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A shear deformable cylindrical shell model based on couple stress theory

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Abstract

In this paper, formulation of the thin cylindrical shell via the modified couple stress theory by taking account of shear deformation and rotary inertia is obtained. To do this, the study developed the first shear deformable cylindrical shell theory by considering the size effects via the couple stress theory and the equations of motion of shell with classical and non-classical boundary conditions were extracted through Hamilton’s principle. In the end, as an example, free vibrations of the single-walled carbon nanotube (SWCNT) were investigated. Here, the SWCNT was modeled as a simply supported shell, and the Navier procedure was used to solve the vibration problem. The results of the new model were compared with those of the classical theory, pointing to the conclusion that the classical model is a special case of the modified couple stress theory. The findings also demonstrate that the rigidity of the nano-shell in the modified couple stress theory compared with that in the classical theory is greater, resulting in the increase in natural frequencies. In addition, the effect of the material length scale parameter on the vibrations of the nano-shell in different lengths and thickness was investigated.

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References

  1. Mattia D., Gogotsi Y.: Review: static and dynamic behavior of liquids inside carbon nanotubes. Microfluid Nanofluid 5(3), 289–305 (2008). doi:10.1007/s10404-008-0293-5

    Article  Google Scholar 

  2. Gao Y., Bando Y.: Nanotechnology: carbon nanothermometer containing gallium. Nature 415(6872), 599–600 (2002)

    Article  Google Scholar 

  3. Hummer G., Rasaiah J.C., Noworyta J.P.: Water conduction through the hydrophobic channel of a carbon nanotube. Nature 414(6860), 188–190 (2001)

    Article  Google Scholar 

  4. Shayan-Amin S., Dalir H., Farshidianfar A.: Molecular dynamics simulation of double-walled carbon nanotube vibrations: comparison with continuum elastic theories. J. Mech. 25(04), 337–343 (2009). doi:10.1017/S1727719100002823

    Article  Google Scholar 

  5. Tsai J.-L., Tu J.-F.: Characterizing mechanical properties of graphite using molecular dynamics simulation. Mater. Des. 31(1), 194–199 (2010). doi:10.1016/j.matdes.2009.06.032

    Article  Google Scholar 

  6. Eringen A.C.: On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves. J. Appl. Phys. 54, 4703–4710 (1983)

    Article  Google Scholar 

  7. Yang F., Chong A.C.M., Lam D.C.C., Tong P.: Couple stress based strain gradient theory for elasticity. Int. J. Solids Struct. 39(10), 2731–2743 (2002). doi:10.1016/S0020-7683(02)00152-X

    Article  MATH  Google Scholar 

  8. Lam D.C.C., Yang F., Chong A.C.M., Wang J., Tong P.: Experiments and theory in strain gradient elasticity. J. Mech. Phys. Solids 51(8), 1477–1508 (2003). doi:10.1016/S0022-5096(03)00053-X

    Article  MATH  Google Scholar 

  9. Gurtin M.E., Weissmüller J., Larché F.: A general theory of curved deformable interfaces in solids at equilibrium. Philos. Mag. A 78(5), 1093–1109 (1998). doi:10.1080/01418619808239977

    Article  Google Scholar 

  10. Toupin R.A.: Elastic materials with couple-stresses. Arch. Ration. Mech. Anal. 11(1), 385–414 (1962). doi:10.1007/bf00253945

    Article  MATH  MathSciNet  Google Scholar 

  11. Mindlin R.D.: Micro-structure in linear elasticity. Arch. Ration. Mech. Anal. 16(1), 51–78 (1964). doi:10.1007/bf00248490

    Article  MATH  MathSciNet  Google Scholar 

  12. Mindlin R.D., Tiersten H.F.: Effects of couple-stresses in linear elasticity. Arch. Ration. Mech. Anal. 11(1), 415–448 (1962). doi:10.1007/bf00253946

    Article  MATH  MathSciNet  Google Scholar 

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

    MATH  Google Scholar 

  14. Mindlin R.D.: Second gradient of strain and surface-tension in linear elasticity. Int. J. Solids Struct. 1(4), 417–438 (1965). doi:10.1016/0020-7683(65)90006-5

    Article  Google Scholar 

  15. Fleck N.A., Hutchinson, J.W.: Strain gradient plasticity. In: Hutchinson, J.W., Wu, T.Y. (Eds.). Advances in Applied Mechanics, vol. 33, pp. 295–361 (1997)

  16. Ma H.M., Gao X.L., Reddy J.N.: A microstructure-dependent Timoshenko beam model based on a modified couple stress theory. J. Mech. Phys. Solids 56(12), 3379–3391 (2008). doi:10.1016/j.jmps.2008.09.007

    Article  MATH  MathSciNet  Google Scholar 

  17. Fu Y., Zhang J.: Modeling and analysis of microtubules based on a modified couple stress theory. Phys. E Low Dimens. Syst. Nanostruct. 42(5), 1741–1745 (2010). doi:10.1016/j.physe.2010.01.033

    Article  Google Scholar 

  18. Tsiatas G.C.: A new Kirchhoff plate model based on a modified couple stress theory. Int. J. Solids Struct. 46(13), 2757–2764 (2009). doi:10.1016/j.ijsolstr.2009.03.004

    Article  MATH  Google Scholar 

  19. Xinping Z., Lin W.: Vibration and stability of micro-scale cylindrical shells conveying fluid based on modified couple stress theory. Micro Nano Lett. 7(7), 679–684 (2012). doi:10.1049/mnl.2012.0184

    Article  Google Scholar 

  20. Zhou X., Wang L., Qin P.: Free vibration of micro- and nano-shells based on modified couple stress theory. J. Comput. Theor. Nanosci. 9(6), 814–818 (2012). doi:10.1166/jctn.2012.2101

    Article  Google Scholar 

  21. Leissa, A.W.: Vibration of Shells. Published for the Acoustical Society of America through the American Institute of Physics (1993)

  22. Eringen, A.C.: Mechanics of Continua. R. E. Krieger Pub. Co., Huntington, New York (1980)

  23. Zhao J., Pedroso D.: Strain gradient theory in orthogonal curvilinear coordinates. Int. J. Solids Struct. 45(11–12), 3507–3520 (2008). doi:10.1016/j.ijsolstr.2008.02.011

    Article  MATH  Google Scholar 

  24. Soldatos K.P., Hadjigeorgiou V.P.: Three-dimensional solution of the free vibration problem of homogeneous isotropic cylindrical shells and panels. J. Sound Vib. 137(3), 369–384 (1990). doi:10.1016/0022-460X(90)90805-A

    Article  MATH  Google Scholar 

  25. Alibeigloo A., Shaban M.: Free vibration analysis of carbon nanotubes by using three-dimensional theory of elasticity. Acta Mech. 224(7), 1415–1427 (2013). doi:10.1007/s00707-013-0817-2

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Faculty of Engineering, Shahrekord University, Shahrekord, Iran

    Hamid Zeighampour & Yaghoub Tadi Beni

  2. Nanotechnology Research Center, Shahrekord University, Shahrekord, Iran

    Yaghoub Tadi Beni

Authors
  1. Hamid Zeighampour
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  2. Yaghoub Tadi Beni
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Correspondence to Yaghoub Tadi Beni.

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Zeighampour, H., Beni, Y.T. A shear deformable cylindrical shell model based on couple stress theory. Arch Appl Mech 85, 539–553 (2015). https://doi.org/10.1007/s00419-014-0929-8

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  • DOI: https://doi.org/10.1007/s00419-014-0929-8

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