Deep-hole drillstring vibrations are an important cause of premature failure of drillstring components and drilling inefficiency. PDC bits are more susceptible to the stick-slip phenomenon characterized by intense RPM fluctuations of the drill bit. Based on full scale laboratory drilling tests and numerical simulations, this paper aims at understanding the dynamic behaviour of the bit/rock interaction and assessing how stick-slip depends on bit design (bit profile and diameter; cutters geometry and set up). It is generally assumed in the scientific and technical literature that forces acting on PDC cutters do not depend on cutter velocity. However, an extensive single cutter experimental program shows that these forces are rate-dependent for four tested carbonate rocks. This rate-effect is associated with the dynamic shearing of a layer of crushed rock carried away beneath the moving cutter. Based on these experiments, a semi-empirical rate-dependent cutter-rock interaction model is developed, implemented in a bit design software and used to predict the velocity signature of various PDC bit designs. These predictions fit well with numerous experimental results obtained using a full scale drilling bench. In particular, the model accounts for the negative damping effect, considered as the main source of stick-slip, and for important operational trends like an increasing risk of stick-slip in harder rocks, at higher weights-on-bit or for advanced cutter wear. The bit-rock interaction model is used as the boundary condition of a drillstring vibration software designed to perform a time-domain analysis of lumped systems. Simulations show that the bit design significantly impacts the risk of stick-slip. As a result, general optimization guidelines are suggested in order to improve PDC bit design.