There is a practical need to elevate both the indentation strength and level of energy absorption of engineering foams by the addition of a stiff and strong face sheet for applications such as packaging and crash mitigation. In this study, the enhancement in plane strain indentation resistance of a polyvinyl chloride (PVC) foam by the presence of a polycarbonate (PC) face sheet is determined by experiment, finite element analysis and by an analytical model. Plane strain indentation is by a flat-bottom punch or by a cylindrical roller, and the strain distribution within the PC face sheet and in the foam substrate are measured by digital image correlation. With increasing indent depth, the face sheet bends and stretches elastically and then plastically until face sheet or substrate fail. The generation of membrane tension in the face sheet plays a major role in supporting the indentation load when the indent depth exceeds the thickness of the face sheet, and leads to a strong hardening behaviour beyond the initial collapse load for indentation. Finite element predictions of the full indentation response are based upon the measured tensile and compressive responses of the PVC foam and PC layer. An analytical model is developed by matching the stretching response of the PC face sheet to the indentation response of the underlying foam, with due consideration for load diffusion from membrane tension of the PC face sheet into the underlying foam substrate. The indentation model is calibrated by ancillary finite element simulations of the load diffusion problem, and they emphasise the role of a shear lag zone in dictating the large indentation resistance. The indentation response of the bi-layer is also compared with that of a sandwich beam in 3-point bending. Experiments, finite element simulations and an additional analytical model for indentation of the sandwich beam in 3-point bending reveal that strong hardening of the post-yield load versus displacement response is now absent, in contrast to that of the bi-layer. The lack of hardening in 3-point bending is traced to the relatively low value of plastic bending moment of the beam section.