The interfacial mechanics and reinforcement by the graphene sheets in polymer matrix nanocomposites are important to their understanding. However, the methods available for their investigation remain a challenge. Here we report on a novel study in which the mechanical responses of a nano-sandwich model structure made of a single graphene sheet sandwiched between ultrathin polymer layers are determined using a nano-bubble inflation method. The stress-strain behavior of the graphene nano-sandwich shows that significant reinforcement is obtained at small strains and that the method also provides a measurement of the interfacial shear strength. In addition, the study provides data related to internal stresses that develop between the graphene layer and the polymer sandwich faces.