Grain Boundary Sliding

Abstract

During plastic deformation at elevated temperatures, grains move relative to each other which is referred to as grain boundary sliding (GBS). The amount of GBS is proportional to the creep strain with a proportionality constant that is known from finite element analyses, and found to agree with experiments for Cu. The most import effect of GBS is that it gives rise to the initiation of creep cavities, Chap. 10. GBS is also the main mechanism for superplasticity. A basic model for superplasticity is presented.

9.1 General

During creep neighboring grains are displaced along the grain boundaries (GB) relative to each other when they are exposed to shear stresses. This is referred to as grain boundary sliding (GBS). GBS can easily be observed by metallography by introducing scribe lines or a micro grid before the test. The principle is illustrated in Fig. 9.1. Using scribe lines on a polished and etched surface, the shear offset under application of stress can be observed and measured where the lines cross the grain boundary (GB).

Fig. 9.1

Schematic illustration of observation and measurement of GBS; a before test; b after test

The appearance of GBS in a micrograph is shown in Fig. 9.2.

Fig. 9.2

SEM observations of GBS in Cu-OFP after 507 h in a creep test at 125ºC, 47 MPa. The strain was 20.8%. The grain boundary goes from northeast to southwest. The scribe line is almost perpendicular to the grain boundary. Reprinted from [1] with permission of Elsevier

The displacement of the scribe line in Fig. 9.2 was about 5 µm, which is the amount of GBS. For a flat GB, the sliding itself experiences little resistance but significant stresses appear at the triple points in the grain boundary corners, which have to be relaxed by creep deformation. Sometimes the stresses are large enough to initiate micro-cracks at the triple points. This is illustrated in Fig. 9.3.

Fig. 9.3

SEM observation of a micro-crack at a triple point after GBS. The same specimen as in Fig. 9.2 [1]

GBS can also result in grains moving perpendicular to the surface. This has the consequence that the specimen surface appears wavy. This has often been observed for pure Al, see for example [2]. However, using the GB offset technique illustrated in Fig. 9.1, the GBS events are only found locally at a limited number of GBs [1, 3]. Unfortunately, only a limited number of studies where a systematic measurement of the GB offsets has been performed are available.

What is causing GBS is not understood in detail. From TEM observations on Al, Kokawa et al. have suggested that it is only random grain boundaries that slide. Lattice dislocations move into the GBs and introduce the sliding. Ordered GBs (coincidence sites) contain extrinsic GB dislocations but they do not contribute to the sliding [4].

One sometimes distinguishes between two types of GBS: Rachinger sliding and Lifshitz sliding that occur during dislocation creep and diffusion creep, respectively [5]. Unfortunately, it has turned out to be difficult to distinguish between dislocation creep and diffusion creep experimentally even if the principles are straightforward. There are numerous scientific papers discussing this issue. The simple principle that will be followed in this book is that it is Rachinger sliding when we discuss dislocation creep and Lifshitz sliding when diffusion creep is analyzed but without being explicit about the type of sliding.

GBS gives a contribution to the overall creep strain. In general, this contribution is expected to be limited. The model analysis in Sect. 9.3 suggests that the contribution is about 15% if all the grain boundaries are active in GBS. However, since only a limited number of GBs give GBS offsets, the effective contribution is likely to be much smaller. However, there is one main exception, superplasticity. The main mechanism for superplastic deformation is believed to be GBS of a fine grained structure, where more or less all grain boundaries participate. This means that the deformation takes place by the sliding of grains against each other without the grains being elongated. In this way large elongation values can be obtained during superplasticity. Empirical modeling of superplasticity will be discussed in Sect. 9.2. In Sect. 9.3, a basic model for grain boundary sliding is presented. This result is used in Sect. 9.4 to find a basic model for superplasticity. Another case where GBS plays a major role is for nano-crystalline materials. The reason is the same as for superplastic materials. The GBs constitute a large fraction of the nano-crystalline structures, and GBS is an important deformation mechanism [6]. However, it has turned out that the behavior is complex and the topic will not be dealt with here.

During creep, the formation and growth of cavities generate creep damage and often initiate failure. In the past, it was believed that creep cavities were nucleated due to the presence of large local stresses. However, detailed analysis demonstrated that this would require very high stresses and in addition these high stresses would relax very quickly during creep. Nowadays, most scientists are convinced that GBS is the main mechanism for forming cavities. For example, it can quantitatively explain the strain dependence of the number of creep cavities and why creep cavities can be formed at low creep stresses. Initiation and growth of cavities will be discussed in Chap. 10. The required model for GBS will be presented in Sect. 9.3.

9.2 Empirical Modeling of GBS During Superplasticity

Discussion of various aspects of GBS can be found in many papers. However, the number of direct measurements of GBS is limited. Results for copper will be presented in Sect. 9.3 together with basic modeling. The only area where numerous measurements can be found is for superplasticity, where stress strain curves and creep rates have been determined. In general, it is assumed that superplasticity is controlled by GBS.

Superplasticity is a mechanism where elongations of several hundred percent can be achieved. This makes it possible to produce deeper drawings and more complex shapes than in ordinary sheet pressing. There are a number of requirements on the alloy to enable superplasticity [7]. The grain size must be fine, less than 10 µm and equiaxed. The pressing must be performed at temperatures above half the absolute melting point (>T_m/2). The strain rate should lie in the interval 1 × 10⁻⁵ to 1 × 10⁻¹ 1/s. The lower limit is to ensure that pressings can be carried out in a reasonable time. The upper limit is to prevent damage formation such as the development of cavities. Finally the strain rate sensitive m_r

$$ m_{{\text{r}}} = \frac{\partial \ln \sigma }{{\partial \ln \dot{\varepsilon }}} $$

should be about 0.5. Under stationary condition, m_r is the inverse of the stress exponent n_N

$$ m_{{\text{r}}} = 1/n_{{\text{N}}} $$

(9.1)

The choice of temperature is critical. If the temperature is too low, climb will be slow and the pressing would require a long time. A too high temperature will initiate grain growth that will destroy the superplastic properties. Often a two-phase structure is used to prevent grain growth. An alternative is to have a fine distribution of particles that acts as grain refiner. Both these alternatives are associated with the risk that abnormal grain growth is initiated implying that some very large grains are created, which is totally unacceptable [8].

Superplastic formed parts are nowadays used in many applications [9]. High strength alloys are typically difficult to form with conventional techniques because of limited ductility. Then superplastic forming can be quite helpful. In particular, applications in aero planes, trains and cars are common. Special high strength aluminum alloys probably cover most of the market. However, there are many components produced in Mg, Ti and Ni base alloys as well. The total number of commonly used alloys for superplastic forming is not very large. Barnes lists 13 alloys [9].

After superplastic deformation, the grain shape is still equiaxed. The only imaginable mechanism that can accomplish this is GBS. Detailed measurements of the amount of GBS during superplasticity confirm that GBS can account for almost all of the strain [10]. Creep deformation inside the grains must also take place to accommodate local strains. First, bulk dislocations moving towards the GBs are the basis of GBS. Secondly, the grains must constantly adapt their shape during the deformation and this takes place by intragranular creep.

A number of authors have used empirical relations to describe the strain rate during superplasticity as a function of temperature, stress and grain size, see for example [11, 12]. The most common form is

$$ \dot{\varepsilon } = A\frac{{bD_{{{\text{GB}}}} }}{{Gk_{{\text{B}}} T}} \left(\frac{b}{{d_{{\text{g}}} }} \right)^{2} \sigma^{2} $$

(9.2)

where $\dot{\varepsilon }$ is the strain rate, σ the applied stress, b burgers vector, G the shear modulus, d_g the grain size, D_GB the grain boundary diffusion coefficient and A a dimensionless factor. For GBS it is natural to assume that it is grain boundary diffusion that supplies the vacancies although the contribution from the lattice dislocation is also of importance. Eq. (9.2) gives a stress exponent of 2 and exponent of −2 for the grain size. It is not possible to intuitively understand the values of these exponents. However, it will be explained in Sect. 9.4 that detailed modeling actually gives these exponents.

Data for a superplastic Zn22%Al alloy is illustrated in Fig. 9.4.

Fig. 9.4

Creep rate versus stress for at Zn22%Al alloy at 190 °C at the three grain sizes 1.3, 2.6. and 3.9 µm. Data from [13]. The grain sizes in [13] have been transferred to linear intercept values

In Fig. 9.4, there are two different levels of the stress exponent. At low stresses, the stress exponent is 4, at higher stresses 2. It is the high stress range where the superplastic behavior appears. The lower stress range is nowadays assumed to be due to the presence of impurities [5]. This range will be discussed further in Sect. 9.4. In the superplastic range the grain size dependence has an exponent of about −2 again in accordance with Eq. (9.2). A comparison of the data with modeling will be presented in Sect. 9.4.

9.3 Grain Boundary Sliding in Copper

Crossman and Ashby [14] formulated a model for the contribution from GBS to the overall creep strain. The basic steps in this model will be followed. The inverse relation namely the amount of GBS generated by an amount of creep strain is equally interesting. The relative displacement u_GB of neighboring grains is controlled by viscous flow. If the GB is exposed to a shear stress, the displacement rate can then be expressed as

$$ \frac{{du_{{{\text{GB}}}} }}{dt} = \frac{{\delta_{{{\text{GB}}}} }}{{\eta_{GB} }}\tau $$

(9.3)

where δ is the width of the grain boundaries (taken as 2 b, where b is burgers’ vector; a common assumption) and τ is the shear stress acting on the grain boundary. η_GB is the viscosity of a flat grain boundary

$$ \eta_{{{\text{GB}}}} = \frac{{k_{{\text{B}}} T}}{{8bD_{{{\text{GB}}}} }} $$

(9.4)

k_B is Boltzmann’s constant, T the absolute temperature and D_GB the grain boundary diffusion coefficient. If ledges with the height h_L are present, the viscosity in Eq. (9.4) for a flat GB is increased by a factor of (h_L/b)² [15]

$$ \eta_{{{\text{GB}}}} = \frac{{k_{{\text{B}}} Th_{{\text{L}}}^{2} }}{{8b^{3} D_{{{\text{GB}}}} }} $$

(9.5)

Presence of a distribution of particles also increases the viscosity [15]

$$ \eta_{{{\text{GB}}}} = \frac{{k_{{\text{B}}} Tf_{A} d_{{{\text{part}}}}^{2} }}{{8b^{3} D_{{{\text{GB}}}} }} $$

(9.6)

where f_A and d_part are the area fraction and diameter of particles in the boundary. Equations (9.5)–(9.7) thus represent three different types of GBs. A finite element analysis was performed for hexagonal grains with sliding boundaries and grains following power-law creep [14]. The sliding of the boundaries is so fast that in general they can be considered as flaws. The overall creep rate could be described by introducing a stress enhancement factor f_c

$$ \dot{\varepsilon } = \dot{\varepsilon }_{0} \left( {f_{{\text{c}}} \frac{\sigma }{{\sigma_{0} }}} \right)^{n} $$

(9.7)

σ is the applied stress and n is the creep exponent. $\dot{\varepsilon }_{0}$ and σ₀ are constants. Crossman and Ashby [14] gave a value of f_c = 1.1. Ghahremani refined the analysis and found a value of f_c = 1.16–1.3 [16]. Also Hsia et al. [17] repeated the analysis and got f_c = 1.17 for the same geometry. In [14, 16] also the contribution to the overall displacement rate was assessed

$$ \phi = \frac{{\dot{U}_{{{\text{GBS}}}} }}{{\dot{U}_{{{\text{All}}}} }} $$

(9.8)

$\dot{U}_{{{\text{All}}}}$ is the total displacement rate, which must be precisely defined in relation to the grain structure. The ϕ values found were from 0.15 (n_N = 1) to 0.33 (n_N = ∞) depending on the creep exponent [16]. Both $\dot{U}_{{{\text{All}}}}$ and $\dot{U}_{{{\text{GBS}}}}$ are proportional to the creep rate $\dot{\varepsilon }$. The finite element analysis [14] shows that the overall displacement rate can be expressed as

$$ \dot{U}_{All} = \frac{{3d_{{\text{g}}} \dot{\varepsilon }}}{2\xi } $$

(9.9)

where d_g is the linear intercept grain size and ξ is a factor that gives the relation to the side length a_hex of the hexagonal grains [1].

$$ \xi = d_{g} /a_{{{\text{hex}}}} = \pi /4\tan (\pi /6) = 1.36 $$

(9.10)

It was early on recognized that the displacement due to GBS is proportional to the creep strain [18]

$$ u_{{{\text{GBS}}}} = C_{{\text{s}}} \varepsilon $$

(9.11)

From Eqs. (9.8), (9.9), and (9.11), the constant C_s can be expressed as

$$ C_{{\text{s}}} = \dot{U}_{{{\text{GBS}}}} /\dot{\varepsilon } = \frac{3\phi }{{2\xi }}d_{{\text{g}}} $$

(9.12)

The model results above will now be compared with experiments for oxygen free copper with P (Cu-OFP) and without P (Cu-OF). Observed values from three investigations are shown in Fig. 9.5.

Fig. 9.5

Observed displacements at grain boundaries as a function of strain in Cu-OF and Cu-OFP. Data from [1, 19, 20]. Redrawn from [1] with permission of Elsevier

The values in Fig. 9.5 represent three types of tests: creep at constant load [19], creep at constant loading rate [1] and slow strain rate tests at constant strain rate [20]. Pettersson’s values increase faster with strain in particular in comparison to the data from [19]. Considering that a wide range of temperatures and strain rates are covered, it is not surprising that the values differ. But they are clearly of the same order.

In Fig. 9.6, the displacement in Fig. 9.5 are divided by the strain to obtain the values for the constant C_s in Eq. (9.11).

Fig. 9.6

Observed displacements at grain boundaries as a function of strain in Cu-OF and Cu-OFP divided by the creep strain giving the constant C_s, Eq. (9.11). Data from [1, 19, 20]. Redrawn from [1] with permission of Elsevier

The constant C_s depends on the strain. C_s is higher at lower strains in all three studies. There is a tendency that the rate of decrease is slower at higher strains. This decrease in slope has also been found for austenitic stainless steels, although the absolute value of C_s is smaller [21]. A complicating factor is that at large strains new grains are frequently formed [22].

The observed values for C_s are compared to the model in Eq. (9.12) in Fig. 9.7. Since the model does not take the strain dependence into account, experimental values for all strains are included in the Figure. The model values are at the lower end of the slow strain rate data, but at the upper end for the other data. One complication is that both large and fine grains were present in the copper in [1]. Since the model values are proportional to the grain size, the use of an average grain size does not take this variation into account. Results on stainless steels demonstrate that the effect is weaker than the model suggests. If this is the case also for copper, the C_s values would be underestimated in [20] where grain size is smaller and overestimated in [19] where the grain size is larger. This is consistent with the results in Fig. 9.7.

Fig. 9.7

Comparison of modeled and observed values for the constant C_s, Eq. (9.11). Modeled values according to Eq. (9.12). Data from [1, 19, 20]. Redrawn from [1] with permission of Elsevier

Although there is considerable variation, a general approximate value of 50 µm for C_s seems reasonable, taking into account that it covers a range of temperatures, strain rates and three testing techniques. No significant difference between Cu-OF and Cu-OFP has been observed.

9.4 Superplasticity

As was explained in Sect. 9.2, studies of superplasticity show that the strain rate during this process can be described by an empirical equation given in Eq. (9.2). According to this equation, the strain rate is proportional to the grain boundary diffusion coefficient D_GB, it has a stress exponent of 2 and the exponent for the grain size dependence is −2. The controlling mechanism is well established. Intuitively this is obvious because an equiaxed grain structure remains equiaxed even after larger strain, and this would be difficult to reconcile unless GBS is the controlling mechanism. The fact that GBS is highly active has also been demonstrated in metallographic studies [10].

In spite of the fact that the mechanisms are well established and the process is well described, no basic model of the creep strain rate during superplasticity has been found in the literature. An attempt will be made here is to formulate such a model.

Grain boundaries (GBs) are normally considered as good sources and sinks for dislocations. For superplastic alloys that are fine grained, the recovery of dislocations at the grain boundaries must be taken into account. This is referred to as GB annihilation recovery. The rate for this mechanism can be represented as

$$ \frac{d\rho }{{dt}} = - \frac{{2\rho v_{{{\text{cl}}}} }}{{d_{{\text{g}}} }} = - \frac{2\rho }{{d_{{\text{g}}} }}M_{{{\text{cl}}}}^{{{\text{GB}}}} b\sigma $$

(9.13)

where ρ is the dislocation density, v_cl the climb velocity and $M_{{{\text{cl}}}}^{{{\text{GB}}}}$ the climb mobility. A dislocation has to travel a distance of half the grain size to reach a GB and to be annihilated. Since GBS is the controlling mechanism, the dislocations are active close to the GB. Consequently, the climb mobility $M_{{{\text{cl}}}}^{{{\text{GB}}}}$ involves GB diffusion. The effective contribution from GB to the diffusion coefficient can be expressed as

$$ D_{{{\text{GBeff}}}} = \frac{{\pi \delta_{{{\text{GB}}}} }}{{d_{{\text{g}}} }}D_{{{\text{GB}}}} = \frac{{\pi \delta_{{{\text{GB}}}} }}{{d_{{\text{g}}} }}D_{{{\text{GB}}0}} e^{{ - \frac{{Q_{{{\text{GB}}}} }}{{R_{{\text{G}}} T}}}} $$

(9.14)

where δ_GB is the width of the grain boundary (taken as 2 b), D_GB is the GB diffusion coefficient, D_GB0 the pre-exponential factor of D_GB and Q_GB the activation energy for grain boundary diffusion. To find $M_{{{\text{cl}}}}^{{{\text{GB}}}}$ Eq. (9.14) for the effective grain boundary diffusion coefficient should replace the lattice diffusion coefficient in the expression for the climb mobility in Eq. (2.34)

$$ M_{{{\text{cl}}}}^{{{\text{GB}}}} = \frac{{\pi \delta_{{{\text{GB}}}} }}{{d_{{\text{g}}} }}\frac{{D_{{{\text{GB}}}} b}}{{k_{{\text{B}}} T}}e^{{\frac{{\sigma b^{3} }}{{k_{{\text{B}}} T}}}} $$

(9.15)

The contribution to the recovery in Eq. (9.14), can now be added to the total expression for time derivative in Eq. (2.17)

$$ \frac{d\rho }{{dt}} = \frac{{m_{{\text{T}}} }}{{bc_{L} }}\rho^{1/2} \dot{\varepsilon } - \omega \rho \dot{\varepsilon } - 2\tau_{{\text{L}}} M_{{{\text{cl}}}} \rho^{2} - \frac{2\rho }{{d_{{\text{g}}} }}M_{{{\text{cl}}}}^{{{\text{GB}}}} b\sigma $$

(9.16)

In Eq. (9.16), a time derivative instead of the strain derivative in Eq. (2.15) is used. This is achieved by multiplying the equation by the strain rate. There are three recovery terms (with minus sign) on the right hand side of Eq. (9.16): dynamic, static and GB annihilation recovery. The role of the dynamic recovery is small and that term is dropped in Eq. (9.16).

Considering stationary conditions where the time derivative disappears, an expression for the stationary creep rate can be obtained from Eq. (9.16)

$$ \dot{\varepsilon }_{\sec } = (2\tau_{{\text{L}}} M_{{{\text{cl}}}} \rho^{3/2} + \frac{{2\rho^{1/2} }}{{d_{{\text{g}}} }}M_{{{\text{cl}}}}^{{{\text{GB}}}} b\sigma )/(\frac{{m_{{\text{T}}} }}{{bc_{L} }}) $$

(9.17)

To transfer Eq. (9.17) to stresses, the Taylor Eq. (2.29) is used

$$ \alpha m_{{\text{T}}} Gb\rho^{1/2} = \sigma - \sigma_{i} $$

(9.18)

where α is the deformation hardening constant and σ_i the internal stress from other contributions such as precipitation hardening. Inserting Eqs. (9.15) and (9.18) into (9.17) gives

$$ \dot{\varepsilon }_{\sec } = \left( {2\tau_{{\text{L}}} M_{{{\text{cl}}}} \left( {\frac{{(\sigma - \sigma_{i} )}}{{\alpha m_{{\text{T}}} Gb}}} \right)^{3} + \frac{{2(\sigma - \sigma_{i} )^{2} }}{{(\alpha m_{{\text{T}}} Gb)}}\frac{{\pi \delta_{{{\text{GB}}}} }}{{d_{{\text{g}}}^{2} }}\frac{{D_{{{\text{GB}}}} b^{2} }}{{k_{{\text{B}}} T}}e^{{\frac{{\sigma b^{3} }}{{k_{{\text{B}}} T}}}} } \right)/\left( {\frac{{m_{{\text{T}}} }}{{bc_{L} }}} \right) $$

(9.19)

Equation (9.19) includes contributions from both creep in the bulk and GBS. To analyze the contribution from GBS, it is given separately

$$ \dot{\varepsilon }_{{\sec }}^{{{\text{GBS}}}} = \frac{{2bc_{{\text{L}}} }}{{\alpha m_{{\text{T}}} ^{2} }}\frac{{(\sigma - \sigma _{i} )^{2} }}{G}\frac{{\pi \delta _{{{\text{GB}}}} }}{{d_{{\text{g}}} ^{2} }}\frac{{D_{{{\text{GB}}}} b}}{{k_{{\text{B}}} T}} $$

(9.20)

Equation (9.20) gives a stress exponent of 2 and exponent of −2 for the grain size dependence. The creep rate in this equation is proportional to the grain boundary diffusion coefficient. These features are the same as in the empirical Eq. (9.2). Equation (9.20) is compared to experimental data for the eutectoid alloy Zn22%Al in Fig. 9.8.

Fig. 9.8

Creep rate during superplasticity in Zn22Al for the five grain sizes 1.2, 1.8, 2.5, 3.3 and 4.3 µm at 230 °C. Model according to Eq. (9.20). Data from [13]

The creep rate is shown as a function of stress for five grain sizes at 230 °C. In the model a grain boundary diffusion coefficient for Zn is used [23]. The experimental curves have two slopes. The slope at the lower range of stresses is believed to be due to the presence of impurities [5]. This could be handled in the model by introducing a small internal stress in the same way as was done for aluminum for the Peierls stress, see Sect. 2.7. However, since the magnitude of the internal stress would not be known, it would be meaningless to take it into account. The part with the higher slope at low stresses is ignored. In the GBS range for larger stresses, both the stress and grain size dependencies are well represented by the model.

In Fig. 9.9 the corresponding data for Zn22Al are presented at 190 °C. There is a difference between Figs. 9.8 and 9.9. In the latter Figure, there is a third stress range at higher stress where the slope of the curves increases. This is believed to be due to ordinary lattice diffusion controlled creep that is starting to contribute. In this case, Eq. (9.19) is used in the model where both GBS and creep by lattice dislocations are taken into account. The lattice diffusion coefficient for Zn is used [24].

Fig. 9.9

Creep rate during superplasticity in Zn22Al for the three grain sizes 1.3, 2.6 and 3.9 µm at 190 °C. Model according to Eq. (9.19). Experimental data from [13]

Again both the stress and grain size dependence can be represented by the model. From Figs. 9.8 and 9.9, it is apparent that the experimental stress exponent is slightly larger than 2 and the absolute value of the exponent for the grain size dependence is somewhat larger than 2. This was already observed in the original work [13]. It has also been found for other superplastic alloys [11].

To illustrate the temperature dependence of the GBS rate, results at 130 °C are illustrated in Fig. 9.10.

Fig. 9.10

Creep rate during superplasticity in Zn22Al for the grain size 2.5 µm at 130 °C. Model according to Eq. (9.19). Experimental data from [13]

It can be seen that the model yields values that are higher than the experimental values. This suggests that the grain boundary diffusion coefficient is slightly overestimated in the model. However, the stress dependence is well described.

It has been shown in Figs. 9.9 and 9.10 that there is a transition from GBS to lattice creep at high stresses. Langdon has suggested that this corresponds to a situation where the grain size goes from being smaller than the subgrain size to being bigger. If the grain size is smaller than the nominal subgrain size according to Eq. (9.21), no subgrains are present. The subgrain size d_sub can be directly related to the dislocation stress, Eq. (2.18)

$$ d_{{{\text{sub}}}} = \frac{{K_{{{\text{sub}}}} Gb}}{{\sigma_{{{\text{disl}}}} }} $$

(9.21)

K_sub is a non-dimensional constant that typically takes values between 10 and 20. Since no value has been found for Zn, the value 18 for Al has been used. The idea that the subgrain size is of importance for GBS is natural. In alloys with normal grain sizes, dislocations can move 1–3 subgrain diameters. If the grain size is smaller than the subgrain size, the dislocations are clearly influenced. In particular, the recovery is affected. Annihilation at GBs becomes of importance. In addition, the recovery at the subgrains disappears which is equivalent to ordinary static recovery in the bulk.

In Fig. 9.11 the subgrain size is shown as a function of stress. The range of grain sizes from [13] is also presented. The grain size is equal to the subgrain size for stresses between 50 and 200 MPa. This is in good qualitative agreement with the transitions in Figs. 9.9 and 9.10. The equivalence of the grain size and the subgrain size cannot be considered as a general principle since the exact position of the transition depends for example on the ratio between the diffusion coefficients for grain boundary and lattice diffusion. So the precise position must be analyzed for each specific experimental case.

Fig. 9.11

Subgrain size for Zn22Al versus stress. The range of grain size from [13] is included for reference

9.5 Summary

• Grain boundary sliding (GBS) is believed to constitute the main mechanism for nucleation of creep cavities and to some extent growth of cavities, see Chap. 10. GBS is therefore of considerable scientific and technical significance.

• According to finite element simulations, the amount of GBS is directly proportional to the overall creep strain with a constant C_s that is known.

• Detailed measurements of GBS have been performed for copper with three techniques; creep at constant load, creep at constant loading rate, and slow strain rate tests at constant strain rate over a range of temperatures and strain rates. In spite of the varying testing conditions, the measured values for C_s are in close agreement with the theoretical value of 50 µm.

• A basic model for superplasticity is presented. It can successfully describe published data for Al22Zn. The form of the basic model is not very different from an earlier presented empirical model, but in the basic derivation of the model, parameter values could be fixed.