Structural Engineering and Mechanics

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Finite element vibration analysis of nanoshell based on new cylindrical shell element

  • Soleimani, Iman (Mechanical Engineering Department, Shahrekord University) ;
  • Beni, Yaghoub T. (Faculty of Engineering, Shahrekord University) ;
  • Dehkordi, Mohsen B. (Faculty of Engineering, Shahrekord University)
  • Received : 2017.04.20
  • Accepted : 2017.10.12
  • Published : 2018.01.10
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Abstract

In this paper, using modified couple stress theory in place of classical continuum theory, and using shell model in place of beam model, vibrational behavior of nanotubes is investigated via the finite element method. Accordingly classical continuum theory is unable to correctly compute stiffness and account for size effects in micro/nanostructures, higher order continuum theories such as modified couple stress theory have taken on great appeal. In the present work the mass-stiffness matrix for cylindrical shell element is developed, and by means of size-dependent finite element formulation is extended to more precisely account for nanotube vibration. In addition to modified couple stress cylindrical shell element, the classical cylindrical shell element can also be defined by setting length scale parameter to zero in the equations. The boundary condition were assumed simply supported at both ends and it is shown that the natural frequency of nano-scale shell using the modified coupled stress theory is larger than that using the classical shell theory and the results of Ansys. The results have indicated using the modified couple stress cylindrical shell element, the rigidity of the nano-shell is greater than that in the classical continuum theory, which results in increase in natural frequencies. Besides, in addition to reducing the number of elements required, the use of this type of element also increases convergence speed and accuracy.

Keywords

References

  1. Abadyan, M.R., Tadi Beni, Y. and Noghrehabadi, A. (2011), "Investigation of elastic boundary condition on the pull-in instability of beam-type NEMS under van der Waals attraction", Procedia Eng., 10, 1724-1729. https://doi.org/10.1016/j.proeng.2011.04.287
  2. Akgoz, B. and Civalek, O. (2013), "Free vibration analysis of axially functionally graded tapered Bernoulli-Euler micro beams based on the modified couple stress theory", Compos. Struct., 98, 314-322. https://doi.org/10.1016/j.compstruct.2012.11.020
  3. Alibeigloo, A. and Shaban, M. (2013), "Free vibration analysis of carbon nanotubes by using three-dimensional theory of elasticity", Acta Mechanic, 224(7), 1415-1427. https://doi.org/10.1007/s00707-013-0817-2
  4. Ansari, R., Ajori, S. and Arash, B. (2012), "Vibrations of single- and double-walled carbon nanotubes with layer wise boundary conditions: A molecular dynamics study", Curr. Appl. Phys., 12, 707-711. https://doi.org/10.1016/j.cap.2011.10.007
  5. Arash, B. and Ansari, R. (2010), "Evaluation of nonlocal parameter in the vibrations of single-walled carbon nanotubes with initial strain", Physica E., 42, 2058-2064. https://doi.org/10.1016/j.physe.2010.03.028
  6. Berrabah, H.M., Tounsi, A., Semmah, A. and Adda Bedia, E.A. (2013), "Comparison of various refined nonlocal beam theories for bending, vibration and buckling analysis of nano beams", Struct. Eng. Mech., 48(3), 351-365. https://doi.org/10.12989/sem.2013.48.3.351
  7. Chong, C.M. and Lam, D.C.C. (1999), "Strain gradient plasticity effect in indentation hardness of polymers", J. Mater. Res., 14, 4103-4110. https://doi.org/10.1557/JMR.1999.0554
  8. Chyuan, S.W. (2008), "Computational simulation for MEMS comb drive levitation using FEM", J. Electrost., 66, 361-365. https://doi.org/10.1016/j.elstat.2008.03.005
  9. Ebrahimi, N. and Tadi Beni, Y. (2016), "Electro-mechanical vibration of nanoshells using consistent size-dependent piezoelectric theory", Steel Compos. Struct., 22(6), 1301-1336. https://doi.org/10.12989/scs.2016.22.6.1301
  10. Eringen, A.C. (1980), Mechanics of Continua, R.E. Krieger Pub. Co.
  11. Fattahian Dehkordi, S. and Tadi Beni, Y. (2017), "Electro-mechanical free vibration of single-walled piezoelectric/flexoelectric nano cones using consistent couple stress theory", Int. J. Mech. Sci., 128-129, 125-139. https://doi.org/10.1016/j.ijmecsci.2017.04.004
  12. Fleck, N.A. and Hutchinson, J.W. (1997), "Strain gradient plasticity", Eds. John, W.H. & Theodore, Y.W., Adv. Appl. Mech., 295-361.
  13. Ghayesh, M.H., Farokhi, H. and Amabili, M. (2013), "Nonlinear dynamics of a micro scale beam based on the modified couple stress theory", Compos. Part B, 50, 318-324. https://doi.org/10.1016/j.compositesb.2013.02.021
  14. Ji, B. and Chen, W. (2009), "Measuring material length parameter with a new solution of microbend beam in couple stress elasto-plasticity", Struct. Eng. Mech., 33(2), 257-260. https://doi.org/10.12989/sem.2009.33.2.257
  15. Kang, X. and Xi, X.F. (2007), "Size effect on the dynamic characteristic of a micro beam based on Cosserat theory", J. Mech. Strength., 29(1), 1-4.
  16. Kheibari, F. and Tadi Beni, Y. (2017), "Size dependent electro-mechanical vibration of single-walled piezoelectric nanotubes using thin shell model", Mater. Des., 114, 572-583. https://doi.org/10.1016/j.matdes.2016.10.041
  17. Kocaturk, T. and Akbas, S.D. (2013), "Wave propagation in a micro beam based on the modified couple stress theory", Struct. Eng. Mech., 46(3), 417-431. https://doi.org/10.12989/sem.2013.46.3.417
  18. Koiter, W.T. (1964), "Couple stresses in the theory of elasticity", Proc. Koninklijke Nederl. Akaad. van Wetensch, 67, 17-44.
  19. Kong, S., Zhou, S., Nie, Z. and Wang, K. (2008), "The size-dependent natural frequency of Bernoulli-Euler micro-beams", Int. J. Eng. Sci., 46, 427-437. https://doi.org/10.1016/j.ijengsci.2007.10.002
  20. Lam, D.C., Yang, F., Chong, A.C.M., Wang, J. and Tong, P. (2003), "Experiments and theory in strain gradient elasticity", J. Mech. Phys. Solid., 51, 1477-1508. https://doi.org/10.1016/S0022-5096(03)00053-X
  21. Li, C. (2013), "Size-dependent thermal behaviors of axially traveling nano beams based on a strain gradient theory", Struct. Eng. Mech., 48(3), 415-434. https://doi.org/10.12989/sem.2013.48.3.415
  22. Li, C. (2014a), "A nonlocal analytical approach for torsion of cylindrical nanostructures and the existence of higher-order stress and geometric boundaries", Compos. Struct., 118, 607-621. https://doi.org/10.1016/j.compstruct.2014.08.008
  23. Li, C. (2014b), "Torsional vibration of carbon nanotubes: Comparison of two nonlocal models and a semi-continuum model", Int. J. Mech. Sci., 82, 25-31. https://doi.org/10.1016/j.ijmecsci.2014.02.023
  24. Li, C., Li, S., Yao L.Q. and Zhu, Z.K. (2015b), "Nonlocal theoretical approaches and atomistic simulations for longitudinal free vibration of nanorods/nanotubes and verification of different nonlocal models", Appl. Math. Model., 39, 4570-4585. https://doi.org/10.1016/j.apm.2015.01.013
  25. Li, C., Lim, C.W. and Yu, J.L. (2011a), "Dynamics and stability of transverse vibrations of nonlocal nanobeams with a variable axial load", Smart Mater. Struct., 20(1), 15-23.
  26. Li, C., Lim, C.W., Yu, J.L. and Zeng, Q.C. (2011b), "Analytical solutions for vibration of simply supported nonlocal nanobeams with an axial force", Int. J. Struct. Stab. Dyn., 11(2), 257-271. https://doi.org/10.1142/S0219455411004087
  27. Li, C., Lim, C.W., Yu, J.L. and Zeng, Q.C. (2011c), "Transverse vibration of pre-tensioned nonlocal nanobeams with precise internal axial loads", Sci. China Technol. Sci., 54(8), 2007-2013. https://doi.org/10.1007/s11431-011-4479-9
  28. Li, C., Yao, L.Q., Chen, W.Q. and Li, S. (2015a), "Comments on nonlocal effects in nano-cantilever beams", Int. J. Eng. Sci., 87, 47-57. https://doi.org/10.1016/j.ijengsci.2014.11.006
  29. Lim, C.W., Li, C. and Yu, J.L. (2012), "Free torsional vibration of nanotubes based on nonlocal stress theory", J. Sound Vib., 331, 2798-2808. https://doi.org/10.1016/j.jsv.2012.01.016
  30. Liu, J.J., Li, C., Yang, C.J. Shen, J.P. and Xie, F. (2016), "Dynamical responses and stabilities of axially moving nanoscale beams with time-dependent velocity using a nonlocal stress gradient theory", J. Vib. Control, 1077546316629013.
  31. Ma, H.M., Gao, X.L. and Reddy, J.N. (2008), "A microstructure-dependent Timoshenko beam model based on a modified couple stress theory", J. Mech. Phys. Solid., 56, 3379-3391. https://doi.org/10.1016/j.jmps.2008.09.007
  32. Mehralian, F. and Tadi Beni, Y. (2016), "Size-dependent torsional buckling analysis of functionally graded cylindrical shell", Compos. Part B: Eng., 94, 11-25. https://doi.org/10.1016/j.compositesb.2016.03.048
  33. Metz, P., Alici, G. and Spinks, G.M. (2006), "A finite element model for bending behavior of conducting polymer electromechanical actuators", Sens. Actuats. A, 130, 1-11.
  34. Mindlin, R.D. (1964), "Micro-structure in linear elasticity", Arch. Rat. Mech. Anal., 16, 51-78.
  35. Mindlin, R.D. and Tiersten, H.F. (1962), "Effects of couple-stresses in linear elasticity", Arch. Rat. Mech. Anal., 11, 415-448. https://doi.org/10.1007/BF00253946
  36. Mohammadi Dashtaki, P. and Tadi Beni, Y. (2014), "Effects of Casimir force and thermal stresses on the buckling of electrostatic nano-bridges based on couple stress theory", Arab. J. Sci. Eng., 39, 5753-5763. https://doi.org/10.1007/s13369-014-1107-6
  37. Mohammadimehr, M., Alimirzaei, S. (2016), "Nonlinear static and vibration analysis of Euler-Bernoulli composite beam model reinforced by FG-SWCNT with initial geometrical imperfection using FEM", Struct. Eng. Mech., 59(3), 431-454. https://doi.org/10.12989/sem.2016.59.3.431
  38. Noghrehabadi, A., Tadi Beni, Y., Koochi, A., Kazemi, A., Yekrangi, A., Abadyan, M. and Noghrehabadi, M. (2011), "Closed-form approximations of the pull-in parameters and stress field of electrostatic cantilever nano-actuators considering van der Waals attraction", Procedia Eng., 10, 3750-3756. https://doi.org/10.1016/j.proeng.2011.04.613
  39. Pradhan, S.C. and Phadikar, J.K. (2009), "Bending, buckling and vibration analyses of nonhomogeneous nanotubes using GDQ and nonlocal elasticity theory", Struct. Eng. Mech., 33(2), 193-213. https://doi.org/10.12989/sem.2009.33.2.193
  40. Reddy, J.N. and Berry, J. (2012), "Nonlinear theories of axisymmetric bending of functionally graded circular plates with modified couple stress", Compos. Struct., 94, 3664-3668. https://doi.org/10.1016/j.compstruct.2012.04.019
  41. Sahmani, S. and Ansari, R. (2013), "On the free vibration response of functionally graded higher-order shear deformable micro plates based on the strain gradient elasticity theory", Compos. Struct., 95, 430-442. https://doi.org/10.1016/j.compstruct.2012.07.025
  42. Simsek, M. (2014), "Nonlinear static and free vibration analysis of micro beams based on the non-linear elastic foundation using modified couple stress theory and he's variational method", Compos. Struct., 112, 264-272 https://doi.org/10.1016/j.compstruct.2014.02.010
  43. Simsek, M. and Reddy, J.N. (2013), "Bending and vibration of functionally graded micro beams using a new higher order beam theory and the modified couple stress theory", Int. J. Eng. Sci., 64, 37-53. https://doi.org/10.1016/j.ijengsci.2012.12.002
  44. Simsek, M., Kocaturk, T. and Akbas, S. (2013), "Static bending of a functionally graded micro scale Timoshenko beam based on the modified couple stress theory", Compos. Struct., 95, 740-747. https://doi.org/10.1016/j.compstruct.2012.08.036
  45. Tadi Beni, Y. (2016), "Size-dependent analysis of piezoelectric nanobeams including electro-mechanical coupling", Mech. Res. Commun., 75, 67-80. https://doi.org/10.1016/j.mechrescom.2016.05.011
  46. Tadi Beni, Y. (2016), "Size-dependent electromechanical bending, buckling, and free vibration analysis of functionally graded piezoelectric nanobeams", J. Intel. Mater. Syst. Struct., 27(16), 2199-2215. https://doi.org/10.1177/1045389X15624798
  47. Tadi Beni, Y. and Abadyan, M. (2013), "Use of strain gradient theory for modeling the size-dependent pull-in of rotational nano-mirror in the presence of molecular force", Int. J. Modern Phys. B, 27(18), 1350083. https://doi.org/10.1142/S0217979213500835
  48. Tadi Beni, Y., Karimipour, I. and Abadyan, M. (2015), "Modeling the instability of electrostatic nano-bridges and nano-cantilevers using modified strain gradient theory", Appl. Math. Model., 39, 2633-2648. https://doi.org/10.1016/j.apm.2014.11.011
  49. Tadi Beni, Y., Koochi, A. and Abadyan, M. (2014), "Using modified couple stress theory for modeling the size dependent pull-in instability of torsional nano-mirror under Casimir force", Int. J. Opto Mech., 8, 47-71.
  50. Taghizadeh, M., Ovesy, H.R. and Ghannadpour, S.A.M. (2015), "Nonlocal integral elasticity analysis of beam bending by using finite element method", Struct. Eng. Mech., 54(4), 755-769. https://doi.org/10.12989/sem.2015.54.4.755
  51. Tajalli, S.A., Moghimi Zand, M. and Ahmadian, M.T. (2009), "Effect of geometric nonlinearity on dynamic pull-in behavior of coupled-domain microstructures based on classical and shear deformation plate theories", Eur. J. Mech. A Solid., 28, 916-925. https://doi.org/10.1016/j.euromechsol.2009.04.003
  52. Toupin, R.A. (1962), "Elastic materials with couple stresses", Arch. Rat. Mech. Anal., 11, 385-414. https://doi.org/10.1007/BF00253945
  53. Wang, Y.G., Lin, W.H. and Liu, N. (2013), "Nonlinear free vibration of a micro scale beam based on modified couple stress theory", Physica E: Low-dimens. Syst. Nanostruct., 47, 80-85. https://doi.org/10.1016/j.physe.2012.10.020
  54. Wu, D.H., Chien, W.T., Yang, C.J. and Yen, Y.T. (2005), "Coupled- field analysis of piezoelectric beam actuator using FEM", Sens. Actuators. A, 118, 171 -176. https://doi.org/10.1016/j.sna.2004.04.017
  55. Yang, F., Chong, A.C.M., Lam, D.C.C. and Tong, P. (2002), "Couple stress Based Strain gradient theory for elasticity", Int. J. Solid. Struct., 39, 2731-2743. https://doi.org/10.1016/S0020-7683(02)00152-X
  56. Yang, J., Jia, X.L. and Kitipornchai, S. (2008), "Pull-in instability of nano-switches using nonlocal elasticity theory", J. Phys. D, Appl. Phys., 41, 035103. https://doi.org/10.1088/0022-3727/41/3/035103
  57. Zeighampour, H. and Tadi Beni, Y. (2014), "Size-dependent vibration of fluid-conveying double-walled carbon nanotubes using couple stress shell theory", Physica E: Low-dimens. Syst. Nanostruct., 61, 28-39. https://doi.org/10.1016/j.physe.2014.03.011
  58. Zeighampour, H. and Tadi Beni, Y. (2015), "A shear deformable conical shell formulation in the framework of couple stress theory", Acta Mechanica, 226, 2607-2629. https://doi.org/10.1007/s00707-015-1318-2
  59. Zeighampour, H. and Tadi Beni, Y. (2015), "A shear deformable cylindrical shell model based on couple stress theory", Arch. Appl. Mech., 85, 539-553. https://doi.org/10.1007/s00419-014-0929-8
  60. Zhang, B., He, Y., Liu, D., Gan, Z. and Shen, L. (2014), "Non-classical Timoshenko beam element based on the strain gradient elasticity theory", Finite Elem. Anal. Des., 79, 22-39. https://doi.org/10.1016/j.finel.2013.10.004
  61. Zhao, J. and Pedroso, D. (2008), "Strain gradient theory in orthogonal curvilinear coordinates", Int. J. Solid. Struct., 45, 3507-3520. https://doi.org/10.1016/j.ijsolstr.2008.02.011
  62. Zhou, S.J. and Li, Z.Q (2001), "Length scales in the static and dynamic torsion of a circular cylindrical micro-bar", J. Shandong Univ. Technol., 31(5), 401-407.

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