Abstract The cyclic deformation behaviour of copper and 2.25Cr–1Mo steel was investigated at room temperature. In the low-cycle fatigue regime, the cyclic deformation behaviour can be represented by cyclic stress–strain curves, cyclic hardening or softening curves and stress–strain hysteresis loops. It is possible to describe all these curves and the hysteresis loops by analytical relationships or alternatively by constitutive equations. The latter can be related with physical processes, e.g. by taking into account the dislocation structure. Several phenomenological and physically based relations for the description of the cyclic deformation behaviour are proposed and discussed in this paper.