Indentation response of two-dimensional materials mounted on different substrates.

Indentation behavior of two-dimensional (2D) materials mounted on the substrates with varying stiffness is investigated using finite element method (FEM). The primary objective of present work is to understand the effect of the ratio of the elastic modulus of 2D-material to that of the substrate ( λ ) on the indentation deformation of the composite structure (2D-material/substrate). It is found that with the increase of λ , the deformations of both 2D-material and substrate become less localized; the contact area rapidly decreases with λ , which cannot be properly predicted by the classic indentation models. Therefore, for a large value of λ (e.g., λ  > 10 4 ), the existing indentation models are not able to properly describe the indentation response of 2D-material/substrate. In addition, due to an ultra-low thickness of 2D-material, the indentation behavior of 2D-material/substrate is mainly controlled by the substrate properties when λ is lower than 10 3 ; whereas the ratio of the strain energy of 2D-maerial to that of the overall strain energy rapidly increases with λ , which will reach the peak value of 1/3 at λ  = 10 6 . Therefore, for a high λ , it is possible to determine the elastic modulus of 2D-material from the overall indentation response of 2D-material/substrate by the inverse analysis with the help of FEM. [ABSTRACT FROM AUTHOR]