The nonlinear mechanical response of monolayer graphene on polyethylene terephthalate (PET) is characterised using in-situ Raman spectroscopy and atomic force microscopy. While interfacial stress transfer leads to tension in graphene as the PET substrate is stretched, retraction of the substrate during unloading imposes compression in the graphene. Two interfacial failure mechanisms, shear sliding under tension and buckling under compression, are identified. Using a nonlinear shear-lag model, the interfacial shear strength is found to range between 0.46 and 0.69 MPa. The critical strain for onset of interfacial sliding is approximate to 0.3%, while the maximum strain that can be transferred to graphene ranges from 1.2% to 1.6% depending on the interfacial shear strength and graphene size. Beyond a critical compressive strain of around -0.7%, buckling ridges are observed after unloading. The results from this work provide valuable insight and design guidelines for a broad spectrum of applications of graphene and other 2D nanomaterials, such as flexible and stretchable electronics, strain sensing, and nanocomposites.