Fig. 5: Predicted time evolution curves of (a) the local normal stressσ n; (b) σ n in the 1st cycle; (c) σ n in the 40th cycle (d) the number density of cavities ρ and cavitated GB fraction f .
When cyclic loading continues, the magnitude of relaxedσ n in tensile hold decreases, whereas that in compressive hold increases. This is as expected because Eq.(2) includes an exponential decay function, indicating that an increased initial stress level would cause a faster stress relaxation over a fixed period of time. When the steady state is reached, the magnitude of relaxedσ n in the tensile and compressive holds becomes identical. This can be clearly seen in Fig. 5c for the 40th cycle, where the tensile and compressive holds introduce 23 MPa stress relaxation, respectively. This means that the net stress change is 0 MPa in the steady state, being consistent with the observation in Fig. 5a.
Since nucleation rate is highly sensitive to σ n, the maximum σ n is key to cavity nucleation (i.e. the number density of cavities ρ ). The model predicted ρevolution as well as cavitated GB fraction f during 50 cycles of creep-fatigue loading are shown in Fig. 5d. The ρ curve in black shows that most of the cavities are nucleated in the tensile hold of the 1st cycle, while the following compressive hold contributes to the major part of cavity sintering. After the first two cycles, the change inρ becomes negligible. By comparison, the f curve in red with its value read from the right axis (Fig. 5d) shows an overall increasing trend. This implies that the increased f value in later creep-fatigue cycles is most likely related to the cavity growth rather than the nucleation of new cavities.
To show this more evidently, two enlarged views of the f curve exhibiting a zigzag shape are highlighted in Fig. 5d. The zigzag shape can be explained by the fact that cavities nucleate and grow up whenσ n>0, and they shrink whenσ n≤0. The zigzag characteristic is more noticeable in the 2nd cycle than the 43th cycle, indicating that the cavity radius change rate reduces with the cycling. This implies that the cavity sintering is less likely to occur in the later cycles of creep-fatigue loading under the present waveform condition (i.e.t t>t c in Fig. 4). Since the model predicts that the later creep-fatigue cycles neither nucleate many cavities (black curve in Fig. 5d) nor cause the already nucleated cavities to be sintered (red curve in Fig. 5d), it is appropriate to reduce fatigue cycles from 50 to 10 to save the computational cost.
In the work by Min and Raj 18, one purpose-designed creep-fatigue cycle at 625 ˚C was employed to create the increased fatigue crack growth rate from 0.05 to 0.17 mm/cycle on the Type 316 stainless steel. This specific load-waveform included a 15 mins stress hold under compression (-382 MPa), followed by a rapid load reversal from compression to tension (382 MPa) with a strain rate of 2×10-4 s-1, and then applying a 1 hr stress hold under tension. The cavitation damage at grain boundaries was responsible for the increased crack growth rate. They further showed that imposing more than one fatigue cycle could not generate a further increase in the crack growth rate. Therefore, our model prediction in terms of the importance of the 1st fatigue cycle agrees with the experimental observation.
According to the predicted σ n with a magnitude of 761 MPa (for a tensile hold stress Δσ t of 160 MPa in Fig. 5a) in the present work, the cavity nucleation rate would be 2.1×10-10mm-2.s-1. When the magnitude ofσ n increases to 1016 MPa, by applying a higher tensile stress hold of 200 MPa, the nucleation rate would change to 4.0×10-1 mm-2.s-1. Recall the work by Raj (1978), the predicted cavity nucleation rate was found to vary from 1014 to 10-16mm-2.s-1, if the local normal stressσ n is decreased by an order of magnitude from 103 MPa to 102 MPa. Thus, our model prediction regarding the nucleation rate is as expected. It is well recognised (e.g. Evans 15) that cavity nucleation under the GB sliding mechanism would require a high stress concentration (in the order of 103 to 104 MPa) particularly associated with the classical nucleation theory. Also, the nucleation rate (and cavity density) is strongly dependent on the material parameters 19, 40.
In this context, it is the predicted trend, rather than the specific value of cavity number density, that provides an important but missing guideline in terms of optimising the load-waveform design to produce more creep cavitation damage with reduced creep-fatigue experimental cost.

3.2 Effect of loading sequence

Creep-fatigue test can start either with the initial compression or tension. Fig. 6a compares these two loading sequences in terms of theσ n evolution. The load waveform parameters and temperature used here are the same as that listed in Table 2, i.e.t t>t c. It can be seen that the test with initial compression has a higher maximumσ n than that with initial tension. This is because the pre-compression period can relax theσ n, contributing to the higher maximumσ n in the tension phase. The difference of maximum σ n between the two loading conditions is 12 MPa, which is in line with the relatively shortt c of 20 s. The number density of cavitiesρ after 10 cycles was calculated as 2.6×10-9mm-2 for the test with initial compression, which is approximately one order of magnitude higher than that with initial tension (ρ =4.9×10-10 mm-2), Fig. 6b.
Now let’s consider a completely different scenario in which the compressive hold is prolonged, i.e.t c>t t. To this end, the σ n evolution for a creep-fatigue loading with t t=20 s andt c=100 s was calculated. The test with initial compression is compared with that with initial tension. For both cases, the σ n increases cycle by cycle, and the maximumσ n appears at the last cycle, Fig. 6c. The test with initial compression has a higher maximum σ nthan that with initial tension. At the 1st cycle, the stress difference is 54 MPa, reducing to 23 MPa at the 10th cycle (the last cycle of calculation). This implies that the maximum σ ndifference between the two loading conditions will eventually diminish to 0 MPa if the cyclic loading continues.