Acta Mechanica Solida Sinica

, Volume 30, Issue 4, pp 416–424 | Cite as

Nonlinear vibration analysis of fractional viscoelastic Euler—Bernoulli nanobeams based on the surface stress theory

  • M. Faraji Oskouie
  • R. Ansari
  • F. Sadeghi


The nonlinear vibrations of viscoelastic Euler—Bernoulli nanobeams are studied using the fractional calculus and the Gurtin—Murdoch theory. Employing Hamilton’s principle, the governing equation considering surface effects is derived. The fractional integro-partial differential governing equation is first converted into a fractional—ordinary differential equation in the time domain using the Galerkin scheme. Thereafter, the set of nonlinear fractional time-dependent equations expressed in a state-space form is solved using the predictor—corrector method. Finally, the effects of initial displacement, fractional derivative order, vis-coelasticity coefficient, surface parameters and thickness-to-length ratio on the nonlinear time response of simply-supported and clamped-free silicon viscoelastic nanobeams are investigated.


Fractional calculus Viscoelastic nanobeam Nonlinear vibrations 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    J.T. Machado, A probabilistic interpretation of the fractional-order differentiation, Fract. Calc. Appl. Anal. 6 (2003) 73–80.MathSciNetzbMATHGoogle Scholar
  2. 2.
    N. Heymans, Dynamic measurements in long-memory materials: fractional calculus evaluation of approach to steady state, J. Vib. Control 14 (2008) 1587–1596.CrossRefGoogle Scholar
  3. 3.
    J.J. De Espíndola, C.A. Bavastri, E.M.D.O. Lopes, Design of optimum systems of viscoelastic vibration absorbers for a given material based on the fractional calculus model, J. Vib. Control 14 (2008) 1607–1630.MathSciNetCrossRefGoogle Scholar
  4. 4.
    R.L. Magin, M. Ovadia, Modeling the cardiac tissue electrode interface using fractional calculus, J. Vib. Control 14 (2008) 1431–1442.CrossRefGoogle Scholar
  5. 5.
    J.T. Machado, A.C. Costa, M.D. Quelhas, Fractional dynamics in DNA, Commun. Nonlinear Sci. Numer. Simul. 16 (2011) 2963–2969.CrossRefGoogle Scholar
  6. 6.
    M.P. Lazarević, Finite time stability analysis of PD α fractional control of robotic time-delay systems, Mech. Res. Commun. 33 (2006) 269–279.MathSciNetCrossRefGoogle Scholar
  7. 7.
    J. Cervera, A. Baños, Automatic loop shaping in QFT using CRONE structures, J. Vib. Control 14 (2008) 1513–1529.MathSciNetCrossRefGoogle Scholar
  8. 8.
    B.M. Vinagre, I. Podlubny, A. Hernandez, V. Feliu, Some approximations of fractional order operators used in control theory and applications, Fract. Calc. Appl. Anal. 3 (2000) 231–248.MathSciNetzbMATHGoogle Scholar
  9. 9.
    G.S. Frederico, D.F. Torres, Fractional conservation laws in optimal control theory, Nonlinear Dyn. 53 (2008) 215–222.MathSciNetCrossRefGoogle Scholar
  10. 10.
    A.J. Calderón, B.M. Vinagre, V. Feliu, Fractional order control strategies for power electronic buck converters, Signal Process. 86 (2006) 2803–2819.CrossRefGoogle Scholar
  11. 11.
    R. Panda, M. Dash, Fractional generalized splines and signal processing, Signal Process. 86 (2006) 2340–2350.CrossRefGoogle Scholar
  12. 12.
    B.M. Vinagre, V. Feliu, Modeling and control of dynamic system using fractional calculus: application to electrochemical processes and flexible structures, in: Proceedings of the Forty-first IEEE Conference on Decision and Control, 1, 2002, pp. 214–239.Google Scholar
  13. 13.
    B. Wang, M.A. Dündar, R. Nötzel, F. Karouta, S. He, R.W. van der Heijden, Photonic crystal slot nanobeam slow light waveguides for refractive index sensing, Appl. Phys. Lett. 97 (2010) 151105.CrossRefGoogle Scholar
  14. 14.
    B.B. On, E. Altus, Stochastic surface effects in nanobeam sensors, Probab. Eng. Mech. 25 (2010) 228–234.CrossRefGoogle Scholar
  15. 15.
    J.S. Duan, R. Rach, A.M. Wazwaz, Solution of the model of beam-type micro-and nano-scale electrostatic actuators by a new modified Adomian decomposition method for nonlinear boundary value problems, Int. J. Non-Linear Mech. 49 (2013) 159–169.CrossRefGoogle Scholar
  16. 16.
    P.B. Deotare, L.C. Kogos, I. Bulu, M. Loncar, Photonic crystal nanobeam cavities for tunable filter and router applications, IEEE J. Sel. Top. Quantum Electron. 19 (2013) 3600210–3600210.CrossRefGoogle Scholar
  17. 17.
    W.S. Fegadolli, N. Pavarelli, P. O’Brien, S. Njoroge, V.R. Almeida, A. Scherer, Thermally controllable silicon photonic crystal nanobeam cavity without surface cladding for sensing applications, ACS Photonics 2 (2015) 470–474.CrossRefGoogle Scholar
  18. 18.
    S. Bauer, A. Pittrof, H. Tsuchiya, P. Schmuki, Size-effects in TiO2 nanotubes: diameter dependent anatase/rutile stabilization, Electrochem. Commun. 13 (2011) 538–541.CrossRefGoogle Scholar
  19. 19.
    S. Xiao, W. Hou, Studies of size effects on carbon nanotubes’ mechanical properties by using different potential functions, Fuller. Nanotub. Carbon Nonstruct. 14 (2006) 9–16.CrossRefGoogle Scholar
  20. 20.
    R. Chowdhury, S. Adhikari, C.Y. Wang, F. Scarpa, A molecular mechanics approach for the vibration of single-walled carbon nanotubes, Comput. Mater. Sci. 48 (2010) 730–735.CrossRefGoogle Scholar
  21. 21.
    C.T. Sun, H. Zhang, Size-dependent elastic moduli of platelike nanomaterials, J. Appl. Phys. 93 (2003) 1212–1218.CrossRefGoogle Scholar
  22. 22.
    H. Zhang, C.T. Sun, Nanoplate model for platelike nanomaterials, AIAA J. 42 (2004) 2002–2009.CrossRefGoogle Scholar
  23. 23.
    R. Ansari, M. Faraji Oskouie, R. Gholami, F. sadeghi, Thermo-electro-mechanical vibration of postbuckled piezoelectric Timoshenko nanobeams based on the nonlocal elasticity theory, Compos. Part B Eng. 89 (2016) 316–327.CrossRefGoogle Scholar
  24. 24.
    R. Ansari, M. Faraji Oskouie, R. Gholami, Size-dependent geometrically nonlinear free vibration analysis of fractional viscoelastic nanobeams based on the nonlocal elasticity theory, Phys. E Low-dimens. Syst. Nanostruct. 75 (2016) 266–271.CrossRefGoogle Scholar
  25. 25.
    B. Zhang, Y. He, D. Liu, L. Shen, J. Lei, Free vibration analysis of four-unknown shear deformable functionally graded cylindrical microshells based on the strain gradient elasticity theory, Compos. Struct. 119 (2015) 578–597.CrossRefGoogle Scholar
  26. 26.
    M. Tahani, A.R. Askari, Y. Mohandes, B. Hassani, Size-dependent free vibration analysis of electrostatically pre-deformed rectangular micro-plates based on the modified couple stress theory, Int. J. Mech. Sci. 94 (2015) 185–198.CrossRefGoogle Scholar
  27. 27.
    M. Mohammadi, M.F. Mahani, An analytical solution for buckling analysis of size-dependent rectangular micro-plates according to the modified strain gradient and couple stress theories, Acta Mech. 226 (2015) 3477–3493.MathSciNetCrossRefGoogle Scholar
  28. 28.
    R. Ansari, T. Pourashraf, R. Gholami, An exact solution for the nonlinear forced vibration of functionally graded nanobeams in thermal environment based on surface elasticity theory, Thin-Walled Struct. 93 (2015) 169–176.CrossRefGoogle Scholar
  29. 29.
    M.E. Gurtin, A.I. Murdoch, A continuum theory of elastic material surfaces, Arch. Ration. Mech. Anal. 57 (1975) 291–323.MathSciNetCrossRefGoogle Scholar
  30. 30.
    M.E. Gurtin, A.I. Murdoch, Surface stress in solids, Int. J. Solids Struct. 14 (1978) 431–440.CrossRefGoogle Scholar
  31. 31.
    Z.Q. Wang, Y.P. Zhao, Z.P. Huang, The effects of surface tension on the elastic properties of nano structures, Int. J. Eng. Sci. 48 (2010) 140–150.CrossRefGoogle Scholar
  32. 32.
    Q. Ren, Y.P. Zhao, Influence of surface stress on frequency of microcantilever-based biosensors, Microsyst. Technol. 10 (2004) 307–314.CrossRefGoogle Scholar
  33. 33.
    L.H. He, C.W. Lim, B.S. Wu, A continuum model for size-dependent deformation of elastic films of nano-scale thickness, Int. J. Solids Struct. 41 (2004) 847–857.CrossRefGoogle Scholar
  34. 34.
    G.F. Wang, X.Q. Feng, Effects of surface elasticity and residual surface tension on the natural frequency of microbeams, Appl. Phys. Lett. 90 (2007) 23.Google Scholar
  35. 35.
    R. Ansari, S. Sahmani, Bending behavior and buckling of nanobeams including surface stress effects corresponding to different beam theories, Int. J. Eng. Sci. 49 (2011) 1244–1255.CrossRefGoogle Scholar
  36. 36.
    Z. Wang, Y. Zhao, Self-instability and bending behaviors of nano plates, Acta Mech. Solida Sin. 22 (2009) 630–643.MathSciNetCrossRefGoogle Scholar
  37. 37.
    S. Sahmani, M.M. Aghdam, M. Bahrami, On the free vibration characteristics of postbuckled third-order shear deformable FGM nanobeams including surface effects, Compos. Struct. 121 (2015) 377–385.CrossRefGoogle Scholar
  38. 38.
    S.M. Pourkiaee, S.E. Khadem, M. Shahgholi, Nonlinear vibration and stability analysis of an electrically actuated piezoelectric nanobeam considering surface effects and intermolecular interactions, J. Vib. Control (2015) 1077546315603270.Google Scholar
  39. 39.
    R. Ansari, V. Mohammadi, M.F. Shojaei, R. Gholami, S. Sahmani, Surface stress effect on the postbuckling and free vibrations of axisymmetric circular Mindlin nanoplates subject to various edge supports, Compos. Struct. 112 (2014) 358–367.CrossRefGoogle Scholar
  40. 40.
    B. Gheshlaghi, S.M. Hasheminejad, Surface effects on nonlinear free vibration of nanobeams, Compos. Part B Eng. 42 (2011) 934–937.CrossRefGoogle Scholar
  41. 41.
    S.S. Rao, Vibration of Continuous Systems, John Wiley & Sons, 2007.Google Scholar
  42. 42.
    L.L. Ke, J. Yang, S. Kitipornchai, Nonlinear free vibration of functionally graded carbon nanotube-reinforced composite beams, Compos. Struct. 92 (2010) 676–683.CrossRefGoogle Scholar
  43. 43.
    H. Schiessel, R. Metzler, A. Blumen, T.F. Nonnenmacher, Generalized viscoelastic models: their fractional equations and solutions, J. Phys. A Math. Gen. 28 (1995) 6567–6584.CrossRefGoogle Scholar
  44. 44.
    M. Cajić, D. Karličić, M. Lazarević, Nonlocal vibration of a fractional order viscoelastic nanobeam with attached nanoparticle, Theor. Appl. Mech. 42 (3) (2015) 167–190.CrossRefGoogle Scholar
  45. 45.
    Y.A. Rossikhin, Reflections on two parallel ways in the progress of fractional calculus in mechanics of solids, Appl. Mech. Rev. 63 (1) (2010) 010701.CrossRefGoogle Scholar
  46. 46.
    W. Grzesikiewicz, A. Wakulicz, A. Zbiciak, Non-linear problems of fractional calculus in modeling of mechanical systems, Int. J. Mech. Sci. 70 (2013) 90–98.CrossRefGoogle Scholar
  47. 47.
    R. Ansari, M.F. Oskouie, F. Sadeghi, M. Bazdid-Vahdati, Free vibration of fractional viscoelastic Timoshenko nanobeams using the nonlocal elasticitytheory, Phys. E Low-dimens. Syst. Nanostruct. 74 (2015) 318–327.CrossRefGoogle Scholar
  48. 48.
    R. Ansari, V. Mohammadi, M. Faghih Shojaei, R. Gholami, S. Sahmani, On the forced vibration analysis of Timoshenko nanobeams based on the surface stress elasticity theory, Compos. Part B Eng. 60 (2014) 158–166.CrossRefGoogle Scholar
  49. 49.
    P. Lu, L.H. He, H.P. Lee, C. Lu, Thin plate theory including surface effects, Int. J. Solids Struct. 43 (2006) 4631–4647.CrossRefGoogle Scholar
  50. 50.
    C.F. Lu, C.W. Lim, W.Q. Chen, Size-dependent elastic behavior of FGM ultra-thin films based on generalized refined theory. 46 (2009) 1176–1185.Google Scholar
  51. 51.
    K. Sun, X. Wang, J.C. Sprott, Bifurcations and chaos in fractional-order simplified Lorenz system, Int. J. Bifurc. Chaos 20 (2010) 1209–1219.MathSciNetCrossRefGoogle Scholar
  52. 52.
    K. Diethelm, An algorithm for the numerical solution of differential equations of fractional order, Electron. Trans. Numer. Anal. 5 (1997) 1–6.MathSciNetzbMATHGoogle Scholar
  53. 53.
    K. Diethelm, N.J. Ford, Analysis of fractional differential equations, J. Math. Anal. Appl. 265 (2002) 229–248.MathSciNetCrossRefGoogle Scholar
  54. 54.
    K. Diethelm, N.J. Ford, A.D. Freed, A predictor–corrector approach for the numerical solution of fractional differential equations, Nonlinear Dyn. 29 (2002) 3–22.MathSciNetCrossRefGoogle Scholar

Copyright information

© The Chinese Society of Theoretical and Applied Mechanics and Technology 2017

Authors and Affiliations

  1. 1.Department of Mechanical EngineeringUniversity of GuilanRashtIran

Personalised recommendations