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 previously6,17,36,51.
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 function55–58.