Consistent evaluation of continuum scale properties of two-dimensional materials: a case study on graphene

We handshake statistical mechanics with continuum mechanics to develop a methodology for consistent evaluation of the continuum scale properties of two-dimensional materials. The methodology is tested on pristine graphene. Our scope is kept limited to elastic modulus, E, which has been reported to vary between 0.912 TPa and 7 TPa, Poisson's ratio, ν, which has been reported to vary from being negative to a value as large as 0.46, and effective thickness, q, whose value varies between 0.75 Å and 3.41 Å. Such a large scatter arises due to inconsistent evaluation of these properties and making assumptions that may not be valid at atomistic scales. Our methodology combines three separate methods: uniaxial tension, equibiaxial tension, and flexural out-of-plane free vibrations of simply supported sheets, which, when used in tandem in molecular dynamics, can provide consistent values of E, ν and q. The only assumption made in the present study is the validity of the continuum scale thin plate vibration equation to represent the free vibrations of a graphene sheet. Our results suggest that-(i) graphene is auxetic in nature, (ii) E decreases with increasing size and temperature, and (iii) the effective thickness q increases with increasing size and temperature. Further, a robustness study of the computed mechanical properties shows consistent results, with differences varying between 1.4% and 6%.