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


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


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


Mass matrix

\(\omega \)

Linear frequency

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

Galerkin basis function

\(\mathbb {R}\)

Residual vector

\(\mathbb {q}\)

Mode shapes

\({\Omega }\)

Nonlinear frequency


Time period


Nonlinear residual



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

Tangent modulus

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



Lengths of single-layered GS in the armchair direction


Lengths of single-layered GS in the zigzag direction

\(\Phi \)

Deformation map

\(\Pi _{s}\)

Internal energy

\(\Pi _{e}\)

Energy due to elastic foundation


First-order deformation gradient


Second-order deformation gradient


Lattice vector in the reference configuration


Deformed one in the current configuration

\(V_{IJ} \)

Interatomic potential function


First Piola–Kirchhoff stress tensor


Higher-order stress tensor

\(V_\mathrm{R} \)

Repulsive and attractive pair term

\(V_\mathrm{A} \)

Attractive pair term


Bond length between atoms I and K

\(\theta _{IJK}\)

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


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


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



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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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