This work is focused on the Newton‐Krylov technique for computing the steady cyclic states of evolution problems in non‐linear mechanics with space‐time periodicity conditions. This kind of problems can be faced, for instance, in the modeling of a rolling tyre with a periodic tread pattern, where the cyclic state satisfies "rolling" periodicity condition, including shifts both in time and space. The Newton‐Krylov method is a combination of a Newton nonlinear solver with a Krylov linear solver, looking for the initial state, which provides the space‐time periodic solution. The convergence of the Krylov iterations is proved to hold in presence of an adequate preconditioner. After preconditioning, the Newton‐Krylov method can be also considered as an observer‐controller method, correcting the transient solution of the initial value problem after each period. Using information stored while computing the residual, the Krylov solver computation time becomes negligible with respect to the residual computation time. The method has been analyzed and tested on academic applications and compared to the standard evolution (fixed point) method. Finally, it has been implemented into the Michelin industrial code, applied to a full 3D rolling tyre model.