**6. Discussion**

With the development of the *shear sliding* and *shear crack models* for grain boundary sliding (GBS), it is now possible to compute the displacements across grain boundaries in a quantitative way and the results are in acceptable agreement with experiments. This has given a dramatic improvement in the understanding of GBS. The previous observations that the amount of grain boundary sliding is proportional to the creep strain are reproduced by the model. According to the shear sliding model the displacement is proportional to the grain size: This has been verified experimentally for small grain sizes, but the grain size dependence is probably exaggerated for larger grain sizes. A detailed comparison between the shear crack model and experiments is more difficult, since in none of the published results on GBS, full details of the particle structure are presented.

It is assumed in general that nucleation of creep cavities is based on GBS. The new models for GBS have made it possible to set up realistic models for nucleation. Many models for nucleation developed in the past are strongly stress dependent and suggest that the amount of cavitation would increase with stress, which is clearly at variance with most observations on creep. This applies for example to models based on classical nucleation theory. Using Lim's model for substructure induced cavitation [40], it has now been demonstrated both for commercial copper and austenitic stainless steel that the creep stresses in the substructure are sufficiently high to nucleate cavities. Since the stresses are stationary, the problem with rapid relaxation in particular at high stresses is avoided. If particles instead of subgrain corners constrain the grain boundary dislocations, the outcome of the model is essentially the same. The model is consequently applicable to cavity nucleation at particles as well.

With the help of the *double ledge model* it can be explained why the nucleation rate is proportional to the creep strain rate. According to the model nucleation takes place when subgrain boundaries meet subgrain corners on the other side of a sliding grain boundary. The displacement rate is proportional to creep strain rate according to the GBS models. As a consequence the suboundaries will meet the subgrain corners at the same rate and this explains why the nucleation rate is proportional to the strain rate.

Harris developed a model for nucleation around particles many years ago [47, 48]. He considered the relation between particle sizes and the GBS displacement. If the particles are sufficiently large it is assumed that they will prevent GBS. He formulated a criterion for the critical particle size. If GBS is prevented significant stresses are formed at the particles. If the critical particle size is exceeded, Harris proposed that cavity nucleation can take place. With the event of the new models for GBS it has now for the first been possible to test Harris' ideas. It turns out that both the critical particle radius and the number of cavities that can be estimated from the particle distributions are in agreement with observations for austenitic stainless steels. In this way a method for estimating the critical particle size for nucleation has been established.

temperature and stress the value from **Figure 12** is chosen if the (ductile) rupture time is

**Figure 14.** Comparison of model rupture curves based on both dislocation creep (ductile rupture) and cavitation (brittle rupture) with experiments [55] for 18Cr12NiTi (321H). Model predictions and experiments at temperatures between

When brittle rupture is taken into account when modelling the creep rupture curves, there is

With the development of the *shear sliding* and *shear crack models* for grain boundary sliding (GBS), it is now possible to compute the displacements across grain boundaries in a quantitative way and the results are in acceptable agreement with experiments. This has given a dramatic improvement in the understanding of GBS. The previous observations that the amount of grain boundary sliding is proportional to the creep strain are reproduced by the model. According to the shear sliding model the displacement is proportional to the grain size: This has been verified experimentally for small grain sizes, but the grain size dependence is probably exaggerated for larger grain sizes. A detailed comparison between the shear crack model and experiments is more difficult, since in none of the published results on GBS, full details of

It is assumed in general that nucleation of creep cavities is based on GBS. The new models for GBS have made it possible to set up realistic models for nucleation. Many models for nucleation developed in the past are strongly stress dependent and suggest that the amount of cavitation would increase with stress, which is clearly at variance with most observations on creep. This applies for example to models based on classical nucleation theory. Using Lim's model for substructure induced cavitation [40], it has now been demonstrated both

shorter than the (brittle) rupture time in **Figure 13** and vice versa.

**6. Discussion**

600°C and 775°C with 25°C interval.

36 Study of Grain Boundary Character

the particle structure are presented.

an improvement in particular at high temperatures and low stresses.

GBS does not occur on every grain boundary. In fact only on a limited number of GBS events is observed in a material that has been creep exposed [19]. As a consequence cavity nucleation only occurs on some grain boundaries. The presence of cavities show a large statistical variation, see for example [39]. The models in the present paper represent grain boundaries where cavities will be formed. These are also the grain boundaries that control the rupture of the material. The models predict a fairly high nucleation rate that represents the most active grain boundaries and not an average over all grain boundaries.

Models for cavity growth have been available for a long time. Unfortunately these models in general generate growth rates that are much higher than the observed ones. This might be the reason why very few quantitative comparisons have been made between experiments and models in the literature. A major step forward was the introduction of constrained growth [17]. Then the cavities were not allowed to grow faster than the surrounding creeping material. A quantitative model for constrained growth was given by Rice [18]. Still the growth rates tended to be higher than the observed ones. Only recently it has been recognised that some minor approximations in the work of Rice were essential to correct [52]. With these new corrections satisfactory predictions for cavity growths in austenitic stainless have been achieved.

The recent development has implied that quantitative models are now available for grain boundary sliding, for the thermodynamic feasibility of cavity nucleation controlled by subboundaries, the nucleation rate and constrained cavity growth, that is, for all of the involved main processes. Each model is of great significance in itself, but by using combinations of them, the development of cavities can be fully understood.
