Peridynamics for predicting thermal expansion coefficient of graphene

This study presents an investigation of thermal fluctuations of a graphene layer by using peridynamics (PD). The stored energy in the graphene layer due to the fluctuations is expressed in a quadratic form in terms of the stiffness matrix under von Karman assumptions. The Gibbs free energy of the graphene layer related to the partition function is calculated using the Gaussian integrals. However, the partition function requires the evaluation of the determinant of stiffness matrix appearing in the energy expression. Although conceptually very attractive, computing the determinant of an extremely large stiffness matrix whose size is dictated by the characteristic length scale poses computational challenges. Therefore, the PD form of the stiffness matrix is constructed by using two levels of discretization in order to evaluate its determinant accurately without any computational challenges. The derivatives of the partition function permit the determination of several thermodynamic quantities such as the thermal expansion coefficient and its dependence on temperature. This approach enables the exploration of the effect of different geometries, boundary conditions and the nature of loading conditions as well as heterogeneous material properties on the fluctuations.