5.3. Predictive ability of the novel damage constitutive model

The variation of minimum creep rate and rupture time with applied stress predicted by the novel damage constitutive model (Eq. and Eq.) at different temperature are shown in Figure 4. The predictions are performed using experimental results of specimens 1-12. Post-audit validation indicates that all experimental data points fall near the prediction curve, and satisfactory interpolation and extrapolation abilities can be achieved by the novel model. Combined with the intergranular and transgranular fracture modes determined by the SEM fracture morphology, an approximate fracture mode split line is also delineated in Figure 4.

FIGURE 4 Predictions of (A) minimum creep rate and (B) rupture time.

After all constants are obtained, the damage evolution curve, i.e. Eq., can be determined by integrating Eq.. Theoretically, the creep strain curve can be obtained by integrating Eq. after introducing Eq. into it, i.e.

However, Eq. cannot be easily integrated to get a closed-form solution for creep deformation. Therefore, the fourth order Runge-Kutta method is adopted to obtain creep deformation curve, just like other researchers did previously^6,17,36,51.

Creep deformation and damage evolution predicted by the novel damage constitutive model with experimental data are shown in Figure 5. When creep time reaches the rupture time , damage and failure occurs. The definition of constant makes the critical experimental damage always unity as shown in Figure 5C and Figure 5D. It can be seen that the prediction results of creep deformation and damage evolution almost fall in the scatter band, including the experimental results of interpolation and extrapolation. It is worth noting that the results of specimens 13-18 exhibited in Figure 5 rely on the stress and temperature dependence fitting function of constants and in Section 6.1 and Section 6.2. Essentially, CDM-based creep damage constitutive model is derived deterministically where scatter is not considered. The best fit line through the creep scattered data is the target of derivation. Then, the model can be calibrated to represent 50% reliability of deformation and damage. The scatter of creep data is inevitable, and this scatter can span across decades of creep rupture time in some cases. Since the CDM-based creep damage constitutive model cannot consider scatter, so the aim of the model is to model the average creep deformation and damage evolution under a certain condition, rather than focusing on fitting a single creep curve. This inherent property of the CDM-based damage constitutive model makes themselves have convincing extrapolation capability that classical plastic theory (CPT)-based model does not have, such as θ projection method⁵².

To consider the scatter of creep data, some investigations have attempted to introduce stochastic processes into CDM-based creep damage constitutive model. Harlow presented a probabilistic K-R model which takes the uncertainty of model constants into account by using probability density function and probability theory⁵³. Penny also proposed a probabilistic K-R model, and Monte Carto simulation method was adopted to deal with the randomness of material and geometrical parameters⁵⁴. Recently, Hossain and Steward have made a lot of efforts to simulate the uncertainty of creep deformation and damage evolution predicted by Sinh model where the randomness of material constants are achieved using Monte Carlo method and probability distribution function^55–58.