Multipole Trefftz method implementation in free vibration of a graphene sheet with multiple vacancy defects

•The vacancy defects can affect mechanical properties of graphene sheets.•Multipole Trefftz method is employed to analyze the problem.•The natural frequencies are determined using the SVD technique.•The eccentricity and number of holes have significant effects on the frequencies.

The single and multiple vacancy defects in a single layer graphene sheet can affect its mechanical properties. Also, these defects can be inverted to greater holes as amorphous construction when the graphene sheet is working in room temperature. In some cases, these defects can be modelled as circular holes. In this article, multipole Trefftz method as well as translational addition theorem are employed to consider effects of circular defects on behavior of freely vibrating circular graphene sheet. Nonlocal thin plate theory is used to solve the problem. According to the direct-searching approach, the natural frequencies are determined as the minimum singular value of the truncated finite system using the singular value decomposition (SVD) technique. The computational efficiency and convergence of the present analytical method are examined. Accuracy and stability of results are validated by literature. Effects of boundary conditions, geometrical properties and nonlocal parameter on vibrational modes are investigated. Results reveal that the eccentricity, nonlocality and number of holes have significant effects on the natural frequencies.

Graphical abstractGeometrical defects have significant influence on free vibrations of annular graphene sheet.Figure optionsDownload full-size imageDownload as PowerPoint slide