**2. Grain boundary sliding**

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 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 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 depen-

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 [9]. The condensation of vacancies can be treated with the help of the classical nucleation theory [10]. It is shown that cavity nucleation would be a very rare event at low stresses but becomes frequent above a certain threshold stress [11]. 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.

the material fail.

20 Study of Grain Boundary Character

the necessary early warning.

dence is nowadays referred to as the omega method [8].

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 [20] and Intrater and Machlin [21] 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 [22]. 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. [23].

**Figure 1.** Illustration of grain boundary sliding (GBS) for a copper specimen that has been exposed to 3.3% creep strain during 307 h at 125°C [19]. The grain boundary lies in the southwest-northeast direction. It is crossed by a major scratch, which makes it possible to measure GBS. The grain to the left has moved downwards by 0.8 μm relative to the grain at the right and that is the amount of GBS.

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. [24]. The first ones to observe this relation were McLean and Farmer [25].

$$
\mu\_{\rm{Class}} = \mathcal{C}\_{\rm{s}}(\boldsymbol{\varepsilon}) \boldsymbol{\varepsilon}.\tag{1}
$$

*C*s (ε) is a constant that is dependent on the creep strain ε. At the same time the nucleation rate of cavities \_\_\_ *dn dt* is also proportional to the creep strain rate *ε* . .

$$\frac{dn}{dt} = B\dot{\varepsilon}.\tag{2}$$

*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

**Figure 2.** Observed displacements at grain boundaries in copper as a function of strain [19]. Data from Refs. [29, 30] are also shown.

coworkers [26, 27]. For a review, see Ref. [28]. 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 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. [24]. The first ones to observe this relation were McLean and

**Figure 1.** Illustration of grain boundary sliding (GBS) for a copper specimen that has been exposed to 3.3% creep strain during 307 h at 125°C [19]. The grain boundary lies in the southwest-northeast direction. It is crossed by a major scratch, which makes it possible to measure GBS. The grain to the left has moved downwards by 0.8 μm relative to the grain at

(ε) is a constant that is dependent on the creep strain ε. At the same time the nucleation 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

. .

(ε)ε. (1)

*dt* <sup>=</sup> *<sup>B</sup>ε*˙. (2)

Farmer [25].

of cavities \_\_\_ *dn*

*dt*

the right and that is the amount of GBS.

22 Study of Grain Boundary Character

*C*s

*u*GBS = *Cs*

\_\_\_ *dn*

is also proportional to the creep strain rate *ε*

The displacement *u*GBS increases as of function of strain in agreement with Eq. (1). *C*<sup>s</sup> 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*<sup>s</sup> values do not vary very much.

To investigate the influence of GBS on the total strain, Crossman and Ashby [31] 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 [32] 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

$$
\dot{\varepsilon} = \dot{\varepsilon}\_0 \left( \frac{\sigma}{\sigma\_0} \right)^n. \tag{3}
$$

σ is the applied stress and *n* is the creep exponent. *ε* . 0 and *σ*<sup>0</sup> are constants. The percentage creep rate due to grain boundary sliding φ was determined

$$
\phi = \frac{\dot{u}\_{\text{cas}}}{\dot{u}\_{\text{all}}}.\tag{4}
$$

*u* . all is the total displacement rate. φ was found to take values from 0.15 (*n* = 1) to 0.33 (*n* = ∞) in Ref. [32]. *u* . all can be expressed in terms of the creep rate *ε* .

$$
\dot{u}\_{\text{all}} = \frac{3 \, d\_{\text{in}} \, \dot{\varepsilon}}{2 \xi} \, \tag{5}
$$

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 *u* . all. By combining Eqs. (1), (4) and (5) we find the values of the GBS parameter *C*<sup>s</sup> in Eq. (1)

$$\mathbf{C}\_s = \left. \dot{u}\_{\rm Cas} \right| \dot{\varepsilon} = \frac{3\phi}{2\zeta} d\_{\rm lin} \,. \tag{6}$$

Eq. (6) is referred to the *shear sliding model*. Eq. (6) is compared with experimental results for copper in **Figure 3**. The *C*<sup>s</sup> values according to Eq. (6) for the individual tests in **Figure 2** have

**Figure 3.** Comparison of modelled Eq. (6) and observed displacements at grain boundaries in copper divided by the creep strain, cf. Eq. (1) [19]. Data from [29, 30] are also shown.

been evaluated in [19]. The model values are about *C*<sup>s</sup> ≈ 50 μm. These values are slightly high for the creep tests [30], but in range for slow strain tests [29] and constant stress rate tests [19].

σ is the applied stress and *n* is the creep exponent. *ε*

*<sup>φ</sup>* <sup>=</sup> *<sup>u</sup>*

*u*

*Cs* = *u*

creep strain, cf. Eq. (1) [19]. Data from [29, 30] are also shown.

the values of the GBS parameter *C*<sup>s</sup>

copper in **Figure 3**. The *C*<sup>s</sup>

size. The factor 3/2 depends on the definition of *u*

*u* .

Ref. [32]. *u*

.

24 Study of Grain Boundary Character

creep rate due to grain boundary sliding φ was determined

all can be expressed in terms of the creep rate *ε*

. 0 and *σ*<sup>0</sup>

.

values according to Eq. (6) for the individual tests in **Figure 2** have

. \_\_\_\_ GBS *u* . all

all is the total displacement rate. φ was found to take values from 0.15 (*n* = 1) to 0.33 (*n* = ∞) in

all <sup>=</sup> <sup>3</sup> *<sup>d</sup>*lin *<sup>ε</sup>* . \_\_\_\_\_

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

.

. <sup>=</sup> \_ 3*φ*

Eq. (6) is referred to the *shear sliding model*. Eq. (6) is compared with experimental results for

**Figure 3.** Comparison of modelled Eq. (6) and observed displacements at grain boundaries in copper divided by the

.

in Eq. (1)

. GBS/ *<sup>ε</sup>* are constants. The percentage

. (4)

<sup>2</sup>*<sup>ξ</sup>* , (5)

all. By combining Eqs. (1), (4) and (5) we find

<sup>2</sup>*<sup>ξ</sup> d*lin . (6)

For materials with particles in the grain boundaries Riedel has derived a model corresponding to Eq. (1) [33]. 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 [38] are not included in the comparison, Ave. 2. It can be seen from **Figure 4** that *C*<sup>s</sup> values of the correct order are predicted.

From **Figures 3** and **4** it is evident that the shear sliding and shear crack models can describe the experimental data for fcc metals with reasonable precision.

**Figure 4.** Modelling and experimental GBS displacement as a function of creep strain for different types of austenitic stainless steels, from Ref. [24]. Experimental data Ref. [34–38].
