Rock is a kind of heterogeneous natural engineering material. Rocks in geotechnical engineering are often in a cyclic loading state. Investigation into the localization evolution law of rock deformation under cyclic loading can pro- vide theoretical support for the stability of geotechnical engineering structures. The study simplified the cyclic loading conditions as one constant amplitude cyclic loading condition and one graded constant amplitude cyclic condition. The tests of red sandstone specimens were carried out under the two cyclic loading conditions. The deformation images of specimens during the tests were collected by CCD camera,and based on the digital speckle correlation method,the dis- placements evolution laws of deformation localization bands of specimens during constant amplitude cyclic loading were analyzed. The analysis finds that during the whole process of cyclic loading,the displacements of localization bands fluctuate upward with the loading and unloading stress. There is a time lag for the displacements relative to the stress change,and with the increase of cyclic number,there is a cumulative effect for the displacements. In the cycle,the val- ues of lag time at the vertexes of the displacements of localization bands fluctuate steadily at the early stage of cycle and fluctuate upward at the later stage of cycle relative to the vertexes of loading stress. The values of lag time at the bottom point of the displacements of localization bands fluctuate steadily with the increase of cyclic number relative to the bottom point of unloading stress. When loading or unloading to the same stress in the whole cycle,the displace- ments of localization bands increase with the increase of cyclic number respectively. The displacements of unloading process are greater than those of loading process under the same stress during each cycle. With the increase of cyclic number,the displacements of localization bands experience slow evolution stage and accelerated evolution stage obvi- ously. Before the failure of specimens,the displacement values of localization bands at the bottom point of unloading will be evidently close to the displacement values at the vertex of loading,and this phenomenon can be used as a pre- cursor information of the instability failure of rock specimens or rock structures under cyclic loading station. The re- search in this paper can provide a new characteristic mechanical monitoring quantity for the instability failure of geotechnical engineering materials or structures under cyclic loading. That is,in a particular stress environment,when the deformation value of low stress state and relative high stress state both increase with time and the increase rate of the deformation value corresponding to low stress is obviously greater than that corresponding to high stress,the failure of geotechnical materials or structures will occur.