A Review on the Application of Nonlocal Elastic Models in Modeling of Carbon Nanotubes and Graphenes

  • Behrouz ArashEmail author
  • Quan Wang
Part of the Springer Series in Materials Science book series (SSMATERIALS, volume 188)


Recent research studies on the application of the nonlocal continuum theory in modeling of carbon nanotubes and graphene sheets are reviewed, and substantial nonlocal continuum models proposed for static and dynamic analyses of the nano-materials are introduced. The superiority of the nonlocal continuum theory to its local counterpart, and the necessity of calibration of the small-scale parameter as the key parameter revealing small-scale effects are discussed. The nonlocal beam, plate, and shell models are briefly presented and potential areas for future research are recommended. It is intended to provide an introduction to the development of the nonlocal continuum theory in modeling the nano-materials, survey the different nonlocal continuum models, and motivate further applications of the nonlocal continuum theory to nano-material modeling.


Carbon nanotubes Graphene sheets Nonlocal continuum theory Modeling and simulations Small scale effect 



This research was undertaken, in part, thanks to funding from the Canada Research Chairs Program (CRC) and the National Science and Engineering Research Council (NSERC).


  1. Adali S (2008) Variational principles for multi-walled carbon nanotubes undergoing buckling based on nonlocal elasticity theory. Phys Lett A 372(35):5701–5705ADSzbMATHGoogle Scholar
  2. Amara K, Tounsi A, Mechab I, Adda-Bedia EA (2010) Nonlocal elasticity effect on column buckling of multiwalled carbon nanotubes under temperature field. Appl Math Model 34(12):3933–3942MathSciNetzbMATHGoogle Scholar
  3. Ansari R, Arash B (2013) Nonlocal Fl[u-umlaut]gge shell model for vibrations of double-walled carbon nanotubes with different boundary conditions. J Appl Mech 80(2):021006–021012Google Scholar
  4. Ansari R, Rajabiehfard R, Arash B (2010a) Nonlocal finite element model for vibrations of embedded multi-layered graphene sheets. Comput Mater Sci 49(4):831–838Google Scholar
  5. Ansari R, Sahmani S, Arash B (2010b) Nonlocal plate model for free vibrations of single-layered graphene sheets. Phys Lett A 375(1):53–62ADSGoogle Scholar
  6. Ansari R, Arash B, Rouhi H (2011) Vibration characteristics of embedded multi-layered graphene sheets with different boundary conditions via nonlocal elasticity. Compos Struct 93(9):2419–2429Google Scholar
  7. Antonelli GA, Maris HJ, Malhotra SG, Harper JME (2002) Picosecond ultrasonics study of the vibrational modes of a nanostructure. J Appl Phys 91(5):3261–3267ADSGoogle Scholar
  8. Arash B, Ansari R (2010) Evaluation of nonlocal parameter in the vibrations of single-walled carbon nanotubes with initial strain. Phys E 42(8):2058–2064Google Scholar
  9. Arash B, Wang Q (2011) Vibration of single- and double-layered graphene sheets. J Nanotechnol Eng Med 2(1):011012–011017Google Scholar
  10. Arash B, Wang Q (2012) A review on the application of nonlocal elastic models in modeling of carbon nanotubes and graphenes. Comput Mater Sci 51(1):303–313Google Scholar
  11. Arash B, Wang Q, Liew KM (2012) Wave propagation in graphene sheets with nonlocal elastic theory via finite element formulation. Comput Methods Appl Mech Eng 223–224:1–9MathSciNetGoogle Scholar
  12. Aydogdu M (2009a) Axial vibration of the nanorods with the nonlocal continuum rod model. Physica E 41(5):861–864ADSGoogle Scholar
  13. Aydogdu M (2009b) A general nonlocal beam theory: Its application to nanobeam bending, buckling and vibration. Phys E 41(9):1651–1655Google Scholar
  14. Aydogdu M, Filiz S (2011) Modeling carbon nanotube-based mass sensors using axial vibration and nonlocal elasticity. Physica E 43(6):1229–1234ADSGoogle Scholar
  15. Behfar K, Naghdabadi R (2005) Nanoscale vibrational analysis of a multi-layered graphene sheet embedded in an elastic medium. Compos Sci Technol 65(7–8):1159–1164Google Scholar
  16. Bodily BH, CTS (2003) Structural and equivalent continuum properties of single-walled car-bon nanotubes. Int J Mater Prod Technol 18(4–6):381–397Google Scholar
  17. Brauns EB, Madaras ML, Coleman RS, Murphy CJ, Berg MA (2002) Complex local dynamics in DNA on the picosecond and nanosecond time scales. Phys Rev Lett 88(15):158101ADSGoogle Scholar
  18. Bunch JS, van der Zande AM, Verbridge SS, Frank IW, Tanenbaum DM, Parpia JM, Craighead HG, McEuen PL (2007) Electromechanical resonators from graphene sheets. Science 315(5811):490–493. doi: 10.1126/science.1136836 ADSGoogle Scholar
  19. ChasteJ EichlerA, MoserJ CeballosG, RuraliR BachtoldA (2012) A nanomechanical mass sensor with yoctogram resolution. Nat Nano 7(5):301–304Google Scholar
  20. Chiu H-Y, Hung P, Postma HWC, Bockrath M (2008) Atomic-scale mass sensing using carbon nanotube resonators. Nano Lett 8(12):4342–4346. doi: 10.1021/nl802181c ADSGoogle Scholar
  21. Duan WH, Wang CM, Zhang YY (2007) Calibration of nonlocal scaling effect parameter for free vibration of carbon nanotubes by molecular dynamics. J Appl Phys 101(2):024305–024307ADSGoogle Scholar
  22. Duan WH, Wang Q, Wang Q, Liew KM (2010) Modeling the instability of carbon nanotubes: from continuum mechanics to molecular dynamics. J Nanotechnol Eng Med 1(1):011001–011010Google Scholar
  23. Duan WH, Gong K, Wang Q (2011) Controlling the formation of wrinkles in a single layer graphene sheet subjected to in-plane shear. Carbon 49(9):3107–3112Google Scholar
  24. Eringen AC (1976) Nonlocal polar field models. Academic, New YorkGoogle Scholar
  25. Eringen AC (1983) On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves. J Appl Phys 54(9):4703–4710ADSGoogle Scholar
  26. Falvo MR, Clary GJ, Taylor RM, Chi V, Brooks FP, Washburn S, Superfine R (1997) Bending and buckling of carbon nanotubes under large strain. Nature 389(6651):582–584ADSGoogle Scholar
  27. Fasolino A, Los JH, Katsnelson MI (2007) Intrinsic ripples in graphene. Nat Mater 6(11):858–861ADSGoogle Scholar
  28. Filiz S, Aydogdu M (2010) Axial vibration of carbon nanotube heterojunctions using nonlocal elasticity. Comput Mater Sci 49(3):619–627Google Scholar
  29. Gao Y, Hao P (2009) Mechanical properties of monolayer graphene under tensile and compressive loading. Phys E 41(8):1561–1566Google Scholar
  30. Gibson RF, Ayorinde EO, Wen Y-F (2007) Vibrations of carbon nanotubes and their composites: A review. Compos Sci Technol 67(1):1–28Google Scholar
  31. Hao MJ, Guo XM, Wang Q (2010) Small-scale effect on torsional buckling of multi-walled carbon nanotubes. Eur J Mech A Solids 29(1):49–55MathSciNetGoogle Scholar
  32. He XQ, Kitipornchai S, Liew KM (2005) Resonance analysis of multi-layered graphene sheets used as nanoscale resonators. Nanotechnology 16(10):2086ADSGoogle Scholar
  33. Heireche H, Tounsi A, Benzair A, Maachou M, Adda Bedia EA (2008) Sound wave propagation in single-walled carbon nanotubes using nonlocal elasticity. Physica E 40(8):2791–2799ADSGoogle Scholar
  34. Hernández E, Goze C, Bernier P, Rubio A (1998) Elastic properties of C and B_{x}C_{y}N_{z} composite nanotubes. Phys Rev Lett 80(20):4502–4505ADSGoogle Scholar
  35. Hu Y-G, Liew KM, Wang Q, He XQ, Yakobson BI (2008) Nonlocal shell model for elastic wave propagation in single- and double-walled carbon nanotubes. J Mech Phys Solids 56(12):3475–3485ADSzbMATHGoogle Scholar
  36. Iijima S (1991) Helical microtubules of graphitic carbon. Nature 354(6348):56–58ADSGoogle Scholar
  37. Iijima S, Brabec C, Maiti A, Bernholc J (1996) Structural flexibility of carbon nanotubes. J Chem Phys 104(5):2089–2092ADSGoogle Scholar
  38. Khademolhosseini F, Rajapakse RKND, Nojeh A (2010) Torsional buckling of carbon nanotubes based on nonlocal elasticity shell models. Comput Mater Sci 48(4):736–742Google Scholar
  39. Kiani K (2010) Longitudinal and transverse vibration of a single-walled carbon nanotube subjected to a moving nanoparticle accounting for both nonlocal and inertial effects. Phys E 42(9):2391–2401Google Scholar
  40. Kiani K, Mehri B (2010) Assessment of nanotube structures under a moving nanoparticle using nonlocal beam theories. J Sound Vib 329(11):2241–2264ADSGoogle Scholar
  41. Kitipornchai S, He XQ, Liew KM (2005) Continuum model for the vibration of multilayered graphene sheets. Phys Rev B 72(7):075443ADSGoogle Scholar
  42. Krishnan A, Dujardin E, Ebbesen TW, Yianilos PN, Treacy MMJ (1998) Young’s modulus of single-walled nanotubes. Phys Rev B 58(20):14013–14019ADSGoogle Scholar
  43. Lau K-t GuC, Hui D (2006) A critical review on nanotube and nanotube/nanoclay related polymer composite materials. Compos B Eng 37(6):425–436Google Scholar
  44. Lee H-L, Chang W-J (2009) Vibration analysis of a viscous-fluid-conveying single-walled carbon nanotube embedded in an elastic medium. Phys E 41(4):529–532Google Scholar
  45. Lee H-L, Hsu J-C, Chang W-J (2010) Frequency shift of carbon-nanotube-based mass sensor using nonlocal elasticity theory. Nanoscale Res Lett 5(11):1774–1778. doi: 10.1007/s11671-010-9709-8 ADSGoogle Scholar
  46. Li C, Chou T-W (2003a) A structural mechanics approach for the analysis of carbon nanotubes. Int J Solids Struct 40(10):2487–2499zbMATHGoogle Scholar
  47. Li C, Chou T-W (2003b) Single-walled carbon nanotubes as ultrahigh frequency nanomechanical resonators. Physical Review B 68(7):073405ADSGoogle Scholar
  48. Li C, Chou T-W (2006) Elastic wave velocities in single-walled carbon nanotubes. Phys Rev B 73(24):245407ADSGoogle Scholar
  49. Li R, Kardomateas GA (2007a) Thermal buckling of multi-walled carbon nanotubes by nonlocal elasticity. J Appl Mech 74(3):399–405zbMATHGoogle Scholar
  50. Li R, Kardomateas GA (2007b) Vibration characteristics of multiwalled carbon nanotubes embedded in elastic media by a nonlocal elastic shell model. J Appl Mech 74(6):1087–1094Google Scholar
  51. Liew KM, Wang Q (2007) Analysis of wave propagation in carbon nanotubes via elastic shell theories. Int J Eng Sci 45(2–8):227–241Google Scholar
  52. Liew KM, Wong CH, He XQ, Tan MJ, Meguid SA (2004) Nanomechanics of single and multiwalled carbon nanotubes. Phys Rev B 69(11):115429ADSGoogle Scholar
  53. Liew KM, He XQ, Kitipornchai S (2006) Predicting nanovibration of multi-layered graphene sheets embedded in an elastic matrix. Acta Mater 54(16):4229–4236Google Scholar
  54. Lu Q, Huang R (2009) Nonlinear mechanics of single-atomic-layer graphene sheets. Int J Appl Mech 01(03):443–467Google Scholar
  55. Mohammadimehr M, Saidi AR, Ghorbanpour Arani A, Arefmanesh A, Han Q (2010) Torsional buckling of a DWCNT embedded on winkler and pasternak foundations using nonlocal theory. J Mech Sci Technol 24(6):1289–1299Google Scholar
  56. Murmu T, Adhikari S (2010) Nonlocal effects in the longitudinal vibration of double-nanorod systems. Phys E 43(1):415–422Google Scholar
  57. Murmu T, Pradhan SC (2009a) Buckling analysis of a single-walled carbon nanotube embedded in an elastic medium based on nonlocal elasticity and Timoshenko beam theory and using DQM. Phys E 41(7):1232–1239Google Scholar
  58. Murmu T, Pradhan SC (2009b) Buckling of biaxially compressed orthotropic plates at small scales. Mech Res Commun 36(8):933–938zbMATHGoogle Scholar
  59. Murmu T, Pradhan SC (2009c) Thermo-mechanical vibration of a single-walled carbon nanotube embedded in an elastic medium based on nonlocal elasticity theory. Comput Mater Sci 46(4):854–859Google Scholar
  60. Murmu T, Pradhan SC (2009d) Small-scale effect on the free in-plane vibration of nanoplates by nonlocal continuum model. Phys E 41(8):1628–1633Google Scholar
  61. Narendar S, Gopalakrishnan S (2009) Nonlocal scale effects on wave propagation in multi-walled carbon nanotubes. Comput Mater Sci 47(2):526–538Google Scholar
  62. Novoselov KS, Geim AK, Morozov SV, Jiang D, Zhang Y, Dubonos SV, Grigorieva IV, Firsov AA (2004) Electric field effect in atomically thin carbon films. Science 306(5696):666–669. doi: 10.1126/science.1102896 ADSGoogle Scholar
  63. Parnes R, Chiskis A (2002) Buckling of nano-fibre reinforced composites: a re-examination of elastic buckling. J Mech Phys Solids 50(4):855–879ADSzbMATHGoogle Scholar
  64. Peddieson J, Buchanan GR, McNitt RP (2003) Application of nonlocal continuum models to nanotechnology. Int J Eng Sci 41(3–5):305–312Google Scholar
  65. Pradhan SC (2009) Buckling of single layer graphene sheet based on nonlocal elasticity and higher order shear deformation theory. Phys Lett A 373(45):4182–4188ADSzbMATHGoogle Scholar
  66. Pradhan SC, Kumar A (2010) Vibration analysis of orthotropic graphene sheets embedded in Pasternak elastic medium using nonlocal elasticity theory and differential quadrature method. Comput Mater Sci 50(1):239–245Google Scholar
  67. Pradhan SC, Kumar A (2011a) Buckling analysis of single layered graphene sheet under biaxial compression using nonlocal elasticity theory and DQ method. J Comput Theory Nanosci 8(7):1325–1334Google Scholar
  68. Pradhan SC, Kumar A (2011b) Vibration analysis of orthotropic graphene sheets using nonlocal elasticity theory and differential quadrature method. Compos Struct 93(2):774–779Google Scholar
  69. Pradhan SC, Murmu T (2009) Small scale effect on the buckling of single-layered graphene sheets under biaxial compression via nonlocal continuum mechanics. Comput Mater Sci 47(1):268–274Google Scholar
  70. Pradhan SC, Murmu T (2010a) Small scale effect on the buckling analysis of single-layered graphene sheet embedded in an elastic medium based on nonlocal plate theory. Phys E 42(5):1293–1301Google Scholar
  71. Pradhan SC, Murmu T (2010b) Application of nonlocal elasticity and DQM in the flapwise bending vibration of a rotating nanocantilever. Phys E 42(7):1944–1949Google Scholar
  72. Pradhan SC, Phadikar JK (2009) Nonlocal elasticity theory for vibration of nanoplates. J Sound Vib 325(1–2):206–223ADSGoogle Scholar
  73. Qian D, Wagner GJ, Liu WK, Yu M-F, Ruoff RS (2002) Mechanics of carbon nanotubes. Appl Mech Rev 55(6):495–533ADSGoogle Scholar
  74. Reddy JN (2007) Nonlocal theories for bending, buckling and vibration of beams. Int J Eng Sci 45(2–8):288–307zbMATHGoogle Scholar
  75. Sánchez-Portal D, Artacho E, Soler JM, Rubio A, Ordejón P (1999) Ab initio structural, elastic, and vibrational properties of carbon nanotubes. Phys Rev B 59(19):12678–12688ADSGoogle Scholar
  76. Schedin F, Geim AK, Morozov SV, Hill EW, Blake P, Katsnelson MI, Novoselov KS (2007) Detection of individual gas molecules adsorbed on graphene. Nat Mater 6(9):652–655ADSGoogle Scholar
  77. Shen H-S (2010a) Buckling and postbuckling of radially loaded microtubules by nonlocal shear deformable shell model. J Theor Biol 264(2):386–394ADSGoogle Scholar
  78. Shen H-S (2010b) Nonlocal shear deformable shell model for bending buckling of microtubules embedded in an elastic medium. Phys Lett A 374(39):4030–4039ADSzbMATHGoogle Scholar
  79. Shen H-S, Zhang C-L (2010) Torsional buckling and postbuckling of double-walled carbon nanotubes by nonlocal shear deformable shell model. Compos Struct 92(5):1073–1084MathSciNetGoogle Scholar
  80. Shen L, Shen H-S, Zhang C-L (2010) Nonlocal plate model for nonlinear vibration of single layer graphene sheets in thermal environments. Comput Mater Sci 48(3):680–685Google Scholar
  81. Şimşek M (2010) Vibration analysis of a single-walled carbon nanotube under action of a moving harmonic load based on nonlocal elasticity theory. Physica E 43(1):182–191ADSGoogle Scholar
  82. Sirtori C (2002) Applied physics: bridge for the terahertz gap. Nature 417(6885):132–133ADSGoogle Scholar
  83. Soltani P, Taherian MM, Farshidianfar A (2010) Vibration and instability of a viscous-fluid-conveying single-walled carbon nanotube embedded in a visco-elastic medium. J Phys D Appl Phys 43(42):425401ADSGoogle Scholar
  84. Song J, Shen J, Li XF (2010) Effects of initial axial stress on waves propagating in carbon nanotubes using a generalized nonlocal model. Comput Mater Sci 49(3):518–523Google Scholar
  85. Stankovich S, Dikin DA, Dommett GHB, Kohlhaas KM, Zimney EJ, Stach EA, Piner RD, Nguyen ST, Ruoff RS (2006) Graphene-based composite materials. Nature 442(7100):282–286ADSGoogle Scholar
  86. Sudak LJ (2003) Column buckling of multiwalled carbon nanotubes using nonlocal continuum mechanics. J Appl Phys 94(11):7281–7287ADSGoogle Scholar
  87. Sun C, Liu K (2007) Vibration of multi-walled carbon nanotubes with initial axial loading. Solid State Commun 143(4–5):202–207ADSGoogle Scholar
  88. Thostenson ET, Ren Z, Chou T-W (2001) Advances in the science and technology of carbon nanotubes and their composites: a review. Compos Sci Technol 61(13):1899–1912Google Scholar
  89. Wagner HD, Lourie O, Feldman Y, Tenne R (1998) Stress-induced fragmentation of multiwall carbon nanotubes in a polymer matrix. Appl Phys Lett 72(2):188–190ADSGoogle Scholar
  90. Wang Q (2005) Wave propagation in carbon nanotubes via nonlocal continuum mechanics. J Appl Phys 98(12):124301–124306ADSGoogle Scholar
  91. Wang L (2009) Vibration and instability analysis of tubular nano- and micro-beams conveying fluid using nonlocal elastic theory. Phys E 41(10):1835–1840Google Scholar
  92. Wang L, Hu H (2005) Flexural wave propagation in single-walled carbon nanotubes. Phys Rev B 71(19):195412ADSGoogle Scholar
  93. Wang Q, Liew KM (2007) Application of nonlocal continuum mechanics to static analysis of micro- and nano-structures. Phys Lett A 363(3):236–242ADSGoogle Scholar
  94. Wang Q, Varadan VK (2006a) Wave characteristics of carbon nanotubes. Int J Solids Struct 43(2):254–265zbMATHGoogle Scholar
  95. Wang Q, Varadan VK (2006b) Vibration of carbon nanotubes studied using nonlocal continuum mechanics. Smart Mater Struct 15(2):659ADSGoogle Scholar
  96. Wang Q, Varadan VK (2007) Application of nonlocal elastic shell theory in wave propagation analysis of carbon nanotubes. Smart Mater Struct 16(1):178ADSGoogle Scholar
  97. Wang Q, Wang CM (2007) The constitutive relation and small scale parameter of nonlocal continuum mechanics for modelling carbon nanotubes. Nanotechnology 18(7):075702ADSGoogle Scholar
  98. Wang X, Yang HK, Dong K (2005) Torsional buckling of multi-walled carbon nanotubes. Mater Sci Eng, A 404(1–2):314–322Google Scholar
  99. Wang CM, Zhang YY, Sai Sudha R, Kitipornchai S (2006a) Buckling analysis of micro- and nano-rods/tubes based on nonlocal Timoshenko beam theory. J Phys D Appl Phys 39(17):3904ADSGoogle Scholar
  100. Wang Q, Varadan VK, Quek ST (2006b) Small scale effect on elastic buckling of carbon nanotubes with nonlocal continuum models. Phys Lett A 357(2):130–135ADSGoogle Scholar
  101. Wang Q, Zhou GY, Lin KC (2006c) Scale effect on wave propagation of double-walled carbon nanotubes. Int J Solids Struct 43(20):6071–6084zbMATHGoogle Scholar
  102. Wang CM, Zhang YY, He XQ (2007) Vibration of nonlocal Timoshenko beams. Nanotechnology 18(10):105401ADSGoogle Scholar
  103. Wang Y-Z, Li F-M, Kishimoto K (2010a) Scale effects on thermal buckling properties of carbon nanotube. Phys Lett A 374(48):4890–4893ADSzbMATHGoogle Scholar
  104. Wang Y-Z, Li F-M, Kishimoto K (2010b) Wave propagation characteristics in fluid-conveying double-walled nanotubes with scale effects. Comput Mater Sci 48(2):413–418Google Scholar
  105. Wang Y-Z, Li F-M, Kishimoto K (2010c) Scale effects on the longitudinal wave propagation in nanoplates. Phys E 42(5):1356–1360Google Scholar
  106. Xie GQ, Han X, Liu GR, Long SY (2006) Effect of small size-scale on the radial buckling pressure of a simply supported multi-walled carbon nanotube. Smart Mater Struct 15(4):1143ADSGoogle Scholar
  107. Yakobson BI, Brabec CJ, Bernholc J (1996) Nanomechanics of carbon tubes: instabilities beyond linear response. Phys Rev Lett 76(14):2511–2514ADSGoogle Scholar
  108. Yakobson BI, Campbell MP, Brabec CJ, Bernholc J (1997) High strain rate fracture and C-chain unraveling in carbon nanotubes. Comput Mater Sci 8(4):341–348Google Scholar
  109. Yan Y, Wang WQ, Zhang LX (2010) Nonlocal effect on axially compressed buckling of triple-walled carbon nanotubes under temperature field. Appl Math Model 34(11):3422–3429MathSciNetzbMATHGoogle Scholar
  110. Yang J, Jia XL, Kitipornchai S (2008) Pull-in instability of nano-switches using nonlocal elasticity theory. J Phys D Appl Phys 41(3):035103ADSGoogle Scholar
  111. Yang J, Ke LL, Kitipornchai S (2010) Nonlinear free vibration of single-walled carbon nanotubes using nonlocal Timoshenko beam theory. Phys E 42(5):1727–1735Google Scholar
  112. Zhang YQ, Liu GR, Wang JS (2004) Small-scale effects on buckling of multiwalled carbon nanotubes under axial compression. Phys Rev B 70(20):205430ADSGoogle Scholar
  113. Zhang YQ, Liu GR, Xie XY (2005) Free transverse vibrations of double-walled carbon nanotubes using a theory of nonlocal elasticity. Phys Rev B 71(19):195404ADSGoogle Scholar
  114. Zhang YQ, Liu GR, Han X (2006) Effect of small length scale on elastic buckling of multi-walled carbon nanotubes under radial pressure. Phys Lett A 349(5):370–376ADSGoogle Scholar
  115. Zhang YY, Wang CM, Duan WH, Xiang Y, Zong Z (2009a) Assessment of continuum mechanics models in predicting buckling strains of single-walled carbon nanotubes. Nanotechnology 20(39):395707ADSGoogle Scholar
  116. Zhang YY, Wang CM, Tan VBC (2009b) Assessment of Timoshenko beam models for vibrational behavior of single-walled carbon nanotubes using molecular dynamics. Adv Appl Math Mech 1(1):89–106MathSciNetGoogle Scholar
  117. Zhang Y, Wang CM, Challamel N (2010) Bending, buckling, and vibration of micro/nanobeams by hybrid nonlocal beam model. J Eng Mech 136(5):562–574Google Scholar
  118. Zhen Y, Fang B (2010) Thermal–mechanical and nonlocal elastic vibration of single-walled carbon nanotubes conveying fluid. Comput Mater Sci 49(2):276–282Google Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  1. 1.Department of Mechanical and Manufacturing EngineeringUniversity of ManitobaWinnipegCanada

Personalised recommendations