Graphene and carbon nanotubes are promising materials for nanoelectromechanical systems. Among other aspects, a proper understanding of the sliding dynamics of parallel graphene sheets or concentric nanotubes is of crucial importance for the design of nano-springs. Here, we analytically investigate the sliding dynamics between two parallel, rigid graphene sheets. In particular, the analysis focuses on configurations in which the distance between the sheets is kept constant and lower than the equilibrium interlayer spacing of graphite (unstable configurations). The aim is to understand how the interlayer force due to van der Waals interactions along the sliding direction changes with the geometrical characteristics of the configuration, namely size and interlayer spacing. Results show metastable equilibrium positions with completely faced sheets, namely a null force along the sliding direction, whereas net negative/positive forces arise when the sheets are approaching/leaving each other. This behavior resembles a molecular spring, being able to convert kinetic into potential energy (van der Waals potential), and viceversa. The amplitude of both storable energy and entrance/exit forces is found to be proportional to the sheet size, and inversely proportional to their interlayer spacing. This model could also be generalized to describe the behavior of configurations made of concentric carbon nanotubes, therefore allowing a rational design of some elements of carbon-based nanoelectromechanical systems.