A mathematical model for the performance of a tunnel boring machine (TBM) is presented. In particular an equation for the TBM cutter wear is proposed. The rock mechanics are described via simple strength parameters and the Cerchar abrasivity index (CAI). Thus one obtains a complete description of TBM tunnelling with instantaneous and accumulated values for the tunnelling rate, the tool consumption, the costsetc. as output parameters when the tunnelling proceeds through rock with variable properties (strength and CAI). Two different types of cutter tools are considered, namely wedge-shaped tools and constant wear flat tools. For wedge-shaped tools the tunnelling costs are shown to be minimized if old tools are replaced by new tools when a certain wear flat tool has been reached. Similarly there is an optimum design (=wear flat) for the constant wear flat tools.