Fig. 8: The dependence of final ρ on (a) Δσ t and (b) Δσ c witht t= 60/100 s; the evolution of ρ in (c) and f in (d) as a function of fatigue cycles calculated with different stress magnitudes of Δσ c=30/60/140/170 MPa but under the same tensile hold time of t t=60 s.
In terms of the effect of Δσ c on the finalρ , a positive correlation is found between the two until Δσ c reaching the breakdown value, Fig. 8b. Such breakdown of the positive correlation can be seen when Δσ c>120 MPa under the condition oft t=60 s. This is followed by a sharp drop of final ρ when Δσ c>160 MPa. With the increase of tensile hold time to t t=100 s, the breakdown occurs at a higher compressive stress of when Δσ c>270 MPa.
To explain this interesting phenomenon, the change of ρ andf as a function of cycles are presented in Fig. 8c and 8d, respectively. Four different stress levels of Δσ c=30/60/140/170 MPa are considered, and they represent different characteristic regions as highlighted by using different colours in Fig. 8b. Comparing the values of ρ at the end of the 1st cycle, the test with Δσ c=170 MPa has the largest ρ of 3.6×10-9mm-2, whereas that with Δσ c=30 MPa shows the smallest ρ of 0.7×10-9mm-2, Fig. 8c. This indicates that a higher Δσ c helps to produce more creep cavities in the subsequent tension phase. However, in the next compression phase of the 2nd cycle, these nucleated cavities would be sintered to different extents depending on the stress level of Δσ c. For example, the ρ value reduction is found to be -0.2×10-9, -0.3×10-9, -1.9×10-9 and -3.6×10-9mm-2 for Δσ c=30, 60, 140 and 170 MPa, respectively, Fig. 8c. Thus, the competing effect of Δσ c on ρ through the cavity nucleation and sintering leads to the fact that an intermediate level of Δσ c is more likely to create more cavities.
For the same reason, the breakdown of the positive correlation between the final ρ and Δσ c as revealed in Fig. 8b, can be explained by the sintering effect that becomes predominant when the compressive stress is sufficiently large. In terms of the cavitated GB fraction f , its value increases with the number of cycles, Fig. 8d. The zigzag shape in the f curves reflect the alternating nucleation and sintering events. In summary, increasing the stress level of Δσ t is more effective in enhancing the final ρ when compared to Δσ c. The underlying mechanism is the side effect associated with the cavity sintering due to the increased Δσ c.

3.5 Effect of stress variation rate

After elaborating the effects of four input parameters (t t, t c, Δσ t and Δσ c), we now consider the effect of stress variation rate , Fig. 4. To limit the effect of hold time on the prediction results, botht c and t t are set to 0 s for calculations. Fig. 9a shows the effect of on the final ρ andf after 10 cycles. There is an optimum value of (~0.4 MPa/s) that leads to the maximum final ρand f . To explain its presence, the ρ values as calculated at the end of 1st cycle (i.e. the moment of finishing the tension phase) are presented in Fig. 9b, while the minimum σ n as calculated in the 2nd cycle (i.e. the following compression phase) are shown in Fig. 9c. Note that the magnitude of ρ at the end of 1st cycle can be regarded as the evaluation of the nucleation ability in the considered load-waveform shape. Equally, the absolute value of the minimum σ n in the 2nd cycle can be regarded as the evaluation of the sintering ability.