Consistent evaluation of continuum scale properties of Graphene

We handshake statistical mechanics with continuum mechanics to develop a methodology for consistent evaluation of the continuum scale properties of graphene. The scope is kept limited to elastic modulus, $E$, which has been reported to vary between 0.912 TPa to 7 TPa, Poisson's ratio, $\nu$, 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 \AA to 3.41 \AA. 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 MD, can provide consistent values of $E, \nu$ 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 long graphene sheet. Our results suggest that -- (i) graphene is auxetic with its Poisson's ratio increasing with increasing temperature, (ii) with increasing temperature, $E$ decreases, and (iii) the effective thickness increases with temperature.