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NON-LOCAL MODELS FOR VIBRATIONS OF GRAPHENE SHEETS
Last modified: 2017-11-30
Abstract
The study of the modes of vibration of single-layer graphene sheets (SLGSs) is important to understand the behaviour of this relatively new material. SLGSs are very thin structures; if they experience vibrations with amplitudes that are larger than about a tenth of their thickness, then geometrical non-linear effects should not be neglected. Furthermore, due to the small dimensions of SLGSs, classical continuum theories may fail to provide accurate models. The application of Eringen's non-local elasticity to investigate the linear and non-linear modes of vibration of SLGSs is examined in this communication.
Two models are compared, both based on Kirchhoff's thin plate theory, complemented with Von Kármán's geometric non-linear terms. The first model is completely displacement based; the ordinary differential equations of motion, with generalized displacements as unknowns, are derived using Hamilton’s principle and Galerkin’s method. In the second model, an Airy stress function is employed, leading to an equation of motion where the transverse displacement and the Airy stress function are unknown. This equation is complemented by a compatibility equation, from which the Airy stress function is obtained as a function of the transverse displacement. Finally, applying Galerkin’s method, one obtains a Duffing type equation solely on the transverse displacement.
A linear analysis is performed first, starting with successful comparisons between our data and data published, followed by a study of the size and non-local effects on the linear natural frequencies and mode shapes of SLGS. Isotropic and orthotropic models are compared. Although the latter is a more accurate representation of SLGSs, it is concluded that the former is a rather accurate simplification that can be used to study of modes of vibration.
In the non-linear regime, the harmonic balance method is applied to both models, and frequency domain models with several unknowns – because they stem from a multi-degree of freedom model and/or because several harmonics are considered - are solved by an arc-length continuation method. To carry out the non-linear analysis with the non-local, displacement based model, it is necessary to neglect some terms in the equations of motion. It is verified that this approach leads to erroneous frequencies of vibration in the non-linear regime, with the error increasing with the non-local parameter and with the degree of the non-linearity. The model that employs the Airy stress function, on the other hand, conducts to results which are in very good agreement with the ones of other authors. Using the latter model and a multi-harmonic approach, the backbone curves of SLGSs are obtained for various non-local parameters and the influence of the small-scale effect is analysed on each individual harmonic considered in the periodic solution.
Two models are compared, both based on Kirchhoff's thin plate theory, complemented with Von Kármán's geometric non-linear terms. The first model is completely displacement based; the ordinary differential equations of motion, with generalized displacements as unknowns, are derived using Hamilton’s principle and Galerkin’s method. In the second model, an Airy stress function is employed, leading to an equation of motion where the transverse displacement and the Airy stress function are unknown. This equation is complemented by a compatibility equation, from which the Airy stress function is obtained as a function of the transverse displacement. Finally, applying Galerkin’s method, one obtains a Duffing type equation solely on the transverse displacement.
A linear analysis is performed first, starting with successful comparisons between our data and data published, followed by a study of the size and non-local effects on the linear natural frequencies and mode shapes of SLGS. Isotropic and orthotropic models are compared. Although the latter is a more accurate representation of SLGSs, it is concluded that the former is a rather accurate simplification that can be used to study of modes of vibration.
In the non-linear regime, the harmonic balance method is applied to both models, and frequency domain models with several unknowns – because they stem from a multi-degree of freedom model and/or because several harmonics are considered - are solved by an arc-length continuation method. To carry out the non-linear analysis with the non-local, displacement based model, it is necessary to neglect some terms in the equations of motion. It is verified that this approach leads to erroneous frequencies of vibration in the non-linear regime, with the error increasing with the non-local parameter and with the degree of the non-linearity. The model that employs the Airy stress function, on the other hand, conducts to results which are in very good agreement with the ones of other authors. Using the latter model and a multi-harmonic approach, the backbone curves of SLGSs are obtained for various non-local parameters and the influence of the small-scale effect is analysed on each individual harmonic considered in the periodic solution.