The mechanism and criterion of crack initiation and propagation of rocks were investigated by many researchers. And the creep behaviour of rocks was also theoretically and experimentally studied by some scientists and engineers. The characteristics of crack initiation and propagation of rocks under creep condition, however, are very important for rock engineering and still not paid enough attention by researchers. In this paper, the criterion and mechanism of crack initiation and propagation under creep condition were investigated using specimens collected from sandstone rock formations outcropping in the Emei Mountain, the Sichuan Province of China. Cuboid specimens under three point bending were used in this investigation. All specimens were classified into four sorts and used for Mode-I fracture or creep fracture tests. The experimental result shows that due to creep deformation, rock crack will inevitably initiate and propagate under a load of KI, which is less than fracture toughness KIC but not less than a constant (marked as KIC2). KIC2 indicates the ability of rock to resist crack initiation and propagation under creep conditions and is less than fracture toughness KIC, defined as creep fracture toughness in this paper. KIC2 should be considered as an important parameter on design and computation of rock engineering. The microstructural mechanism for crack initiation and propagation of rock materials under creep condition was introduced based on competitive model between softening effect and hardening effect, and the validity of test result was explained. The test result was also verified in rheological theory. When KI is more than KIC2 but less than KIC, rock crack will initiate and propagate after a time interval of sustained loading under creep condition. In order to find the relation between duration of sustained loading, which can lead to crack initiation and propagation, and the initial stress intensity factor KI, an unequal-interval time sequence forecasting and predicting model was introduced, and the relation was obtained for homogeneous and isotropic fine-grained red sandstone. Finally a modified fracture toughness formula was given, in which the influence of fracture process zone(FPZ) was fully considered.