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A Nonlinear Microbeam Model Based on Strain Gradient Elasticity Theory

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Abstract

A micro scale nonlinear beam model based on strain gradient elasticity is developed. Governing equations of motion and boundary conditions are obtained in a variational framework. As an example, the nonlinear vibration of microbeams is analyzed. In a beam having a thickness to length parameter ratio close to unity, the strain gradient effect on increasing the natural frequency is predominant. By increasing the beam thickness, this effect decreases and geometric nonlinearity plays the main role on increasing the natural frequency. For some specific ratios, both geometric nonlinearity and size effects have a significant role on increasing the natural frequency.

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References

  1. Fleck, N.A., Muller, G.M. and Ashby, M.F., Strain gradient plasticity: theory and experiment. Acta Metallurgical et Materialia, 1994, 42: 475–487.

    Article  Google Scholar 

  2. McFarland, A.W. and Colton, J.S., Role of material microstructure in plate stiffness with relevance to microcantilever sensors. Journal of Micromechanics and Microengineering, 2005, 15: 1060–1067.

    Article  Google Scholar 

  3. Ma, H.M., Gao, X.L. and Reddy, J.N., A microstructure-dependent Timoshenko beam model based on a modified couple stress theory. Journal of the Mechanics and Physics of solids, 2008, 56: 3379–3391.

    Article  MathSciNet  Google Scholar 

  4. Koiter, W.T., Couple stresses in the theory of elasticity, Parts I & II. Proceedings of the Koninklijke Nederlandse Akademie van Wetenschappen, Series B, 1964, 67: 17–44.

    MATH  Google Scholar 

  5. Anthoine, A., Effect of couple-stresses on the elastic bending of beams. International Journal of Solids and Structures, 2000, 37: 1003–1018.

    Article  Google Scholar 

  6. Park, S.K. and Gao, X.L., Bernoulli-Euler beam model based on a modified couple stress theory. Journal of Micromechanics and Microengineering, 2006, 16: 2355–2359.

    Article  Google Scholar 

  7. Kong, S.L., Zhou, S.J., Nie, Z.F. and Wang, K., The size-dependent natural frequency of Bernoulli-Euler microbeams. International Journal of Engineering Science, 2008, 46: 427–437.

    Article  Google Scholar 

  8. Yang, F., Chong, A.C.M. and Lam, D.C.C., Couple stress based strain gradient theory for elasticity. International Journal of Solids and Structures, 2002, 39: 2731–2743.

    Article  Google Scholar 

  9. Asghari,M., Kahrobaiyan,M.H., Rahaeifard,M. and Ahmadian,M.T., Investigation of the size effects in Timoshenko beams based on the couple stress theory. Archive of Applied Mechanics, doi:10.1007/s00419-010-0452-5.

    Article  MATH  Google Scholar 

  10. Asghari, M., Ahmadian, M.T., Kahrobaiyan, M.H. and Rahaeifard, M., On the size-dependent behavior of functionally graded microbeams. Materials & Design, 2010, 31: 2324–2329.

    Article  Google Scholar 

  11. Asghari, M., Rahaeifard, M., Kahrobaiyan, M.H. and Ahmadian, M.T., The modified couple stress functionally graded Timoshenko beam formulation. Materials & Design, 2011, 32: 1435–1443.

    Article  Google Scholar 

  12. Xia, W., Wang, L. and Yin, L., Nonlinear non-classical microscale beams: Static bending, postbuckling and free vibration. International Journal of Engineering Science, 2010, 48: 2044–2053.

    Article  MathSciNet  Google Scholar 

  13. Asghari, M., Kahrobaiyan, M.H. and Ahmadian, M.T., A nonlinear Timoshenko beam formulation based on the modified couple stress theory. International Journal of Engineering Science, 2010, 48: 1749–1761.

    Article  MathSciNet  Google Scholar 

  14. Ke,L.L., Wang,Y.S., Yang,J. and Kitipornchai,S., Nonlinear free vibration of size-dependent functionally graded microbeams. International Journal of Engineering Science, 2011, doi:10.1016/j.ijengsci.2010.12.008.

    Article  MathSciNet  Google Scholar 

  15. Ke, L.L. and Wang, Y.S., Size effect on dynamic stability of functionally graded microbeams based on a modified couple stress theory. Composite Structures, 2011, 93: 342–350.

    Article  Google Scholar 

  16. Peddieson, J., Buchanan, G.R. and McNitt, R.P., Application of nonlocal continuum models to nanotechnology. International Journal of Engineering Science, 2003, 41: 305–312.

    Article  Google Scholar 

  17. Wang, Q., Wave propagation in carbon nanotubes via nonlocal continuum mechanics. Journal of Applied Physics, 2005, 98: 1–6.

    Article  Google Scholar 

  18. Reddy, J.N., Nonlocal theories for bending, buckling and vibration of beams. International Journal of Engineering Science, 2007, 45: 288–307.

    Article  Google Scholar 

  19. Ke, L.L., Xiang, Y., Yang, J. and Kitipornchai, S., Nonlinear free vibration of embedded double-walled carbon nanotubes based on nonlocal Timoshenko beam theory. Computational Materials Science, 2009, 47: 409–417.

    Article  Google Scholar 

  20. Vardoulakis, I., Exadaktylos, G. and Kourkoulis, S.K., Bending of marble with intrinsic length scales: a gradient theory with surface energy and size effects. Journal de Physics IV, 1998, 8: 399–406.

    Google Scholar 

  21. Vardoulakis, I. and Sulem, J., Bifurcation Analysis in Geomechanics. London: Blackie/Chapman & Hall, 1995.

    Google Scholar 

  22. Papargyri-Beskou, S., Tsepoura, K.G., Polyzos, D. and Beskos, D.E., Bending and stability analysis of gradient elastic beams. International Journal of Solids and Structures, 2003, 40: 385–400.

    Article  Google Scholar 

  23. Lazopoulos, K.A. and Lazopoulos, A.K., Bending and buckling of thin strain gradient elastic beams. European Journal of Mechanics, A/Solids, 2010, 29: 837–843.

    Article  Google Scholar 

  24. Yi, D., Wang, T.C. and Chen, S., New strain gradient theory and analysis. Acta Mechanica Solida Sinica, 2009, 22(1): 45–52.

    Article  Google Scholar 

  25. Kong, S.L., Zhou, S.J., Nie, Z.F. and Wang, K., Static and dynamic analysis of microbeams based on strain gradient elasticity theory. International Journal of Engineering Science, 2009, 47: 487–498.

    Article  MathSciNet  Google Scholar 

  26. Lam, D.C.C., Yang, F. and Chong, A.C.M., Experiments and theory in strain gradient elasticity. Journal of the Mechanics and Physics of solids, 2003, 51: 1477–1508.

    Article  Google Scholar 

  27. Wang, B., Zhao, J. and Zhou, S., A micro scale Timoshenko beam model based on strain gradient elasticity theory. European Journal of Mechanics, A/Solids, 2010, 29: 591–599.

    Article  Google Scholar 

  28. Yin, L., Qian, Q. and Wang, L., Strain gradient beam model for dynamics of microscale pipes conveying fluid. Applied Mathematical Modelling, 2011, 35: 2864–2873.

    Article  MathSciNet  Google Scholar 

  29. Kahrobaiyan,M.H., Asghari,M., Rahaeifard,M. and Ahmadian,M.T., A nonlinear strain gradient beam formulation. International Journal of Engineering Science, 2011, doi:10.1016/j.ijengsci.2011.01.006.

    Article  MathSciNet  Google Scholar 

  30. Mindlin, R.D., Microstructure in linear elasticity. Archive for Rational Mechanics and Analysis, 1964, 16: 51–78.

    Article  MathSciNet  Google Scholar 

  31. Giannakopoulos, A.E., Amanatidou, E. and Aravas, N., A reciprocity theorem in linear gradient elasticity and the corresponding Saint-Venant principle. International Journal of Solids and Structures, 2006, 43: 3875–3894.

    Article  MathSciNet  Google Scholar 

  32. Reddy,J.N., An Introduction to Nonlinear Finite Element Analysis. Oxford University Press, 2004.

    Chapter  Google Scholar 

  33. Reddy, J.N., Applied Functional Analysis and Variational Methods in Engineering. McGraw-Hill, New York, 1986.

    MATH  Google Scholar 

  34. Nayfeh, A.H. and Mook, D.T., Nonlinear Oscillations. Wiley, New York, 1979.

    MATH  Google Scholar 

  35. Nayfeh, A.H., Introduction to Perturbation Techniques, Wiley, New York, 1981.

    MATH  Google Scholar 

  36. Liao, S.J., Beyond Perturbation: Introduction to Homotopy Analysis Method. Chapman & Hall/CRC Press, Boca Raton, 2003.

    Book  Google Scholar 

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Authors and Affiliations

  1. Department of Mechanical Engineering, Islamic Azad University, Bandar Anzali Branch, Anzali, Iran

    Farshid Rajabi

  2. Department of Mechanical Engineering, Faculty of Engineering, University of Guilan, Rasht, Iran

    Shojaa Ramezani

Authors
  1. Farshid Rajabi
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  2. Shojaa Ramezani
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Correspondence to Shojaa Ramezani.

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Rajabi, F., Ramezani, S. A Nonlinear Microbeam Model Based on Strain Gradient Elasticity Theory. Acta Mech. Solida Sin. 26, 21–34 (2013). https://doi.org/10.1016/S0894-9166(13)60003-8

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  • DOI: https://doi.org/10.1016/S0894-9166(13)60003-8

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