Fig. 13 Summary of mean values of the maximum strain and the recovery strain during stable stage under different unloading conditions.
Thus, based on ductile exhaustion theory, the damage per cycle of cyclic creep can be expressed as:
\(\text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }\frac{\text{dD}}{\text{dN}}=\frac{1}{\varepsilon^{f}}\frac{d\varepsilon^{\text{act}}}{\text{dN}}=\frac{1}{\varepsilon^{f}}\frac{d(\varepsilon^{r}+\varepsilon^{c}-\varepsilon^{\text{re}})}{\text{dN}}\)
where \(\varepsilon^{f}\) is the fracture strain of failed specimen, the sum of \(\varepsilon^{r}\) and \(\varepsilon^{c}\) is the maximum strain, and \(\varepsilon^{\text{re}}\) is the total recovery strain as mentioned above. It should be noted that the ductility\(\varepsilon^{f}\) is assumed to follow a power law relationship with the mean actual damage per cycle \(\varepsilon_{m}^{\text{act}}\) during the stable stage, thus a simple prediction equation can be obtained as:
\(N_{f}=\alpha{(\varepsilon_{m}^{\text{act}})}^{\beta}\)
where \(\alpha\) and \(\beta\) are fitting constants, which depends on material type. As shown in Fig. 14 (a), the relationship betweenNf and \(\varepsilon_{m}^{\text{act}}\) under different unloading conditions can be well expressed by Eq. (11) in the double logarithmic coordinates, where\(\alpha\)=14.431,\(\ \beta\)=-0.577. Comparison of predictions and experimental results by modified approach is shown by black dots in Fig. 14 (b). It can be observed that all the accuracy of the estimation results is vividly improved, which fall within a scatter band of ±1.5.
It can be further speculated that, if a creep-fatigue test under stress control meets the condition that the stable stage occupies most of fatigue life, the actual damage of the stable stage obtained by strain classification method can still estimate the cycles to failure, regardless of the changed test conditions, such as peak stress, stress ratio, loading and unloading rate and the duration of peak or valley stress. The data of bainite 2.25Cr1MoV steel with short-term duration of hold stress and negative stress ratio at 455 °C is used to check the suitability of the proposed model34. The blue square points of Fig. 14 (b) presents the comparison of predicted and experimental results, where all the predicted results fall within a scatter band of ±1.5. Therefore, the mean actual damage can excellently reflect the essential strain accumulation process under various stress controlled test conditions, and it can be used as a single parameter of a prediction model with a satisfactory accuracy.