Nanoindentation of circular multilayer graphene allotropes

Nanoindentaion has been proposed as an efficient technique to measure mechanical single-layer two-dimensional(2D) materials via combining the membrane theory with the indentation data. However, for multilayered structures of 2D materials, significant discrepancy exists between the Young's modulus obtained from the existing membrane model and those from other methods.Here we develop a multilayer indentation model by taking the multilayer effect into account in the previous membrane model.We show that the present model can accurately predict the Young's modulus of multilayered 2D carbon materials. For few layer graphene and twin graphene structures, the deviation of the Young's moduli obtained by the present model are both within a reasonable range, while the error caused by the direct use of the previous single-layer membrane model increases with the number of layers. The present model provides an efficient tool to extract the mechanical properties of 2D materials from the nanoindentation data of their multilayered structures.