Drill strings are subjected to complex coupled dynamics. Therefore, accurate dynamic modeling, which can represent the physical behavior of real drill strings, is of great importance for system analysis and control. The most widely used dynamic models for such systems are the lumped element models, which neglect the system distributed feature. In this paper, a dynamic model called neutral-type time delay model is modified to investigate the coupled axial–torsional vibrations in drill strings. This model is derived directly from the distributed parameter model by employing the d'Alembert method. Coupling of axial and torsional vibration modes occurs in the bit–rock interface. For the first time, the neutral-type time delay model is combined with a bit–rock interaction model that regards cutting process in addition to frictional contact. Moreover, mistakes made in some of the related previous studies are corrected. The resulting equations of motion are in terms of neutral-type delay differential equations with two constant delays, related to the oscillatory behavior of the system, and a state-dependent delay, induced by the bit–rock interaction. Illustrative simulation results are presented for a representative drill string, which demonstrates intense axial and torsional vibrations that may lead to system failure without a controller.