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