On the Free Vibrations of Piezoelectric Carbon Nanotube-Reinforced Microbeams: A Multiscale Finite Element Approach

  • M. Ahmadi
  • R. Ansari
  • H. RouhiEmail author
Research Paper


In this article, the vibrational behavior of beam-type microstructures made of carbon nanotube-reinforced composite is studied based on a finite element approach accounting for micro-/nanoscale effects. It is considered that the surface of microbeams is perfectly bonded with a piezoelectric actuator layer. First, the random distribution of CNTs into the polymer matrix is modeled using a three-phase representative volume element (RVE), and the properties of CNT-reinforced polymer are determined for various volume fractions of CNT. In the selected RVE, the interphase region formed due to the interaction between CNTs and the matrix is taken into account. In the next step, natural frequencies of composite piezoelectric microbeams subject to different end conditions are calculated. The influences of CNT volume fraction, interphase, boundary conditions and geometrical properties on the results are investigated.


Carbon nanotube-reinforced microbeam Multiscale approach Piezoelectric layer Free vibration 


  1. Ansari R, Hassanzadeh Aghdam MK (2016) Micromechanics-based viscoelastic analysis of carbon nanotube-reinforced composites subjected to uniaxial and biaxial loading. Compos Part B Eng 90:512–522CrossRefGoogle Scholar
  2. Ansari R, Rouhi H, Sahmani S (2011a) Calibration of the analytical nonlocal shell model for vibrations of double-walled carbon nanotubes with arbitrary boundary conditions using molecular dynamics. Int J Mech Sci 53:786–792CrossRefGoogle Scholar
  3. Ansari R, Rouhi H, Sahmani S (2011b) Thermal effect on axial buckling behavior of multi-walled carbon nanotubes based on nonlocal shell model. Physica E 44:373–378CrossRefGoogle Scholar
  4. Ansari R, Gholami R, Rouhi H (2012) Vibration analysis of single-walled carbon nanotubes using different gradient elasticity theories. Compos Part B Eng 43:2985–2989CrossRefGoogle Scholar
  5. Ansari R, Shahabodini A, Rouhi H, Alipour A (2013a) Thermal buckling analysis of multi-walled carbon nanotubes through a nonlocal shell theory incorporating interatomic potentials. J Therm Stress 36:56–70CrossRefGoogle Scholar
  6. Ansari R, Shahabodini A, Rouhi H (2013b) A thickness-independent nonlocal shell model for describing the stability behavior of carbon nanotubes under compression. Compos Struct 100:323–331CrossRefGoogle Scholar
  7. Ansari R, Rouhi H, Mirnezhad M (2014a) A hybrid continuum and molecular mechanics model for the axial buckling of chiral single-walled carbon nanotubes. Curr Appl Phys 14:1360–1368CrossRefGoogle Scholar
  8. Ansari R, Rouhi H, Sahmani S (2014b) Free vibration analysis of single- and double-walled carbon nanotubes based on nonlocal elastic shell models. J Vib Control 20:670–678MathSciNetCrossRefzbMATHGoogle Scholar
  9. Ansari R, Faghih Shojaei M, Mohammadi V, Gholami R, Sadeghi F (2014c) Nonlinear forced vibration analysis of functionally graded carbon nanotube-reinforced composite Timoshenko beams. Compos Struct 113:316–327CrossRefGoogle Scholar
  10. Ansari R, Mirnezhad M, Rouhi H (2015) Torsional buckling analysis of chiral multi-walled carbon nanotubes based on an accurate molecular mechanics model. Acta Mech 226:2955–2972MathSciNetCrossRefzbMATHGoogle Scholar
  11. Arash B, Wang Q, Varadan VK (2014) Mechanical properties of carbon nanotube/polymer composites. Sci Rep 4:6479CrossRefGoogle Scholar
  12. Bellucci S (2005) Carbon nanotubes: physics and applications. Phys Status Solidi (c) 2:34–47CrossRefGoogle Scholar
  13. Chalioris CE, Karayannis CG, Angeli GM, Papadopoulos NA, Favvata MJ, Providakis CP (2016) Applications of smart piezoelectric materials in a wireless admittance monitoring system (WiAMS) to structures: tests in RC elements. Case Stud Constr Mater 5:1–18Google Scholar
  14. Charlier JC, Roche S (2007) Electronic and transport properties of nanotubes. Rev Mod Phys 79:677–732CrossRefGoogle Scholar
  15. Coleman JN, Khan U, Blau WJ, Gun’ko YK (2006) Small but strong: a review of the mechanical properties of carbon nanotube–polymer composites. Carbon 44:1624–1652CrossRefGoogle Scholar
  16. Demczyk BG, Wang YM, Cumings J, Hetman M, Han W, Zettl A, Ritchie RO (2002) Direct mechanical measurement of the tensile strength and elastic modulus of multiwalled carbon nanotubes. Mater Sci Eng A 334:173–178CrossRefGoogle Scholar
  17. Dinçkal Ç (2016) Free vibration analysis of carbon nanotubes by using finite element method. Iran J Sci Technol Trans Mech Eng 40:43–55CrossRefGoogle Scholar
  18. Esawi AMK, Morsi K, Sayed A, Taher M, Lanka S (2010) Effect of carbon nanotube (CNT) content on the mechanical properties of CNT-reinforced aluminium composites. Compos Sci Technol 70:2237–2241CrossRefGoogle Scholar
  19. Ghorbanpour Arani A, Haghparast E, Heidari Rarani M, Khoddami Maraghi Z (2015) Strain gradient shell model for nonlinear vibration analysis of visco-elastically coupled Boron Nitride nano-tube reinforced composite micro-tubes conveying viscous fluid. Comput Mater Sci 96:448–458CrossRefGoogle Scholar
  20. Ghorbanpour Arani A, Mosayyebi M, Kolahdouzan F, Jamali M (2016) Refined zigzag theory for vibration analysis of viscoelastic functionally graded carbon nanotube reinforced composite microplates integrated with piezoelectric layers. Proc IMechE Part G J Aerosp Eng. Google Scholar
  21. Giannopoulos GI, Georgantzinos SK, Anifantis NK (2010) A semi-continuum finite element approach to evaluate the Young’s modulus of single-walled carbon nanotube reinforced composites. Compos Part B Eng 41:594–601CrossRefGoogle Scholar
  22. Iijima S (1991) Helical microtubes of graphitic carbon. Nature 354:56–58CrossRefGoogle Scholar
  23. Ke LL, Yang J, Kitipornchai S (2010) nonlinear free vibration of functionally graded carbon nanotube-reinforced composite beams. Compos Struct 92:676–683CrossRefGoogle Scholar
  24. Ke LL, Yang J, Kitipornchai S (2013) Dynamic stability of functionally graded carbon nanotube-reinforced composite beams. Mech Adv Mater Struct 20:28–37CrossRefGoogle Scholar
  25. Legoas SB, Coluci VR, Braga SF, Coura PZ, Dantas SO, Galvão DS (2004) Gigahertz nanomechanical oscillators based on carbon nanotubes. Nanotechnology 15:S184CrossRefGoogle Scholar
  26. Lei ZX, Zhang LW, Liew KM (2016) Parametric analysis of frequency of rotating laminated CNT reinforced functionally graded cylindrical panels. Compos Part B Eng 90:251–266CrossRefGoogle Scholar
  27. Li K, Gao XL, Roy AK (2006) Micromechanical modeling of viscoelastic properties of carbon nanotube-reinforced polymer composites. Mech Adv Mater Struct 13:317–328CrossRefGoogle Scholar
  28. Mehar K, Panda SK, Dehengia A, Kar VR (2016) Vibration analysis of functionally graded carbon nanotube reinforced composite plate in thermal environment. J Sandw Struct Mater 18:151–173CrossRefGoogle Scholar
  29. Odegard GM, Clancy TC, Gates TS (2005) Modeling of the mechanical properties of nanoparticle/polymer composites. Polymer 46:553–562CrossRefGoogle Scholar
  30. Paradise M, Goswami T (2007) Carbon nanotubes: production and industrial applications. Mater Des 28:1477–1489CrossRefGoogle Scholar
  31. Phung-Van P, Lieu QX, Nguyen-Xuan H, Abdel Wahab M (2017) Size-dependent isogeometric analysis of functionally graded carbon nanotube-reinforced composite nanoplates. Compos Struct 166:120–135CrossRefGoogle Scholar
  32. Pop E, Mann D, Wang Q, Goodson K, Dai H (2006) Thermal conductance of an individual single-wall carbon nanotube above room temperature. Nano Lett 6:96–100CrossRefGoogle Scholar
  33. Quakad M, Sedighi HM (2016) Rippling effect on the structural response of electrostatically actuated single-walled carbon nanotube based NEMS actuators. Int J Non-Linear Mech 87:97–108CrossRefGoogle Scholar
  34. Rezaei MP, Zamanian M (2017) A two-dimensional vibration analysis of piezoelectrically actuated microbeam with nonideal boundary conditions. Physica E 85:285–293CrossRefGoogle Scholar
  35. Rokni H, Milani AS, Seethaler RJ (2015) Size-dependent vibration behavior of functionally graded CNT-Reinforced polymer microcantilevers: modeling and optimization. Eur J Mech A/Solids 49:26–34MathSciNetCrossRefzbMATHGoogle Scholar
  36. Selmi A, Friebel C, Doghri I, Hassis H (2007) Prediction of the elastic properties of single walled carbon nanotube reinforced polymers: a comparative study of several micromechanical models. Compos Sci Technol 67:2071–2084CrossRefGoogle Scholar
  37. Shokrieh MM, Rafiee R (2010) A review of the mechanical properties of isolated carbon nanotubes and carbon nanotube composites. Mech Compos Mater 46:155–172CrossRefGoogle Scholar
  38. Sinnott SB, Andrews R (2001) Carbon nanotubes: synthesis, properties, and applications. Crit Rev Solid State Mater Sci 26:145–249CrossRefGoogle Scholar
  39. Tang ZK, Zhang L, Wang N, Zhang XX, Wen GH, Li GD, Wang JN, Chan CT, Sheng P (2001) Superconductivity in 4 angstrom single-walled carbon nanotubes. Science 292:2462–2465CrossRefGoogle Scholar
  40. Tsai JL, Tzeng SH, Chiu YT (2010) Characterizing elastic properties of carbon nanotubes/polyimide nanocomposites using multi-scale simulation. Compos Part B 41:106–115CrossRefGoogle Scholar
  41. Uchino K (ed) (2017) Advanced piezoelectric materials, 2nd edn. Science and Technology. Paperback ISBN: 9780081012543. Imprint: Woodhead Publishing. Published date: 1st July 2017Google Scholar
  42. Wan H, Delale F, Shen L (2005) Effect of CNT length and CNT-matrix interphase in carbon nanotube (CNT) reinforced composites. Mech Res Commun 32:481–489CrossRefzbMATHGoogle Scholar
  43. Yang WD, Wang X (2016) Nonlinear pull-in instability of carbon nanotubes reinforced nano-actuator with thermally corrected Casimir force and surface effect. Int J Mech Sci 107:34–42CrossRefGoogle Scholar
  44. Yang WD, Yang FP, Wang X (2016) Coupling influences of nonlocal stress and strain gradients on dynamic pull-in of functionally graded nanotubes reinforced nano-actuator with damping effects. Sens Actuators A Phys 248:10–21CrossRefGoogle Scholar
  45. Yang WD, Fang CQ, Wang X (2017) Nonlinear dynamic characteristics of FGCNTs reinforced microbeam with piezoelectric layer based on unifying stress-strain gradient framework. Compos Part B 111:372–386CrossRefGoogle Scholar
  46. Yu MF, Lourie O, Dyer MJ, Moloni K, Kelly TF, Ruoff RS (2000) Strength and breaking mechanism of multiwalled carbon nanotubes under tensile load. Science 287:637–640CrossRefGoogle Scholar
  47. Zare J, Shateri A (2017) Instability threshold of rippled carbon nanotube nanotweezers in the low vacuum gas flow incorporating Dirichlet and Neumann modes of Casimir energy. Physica E 90:67–75CrossRefGoogle Scholar

Copyright information

© Shiraz University 2018

Authors and Affiliations

  1. 1.Department of Mechanical EngineeringUniversity of GuilanRashtIran
  2. 2.Department of Engineering Science, Faculty of Technology and Engineering, East of GuilanUniversity of GuilanRudsar-VajargahIran

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