Axial Vibration of Strain Gradient Micro-rods

  • Ömer Civalek
  • Bekir Akgöz
  • Babür Deliktaş
Reference work entry


In this chapter, size-dependent axial vibration response of micro-sized rods is investigated on the basis of modified strain gradient elasticity theory. On the contrary to the classical rod model, the developed nonclassical micro-rod model includes additional material length scale parameters and can capture the size effect. If the additional material length scale parameters are equal to zero, the current model reduces to the classical one. The equation of motion together with initial conditions, classical and nonclassical corresponding boundary conditions, for micro-rods is derived by implementing Hamilton’s principle. The resulting higher-order equation is analytically solved for clamped-free and clamped-clamped boundary conditions. Finally, some illustrative examples are presented to indicate the influences of the additional material length scale parameters, size dependency, boundary conditions, and mode numbers on the natural frequencies. It is found that size effect is more significant when the micro-rod diameter is closer to the additional material length scale parameter. In addition, it is observed that the difference between natural frequencies evaluated by the present and classical models becomes more considerable for both lower values of slenderness ratio and higher modes.


Micro-rod Size dependency Axial vibration Small-scale effect Modified strain gradient theory Length scale parameter Higher-order rod model Natural frequency 



This study has been supported by The Scientific and Technological Research Council of Turkey (TÜBİTAK) with Project No: 112 M879. This support is gratefully acknowledged.


  1. B. Akgöz, Ö. Civalek, Int. J. Eng. Sci. 49, 1268 (2011)CrossRefGoogle Scholar
  2. B. Akgöz, Ö. Civalek, Compos. Part B 55, 263 (2013)CrossRefGoogle Scholar
  3. B. Akgöz, Ö. Civalek, J. Vib. Control. 20, 606 (2014)MathSciNetCrossRefGoogle Scholar
  4. B.S. Altan, E.C. Aifantis, J. Mech. Behav. Mater. 8, 231 (1997)CrossRefGoogle Scholar
  5. B.S. Altan, H. Evensen, E.C. Aifantis, Mech. Res. Commun. 23, 35 (1996)CrossRefGoogle Scholar
  6. A. Anthoine, Int. J. Solids Struct. 37, 1003 (2000)CrossRefGoogle Scholar
  7. M. Aydogdu, Phys. E. 41, 861 (2009)CrossRefGoogle Scholar
  8. A.C.M. Chong, D.C.C. Lam, J. Mater. Res. 14, 4103 (1999)CrossRefGoogle Scholar
  9. E. Cosserat, F. Cosserat, Theory of Deformable Bodies (Trans. by D.H. Delphenich) (Scientific Library, A. Hermann and Sons, Paris, 1909)Google Scholar
  10. A.C. Eringen, J. Appl. Phys. 54, 4703 (1983)CrossRefGoogle Scholar
  11. A.C. Eringen, E.S. Suhubi, Int. J. Eng. Sci. 2, 189 (1964)CrossRefGoogle Scholar
  12. W. Faris, A.H. Nayfeh, Commun. Nonlinear Sci. Numer. Simul. 12, 776 (2007)CrossRefGoogle Scholar
  13. N.A. Fleck, J.W. Hutchinson, J. Mech. Phys. Solids 41, 1825 (1993)MathSciNetCrossRefGoogle Scholar
  14. N.A. Fleck, G.M. Muller, M.F. Ashby, J.W. Hutchinson, Acta Metall. Mater. 42, 475 (1994)CrossRefGoogle Scholar
  15. Y.Q. Fu, H.J. Du, S. Zhang, Mater. Lett. 57, 2995 (2003)CrossRefGoogle Scholar
  16. U. Güven, Comptes Rendus Mécanique 342, 8 (2014)CrossRefGoogle Scholar
  17. M.A. Haque, M.T.A. Saif, Acta Mater. 51, 3053 (2003)CrossRefGoogle Scholar
  18. M.H. Kahrobaiyan, M. Asghari, M.T. Ahmadian, Int. J. Eng. Sci. 44, 66–67 (2013)Google Scholar
  19. M.H. Kahrobaiyan, M. Rahaeifard, M.T. Ahmadian, Appl. Math. Model. 35, 5903 (2011a)MathSciNetCrossRefGoogle Scholar
  20. M.H. Kahrobaiyan, S.A. Tajalli, M.R. Movahhedy, J. Akbari, M.T. Ahmadian, Int. J. Eng. Sci. 49, 856 (2011b)CrossRefGoogle Scholar
  21. W.T. Koiter, Proc. K. Ned. Akad. Wet. B 67, 17 (1964)Google Scholar
  22. S. Kong, S. Zhou, Z. Nie, K. Wang, Int. J. Eng. Sci. 46, 427 (2008)CrossRefGoogle Scholar
  23. S. Kong, S. Zhou, Z. Nie, K. Wang, Int. J. Eng. Sci. 47, 487 (2009)CrossRefGoogle Scholar
  24. D.C.C. Lam, F. Yang, A.C.M. Chong, J. Wang, P. Tong, J. Mech. Phys. Solids 51, 1477 (2003)CrossRefGoogle Scholar
  25. A.K. Lazopoulos, Int. J. Mech. Sci. 58, 27 (2012)CrossRefGoogle Scholar
  26. K.A. Lazopoulos, A.K. Lazopoulos, Eur. J. Mech. A/Solids 29, 837 (2010)CrossRefGoogle Scholar
  27. X. Li, B. Bhushan, K. Takashima, C.W. Baek, Y.K. Kim, Ultramicroscopy 97, 481 (2003)CrossRefGoogle Scholar
  28. J. Lou, P. Shrotriya, S. Allameh, T. Buchheit, W.O. Soboyejo, Mater. Sci. Eng. A 441, 299 (2006)CrossRefGoogle Scholar
  29. H.M. Ma, X.L. Gao, J.N. Reddy, J. Mech. Phys. Solids 56, 3379 (2008)MathSciNetCrossRefGoogle Scholar
  30. R.D. Mindlin, Arch. Ration. Mech. Anal. 16, 51 (1964)CrossRefGoogle Scholar
  31. R.D. Mindlin, Int. J. Solids Struct. 1, 417 (1965)CrossRefGoogle Scholar
  32. R.D. Mindlin, H.F. Tiersten, Arch. Ration. Mech. Anal. 11, 415 (1962)CrossRefGoogle Scholar
  33. Y. Moser, M.A.M. Gijs, Miniaturized flexible temperature sensor. J. Microelectromech. Syst. 16, 1349 (2007)CrossRefGoogle Scholar
  34. F. Najar, S. Choura, S. El-Borgi, E.M. Abdel-Rahman, A.H. Nayfeh, J. Micromech. Microeng. 15, 419 (2005)CrossRefGoogle Scholar
  35. S. Narendar, S. Ravinder, S. Gopalakrishnan, Int. J. Nano Dimens. 3, 1 (2012)Google Scholar
  36. S. Papargyri-Beskou, D. Polyzos, D.E. Beskos, Struct. Eng. Mech. 15, 705 (2003a)CrossRefGoogle Scholar
  37. S. Papargyri-Beskou, K.G. Tsepoura, D. Polyzos, D.E. Beskos, Int. J. Solids Struct. 40, 385 (2003b)CrossRefGoogle Scholar
  38. S.K. Park, X.L. Gao, J. Micromech. Microeng. 16, 2355 (2006)CrossRefGoogle Scholar
  39. J. Peddieson, G.R. Buchanan, R.P. McNitt, Int. J. Eng. Sci. 41, 305 (2003)CrossRefGoogle Scholar
  40. S.S. Rao, Vibration of Continuous Systems (Wiley Inc, Hoboken, 2007)Google Scholar
  41. J.N. Reddy, Int. J. Eng. Sci. 45, 288–307 (2007a)CrossRefGoogle Scholar
  42. J.N. Reddy, Theory and Analysis of Elastic Plates and Shells, 2nd edn. (Taylor & Francis, Philadelphia, 2007b)Google Scholar
  43. S.D. Senturia, Microsystem Design (Kluwer Academic Publishers, Boston, 2001)Google Scholar
  44. R.A. Toupin, Arch. Ration. Mech. Anal. 11, 385 (1962)CrossRefGoogle Scholar
  45. K.G. Tsepoura, S. Papargyri-Beskou, D. Polyzos, D.E. Beskos, Arch. Appl. Mech. 72, 483 (2002)CrossRefGoogle Scholar
  46. I. Vardoulakis, J. Sulem, Bifurcation Analysis in Geomechanics, Blackie (Chapman and Hall, London, 1995)Google Scholar
  47. C.M. Wang, S. Kitipornchai, C.W. Lim, M. Eisenberger, J. Eng. Mech. 134, 475 (2008)CrossRefGoogle Scholar
  48. B. Wang, J. Zhao, S. Zhou, Eur. J. Mech. A/Solids 29, 591 (2010)CrossRefGoogle Scholar
  49. F. Yang, A.C.M. Chong, D.C.C. Lam, P. Tong, Int. J. Solids Struct. 39, 2731 (2002)CrossRefGoogle Scholar

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Authors and Affiliations

  1. 1.Civil Engineering Department, Division of MechanicsAkdeniz UniversityAntalyaTurkey
  2. 2.Faculty of Engineering-Architecture, Department of Civil EngineeringUludag UniversityBursaTurkey

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