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

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 defor...

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Bibliographic Details
Published in:Acta mechanica 2019-12, Vol.230 (12), p.4157-4174
Main Authors: Gholami, Y., Shahabodini, A., Ansari, R., Rouhi, H.
Format: Article
Language:English
Subjects:
Analysis
Atomic structure
Boundary conditions
Classical and Continuum Physics
Control
Differential equations
Discretization
Dynamical Systems
Elastic deformation
Elastic foundations
Elastic media
Engineering
Engineering Fluid Dynamics
Engineering Thermodynamics
Galerkin method
Graphene
Graphite
Heat and Mass Transfer
Mathematical models
Nonlinear analysis
Nonlinear equations
Nonlinear response
Original Paper
Shear stress
Sheets
Solid Mechanics
Theoretical and Applied Mechanics
Vibration
Vibration analysis
Vibration control
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Summary: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.
ISSN:0001-5970
1619-6937
DOI:10.1007/s00707-019-02490-z