Model composite specimens have been prepared consisting of a graphene monolayer sandwiched between two thin layers of polymer on the surface of a poly(methyl methacrylate) beam. It has been found that well-defined Raman spectra can be obtained from the single graphene atomic layer and that stress-induced Raman band shifts enable the strain distribution in the monolayer to be mapped with a high degree of precision. It has been demonstrated that the distribution of strain across the graphene monolayer is relatively uniform at levels of applied strain up to 0.6% but that it becomes highly nonuniform above this strain. The change in the strain distributions has been shown to be due to a fragmentation process due to the development of cracks, most likely in the polymer coating layers, with the graphene remaining intact. The strain distributions in the graphene between the cracks are approximately triangular in shape, and the interfacial shear stress in the fragments is only about 0.25 MPa, which is an order of magnitude lower than the interfacial shear stress before fragmentation. This relatively poor level of adhesion between the graphene and polymer layers has important implications for the use of graphene in nanocomposites, and methods of strengthening the graphene-polymer interface are discussed.