Fig. 1: Schematic diagrams of a cavity that (a) grows by the surface diffusion limited GB sliding; (b) shrinks by the unconstrained GB diffusion.
In the growth model 34, the cavity has been assumed to have a crack-like shape, Fig. 1a. The growth process is driven by the accumulation of vacancies at the cavity tip, leading to the phenomenon similar to a crack-tip continuous extension. Meanwhile, the surface diffusion is not very much faster than GB diffusion under low stress, while the surface diffusivity becomes much higher than that of GB under high stress. Refer to Appendix B for the derivation of stress level that separates the two conditions.
The reason for ignoring the creep effect on the early-stage growth can be justified as follows. The growth mechanism map proposed by Miller, Hamm 26 informs that the cavity radius change is controlled by unconstrained GB diffusion when the ratio of cavity spacing (λ) to its diameter (2r ) is greater than 10. In the case of early-stage cavitation, the value of λ/2r falls into the range of 200 to 3000, given that the value range of r is 3 to 5 nm3 and that of λ is 1 to 10 μm 23. Therefore, the unconstrained GB diffusion is the rate-limiting factor for the early-stage growth.
Because of the analogy of shrinkage to growth, shrinkage occurs by the GB vacancy diffusion that pushes vacancies out of the cavity under the compressive σ n. The reversed direction of GB sliding, coupled with the reduced sharpness of the cavity tip, causes a more uniform distribution of vacancies within the cavity, as shown in Fig. 1b. Since the vacancies are no longer concentrated at the tip, more surface areas become the diffusional interface. Ultimately, the cavity shape changes back to sphere, that is the equilibrium shape when the surface diffusion rate is fast enough 23.
To calculate the shrinkage rate, equation proposed by Riedel35 for cavity growth under the unconstrained GB diffusion is adapted:
When the cavity radius r becomes larger than λ/4.24, is set to zero to prevent the unrealistic positive value under compression. This upper limit value has been used to define the largest cavity size. The assumption behind the Riedel model is that cavity is not perfectly spherical, which aligns very well with the considered crack-like cavities in the present work. If the cavity radius r becomes less than the critical nuclei size defined by the value of 2γ/σ n 14, cavities are considered as sintered 12, 36.

2.2 Material parameters and temperature effect

Except for the surface diffusion factor D S, all the material parameters have been sourced from the authoritative literature listed in Table 1. D S values in the temperature range of 500-600 ˚C are not available; hence the diffusivity ratio ofδ SD S/δD B=0.001 has been used for calculations. This meets the criterion ofδ SD S/δD B<<1 for a crack-like cavity 34. Justification of theD S value selection through a sensitivity study is described in Appendix B. In this context, our model has been established based on the state-of-the-art mechanistic understanding of cavitation and the prediction results are reliable (no approximation or extrapolation).
Creep database for Type 316H stainless steel 29 has been used to derive the power-law creep related parameters (1/ƞ p and n, Table 1) for two reasons. First, this material was subjected to 65,015 h in service at temperatures between 490 and 530 ˚C, prior to creep testing 37. Thus, it is unlikely that GB particle evolution occurs during the test (hence a fixed f b value). Second, the operating pressure of less than 20 MPa would be too low to generate any noticeable creep cavities in this ex-service material, and hence no need to consider pre-existing cavities in our calculations. Note that the maximum allowable carbon content for Type 316 stainless steel is <0.08 wt.% according to 38. This means that the chosen material with a carbon content of 0.06 wt.%39 falls into the category of Type 316 stainless steel.
Table 1. Material parameters in the proposed cavity nucleation and sintering model for Type 316 stainless steel