The buckling and the post-buckling of bilayer graphene sheets (BLGSs) on an elastic foundation subjected to an axial compressive load in plane-strain condition are studied. The solutions for the buckling strain and the characteristic wavelength at the onset of the buckling are obtained by using proposed continuum models describing deformations of BLGSs. The buckling analysis reveals that the buckling of BLGSs on an elastic foundation could be categorized into four. When the foundation is very soft, the Euler beam kinematics prevails in the entire BLGSs. On the stiffer foundation, the shear deformation on the cross-section of BLGSs occurs, and the characteristic wavelength changes sensitively as the stiffness of the foundation varies. With the foundation whose stiffness is almost the same as the shear modulus of the graphene interlayer, the buckling strain and the characteristic wavelength are nearly linear versus the elastic modulus of the foundation in logarithmic scale. When the foundation is much stiffer than the rigidities of the graphene interlayer, detachment between graphene layers could occur. Finally, we conduct the post-buckling analysis by using the proposed finite element model and obtain various morphology evolutions which contain detachment between graphene layers, period doubling, localization, period switching, and folding. Also, we present a diagram includes distinct deformation patterns according to the stiffness of the foundation and the magnitude of the applied compressive strain. (C) 2019 Published by Elsevier Ltd.