Primary resonance of an Euler-Bernoulli nano-beam modelled with second strain gradient

Document Type : Research Article


1 Assistant Professor, School of Mechanical Engineering, Shiraz University, Shiraz, Islamic Republic of Iran

2 Phd. student, School of Mechanical Engineering, Tehran University, Tehran, Islamic Republic of Iran


In the present manuscript, the second strain gradient (SSG) is utilized to investigate the primary resonance of a nonlinear Euler-Bernoulli nanobeam is analyzed in this paper for the first time. To that end, the second strain gradient theory, a higher-order continuum theory capable of taking the size effects into account, is utilized and the governing equation of the motion for
an Euler-Bernoulli nanobeam is derived with sixteen higher-order material constants. Then by implementing the Galerkin’s method,the Duffing equation for the vibration of a hinged-hinged nanobeam is obtained and its primary resonance is studied utilizing the method of multiple scales. The size effects and impact of various system parameters on the amplitude of the response are then investigated for three different materials and the results are compared to that
of the first strain gradient and classical theories. The results of this manuscript clearly shows that the nonlinear vibration of a second strain gradient nanobeam is size-dependent and although the difference between the results obtained by the second strain gradient theory and the first strain gradient theory is negligible for thicker beams, as the thickness decreases, the difference becomes more prominent. Also, the effects of nonlinearity on the forced vibration nonlinear response of an SSG beam are investigated and some observations are reported.


  • The second strain gradient theory is used in order to capture size effects.
  • Primary resonance behavior of a nonlinear nanobeam made of three different materials is investigated.
  • The size effects and influence of the excitation amplitude on the response of the system is studied.
  • Having multiple answers (up to three) and the jump phenomenon are important characteristics of nonlinear vibrating systems.


[1] L.J. Currano, M. Yu, B. Balachandran, Latching in a MEMS shock sensor: Modeling and experiments, Sensors and Actuators A: Physical, 159 (2010) 41-50.
[2] L. Li, Z.J. Chew, Microactuators: Design and technology, in:  Smart Sensors and MEMs, Elsevier, 2018, pp. 305-348.
[3] Z. Djurić, I. Jokić, A. Peleš, Fluctuations of the number of adsorbed molecules due to adsorption–desorption processes coupled with mass transfer and surface diffusion in bio/chemical MEMS sensors, Microelectronic Engineering, 124 (2014) 81-85.
[4] S. Kong, S. Zhou, Z. Nie, K. Wang, The size-dependent natural frequency of Bernoulli–Euler micro-beams, International Journal of Engineering Science, 46 (2008) 427-437.
[5] W.T. Koiter, Couple-stresses in the theory of elasticity, I and II, Prec, Roy. Netherlands Acad. Sci. B, 67  0964.
[6] R.D. Mindlin, H.F. Tiersten, Effects of couple-stresses in linear elasticity, Archive for Rational Mechanics and analysis, 11 (1962) 415-448.
[7] W. Su, S. Liu, Vibration analysis of periodic cellular solids based on an effective couple-stress continuum model, International Journal of Solids and Structures, 51 (2014) 2676-2686.
[8] F.A.C.M. Yang, A.C.M. Chong, D.C.C. Lam, P. Tong, Couple stress based strain gradient theory for elasticity, International Journal of Solids and Structures, 39 (2002) 2731-2743.
[9] S.K. Park, X.L. Gao, Bernoulli–Euler beam model based on a modified couple stress theory, Journal of Micromechanics and Microengineering, 16 (2006) 2355.
[10] M. Asghari, M.H. Kahrobaiyan, M.T. Ahmadian, A nonlinear Timoshenko beam formulation based on the modified couple stress theory, International Journal of Engineering Science, 48 (2010) 1749-1761.
[11] M. Asghari, M.T. Ahmadian, M.H. Kahrobaiyan, M. Rahaeifard, On the size-dependent behavior of functionally graded micro-beams, Materials & Design (1980-2015), 31 (2010) 2324-2329.
[12] M. Asghari, M. Rahaeifard, M.H. Kahrobaiyan, M.T. Ahmadian, The modified couple stress functionally graded Timoshenko beam formulation, Materials & Design, 32 (2011) 1435-1443.
[13] R.D. Mindlin, Second gradient of strain and surface-tension in linear elasticity, International Journal of Solids and Structures, 1 (1965) 417-438.
[14] S.M.H. Karparvarfard, M. Asghari, R. Vatankhah, A geometrically nonlinear beam model based on the second strain gradient theory, International Journal of Engineering Science, 91 (2015) 63-75.
[15] S.A. Momeni, M. Asghari, The second strain gradient functionally graded beam formulation, Composite Structures, 188 (2018) 15-24.
[16] H. Shodja, F. Ahmadpoor, A. Tehranchi, Calculation of the additional constants for fcc materials in second strain gradient elasticity: behavior of a nano-size Bernoulli-Euler beam with surface effects, Journal of applied mechanics, 79 (2012).
[17] R.D. Mindlin, N.N. Eshel, On first strain-gradient theories in linear elasticity, International Journal of Solids and Structures, 4 (1968) 109-124.
[18] R. Ansari, R. Gholami, M.F. Shojaei, V. Mohammadi, S. Sahmani, Size-dependent bending, buckling and free vibration of functionally graded Timoshenko microbeams based on the most general strain gradient theory, Composite Structures, 100 (2013) 385-397.
[19] D.C.C. Lam, F. Yang, A.C.M. Chong, J. Wang, P. Tong, Experiments and theory in strain gradient elasticity, Journal of the Mechanics and Physics of Solids, 51 (2003) 1477-1508.
[20] M.H. Kahrobaiyan, M. Asghari, M. Rahaeifard, M.T. Ahmadian, A nonlinear strain gradient beam formulation, International Journal of Engineering Science, 49 (2011) 1256-1267.
[21] H. Mohammadi, M. Mahzoon, Thermal effects on postbuckling of nonlinear microbeams based on the modified strain gradient theory, Composite Structures, 106 (2013) 764-776.
[22] H. Mohammadi, S. Sepehri, Analyzing dynamical snap-through of a size dependent nonlinear micro-resonator via a semi-analytic method, Journal of Theoretical and Applied Vibration and Acoustics, 4 (2018) 19-36.
[23] F. Amiot, An Euler–Bernoulli second strain gradient beam theory for cantilever sensors, Philosophical Magazine Letters, 93 (2013) 204-212.
[24] S. Khakalo, J. Niiranen, Form II of Mindlin's second strain gradient theory of elasticity with a simplification: For materials and structures from nano-to macro-scales, European Journal of Mechanics-A/Solids, 71 (2018) 292-319.
[25] R.A. Toupin, Elastic materials with couple-stresses, Archive for Rational Mechanics and Analysis, 11 (1962) 385-414.
[26] W. Thomson, Theory of vibration with applications, CrC Press, 2018.
[27] A.H. Nayfeh, D.T. Mook, Nonlinear oscillations, John Wiley & Sons, 2008.
[28] R. Vatankhah, M.H. Kahrobaiyan, A. Alasty, M.T. Ahmadian, Nonlinear forced vibration of strain gradient microbeams, Applied Mathematical Modelling, 37 (2013) 8363-8382.