The introduction of creep fracture constraint parameter offers the possibility of accurately obtaining the CCG rate for specimens/structures subject to different constraint levels through the establishment of constraint-dependent CCG rate equation. However, experimental evidence indicates that the existing constraint-dependent CCG rate equation is generally material- and stress regime-dependent. It is noted that the dependencies may be attributed to the intrinsic difference in creep failure mechanisms of materials, which has not been fully considered by previous constraint parameters. To cope with the dependencies, a parameter, Kz, for characterizing the intensity of constraint effect is proposed in this work based on the representative rupture stress, which could better reflect the operative creep failure mechanism sustained by the material. To justify the parameter, finite element analysis is carried out for five different materials based on experimental data available in the literature. Results show that, compared to the existing constraint parameters, R* and Ac, the Kz is capable of characterizing the crack-tip constraint level under the influence of creep failure mechanism. Furthermore, a generalized constraint-dependent CCG rate equation is established based on the Kz, which overcomes the material and stress regime dependencies of the existing equation. Finally, to clarify the advantage of the Kz, further comparison between the new parameter and the existing parameters is made.