FIGURE 3 Fracture appearances of creep ruptured specimens at (A, E) 650℃/820MPa, (B, F) 650℃/770MPa, (C, G) 650℃/720MPa, (D, H) 650℃/670MPa, (I, L) 650℃/615MPa, (J, M) 650℃/595MPa, and (K, N) 650℃/515MPa
5.2. Determination of the material constants
From Eq. and Eq., it can be determined that the constants needed to be obtained are , , , , , , , and . The method to obtain those constants is described as follows:
5.2.1. and
The ultimate tensile stress of 1440MPa and 1255MPa at 600℃ and 650℃ respectively, are determined by tensile tests. In order to determine the threshold stress, a standard linear extrapolation method was introduced by Mishra38,39, Huang40 and Langdon45 et al. And this method has been successfully used to determine the threshold stress of Grade 91 steel46,47, ferritic-martensitic steel48, Fe-Cr-Ni alloy44, Grade 92 steel49, nickel-based superalloy50, etc. Using the same method, the threshold stresses at 600℃ and 650℃ were determined to be 593 MPa and 309 MPa in recent study28, respectively. It should be pointed out that the threshold stress is closely related to microstructural evolution and stability.
5.2.2. , , , and
When damage is zero, Eq. is simplify to Eq.. is the apparent activation energy of TTC relations, which can be calculated from the slop of diagram at constant .
Taking the logarithm of Eq. and Eq. gives:
Therefore, constants , and , are the slope and y-intercept in the — and — diagrams, respectively. Full details are given in reference [28].
5.2.3.
To ensure when failure occurs, the definition of constant here is consistent with the definition of Haque20,21, i.e. . Constant is determined directly from the experimental data. It can be inferred that constant has dependence on stress and temperature.
5.2.4.
So far, all constants have been determined except for constant . As mentioned above, the constant can be determined by the least square fitting of experimental damage Eq. and analytical damage Eq.. In this study, the constant is obtained by minimizing the constructed mean square error (MSE) function:
where is the analytical damage, is the experimental damage, and is the number of test data.
In order to verify the interpolation and extrapolation abilities of the novel model, specimens 1-12 are used to calibrate the novel damage constitutive model. The obtained constants , , , , , and are summarized in Table 3. The material constants and exhibit the dependence on stress and temperature. The constant is determined directly using Eq. experimentally. The constant is determined by the least square fitting of experimental damage Eq. and analytical damage Eq.. The calibrated constants and are listed in Table 4.
TABLE 3 Values of material constants