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An examination of the use of the Modified Jogged-Screw model for predicting creep behavior in Zircaloy-4
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Acta Materialia 61 (2013) 4452–4460 www.elsevier.com/locate/actamat
An examination of the use of the Modified Jogged-Screw model for predicting creep behavior in Zircaloy-4 B.M. Morrow a,⇑, R.W. Kozar b, K.R. Anderson b, M.J. Mills a a
The Ohio State University, 2041 College Rd., 477 Watts Hall, Columbus, OH 43210, USA b Bettis Laboratory, Bechtel Marine Propulsion Corp., West Mifflin, PA 15122, USA
Received 20 September 2012; received in revised form 2 April 2013; accepted 7 April 2013 Available online 6 May 2013
Abstract The alloy Zircaloy-4 is used primarily as a structural material in nuclear reactors. A thorough understanding of the thermal creep response of such materials is essential to enable accurate forecasting of material behavior during both service and long-term storage. To this end, the Modified Jogged-Screw model, originally developed to predict steady-state creep rates in c-TiAl, has been applied to Zircaloy-4. This work provides a more complete database for model validation through creep testing of several Zircaloy-4 specimens over a range of stress and temperature conditions. Bright-field scanning transmission electron microscopy was used to observe the substructure and quantify the model parameters important to the Modified Jogged-Screw model. These observations were used to confirm the applicability of the proposed model and assess the sensitivity of the model parameters to test conditions. The increased availability of statistics for model parameters as a result of substructural observation will elucidate the strengths and weaknesses of the model as previously proposed. Ó 2013 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. Keywords: Creep; Screw dislocation; Mechanical properties; Mechanistic modeling; STEM
1. Introduction Zirconium alloys serve an important role as cladding material for nuclear fuel, separating the fuel (and various fission products) from the moderator or coolant. Zirconium alloys are widely used for cladding due to a combination of good thermal conductivity, corrosion resistance and structural integrity at high temperature [1]. They are especially important, however, due to their low thermal neutron capture cross-section [2], which allows neutrons to pass more easily from the fuel to the moderator. During service, heat and pressure produce conditions favorable for creep of the material. Cladding temperatures of 560– ⇑ Corresponding author. Present address: Los Alamos National Laboratory, P.O. Box 1663, MS G755, Los Alamos, NM 87545, USA. Tel.: +1 505 665 9224; fax: +1 505 667 8021. E-mail addresses: [email protected] (B.M. Morrow), millsmj@mse. osu.edu (M.J. Mills).
625 K and pressures of 10–16 MPa are typical [3–5], and partial or complete collapse of cladding due to creep, known as “creepdown”, has been observed in the past [6– 8]. Furthermore, residual decay heat, which is present in significant levels for many years after a fuel assembly is removed from service, may facilitate thermal creep during dry storage. Creep is regarded by many as the leading concern about the feasibility of permanent dry storage [9], during which temperatures near 723 K and stresses of 100 MPa can be present [10]. Some estimates expect decay heat during dry storage to initially increase from 575 K, peak 10 years after emplacement, then decrease continuously over a long period of time, still being at 425 K after 1000 years [9]. For reference, the melting temperature of Zircaloy-4 is 2034 K [11], though the material becomes significantly weaker as temperatures approach the beta transition temperature, 862 K [12,13]. Thermal creep behavior in non-irradiated materials can serve as an important cornerstone in understanding the material response during
1359-6454/$36.00 Ó 2013 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.actamat.2013.04.014
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and after irradiation. Accurate prediction of thermal creep response is essential for the safe use of the material during both service and long-term storage. The Modified Jogged-Screw (MJS) model was initially developed to describe the creep behavior of c-TiAl alloys, and represents a reconsideration of the original joggedscrew creep model proposed by Barrett and Nix [14]. The original jogged-screw model asserts that the steady-state creep behavior of certain alloys is controlled by the motion of jogs on screw dislocations. Such dislocation behavior has been observed in various materials in the past [14– 18]. Because the jogs are edge in character, dislocation climb is required in order for jogs to translate with the rest of the gliding dislocation. The balance between applied stress and the chemical drag imposed by the diffusion of vacancies to and from the jog segments leads to a constant creep rate. The Barrett and Nix model proposed that jogs on screw dislocations are formed primarily from dislocation intersections, producing jogs of one Burgers vector height. Zirconium is known to prefer prismatic hai-type slip, and hc + ai-type slip is observed only occasionally [19]; therefore intersection of screw dislocations on different prism planes is a likely formation mechanism of the small jogs described by the Barrett and Nix model. The MJS model made several changes to the original model, including allowing for taller jogs that have been observed in the past and are believed to form through either growth of smaller jogs or formation of tall jogs directly through multiple cross-slip events [20]. After successful application of the MJS model to titanium alloys [17,20,21], the model was adapted to describe the creep behavior of zirconium alloys, due to similarities between elemental zirconium and titanium, and observed similarities in microstructural features [18]. The model predictions of Moon et al. showed good agreement in steadystate strain rates with experimental data; however, the model was calibrated and validated using a relatively small amount of creep data for Zircaloy-4 (one specimen for each of four temperatures). Because of this, several assumptions were necessary to determine how model parameters changed with stress and temperature. Expanding the applicability of the MJS model to conditions outside of the stress–temperature regime tested in Ref. [18] is a focus of the current study. Additionally, the model utilized jog heights that were calculated theoretically rather than based on experimental investigation [17,18]; differences between predicted and observed jog heights have been observed previously [17]. In prior work utilizing the MJS model, single values for model parameters were assumed without consideration of a distribution of values that would be expected in real materials. Thus, additional testing and microstructural examinations are necessary to characterize the active deformation mechanism. It is unclear how well the previous assumptions will hold when subjected to a broader range of testing conditions, including the power law breakdown regime. This work attempts to address these issues by examining a broader range of stresses at each testing
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temperature, and using a statistical approach to the determination of model parameters. 2. Experimental procedures 2.1. Mechanical test setup Creep tests were performed on round tensile specimens of non-irradiated Zircaloy-4 that were machined from processed plate material. The alloy was rolled, then heated and quenched from the b phase-field, yielding a mostly isotropic structure [22,23]. The alloy composition for Zircaloy4 is shown in Table 1. Each sample had a gage diameter of 5.08 mm and a gage length of 60.96 mm. Samples were loaded using a lever arm creep frame at a 20:1 load ratio. All tests were loaded in uniaxial tension. A load cell was used to directly measure the force applied to each sample. A three-zone furnace was used to heat the samples to the desired temperature. Extensometers and linear variable displacement transducers (LVDTs) were used to determine strain in the gage section. An independent thermocouple measured temperature at the surface of each sample. Testing was performed at a temperature low enough that oxidation was not expected to be a concern, allowing the samples to be tested in a laboratory air atmosphere. After testing, samples were cooled under load to preserve the dislocation substructure. The test conditions, resultant creep strains and minimum (steady-state) strain rates for the samples used in this work are shown in Table 2. 2.2. Specimen preparation Specimens were cut from the center of the gage section at a 45° angle to the loading axis in order to increase the probability that a principal glide plane will lie in the plane of the thin foil, facilitating analysis of dislocations. The samples were then thinned to 100 lm using 1200 grit SiC paper, and cut into 3 mm disks. The samples were prepared for transmission electron microscopy (TEM) analysis in a dual-jet electropolisher using a solution of 940 ml of methanol, 60 ml of sulfuric acid and 2.4 ml of hydrofluoric acid, at a potential of 21 V and a temperature of 40 to 45 °C. Some hydride formation occurred on the surfaces of the prepared foils due to this technique, but these were generally easily distinguishable from dislocations. 2.3. Dislocation substructure analysis techniques An FEI Tecnai F20 transmission electron microscope, using bright-field (BF) and/or dark-field (DF) imaging in scanning transmission electron microscopy (STEM) mode, Table 1 Nominal compositions of Zircaloy-4. Sn (wt.%)
C (ppm)
Fe (wt.%)
Cr (wt.%)
Zr
1.3–1.5
150
0.18–0.24
0.07–0.13
Balance
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Table 2 Test matrix of samples used in this work. Temperature (K)
Stress (MPa)
Strain (%)
Strain rate (s1)
533 533 589 644 644 644 644 644 644 644 700 700 700 700 700 783 866 866
196.5 219.2 203.5 156.4 156.4 165.5 165.5 165.5 165.5 178.7 99.9 126.5 126.5 146.1 173.4 59.2 26.8 49.0
1.07 1.78 4.41 0.70 1.24 1.43 2.65 4.44 6.72 2.32 0.87 2.00 2.19 4.27 4.38 1.29 1.21 2.50
1.98 109 3.18 109 2.00 108 4.90 108 2.23 108 8.00 109 6.95 109 8.00 109 8.00 109 4.86 108 9.14 109 1.55 107 1.78 108 8.50 108 1.10 106 9.04 109 3.04 108 2.50 107
was used for all microscopy related to this work. STEM imaging was used to image the substructure because it provides many advantages over conventional TEM (CTEM). The convergence of the beam creates a range of incident angles for the electrons, leading to “relaxed” diffraction conditions [24,25]. This results in a more complete sampling of the substructure, as defects will display contrast, often in the absence of a strict two-beam condition. As a result, this technique makes it possible to sample essentially all of the dislocations present in the substructure at the same time. Additionally, STEM imaging can reduce the effect of bend and thickness contours, and allow for imaging in thicker regions than would be possible using CTEM. A more thorough investigation of the use of STEM diffraction contrast imaging is given by Phillips et al. [26]. Multiple foils from each test condition were examined, and often several grains from each foil were used for imaging. Tilting experiments were performed, including g b analysis and line trace analysis on selected dislocations, to determine the type and character of the dislocations present in the substructure. The MJS model is based on changes in characteristics of jogged-screw dislocations with different stress and temperature conditions. Three different behaviors are anticipated: jog dragging (for small jogs), dipole dragging (for medium-sized jogs) and dipole bypass (for jogs of sufficient height that the shear stress forces the dipole segments to bypass). When jogs exhibit dipole bypass, they are assumed to become immobile and act as dislocation sources [18]. A schematic of these jogged-screw behaviors is shown in Fig. 1. The three model parameters that are measurable using TEM are dislocation density, jog spacing and jog height. Fig. 1 also shows a schematic of jog spacing, l and jog height, h. Image processing and analysis were performed using Adobe Photoshop with the Fovea Pro image analysis plug-in from Reindeer Graphics. Dislocation density can be determined through direct microstructural observation using either of two methods.
Fig. 1. Schematic of jogged-screw dislocation behavior as a function of height. For small heights, the diffusion of vacancies to or from the jog is sufficient for the jog to move along with the screw segments (jog dragging). For medium-sized jogs, the mobility of the jog makes it more favorable for a dipole to form (dipole dragging). For very tall jogs, the two dipole segments may move past each other, acting as independent single-ended dislocation sources (dipole bypass).
The method used in this work involves measuring the number of dislocations per area, and is described by: qmeasured ¼
0:5j Amicrograph
ð1Þ
where j represents the total number of dislocation endpoints (intersections with the foil surfaces) present in a micrograph and Amicrograph is the measured area of the micrograph. A second method measures the dislocation line length per volume (see Kruml et al. [27]). Due to the large area of inspection afforded by STEM imaging, and given the many testing conditions being studied, the more rigorous line length per volume technique would be prohibitively time-consuming to perform. The end point per area method is a viable option to address this concern; however, a correction factor of between one (for arrays of parallel dislocations) and two (for random dislocations) must be applied to correct for an under-sampling of dislocations [28]. This was confirmed through a comparison of the two methods for selected specimens in this work. Additionally, it has been reported in the past that, for very closely spaced dislocations, strain field overlap could result in an under-sampling of dislocations by as much as 30% when using BF STEM images to measure dislocation density [24]. The dislocation densities in this work are not typically high enough for this to be an issue. There are several factors that lead to apparent error in dislocation density measurements. One is inhomogeneity in the dislocation distribution [27], leading to variability in measurements. This concern is minimized by using STEM imaging, which allows for much larger areas to be imaged and, through use of multi-beam diffraction conditions, allows nearly all the dislocations present to appear
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in contrast. As a result, a much larger sampling of the microstructure can be obtained than would be possible using conventional TEM imaging. By sampling many random grains from several different foils for each condition, sampling error is also reduced. The MJS model uses mobile dislocation density to calculate strain rates. Some judgement was necessary to exclude dislocations believed to be immobile (e.g. involved in dislocation networks or dense tangles and grain boundaries), creating another potential source of error. Indeed, there are many factors that may contribute in varying degrees to the overall error in the average dislocation density measurements and, due to this complexity, such error has been estimated [27]. For the polycrystalline material being studied in this work, it is reasonable to expect some inherent area-to-area variability in the measured dislocation density. As such, total error (combining both accuracy and variability of measurements) would be impossible to accurately determine. In this work, standard deviation is used as a measure of variability in observed average dislocation density. Jog spacing is defined as the linear distance between consecutive jogs on a screw dislocation (see Fig. 1). For jog spacing measurements, the TEM foil was oriented so that dislocations could be viewed using a beam direction as close to normal to the glide plane of the screw dislocations as possible, making the screw segments appear as classic cusps of characteristic length, which could then be measured. Generally speaking, this resulted in a beam direction that was close to parallel with the jog line direction (i.e. viewing the jogs edge-on). However, in order to capture the characteristic jog spacings, slight deviations from this ideal beam direction were necessary if the line direction of the jog was not perpendicular to the glide planes of the screw segments. A range of values was recorded for each sample, with the characteristic length being set to the average value for each sample. Unless otherwise indicated, error was determined by examining the distribution of values and using a standard 95% confidence interval. Reported error values were expected to be a combination of the error from the measurement technique and the actual deviation in the jog spacing present in the microstructure. Jog spacings were measured in many differ-
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ent grains for each sample, so any error stemming from the measurement method is expected to be relatively minor. Jog height is the length of a jog segment on a screw dislocation (see Fig. 1). Jog height measurements prove to be much more difficult to obtain, because foils must be oriented so that measurements could be taken for jogs that appear in profile so that the screw segments appear straight. With the fixed tilt range of the TEM, only a relatively small number of grains were oriented to allow for this measurement. As was the case for jog spacing, an average of all measured jog height values for each sample was used to represent the characteristic measured value. Unless otherwise indicated, error was determined by examining the distribution of values and using a standard 95% confidence interval. Reported error values are expected to be a combination of the error from the measurement technique and the actual deviation in the jog height present in the microstructure. Due to the difficulty of measuring jog heights, the error as a result of the measurement technique may be the more significant factor, as will be discussed in Section 3.4. As an alternative to measurement, the jog heights will also be theoretically calculated for comparison with the measured values (discussed in Section 3.4). 3. Results and discussion 3.1. Overall dislocation structure The substructure contained predominantly hai-type screw dislocations, with relatively few hc + ai dislocations being observed. Many grains exhibited multiple active slip systems, evidenced by the fact that only some of the dislocations became invisible for a given invisibility criterion (g b = 0). Examples of BF STEM images used to characterize the microstructure and quantify model parameters are shown in Fig. 2. In general, as stress increases or temperature decreases, dislocations were observed to be more tortuous. As temperature increases or stress decreases, dislocations exhibited a more gently curved morphology, with a more classic cusped screw structure. At the highest temperatures investigated, dislocation network formation (Fig. 3) was common. At high stress levels, densely packed
Fig. 2. Examples of BF STEM micrographs of Zircaloy-4. Similar images were used to characterize the microstructure and quantify the MJS model parameters.
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Fig. 3. Dislocation networks formed at high temperatures and low stresses in Zircaloy-4. Such networks may indicate a transition at high temperatures toward a different dislocation mechanism, possibly involving general climb of dislocations.
dislocation structures are sometimes observed. Neither networks nor dense dislocation structures are accounted for in the MJS model, which may indicate a transition to a different dominant deformation mechanism involving extensive dislocation interaction. The extreme bounds of the jogged-screw regime are not discussed further here, but all specimens in this work are believed to lie within the temperature and stress regime where the jogged-screw mechanism is operative/controlling. 3.2. Dislocation density In the past, the Taylor relation has been successfully applied to describe the mobile dislocation density, q, in zirconium alloys as a function of stress [1,18,29]: s 2 q¼ ð2Þ aGb where s is the shear stress, a is the Taylor factor (a constant), G is the temperature-dependent shear modulus [30] and b is the length of the Burgers vector. Fig. 4 shows the average dislocation density measured in each sample. The measured dislocation density will be lower than the volumetric dislocation density by a factor of approximately two [28], as previously discussed. Taking differences in measurement technique into account, the dislocation density values in Fig. 4 fall in line with previously reported data for Zircaloy-4 [18,29], and a Taylor factor (a) of 2.16 fits the data well. The Taylor curve shown in Fig. 4 assumes a temperature of 644 K (to fix the temperature dependence of G), and as such will change slightly with temperature. For the purposes of the MJS model, Eq. (2) was used, including the temperature dependence of the shear modulus. The dashed line in Fig. 4 shows the dislocation density measured in an unstrained specimen, which should serve as a lower bound at low stresses, even though the Taylor prediction trends toward zero. It is important to
Fig. 4. Measured average dislocation density as a function of normalized stress, and the Taylor-predicted dislocation density utilized by the MJS model.
note that the accumulated creep strains of the tests in this work vary from sample to sample (Table 2). It is reasonable to expect that the dislocation density should increase as a function of strain, as has been observed by several authors in the past [27,31,32], especially during primary creep. However, it is also reasonable to assume a constant structure during steady-state creep [18,33–35]. All tests used in this work were at or very near to minimum creep rate conditions when interrupted for substructure evaluation, as determined by a plot of the strain rate as a function of strain. In this manner, an attempt was made to minimize the variation in structure as a function of strain over the range of testing conditions; however, a more detailed examination of structure evolution with strain would prove useful in the future.
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Fig. 5. Measured average jog spacing as a function of stress, and the Orowan-predicted jog spacing utilized by the MJS model.
3.3. Jog spacing The jogged-screw model states that steady-state dislocation velocities are achieved when the drag force on screw dislocations due to non-conservative motion of jogs is offset by the line tension of the dislocation caused by an applied shear stress [14]. If it is assumed that the dragging force is equal to sb l [14] and an isotropic line tension of 0.5 Gb2 [36], the resultant jog spacing, l, is inversely proportional to shear stress: l ¼ As1
ð3Þ
where A is a constant and s is the shear stress. This equation describes the average equilibrium jog spacing, which has been determined through simulations in previous work [20]. The average jog spacings measured in this work are shown in Fig. 5.
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The jog spacing relationship used in the jogged-screw model (Eq. (3)) is shown in Fig. 5, with A equal to 35.6 N m1. The experimental data shown in Fig. 5 falls below model predictions at low stresses. This is most likely due to a limitation imposed by the transparent thickness of the TEM foils (typically a few hundred nanometers or less), which may truncate dislocations, resulting in average measured jog spacings that are less than those theoretically expected. Additionally, it is reasonable to suspect that localized variations in stress from grain to grain (due to orientations and stress concentrators within the material) will lead to a range of jog spacings instead of a single value. Jog spacing data can be arranged into histograms to show the relative distribution of measured jog spacings in each specimen (Fig. 6a and b). Fig. 6a and b shows histograms for high stress and low stress conditions, respectively. If foil thickness is affecting the observation of large jog spacings, the histogram should show a clear truncation at high spacings, as seen in Fig. 6b. This may explain the “over-prediction” of jog spacing at small stress seen in Fig. 5. As a result, the previously proposed model-predicted form has been used, leaving the underlying theoretical approach intact, instead of modifying the MJS model to completely reflect the empirical data. A normal (Gaussian) and lognormal distribution for each histogram is shown in Fig. 6 for comparative purposes. Any of a number of common fitting routines could be employed. At present, it is unclear which distribution is most appropriate, though a Gaussian type has been suggested in the past [37]. Inhomogeneity of jog spacing is not currently accounted for by the jogged-screw model, but information on the distribution of model parameters can provide valuable insight on the characteristic values currently used in the model, and how these representative values change with environmental conditions. 3.4. Jog height Jog height is one of the most important model parameters, but also the most challenging to measure. Critical jog
Fig. 6. Example histogram of (a) measured jog spacings for Zircaloy-4 tested at 644 K/178.7 MPa, (b) measured jog spacings for Zircaloy-4 tested at 866 K/26.8 MPa and (c) measured jog heights for Zircaloy-4 tested at 644 K/178.7 MPa. Normal and lognormal fits to each histogram are shown as examples of data that could be used in future work to describe such characteristic lengths in the MJS model.
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Fig. 8. Schematic of jog tilting phenomena, as proposed by Karthikeyan et al. [38]. Under this scenario, tall jogs tend to tilt, dispersing unit jogs over a greater area, reducing the diffusion gradient necessary for jog motion. Fig. 7. Measured average jog height as a function of stress, and the modelpredicted jog height necessary to achieve experimentally observed strain rates.
height has been described in previous work by the equation [18]: hd ¼ Lsa
ð4Þ
where L and a are constant values. The measured average jog height as a function of stress is shown in Fig. 7. As seen in Fig. 7, the measured jog height can be approximated using the general form shown in Eq. (4). It should be noted, however, that the MJS model does not use a measured average jog height directly in strain rate predictions. Instead, a “critical” jog height is used. The critical jog height is defined as the height at which the stress required for jog dragging and dipole bypass are equal. This concept is discussed in detail in the literature [17,18,20]. Critical jog heights were calculated for the creep conditions of the experimental data and are shown in Fig. 7. The model-predicted critical jog height in this work can be described by Eq. (4), where s is in pascals, and L and a are 2.11 102 m Pa1 and 0.865, respectively. The values predicted by Eq. (4) are smaller in magnitude than those directly observed in the microstructure (see Fig. 7). Jogs taller than predicted by the model have been observed previously [17,18]. There are several possible reasons for this apparent discrepancy, one of which is measurement bias. Dislocations will appear with some finite thickness in TEM micrographs (in this work, the lower limit is 3 nm using BF STEM). Any jog heights smaller than this limit cannot be clearly resolved. Additionally, since smaller jogs are more difficult to discern than taller ones, the smaller jogs may be under-represented during the observation and analysis of the substructure, leading to an average jog height that is somewhat biased towards larger jogs. Additionally, it has been proposed by Karthikeyan et al. [38] that tall jogs consisting of many unit jogs stacked on top of each other may not remain perpen-
dicular to the glide planes but, rather, “tilt”, thereby dispersing the unit jogs over a greater length of dislocation. This may spread out the necessary vacancy diffusion field, leading to the accommodation and eased dragging of taller jogs than would otherwise be predicted in the model. A schematic of such a scenario is shown in Fig. 8. The reconciliation of model-predicted and measured jog heights would be advantageous, given the relative difficulty of direct measurement of jog heights. As with jog spacings, jog height measurements for each sample can be compiled into histograms. An example of one such histogram, along with an associated lognormal distribution, is shown in Fig. 6c. These distributions carry a wealth of information useful to modeling efforts. The lack of any observed jogs of very small height present in the histograms underscores some of the arguments above regarding the discrepancies between model-predicted and measured values. Additionally, based on previous arguments, jogs are expected to be small when formed, then grow as kink/cross-link interactions proceed, and finally transition into a dipole bypass condition and become immobile [17,18,20]. If this were true, a sharp cutoff in the jog height histogram would be expected at some critical value, with very few, if any, taller jogs. This does not seem to be the case, however, given the very large jogs observed in the microstructure. This suggests that the majority of jog formation may not be due to dislocation intersections, as proposed by Barrett and Nix [14], but caused through a multiple cross-slip process, as has been predicted and observed for similar materials (with strong lattice friction on screw dislocations) in previous studies [39–41]. 3.5. Application of the MJS model to the data The strain rate predicted by the MJS model represents a form of the Orowan equation, which relates strain rate to density and velocity of dislocations in the form:
B.M. Morrow et al. / Acta Materialia 61 (2013) 4452–4460
c_ ¼ qm bv
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ð5Þ
where c_ is the shear strain rate, qm is the mobile dislocation density, and v is the velocity of mobile dislocations. Accounting for the velocity of jogged-screw dislocations, the MJS model equation for steady-state creep rates is then [18]: ! pffiffiffi 2 2pDs r 2 rXl sin h ð6Þ _ ¼ 2aGb 2hkT dðhÞ where h and l are jog height and spacing, respectively, Ds is the coefficient of self-diffusion, r is the applied stress, X is the atomic volume, k is the Boltzmann constant, T is the absolute temperature, and d(h) is a logarithmic function of jog height [18]: h dðhÞ ¼ 0:17 þ 4:528ln : ð7Þ b A detailed description of how the model was derived can be found elsewhere [17,18,20]. The equation for diffusion coefficient, Ds, is: Qc Ds ¼ D0 exp ð8Þ RT where Qc is the activation energy for creep (270 kJ mol1) and D0 is the pre-exponential term (5 104 cm2 s1) [18,42,43]. Using the model parameter relationships derived from experimental data discussed above, the strain rate can be calculated as a function of stress. Modelpredicted strain rates as a function of stress for three representative temperatures are shown in Fig. 9. The model-predicted strain rates show a good agreement with the experimental data. They can also be normalized to remove the temperature dependence of the jogged-screw model, for comparison with experimental strain rates across a wide
Fig. 10. Model-predicted strain rates compared to experimental data from the current work. Strain rates are normalized to account for differences in testing conditions.
range of stress–temperature conditions, as shown in Fig. 10. The current model provides a reasonable fit with experimental data for strain rate in Zircaloy-4, as seen in Fig. 10. Due to the hyperbolic sine present in Eq. (6), the model can predict strain rates into the breakdown regime (r/G P 2 103) as well as in the power law regime (n = 5). Historical experimental data can be found in the literature [44], but, as pointed out by Hayes and Kassner, variations in alloy chemistry and thermomechanical history can influence creep strength, which makes such data difficult to compare directly with the current work. In the future, it should be feasible for the model to incorporate compositional or processing parameters to predict creep behavior more accurately. Despite promising results, there are several possible improvements to be pursued with respect to the MJS model. Significant effects of alloying have been observed for several alloying elements, yielding noticeable changes in strain rates [44–47]. However, very few studies have examined the effect of alloying elements on microstructural features, thus the underlying cause of the differences is presently only inferred. Additionally, changes in substructure with varying strain levels are, for the most part, untreated in the literature. A thorough characterization of the substructure with regard to these factors would aid in the development of a more robust model for strain rates during creep. 4. Conclusions
Fig. 9. MJS model-predicted strain rates as a function of stress, compared to experimental data from the current work.
The creep behavior and dislocations substructures of Zircaloy-4 were examined over a range of stress and temperature conditions. The substructure was dominated by
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hai-type screw dislocations. STEM observations of dislocations showed jogs pinning the dislocations at intervals along dislocation lines. The general form of equations utilized in Ref. [18] to describe model parameters were validated, though slightly different constants were used to more closely match the experimental parametric data. Combining these relationships into the steady-state strain rate model (Eq. (6)) yields reasonable agreement with experimental creep data (Fig. 9). The stress exponent, n, predicted by the model in the power law regime (r/G 6 2 103) was 4–5, within the acceptable range for five power law creep [44]. The predicted stress exponent shows good agreement with experimental observations in the power law breakdown regime, as well as the result of the hyperbolic sine function in Eq. (6). The “breakdown” at high stress merely states a deviance from the empirical power law, and is not necessarily an indication that a different deformation mechanism is active. In the future, emphasis should be placed on the effects of chemistry on strain rates. Also, a study of the substructural development and evolution of model parameters as a function of strain levels could provide many insights into the jogged-screw mechanism and how model parameters evolve during deformation. Finally, a more thorough understanding of the boundaries of the stress–temperature regime where the jogged-screw model is active would improve understanding of the mechanism and aid deployment of the MJS model to describe creep behavior of zirconium alloys. References [1] Dunlop J, Bre´chet Y, Legras L, Estrin Y. Mater Sci Eng A 2007;443: 77–86. [2] Baum E, Knox H, Miller T. Nuclides and isotopes: chart of the nuclides. 16th ed. Knolls Atomic Power Laboratory; 2002. [3] Franklin D, Lucas G, Bement A. Creep of zirconium alloys in nuclear reactors. American Society of Testing and Materials; 1983. [4] Charit I, Murty K. J Nucl Mater 2008;374:354–63. [5] Sinha R, Sinha S, Madhusoodanan K. J Nucl Mater 2008;383:14–21. [6] Franklin D, Adamson R. J Nucl Mater 1988;159:12–21. [7] US Department of Energy. DOE fundamentals handbook: material science. Technical Report DOE-HDBK-1017/2-93 1993. [8] Soniak A, L’Hullier N, Mardon JP, Rebeyrolle V, Bouffioux P, Bernaudat C. ASTM Spec Tech Publ 2002;1423:837–62. [9] Murty K. J Miner Metals Mater Soc 2000;52:34–8. [10] Hayes T, Rosen R, Kassner M. J Nucl Mater 2006;353:109–18. [11] Hayward P, George I. J Nucl Mater 1999;273:294–301. [12] Rosinger H, Bera P, Clendening W. The steady-state creep of Zircaloy-4 fuel cladding from 940 to 1873 K. Technical Report AECL-6193, Atomic Energy of Canada Engineering Company, Whiteshell Nuclear Research Establishment; 1978.
[13] Nuclear Energy Agency. Nuclear fuel behaviour in loss-of-coolant accident (LOCA) conditions. Technical Report 6846; 2009. [14] Barrett C, Nix W. Acta Metall 1965;13:1247–58. [15] Hasegawa T, Sato H, Karashima S. Trans Jpn Inst Met 1970;11: 231–8. [16] Karashima S, Iikubo T, Watanabe T, Oikawa H. Trans Jpn Inst Met 1971;12:369–74. [17] Viswanathan G, Vasudevan V, Mills M. Acta Mater 1999;47: 1399–411. [18] Moon J, Cantonwine P, Anderson K, Karthikeyan S, Mills M. J Nucl Mater 2006;353:177–89. [19] Partridge PG. Metall Rev 1967;12:169–94. [20] Karthikeyan S, Viswanathan G, Mills M. Acta Mater 2004;52: 2577–89. [21] Karthikeyan S, Viswanathan G, Gouma P, Vasudevan V, Kim YW, Mills M. Mater Sci Eng A 2002;329-331:621–30. [22] Kearns J. Thermal expansion and preferred orientation in Zircaloy. Technical Report WAPD-TM-472, Bettis Atomic Power Laboratory; 1965. [23] Schemel J. Trans Metall Soc AIME 1961;221:1129–36. [24] Norfleet D, Dimiduk D, Polasik S, Uchic M, Mills M. Acta Mater 2008;56:2988–3001. [25] Maher D, Joy D. Ultramicroscopy 1976;1:239–53. [26] Phillips P, Mills M, De Graef M. Philos Mag 2011. iFirst:1–21. [27] Kruml T, Paidar V, Martin J. Intermetallics 2000;8:729–36. [28] Hull D, Bacon D. Introduction to dislocations. 4th ed. Butterworth Heinemann; 2001. [29] Ecob R, Donaldson A. J Nucl Mater 1985;132:110–25. [30] Nam C, Choi BK, Lee MH, Jeong YH. J Nucl Mater 2002;305:70–6. [31] Lu M, Hemker K. Metall Mater Trans A 1998;29:99–104. [32] Daehn G, Brehm H, Lee H, Lim BS. Mater Sci Eng A 2004;387389:576–84. [33] Daehn G. Acta Mater 2001;49:2017–26. [34] Rudolf T, Skrotzki B, Eggeler G. Mater Sci Eng A 2001;319321:815–9. [35] da C Andrade E. Proc R Soc London A 1914;90:329–42. [36] Hirth J, Lothe J. Theory of dislocations. New York: John Wiley and Sons Inc.; 1982. [37] Viswanathan G, Karthikeyan S, Hayes R, Mills M. Metall Mater Trans A 2002;33A:329–36. [38] Karthikeyan S. Unpublished work. [39] Girshick A, Bratkovsky A, Pettifor D, Vitek V. Philos Mag 1998;77:981–97. [40] Woodward C, Rao S. Philos Mag 2004;84:401–13. [41] Neeraj T, Savage M, Tatalovich J, Kovarik L, Hayes R, Mills M. Philos Mag 2005;85:279–95. [42] Lyashenko V, Bykov V, Pavlinov L. Phys Met Metallog USSR 1959;8:40–6. [43] Hayes T, Kassner M, Rosen R. Metall Mater Trans A 2002;33A: 337–43. [44] Hayes T, Kassner M. Metall Mater Trans A 2006;37A:2389–96. [45] Hayes R, Viswanathan G, Mills M. Acta Mater 2002;50: 4953–63. [46] Kim HG, Kim YH, Choi BK, Jeong YH. J Nucl Mater 2006;359: 268–73. [47] McInteer W, Baty D, Stein K. ASTM Spec Tech Publ 1988;1023: 621–40.
Acta Materialia 61 (2013) 4452–4460 www.elsevier.com/locate/actamat
An examination of the use of the Modified Jogged-Screw model for predicting creep behavior in Zircaloy-4 B.M. Morrow a,⇑, R.W. Kozar b, K.R. Anderson b, M.J. Mills a a
The Ohio State University, 2041 College Rd., 477 Watts Hall, Columbus, OH 43210, USA b Bettis Laboratory, Bechtel Marine Propulsion Corp., West Mifflin, PA 15122, USA
Received 20 September 2012; received in revised form 2 April 2013; accepted 7 April 2013 Available online 6 May 2013
Abstract The alloy Zircaloy-4 is used primarily as a structural material in nuclear reactors. A thorough understanding of the thermal creep response of such materials is essential to enable accurate forecasting of material behavior during both service and long-term storage. To this end, the Modified Jogged-Screw model, originally developed to predict steady-state creep rates in c-TiAl, has been applied to Zircaloy-4. This work provides a more complete database for model validation through creep testing of several Zircaloy-4 specimens over a range of stress and temperature conditions. Bright-field scanning transmission electron microscopy was used to observe the substructure and quantify the model parameters important to the Modified Jogged-Screw model. These observations were used to confirm the applicability of the proposed model and assess the sensitivity of the model parameters to test conditions. The increased availability of statistics for model parameters as a result of substructural observation will elucidate the strengths and weaknesses of the model as previously proposed. Ó 2013 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. Keywords: Creep; Screw dislocation; Mechanical properties; Mechanistic modeling; STEM
1. Introduction Zirconium alloys serve an important role as cladding material for nuclear fuel, separating the fuel (and various fission products) from the moderator or coolant. Zirconium alloys are widely used for cladding due to a combination of good thermal conductivity, corrosion resistance and structural integrity at high temperature [1]. They are especially important, however, due to their low thermal neutron capture cross-section [2], which allows neutrons to pass more easily from the fuel to the moderator. During service, heat and pressure produce conditions favorable for creep of the material. Cladding temperatures of 560– ⇑ Corresponding author. Present address: Los Alamos National Laboratory, P.O. Box 1663, MS G755, Los Alamos, NM 87545, USA. Tel.: +1 505 665 9224; fax: +1 505 667 8021. E-mail addresses: [email protected] (B.M. Morrow), millsmj@mse. osu.edu (M.J. Mills).
625 K and pressures of 10–16 MPa are typical [3–5], and partial or complete collapse of cladding due to creep, known as “creepdown”, has been observed in the past [6– 8]. Furthermore, residual decay heat, which is present in significant levels for many years after a fuel assembly is removed from service, may facilitate thermal creep during dry storage. Creep is regarded by many as the leading concern about the feasibility of permanent dry storage [9], during which temperatures near 723 K and stresses of 100 MPa can be present [10]. Some estimates expect decay heat during dry storage to initially increase from 575 K, peak 10 years after emplacement, then decrease continuously over a long period of time, still being at 425 K after 1000 years [9]. For reference, the melting temperature of Zircaloy-4 is 2034 K [11], though the material becomes significantly weaker as temperatures approach the beta transition temperature, 862 K [12,13]. Thermal creep behavior in non-irradiated materials can serve as an important cornerstone in understanding the material response during
1359-6454/$36.00 Ó 2013 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.actamat.2013.04.014
B.M. Morrow et al. / Acta Materialia 61 (2013) 4452–4460
and after irradiation. Accurate prediction of thermal creep response is essential for the safe use of the material during both service and long-term storage. The Modified Jogged-Screw (MJS) model was initially developed to describe the creep behavior of c-TiAl alloys, and represents a reconsideration of the original joggedscrew creep model proposed by Barrett and Nix [14]. The original jogged-screw model asserts that the steady-state creep behavior of certain alloys is controlled by the motion of jogs on screw dislocations. Such dislocation behavior has been observed in various materials in the past [14– 18]. Because the jogs are edge in character, dislocation climb is required in order for jogs to translate with the rest of the gliding dislocation. The balance between applied stress and the chemical drag imposed by the diffusion of vacancies to and from the jog segments leads to a constant creep rate. The Barrett and Nix model proposed that jogs on screw dislocations are formed primarily from dislocation intersections, producing jogs of one Burgers vector height. Zirconium is known to prefer prismatic hai-type slip, and hc + ai-type slip is observed only occasionally [19]; therefore intersection of screw dislocations on different prism planes is a likely formation mechanism of the small jogs described by the Barrett and Nix model. The MJS model made several changes to the original model, including allowing for taller jogs that have been observed in the past and are believed to form through either growth of smaller jogs or formation of tall jogs directly through multiple cross-slip events [20]. After successful application of the MJS model to titanium alloys [17,20,21], the model was adapted to describe the creep behavior of zirconium alloys, due to similarities between elemental zirconium and titanium, and observed similarities in microstructural features [18]. The model predictions of Moon et al. showed good agreement in steadystate strain rates with experimental data; however, the model was calibrated and validated using a relatively small amount of creep data for Zircaloy-4 (one specimen for each of four temperatures). Because of this, several assumptions were necessary to determine how model parameters changed with stress and temperature. Expanding the applicability of the MJS model to conditions outside of the stress–temperature regime tested in Ref. [18] is a focus of the current study. Additionally, the model utilized jog heights that were calculated theoretically rather than based on experimental investigation [17,18]; differences between predicted and observed jog heights have been observed previously [17]. In prior work utilizing the MJS model, single values for model parameters were assumed without consideration of a distribution of values that would be expected in real materials. Thus, additional testing and microstructural examinations are necessary to characterize the active deformation mechanism. It is unclear how well the previous assumptions will hold when subjected to a broader range of testing conditions, including the power law breakdown regime. This work attempts to address these issues by examining a broader range of stresses at each testing
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temperature, and using a statistical approach to the determination of model parameters. 2. Experimental procedures 2.1. Mechanical test setup Creep tests were performed on round tensile specimens of non-irradiated Zircaloy-4 that were machined from processed plate material. The alloy was rolled, then heated and quenched from the b phase-field, yielding a mostly isotropic structure [22,23]. The alloy composition for Zircaloy4 is shown in Table 1. Each sample had a gage diameter of 5.08 mm and a gage length of 60.96 mm. Samples were loaded using a lever arm creep frame at a 20:1 load ratio. All tests were loaded in uniaxial tension. A load cell was used to directly measure the force applied to each sample. A three-zone furnace was used to heat the samples to the desired temperature. Extensometers and linear variable displacement transducers (LVDTs) were used to determine strain in the gage section. An independent thermocouple measured temperature at the surface of each sample. Testing was performed at a temperature low enough that oxidation was not expected to be a concern, allowing the samples to be tested in a laboratory air atmosphere. After testing, samples were cooled under load to preserve the dislocation substructure. The test conditions, resultant creep strains and minimum (steady-state) strain rates for the samples used in this work are shown in Table 2. 2.2. Specimen preparation Specimens were cut from the center of the gage section at a 45° angle to the loading axis in order to increase the probability that a principal glide plane will lie in the plane of the thin foil, facilitating analysis of dislocations. The samples were then thinned to 100 lm using 1200 grit SiC paper, and cut into 3 mm disks. The samples were prepared for transmission electron microscopy (TEM) analysis in a dual-jet electropolisher using a solution of 940 ml of methanol, 60 ml of sulfuric acid and 2.4 ml of hydrofluoric acid, at a potential of 21 V and a temperature of 40 to 45 °C. Some hydride formation occurred on the surfaces of the prepared foils due to this technique, but these were generally easily distinguishable from dislocations. 2.3. Dislocation substructure analysis techniques An FEI Tecnai F20 transmission electron microscope, using bright-field (BF) and/or dark-field (DF) imaging in scanning transmission electron microscopy (STEM) mode, Table 1 Nominal compositions of Zircaloy-4. Sn (wt.%)
C (ppm)
Fe (wt.%)
Cr (wt.%)
Zr
1.3–1.5
150
0.18–0.24
0.07–0.13
Balance
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Table 2 Test matrix of samples used in this work. Temperature (K)
Stress (MPa)
Strain (%)
Strain rate (s1)
533 533 589 644 644 644 644 644 644 644 700 700 700 700 700 783 866 866
196.5 219.2 203.5 156.4 156.4 165.5 165.5 165.5 165.5 178.7 99.9 126.5 126.5 146.1 173.4 59.2 26.8 49.0
1.07 1.78 4.41 0.70 1.24 1.43 2.65 4.44 6.72 2.32 0.87 2.00 2.19 4.27 4.38 1.29 1.21 2.50
1.98 109 3.18 109 2.00 108 4.90 108 2.23 108 8.00 109 6.95 109 8.00 109 8.00 109 4.86 108 9.14 109 1.55 107 1.78 108 8.50 108 1.10 106 9.04 109 3.04 108 2.50 107
was used for all microscopy related to this work. STEM imaging was used to image the substructure because it provides many advantages over conventional TEM (CTEM). The convergence of the beam creates a range of incident angles for the electrons, leading to “relaxed” diffraction conditions [24,25]. This results in a more complete sampling of the substructure, as defects will display contrast, often in the absence of a strict two-beam condition. As a result, this technique makes it possible to sample essentially all of the dislocations present in the substructure at the same time. Additionally, STEM imaging can reduce the effect of bend and thickness contours, and allow for imaging in thicker regions than would be possible using CTEM. A more thorough investigation of the use of STEM diffraction contrast imaging is given by Phillips et al. [26]. Multiple foils from each test condition were examined, and often several grains from each foil were used for imaging. Tilting experiments were performed, including g b analysis and line trace analysis on selected dislocations, to determine the type and character of the dislocations present in the substructure. The MJS model is based on changes in characteristics of jogged-screw dislocations with different stress and temperature conditions. Three different behaviors are anticipated: jog dragging (for small jogs), dipole dragging (for medium-sized jogs) and dipole bypass (for jogs of sufficient height that the shear stress forces the dipole segments to bypass). When jogs exhibit dipole bypass, they are assumed to become immobile and act as dislocation sources [18]. A schematic of these jogged-screw behaviors is shown in Fig. 1. The three model parameters that are measurable using TEM are dislocation density, jog spacing and jog height. Fig. 1 also shows a schematic of jog spacing, l and jog height, h. Image processing and analysis were performed using Adobe Photoshop with the Fovea Pro image analysis plug-in from Reindeer Graphics. Dislocation density can be determined through direct microstructural observation using either of two methods.
Fig. 1. Schematic of jogged-screw dislocation behavior as a function of height. For small heights, the diffusion of vacancies to or from the jog is sufficient for the jog to move along with the screw segments (jog dragging). For medium-sized jogs, the mobility of the jog makes it more favorable for a dipole to form (dipole dragging). For very tall jogs, the two dipole segments may move past each other, acting as independent single-ended dislocation sources (dipole bypass).
The method used in this work involves measuring the number of dislocations per area, and is described by: qmeasured ¼
0:5j Amicrograph
ð1Þ
where j represents the total number of dislocation endpoints (intersections with the foil surfaces) present in a micrograph and Amicrograph is the measured area of the micrograph. A second method measures the dislocation line length per volume (see Kruml et al. [27]). Due to the large area of inspection afforded by STEM imaging, and given the many testing conditions being studied, the more rigorous line length per volume technique would be prohibitively time-consuming to perform. The end point per area method is a viable option to address this concern; however, a correction factor of between one (for arrays of parallel dislocations) and two (for random dislocations) must be applied to correct for an under-sampling of dislocations [28]. This was confirmed through a comparison of the two methods for selected specimens in this work. Additionally, it has been reported in the past that, for very closely spaced dislocations, strain field overlap could result in an under-sampling of dislocations by as much as 30% when using BF STEM images to measure dislocation density [24]. The dislocation densities in this work are not typically high enough for this to be an issue. There are several factors that lead to apparent error in dislocation density measurements. One is inhomogeneity in the dislocation distribution [27], leading to variability in measurements. This concern is minimized by using STEM imaging, which allows for much larger areas to be imaged and, through use of multi-beam diffraction conditions, allows nearly all the dislocations present to appear
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in contrast. As a result, a much larger sampling of the microstructure can be obtained than would be possible using conventional TEM imaging. By sampling many random grains from several different foils for each condition, sampling error is also reduced. The MJS model uses mobile dislocation density to calculate strain rates. Some judgement was necessary to exclude dislocations believed to be immobile (e.g. involved in dislocation networks or dense tangles and grain boundaries), creating another potential source of error. Indeed, there are many factors that may contribute in varying degrees to the overall error in the average dislocation density measurements and, due to this complexity, such error has been estimated [27]. For the polycrystalline material being studied in this work, it is reasonable to expect some inherent area-to-area variability in the measured dislocation density. As such, total error (combining both accuracy and variability of measurements) would be impossible to accurately determine. In this work, standard deviation is used as a measure of variability in observed average dislocation density. Jog spacing is defined as the linear distance between consecutive jogs on a screw dislocation (see Fig. 1). For jog spacing measurements, the TEM foil was oriented so that dislocations could be viewed using a beam direction as close to normal to the glide plane of the screw dislocations as possible, making the screw segments appear as classic cusps of characteristic length, which could then be measured. Generally speaking, this resulted in a beam direction that was close to parallel with the jog line direction (i.e. viewing the jogs edge-on). However, in order to capture the characteristic jog spacings, slight deviations from this ideal beam direction were necessary if the line direction of the jog was not perpendicular to the glide planes of the screw segments. A range of values was recorded for each sample, with the characteristic length being set to the average value for each sample. Unless otherwise indicated, error was determined by examining the distribution of values and using a standard 95% confidence interval. Reported error values were expected to be a combination of the error from the measurement technique and the actual deviation in the jog spacing present in the microstructure. Jog spacings were measured in many differ-
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ent grains for each sample, so any error stemming from the measurement method is expected to be relatively minor. Jog height is the length of a jog segment on a screw dislocation (see Fig. 1). Jog height measurements prove to be much more difficult to obtain, because foils must be oriented so that measurements could be taken for jogs that appear in profile so that the screw segments appear straight. With the fixed tilt range of the TEM, only a relatively small number of grains were oriented to allow for this measurement. As was the case for jog spacing, an average of all measured jog height values for each sample was used to represent the characteristic measured value. Unless otherwise indicated, error was determined by examining the distribution of values and using a standard 95% confidence interval. Reported error values are expected to be a combination of the error from the measurement technique and the actual deviation in the jog height present in the microstructure. Due to the difficulty of measuring jog heights, the error as a result of the measurement technique may be the more significant factor, as will be discussed in Section 3.4. As an alternative to measurement, the jog heights will also be theoretically calculated for comparison with the measured values (discussed in Section 3.4). 3. Results and discussion 3.1. Overall dislocation structure The substructure contained predominantly hai-type screw dislocations, with relatively few hc + ai dislocations being observed. Many grains exhibited multiple active slip systems, evidenced by the fact that only some of the dislocations became invisible for a given invisibility criterion (g b = 0). Examples of BF STEM images used to characterize the microstructure and quantify model parameters are shown in Fig. 2. In general, as stress increases or temperature decreases, dislocations were observed to be more tortuous. As temperature increases or stress decreases, dislocations exhibited a more gently curved morphology, with a more classic cusped screw structure. At the highest temperatures investigated, dislocation network formation (Fig. 3) was common. At high stress levels, densely packed
Fig. 2. Examples of BF STEM micrographs of Zircaloy-4. Similar images were used to characterize the microstructure and quantify the MJS model parameters.
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Fig. 3. Dislocation networks formed at high temperatures and low stresses in Zircaloy-4. Such networks may indicate a transition at high temperatures toward a different dislocation mechanism, possibly involving general climb of dislocations.
dislocation structures are sometimes observed. Neither networks nor dense dislocation structures are accounted for in the MJS model, which may indicate a transition to a different dominant deformation mechanism involving extensive dislocation interaction. The extreme bounds of the jogged-screw regime are not discussed further here, but all specimens in this work are believed to lie within the temperature and stress regime where the jogged-screw mechanism is operative/controlling. 3.2. Dislocation density In the past, the Taylor relation has been successfully applied to describe the mobile dislocation density, q, in zirconium alloys as a function of stress [1,18,29]: s 2 q¼ ð2Þ aGb where s is the shear stress, a is the Taylor factor (a constant), G is the temperature-dependent shear modulus [30] and b is the length of the Burgers vector. Fig. 4 shows the average dislocation density measured in each sample. The measured dislocation density will be lower than the volumetric dislocation density by a factor of approximately two [28], as previously discussed. Taking differences in measurement technique into account, the dislocation density values in Fig. 4 fall in line with previously reported data for Zircaloy-4 [18,29], and a Taylor factor (a) of 2.16 fits the data well. The Taylor curve shown in Fig. 4 assumes a temperature of 644 K (to fix the temperature dependence of G), and as such will change slightly with temperature. For the purposes of the MJS model, Eq. (2) was used, including the temperature dependence of the shear modulus. The dashed line in Fig. 4 shows the dislocation density measured in an unstrained specimen, which should serve as a lower bound at low stresses, even though the Taylor prediction trends toward zero. It is important to
Fig. 4. Measured average dislocation density as a function of normalized stress, and the Taylor-predicted dislocation density utilized by the MJS model.
note that the accumulated creep strains of the tests in this work vary from sample to sample (Table 2). It is reasonable to expect that the dislocation density should increase as a function of strain, as has been observed by several authors in the past [27,31,32], especially during primary creep. However, it is also reasonable to assume a constant structure during steady-state creep [18,33–35]. All tests used in this work were at or very near to minimum creep rate conditions when interrupted for substructure evaluation, as determined by a plot of the strain rate as a function of strain. In this manner, an attempt was made to minimize the variation in structure as a function of strain over the range of testing conditions; however, a more detailed examination of structure evolution with strain would prove useful in the future.
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Fig. 5. Measured average jog spacing as a function of stress, and the Orowan-predicted jog spacing utilized by the MJS model.
3.3. Jog spacing The jogged-screw model states that steady-state dislocation velocities are achieved when the drag force on screw dislocations due to non-conservative motion of jogs is offset by the line tension of the dislocation caused by an applied shear stress [14]. If it is assumed that the dragging force is equal to sb l [14] and an isotropic line tension of 0.5 Gb2 [36], the resultant jog spacing, l, is inversely proportional to shear stress: l ¼ As1
ð3Þ
where A is a constant and s is the shear stress. This equation describes the average equilibrium jog spacing, which has been determined through simulations in previous work [20]. The average jog spacings measured in this work are shown in Fig. 5.
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The jog spacing relationship used in the jogged-screw model (Eq. (3)) is shown in Fig. 5, with A equal to 35.6 N m1. The experimental data shown in Fig. 5 falls below model predictions at low stresses. This is most likely due to a limitation imposed by the transparent thickness of the TEM foils (typically a few hundred nanometers or less), which may truncate dislocations, resulting in average measured jog spacings that are less than those theoretically expected. Additionally, it is reasonable to suspect that localized variations in stress from grain to grain (due to orientations and stress concentrators within the material) will lead to a range of jog spacings instead of a single value. Jog spacing data can be arranged into histograms to show the relative distribution of measured jog spacings in each specimen (Fig. 6a and b). Fig. 6a and b shows histograms for high stress and low stress conditions, respectively. If foil thickness is affecting the observation of large jog spacings, the histogram should show a clear truncation at high spacings, as seen in Fig. 6b. This may explain the “over-prediction” of jog spacing at small stress seen in Fig. 5. As a result, the previously proposed model-predicted form has been used, leaving the underlying theoretical approach intact, instead of modifying the MJS model to completely reflect the empirical data. A normal (Gaussian) and lognormal distribution for each histogram is shown in Fig. 6 for comparative purposes. Any of a number of common fitting routines could be employed. At present, it is unclear which distribution is most appropriate, though a Gaussian type has been suggested in the past [37]. Inhomogeneity of jog spacing is not currently accounted for by the jogged-screw model, but information on the distribution of model parameters can provide valuable insight on the characteristic values currently used in the model, and how these representative values change with environmental conditions. 3.4. Jog height Jog height is one of the most important model parameters, but also the most challenging to measure. Critical jog
Fig. 6. Example histogram of (a) measured jog spacings for Zircaloy-4 tested at 644 K/178.7 MPa, (b) measured jog spacings for Zircaloy-4 tested at 866 K/26.8 MPa and (c) measured jog heights for Zircaloy-4 tested at 644 K/178.7 MPa. Normal and lognormal fits to each histogram are shown as examples of data that could be used in future work to describe such characteristic lengths in the MJS model.
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Fig. 8. Schematic of jog tilting phenomena, as proposed by Karthikeyan et al. [38]. Under this scenario, tall jogs tend to tilt, dispersing unit jogs over a greater area, reducing the diffusion gradient necessary for jog motion. Fig. 7. Measured average jog height as a function of stress, and the modelpredicted jog height necessary to achieve experimentally observed strain rates.
height has been described in previous work by the equation [18]: hd ¼ Lsa
ð4Þ
where L and a are constant values. The measured average jog height as a function of stress is shown in Fig. 7. As seen in Fig. 7, the measured jog height can be approximated using the general form shown in Eq. (4). It should be noted, however, that the MJS model does not use a measured average jog height directly in strain rate predictions. Instead, a “critical” jog height is used. The critical jog height is defined as the height at which the stress required for jog dragging and dipole bypass are equal. This concept is discussed in detail in the literature [17,18,20]. Critical jog heights were calculated for the creep conditions of the experimental data and are shown in Fig. 7. The model-predicted critical jog height in this work can be described by Eq. (4), where s is in pascals, and L and a are 2.11 102 m Pa1 and 0.865, respectively. The values predicted by Eq. (4) are smaller in magnitude than those directly observed in the microstructure (see Fig. 7). Jogs taller than predicted by the model have been observed previously [17,18]. There are several possible reasons for this apparent discrepancy, one of which is measurement bias. Dislocations will appear with some finite thickness in TEM micrographs (in this work, the lower limit is 3 nm using BF STEM). Any jog heights smaller than this limit cannot be clearly resolved. Additionally, since smaller jogs are more difficult to discern than taller ones, the smaller jogs may be under-represented during the observation and analysis of the substructure, leading to an average jog height that is somewhat biased towards larger jogs. Additionally, it has been proposed by Karthikeyan et al. [38] that tall jogs consisting of many unit jogs stacked on top of each other may not remain perpen-
dicular to the glide planes but, rather, “tilt”, thereby dispersing the unit jogs over a greater length of dislocation. This may spread out the necessary vacancy diffusion field, leading to the accommodation and eased dragging of taller jogs than would otherwise be predicted in the model. A schematic of such a scenario is shown in Fig. 8. The reconciliation of model-predicted and measured jog heights would be advantageous, given the relative difficulty of direct measurement of jog heights. As with jog spacings, jog height measurements for each sample can be compiled into histograms. An example of one such histogram, along with an associated lognormal distribution, is shown in Fig. 6c. These distributions carry a wealth of information useful to modeling efforts. The lack of any observed jogs of very small height present in the histograms underscores some of the arguments above regarding the discrepancies between model-predicted and measured values. Additionally, based on previous arguments, jogs are expected to be small when formed, then grow as kink/cross-link interactions proceed, and finally transition into a dipole bypass condition and become immobile [17,18,20]. If this were true, a sharp cutoff in the jog height histogram would be expected at some critical value, with very few, if any, taller jogs. This does not seem to be the case, however, given the very large jogs observed in the microstructure. This suggests that the majority of jog formation may not be due to dislocation intersections, as proposed by Barrett and Nix [14], but caused through a multiple cross-slip process, as has been predicted and observed for similar materials (with strong lattice friction on screw dislocations) in previous studies [39–41]. 3.5. Application of the MJS model to the data The strain rate predicted by the MJS model represents a form of the Orowan equation, which relates strain rate to density and velocity of dislocations in the form:
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c_ ¼ qm bv
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ð5Þ
where c_ is the shear strain rate, qm is the mobile dislocation density, and v is the velocity of mobile dislocations. Accounting for the velocity of jogged-screw dislocations, the MJS model equation for steady-state creep rates is then [18]: ! pffiffiffi 2 2pDs r 2 rXl sin h ð6Þ _ ¼ 2aGb 2hkT dðhÞ where h and l are jog height and spacing, respectively, Ds is the coefficient of self-diffusion, r is the applied stress, X is the atomic volume, k is the Boltzmann constant, T is the absolute temperature, and d(h) is a logarithmic function of jog height [18]: h dðhÞ ¼ 0:17 þ 4:528ln : ð7Þ b A detailed description of how the model was derived can be found elsewhere [17,18,20]. The equation for diffusion coefficient, Ds, is: Qc Ds ¼ D0 exp ð8Þ RT where Qc is the activation energy for creep (270 kJ mol1) and D0 is the pre-exponential term (5 104 cm2 s1) [18,42,43]. Using the model parameter relationships derived from experimental data discussed above, the strain rate can be calculated as a function of stress. Modelpredicted strain rates as a function of stress for three representative temperatures are shown in Fig. 9. The model-predicted strain rates show a good agreement with the experimental data. They can also be normalized to remove the temperature dependence of the jogged-screw model, for comparison with experimental strain rates across a wide
Fig. 10. Model-predicted strain rates compared to experimental data from the current work. Strain rates are normalized to account for differences in testing conditions.
range of stress–temperature conditions, as shown in Fig. 10. The current model provides a reasonable fit with experimental data for strain rate in Zircaloy-4, as seen in Fig. 10. Due to the hyperbolic sine present in Eq. (6), the model can predict strain rates into the breakdown regime (r/G P 2 103) as well as in the power law regime (n = 5). Historical experimental data can be found in the literature [44], but, as pointed out by Hayes and Kassner, variations in alloy chemistry and thermomechanical history can influence creep strength, which makes such data difficult to compare directly with the current work. In the future, it should be feasible for the model to incorporate compositional or processing parameters to predict creep behavior more accurately. Despite promising results, there are several possible improvements to be pursued with respect to the MJS model. Significant effects of alloying have been observed for several alloying elements, yielding noticeable changes in strain rates [44–47]. However, very few studies have examined the effect of alloying elements on microstructural features, thus the underlying cause of the differences is presently only inferred. Additionally, changes in substructure with varying strain levels are, for the most part, untreated in the literature. A thorough characterization of the substructure with regard to these factors would aid in the development of a more robust model for strain rates during creep. 4. Conclusions
Fig. 9. MJS model-predicted strain rates as a function of stress, compared to experimental data from the current work.
The creep behavior and dislocations substructures of Zircaloy-4 were examined over a range of stress and temperature conditions. The substructure was dominated by
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hai-type screw dislocations. STEM observations of dislocations showed jogs pinning the dislocations at intervals along dislocation lines. The general form of equations utilized in Ref. [18] to describe model parameters were validated, though slightly different constants were used to more closely match the experimental parametric data. Combining these relationships into the steady-state strain rate model (Eq. (6)) yields reasonable agreement with experimental creep data (Fig. 9). The stress exponent, n, predicted by the model in the power law regime (r/G 6 2 103) was 4–5, within the acceptable range for five power law creep [44]. The predicted stress exponent shows good agreement with experimental observations in the power law breakdown regime, as well as the result of the hyperbolic sine function in Eq. (6). The “breakdown” at high stress merely states a deviance from the empirical power law, and is not necessarily an indication that a different deformation mechanism is active. In the future, emphasis should be placed on the effects of chemistry on strain rates. Also, a study of the substructural development and evolution of model parameters as a function of strain levels could provide many insights into the jogged-screw mechanism and how model parameters evolve during deformation. Finally, a more thorough understanding of the boundaries of the stress–temperature regime where the jogged-screw model is active would improve understanding of the mechanism and aid deployment of the MJS model to describe creep behavior of zirconium alloys. References [1] Dunlop J, Bre´chet Y, Legras L, Estrin Y. Mater Sci Eng A 2007;443: 77–86. [2] Baum E, Knox H, Miller T. Nuclides and isotopes: chart of the nuclides. 16th ed. Knolls Atomic Power Laboratory; 2002. [3] Franklin D, Lucas G, Bement A. Creep of zirconium alloys in nuclear reactors. American Society of Testing and Materials; 1983. [4] Charit I, Murty K. J Nucl Mater 2008;374:354–63. [5] Sinha R, Sinha S, Madhusoodanan K. J Nucl Mater 2008;383:14–21. [6] Franklin D, Adamson R. J Nucl Mater 1988;159:12–21. [7] US Department of Energy. DOE fundamentals handbook: material science. Technical Report DOE-HDBK-1017/2-93 1993. [8] Soniak A, L’Hullier N, Mardon JP, Rebeyrolle V, Bouffioux P, Bernaudat C. ASTM Spec Tech Publ 2002;1423:837–62. [9] Murty K. J Miner Metals Mater Soc 2000;52:34–8. [10] Hayes T, Rosen R, Kassner M. J Nucl Mater 2006;353:109–18. [11] Hayward P, George I. J Nucl Mater 1999;273:294–301. [12] Rosinger H, Bera P, Clendening W. The steady-state creep of Zircaloy-4 fuel cladding from 940 to 1873 K. Technical Report AECL-6193, Atomic Energy of Canada Engineering Company, Whiteshell Nuclear Research Establishment; 1978.
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