Possessing the unique and highly valuable properties, graphene sheets (GSs) have attracted increasing attention including that from the building engineer due to the fact that Graphene can be utilized to reinforce concrete and other building materials. In this work, the nonlocal elastic theory and classical plate theory (CLPT) are used to derive the goving equations. The element-free framework for analyzing the buckling behaviors of double layer circular graphene sheets (DLCGSs) relying on an elastic medium is proposed. Pastak-type model is adopted to describe the elastic medium. Accordingly, the influences of boundary conditions, size of GSs and nonlocal parameters on the buckling behavior of DLCGSs are investigated. The results show that the OP buckling modes are only sensible to the van der Waals forces.