene sheets in thermal environments. Comput. Mater. Sci. 48, 680–685 (2010)

    Article  Google Scholar 

  165. Shen, H.S., Xu, Y.M., Zhang, C.L.: Prediction of nonlinear vibration of bilayer graphene sheets in thermal environments via molecular dynamics simulations and nonlocal elasticity. Comput. Methods Appl. Mech. Eng. 267, 458–470 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  166. Jomehzadeh, E., Saidi, A.R.: A study on large amplitude vibration of multilayered graphene sheets. Comput. Mater. Sci. 50, 1043–1051 (2011)

    Article  Google Scholar 

  167. Zhou, Z.G., Du, S.Y., Wu, L.Z.: Investigation of anti-plane shear behavior of a Griffith permeable crack in functionally graded piezoelectric materials by use of the non-local theory. Compos. Struct. 78, 575–83 (2007)

    Article  Google Scholar 

  168. Zhou, Z.G., Wang, B.: The scattering of harmonic elastic anti-plane shear waves by a Griffith crack in a piezoelectric material plane by using the non-local theory. Int. J. Eng. Sci. 40, 303–317 (2002)

    Article  Google Scholar 

  169. Wang, K.F., Wang, B.L.: The electromechanical coupling behavior of piezoelectric nanowires: Surface and small-scale effects. EPL 97, 66005 (2012)

    Article  Google Scholar 

  170. Arani, A.G., Amir, S., Shajari, A.R.: Electro-thermo-mechanical buckling of DWBNNTs embedded in bundle of CNTs using nonlocal piezoelasticity cylindrical shell theory. Compos. Part B 43, 195–203 (2012)

    Article  Google Scholar 

  171. Liu, C., Ke, L.L., Wang, Y.S.: Buckling and postbuckling of size-dependent piezoelectric Timoshenko nanobeams subject to thermo-electro-mechanical loadings. Int. J. Struct. Stab. Dy. 14, 1350067. (2014). doi:10.1142/S0219455413500673

    Article  MathSciNet  Google Scholar 

  172. Ke, L.L., Wang, Y.S.: Thermoelectric-mechanical vibration of piezoelectric nanobeams based on the nonlocal theory. Smart Mater. Struct. 21, 025018 (2012)

    Article  Google Scholar 

  173. Ke, L.L., Wang, Y.S., Wang, Z.D.: Nonlinear vibration of the piezoelectric nanobeams based on the nonlocal theory. Comput. Struct. 94, 2038–2047 (2012)

    Article  Google Scholar 

  174. Arani, A.G., Amir, S., Shajari, A.R., et al.: Electro-thermal non-local vibration analysis of embedded DWBNNTs. P. I. Mech. Eng. C-J. Mec. 2269, 1410–1422 (2012)

    Article  Google Scholar 

  175. Liu, C., Ke, L.L., Wang, Y.S., et al.: Thermo-electro-mechanical vibration of piezoelectric nanoplates based on the nonlocal theory. Comp. Struct. 106, 167–174 (2013)

    Article  Google Scholar 

  176. Alaimo, A., Bruno, M., Milazzo, A., et al.: Nonlocal model for a magneto-electro-elastic nanoplate. AIP Conf. Proc. 1558, 1208–1211 (2013)

    Article  Google Scholar 

  177. 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)

    Article  Google Scholar 

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

    Article  Google Scholar 

  179. Adhikari, S., Murmu, T., McCarthy, M.A.: Dynamic finite element analysis of axially vibrating nonlocal rods. Finite Elem. Anal. Des. 63, 42–50 (2013)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  Google Scholar 

  181. Shenoy, V.B.: Nanomechanics. In: Lakhtakia, A. (ed.) Handbook of nanotechnology: Nanometer structure theory, modeling, and simulation. SPIE Press, Washington, USA (2004). Chap. 7

    Google Scholar 

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

    Article  Google Scholar 

  183. Juntarasaid, C., Pulngern, T., Chucheepsakul, S.: Bending and buckling of nanowires including the effects of surface stress and nonlocal elasticity. Physica E 46, 68–76 (2012)

  184. Lee, H.L., Chang, W.J.: Surface and small-scale effects on vibration analysis of a nonuniform nanocantilever beam. Physica E 43, 466–469 (2010)

    Article  Google Scholar 

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

    Article  Google Scholar 

  186. Lei, X.W., Natsuki, T., Shi, J.X., et al.: Surface effects on the vibrational frequency of double-walled carbon nanotubes using the nonlocal Timoshenko beam model. Compos. Part B: Eng. 43, 64–69 (2011). 9

    Article  Google Scholar 

  187. Elishakoff, I., Soret, C.: Nonlocal refined theory for nanobeams with surface effects. Bull. Georgian Natl. Acad. Sci. 6, 59–67 (2012)

    MathSciNet  MATH  Google Scholar 

  188. Eltaher, M.A., Mahmoud, F.F., Assie, A.E., et al.: Coupling effects of nonlocal and surface energy on vibration analysis of nanobeams. Appl. Math. Comput. 224, 760–774 (2013)

    Article  MathSciNet  Google Scholar 

  189. Xu, X.J., Deng, Z.C.: Surface effects of adsorption-induced resonance analysis on micro/nanobeams via nonlocal elasticity. Appl. Math. Mech. 34, 37–44 (2013)

    Article  MathSciNet  Google Scholar 

  190. Wang, K.F., Wang, B.L.: Vibration of nanoscale plates with surface energy via nonlocal elasticity. Physica E 44, 448–453 (2011)

  191. Wang, K.F., Wang, B.L.: Combining effects of surface energy and non-local elasticity on the buckling of Nanoplates. Micro Nano Lett. 6, 941–943 (2011)

    Article  Google Scholar 

  192. Gheshlaghi, B., Hasheminejad, S.M.: Vibration analysis of piezoelectric nanowires with surface and small scale effects. Curr. Appl. Phys. 12, 1096–1099 (2012)

    Article  Google Scholar 

  193. Hosseini-Hashemi, S., Nahas, I., Fakher, M., et al.: Surface effects on free vibration of piezoelectric functionally graded nanobeams using nonlocal elasticity. Acta Mech. 225, 1555–1564 (2014)

  194. Arani, A.G., Roudbari, M.A.: Nonlocal piezoelastic surface effect on the vibration of visco-Pasternak coupled boron nitride nanotube system under a moving nanoparticle. Thin Solid Films 542, 232–241 (2013)

    Article  Google Scholar 

  195. Sumigawa, T., Hirakata, H., Takemura, M., et al.: Disappearance of stress singularity at interface edge due to nanostructured thin film. Eng. Fract. Mech. 75, 3073–3083 (2008)

    Article  Google Scholar 

  196. Kitamura, T., Hirakata, H., Itsuji, T.: Effect of residual stress on delamination from interface edge between nano-films. Eng. Fract. Mech. 70, 2089–2101 (2003)

    Article  Google Scholar 

  197. Hirakata, H., Takahashi, Y., Kitamura, T.: Role of plasticity on interface crack initiation from a free edge and propagation in a nano-component. Int. J. Fract. 145, 261–271 (2007)

    Article  Google Scholar 

  198. Hirakata, H., Kitamura, T., Yamamoto, Y.: Evaluation of interface strength of micro-dot on substrate by means of AFM. Int. J. Solids Struct. 41, 3243–3253 (2004)

    Article  Google Scholar 

  199. Takahashi, Y., Hirakata, H., Kitamura, T.: Quantitative evaluation of plasticity of a ductile nano-component. Thin Solid Films 516, 1925–1930 (2008)

    Article  Google Scholar 

  200. Kitamura, T., Hirakata, H., Satake, Y.: Applicability of fracture mechanics on brittle delamination of nanoscale film edge. JSME Int. J. Ser. A 47, 106–112 (2004)

    Article  Google Scholar 

  201. Hirakata, H., Nishijima, O., Fukuhara, N., et al.: Size effect on fracture toughness of freestanding copper nano-films. Mater. Sci. Eng. A 528, 8120–8127 (2011)

    Article  Google Scholar 

  202. Kitamura, T., Hirakata, H., Sumigawa, T., Shimada, T.: Fracture Nanomechanics. CRC Press, Boca Raton (2011)

    Google Scholar 

Download references

Acknowledgments

The project was supported by the National Natural Science Foundation of China (Grant 11372086) and the Natural Science Foundation of Guangdong Province of China (Grant 2014A030313696).

Author information

Authors and Affiliations

  1. Graduate School at Shenzhen, Harbin Institute of Technology, Harbin, 150001, China

    K. F. Wang & B. L. Wang

  2. Department of Mechanical Engineering, Kyoto University, Kyoto, Japan

    T. Kitamura

Authors
  1. K. F. Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. B. L. Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. T. Kitamura
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to K. F. Wang.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Wang, K.F., Wang, B.L. & Kitamura, T. A review on the application of modified continuum models in modeling and simulation of nanostructures. Acta Mech. Sin. 32, 83–100 (2016). https://doi.org/10.1007/s10409-015-0508-4

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10409-015-0508-4

Keywords

Search

Navigation