There has recently been a significant growth in interest in determining the mechanical properties of living human cells. Much of this work has centred around the application of the Hertz Contact Stress model to force/displacement curves that are obtained from atomic force microscopy. However, it is widely recognised that the conventional Hertzian approach, that is based upon a linear elastic model for the stress strain relationship of the material, is inadequate for accurately modelling and describing the behaviour of living cells under load. This paper presents a new non-linear extension of the Hertzian model and shows results that provide excellent agreement with experimentally obtained force/displacement data from a range of different cell types. Furthermore the paper goes on to show the way in which the model can be used as a tool to distinguish between different populations of cells of different types, morphologies, or pathological status.