Basic modelling of tertiary creep of copper
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Mechanisms that are associated with acceleration of the creep rate in the tertiary stage such as microstructure degradation, cavitation, necking instability and recovery have been known for a long time. Numerous empirical models for tertiary creep exist in the literature, not least to describe the development of creep damage, which is vital for understanding creep rupture. Unfortunately, these models almost invariably involve parameters that are not accurately known and have to be fitted to experimental data. Basic models that take all the relevant mechanisms into account which makes them predictive have been missing. Only recently, quantitative basic models have been developed for the recovery of the dislocation structure during tertiary creep and for the formation and growth of creep cavities. These models are employed in the present paper to compute the creep strain versus time curves for copper including tertiary creep without the use of any adjustable parameters. A satisfactory representation of observed tertiary creep has been achieved. In addition, the role of necking is analysed with both uniaxial and multiaxial methods.
Creep deformation is usually measured by exposing tensile specimens to a constant load or sometimes to a constant stress and recording the elongation of the specimens as a function of time. The result is given as the creep strain versus time, referred to as a creep curve. A creep curve is in most cases characterised by three stages: primary, secondary and tertiary. Creep deformation is induced by the generation, motion and annihilation of dislocations. The creep rate decreases during primary stage, reaches a steady value in secondary creep, accelerates during tertiary creep and terminates at rupture. For many materials, the high initial creep rate is due to a low starting dislocation density. Due to work hardening, new dislocations are generated and the dislocation density increases, leading to a decrease in the creep rate. At the same time, the recovery due to the annihilation of dislocations starts to become of importance. When achieving a balance between recovery and work hardening, the strain rate is approximately constant and the secondary creep stage is reached. It is also referred to as stationary creep. During tertiary creep, a modification of the microstructure occurs, leading to acceleration of the creep rate.
The scientific literature has to a large extent been focusing on the secondary stage due to its direct relation to the operating deformation mechanisms. Much less attention has been paid to primary creep and tertiary creep. Technically, both primary creep and tertiary creep are of utmost importance. For fcc alloys that cover a large fraction of the technically used creep-resistant materials, the creep strain in the primary stage is often of the same order as that in the secondary stage. Tertiary creep is also of major technical significance since it controls creep rupture.
The increase in the creep rate in the tertiary stage due to changes in the microstructure is referred to as the formation of creep damage [1, 2, 3]. There are a number of creep damage mechanisms including particle coarsening, subgrain growth, cavitation and recovery of the dislocation structure, which can all accelerate the creep rate during tertiary creep. In addition, the creep rate is influenced by the necking instability. Failure induced by microstructure degradation has been commonly observed in creep-resistant martensitic steels, which have a complex microstructure. During long-term creep, fine carbonitrides (e.g. M2X and MX) coarsen and dissolve and new brittle phases (e.g. Z-phase, Laves phase, M6X carbides) are formed. The absence of fine particles reduces the creep strength. In addition, the new coarser phases can serve as sites for crack nucleation that lowers the creep strength further [4, 5, 6].
Good models for particle coarsening  and subgrain growth  have been available for quite some time. However, for nucleation and growth of cavities, only empirical models involving adjustable parameters had been documented. Only recently, basic quantitative models for nucleation and growth of cavities have appeared [9, 10, 11]. The same applies to the modification of the dislocation structure [12, 13].
As a consequence, modelling of creep damage has almost invariably disregarded some important mechanisms and compensated this by using adjustable parameters. It was demonstrated in  that by just involving two adjustable parameters, a wide range of creep curves in the tertiary stage could be represented. There are good empirical models for describing tertiary creep. In particular, what is now usually referred to as the omega model can be mentioned [15, 16, 17]. However, with two adjustable parameters, a good empirical model is not essential. Many mathematical expressions can be used . This implies that creep damage models with two or more adjustable parameters are not predictive and cannot be used to identify the operating mechanisms. It is therefore of vital importance that basic equations without adjustable parameters are employed.
Oxygen-free copper alloyed with 50 ppm phosphorus (Cu-OFP) has been selected as canister material for storing spent nuclear fuel in Sweden due to its excellent corrosion resistance and high ductility . During storage, the spent nuclear fuel will release heat while decaying, increasing the temperature up to 100 °C in the canister. Both hydrostatic pressure and swelling pressure from clay will impact the copper canisters, which will be exposed to creep as a consequence. The copper canisters are expected to stay intact for 100000 years. In order to predict the creep damage under such long times, it is critical to use fundamental models based only on physical phenomena . Cu-OFP is the material that will be investigated in the present paper.
The creep mechanisms at low temperatures (below 0.3 Tm, melting point) can be quite different from those at high temperatures. For many materials, logarithmic creep form is more appropriate than power law creep to describe the deformation behaviour. This applies, for example, to austenitic stainless steels, where creep never leaves the primary stage [20, 21]. However, for copper, this is not the case. A large number of creep tests have been performed for Cu-OFP at 75 °C (0.25 Tm), and the creep curves are quite similar to those achieved at high temperatures, where a well-developed and long-duration steady state is observed . The mechanisms for low-temperature creep are not yet fully established. It has been suggested that low-temperature creep is controlled by glide and cross slip . Dislocation climb was not considered to be active due to the low estimated climb rate at low temperatures. However, if the increase in the climb rate from the increase in vacancy concentration due to plastic deformation is taken into account, the observed creep rates at ambient temperatures for aluminium and copper can be accurately accounted for . If climb is the operating mechanism, the observed extended secondary stage can be explained directly.
In Cu-OFP, changes in the dislocation structure could provide microstructure degradation. Accelerated recovery and an associated decrease in dislocation density as main creep damage mechanism were reported from experimental results [5, 24] and computation . The nucleation of cavities followed by growth and interlinkage is believed to play an important role in creep failure of metals [9, 10]. Necking is known as a macroscale deformation inhomogeneity. When the material is plastically unstable, even small defects can promote localised deformation . At some point during creep testing, strain localised in a small region takes place and necking appears. Studies on the effect of an initial defect on creep deformation have been carried out [27, 28, 29, 30].
Dislocation recovery mechanism has been used to describe the three stages of creep deformation. Fundamental dislocation models based on this mechanism for primary creep and secondary creep were formulated . It has been demonstrated that it can be used to describe the primary creep and secondary creep of Cu-OFP and also slow strain rate tensile tests under both uniaxial and multiaxial stress states [31, 32]. The purpose of the present paper is to model tertiary creep of Cu-OFP taking the relevant microstructure processes into account without involving adjustable parameters. The basis is a model that takes substructure development during creep into account. It was derived originally for cold-worked materials and will be employed to simulate accelerated recovery . In order to evaluate the effect of necking on tertiary creep of Cu-OFP, a small imperfection is artificially introduced to the specimen in computation according to the method proposed by Burke and Nix . Influence of cavitation is also considered when describing tertiary creep. The modelled results will be compared with experimental data for Cu-OFP.
Accelerated recovery model
A dislocation model was developed in [14, 32] that could describe primary creep and secondary creep of copper. Some parts in the model were taken from the literature for granted but have been precisely derived recently . It is believed that the dislocation model is general. Its validity has been demonstrated also for austenitic stainless steels  and for aluminium alloys . By applying the model, it has been shown that the recovery during tertiary creep can be analysed by taking the role of the substructure into account [12, 13].
In copper and many other materials, a cell structure is formed during deformation. Already after 10% strain, the majority of the dislocations can be found in the cell boundaries , and after 20% strain, virtually all dislocations are in the cell boundaries . In the model, only the dislocations in the cell walls are taken into account to avoid an excessive number of parameters. This assumption is also consistent with X-ray measurements done by Straub et al. , where the strength contribution from the cell interior is less than 10 MPa for pure copper.
Equation (1) has the same format as the basic model for homogeneous dislocations . Since unbalanced dislocations cannot combine with dislocations of opposite Burgers vector, they are not exposed to static recovery. This is the reason for the absence of the static recovery term in Eq. (2). Both unbalanced and balanced dislocations are subjected to dynamic recovery.
A back stress is introduced to model the creep curves. In a number of publications in the past, a back stress has been considered as an intrinsic property that could be measured, for example, in a stress change experiment. With the help of dislocation dynamics simulations, the back stress from dislocations can be computed directly. It turns out that computed back stress is almost identical to the applied stress. This is also the case after a stress drop test. The change in the back stress takes place in less than 1 ms. The difference between the applied stress and the back stress is less than 1/500 of the applied stress . Thus, an intrinsic back stress is not very meaningful to use in modelling, which has been realised by a number of authors, see, for example . Although the back stress cannot be measured, it can be introduced if it is properly defined.
This approach suggests that if you know the stress dependence of the secondary creep rate, the strain dependence of the creep rate for the creep curve can be derived, provided the variation of the dislocation density is known.
Since creep cavitation gives rise to a loss of the load carrying cross section, it can give a contribution to tertiary creep. There are numerous models for nucleation and growth of creep cavities in the literature, but practically all of them involve parameters that are not accurately known and have to be fitted to the observations. Only recently, basic models for nucleation and growth of creep cavities have been formulated by He and Sandström [9, 10, 49]. For a review, see . The equations that are used to compute the amount of cavitation are listed and discussed here.
At sufficiently large strain during creep, a plastic instability takes place leading to the formation of a waist on the specimen that is usually referred to as necking. Necking is initiated by a geometrical imperfection or a material inhomogeneity in the specimen. Once the waist has been formed, its continuous growth does not depend on how it was initiated. In this analysis, the effect of a geometric defect on the creep deformation is studied.
When a waist is introduced, the stress state changes from uniaxial to multiaxial. Finite element (FEM) computations were performed to analyse the influence of multiaxiality. Due to limitation in the FEM software, only the secondary stage could be simulated. Severe necking was obtained. Experimental necking profiles were taken from ruptured creep specimens.
Accelerated recovery results
Severe necking was observed on the Cu-OFP specimens after the creep tests, implying that the necking effect should be taken into account when modelling tertiary creep. An initial nonuniform cross-sectional area was introduced to investigate the necking effect on creep deformation of Cu-OFP.
Hart’s criterion was applied to the experimental data to determine the onset of the unstable deformation. For all test conditions, the unstable deformation starts very close to the inflection point of the strain versus time curve. A plus marker is given in Fig. 6b indicating the necking starting point calculated by Hart’s criterion. The FEM modelling suggests that the necking starts at a very late stage of creep, almost at the failure strain, resulting in a steep rise of the creep curve. The development of the neck is initially obviously quite a slow process.
There is an extensive literature on the formation of creep damage and its influence on tertiary creep. As pointed out in the introduction, these models are almost invariably empirical. One model that has been used frequently is the one that Riedel presented in his book on creep fracture . He derives the creep damage based on cavity formation. He assumes that the nucleation rate is proportional to the creep strain and that the volume growth rate is linear in time t. Also assuming that only secondary creep is of importance, he found that the area fraction of cavities is proportional to t5/3. This covers the main development of the cavitation in a simple form. However, there are shortcomings. The nucleation rate constant is handled as an adjustable parameter. The effect of constrained growth and possible overlap between cavities is neglected, typically significantly overestimating the amount of cavitation. Today, there is no need to make these simplifications, since the additional effects can be taken into account without much computational effort .
The concept that dynamic recovery plays an important role during tertiary creep is relatively new but well established. Creep tests of 24% cold-worked Cu-OFP were performed at 75 °C . For all the creep tests, the creep strain versus time curves were dominated by a continuously increasing strain rate, i.e. by tertiary creep. The creep curves could elegantly be reproduced by assuming that dynamic recovery according to a model similar to the one in the present paper was the dominating creep damage mechanisms. Very limited cavitation was observed. This was not surprising since the reduction in area at rupture was almost exactly 90% (89–91%).
For the specimens in the present investigation, the reduction in area at rupture was also very high (90–92%) indicating fully ductile rupture. This indicates that cavitation is of little importance for the failure. This was indeed confirmed by metallographic investigations and modelling. In both cases, the area fraction of cavities in the grain boundaries was less than 0.5%. This makes it natural to assume that dynamic recovery of the substructure is the dominating mechanism for tertiary creep until significant necking quickly develops at the very end of the creep life.
The effects of accelerated recovery, cavitation and necking on tertiary creep have been analysed in the present paper. Accelerated recovery gives the largest contribution to tertiary creep for the investigated alloys. In the model, two distinct sets of dislocations (balanced and unbalanced) are involved. The balanced dislocations are exposed to static recovery, and its density remains approximately constant during secondary creep. At the same time, the unbalanced dislocation density continuously increases and gives rise to a major back stress that matches the continuous increase in the true applied stress. At the end of the secondary stage, the true applied stress increases faster than the back stress. This means that the effective stress rises at the end of the secondary stage, resulting in the increase in the creep rate in the tertiary stage.
Basic models are now available for the nucleation and growth of creep cavities. With these models, the cavitated area fraction of grain boundaries and its influence on the creep curves can be predicted. For the investigated copper alloys, the predicted area fraction at failure was less than 0.5%. Consequently, cavitation had no significant influence on tertiary creep.
Uniaxial and multiaxial models for necking have been considered. Hart’s criterion  indicates that an instability that would give rise to necking is formed directly at the end of secondary creep. This has also well known for other types of materials. For example, Lim et al.  found this for 9% Cr steels. The uniform strain calculated from uniaxial and multiaxial simulations is almost the same. The analyses suggest that significant necking only appears very close to the failure strain. This is also what is found in . Necking simulations give a rapid increase in the strain near rupture in agreement with the observations. Figures 4a and 6b show a comparison of modelled and experimental creep curves for the same case. In both figures, primary creep and secondary creep can be well reproduced. Accelerated recovery exerts the influence on the entire tertiary stage while necking contributes to the very end of tertiary creep. The radii in the neck can be predicted quite well with the help of FEM computations. The model results lie within 10% of the observed values. In spite of the fact that Hart’s criterion gives an early start of the necking, the necking is not really developed until close to failure.
It is expected that the model for tertiary creep is also applicable to other fcc materials than copper. The basic dislocation model has been demonstrated to be valid also for austenitic stainless steels  and aluminium . Due to lack of data, it has not been possible to verify that tertiary creep can be represented for these materials. However, the model cannot be used for martensitic 9 and 12% Cr steels. Tertiary creep is frequently studied in these materials, because of their extensive use in fossil-fired power plants. Tertiary creep in these materials typically show a linear increase in the creep rate with creep strain, see, for example . This behaviour follows what is often referred to as the omega model [15, 17, 61]. Although this empirical model has been known for a long time and the mechanisms involved are well established, it has not yet been derived from basic principles.
The most important contribution to creep curves of copper comes from the dislocation structure. A dislocation model is presented that can be used to compute this contribution.
- To be able to describe the contribution from the dislocation structure to the both secondary creep and tertiary creep, there are three important requirements.
The dislocations in the cell walls must be taken into account.
Both balanced and unbalanced dislocations must be considered.
Both dynamic recovery and static recovery must be covered in the equations for the dislocation densities.
Cavitation often plays an important role in tertiary creep. At the considered temperature 75 °C, the cavitated area fraction is less than 0.5% and the contribution from cavitation can be ignored.
The influence of necking on the creep curve has been analysed with uniaxial assumptions as well as with multiaxial methods. It turns that both approaches predict that pronounced necking does not take place until the failure strain has almost been reached. The uniaxial computations give a necking that is narrower than the observed ones. However, the multiaxial approach using FEM predicts a necking that is in good accordance with experiments with computed neck radii within about 10% of the observed values.
The authors would like to thank Svensk Kärnbränslehantering AB (Swedish Nuclear Fuel and Waste Management Company, SKB) for financial support [Contract Number 16884] and the China Scholarship Council [No. 201307040027] for funding a stipend.
Compliance with ethical standards
Conflict of interest
The authors declare that they have no conflict of interest.
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