Finite element modeling of the indentation behavior of two-dimensional materials

Abstract

The finite element method is used to investigate the indentation behavior of two-dimensional (2D) materials mounted on a substrate. The overall indentation response of the composite structure of 2D-material/substrate is highly sensitive to the elastic modulus ratio of the 2D-material to its substrate ( $$\lambda $$ ). When $$\lambda $$ is small (e.g., $$\lambda < 100$$ ), the overall indentation load–displacement relationship agrees with the classic indentation model (e.g., the Hertz model), whereas with a large $$\lambda $$ (e.g., $$\lambda \ge 10^{3}$$ ), the indentation behavior of the composite structure will deviate from the manner predicted by the classic indentation model. In addition, with a small $$\lambda $$ , the overall indentation modulus of the composite structure is very close to that of the pure substrate (i.e., the 2D-material has a very weak contribution to the overall indentation modulus), and thus, the elastic modulus of the 2D-material cannot be effectively determined from the overall indentation modulus. The contribution of the 2D-material rapidly increases with $$\lambda $$ , and when $$\lambda > 10^{4}$$ , it is possible to accurately determine the elastic modulus of the 2D-material from the overall indentation response by the inverse analysis.