8. Cells and Subgrains. The Role of Cold Work

Tangles of dislocations are formed in virtually all alloys during plastic deformation. With increasing strain the tangles form boundaries that divide the materials into micrometer sized cells or subgrains, Fig. 8.1. With increasing temperature and strain the boundaries become better developed and thinner. At high temperatures the boundaries consist of regular networks of dislocations, and are then referred to as subboundaries or subgrain boundaries. At lower temperatures the boundaries are made up of loose tangles that are called cell boundaries. Expressed in another way subgrains are formed in the creep range and cells in the work hardening range [1], although there is no sharp transition. For a definition of the work hardening and the creep range, see Sect. 3.​4. Both cells and subgrains are referred to as substructure. In most materials the substructure is well developed already at modest strains. This means that the substructure can be observed in tensile and creep tests. However in some alloys, for example Al–Mg alloys, the development of substructure is delayed to higher strains [2].

The microstructure of the copper O F P cell reveals a dark, marbled-like structure with shaded lines in between, following a 4% cold working process.

There are excellent reviews on substructure formation in the literature [2, 4, 5]. Many results are similar for cells and subgrains so there is no need in general to make a clear distinction between them. For example, both the cell and subgrain sizes are related to the applied stress in the same way. One question that appeared early on was if the substructure contributed to the creep strength [5, 6]. In a number of investigations it has been shown that strength contribution from the dislocations in the subgrain interiors could account for the full creep strength in single phase alloys, see for example Orlova’s paper [7]. However, with the event of Mughrabi’s composite model where the subboundaries are considered as hard zones, it is clear that there are long range stresses from the subboundaries [8]. In the composite model, the strength is taken as a weighted average of the “hard” boundaries and the “soft” subgrain interiors. In a single phase alloy, the subgrain size is fully controlled by the applied stress and there is no way of varying the strength contribution from the subgrains [5]. However, the presence of particles can stabilize the subgrain size. In this way a major contribution to the creep strength from subgrains stabilized by M₂₃C₆ carbides is obtained in modern creep resistant 9–12% Cr steels [9].

There are many investigations on the formation of substructure but few of them are quantitative. Notable exceptions are work by Blum and Straub and coworkers who measured the development of the subgrain size during creep in martensitic steels [10‐12]. These results could be combined with a basic model for the influence of particles on subgrain growth [13] to understand the long term behavior of 9–12% Cr steels [9].

Dislocations with burgers vectors b and −b in a slip system are moving in opposite directions in an applied stress field inside a cell. Dislocations with b and −b end up at opposite sides of the cells. If a cell boundary is considered, dislocations with b are found on one side of the boundary and those with −b on the other side. This means that the cell boundaries become polarized. It has also the consequence that there is a boundary between the dislocations with b and −b and recovery of them cannot take place. These dislocations are also referred to as unbalanced because in the region with b dislocations there are no −b dislocations. This should be contrasted with balanced dislocations where dislocations with opposite burgers vectors are present. Models for the formation of substructure is presented in Sect. 8.2.

When unbalanced dislocations are present static recovery is slowed down since dislocation with opposite burgers vectors cannot meet and annihilate. The unbalanced dislocations are of importance for several properties. Modeling tertiary creep of copper has demonstrated that the recovery rate of the substructure gives a main contribution to the increasing strain rate [14]. This is likely to be the case for other ductile alloys as well. This will be further discussed in Sect. 12.​4. Cold work can reduce the creep rate by many orders of magnitude. Taking balanced and unbalanced dislocations in the subgrain walls into account has made it possible to explain this quantitatively for copper [15]. The role of cold work is discussed in Sect. 8.3.

Most creep tests are performed at constant load. For example, when the creep rate is plotted versus stress, usually the engineering stress, i.e. the nominal stress is used, not the true stress. At high temperatures when the creep exponent is about 5 this is not so critical, but at lower temperatures in the power-law break down regime where the creep exponent can be 30–50, the difference between using the engineering and the true stress is huge, which can easily be demonstrated. It turned out for copper that the engineering stress is still the relevant quantity. It took many years to explain this feature, but by considering the role of the substructure it was possible, Sect. 8.4 [16].

The influence of cold deformation on the creep rate and creep rupture is a classical problem. During primary creep of annealed material, the dislocation density is normally raised from a low value to a stationary one when the secondary stage is reached. This is a direct outcome of the creep recovery theory and it is well described by the basic dislocation equation used in this book, Eq. (4.​5). On the other hand for a cold deformed material, the initial dislocation density is high. If Eq. (4.​5) is applied, the dislocation density would be reduced during primary creep and the same stationary dislocation density as for an annealed material would be found and no effect of the cold deformation would remain. This is in direct at variance with observations for example for fcc alloys. For a number of austenitic stainless steels the creep strength can be increased significantly [28‐31]. A review is given in [32]. If the temperature is too high or the strain is too large the effect of cold work disappears. The reason is that the dislocation structure is not sufficiently stable under such conditions and recrystallization may appear.

In this section the influence of cold work on the creep of Cu-OFP will be analyzed. Results for creep rupture data are shown in Fig. 8.7.

A scatter plot of stress versus rupture time plots 4 decreasing trends labeled 12% and 24% cold work tension, 12% cold work compression, and 0% cold work forged, major data points are plotted on the line.

Values for 0, 12 and 24% cold work are compared. It is evident that the cold work has a dramatic effect on the rupture time. For 12% cold work the rupture time is increased by more than three orders of magnitude. For 24% cold work the rupture time is raised by six orders of magnitude. This effect is only observed if the cold work is performed in tension. If the cold work is in compression only quite a small increase in the rupture time is found. The creep testing was carried out in tension. Thus if the deformation direction is reversed between the cold work and creep testing only a limited effect is observed.

With increasing cold deformation, the creep ductility is practically always reduced. This is clearly found for Cu-OFP, Fig. 8.8.

A scatter plot of rupture elongation percent versus rupture time has 4 legends labeled as 12% and 24% cold work tension, 12% cold work compression, and 0% cold work forged with higher elongation percent.

For Cu-OFP without cold work the rupture elongation is typically quite high, above 40%. For 12% cold deformation in tension the rupture elongation is still high, 30% and above. For 24% cold deformation, the rupture elongation is a little bit above 10%. It is interesting to note that the creep ductility of Cu-OFP deformed in compression is low in spite of the small increase in the rupture time.

It has now been found that the role of the substructure must be taken into account to understand the influence of cold working [15]. This has also been suggested in the past but without any basic analysis that could predict the magnitude of the effect [4, 33]. As described in Sects. 8.1 and 8.2 a cell structure is formed in practically all alloys during deformation at ambient temperatures. A large fraction of the dislocations moves to the cell boundaries and in this way they create the cell structure. During the deformation all dislocations do not behave in the same way. Dislocations with opposite burgers vectors move in opposite directions in a given stress field. This can be seen from the Peach-Koehler formula for the force F on a dislocation with direction ξ and burgers vector b

If the direction of the burgers vector is changed to the opposite one (−b), this is the same as changing the sign of the burgers vector. For this reason, burgers vectors of opposite directions will also be referred to as burgers vectors of opposite signs. From Eq. (8.16), it can be seen that if the direction of the burgers vector is changed to the opposite one, the sign and the direction of the force is also changed. Dislocations of opposite signs on the same glide plane move to different ends of the cell. With opposite signs at different ends of the cell, the dislocations are said to be polarized. Not all dislocations are polarized. It is assumed that the outer layers of the boundaries are polarized.

The polarization of dislocations has a pronounced effect on the recovery. Since dislocations with opposite burgers vectors cannot be found amongst polarized dislocations, static recovery is not possible. Polarized dislocations are referred to as unbalanced since dislocations with opposite burgers vector are not present. For unpolarized dislocations, dislocations with opposite burgers vectors can be found and they are therefore referred to as balanced.

In the model for the development of the cell structure, the following dislocation densities in the cell boundaries are taken into account: balanced dislocation density ρ_bnd, the unbalanced dislocation density ρ_bnde, and the density of the locks ρ_lock [15]. Most of the dislocations are in the boundaries, and the content in the cell interiors is neglected. The equation for the balanced dislocation density ρ_bnd is almost identical to Eq. (2.​17)

Work hardening, dynamic recovery and static recovery are considered. The only difference is the introduction of the factor k_bnd. It takes into account since Eq. (8.3) is modified in comparison to the ordinary version of the Taylor equation. For the unbalanced content ρ_bnde, the equation corresponding to (8.17) is

There are two significant differences between (8.17) and (8.18). There is no static recovery term in Eq. (8.18). Unbalanced dislocations cannot annihilate by combining with dislocations of opposite signs, since such dislocations are not present. The other difference is that both balanced and unbalanced contribute to the generation of the unbalanced content since both types move across the cell interiors.

The traditional view is that dynamic recovery is due to dislocations coming sufficiently close that they can combine with dislocations of opposite sign and annihilate [34]. This assumption tends to overestimate the recovery rate, see Sect. 2.​3.​2. In addition, the mechanisms for dynamic and static recovery would be similar although their temperature and time dependencies are quite different. Argon has instead suggested that the dynamic recovery is due to the interaction between dislocations generated during work hardening and the cell boundaries [35]. It is known experimentally that spurting dislocations are moving a distance of about three cell diameters [36] and consequently they will pass through more than one cell boundary. During this passage boundary dislocations will be removed. When the dislocations hit the boundaries low energy configurations will be formed and this is part of the dynamic recovery process. Some of these low energy configurations are locks that are dominated by Cottrell-Lomer locks. They are created when partial dislocations cross. The formation of locks are assumed to be controlled by the following equation

This equation describes how both balanced and unbalanced dislocations contribute to the formation of locks. Dynamic recovery influences the number of locks since spurting dislocations passing through the boundaries remove locks. The locks can also be eliminated by static recovery since this process reduces the energy even for complex dislocation configurations.

In Sect. 3.​3, experimental stress strain curves for Cu-OFP were accurately reproduced using Eq. (2.​17) assuming a homogenous distribution of dislocations. If now the dislocations are considered to be located in the cell boundaries and Eqs. (8.17)–(8.19) are used, the stress should be computed with the help of Eq. (8.3) with ρ_bound equal to the sum of ρ_bnd, ρ_bnde and ρ_lock. The results for the stress strain curves should be the same. With k_bnd = \(\sqrt 2\) and k_bnde = \(\sqrt 2\) this is the case. The value of k_lock should be considerably smaller than the value for k_bnd. A value of k_lock = 0.1 has been assumed. With Eq. (8.14), this gives a value for the separation distance of the dislocations in the cell walls of about 20 nm that is in accordance with experiments for several materials, see Sect. 8.2. The selection of k_lock also affects the values k_bnd and k_bnde. Values k_bnd = 1.7, k_bnde = 0.2 and k_lock = 0.1 reproduce the stress strain curves [15].

The results for the influence of cold work in Fig. 8.7 will now be analyzed. 12% and 24% cold deformation at ambient temperature gives stresses of 154 and 191 MPa, respectively. Assuming the dislocations are located in cell boundaries in agreement with observations [3], and using the modified Taylor Eq. (8.3), this gives total densities of dislocations in the cell walls of 8.7 × 10¹⁴ and 1.5 × 10¹⁵ 1/m². The development of the dislocation densities according to in Eqs. (8.17)–(8.19) is shown in Fig. 8.9. In this case the balanced dislocations dominate the total content. It should be emphasized that the stresses from the cold work are much higher than even the dramatic increase in creep strength demonstrated in Fig. 8.7. Extensive recovery is taken place but not to such an extent that the stationary state for annealed material is reached.

A line graph of dislocation density versus strain has 3 increasing lines labeled balanced wall, unbalanced wall, and locks for copper O F P. The line for the balanced wall depicts a higher dislocation density, and the unbalanced wall is the lowest.

The key to understanding the influence of cold work is the unbalanced dislocation density ρ_bnde that is not exposed to static recovery. It is assumed to give rise to a back stress that reduces the creep rate

If the secondary creep rate for undeformed material is \(\dot{\varepsilon }_{\sec } (\sigma )\), the corresponding value for cold worked material iswhere σ is the applied creep stress. It might be thought that the back stress could be measured in a stress drop test by reducing the applied stress until the creep rate vanishes. However, this is not possible. The stress drop is in general based on the assumption that the dislocation structure is essentially unchanged after the reduction in stress. With dislocation dynamics it has been demonstrated that the dislocation structure is adapted to the new stress level within milliseconds [37]. This applies both to the dislocations in the cell interior and in the cell walls. On the other hand it takes a longer time before the cell size corresponds to the new stress level. Thus the dislocation structure after a stress drop neither represents the old stress level nor the new one. Back stresses can be quite useful in modeling, but to measure them would require quite a sophisticated analysis to interpret the results.

The stress dependence of the secondary creep rate according to Eq. (4.​3) is given bywhich is inserted into Eq. (8.21)

To handle primary creep, Eqs. (4.​6) and (4.​7) are appliedσ_disl is the stress created by the dislocations. In the secondary stage this stress takes the value σ_dislsec. Thus, in the secondary stage, Eq. (8.25) is identical to Eq. (8.21).

The use of Eq. (8.25) with no cold work present has been illustrated for Cu-OFP in Fig. 4.​10. It was demonstrated that the primary creep could be well reproduced and that both the experimental and the model results followed the ϕ-model.

Two examples of creep-strain curves for 12% cold-work Cu-OFP are shown in Fig. 8.10.

2 line graphs, strain versus time, have 2 increasing lines for model and experiment, a. at 250 megapascals, the lines move straight up below 0.1 then increase up to strain value 0.2. b. at 192 Megapascals the lines move straight up to strain 0.05 then increase up to strain value 0.2.

Both the experimental and model curves show distinct primary and secondary stages. The model exaggerates somewhat the size of the primary stage and reaches the secondary stage too soon. The model accurately reproduces the creep rate in the secondary stage in spite of the fact that the creep rate is three orders of magnitude lower than without cold work. This would not be possible unless the recovery rate of the unbalanced dislocations would be much lower than for the other types of dislocations. The amount of tertiary creep is very limited in the experimental data and the tertiary stage appears late in the test. It is probably caused by necking [14]. Since necking is not taken into account in the creep model, it is then natural that tertiary creep is absent in the model curves.

For one of the cases in Fig. 8.10, the creep rate as a function of time is given in Fig. 8.11.

A line graph of creep rate versus time has 2 decreasing lines steep for the model and fluctuating for the experiment at 12% C W, 75 degrees Celsius, and 192 megapascals.

In the same way as in Fig. 4.​10b the experiment and the model obeys the ϕ-model at least approximately. The drop in strain rate with increasing time is however much more dramatic in Fig. 8.11 in comparison to Fig. 4.​10b.

Creep strain curves for 24% cold deformed Cu-OFP are illustrated in Fig. 8.12.

2 line graphs, a. strain from 0 to 0.1 versus time from 0 to 150 h for the experiment, and model at 235 megapascal has 2 increasing lines depicting a steep increase after 150 h. b. strain from 0 to 0.06 versus time from 0 to 8000 h for the experiment and model at 223 megapascal depicts a steep increase after 7000 h.

It is immediately evident that that creep strain curves for 24% cold deformed material are very different from those of 12% cold deformed. In Fig. 8.12, primary and secondary creep is only present to a limited extent and tertiary creep is totally dominating. It is striking that the model can reproduce the creep strain curves also in this case. The cell sizes are smaller, the boundaries are narrower and the dislocations in the walls are closer for 24% cold deformed materials in comparison to 12% cold deformed, see Figs. 8.5 and 8.6. It is believed that the continuously increasing creep rate is due to enhanced recovery [14]. Tertiary creep will be further discussed in Chap. 12.

It can be concluded that by taking the back stress from the unbalanced dislocations into account, Eq. (8.12), the main features for cold deformed Cu-OFP can be well described. Thus, the reduction of the creep rate by three and six orders of magnitude for 12 and 24% can be modeled. The whole creep curves can be reproduced in a reasonable way. In the argument above the model was analyzed for primary and secondary creep. It will be seen in Chap. 12 on tertiary creep that Eq. (8.25) is also valid for tertiary creep. This is also clearly demonstrated in Fig. 8.12.

In the analysis above it has been assumed that the substructure is stabilized by the presence of unbalanced dislocation. An alternative way is to use particles to stabilize the substructure. This is extensively utilized for modern 9%Cr steels [38]. For the influence of cold work on the creep rate of austenitic stainless steels it has been suggested that particles can lock the substructure and prevent that the effect of cold work is lost [28, 29]. However, no detailed analysis of the role of the particles has been performed.

Cu-OFP close to ambient temperatures show creep curves that have the same appearance as at much higher temperatures with distinct primary, secondary and tertiary creep. One example is given in Fig. 8.13.

A line graph of strain 0 to 0.4 versus time 0 to 1400 h has increasing 3 lines labeled as experiment, model, and constant true rate. The experiment and model show a steep increase from 0 to 1200 h, while the constant true rate maintains steady strain till 600 h, then sharply rises to 0.4.

The cusp on the experimental curve is due to the necessity to reload the creep machine. Thus, the creep curves have many characteristics in common with typical creep curves at higher temperatures at about half the melting point. However, there is one aspect that is different. It is in general assumed that the true creep rate is constant in the secondary stage. This is frequently a starting point in stress analysis with finite element programs. To check if that is the case for the creep curve in Fig. 8.13, it is assumed for simplicity that the creep rate can be described with a Norton equationwhere A₀ is a constant and σ₀ is the nominal applied stress. The stress exponent n_N is about 70 for the case in Fig. 8.13. The factor e^ε takes into account the increase in the true stress when the specimen cross section is reduced during straining. A₀ is chosen so the Norton expression crosses the experimental at 600 h, which is about half the rupture time. Equation (8.26) is now integrated starting with ε₀ = 0.17 to simulate the influence of primary creep. The result is included in Fig. 8.13. It is obvious that Eq. (8.26) cannot represent the creep curve in Fig. 8.13. This conclusion is in no way affected by the choice of parameter values in Eq. (8.26).

The creep exponent exp(n_N ε) in Eq. (8.26) has a dramatic effect on the strain rate giving a creep curve with rapidly increasing slope that is fully inconsistent with observations. The effect is in fact quite large. For example, for ε = 0.1, exp(n_N ε) is equal to 1100. This enormous increase has never been observed in creep curves and one can conclude that the simple assumption of a constant true strain is strongly at variance with observations.

Instead there must be a back stress that prevents the rapid increase in the strain rate. The back stress must be built up in the dislocation structureσ_disl is given by Eq. (8.3). σ₀ is again the nominal applied stress. It is related to the true applied stress σ as

The stress σ_creep that drives the creep deformation is given by

In the second equality, Eq. (8.27) has been applied. Thus by applying Eq. (4.​6), the creep rate is given by

It is interesting to compare this equation with the simplified version in Eq. (4.​9). In Eq. (4.​9), the applied stress is the nominal one but at the same time the full impact of substructure on σ_disl is not included. For primary and secondary creep these differences are not very important. However, Eq. (4.​9) cannot describe tertiary creep contrary to Eq. (8.30), which will be explained now.

The development of the balanced and unbalanced dislocation densities ρ_bnd and ρ_bnde for the case in Fig. 8.13 is illustrated in Fig. 8.14. Equations (8.17) and (8.18) are used. The small contribution from ρ_lock is neglected in this case. Since the relation between ρ_bnd and ρ_bnde is not known, the relation between k_bnd and k_bnde cannot be determined. It is assumed that k_bnd = k_bnde with a value of \(\sqrt 2\) that reproduces the results of Sect. 4.​3.

2 line graphs of dislocation density versus time 0 to 1400 h have 2 increasing lines for the bal ance wall and the unbalanced wall. b. dislocation stress from 0 to 250 megapascal versus strain 0 to 0.4 has 4 increasing lines labeled as all, balanced boundary density, unbalanced boundary density, and applied stress.

As can be seen from Fig. 8.14a, the balanced dislocation density reaches an approximately constant value in the secondary stage whereas the unbalanced content increases continuously even during the secondary stage. The implication for the dislocation stresses is shown in Fig. 8.14b. The total stress from the balanced and the unbalanced dislocation stresses are marked as ‘all’. This total dislocation stress matches the true applied stress σ in the secondary stage. This is the reason why the creep rate does not increase in an uncontrolled way. This balance is possible due to the increase in the unbalanced stress. The difference between the starting value of the applied stress of 150 MPa and the test stress of 175 MPa is the value of the yield strength. With increasing strain the total dislocation stress cannot match the true applied stress anymore. Then the tertiary stage is reached and the creep rate increases.

It can be seen from Fig. 8.13 that tertiary creep is reasonably well represented. This can also be demonstrated by plotting the strain rate as a function of time, Fig. 8.15.

A line graph of strain rate 0 to 10 to the power of negative 6 versus time 0 to 1000 h has 2 decreasing lines between the time 500 h then increases straight up at 1000 h.

Although the cusps in the experimental data do not make a detailed comparison possible, it is evident that the overall picture reproduces both the primary and tertiary stages in a good way.

It can be concluded that the presence of back stress from the unbalanced dislocations, prevents the creep rate from increasing in an uncontrolled way that would be suggested if a constant true strain rate in the secondary would be assumed. In addition, the introduction of this back stress makes it possible to model tertiary creep.

Also stress strain curves seem to be affected by the back stress. One example is illustrated in Fig. 8.16. A stress strain curve for 15% cold worked Cu-OFP is shown. A model curve using Eqs. (8.3), (8.17) and (8.18) is also included in Fig. 8.16a.

2 line graphs, a. stress versus strain has 3 increasing lines labeled Disl. model, stationary stress, and experiment, increase between 180 to 200 M P a over the strains 0 to 0.15. b. dislocation density versus strain has 2 lines labeled bal ance. wall increases, and the unbalanced wall remains stable at all strains.

It is assumed that the stress strain relations are controlled by the same equations as the creep curves in the same way as in other parts of this book. This means that the stress level at higher strains should correspond to the stationary stress in a creep test. This stress is given bywhere σ_stat0 and σ_stat are the nominal and the true stationary stress that give the same strain rate as in the stress strain curve (1 × 10⁻⁴ 1/s at 125 °C). The stationary stress is not identical to the stress strain curve but it is very close to supporting the assumed principle.

The strain dependence of the dislocation densities is given in Fig. 8.16b. The balanced and unbalanced dislocation densities are assumed to be the same at zero strain. Using these assumptions, the values of k_bnd and k_bnde can be determined, see [16] for details. It can be seen from Fig. 8.16b that the unbalanced dislocation density increases with strain and compensates for the increase in the true stationary stress.

It has been seen above that both creep curves and stress strain curves are strongly affected by the back stress from the unbalanced dislocation content. In particular for creep, it was demonstrated above that the effect is huge and cannot be ignored. This is especially important to take into account in stress analysis with finite element methods (FEM). There are two straight forward alternative ways to handle the problem. The first way is to take the back stress into account explicitly. This requires however the development of special software. The other alternative is to replace the true stress σ with σ exp(−ε). This alternative represents no practical problem but there is a psychological barrier because it is not in accordance with what people have been trained to do. However, ignoring it will give rise to large errors.

The dramatic effect of the back stress has only been verified for copper at lower temperatures. There are no reasons to believe that it should not applicable to other materials as well because there is nothing in the derivation that is specific for copper. The open question is to what temperature the effect survives. At a sufficiently high temperature the back stress from the unbalanced dislocations cannot be expected to be stable anymore. This section is mainly taken from [16] where further detail can be found.