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