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Acta Mechanica

, Volume 230, Issue 12, pp 4157–4174 | Cite as

Nonlinear vibration analysis of graphene sheets resting on Winkler–Pasternak elastic foundation using an atomistic-continuum multiscale model

  • Y. Gholami
  • A. Shahabodini
  • R. AnsariEmail author
  • H. Rouhi
Original Paper
  • 79 Downloads

Abstract

Based on the higher-order Cauchy–Born (HCB) rule, an atomistic-continuum multiscale model is proposed to address the large-amplitude vibration problem of graphene sheets (GSs) embedded in an elastic medium under various kinds of boundary conditions. By HCB, a linkage is established between the deformation of the atomic structure and macroscopical deformation gradients without any parameter fitting. The elastic foundation is formulated according to the Winkler–Pasternak model which considers both normal pressure and transverse shear stress effects. The weak form of nonlinear governing equations is derived via a variational approach, namely based on the variational differential quadrature (VDQ) method and Hamilton’s principle. In order to solve the obtained equations, a numerical scheme is adopted in which the generalized differential quadrature (GDQ) method together with a numerical Galerkin technique is utilized for discretization in the space domain, and the time-periodic discretization method is used to discretize in the time domain. The effects of the arrangement of atoms, the Winkler and Pasternak coefficients of the elastic foundation, and boundary conditions on the frequency–response curves of GSs are illustrated. It is revealed that the nonlinear effects on the response of GSs with larger size in armchair direction are less important.

List of symbols

N and M

Total grid points in the \(X_1 \) and \(X_2 \) directions

\(\mathbb {U}^{\mathrm {T}}\)

Displacement variables at grid points

\({\varvec{D}}_{X_{1}}^{(p)}\) and \({\varvec{D}}_{X_{2}}^{(q)}\)

Weighting coefficient matrices of the pth- and qth-order derivatives in the x and y directions

f(x)

A variable function on the domain \(\left[ {x_1,\ldots ,x_N } \right] \)

\(\mathbb {F}, \mathbb {G}\)

Discretized tensors of deformation gradients

\({\mathbb {S}}\)

2D integral operator

\(\mathbf{M}\)

Mass matrix

\(\omega \)

Linear frequency

\({\varvec{\upphi }}\)

Galerkin basis function

\(\mathbb {R}\)

Residual vector

\(\mathbb {q}\)

Mode shapes

\({\Omega }\)

Nonlinear frequency

T

Time period

Res

Nonlinear residual

t

Time

\(\mathbf{M}^{FF}, \mathbf{M}^{FG}\)

Tangent modulus

\(\mathbf{M}^{GF}, \mathbf{M}^{GG}\)

Tensors

a

Lengths of single-layered GS in the armchair direction

b

Lengths of single-layered GS in the zigzag direction

\(\Phi \)

Deformation map

\(\Pi _{s}\)

Internal energy

\(\Pi _{e}\)

Energy due to elastic foundation

\(\mathbf{F}\)

First-order deformation gradient

\(\mathbf{G}\)

Second-order deformation gradient

\(\mathbf{R}\)

Lattice vector in the reference configuration

\(\mathbf{r}\)

Deformed one in the current configuration

\(V_{IJ} \)

Interatomic potential function

\(\mathbf{P}\)

First Piola–Kirchhoff stress tensor

\(\mathbf{Q}\)

Higher-order stress tensor

\(V_\mathrm{R} \)

Repulsive and attractive pair term

\(V_\mathrm{A} \)

Attractive pair term

\(r_{IK}\)

Bond length between atoms I and K

\(\theta _{IJK}\)

Angle between bonds \(I{-}J\) and \(I{-}K\)

\(B_{IJ}\)

Multi-body coupling

\(\forall _c \)

Average area per atom

\(W_0 \)

Strain energy density

\({{\varvec{\upeta }}}\)

Relaxed inner displacement

\(k_\mathrm{w} \)

Winkler coefficient of the foundation

\(k_\mathrm{g}\)

Pasternak shear coefficient of the foundation

\(\mathbf{E}_1, \mathbf{E}_2\)

First- and second-order derivations with respect to the material coordinates

\({\Pi }_\mathrm{T} \)

Kinetic energy

\(\rho \)

Real mass density

\(m_\mathrm{c} \)

Mass of a carbon atom

Notes

References

  1. 1.
    Novoselov, K.S., Geim, A.K., Morozov, S.V., Jiang, D., Zhang, Y., Dubonos, S.V., Grigorieva, I.V., Firsov, A.: Electric field effect in atomically thin carbon films. Science 306, 666–669 (2004)CrossRefGoogle Scholar
  2. 2.
    Lee, C., Wei, X., Kysar, J.W., Hone, J.: Measurement of the elastic properties and intrinsic strength of monolayer graphene. Science 321, 385–388 (2008)CrossRefGoogle Scholar
  3. 3.
    Nair, R.R., Blake, P., Grigorenko, A.N., Novoselov, K.S., Booth, T.J., Stauber, T., Peres, N.M.R., Geim, A.K.: Fine structure constant defines visual transparency of graphene. Science 320, 1308–1308 (2008)CrossRefGoogle Scholar
  4. 4.
    Eringen, A.C.: Nonlocal polar elastic continua. Int. J. Eng. Sci. 10, 1–16 (1972)MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Eringen, A.C., Edelen, D.G.B.: On nonlocal elasticity. Int. J. Eng. Sci. 10, 233–248 (1972)MathSciNetzbMATHCrossRefGoogle Scholar
  6. 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)CrossRefGoogle Scholar
  7. 7.
    Gurtin, M.E., Murdoch, A.I.: A continuum theory of elastic material surface. Arch. Ration. Mech. Anal. 57, 291–323 (1975)MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Gurtin, M.E., Murdoch, A.I.: Surface stress in solids. Int. J. Solids Struct. 14, 431–440 (1978)zbMATHCrossRefGoogle Scholar
  9. 9.
    Ansari, R., Arash, B., Rouhi, H.: Vibration characteristics of embedded multi-layered graphene sheets with different boundary conditions via nonlocal elasticity. Compos. Struct. 93, 2419–2429 (2011)CrossRefGoogle Scholar
  10. 10.
    Ansari, R., Rouhi, H.: Explicit analytical expressions for the critical buckling stresses in a monolayer graphene sheet based on nonlocal elasticity. Solid State Commun. 152, 56–59 (2012)CrossRefGoogle Scholar
  11. 11.
    Farajpour, A., Dehghany, M., Shahidi, A.R.: Surface and nonlocal effects on the axisymmetric buckling of circular graphene sheets in thermal environment. Compos. B 50, 333–343 (2013)CrossRefGoogle Scholar
  12. 12.
    Zhang, L.W., Zhang, Y., Liew, K.M.: Vibration analysis of quadrilateral graphene sheets subjected to an in-plane magnetic field based on nonlocal elasticity theory. Compos. B 118, 96–103 (2017)CrossRefGoogle Scholar
  13. 13.
    Jalaei, M.H., Civalek, O.: A nonlocal strain gradient refined plate theory for dynamic instability of embedded graphene sheet including thermal effects. Compos. Struct. 220, 209–220 (2019)CrossRefGoogle Scholar
  14. 14.
    Kumar, D., Heinrich, C., Waas, A.M.: Buckling analysis of carbon nanotubes modeled using nonlocal continuum theories. J. Appl. Phys. 103, 073521 (2008)CrossRefGoogle Scholar
  15. 15.
    Rouhi, H., Ansari, R.: Nonlocal analytical Flugge shell model for axial buckling of double-walled carbon nanotubes with different end conditions. NANO 7, 1250018 (2012)CrossRefGoogle Scholar
  16. 16.
    Sun, Y.G., Yao, X.H., Liang, Y.J., Han, Q.: Nonlocal beam model for axial buckling of carbon nanotubes with surface effect. EPL 99, 56007 (2012)CrossRefGoogle Scholar
  17. 17.
    Ansari, R., Rouhi, H.: Analytical treatment of the free vibration of single-walled carbon nanotubes based on the nonlocal Flugge shell theory. ASME J. Eng. Mater. Technol. 134, 011008 (2012)CrossRefGoogle Scholar
  18. 18.
    Ghorbanpour Arani, A., Roudbari, M.A.: Nonlocal piezoelastic surface effect on the vibration of visco-Pasternak coupled boron nitride nanotube system under a moving nanoparticle. Thin Solid Films 542, 232–241 (2013)CrossRefGoogle Scholar
  19. 19.
    Ansari, R., Rouhi, H., Arash, B.: Vibrational analysis of double-walled carbon nanotubes based on the nonlocal Donnell shell theory via a new numerical approach. Iran. J. Sci. Technol. Trans. Mech. Eng. 37, 91–105 (2013)Google Scholar
  20. 20.
    Kiani, K., Roshan, M.: Nonlocal dynamic response of double-nanotube-systems for delivery of lagged-inertial-nanoparticles. Int. J. Mech. Sci. 152, 576–595 (2019)CrossRefGoogle Scholar
  21. 21.
    Rouhi, H., Ansari, R., Darvizeh, M.: Size-dependent free vibration analysis of nanoshells based on the surface stress elasticity. Appl. Math. Model. 40, 3128–3140 (2016)MathSciNetzbMATHCrossRefGoogle Scholar
  22. 22.
    Rouhi, H., Ansari, R., Darvizeh, M.: Nonlinear free vibration analysis of cylindrical nanoshells based on the Ru model accounting for surface stress effect. Int. J. Mech. Sci. 113, 1–9 (2016)zbMATHCrossRefGoogle Scholar
  23. 23.
    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)MathSciNetzbMATHCrossRefGoogle Scholar
  24. 24.
    Ansari, R., Rouhi, H., Sahmani, S.: Calibration of the analytical nonlocal shell model for vibrations of double-walled carbon nanotubes with arbitrary boundary conditions using molecular dynamics. Int. J. Mech. Sci. 53, 786–792 (2011)CrossRefGoogle Scholar
  25. 25.
    Batra, R.C., Gupta, S.S.: Wall thickness and radial breathing modes of single-walled carbon nanotubes. ASME J. Appl. Mech. 75, 061010 (2008)CrossRefGoogle Scholar
  26. 26.
    Shah, P.H., Batra, R.C.: In-plane elastic moduli of covalently functionalized single-wall carbon nanotubes. Comput. Mater. Sci. 83, 349–361 (2014)CrossRefGoogle Scholar
  27. 27.
    Arroyo, M., Belytschko, T.: An atomistic-based finite deformation membrane for single layer crystalline films. J. Mech. Phys. Solids 50, 1941–1977 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  28. 28.
    Zhang, P., Huang, Y., Geubelle, P.H., Klein, P.A., Hwang, K.C.: The elastic modulus of single-wall carbon nanotubes: a continuum analysis incorporating interatomic potentials. Int. J. Solid Struct. 39, 3893–3906 (2002)zbMATHCrossRefGoogle Scholar
  29. 29.
    Arroyo, M., Belytschko, T.: Finite crystal elasticity of carbon nanotubes based on the exponential Cauchy–Born rule. Phys. Rev. B 69, 115415 (2004)CrossRefGoogle Scholar
  30. 30.
    Guo, X., Wang, J.B., Zhang, H.W.: Mechanical properties of single-walled carbon nanotubes based on higher order Cauchy–Born rule. Int. J. Solids Struct. 43, 1276–1290 (2006)zbMATHCrossRefGoogle Scholar
  31. 31.
    Stefan, H.: Numerical validation of a concurrent atomistic-continuum multiscale method and its application to the buckling analysis of carbon nanotubes. Comput. Methods Appl. Mech. Eng. 270, 220–246 (2014)MathSciNetzbMATHCrossRefGoogle Scholar
  32. 32.
    Sun, Y., Liew, K.M.: Mesh-free simulation of single-walled carbon nanotubes using higher order Cauchy–Born rule. Comput. Mater. Sci. 42, 444–452 (2008)CrossRefGoogle Scholar
  33. 33.
    Sun, Y., Liew, K.M.: The buckling of single-walled carbon nanotubes upon bending: the higher order gradient continuum and mesh-free method. Comput. Methods Appl. Mech. Eng. 197, 3001–3013 (2008)zbMATHCrossRefGoogle Scholar
  34. 34.
    Sun, Y., Liew, K.M.: Application of the higher-order Cauchy-Born rule in mesh-free continuum and multiscale simulation of carbon nanotubes. Int. J. Numer. Methods Eng. 75, 1238–1258 (2008)zbMATHCrossRefGoogle Scholar
  35. 35.
    Ansari, R., Shahabodini, A., Rouhi, H., Alipour, A.: Thermal buckling analysis of multi-walled carbon nanotubes through a nonlocal shell theory incorporating interatomic potentials. J. Therm. Stress. 36, 56–70 (2013)CrossRefGoogle Scholar
  36. 36.
    Ansari, R., Shahabodini, A., Rouhi, H.: Prediction of the biaxial buckling and vibration behavior of graphene via a nonlocal atomistic-based plate theory. Compos. Struct. 95, 88–94 (2013)CrossRefGoogle Scholar
  37. 37.
    Ansari, R., Shahabodini, A., Rouhi, H.: A thickness-independent nonlocal shell model for describing the stability behavior of carbon nanotubes under compression. Compos. Struct. 100, 323–331 (2013)CrossRefGoogle Scholar
  38. 38.
    Ansari, R., Shahabodini, A., Rouhi, H.: A nonlocal plate model incorporating interatomic potentials for vibrations of graphene with arbitrary edge conditions. Curr. Appl. Phys. 15, 1062–1069 (2015)CrossRefGoogle Scholar
  39. 39.
    Wang, X., Guo, X.: Quasi-continuum model for the finite deformation of single layer graphene sheets based on the temperature-related higher order Cauchy–Born rule. J. Comput. Theor. Nanosci. 10, 154–164 (2013)CrossRefGoogle Scholar
  40. 40.
    Singh, S., Patel, B.P.: Nonlinear elastic properties of graphene sheet under finite deformation. Compos. Struct. 119, 412–421 (2015)CrossRefGoogle Scholar
  41. 41.
    Singh, S., Patel, B.P.: Nonlinear dynamic response of single layer graphene sheets using multiscale modelling. Eur. J. Mech. A/Solids 59, 165–177 (2016)MathSciNetzbMATHCrossRefGoogle Scholar
  42. 42.
    Singh, S., Patel, B.P.: A computationally efficient multiscale finite element formulation for dynamic and postbuckling analyses of carbon nanotubes. Comput. Struct. 195, 126–144 (2018)CrossRefGoogle Scholar
  43. 43.
    Shahabodini, A., Ansari, R., Darvizeh, M.: Multiscale evaluation of the nonlinear elastic properties of carbon nanotubes under finite deformation. J. Ultrafine Grained Nanostruct. Mater. 50, 60–80 (2017)Google Scholar
  44. 44.
    Shahabodini, A., Ansari, R., Darvizeh, M.: Multiscale modeling of embedded graphene sheets based on the higher-order Cauchy–Born rule: nonlinear static analysis. Compos. Struct. 165, 25–43 (2017) CrossRefGoogle Scholar
  45. 45.
    Shahabodini, A., Ansari, R., Darvizeh, M.: Atomistic-continuum modeling of vibrational behavior of carbon nanotubes using the variational differential quadrature method. Compos. Struct. 185, 728–747 (2018)CrossRefGoogle Scholar
  46. 46.
    Khoei, A.R., Jahanshahi, M., Toloui, G.: Validity of Cauchy–Born hypothesis in multi-scale modeling of plastic deformations. Int. J. Solids Struct. 115–116, 224–247 (2017)CrossRefGoogle Scholar
  47. 47.
    Faghih Shojaei, M., Ansari, R.: Variational differential quadrature: a technique to simplify numerical analysis of structures. Appl. Math. Model. 49, 705–738 (2017)MathSciNetCrossRefGoogle Scholar
  48. 48.
    Tersoff, J.: New empirical approach for the structure and energy of covalent systems. Phys. Rev. B 37, 6991 (1988)CrossRefGoogle Scholar
  49. 49.
    Brenner, D.W.: Empirical potential for hydrocarbons for use in simulating the chemical vapor deposition of diamond films. Phys. Rev. B 42, 9458 (1990)CrossRefGoogle Scholar
  50. 50.
    Shu, C.: Differential Quadrature and Its Application in Engineering. Springer, London (2000)zbMATHCrossRefGoogle Scholar
  51. 51.
    Faghih Shojaei, M., Ansari, R., Mohammadi, V., Rouhi, H.: Nonlinear forced vibration analysis of postbuckled beams. Arch. Appl. Mech. 84, 421–440 (2014)zbMATHCrossRefGoogle Scholar
  52. 52.
    Gholami, R., Ansari, R.: A most general strain gradient plate formulation for size-dependent geometrically nonlinear free vibration analysis of functionally graded shear deformable rectangular microplates. Nonlinear Dyn. 84, 2403–2422 (2016)MathSciNetCrossRefGoogle Scholar
  53. 53.
    Keller, H.B.: Numerical Solution of Bifurcation and Nonlinear Eigenvalue Problems, Applications of Bifurcation Theory, (Proceedings Advanced Seminars, University Wisconsin, Madison, WI, 1976), pp. 359–384. Academic Press, New York (1977)Google Scholar

Copyright information

© Springer-Verlag GmbH Austria, part of Springer Nature 2019

Authors and Affiliations

  • Y. Gholami
    • 1
  • A. Shahabodini
    • 2
  • R. Ansari
    • 1
    Email author
  • H. Rouhi
    • 3
  1. 1.Department of Mechanical EngineeringUniversity of GuilanRashtIran
  2. 2.Department of Mechanical EngineeringVali-e-Asr University of RafsanjanRafsanjanIran
  3. 3.Department of Engineering Science, Faculty of Technology and Engineering, East of GuilanUniversity of GuilanRudsar-VajargahIran

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