In force measurements with the atomic force microscope (AFM) often nonelastic deformations of the sample are observed. This is especially the case for AFM experiments on ice. In this paper two theoretical approaches are used to calculate the indentation of the tip. First, in an extended plastic indentation model we assume that the pressure exerted by the tip plastifies the sample material once the pressure exceeds three times the yield stress. The rate of plastification limits the indentation speed. With two parameters this model adequately describes measured force curves on ice in a temperature range of -4 degrees C to -20 degrees C and for driving speeds of 0.8 to 110 mu m/s. The calculation also shows that in general plastic deformation can lead to different force curves depending on the driving speed. For low driving speeds a jump-in and a linear force curve is expected. For fast driving speeds plastic deformation leads to curved force curves with gradually increasing slope. Second, in a hydrodynamic model we assume that the presence of the tip surface leads to an interfacial premelting of the ice. The flow of the liquidlike water out of the gap between tip and bulk ice is limiting the indentation speed of the tip. The hydrodynamic model does not fit measured force curves as good as the extended plastic indentation model and viscosities which are unrealistically high need to be assumed. Hence, we can exclude that a liquidlike layer of a thickness significantly larger than a monolayer exists in AFM experiments on ice.