Thermal vibration of a circular single-layered graphene sheet with simply supported or clamped boundary

The thermal vibration of a Circular Single-Layered Graphene Sheet (CSLGS) is studied by using a circular thin plate model with quantum effects taken into account when the law of energy equipartition is unreliable. The relation between the temperature and the Root of Mean Squared (RMS) amplitude of vibration at any point of the CSLGS with simply supported or clamped boundary is derived first from the circular thin plate model via Bessel functions and modified Bessel functions of the first kind. The RMS amplitude of the CSLGS is constant at any point with the same radius for either quantum case or classical case. The study shows that the RMS amplitude of the CSLGS predicted from the quantum theory is lower than that predicted from the law of energy equipartition. The quantum effect is more important for the thermal vibration in the case of higher order modes, smaller diameter and lower temperature. The relative difference of the maximal RMS amplitude of thermal vibration of the CSLGS decreases monotonically with an increase of temperature.