Based on the meso- and micro-structure change and damage characteristic of Ni-based single crystal superalloys in the creep condition, a two-state-variable creep damage constitutive model considering simultaneously the rafting damage and void damage has been built. The model has been employed to investigate creep damage and crack initiation behaviour of the single crystal compact tension (CT) specimen containing a cylindrical void ahead of crack tip. The influence of the deviation and randomness of the crystallographic orientations has been considered. The analysis results have shown that creep damage and crack initiation behaviour depend strongly on the void location and crystallographic orientation. When the void is close to the notch root, a microcrack will initiate at the surface of the void in the vicinity of the notch. The time of crack initiation is short. But when the void is far from the notch root, the microcrack will initiate at the notch surface. The crack location is related to the crystallographic orientations, and the time of crack initiation is long. With increasing of the deviation of the axial crystallographic orientations of the single crystal CT specimen, the time of crack initiation is decreased obviously. The maximum amplitude variation is 34.7%. While the two uncontrolled crystallographic orientations of the CT specimens vary randomly, the time of crack initiation is longest in 45° and 80° deviations, and the maximum deviation of crack initiation time is up to 3 times.