The stress, strain rate, and process zone with respect to the creep‐crack growth in pure modes I and II are analysed by using damage‐evolution equations. The damage models for the fracture of the process zone are represented using a stress‐ and ductility‐based formulation. According to the stress‐based model, 2 expressions are proposed to describe the stress‐sensitive nature of a multiaxial rupture behaviour. Following the ductility model, new multiaxial failure strain conditions are derived. Both damage‐free and defective creeping solids have been studied. The variations in the creep stress or strain and crack‐tip governing parameter in terms of the creep In‐integral with respect to time and the evolution of creep damage are analysed using a finite element model. The effect of the introduced creep stress intensity factor as a function of the creep time, loading biaxiality, and creep damage models is discussed in detail.