Double Steady-State Creep Behavior in Solid-Solute-Strengthened Zircaloy-4

Abstract

Long-term creep tests on the solid-solute-strengthened zirconium alloy, i.e., Zircaloy-4, revealed the presence of double steady-state creep behavior at 623–673 K. The first steady state manifested as a faster creep strain rate at a strain (ε) of ∼0.05, and the second appeared with a slower creep strain rate at ε > 0.1. Several electron microscopy revealed a change in the dislocation structure during creep from the motion of individual dislocations to the generation of cell structure. Along with this change, the stress exponent differed in each steady state, increasing from 6.7 in the first steady state to 10 in the second steady state. The temperature dependency also changed because the apparent activation energies were 201 and 298 kJ/mol for the first and second steady states, respectively. These results show that pipe diffusion and self-diffusion are the rate-controlling processes in the first and second steady state, respectively. Because such changes in the deformation mechanism during one creep curve are highly unusual, the double steady state is a characteristic feature of high-temperature deformation in Zircaloy-4. For engineering, observation of the second steady state is necessary in estimating the time-to-rupture by means of the Monkman–Grant relationship because creep strain rates at the first steady state deviate from the relationship.

Change of dislocation morphology leads to “double-steady state” in Zr alloy.

1. Introduction

Hexagonal close-packed (HCP) metals and alloys, such as zirconium (Zr) alloys, have been used widely in the nuclear-power industry because these materials have a small absorption cross-section for thermal neutrons and display good mechanical properties at operating temperatures of less than about 673 K. For example, zirconium alloys such as Zircaloy-2 and Zircaloy-4 are used as fuel cladding materials filled with coolant water at high temperatures. Therefore, ensuring the safety of components for secure operation at around 673 K is of crucial importance. In addition, geological disposal of redundant alloys has been considered because the materials are categorized as radioactive waste.^1,2) In such processes, the temperatures of the materials can reach approximately 573 K for ten years, decreasing gradually to 423 K after 1000 years.³⁾ To ensure the long-time reliability of nuclear components, numerous creep tests have been applied to reveal creep mechanisms.^4–7) A deformation mechanism map has been developed by using the modified Dorn equation:⁸⁾   

\begin{equation} \dot{\varepsilon} = A(Gb/kT)(\sigma/E)^{n}(b/d)^{p}D_{0}\exp(-Q/RT), \end{equation} (1)

where $\dot{\varepsilon }$ is the strain rate, A is a dimensionless constant, G is the shear modulus, b is the Burger vector, k is the Boltzmann constant, T is the temperature, σ is the stress, E is Young’s modulus, n is a stress exponent, d is the grain size, p is a grain-size exponent, D₀ is the frequency factor, Q is the activation energy, and R is the gas constant.

Hayes et al. reported that five-power-law creep is dominant at around 673 K with n ≈ 6.4 and Q ≈ 270 kJ/mol.⁶⁾ Because of these parameters, they concluded that creep is rate-controlled by dislocation climb as a recovery process, because the value of Q is similar to that for self-diffusion.⁶⁾ Nam et al. also investigated creep in Zircaloy-4 and also concluded that the dislocation climb is the rate-controlling process at high stresses and that Coble creep becomes dominant when the stress decreases.⁷⁾ Subsequently, the Mills group proposed a modified jogged-screw model using similar creep parameters for Zircaloy-4.^9–11) In this model, a steady state is achieved by dragging a tall jog or dipole; self-diffusion is responsible for rate control of the creep. For this reason, creep is dominated by the motion of individual dislocations with few interactions between dislocations. However, this conclusion was based on data from interrupted creep tests with a small creep strain of a few percent.^6,7,9–11) It is possible, therefore, that conventional creep analyses do not capture the steady state.

In the present study, we performed creep tests until rupture to observe the steady state in Zircloy-4 to permit a detailed discussion of its mechanical response at high temperatures. The tests revealed that creep behavior is much more complicated at around 673 K than is suggested by the conventional results.^6,7,9–11) One creep curve showed the existence of two steady states, the second of which had a different stress exponent. Because of this, it is necessary to reconsider the creep behavior of this alloy to ensure the safety of nuclear-power plants and to discuss this new mechanical response.

2. Experimental Procedures

A rolled sample of Zircaloy-4 was used after heat treatment at 873 K for three hours under a vacuum. After this treatment, the grain size was about 30 µm with equiaxial grains.¹²⁾ The chemical composition of the sample is listed in Table 1. Plate-type creep specimens with a 10 mm gauge length, 3.5 mm width, and 0.9 mm thickness were fabricated by electrical-discharge machining. The stress axis was perpendicular to the rolling direction. Creep tests were performed by using dead-load creep frames. The range of testing temperatures was 573–723 K, as monitored by using an R-type thermocouple attached to the specimen. Displacement was measured by using linear gauges during the tests.

Table 1 Chemical composition of the sample used in this study. Unit is mass%.

Scanning electron microscopy (SEM) in conjunction with electron back-scattered diffraction (EBSD) analyses before and after a creep test conducted at 673 K/55 MPa, were used to elucidate the strain distribution in grains, which showed a clear double steady state. The creep test was interrupted at strains of 0.05 (the first steady state), 0.075 (transition region), and 0.15 (the second steady state). The EBSD samples were polished mechanically and chemically before the tests. Final treatment was performed by electropolishing at room temperature with a solution of 10% perchloric acid and 90% ethanol. The polishing voltage range was 14–20 V.

After the creep test at 673 K/55 MPa, transmission electron microscopy (TEM) was performed at 200 kV to reveal the dislocation structure at the first steady state (ε = 0.05) and the second steady state (ε = 0.15). The samples for TEM were prepared by mechanical grinding followed by twin-jet electropolishing with a solution of 6% perchloric acid, 34% 2-butoxyethanol, and 60% methanol at a temperature of 243 K and a voltage of 30 V. Because the sample preparation method is capable of generating hydrides during the polishing phase, contrasts such as small precipitates were possibly artifacts and could be ignored during the observations.

3. Experimental Results

Figure 1(a) shows the relationships between $\dot{\varepsilon }$ and the time (t) at temperatures of 573 K (open symbols) and 623 K (closed symbols); Fig. 1(b) and 1(c) show the corresponding relationships at 673 and 723 K, respectively. The results of the tests at 573 K and 723 K (open symbols) showed a typical relationship in which the creep strain rate decreases concomitantly with increasing time and, subsequently, when the accelerated creep starts, showed a rapid increase until rupture. However, a characteristic creep behavior was observed at 623 and 673 K. At 623 K and 673 K/40 MPa, the first and second steady states were observed at h ∼ 10³ hours and h ∼ 10⁴ hours, respectively. Although the test is ongoing, difference in creep strain rates at the two steady states is visible. At 673 K/30 MPa, the difference became small because diffusional creep might be activated as shown in Fig. 2 later. In addition, it might be difficult to distinguish the difference between the two creep behaviors because the condition is close to a border where the first steady state does not appear. However, we considered that the first and second steady states were observed at h ∼ 10³ hours and h ∼ 10⁴ hours, respectively. According to Fig. 1, the creep strain rate became stable during the early stage of the creep (5–10% of the time to rupture) then decreased after the stable stage, reaching another steady state at about 50% of the time to rupture. This new behavior is termed “double steady-state creep behavior” in this study.

Fig. 1

Relationship between the creep strain rate and time at (a) 573 K (open symbols) and 623 K (closed circles), (b) 673 K, and (c) 723 K. The open symbols do not show a double steady-state creep behavior, whereas closed symbols do so. A clear first steady state was observed at 623 K and 673 K, but not at 573 K or 723 K.

Fig. 2

Double-logarithmic plot of the creep strain rate and modulus-compensated applied stress at various temperatures. Closed symbols are data for the first steady-state creep rate. Open symbols are data at the second steady-state creep rate. The n value for the first steady state is 6.7; that of the second steady state is 10. The dashed lines at 573 K and 673 K correspond to a power-law-breakdown region. The region with n = 1.2 was also observed at low stresses. Arrows indicate ongoing tests. Some data for 673 K were taken from an earlier study.¹²⁾

The results also showed that the behavior was affected by the test temperature. Thus, the upper limit for the behavior appeared to be less than 723 K because, at this temperature, the first steady state became unclear, as shown in Fig. 1(c). The lower limit for the double steady-state creep behavior was between 573 and 623 K, because the behavior was clearly observed at 623 K and 60 MPa but was not visible at 573 K, as shown in Fig. 1(a).

Because this double steady-state creep behavior is unique and had not been previously reported, the creep parameters, the stress exponent (n), and the apparent activation energy (Q), were determined by using a double-logarithmic plot of the modulus-compensated applied stress (σ/E) and creep strain rate (Fig. 2) and an Arrhenius plot (Fig. 3), respectively. The Young’s modulus was calculated by using the following equation:¹³⁾   

\begin{equation} E\ [\text{MPa}]= 114000 - 59.9T\ [\text{K}]. \end{equation} (2)

Fig. 3

Arrhenius plot for Zircaloy-4 at 623–673 K. The apparent activation energy is 201 kJ/mol at the first steady state (closed circles) and 298 kJ/mol at the second steady state (open circles). Data for 623 K are extrapolated by the lines with n = 6.7 and 10 in Fig. 2.

Figure 2 is a double-logarithmic plot of the creep strain rate and modulus-compensated applied stress at various temperatures. The closed symbols represent the creep strain rate of the first steady state and the open symbols represent the creep strain rate of the second steady state. This separation of the n values is distinct from that of other materials. Furthermore, a region with n = 1.2 might be appeared at 673 K at low stresses. Figure 3 shows the Q values for the regions with n values of 6.7 and 10. Although few data were available for 623 K because of the limitations of the batch, the Q values were evaluated to be 201 and 298 kJ/mol for the regions with n = 6.7 and 10, respectively.

The region with n = 1.2 might be categorized as displaying Coble creep because the parameter is similar to that previously reported.⁶⁾ The Q value of 201 kJ/mol at the first steady state is smaller than that reported in earlier studies in which self-diffusion was activated.^4,6,9–11) Because the value is only 70% of that for self-diffusion, pipe diffusion might be activated at the first steady state. However, self-diffusion has the potential to be activated under these conditions because a larger Q value is observed in the same temperature ranges, as shown in Fig. 3.

The modified jogged-screw model¹¹⁾ might apply for the first steady state; however, the actual thermal-activation process will be discussed later. On the other hand, the newly observed second steady state showed a higher n value at a high strain of, for example, 0.15 at 673 K and 55 MPa, where n = 10 and Q = 298 kJ/mol. The change in the n value reflects a transition in the deformation mode, as shown in Fig. 4. Kernel average misorientation (KAM) maps at each strain level are presented in Fig. 4(b)–(e). Because the KAM map can show the strain distribution in grains, the deformation characteristics of the alloy can be confirmed. Marked strain accumulation was not observed until ε = 0.05, i.e., the first steady state, as shown in Fig. 4(b) and 4(c). On increasing the strain, a significant gradation began to occur inside grains. This indicates that interaction between dislocations became stronger, because this type of gradation has been previously reported: the KAM map corresponds well with the formation of a cell structure.¹⁴⁾

Fig. 4

(a) Relationship between the creep strain rate and creep strain at 673 K/55 MPa. Alphabetic characters show the conditions of the EBSD observations. The KAM maps shows creep strains of (b) 0, (c) 0.05, (d) 0.075, and (e) 0.15. No marked strain accumulation was observed at low strains, whereas a cell-like structure was formed at high strains, as shown by arrows.

TEM observations were performed to confirm the dislocation structures in the first and the second steady states. Bright-field images recorded for the two states are shown in Fig. 5. In the first steady state, no cell structure was observed, although dislocations were distributed in the grains, as shown in Fig. 5(a). Figure 5(b) is a high-magnification image of the region at the center of Fig. 5(a), where straight dislocation arrays¹⁵⁾ were observed. Moon et al.¹⁰⁾ and Morrow et al.^11,16) found a homogeneous distribution of dislocations with no cell structure, even at ε < 0.09. These results mean that the first steady state results from individual dislocation motions without interactions between dislocations. Therefore, the formation of a cell structure was suppressed, which is a similar result to that shown in the KAM map (Fig. 4(c)).

Fig. 5

Bright-field images taken after the creep test at 673 K/55 MPa. Dislocations were linearly aligned with few tangles at ε = 0.05 (a), (b), whereas they were tangled and formed a cellular structure at ε = 0.15 (c), (d). (b) and (d) are high-magnification images corresponding to (a) and (c), respectively.

At the second steady state, an obvious cell structure formed in the grains, as shown in Fig. 5(c) and 5(d). As is the case of the EBSD analyses, the TEM observations revealed an obvious transition of the dislocation structure during one creep curve in Zircaloy-4. The microstructure shows that the second steady state arises through accommodation of interactions between dislocations by self-diffusion, because the Q value was 298 kJ/mol, which differed by only 10% from that of self-diffusion, i.e., 270 kJ/mol.^4,6,9–11) Therefore, a change in the dislocation structures resulted in the unique creep behavior, i.e., the double steady state, in Zircaloy-4 at 623–673 K.

4. Discussion

The double steady state during creep of Zircaloy-4 at 623–673 K resulted from a change in the dislocation structure. The stress exponent increased from 6.7 to 10 with the change in the dislocation structure. The value of Q also increased from 201 kJ/mol to 298 kJ/mol in one creep curve. The modified jog-screw model^9–11) supports a value of 270 kJ/mol in the first steady state. However, the activation energy observed in the first steady state in the present study was lower (201 kJ/mol) than that for self-diffusion, as shown in Fig. 3. Moreover, Barrett and Nix have pointed out the importance of pipe diffusion under conditions where the jog velocity is determined by diffusion in this model.¹⁷⁾ They also mention that the mechanism becomes apparent at ∼0.5 of the melting temperature for metals such as Al, Ag, and Cu that show no phase transformation. For Zr, because a phase transformation occurs at 1135 K, pipe diffusion might be activated at ∼600 K. Therefore, the Q value from Morrow et al.¹¹⁾ was reevaluated in Fig. 6 because they did not report their value and they used a theoretical value of 270 kJ/mol in eq. (3):   

\begin{equation} \dot{\varepsilon} = \left(\frac{2\sqrt{2}\pi D_{s}}{\delta(h)} \right) \left(\frac{\sigma}{2\alpha Gb} \right)^{2} \left[\mathit{sinh}\left(\frac{\sigma \Omega l}{2hkT}\right) \right], \end{equation} (3)

where D_s is the coefficient of self-diffusion, δ(h) is a logarithmic function of the jog height (h), l is the jog spacing, and Ω is the atomic volume.¹¹⁾ However, the figure clearly showed a Q value of 200 kJ/mol (≈ 0.7Q_s, where Q_s is the activation energy of self-diffusion) at 644–700 K when the creep data obtained by Morrow et al.¹¹⁾ was used, which corresponds to that in the present study. This means that the first steady state is rate-controlled by pipe diffusion and not by self-diffusion. Therefore, the coefficient of self-diffusion is replaced by that for pipe diffusion (D_p). For creep analysis, the coefficient of diffusion has been described as the effective diffusion coefficient.   

\begin{equation} D_{\text{eff}} = D_{\text{s}} + (\sigma/G)^{2} D_{\text{p}}. \end{equation} (4)

By substituting eq. (4) into eq. (3), the creep strain rate can be explained in terms of eq. (5):   

\begin{equation} \dot{\varepsilon} = \left(\frac{\sqrt{2}\pi D_{p}}{2 \alpha^{2}b^{2}\delta (h)}\right)\left(\frac{\sigma}{G}\right)^{4} \left[\mathit{sinh} \left(\frac{\sigma \Omega l}{2hkT}\right) \right]. \end{equation} (5)

Then, because σΩl ≪ 2hkT, eq. (5) can be rewritten as follows:   

\begin{equation} \dot{\varepsilon} = A\left(\frac{\text{Gb}}{kT}\right)\left(\frac{\sigma}{G}\right)^{5}D_{p}, \end{equation} (6)

where A is a constant that includes the constants of the material. According to this equation, the modified jog-screw model with pipe diffusion theoretically includes a stress exponent from eq. (6) for pure metals. Although the theoretical value is slightly smaller than that shown in Fig. 2, previous reports give an n value of 6.4 for the modified jog-screw model.^9–11) Therefore, the stress exponent of 6.7 is acceptable for recognizing that the first steady state belongs to a model with pipe diffusion at 623–673 K. The creep tests reported by Morrow et al. were performed at 533 K until ε = 0.02, where creep might not have reached the steady-state region. Because their data at this temperature might not be valid, these were ignored in our analysis.

Fig. 6

Comparison of the Q values obtained in the present study with those obtained by Morrow et al.¹¹⁾ At 623–700 K, the Q value is evaluated to be about 200 kJ/mol, which is only ∼0.7 that for self-diffusion. The data at 533 K are ignored because the creep tests did not reach steady states.

Next, we will discuss the increase of stress exponent in the second steady state, i.e., n = 10. The observation results suggested that the second steady state is brought about by the formation of cell structure accommodated by self-diffusion. In this case, the internal stress increases because of tangling dislocations in grains, meaning that work hardening occurs in the second steady state. The features correspond to the mechanism of five-power-law creep reported by Hayes et al.:⁶⁾   

\begin{equation} \dot{\varepsilon} = A\left(\frac{\text{Gb}}{kT}\right)\left(\frac{\sigma}{G}\right)^{5}D_{s}. \end{equation} (7)

Although the form of the equation is similar to that for the first steady state, the diffusion coefficient is that of self-diffusion in the second steady state. However, there is a remarkable difference between the stress exponents reported by Hayes et al.⁶⁾ and those of the present study. The previous works^6,9–11) included a number of items of creep data where a steady state was not attained because of the very small creep strain of a few percent, suggesting that only the first steady state at around 673 K was observed in the previous studies.^6,9–11) In addition, the stress exponents of alloys become larger than those of metals.¹⁸⁾ Note that an n value of 10 can be acceptable for a five-power-law creep.

Although the double steady state creep behavior results from the change of deformation mechanism during creep, the reason why the behavior is observed in the alloy is remained. It is considered that the HCP structure affects the unique behavior. One of the authors had been reported metals and alloys with the structure show straight dislocation arrays because of the low symmetricity of crystalline structure.^15,19) In this case, dislocations are relatively hard to be tangled with lack of number of activated slip systems unlike in metals with cubic structure. In addition, solid solution promotes planar slip as reported in Ti–6 mass% Al.²⁰⁾ Therefore, solid solution makes dislocations interaction weak in HCP alloys, suppressing the generation of cell structure. It might need an incubation time, i.e., accumulation of strain, to make the dislocation structure at 623–673 K, leading to the double steady state creep behavior in Zircaloy-4.

Finally, the Monkman–Grant relationship²¹⁾ is shown in Fig. 7 to confirm which steady-state creep rate can be used to estimate the time-to-rupture of the material for the purpose of safety evaluation. Open symbols correspond to the minimum creep rate ($\dot{\varepsilon }_{\text{min}}$) containing the second steady-state creep rate, whereas the closed symbols denote the first steady-state creep rate. The figure clearly showed that the former obeys a relationship with a parameter p of −1.1 and K of 0.80:   

\begin{equation} \dot{\varepsilon}_{\text{min}} = K t_{\text{r}}{}^{p}, \end{equation} (8)

where K is constant for the material. On the other hand, the first steady-state creep rate deviated from the relationship shown in Fig. 7. Therefore, the use of the second steady-state creep rate with n = 10 is necessary in estimating the long-term creep lifetime for Zircaloy-4.

Fig. 7

Monkman–Grant relationship for Zircaloy-4. The closed symbols are data at the first steady-state creep rate, whereas the open symbols are data at the second steady-state creep rate and the minimum creep rate.

5. Conclusions

In this study, we performed long-term creep tests on Zircaloy-4 at 573–723 K and we used several microscopy techniques to reveal that the characteristic creep behavior involves a double steady state. This is the result of a change in the deformation mode from individual dislocation motion, based on the modified jogged-screw model with pipe diffusion, to an accommodation of the interaction between dislocations that is rate-controlled by self-diffusion. With this change, the stress exponent changes from 6.7 for the first steady state to 10 for the second steady state. The apparent activation energy changes from 201 kJ/mol in the first steady to 298 kJ/mol in the second steady state. In addition, the second steady-state creep rate is important when the Monkman–Grant relationship is used to estimate the long-term creep lifetime for a solid-solution strengthened Zr alloy.

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