In discrete element method simulations for industrial applications, long, flat, bendable particles are often approximated as rods with circular cross sections. However, such crude approximation of the particle geometry raises questions on the ability to reproduce the material bulk properties. Therefore, we describe a novel discrete element representation of flexible, flat, thin particles, consisting of a triangulated mass-spring model of the particle's neutral plane, with the thickness of the particle incorporated in customised contact models. We validate the model with experiments of packing after gravitational deposition and sinusoidal excitation, where we show that the model correctly predicts the packing density for long, flexible, flat particles. Furthermore, we benchmark the model against a circular cross-sectional flexible rod model. Investigation of the dynamics of densification under vibrational compaction shows that the Kohlrausch-Williams-Watts stretched exponential law is well suited to describe the compaction dynamics of the described particles. The novel foil model provides insights in the bulk behaviour of flexible flat particles, otherwise difficult or even impossible to obtain theoretically or experimentally for random orientations. Applications of the model could include the modelling of fabrics, biological material like leaves, soft plastic, and paper.