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Acta Mechanica Sinica

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30.03.2017 | Research Paper

An analytical symplectic approach to the vibration analysis of orthotropic graphene sheets

verfasst von: Xinsheng Xu, Dalun Rong, C. W. Lim, Changyu Yang, Zhenhuan Zhou

Erschienen in: Acta Mechanica Sinica | Ausgabe 5/2017

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Abstract

A nonlocal continuum orthotropic plate model is proposed to study the vibration behavior of single-layer graphene sheets (SLGSs) using an analytical symplectic approach. A Hamiltonian system is established by introducing a total unknown vector consisting of the displacement amplitude, rotation angle, shear force, and bending moment. The high-order governing differential equation of the vibration of SLGSs is transformed into a set of ordinary differential equations in symplectic space. Exact solutions for free vibration are obtianed by the method of separation of variables without any trial shape functions and can be expanded in series of symplectic eigenfunctions. Analytical frequency equations are derived for all six possible boundary conditions. Vibration modes are expressed in terms of the symplectic eigenfunctions. In the numerical examples, comparison is presented to verify the accuracy of the proposed method. Comprehensive numerical examples for graphene sheets with Levy-type boundary conditions are given. A parametric study of the natural frequency is also included.

Vorheriger Artikel Plate/shell structure topology optimization of orthotropic material for buckling problem based on independent continuous topological variables

Nächster Artikel The vibroacoustic response and sound absorption performance of multilayer, microperforated rib-stiffened plates

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Gibson, R.F., Ayorinde, E.O., Wen, Y.F.: Vibrations of carbon nanotubes and their composites: a review. Compos. Sci. Technol. 67, 1–28 (2007). doi:10.1016/j.compscitech.2006.03.031 CrossRef

Wang, K.F., Wang, B.L., Kitamura, T.: A review on the application of modified continuum models in modeling and simulation of nanostructures. Acta Mech. Sin. 32, 83–100 (2016). doi:10.1007/s10409-015-0508-4 MathSciNetCrossRefMATH

Eringen, A.C.: Nonlocal Continuum Field Theories. Springer, New York (2002)MATH

Eringen, A.C.: On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves. J. Appl. Phys. 54, 4703–4710 (1983). doi:10.1063/1.332803 CrossRef

Arash, B., Wang, Q.: A review on the application of nonlocal elastic models in modeling of carbon nanotubes and graphenes. Comput. Mater. Sci. 51, 303–313 (2012). doi:10.1016/j.commatsci.2011.07.040 CrossRef

Wang, C.Y., Murmu, T., Adhikari, S.: Mechanisms of nonlocal effect on the vibration of nanoplates. Appl. Phys. Lett. 98, 153101 (2011). doi:10.1063/1.3579249 CrossRef

Sobhy, M.: Thermomechanical bending and free vibration of single-layered graphene sheets embedded in an elastic medium. Phys. E 56, 400–409 (2014). doi:10.1016/j.physe.2013.10.017 CrossRef

Kitipornchai, S., He, X.Q., Liew, K.M.: Continuum model for the vibration of multilayered graphene sheets. Phys. Rev. B 72, 075443 (2005). doi:10.1103/PhysRevB.72.075443 CrossRef

Karličić, D., Adhikari, S., Murmu, T., et al.: Exact closed-form solution for non-local vibration and biaxial buckling of bonded multi-nanoplate system. Compos. B Eng. 66, 328–339 (2014). doi:10.1016/j.compositesb.2014.05.029 CrossRef

10.

He, X.Q., Kitipornchai, S., Liew, K.M.: Resonance analysis of multi-layered graphene sheets used as nanoscale resonators. Nanotechnology 16, 2086 (2005). doi:10.1088/0957-4484/16/10/018 CrossRef

11.

Liew, K.M., He, X.Q., Kitipornchai, S.: Predicting nanovibration of multi-layered graphene sheets embedded in an elastic matrix. Acta Mater. 54, 4229–4236 (2006). doi:10.1016/j.actamat.2006.05.016 CrossRef

12.

Reddy, C.D., Rajendran, S., Liew, K.M.: Equilibrium configuration and continuum elastic properties of finite sized graphene. Nanotechnology 17, 864 (2006). doi:10.1088/0957-4484/17/3/042 CrossRef

13.

Pradhan, S.C., Kumar, A.: Vibration analysis of orthotropic graphene sheets embedded in Pasternak elastic medium using nonlocal elasticity theory and differential quadrature method. Comput. Mater. Sci. 50, 239–245 (2010). doi:10.1016/j.commatsci.2010.08.009 CrossRef

14.

Lu, P., Zhang, P.Q., Lee, H.P., et al.: Non-local elastic plate theories. Proc. R. Soc. Lond. A Math. Phys. Sci. 463, 3225–3240 (2007). doi:10.1098/rspa.2007.1903 MathSciNetCrossRefMATH

15.

Tsiatas, G.C., Yiotis, A.J.: A BEM-based meshless solution to buckling and vibration problems of orthotropicplates. Anal. Bound. Elem. 37, 579–584 (2013). doi:10.1016/j.enganabound.2013.01.007 MathSciNetCrossRefMATH

16.

Analooei, H.R., Azhari, M., Heidarpour, A.: Elastic buckling and vibration analyses of orthotropic nanoplates using nonlocal continuum mechanics and spline finite strip method. Appl. Math. Model. 37, 6703–6717 (2013). doi:10.1016/j.apm.2013.01.051 MathSciNetCrossRef

17.

Shen, H.S.: Nonlocal plate model for nonlinear analysis of thin films on elastic foundations in thermal environments. Compos. Struct. 93, 1143–1152 (2011). doi:10.1016/j.compstruct.2010.10.009 CrossRef

18.

Pradhan, S.C., Kumar, A.: Vibration analysis of orthotropic graphene sheets using nonlocal elasticity theory and differential quadrature method. Compos. Struct. 93, 774–779 (2011). doi:10.1016/j.compstruct.2010.08.004 CrossRef

19.

Setoodeh, A.R., Malekzadeh, P., Vosoughi, A.R.: Nonlinear free vibration of orthotropic graphene sheets using nonlocal Mindlin plate theory. P. I. Mech. Eng. C. J. Mech. Eng. Sci. 226, 1896–1906 (2012). doi:10.1177/0954406211428997 CrossRef

20.

Malekzadeh, P., Setoodeh, A.R., Alibeygi Beni, A.: Small scale effect on the free vibration of orthotropic arbitrary straight-sided quadrilateral nanoplates. Compos. Struct. 93, 1631–1639 (2011). doi:10.1016/j.compstruct.2011.01.008 CrossRef

21.

Shahidi, A.R., Anjomshoa, A., Shahidi, S.H., et al.: Fundamental size dependent natural frequencies of non-uniform orthotropic nano scaled plates using nonlocal variational principle and finite element method. Appl. Math. Model. 37, 7047–7061 (2013). doi:10.1016/j.apm.2013.02.015 MathSciNetCrossRef

22.

Behfar, K., Naghdabadi, R.: Nanoscale vibrational analysis of a multi-layered graphene sheet embedded in an elastic medium. Compos. Sci. Technol. 65, 1159–1164 (2005). doi:10.1016/j.compscitech.2004.11.011 CrossRef

23.

Pouresmaeeli, S., Fazelzadeh, S.A., Ghavanloo, E.: Exact solution for nonlocal vibration of double-orthotropic nanoplates embedded in elastic medium. Compos. B Eng. 43, 3384–3390 (2012). doi:10.1016/j.compositesb.2012.01.046 CrossRef

24.

Pradhan, S.C., Phadikar, J.K.: Small scale effect on vibration of embedded multilayered graphene sheets based on nonlocal continuum models. Phys. Lett. A 373, 1062–1069 (2009). doi:10.1016/j.physleta.2009.01.030 CrossRefMATH

25.

Karličić, D., Kozić, P., Pavlović, R.: Free transverse vibration of nonlocal viscoelastic orthotropic multi-nanoplate system (MNPS) embedded in a viscoelastic medium. Compos. Struct. 115, 89–99 (2014). doi:10.1016/j.compstruct.2014.04.002 CrossRef

26.

Pouresmaeeli, S., Ghavanloo, E., Fazelzadeh, S.A.: Vibration analysis of viscoelastic orthotropic nanoplates resting on viscoelastic medium. Compos. Struct. 96, 405–410 (2013). doi:10.1016/j.compstruct.2012.08.051 CrossRef

27.

Karličić, D., Cajić, M., Kozić, P., et al.: Temperature effects on the vibration and stability behavior of multi-layered graphene sheets embedded in an elastic medium. Compos. Struct. 131, 672–681 (2015). doi:10.1016/j.compstruct.2015.05.058 CrossRef

28.

Satish, N., Narendar, S., Gopalakrishnan, S.: Thermal vibration analysis of orthotropic nanoplates based on nonlocal continuum mechanics. Phys. E 44, 1950–1962 (2012). doi:10.1016/j.physe.2012.05.024 CrossRef

29.

Mohammadi, M., Moradi, A., Ghayour, M., et al.: Exact solution for thermo-mechanical vibration of orthotropic mono-layer graphene sheet embedded in an elastic medium. Lat. Am. J. Solids Struct. 11, 437–458 (2014). doi:10.1590/S1679-78252014000300005 CrossRef

30.

Mohammadi, M., Farajpour, A., Goodarzi, M., et al.: Levy type solution for nonlocal thermo-mechanical vibration of orthotropic mono-layer graphene sheet embedded in an elastic medium. J. Solid Mech. 5, 116–132 (2013)

31.

Zhong, W.X.: A New Systematic Methodology for Theory of Elasticity. Publishing House of Dalian University of Technology Dalian, Dalian (1995). (in Chinese)

32.

Yao, W.A., Zhong, W.X., Lim, C.W.: Symplectic Elasticity. World Scientific, Singapore (2009)CrossRefMATH

33.

Timoshenko, S.P., Goodier, J.N.: Theory of Elasticity. McGraw-Hill, New York (1970)MATH

34.

Lim, C.W., Xu, X.S.: Symplectic elasticity: theory and applications. Appl. Mech. Rev. 63, 050802 (2011). doi:10.1115/1.4003700 CrossRef

35.

Zhao, L., Chen, W.Q.: Plane analysis for functionally graded magneto-electro-elastic materials via the symplectic framework. Compos. Struct. 92, 1753–1761 (2010). doi:10.1016/j.compstruct.2009.11.029 CrossRef

36.

Zhao, L., Chen, W.Q., Lü, C.F.: Symplectic elasticity for bi-directional functionally graded materials. Mech. Mater. 54, 32–42 (2012). doi:10.1016/j.mechmat.2012.06.001 CrossRef

37.

Zhou, Z., Xu, X., Leung, A.Y.T., et al.: Stress intensity factors and T-stress for an edge interface crack by symplectic expansion. Eng. Fract. Mech. 102, 334–347 (2013). doi:10.1016/j.engfracmech.2013.03.007 CrossRef

38.

Zhou, Z., Leung, A.Y.T., Xu, X., et al.: Mixed-mode thermal stress intensity factors from the finite element discretized symplectic method. Int. J. Solids Struct. 51, 3798–3806 (2014). doi:10.1016/j.ijsolstr.2014.07.016 CrossRef

39.

Li, R., Zhong, Y., Li, M.: Analytic bending solutions of free rectangular thin plates resting on elastic foundations by a new symplectic superposition method. Proc. R. Soc. Lond. Ser. A 469, 20120681 (2013). doi:10.1098/rspa.2012.0681 MathSciNetCrossRef

40.

Li, R., Wang, B., Li, G., et al.: Analytic free vibration solutions of rectangular thin plates point-supported at a corner. Int. J. Mech. Sci. 96–97, 199–205 (2015). doi:10.1016/j.ijmecsci.2015.04.004 CrossRef

41.

Lim, C.W., Lü, C.F., Xiang, Y., et al.: On new symplectic elasticity approach for exact free vibration solutions of rectangular Kirchhoff plates. Int. J. Eng. Sci. 47, 131–140 (2009). doi:10.1016/j.ijengsci.2008.08.003 MathSciNetCrossRefMATH

42.

Lim, C.W., Cui, S., Yao, W.A.: On new symplectic elasticity approach for exact bending solutions of rectangular thin plates with two opposite sides simply supported. Int. J. Solids Struct. 44, 5396–5411 (2007). doi:10.1016/j.ijsolstr.2007.01.007

43.

Sun, J., Xu, X., Lim, C.W.: Localization of dynamic buckling patterns of cylindrical shells under axial impact. Int. J. Mech. Sci. 66, 101–108 (2013). doi:10.1016/j.ijmecsci.2012.10.012 CrossRef

44.

Andrianarison, O., Benjeddou, A.: Hamiltonian partial mixed finite element-state space symplectic semi-analytical approach for the piezoelectric smart composites and FGM analysis. Acta Mech. 223, 1597–1610 (2012). doi:10.1007/s00707-012-0646-8 MathSciNetCrossRefMATH

45.

Tarn, J.Q., Tseng, W.D.: Exact analysis of curved beams and arches with arbitrary end conditions: a Hamiltonian state space approach. J. Elast. 107, 39–63 (2012). doi:10.1007/s10659-011-9335-4 MathSciNetCrossRefMATH

46.

Zhang, K., Deng, Z., Meng, J., et al.: Symplectic analysis of dynamic properties of hexagonal honeycomb sandwich tubes with plateau borders. J. Sound Vib. 351, 177–188 (2015). doi:10.1016/j.jsv.2015.04.012 CrossRef

47.

Li, R., Li, M., Su, Y., et al.: An analytical mechanics model for the island-bridge structure of stretchable electronics. Soft Matter. 9, 8476–8482 (2013). doi:10.1039/C3SM51476E CrossRef

48.

Ansari, R., Sahmani, S., Arash, B.: Nonlocal plate model for free vibrations of single-layered graphene sheets. Phys. Lett. A 375, 53–62 (2010). doi:10.1016/j.physleta.2010.10.028 CrossRef

49.

Peddieson, J., Buchanan, G.R., McNitt, R.P.: Application of nonlocal continuum models to nanotechnology. Int. J. Eng. Sci. 41, 305–312 (2003). doi:10.1016/S0020-7225(02)00210-0 CrossRef

50.

Reddy, J.N.: Mechanics of Laminated Composite Plates and Shells. CRC Press, Boca Raton (2003)

51.

Chakraverty, S., Behera, L.: Free vibration of rectangular nanoplates using Rayleigh–Ritz method. Phys. E 56, 357–363 (2014). doi:10.1016/j.physe.2013.08.014 CrossRefMATH

52.

Pradhan, S.C., Phadikar, J.K.: Nonlocal elasticity theory for vibration of nanoplates. J. Sound Vib. 325, 206–223 (2009). doi:10.1016/j.jsv.2009.03.007 CrossRefMATH

53.

Leissa, A.W.: The free vibration of rectangular plates. J. Sound Vib. 31, 257–293 (1973). doi:10.1016/S0022-460X(73)80371-2 CrossRefMATH

54.

Hemmasizadeh, A., Mahzoon, M., Hadi, E., et al.: A method for developing the equivalent continuum model of a single layer graphene sheet. Thin Solid Films 516, 7636–7640 (2008). doi:10.1016/j.tsf.2008.05.040 CrossRef

55.

Huang, Y., Wu, J., Hwang, K.C.: Thickness of graphene and single-wall carbon nanotubes. Phys. Rev. B 74, 245413 (2006). doi:10.1103/PhysRevB.74.245413 CrossRef

56.

Lee, C., Wei, X., Kysar, J.W., et al.: Measurement of the elastic properties and intrinsic strength of monolayer graphene. Science 321, 385–388 (2008). doi:10.1126/science.1157996 CrossRef

57.

Li, C., Chou, T.W.: A structural mechanics approach for the analysis of carbon nanotubes. Int. J. Solids Struct. 40, 2487–2499 (2003). doi:10.1016/S0020-7683(03)00056-8 CrossRefMATH

58.

Wang, Q.: Effective in-plane stiffness and bending rigidity of armchair and zigzag carbon nanotubes. Int. J. Solids Struct. 41, 5451–5461 (2004). doi:10.1016/j.ijsolstr.2004.05.002 CrossRefMATH

59.

Shen, L., Shen, H.S., Zhang, C.L.: Nonlocal plate model for nonlinear vibration of single layer graphene sheets in thermal environments. Comput. Mater. Sci. 48, 680–685 (2010). doi:10.1016/j.commatsci.2010.03.006 CrossRef

Titel: An analytical symplectic approach to the vibration analysis of orthotropic graphene sheets
verfasst von: Xinsheng Xu
Dalun Rong
C. W. Lim
Changyu Yang
Zhenhuan Zhou
Publikationsdatum: 30.03.2017
Verlag: The Chinese Society of Theoretical and Applied Mechanics; Institute of Mechanics, Chinese Academy of Sciences
Erschienen in: Acta Mechanica Sinica / Ausgabe 5/2017
Print ISSN: 0567-7718
Elektronische ISSN: 1614-3116
DOI: https://doi.org/10.1007/s10409-017-0656-9

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