At temperatures above 0.4 of the absolute melting temperature T m, materials are exposed to a slow plastic deformation called creep. For steel 0.4 T m is about 500°C. The slow deformation takes place also at constant stress. Many important technical units operate at such high temperatures. This applies, for example, to fossil fired power plants that produce most of the world’s electric power. Another example is gas turbines in aircrafts where the maximum metal temperature exceeds 900°C.
Due to creep the total strain in the material gradually increases. At the same time the microstructure can also change. For example the strength of many materials used at high temperatures is based on the presence of fine particles that slow down the deformation. There is a thermodynamic driving force for coarsening of the particles, since this will reduce the total surface area of the particles and thereby the surface energy. The deformation of the material and changes in the microstructure will decrease the strength of the material and this is referred to as the formation of creep damage. When the creep damage has reached a certain level, failure takes place.
Polycrystalline materials consist of regions with a specific lattice orientation called grains and the boundaries between them grain boundaries. During the creep deformation, small voids called creep cavities are formed at the grain boundaries. The size for the cavities is of the order 1 μm. The creep cavities are continuously nucleated so their number increases with time. Each individual cavity also grows so their radii increase. The driving force is the same as for coarsening of particles, that is, a reduction of the surface energy. In this way, there is a gradually increasing fraction of the grain boundaries that is cavitated. When the area fraction of cavities has reached a critical value, the cavities join and eventually form cracks that make the material fail.
The fact that creep rupture is mainly controlled by the development of grain boundary cavities has created a large technical interest in cavitation. This interest increased even more in the 1980s, when it was recognised that the appearance of the cavitation could be used to estimate the residual life time of fossil fired power plants. The service time of many power plants were approaching the design life and operators were asking whether it would be safe to continue running the plants. Neubauer found that by observation of the cavitation with the help of the replica technique, the residual lifetime could be estimated [1, 2]. The cavitation was subdivided into four classes (and one for undamaged material): individual cavities to a small extent, individual cavities to a large extent, stringers of cavities and finally microcracks. Replicas were taken at welds, pipe bends and other critical positions. The basic idea was that if damage of one class was detected, the damage did at most correspond to the next class at the next inspection. The method was very successful when it was applied to low alloy steels such as 0.5Cr0.5Mo0.25V and 2.25Cr1Mo, which represented the body of materials of the plants at the time. Unfortunately the method is less applicable to today’s materials such as 9 and 12Cr steels, because cavitation appears only at a late stage of life and does not provide the necessary early warning.
The successful technical use of observations of cavitation stimulated a lot of scientific work. This was dominated by empirical approaches to describe the development of the creep damage. The first and perhaps most well-known approach was set up by Kachanow and Rabotnov [3–5]. They simply assumed that the cavities represented voids that reduced the loading capacity. The most interesting feature of the method is that it is consistent with behaviour of creep strain during the tertiary creep, that is, the final stage before rupture [6, 7]. This dependence is nowadays referred to as the omega method .
We will now concentrate on basic models for cavitation. Traditionally cavity nucleation has been modelled either as a process of rupturing atomic bonds or of atomic vacancy condensation. For the former approach the estimated threshold stress is orders of magnitude higher than the applied stress, which makes it physically unrealistic because high stresses will be reduced quickly in a creeping material . The condensation of vacancies can be treated with the help of the classical nucleation theory . It is shown that cavity nucleation would be a very rare event at low stresses but becomes frequent above a certain threshold stress . High stress concentration can be formed at grain boundary ledges, grain boundary triple points and particles. Cavity nucleation at particles can be a result of decohesion of particles from the matrix. In practically all models that have been presented, a high stress concentration is needed. A threshold stress and an incubation time are essential to form a cavity [10, 12]. Contrary to these suggestions nucleation frequently takes place at low stresses and is controlled by strain rather than stress. In agreement with statements in the literature it can be concluded that theories of cavity nucleation have not been fully successful in earlier work [13, 14]. In both these papers excellent reviews are given.
For modelling cavity growth the situation was different but not entirely unproblematic. A diffusion controlled cavity growth was formulated by Hull and Rimmer . The model was later improved in particular by Beere and Speight . It was soon realised that the model gave much larger growth rates than observed experimentally in many cases. Dyson proposed that the cavities should not be able to grow faster than the creep deformation of the surrounding material . This is referred to as constrained cavity growth. Rice developed an explicit model for constrained growth . The idea of constrained growth is now fully accepted. However, the model still tends to overestimate the growth rates. It is now believed to be due to the assumptions in Rice’s derivation. This will be analysed in the present paper.
It is evident from the summary above that the modelling of formation and growth of cavities has met considerable difficulties in the past. In recent years important new developments have taken place. With the help of these developments fundamental quantitative modelling of both nucleation and growth of creep cavities has now been possible to set up. In addition some of the difficulties that have been encountered in the past can be understood. It is the purpose of this paper to review these new developments.
2. Grain boundary sliding
Grain boundary sliding (GBS) occurs when neighbouring grains move with respect to each other in shear. The mechanism is illustrated in Figure 1. During the creep deformation the two grains have moved 0.8 μm with respect to each other. To observe GBS, the specimen surface has to be scratched, for example, with a knife. When the scratches cross a sliding grain boundary, the two parts of the scratches on the different sides of the grain boundary are displaced. This displacement is a direct measure of GBS.
It is generally accepted that a prerequisite for cavity nucleation is grain boundary sliding. Experiments on copper bicrystals have shown that artificially introduced GBS can dramatically increase the amount of cavitation. Chen and Machlin  and Intrater and Machlin  exposed bicrystals of copper to either tensile loading or to a combination of tensile loading and shear. The latter alternative gave much larger number of cavities. It is natural that GBS gives rice to cavitation, since any obstacle at the grain boundary such as a particle will give rise to large stress concentrations. In low alloy steels cavities have frequently been observed around manganese sulphides . Since the interface between the sulphides and the matrix is weak, voids are easily formed there. Some papers also suggest that cavities can be formed at carbides, see, for example, Ref. .
The most convincing argument concerning the central role of GBS in cavity nucleation comes from the creep strain dependence of both GBS and cavity nucleation. It has been observed many times that the displacement u GBS due to GBS is approximately proportional to the creep strain ε, see, for example, Ref. . The first ones to observe this relation were McLean and Farmer .
C s(ε) is a constant that is dependent on the creep strain ε. At the same time the nucleation rate of cavities is also proportional to the creep strain rate .
B is constant. This means that the number of cavities is proportional to the creep strain in the same way as the GBS displacement in Eq. (1). Eq. (2) was first observed by Needham and coworkers [26, 27]. For a review, see Ref. . Eqs. (1) and (2) will be derived below. Eq. (2) would be very difficult to explain unless it assumed that the nucleation is controlled by GBS.
Experiments that give the strain dependence of u GBS are illustrated in Figure 2.
The displacement u GBS increases as of function of strain in agreement with Eq. (1). C s is the slope of the curves. Three types of tests are represented in the figure: tests at constant stress, at constant stress rate and at constant strain rate. In spite of the fact that a range of temperatures, strain rates and test methods is covered, the C s values do not vary very much.
To investigate the influence of GBS on the total strain, Crossman and Ashby  developed a finite element model (FEM) for shear stresses. If a free grain boundary is considered, they found that the sliding rates are very high for typical creep stresses and that the grain boundaries could be considered as flaws in the material with respect to GBS. Later Ghahremani  transferred the model to tensile stresses, which are typically used in creep testing. In both Refs. [31, 32] a Norton equation for the creep strain rate was considered
σ is the applied stress and n is the creep exponent. and σ 0 are constants. The percentage creep rate due to grain boundary sliding φ was determined
is the total displacement rate. φ was found to take values from 0.15 (n = 1) to 0.33 (n = ∞) in Ref. . can be expressed in terms of the creep rate
where d lin is the linear intercept grain size and ξ = 1.36 is a pure geometrical factor that explains how the hexagonal grains studied in Refs. [31, 32] should be related to the measured grain size. The factor 3/2 depends on the definition of . By combining Eqs. (1), (4) and (5) we find the values of the GBS parameter C s in Eq. (1)
Eq. (6) is referred to the shear sliding model. Eq. (6) is compared with experimental results for copper in Figure 3 . The C s values according to Eq. (6) for the individual tests in Figure 2 have been evaluated in . The model values are about C s ≈ 50 μm. These values are slightly high for the creep tests , but in range for slow strain tests  and constant stress rate tests .
For materials with particles in the grain boundaries Riedel has derived a model corresponding to Eq. (1) . The sliding boundary was represented by a shear crack surrounded by creep deforming grains. The model is referred to as the shear crack model. Although the author was not very happy with the model, it turns out that it does not give very different results for austenitic stainless steels in comparison with the shear sliding model.
The two models (shear sliding and shear crack models) are compared with the experimental GBS displacements for different austenitic stainless steels [34–38] in Figure 4. The shear crack model is compared with the average of all the experimental data, Ave. 1. The shear sliding model does not work very well for large grain sizes so data for such grain sizes  are not included in the comparison, Ave. 2. It can be seen from Figure 4 that C s values of the correct order are predicted.
3. Cavity nucleation
3.1. Thermodynamic considerations
The mechanisms for cavitation nucleation have been a puzzle for a long time as explained in the introduction. However part of the explanation came from studies on copper. Pure copper can show extensive cavitation during creep , but the number of particles present is so low that they cannot explain the large number of cavities. Lim suggested that it was the substructure of the dislocations that could nucleate the cavities . He also presented a model that can be used to demonstrate whether a nucleation mechanism is thermodynamically feasible or not. He assumed that pile ups of grain boundary dislocations generate the necessary high stresses for the nucleation. Since these high stresses are stationary as a result of the creep process, it avoids the problem of fast stress relaxation in many models. Lim’s model is fairly complex and details in the model will not be given here. When a cavity is formed the free energy ΔG is changed in a number of ways that are represented by the terms in the following equation 
γs and γGB are the surface and grain boundary energies per unit area. F v = 2π/3 (2–3cos α + cos3 α), F s = 4π(1−cos α), F b = πsin2 α and F v′ = 1.5 F v, where α is half the tip angle of the cavity. The first term in Eq. (7) is the work done by the applied stress. The second and third terms represent the modification in the surface and grain boundary energies. The fourth term is the decrease in the strain energy. ΔG 1 is the change in the line energy of the grain boundary dislocations (GBD). ΔG 2 is the interaction energy between the remaining and the consumed GBD. The strain energy ΔG 3 is the reduction of the strain energy of GBDs outside the cavity. Full details can be found in [19, 40].
Lim’s model has been applied to copper and austenitic stainless. As long as energy is gained when a cavity is formed, that is, ΔG in Eq. (7) is negative, cavitation is possible. From Eq. (7) ΔG is reduced when the applied stress σappl is raised, that is, cavitation becomes more likely which is natural. On the other hand when σappl is reduced cavitation is more difficult. There is minimum stress where cavitation is no longer possible because ΔG becomes positive. This minimum stress is plotted as a function of temperature for copper in Figure 5.
These minimum stresses are compared with design stresses during creep in copper. It is clear that the stresses required for nucleation are well below the stresses that typically appear in the material. This demonstrates that nucleation based on the substructure is a viable process.
From a technical point of view it is well established that the creep ductility of oxygen free pure copper Cu-OF can be very much lower than for phosphorus alloyed copper Cu-OFP. As a consequence the latter material should be used in creep exposed components [28, 41]. It is evident from Figure 5 that much lower stresses are needed in Cu-OF than in Cu-OFP, which makes the cavitation in the former material much more abundant. This is believed to be the main reason for the low creep ductility of Cu-OF.
It has also been verified that the minimum nucleation stresses are below typical design stresses for the common stainless steels 304H (18Cr10Ni), 316 (17Cr12Ni2Mo), 321 (18Cr12NiTi) and 347 (18Cr12NiNb). For example this is illustrated for 347 (18Cr12NiNb) in Figure 6. The design stresses are 10,000 h rupture data. The minimum cavitation stress lies in the interval 35–50 MPa in the interval from 500°C to 750°C. The minimum cavitation stresses are again below the design stresses. The temperature dependence of Lim’s model is probably not fully correct. In general it is thought that the amount of cavitation will increase with temperature. However the temperature dependence of the minimum cavitation stress is weaker than that of the design stress, which suggests the opposite behaviour.
3.2. Strain dependence
Experimentally it has been found many times that the number of cavities is proportional to the creep strain, cf. Eq. (2). To explain this strain dependence, Sandstrom and Wu introduced the double ledge model . They considered a sliding grain boundary with dislocation substructures on both sides of the boundary that moved along with the grains. The substructures consist of subgrains that contain fairly few dislocations in their interior but with well-developed subgrain walls. The positions where the subgrain walls meet at the grain boundary are referred to as subgrain corners. Nucleation was assumed to take place when a subboundary on one side of the boundary hits a subgrain corner on the other side. The nucleation rate can be expressed as
where d sub is the subgrain diameter. The last factor takes into account that one nucleus can be formed in each subgrain on the boundary. The subgrain size is directly related to the applied stress 
The constant K sub is about 20 for austenitic stainless steels and about 11 for copper. The same model can be applied to particles in the grain boundary that are known to contribute to the nucleation. In the model the subgrain corners are replaced by the particles in the grain boundaries with an interparticle distance of λ. Taking both subgrain corners and particles into account, the resulting expression for the nucleation rate is 
The model in Eq. (10) is compared with experimental data for austenitic stainless steels in Figure 7. For three of the experimental data sets TP347 at 550°C and 650°C and TP304 at 727°C, the model gives quite an acceptable description. For TP304XX at 750°C the deviation between model and experiment is larger.
3.3. Particle size
It has been proposed that a critical particle radius exists for nucleation [45, 46]. The radius must exceed a minimum value in order for nucleation to take place. Harris developed a model that related the critical particle size to the GBS velocity [47, 48]. His basic assumption was that particles are not able to stop GBS if the diffusion is fast enough. According to Harris this critical particle radius also represented the minimum radius that could nucleate cavities. Harris gave the following relation between the GBS velocity and the critical particle radius r c
where λ is the interpaticle spacing, γs the surface energy, δ the grain boundary width, D GB the grain boundary self-diffusion coefficient, Ω the atomic volume, k B Boltzmann’s constant and T the absolute temperature. The application of Eq. (11) is illustrated in Figure 8 for different types of austenitic stainless steels .
In Figure 8 the particle parameters are taken from the experimental references. The minimum particles that nucleated cavities in the experiments are chosen for the critical particle radius. The experimental data clearly support Harris’ model.
From the particle size distributions  the number of nucleated cavities can be estimated if the critical particle size is known. The computed number of nuclei is compared with the observed ones for austenitic stainless steels in Figure 9.
4. Cavity growth
4.1. Unconstrained cavity growth model
After the cavities have been nucleated, they start to grow if they exceed a critical size. The main mechanism for the growth is diffusion. Vacancies are transported away from the surfaces of the cavities. The grain boundaries are good sinks for the vacancies. The first model for diffusion controlled growth was presented by Hull and Rimmer . A much more elegant formulation was later given by Beere and Speight  and this is the model that has been used since. Their growth equation can be expressed as
where R the cavity radius in the grain boundary plane, dR/dt its growth rate, σ0 the sintering stress 2γs sin(α)/R, where γs is the surface energy of the cavity per unit area and α the cavity tip angle. D 0 is a grain boundary diffusion parameter, D 0 = δD GBΩ/k B T, where δ is the grain boundary width, D GB the grain boundary self-diffusion coefficient, Ω the atomic volume, k B Boltzmann’s constant and T the absolute temperature. The factor K f was introduced in . It is a function of the cavitated grain boundary area fraction f a = (2R/L)2
From the number of cavities per unit grain boundary area n cav, the cavity spacing L can be determined
4.2. Constrained cavity growth
When diffusion controlled growth models were compared with experimental data, it was evident that the models often strongly exaggerated the growth rate. Dyson found that the predicted growth rate of the cavities many times exceeded the deformation rate of the surrounding material which he considered as unphysical . He suggested that the cavity growth rate should not be larger than the creep rate of the material. This was referred to as constrained growth. Based on this assumption, Rice developed a quantitative model . The result is that in the growth equation, the applied stress is replaced by a reduced stress
The reduced stress is given by
where β is a material constant (β = 1.8 for homogeneous materials) and d the grain diameter. With this approach a growth model that fulfils Dyson’s criterion has been achieved.
Rice based his analysis on a linear viscoplastic model of an opening crack. He and Sandstrom reanalyzed the model and avoided the assumption of linearity . A grain structure with a pillar of height h and width corresponding to the grain size d was set up. In this pillar the creep deformation in the axial (z) direction is given by
and are the creep rates at the reduced and applied stress, respectively. The first term in the middle part of Eq. (17) is the volume growth rate of a cavity multiplied by the number of cavities per unit grain boundary area. The creep displacement of the pillar at the reduced stress is the second term. The final term on the right hand is the displacement in the surrounding material. A finite element analysis was performed to determine the size of the height h. It was found that h ≈ 2R in the investigated cases . If this value for h is used and n cav is replaced by 1/L 2 according to Eq. (14), the following equation is obtained
In general Eq. (19) has to be solved by iteration to find the new value of σred. This new value for σred is lower than that given by Eq. (17). This is illustrated in Figure 10: Both the absolute and relative difference increase with time.
The new constrained growth model is compared to experimental data for austenitic stainless steels in . Some examples are given here in Figure 11. Growth data for 18Cr10Ni steel with and without Nb or Ti are shown. It can be seen that the growth data can be described with fair accuracy. The lower growth rate according Eq. (19) is important in this respect.
5. Brittle creep rupture
Creep rupture is technically very important, because it determines the life of many plants operating at high temperatures. Two main mechanisms are distinguished: ductile rupture and brittle rupture. Ductile rupture is controlled by the exhaustion of the deformation capacity of the material. In this case the usual rupture criterion is that the creep strain reaches a critical value. The deformation takes place by dislocation mechanisms. The faster the dislocations move, the faster rupture occurs. Since ductile rupture does not involve cavitation, it is not reviewed here, but full details can be found elsewhere .
Rupture curves for dislocation creep are illustrated in Figure 12 for the austenitic stainless steel 18Cr12NiTi (321H) at temperatures between 600°C and 775°C. The experimental creep rupture data cover times up to 100,000 h (11 years). The general overall behaviour is well described by the model predictions.
The second process brittle rupture is due to grain boundary decohesion. By far the most important mechanism in this respect is the formation and growth of cavities. It is well established that when the cavitated grain boundary area reaches a certain fraction of about 0.25, brittle rupture takes place . The cavitated area fraction A cav can be computed from Ref. 
A continuous nucleation of cavities takes place. The number of cavities is directly proportional to the creep strain, Eq. (10). Once a cavity has nucleated it starts to grow after an incubation time t i that is a small fraction of the rupture time . The growth is described with Eq. (16) with the reduced stress given by Eq. (19). When A cav has reached 0.25, rupture is assumed to take place.
The model predictions for brittle rupture for 18Cr12NiTi (321H) are shown in Figure 13. The predictions are compared to the same experimental data as in Figure 12. Again the overall time dependence of the rupture strength at different temperatures is well represented. In fact the differences between the model predictions for ductile rupture in Figure 12 and brittle rupture Figure 13 are not very large.
Ductile rupture is assumed to be controlling if the strain exhaustion occurs before A cav = 0.25 has been reached. On the other hand if the cavitation criterion is reached first, brittle rupture takes place. The results for ductile and brittle rupture are combined in Figure 14. For a given temperature and stress the value from Figure 12 is chosen if the (ductile) rupture time is shorter than the (brittle) rupture time in Figure 13 and vice versa.
When brittle rupture is taken into account when modelling the creep rupture curves, there is an improvement in particular at high temperatures and low stresses.
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 , 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.
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 . As a consequence cavity nucleation only occurs on some grain boundaries. The presence of cavities show a large statistical variation, see for example . 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 . Then the cavities were not allowed to grow faster than the surrounding creeping material. A quantitative model for constrained growth was given by Rice . 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 . 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.
Recent development of processes concerning the development of creep cavities has been reviewed. Models have been covered for grain boundary sliding, cavity nucleation and cavity growth.
Based on analysis of previously presented FEM models, a model for the grain boundary displacement during sliding called the shear sliding model has been presented. The model gives a displacement that is proportional to the creep strain. The model can quantitatively reproduce the observed displacements for fcc alloys.
Using Lim’s model for subboundary assisted cavity nucleation, it has been demonstrated that this process is thermodynamically feasible for copper and for austenitic stainless steels. It gives a minimum cavitation stress that is well below stresses in creep exposed components.
According to the double ledge model, nucleation is assumed to take place when subboundaries on one side of a sliding grain boundary meet subgrain corners or particles on the other side. The model gives a nucleation rate that is proportional to the creep rate in good accordance with observations. The model can predict the measured nucleation rates in austenitic stainless steels.
Due to the development of the new models for GBS, it has for the first time been possible to test Harris’ model for cavity nucleation around particles. The model gives a relation between the critical particle radius and the GBS velocity. Data for austenitic stainless steels confirm the validity of the model.
A modified constrained growth model has been presented. It gives lower growth rates than previous models. These lower growth rates have implied that observed growth rates for austenitic stainless steels can now be reproduced.
Alloys can fail by ductile or brittle creep rupture. It is demonstrated that by considering both ductile and brittle rupture, the prediction of creep rupture curves can be improved in particular at high temperatures and low stresses.