Browse > Article
http://dx.doi.org/10.12989/sem.2018.65.1.033

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)
Publication Information
Structural Engineering and Mechanics / v.65, no.1, 2018 , pp. 33-41 More about this Journal
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
modified couple stress theory; FEM; cylindrical shell element; size dependent; Thin Shell Theory;
Citations & Related Records
Times Cited By KSCI : 9  (Citation Analysis)
연도 인용수 순위
1 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.   DOI
2 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   DOI
3 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.   DOI
4 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.   DOI
5 Tadi Beni, Y. (2016), "Size-dependent analysis of piezoelectric nanobeams including electro-mechanical coupling", Mech. Res. Commun., 75, 67-80.   DOI
6 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.   DOI
7 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.   DOI
8 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.   DOI
9 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.
10 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.   DOI
11 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.   DOI
12 Toupin, R.A. (1962), "Elastic materials with couple stresses", Arch. Rat. Mech. Anal., 11, 385-414.   DOI
13 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.   DOI
14 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.   DOI
15 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.   DOI
16 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.   DOI
17 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.   DOI
18 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.   DOI
19 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.   DOI
20 Zeighampour, H. and Tadi Beni, Y. (2015), "A shear deformable cylindrical shell model based on couple stress theory", Arch. Appl. Mech., 85, 539-553.   DOI
21 Zhao, J. and Pedroso, D. (2008), "Strain gradient theory in orthogonal curvilinear coordinates", Int. J. Solid. Struct., 45, 3507-3520.   DOI
22 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.
23 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.   DOI
24 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.   DOI
25 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.
26 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.   DOI
27 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.   DOI
28 Koiter, W.T. (1964), "Couple stresses in the theory of elasticity", Proc. Koninklijke Nederl. Akaad. van Wetensch, 67, 17-44.
29 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.   DOI
30 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.   DOI
31 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.   DOI
32 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.   DOI
33 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.   DOI
34 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.
35 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.   DOI
36 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.   DOI
37 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.   DOI
38 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.   DOI
39 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.   DOI
40 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.   DOI
41 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.   DOI
42 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.   DOI
43 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.   DOI
44 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.   DOI
45 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.   DOI
46 Chong, C.M. and Lam, D.C.C. (1999), "Strain gradient plasticity effect in indentation hardness of polymers", J. Mater. Res., 14, 4103-4110.   DOI
47 Chyuan, S.W. (2008), "Computational simulation for MEMS comb drive levitation using FEM", J. Electrost., 66, 361-365.   DOI
48 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.   DOI
49 Eringen, A.C. (1980), Mechanics of Continua, R.E. Krieger Pub. Co.
50 Fleck, N.A. and Hutchinson, J.W. (1997), "Strain gradient plasticity", Eds. John, W.H. & Theodore, Y.W., Adv. Appl. Mech., 295-361.
51 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.   DOI
52 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.
53 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.   DOI
54 Mindlin, R.D. and Tiersten, H.F. (1962), "Effects of couple-stresses in linear elasticity", Arch. Rat. Mech. Anal., 11, 415-448.   DOI
55 Mehralian, F. and Tadi Beni, Y. (2016), "Size-dependent torsional buckling analysis of functionally graded cylindrical shell", Compos. Part B: Eng., 94, 11-25.   DOI
56 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.
57 Mindlin, R.D. (1964), "Micro-structure in linear elasticity", Arch. Rat. Mech. Anal., 16, 51-78.
58 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.   DOI
59 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.   DOI
60 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.   DOI
61 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.   DOI
62 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.   DOI