An assessment of delayed hydride cracking in zirconium alloy cladding tubes under stress transients

Threshold stress intensity factor K _IH

DHC requires the diffusion of hydrogen to a highly stressed region, the precipitation of hydrides, and then the fracture of hydrides within the fracture process zone.5,13 The fracture toughness of zirconium hydrides with the face centred cubic (fcc) structure ranges from 1 to 3 MPa m^1/2 (0·91–2·73 ksi in.^1/2) from 25 to 300°C (77–572°F),63 as shown in Fig. 2. In comparison, Fig. 2 also shows that the K _IH value for Zr–2·5Nb ranges from 4·3 to 10 MPa m^1/2 (3·91–9·12 ksi in.^1/2) with a 95% confidence limit for a single observation and a mean of 8·2 MPa m^1/2 (7·46 ksi in^1/2) for all tests in the database compiled by Shi and Puls.13 The K _IH value for Zircaloy-2 tubes38,40,44 ranges from 5·2 to 14·2 MPa m^1/2 (4·73–8·29 ksi in.^1/2). Recently, Huang and Ho42 reported that the K _IH values for the Zircaloy-4 plates with 100–168 wppm hydrogen are about 20–21 MPa m^1/2 at 200°C (392°F) and 300°C (572°F). These K _IH values are considerably higher than the K _IH value of 5 MPa m^1/2 reported by Grigoriev and Jakobsson46 for Zircaloy-4 tubes. The reasons for the high K _IH values observed in the Zircaloy-4 studies by Huang and Ho42 are not known; possible reasons include low hydrogen contents, texture effect and incomplete hydride coverage and ligament toughening by Zr grain ligaments at the crack wake. All the K _IH measurements for Zr–2·5Nb,9,12 Zircaloy-237,38,40,44 ^– 46 and Zircaloy-442,46 have been obtained using fracture mechanics specimens with a large crack. For large crack fracture mechanics specimens, the K _IH for zirconium alloys is always larger than the fracture toughness of the hydrides, indicating that the limiting process associated with the DHC threshold is more involved than just the fracture of hydrides near the crack tip or notch tip region. To the knowledge of the author, K _IH threshold values have not been reported for small cracks in zirconium alloys.

OPEN IN VIEWER

Figure 2.

The DHC growth threshold K _IH of Zr–2·5Nb compared to the critical stress intensity factor K _IC for fracture of bulk hydrides. Experimental data are from Shi and Puls13 and Simpson and Cann63 {1 MPa m^1/2 = 0·91 ksi in.^1/2; T (°F) = 1·8[T (K)–273]+32}

Shi and Puls13 elucidated the theoretical value of K _IH for zirconium alloys by considering fracture initiation of hydrides formed ahead of a sharp crack tip and a shallow notch tip14 based on a local critical stress for hydride fracture. The local stress at the crack or notch tip was modelled by summing the contributions from the elastic stress field of a sharp crack or a notch and the compressive stress field induced by the volume change incurred during the precipitation of hydrides in a zirconium matrix with hydrogen in solid solution. The volume increases associated with the formation of the metastable face centred tetragonal hydride and the stable fcc hydride are 12% and 17% respectively.4 Figure 3a illustrates the effects of hydride formation on the near tip stress field of a sharp crack after hydride formation in the crack tip process zone. The von Mises effective stress σ _eff computed from the crack tip elastic stress field, increases with decreasing distance from the crack tip, reaching a value equal to the yield stress σ _y of the matrix in the crack tip plastic zone. Within the crack tip plastic zone, the compressive hydride transformation stress σ _hydride is superimposed and summed with σ _eff to give the local stress σ _local normal to the crack tip hydride. The calculated local stress distributions for a crack tip with a hydride are shown for various K levels in Fig. 3b . The calculation by Shi and Puls13 showed that the local stress near the crack tip was compressive at low K levels [K<2 MPa m^1/2 (1·82 ksi in.^1/2)], but became tensile when K>4 MPa m^1/2 (3·64 ksi in.^1/2). Shi and Puls13 postulated that hydride fracture occurs only when the local stress in the hydride exceeds a critical value. The critical stress criterion for hydride fracture led to an expression for K _IH, which is given by Shi and Puls13 (2) with (3) where E is the elastic modulus, ν is Poisson’s ratio, σ _y is the yield stress of the Zr alloy, t is hydride thickness, is the fracture strength of the hydride and is the transformation strain component normal to the crack in a Mode I crack configuration. Using equation (2) and the pertinent material properties for Zr–2·5Nb and the hydrides, Shi and Puls13 computed the K _IH values for Zr–2·5Nb as a function of temperature. Their predictions of the K _IH values for both irradiated and unirradiated Zr–2·5Nb are compared against experimental data in Fig. 4. The predicted K _IH values are lower than those observed experimentally. Shi and Puls13 attributed the discrepancies between model prediction and experimental data partly to uncertainties of the values of material parameters used in the computation and partly to less than 100% coverage of hydrides in the near tip process zone so that the fracture toughness of the zirconium matrix also contributed to the observed K _IH value. In particular, the hydride fracture strength was not available and was taken to be 7·357×10⁻³ E, which corresponds to fracture strengths of 694 MPa (100·7 ksi) at 25°C (77°F) and 579 MPa (84·0 ksi) at 300°C (572°F). Subsequent work by Shi and Puls64 determined that the fracture strength of hydrides in Zr–2·5Nb was about 648 MPa (94·0 ksi) at 25°C (77°F) and 623 MPa (90·4 ksi) at 300°C (572°F). Thus, the author concludes that part of the discrepancies could have arisen from uncertainties in the material properties used (fracture strength), as indicated by Shi and Puls.13 However, the fracture toughness of the zirconium matrix is considerably higher than that of the hydrides. In the author’s opinion, incomplete hydride coverage, which results in toughening by matrix ligaments and is discussed later (the section on ‘Ligament toughening by zirconium grains’), is the more important explanation for the discrepancy between theory and experiment.

OPEN IN VIEWER

Figure 3.

a Schematic showing crack tip hydride and stresses. b Profiles of σ_local = σ_eff+σ_hydride at crack tip for a hydride 2 μm (7·87×10⁻² mil) thickness, 10 μm (0·39 mil) long and 10 μm (0·39 mil) wide under different K values. Results are from Shi and Puls13 (1 MPa = 0·145 ksi)

OPEN IN VIEWER

Figure 4.

Comparison of theoretical K _IH with experimental data. Data are from Shi and Puls13 (1 MPa m^1/2 = 0·91 ksi in.^1/2)

The effects of a compressive transformation stress field on the K _IH threshold can be modelled based on transformation toughening developed to explain the toughening resulting from the transformation of zirconia (ZrO₂) from the tetragonal structure to the monoclinic structure near the crack tip region.65 In both zirconia and zirconium hydrides, the stress induced phase transformation results in a volume increase and a compressive stress field in the matrix that shields the crack tip from the applied tensile stress. This is illustrated in Fig. 5, which shows a Mode I crack subjected to a remotely applied stress (resulting in K) and a hydride formed at the crack tip. The local stress intensity at the crack tip K _tip is given by65 (4) where K _S is the reduction in the stress intensity factor of the main crack resulting from shielding induced by phase transformation near the crack tip. According to the analysis by McMeeking and Evans65 (5) where ϵ ^T is the transformation strain, E is the elastic modulus, ν is the Poisson’s ratio, V _f is the volume fraction of the transformed phase (e.g. hydrides) and w is the width of the transformation zone. At the onset of hydride fracture, K _tip equals the fracture toughness of the hydride , and K = K _IH, leading to (6) and (7) after equation (5) is substituted into equation (6).

OPEN IN VIEWER

Figure 5.

Crack tip shielding of a crack in an elastic field by the compressive transformation stresses accompanied by hydride formation at a crack tip

Pan et al. 66 utilised equations (5) and (7) and the pertinent material parameters for fcc zirconium hydrides, which are summarised in Table 1, to predict the K _S and K _IH values for Zr–2·5Nb and Zircaloy-2 as a function of temperature. As shown in Fig. 6, the fracture toughness value of zirconium hydrides, taken from Simpson and Cann,63 ranges from 1 to 3 MPa m^1/2 (0·91–2·73 ksi in.^1/2) in the temperature range of 300–600 K (80·6–620·6°F). In contrast, the computed value of the shielding stress intensity factor K _S is −4·34 MPa m^1/2 (−3·95 ksi in.^1/2) at 300 K (80·6°F) and increases linearly with increasing temperature to −2·84 MPa m^1/2 (−2·58 ksi in.^1/2) at 600 K (620·6°F). The calculated K _IH value, which is the sum of K _S and , has a value of 5·36 MPa m^1/2 (4·88 ksi in.^1/2) at 300 K (80·6°F) and increases linearly with temperature to 6·04 MPa m^1/2 (5·50 ksi in.^1/2) at 600 K (620·6°F). The calculated K _IH values are in agreement with the lower bounds of the experimental data for Zr–2·5Nb13 and Zircaloy-2.38,44,50 A substantial number of the experimental K _IH measurements, however, exceed the lower bound and the model calculations based on the crack tip shielding and phase transformation toughening theories. The results indicate that crack tip shielding by hydride formation at the crack tip is significant and is partly responsible for the experimental observation that the K _IH is higher than .

OPEN IN VIEWER

Figure 6.

Theoretical values of K _IH based on the fracture toughness K _IC of hydrides (data are from Simpson and Cann63) and crack tip shielding of near tip stress intensity factor K _S compared against experimental data of K _IH from the literature (data are from Coleman,39 Shi and Puls,13 Huang and Mills,38 Pettersson et al. 44 and Kim et al. 40) {1 MPa m^1/2 = 0·91 ksi in.^1/2; T (°F) = 1·8 [T (K)–273]+32}

Because the K _IH threshold originates from crack tip shielding, the value of K _IH for DHC in zirconium based alloys is expected to vary with the crack length and exhibits different K _IH values for small and large cracks. In general, the K _IH for small cracks is expected to have a lower value than that of large cracks because of a longer crack wake with greater crack-tip shielding. The growth threshold for small cracks and large cracks are related according to the expression given by68 (8) where a _o is the small crack parameter. The a _o parameter is defined based on the dependence of the hydride fracture stress on crack length. For fracture of hydrides by large cracks, linear elastic fracture mechanics is applicable, and the fracture stress increases with decreasing crack length according to a −1/2 power, as shown by the dashed lines to K _IH = 5·38 MPa m^1/2 (4·89 ksi in.^1/2) and 12 MPa m^1/2 (10·9 ksi in.^1/2) in Fig. 7. This range of K _IH values encompasses the lower and upper bound of the growth thresholds observed in Zr–2·5Nb. For hydrides with small or no cracks, fracture occurs at a constant stress value [≈648 MPa (91·0 ksi)] that is relatively independent of the crack length. The transition from a stress controlled process to a K controlled process can be considered to occur at a = a _o, as shown in Fig. 7. The value of a _o can be computed on the basis that the fracture stress corresponds to the hydride fracture strength at a = a _o, leading to (9) where F is the crack geometry correction factor with F = 1·12 for an edge crack with a shallow crack depth (e.g. a/W<0·05). For fcc hydrides, a _o has a value of about 17 and 85 μm (0·67 and 3·35 mil) for K _IH = 5·28 and 12 MPa m^1/2 (4·8 and 10·9 ksi in.^1/2) respectively, when the pertinent fracture toughness and fracture strength values are substituted into equation (9). Figure 7 shows the corresponding threshold stresses as a function of crack length (denoted as solid lines), so for a given crack length, small cracks can propagate at a stress lower than those of large cracks.

OPEN IN VIEWER

Figure 7.

Comparisons of large crack and small crack threshold stresses for delayed hydride cracking in Zr–2·5Nb for K _IH values of 5·38 MPa m^1/2 (4·9 ksi in.^1/2) and 12 MPa m^1/2(10·9 ksi in.^1/2). The threshold stresses for DHC approach the hydride fracture stress, which is 648 MPa (94 ksi) (data are from Shi and Puls17), at various crack lengths, depending on the K _IH value. DHC growth occurs when the cladding stress exceeds the threshold stress at a given crack length. (1 MPa = 0·145 ksi; 1 MPa m^1/2 = 0·91 ksi in.^1/2; 1 μm = 0·0394 mil)

Using the appropriate a _o for various K _IH values at 300 and 600 K (80·6 and 620·6°F), equation (8) was used to calculate the K _IH values for small cracks as a function of crack length, and the results are presented in Fig. 8. The results show that the K _IH value for small cracks in zirconium alloys approaches a very small value at diminishing crack size. On the other hand, hydride fracture occurs only when the hydride length exceeds a critical length. Experimental data indicated that for Zr–2·5Nb, the critical hydride length is on the order of 10–20 μm (0·394–0·787 mil), and the hydride width is about 2 μm (7·87×10⁻² mil) thick.69,70 These hydrides are formed ahead of a crack tip loaded at a stress intensity factor just below the K _IH threshold.70 The crack front is not uniformly hydrided but occurs in a discontinuous, intermittent process or a narrow band of planes parallel to the crack growth plane.69 The smallest crack length for a in equation (8) is therefore about 10–20 μm (0·39–0·79 mil), leading to a lower bound K _IH value of 3·26 and 4 MPa m^1/2 (2·97 and 3·64 ksi in^1/2) for critical crack lengths of 10 and 20 μm (0·39–0·79 mil) respectively. The K _IH value increases with increasing crack length and asymptotically approaches the large crack K _IH value, which is 5·36 MPa m^1/2 (4·88 ksi in.^1/2) at 300 K (80·6°F), but is 6·03 MPa m^1/2 (5·48 ksi in.^1/2) at 600 K (620·6°F).

OPEN IN VIEWER

Figure 8.

Dependence of K _IH on crack length for small cracks. For a crack depth of 10–20 μm (0·39–0·99 mil), K _IH≈3·3–4 MPa m^1/2 (3–3·64 ksi in.^1/2), but its value increases rapidly with the crack length to saturate at the large crack threshold (1 MPa m^1/2 = 0·92 ksi in.^1/2; 1 μm = 0·0394 mil; 300 K = 80·6°F; 600 K = 620·6°F)

It appears that the critical hydride length significantly affects the DHC threshold for small cracks. The existence of a critical hydride length, which was assessed by Shi and Puls,41 has been attributed to a transition of the local stress in the hydride from compressive to tensile as the hydride lengthens. This relation is illustrated in Fig. 9, which shows the local stress at the tip of a crack with a hydride of length L subjected to a stress intensity factor of 10 MPa m^1/2 (9·1 ksi in.^1/2). The local stress is compressive when the hydride length is 2 μm (7·87×10⁻² mil). The local stress becomes slightly tensile when the hydride lengthens to 3·3 μm (0·13 mil), and the local stress increases to above 1000 MPa (145 ksi) and exceeds the fracture strength [648 MPa (94 ksi)] of the hydride when L increases to 10 μm (0·394 mil). Furthermore, the local stress increases with increasing K values for a given hydride length. The results from Shi and Puls41 in Fig. 3b show that at L = 10 μm (0·394 mil), the local stress is compressive for K = 2 MPa m^1/2 (1·82 ksi in.^1/2), and a local tensile stress occurs at the crack tip only when K>4 MPa m^1/2 (3·64 ksi in.^1/2). Experimental evidence on Zr–2·5Nb69 ^– 71 and Zircaloy-237,38,44,45 supporting the concept of a critical hydride length includes (1) direct observations of the presence of intact hydrides at the crack tip stress intensity factor at or just below K _IH in Zr–2·5Nb and (2) the observations of discrete striation spacings and hydride spacings associated with DHC in zirconium alloys.46,47 For Zr–2·5Nb, the critical hydride length is about 10–20 μm (0·394–0·787 mil),70 which was measured directly ahead of a crack tip loaded just below K _IH. In many instances, a series of hydride platelets with a length on the order of 60–100 μm (2·36–3·94 mil) precipitated ahead of the crack tip during DHC in Zr–2·5Nb69 and Zircaloy-2.22,37,38,44,45 In both cases, the length of individual hydrides was about 10–15 μm (0·394–0·59 mil).22,38 These experimental observations suggested that a critical hydride length of 10–20 μm (0·394–0·79 mil) may apply to both Zr–2·5Nb and Zircaloy-2. In addition, the critical hydride length [10–20 μm (0·394–0·787 mil)] is essentially identical to the a _o value. This implies that once a hydride crack forms, its crack length is almost within the continuum crack growth regime (a>a _o); consequently, the small crack regime is fairly limited for DHC in zirconium alloys. Thus, the large crack threshold K _IH values should be adequate for DHC in Zr–2·5Nb and Zircaloy-2. Figure 8 shows the minimum K _IH value for small cracks in Zr–2·5Nb is calculated to be about 3·26–4 MPa m^1/2 (2·97–3·64 ksi in.^1/2) at about 10–20 μm (0·394–0·787mil). The K _IH increases rapidly with crack extension and approaches that of the large crack K _IH threshold after about 100–200 μm (3·94–7·87 mil) crack growth for the lower bound values of K _IH = 5·36 MPa m^1/2 (4·88 ksi in.^1/2) at 300 K (80·6°F) and K _IH = 6 MPa m^1/2 (5·46 ksi in.^1/2) at 625 K (665·6°F). For the upper bound K _IH [ = 12 MPa m^1/2 (10·9 ksi in.^1/2)], the large crack threshold is not reached until after 1000 μm [39·4 mil] crack extension. The difference in the K _IH values for small crack and large cracks gives rise to the resistance curve (R curve) behaviour that has been reported by Yan and Eadie70 and modelled by Jernkvist.72

OPEN IN VIEWER

Figure 9.

Profiles of σ_local = σ_eff+σ_hydride at crack tip for hydrides having different lengths under K = 10 MPa m^1/2 (9·1 ksi in.^1/2). Results are from Shi and Puls13 (1 MPa = 0·145 ksi; 1 μm = 0·0394 mil)

The difference between the experimentally measured large crack K _IH threshold and the estimations of the K _IH model shown in Fig. 6 is largely caused by the presence of additional toughening mechanisms that shield the crack tip stresses from the fracturing of the near tip hydrides. The toughening mechanisms responsible for increasing the K _IH threshold in Zr based cladding are summarised in the following subsections.

Texture toughening

Zirconium hydrides are favoured to form on the {1017} crystalline planes, often referred to as hydride habit planes, which are orientated about 15° from the (0001) planes.40 This tendency results in a dependence of the K _IH threshold value on the crystallographic texture in the zirconium alloy cladding in general and on the basal pole component in particular. Figure 10 shows the K _IH threshold data38 ^– 40,73– 75 compiled by Kim et al. 40 as a function of volume fraction of the basal pole components. The lowest K _IH threshold occurs in specimens with a strong basal texture and the maximum applied stress normal to the basal plane. Under these circumstances, many grains in the near-tip regions are oriented for the formation of fcc zirconium hydrides on the {1017} planes that are aligned normal to the maximum principal stress and can be fractured easily, thereby leading to a low K _IH threshold. In contrast, non-basal grains in the near tip region form hydrides that are not oriented to the maximum principal stress and require a higher applied stress or stress intensity factor to cause hydride fracture at the crack tip, thereby enhancing the K _IH value. Thus, crystallographic orientation or texture of zirconium grains significantly influences the K _IH value for the onset of DHC.

OPEN IN VIEWER

Figure 10.

Threshold stress intensity factor K _IH of Zr–2·5Nb tube with crack growth direction as a function of the basal pole components. Data are compiled by Kim et al. 40 (1 MPa m^1/2 = 0·91 ksi in.^1/2)

Hydrogen content

The value of K _IH for DHC at a given temperature is dependent on the hydrogen content. Figure 11 shows that the K _IH value decreases with increasing hydrogen contents for Zr–2·5Nb at 250°C (482°F),41 and the theoretical curves indicate that in Shi and Puls’s analysis,41 crack tip hydride length must exceed a critical length for hydride cracking to occur, and the maximum length a hydride can attain depends on the hydrogen content in solution. As a result, the K _IH increases with decreasing hydrogen content in solution.

OPEN IN VIEWER

Figure 11.

The threshold stress intensity factor K _IH as a function of hydrogen concentration in solid solution. Experimental data and theoretical model are from Shi and Puls41 (1 MPa m^1/2 = 0·91 ksi in.^1/2; 523 K = 482°F; 2 μm = 0·0787 mil)

The dependence of the critical hydrogen content for the onset of hydride cracking in Zr–2·5Nb is shown as a function of temperature in Fig. 12.7 The critical hydrogen content for crack initiation is larger than the hydrogen solubility limit for hydride dissolution (TSSD), but is less that that for hydride precipitation (TSSP) at a given temperature. For a given hydrogen content, DHC cannot occur at a temperature higher than TSSD because no hydrides can form, even at the crack tip. DHC occurs at a small undercooling below the TSSD, and there is a large hysteresis between the TSSP and TSSD for hydrogen in zirconium. The hysteresis, which is caused by the volume expansion during TSSP, results in different DHC behaviours during heating and cooling of zirconium alloys, as well as variations with hydrogen contents.7 The terminal solid solubility of hydrogen in Zircaloy-2 was measured by McMinn et al. 86 Schofield et al. 87 recently showed that the critical temperatures for the onset and the arrest of DHC in Zircaloy-2 fall below the dissolution solvus temperature and above the precipitation solvus temperature, in accordance with experimental findings for Zr–2·5Nb.7

OPEN IN VIEWER

Figure 12.

Terminal solubility limits for zirconium: TSSD hydride dissolution limit, TSSP hydride precipitation limit and crack initiation temperature. Results are from Sagat and Puls7 {T (°F) = 1·8[T (K)–273]+32}

Irradiation effects

Irradiation appears to exert only a small effect on the K _IH value of Zr–2·5Nb tested at 130–350°C (266–662°F). Sagat et al. 43 reported a small decrease in K _IH as the fluence increased to about 1×10²⁵ n m⁻² (1·08×10²⁶ n ft⁻²) for fast neutrons with energy above 1 MeV at an irradiation temperature in the range of 250–290°C (482–554°F), but no increase upon further increase in fluence up to 8×10²⁵ n m⁻² (8·61×10²⁶ n ft⁻²), as shown in Fig. 13. The temperature corresponding to the data shown in Figure 13 was not reported by Sagat et al.,43 but K _IH testing was conducted at 130–350°C (266–662°F). The average K _IH was 7·5±1·3 MPa m^1/2 (6·82±1·18 ksi in.^1/2) for the unirradiated materials compared to 6·2±0·9 MPa m^1/2 (5·64±0·82 ksi in.^1/2) for the irradiated materials at the 95% confidence level. Similarly, the K _IH value for unirradiated Zircaloy-2 ranges from 7·5 to 13·8 MPa m^1/2 (6·87±1·18 ksi in.^1/2) for 200–300°C (392–572°F), while it ranges from 9·9 to 12·7 MPa m^1/2 (9–11·6 ksi in.^1/2) in the irradiated materials at 200–300°C (392–572°F).37,44 These results suggest that the K _IH value at high fluence (8×10²⁵ n m⁻²) (8·61×10²⁶ n ft⁻²] might be used as a lower bound K _IH for irradiated Zircaloy-2 or Zircaloy-4.

OPEN IN VIEWER

Figure 13.

Dependence of K _IH on irradiation fluence. Data are from Sagat et al. 43 (1 MPa m^1/2 = 0·91 ksi in.^1/2; 1 n m⁻² = 10·8 n ft⁻²)

Temperature dependence of DHC

Although it has a minimal effect of a factor of 2 on the K _IH threshold, temperature exerts significant influence on the growth kinetics of DHC in zirconium based alloys. Figure 14 summarises the complex relationships between crack velocity and reciprocal temperature observed in Zr–2·5Nb7 for Zircaloy-2 cladding tubes.38,46 At temperatures below 318°C (604°F) or the TSSP, DHC velocity V, is dictated by the diffusion of hydrogen to the crack tip to form hydrides and is related to the absolute temperature according to the Arrhenius relation given by12,38 (11) where Q _DHC is the activation energy for DHC, V _o is an empirical constant, T is absolute temperature and R is the universal gas constant. In general, DHC velocity decreases rapidly with decreasing temperatures because the diffusion of hydrogen to the crack tip is slow at low temperatures. At temperatures above 318°C (604°F), the DHC velocity decreases abruptly with increasing temperature for both irradiated and unirradiated materials because of increasing hydrogen solubility that leads to more hydrogen in solid solution and less hydride being formed at the crack tip. Because there is a critical hydride length for DHC to occur, DHC velocity eventually goes to zero when the crack tip hydrides are below the critical length or the temperature exceeds the TSSD, so that no hydrides can form at the crack tip. For hydride formation limited growth, the DHC velocity is given by (12) where Q _h is the activation energy for TSSP, and is a pre-exponent coefficient. As shown in Fig. 14, the temperature range and the experimental database over which DHC velocity decreases with increasing temperature are fairly small for irradiated materials to allow an accurate determination of Q _h. On the other hand, the experimental velocity data span a sufficiently large temperature range to determine Q _h and based on the irradiated materials, which are then utilised for both irradiated and unirradiated materials. The value of Q _h was determined from the slope in a plot of ln V versus reciprocal temperature. Values of V _o, , Q _DHC and Q _h for Zr–2·5Nb are presented in Table 2.

OPEN IN VIEWER

Figure 14.

High temperature crack velocity data for DHC in irradiated and unirradiated Zr–2·5Nb material. Data are from Sagat and Puls,7 Huang and Mills38 and Grigoriev and Jakobsson.46 {T (°F) = 1·8[T (K)–273]+32; 1 m s⁻¹ = 3·28 ft s⁻¹}

It is evident that there are two competing mechanisms that control the dependence of DHC velocity on temperature: TSSD and TSSP.7 Figure 12 shows the TSSD and the TSSP for Zr–2·5Nb. The increase in DHC velocity with decreasing temperature below TSSD is controlled by hydrogen solubility limit and hydride formation beyond the critical size. In this regime, the temperature is sufficiently high, and hydrogen diffusion to the crack tip occurs quickly; a lower temperature provides a higher driving force for TSSP where fracture at the crack tip results in a higher DHC velocity. At temperatures below about 318°C (604°F), the DHC velocity decreases with decreasing temperature because of a lower hydrogen diffusivity and a longer time for hydrogen to move from the bulk to the crack tip. The DHC velocity exhibits a maximum value at about a critical temperature T _C, which is 318°C (604°F) for Zr–2·5Nb, where the precipitation and diffusion kinetics are optimum. The critical temperature at which the maximum DHC velocity occurs is about 30–40°C (54–72°F) below the TSSD.7 Limited experimental data indicated that the critical temperature T _C ranges from 310 to 320°C (590–608°F) and is not sensitive to hydrogen content in the range of 60–170 wt ppm.7

For DHC at a constant temperature, the time to failure t _f can be obtained as (13) where da is the crack increment. The crack velocity V is given by equation (11) for T<T _C, but is given by equation (12) for T _C<T<TSSD and V = 0 for T>TSSD. The time to leakage t _l has been computed on the basis of the crack depth reaching the cladding wall thickness, while the time to fracture t _f has been computed based on the stress intensity factor exceeding a critical stress intensity factor of 70 MPa m^1/2 (63·7 ksi in.^1/2). At leakage, a crack penetrates the entire cladding thickness and internal gases inside the cladding tube can escape outward through the crack surfaces. The corresponding stress intensity factor, however, is less than the fracture toughness (K<K _IC) of the cladding so that cladding fracture does not occur at leakage. After leakage has occurred, subsequent crack growth proceeds axially and causes the stress intensity factor to increase. When K>K _IC, cladding fracture occurs by splitting the cladding tube along the axial direction. A comparison of the computed time to leakage and time to fracture for cladding tubes at 25°C (77°F) is presented in Fig. 15. These calculations have been performed for Zircaloy-2 and Zircaloy-4, assuming that the material constants for Zr–2·5Nb are applicable to the Zircaloy-2 and Zircaloy-4. In these calculations, the stress due to a cladding pressure of 5 MPa (0·73 ksi) was superimposed with a hoop stress of 275 MPa (39·9 ksi) assumed to result from reactor transient loading. The initial crack depth resulting from oxide and hydride blister cracking on the outer surface of a cladding tube was 0·253 mm (9·96 mil), leading to an initial stress intensity factor of 6·5 MPa m^1/2 (5·92 ksi in.^1/2) at the initial stage of DHC. Equations (11)–(13) indicate that temperature is the most significant parameter affecting the time to failure for DHC in zirconium based alloys. The time to leakage and the time to fracture by DHC after the assumed DHC initiation were computed using equations (11)–(13), and the results are presented in Fig. 16. The computed times to leakage and fracture are significantly reduced when the cladding temperature is increased from ambient temperature. The shortest times to leakage and failure are 0·4 and 3·3 h respectively; they both occur at a critical temperature T _c of 318·5°C (605·3°F), where the TSSP and hydrogen diffusion kinetics are high. The leakage time is about one-eighth of the time to fracture. Both the leakage and fracture time increase at temperatures below T _c because the transport of hydrogen to the crack tip is reduced by a slower rate of hydrogen diffusion. At temperatures above T _c, the time to leakage and fracture increase because hydrogen solubility is increased and hydride formation at the crack tip is reduced. The results in Fig. 16 indicate that the optimum temperature for the occurrence of DHC in zirconium alloys is in the temperature range of 280–320°C (536–608°F), near the tip of the nose of the T–t _f curve. The nose of the T–t _f curve is expected to vary with hydrogen content because both the TSSD and TSSP are functions of hydrogen content.

OPEN IN VIEWER

Figure 15.

Computed time to leakage based on crack depth equal cladding wall thickness compared against computed time to fracture based on the stress intensity factor exceeding a critical stress intensity factor of 70 MPa m^1/2 (63·7 ksi in.^1/2) (1 mm = 39·4 mil)

OPEN IN VIEWER

Figure 16.

Computed time to leakage (crack depth = wall thickness) and time to fracture K>K _IC) by DHC in Zr–2·5Nb as a function of temperature. The results show the time to failure (leakage or fracture) decreases with increasing temperature from 0 to 320°C (32–608°F) and increases from 320 to 350°C (608–662°F). No DHC failure occurs at temperatures above 350°C (662°F) because of the absence of hydride formation at the crack tip [T (°F) = 1·8T (°C)+32; 1 MPa m^1/2 = 0·91 ksi in.^1/2]

Texture effects

The DHC velocity at a given stress intensity factor K exhibits an orientation dependence. Results on Zr–2·5Nb showed that the DHC velocity in the longitudinal or axial direction of the cladding is higher than that in the radial direction, as shown in Fig. 17 for various temperatures.40,43 There is, however, a discrepancy in the reported activation energies for the longitudinal and radial directions. Sagat et al. 43 reported that the temperature dependency of DHC velocity and the activation energy in the longitudinal (axial) direction are smaller than those in the radial direction. On the other hand, the results by Kim et al. 40 indicated similar activation energy values but lower DHC velocity values in both the longitudinal and radial directions than those reported by Sagat et al. 43

OPEN IN VIEWER

Figure 17.

Comparison between radial and axial DHC velocity for Zr–2·5Nb pressure tubes. Data are from Sagat et al. 43 and Kim et al. 40 RD: radial direction; AD: axial direction; CL: confidence limits {1 m s⁻¹ = 3·28 ft s⁻¹; T (°F) = 1·8[T (K)–273]+32}

Gas pressure

The cladding stress depends on the temperature and quantity of fission gases in the plenum, which in turn depends on the burnup and fission gas release fraction. The hoop stress in cladding was computed on the basis of the thin wall approximation for a burnup ranging from 25–60 GWd MTU⁻¹ and plenum pressure ranging from 3·41 to 5·51 MPa (0·5–0·8 ksi). The gas pressure P of fission gas and helium is given by48 (14) where P is fission gas pressure in MPa, Mkg is the mass of uranium in one fuel rod in kg, T is absolute temperature in K, FGR is fission gas release, br is burnup in GWd MTU⁻¹ and Fvol is the rod file volume in cm³. The stress σ _p due to the fission gas pressure was computed according to the expression given by48 (15) where D _I is the cladding inside diameter, a is crack depth, t is cladding wall thickness and t _ox is the oxide thickness. The cladding stresses are in the range of 26–48 MPa (3·77–6·96 ksi) and 36–66 MPa (5·22–9·57 ksi) at room temperature respectively, for cladding wall thickness [5·6–49 mm (0·22–1·93 in.)] and corresponding plenum pressures of 3·41–5·5 MPa (0·5–0·8 ksi) with a median crack size [a = 12 μm (0·47 mil)] and the maximum [a = 160 μm (6·3 mil)] and excluding oxide thickness (t _ox = 0), according to Ref. 48. Pescatore et al. 92 gave both the average and maximum hoop stresses for PWR fuels from several sources. At 320°C (608°F), the average hoop stress value ranged from 24 to 62 MPa (3·5–0 ksi), and the maximum value was up to 134 MPa (19·4 ksi) for less than 1% of the total rods.

Pellet–cladding mechanical interaction

The cladding stresses due to fission gas release are relatively low under normal conditions. During power ramp, the cladding stress increases because of mechanical interaction between the fuel pellets and the cladding (Fig. 20). The mechanism of PCMI, which is well established,91 results from the thermal expansion and crack induced volumetric increase of the fuel pellets during thermal transients. The thermal stresses lead to fuel fragmentation and increases in the pellet diameter. Creep of the cladding results in local ridges in the cladding tube as the fuel pellets change to an ‘hourglass’ shape.54 Early analyses of the PCMI stresses were reviewed by Cox,91 who concluded that stresses generated by PCMI during in-reactor power ramp are high, and the corresponding stress intensity factors are comparable to those measured in the laboratory. The stress concentration factors at cladding tube ridges were computed by Ranjan et al. 93 using an elastic axisymmetric shell analysis of the cladding by an asymptotic expansion method that yielded closed form solutions. Elastic concentration factors on the order of 1·1–2·8 were obtained for the circumferential stresses at the ridges, depending on the ridge height and the strain state developed at the ridge.

OPEN IN VIEWER

Figure 20.

Pellet–cladding mechanical interaction results from the thermal expansion and cracking of the UO₂ fuel pellets during power ramp and induces local tensile hoop stresses in the cladding that can potentially propagate an existing crack or defect by DHC

Recent PCMI analyses are elastic–plastic analyses based on FEM with complex constitutive models that include treatment of plasticity, creep, stress relaxation and irradiation effects.53 ^– 61 These FEM analyses are typically two-dimensional (2D) analyses. More recent 3D FEM analyses provided better descriptions of the time dependent evolution of the local transient stresses at various locations in the cladding tubes during power ramp including various hold times at the ramp terminal level.

Using an elastic–viscoplastic constitutive model that treats plasticity, creep, stress relaxation, anisotropic and irradiation effects, Schäffler et al. 57 computed the cladding stresses in a Zircaloy-4 cladding tube during a PWR power ramp transient using simulated loading conditions by imposing a circumferential strain rate of 2×10⁻⁵ s⁻¹ followed by a relaxation period. Figure 21a shows the computed hoop stress in the cladding compared to the corresponding biaxial tensile tests at 350°C (662°F). The cladding reaches as high as 600 MPa (87 ksi), but decreases quickly in the first 20 min and saturates to about 410–450 MPa (59·5–65·3 ksi) after 100 min into the relaxation or strain hold period. This stress transient will be referred to as Type I, which is characterised by a stable, relaxed stress that is higher than that required to meet K>K _IH and cause DHC.

OPEN IN VIEWER

Figure 21.

Stress transients associated with pellet–cladding mechanical interactions: a Type I stress transient with a stabilised stress that leads to K>K _IH and DHC, b Type II stress transient with a peak stress that leads to K>K _IH and DHC over a short duration (130 min) and stabilised stress that results in K<K _IH and crack arrest and c Type II stress transient with a 7 min duration where K>K _IH and DHC growth can occur (1 MPa = 0·145 ksi)

Suzuki and Uetsuka58 developed a fuel performance code, FEMAXI-6, for analysing light water reactor fuel rod behaviours in normal and transient conditions. The FEM based code, which has incorporated thermal and mechanical models for treating thermal conductivity degradation and pellet–clad bonding in high burnup fuel rods, analyses the PCMI stresses induced by swelling in high burnup BWR type fuel rods. Their calculations indicated that the cladding hoop stress transient during a power ramp showed a peak stress of 350 MPa (50·8 ksi), which relaxed quickly and became stable at 260–300 MPa (37·7–43·5 ksi) over a time period of about 120 min. The corresponding cladding temperature was 676 K (712·4°F) at the inner surface and 598 K (572°F) at the outer surface during the ramp. Bonding between the clad and pellets increased the hoop stress by 17% and also increased the biaxial stress state. Suzuki and Uetsuka58 also reported that the peak value in the stress transient was sensitive to the creep model chosen for the stress analysis. A hoop stress as high as 600 MPa (87 ksi) had been obtained depending on the creep model. The results of Suzuki and Uetsuka58 are thus comparable to those reported by Schäffler et al.,57 both in the transient stress levels and the length of the relaxation period.

Using the FEM code TOUTATIS, Brochard et al. 53 computed and reported the cladding stresses for a rod irradiated for two annual PWR operating cycles and subjected to a power increase from 240 to 450 W cm⁻¹ (94·1–177·2 W in.⁻¹) in about 20 min with a hold time of 100 min at the ramp terminal level. The transient inner clad hoop stress during the power ramp is presented in Fig. 21b . The shape of the stress evolution curve is similar to that shown in Fig. 21a , but the stresses are lower. The time period over which the transient stress persists is shorter and lasts for 100 min, which corresponds to the hold time at the ramp terminal level.

More recently, Brochard et al. 54 computed the cladding stresses for a rod irradiated for two annual PWR operating cycles and subjected to a power increase from 200 to 450 W cm⁻¹ (78·7–177·2 W in.⁻¹) in 2·5 min with an unspecified hold time at the ramp terminal level. Their results for the cladding circumference stress at the interpellet level, which are shown in Fig. 21c , indicate that the cladding stress is fairly low at the initial stage of the power ramp when the gap still exists between the fuel pellets and the cladding tube. The cladding stress builds up quickly once the gap between cladding and pellets closes under the combined effect of cladding creep and pellet swelling.54 The cladding circumferential stress reaches a maximum value of 350 MPa (50·8 ksi) and then decreases quickly to a lower value of 210 MPa (30·5 ksi) after 7 min at the ramp terminal level. For this case, the stress concentration factor at the cladding is about 1·34–1·43 based on experimentally measured values of 0·4–0·6 for the coefficient of friction between pellet and cladding. The stress transients shown in Fig. 21b and c will be referred to as Type II. In contrast to the Type I stress transient, Type II meets the condition of K>K _IH for only a short time. Relaxation of the local stress occurs quickly, and the stabilised stress is below that required to exceed K _IH; therefore, Type II transients lead to intermittent crack growth and crack arrest.

The magnitude and the duration of the transient PCMI stress are sensitive to a number of factors,53,54,59 including (1) pellet geometry such as pellet shape, the height to diameter ratio, fuel swelling and pellet fragmentation; (2) thermomechanical loading history during power ramp and the duration of hold time at the ramp terminal level; (3) modelling assumptions such as the axial constraint and the constitutive model utilised to treat time and temperature dependent inelastic flow resulting from plasticity, creep, stress relaxation, swelling and irradiation effects; and (4) cladding characteristics and material parameters such as wall thickness, thermal properties, elastic–viscoplastic properties, pellet–pellet friction and the coefficient of friction at the fuel cladding interface that are used in the computation.

Systematic studies of the sensitivity of the PCMI stress to structural and material parameters were performed by Brochard et al.,53,54 who identified the axial constraint, the pellet height, the pellet fragmentation and the hollow pellet as important variables affecting the PCMI stress. Retel et al. 59 evaluated the separate influences of structural and materials parameter variability on the PCMI in a fuel rod subjected to two PWR cycles with a power equal to about 200 W cm⁻¹ (78·7 W in.⁻¹) followed by a power transient with a maximum power equal to 412 W cm⁻¹ (162·2 W in.⁻¹). In the 3D FEM simulations of this set of baseline ramping conditions, the number of axial and radial pellet cracks, pellet fragment size, relative fragment displacement, degrees of symmetry in the fragment configuration, pellet–pellet friction and pellet–cladding friction were varied systematically. Among these six parameters, Retel et al. 59 found that the pellet–cladding friction coefficient and the number of axial pellet cracks are the most important in increasing the cladding stress. The maximum cladding stress obtained by Retel et al. 59 was in the range of 300–400 MPa (43·5–58 ksi) for a pellet–cladding friction coefficient of 0–0·2, with cladding temperatures of 440°C (824°F) and 366°C (691°F) at inner and outer cladding walls respectively.

Because of the large number of structural and material variables involved, it is difficult to assess the validity of the computed PCMI transient stress from a particular computation unless the computational results are verified by relevant experimental data. Bourreau et al. 56 performed power ramp tests on PWR fuel rods with fuel segments extracted from the same fuel rod. Designed by Connectors International, the original segmented fuel rod contained UO₂ pellets that were initially enriched up to 4·5% in ²³⁵U and irradiated during two cycles in a 900 MW PWR to reach an average burnup of 23·8 GWd MTU⁻¹. Three segments of the original fuel rod were extracted, refabricated into subrods and used in subsequent power ramp tests with four different ramp power histories and hold times at the maximum power or terminal ramp level (TRL). The hold times at the TRL were 0 s (no hold time), 16·5 min and 12 h 19 min. Cladding deformations were measured during the power ramping, and the power ramp tests were modelled using 2D and 3D fuel modelling codes. The experimental results of cladding deformation were correlated with power density and compared against FEM computations. The experimental results indicated that the hold time at the TRL exerted a significant effect on the cladding deformation and stresses in the power ramp tests. For all ramp tests, no rod failure occurred during or after the power ramp experiments. A good correlation was observed between the local power during irradiation and the cladding deformation measured after the ramp tests. Comparisons of post-irradiation measurements performed on the three tubes indicated that the cladding deformation increased with the hold time at TRL. The hold time particularly affected the evolution of secondary ridges. Many features of the power ramp tests, such as the axial power profile on cladding diameter variations during the power transients, were simulated by the provided FEM codes that the analysis used for pre-existing oxide fragmentation. In Bourreau et al.,56 2D modelling satisfactorily reproduced cladding diameter changes off ridges, while 3D modelling correctly simulated primary ridge changes and allowed secondary ridge buildup.

Direct evidence for the presence of a high PCMI stress was provided by the power ramp testing of BWR segmented rods conducted by Hayashi et al. 36 BWR 8×8 fuel assemblies with segmented rods of Zircaloy-2 cladding tubes were irradiated up to five cycles. Ramp tests of 25 segments with burnup ranging from 43 to 61 GWD MTU⁻¹ were then conducted in a test reactor. One segment rod irradiated for three cycles (43 GWd MTU⁻¹) failed by a single step ramp test after 9 min at a terminal ramp power of 614 W cm⁻¹ with a pinhole caused by PCI and SCC. One segment rod irradiated for four cycles (56 GWd MTU⁻¹) failed by a single step ramp test after 149 min at 551 W cm⁻¹ with an outer side axial crack. Nine segment rods were irradiated for five cycles (61 GWd MTU⁻¹). Two of these rods failed by a single step ramp test after 68–100 min at about 421–428 W cm⁻¹, and one failed by a stair ramp test at 446 W cm⁻¹, all with outer axial cracks. According to Hayashi et al.,36 the increase in hydrogen contents and PCMI stresses with increasing burnups led to the formation of radial hydrides at the outer rim and a change in the operative failure mechanisms in the cladding tubes. Specifically, the failure mechanism in the segment rods was PCI/SCC at low burnups (<43 GWd MTU⁻¹). At high burnups (56 and 62 GWD MTU⁻¹), rod failure was caused by a combination of high PCMI stresses and radial hydride assisted DHC growth during ramp tests. Cladding cracks initiated by fracture of radial hydrides at the outer rim and propagated inward by DHC to the inside wall, followed by DHC in the axial directions and finally ductile fracture. Post-test examination of the pellet–cladding interface revealed a bonding layer formed between the pellet and the cladding by mutual diffusion of UO₂ and ZrO₂.36 The observed times to rod failure in the power ramp tests ranged from 9 to 149 min, which are consistent with the durations (7–130 min) of PCMI transient stresses predicted by Brochard et al. 53,54

It is well known that the presence of a large tensile hoop stress during TSSP would align the hydrides along the radial direction of the cladding tube, forming radial hydrides. At certain temperatures, circumferential hydrides in zirconium cladding tubes can reorient under a tensile hoop stress to form radial hydride, providing that the tensile stress exceeds the critical stress for hydride reorientation. Experimental data of the critical stress for hydride reorientation were reviewed by Chung,25 and additional data needs in this area were discussed by Einziger et al. 94 Previously, Chung25 determined empirically that the critical tensile for hydride reorientation is about 100 MPa and its value is relatively insensitive to temperature, as shown in Fig. 22. Subsequently, Ferry and Poinssot95 suggested that the critical stress for hydride reorientation may not be a constant, but its value may increase with decreasing temperature, as shown in Fig. 22. More recently, Chu et al. 96 presented a theoretical analysis that showed the threshold stress for hydride reorientation decreases with increasing temperature and hydrogen content in a complex manner. The results for Zircaloy-4 with 200 wt ppm H₂ are shown in Fig. 22. Daum et al. 97 recently reported that the critical stresses for hydride reorientation of non-irradiated and high burnup Zircaloy-2 are 80±10 and 75±10 MPa respectively. A comparison of the threshold stress boundaries against all of the available experimental data in Fig. 22 indicates that Chung’s empirical threshold stress boundary gives the best agreement, even though a few data points at temperatures of hydride reorientation at temperatures ≧400°C fall below the threshold line. The temperature during in-reactor power ramp is in the range of 300–350°C. Figure 22 shows that hydride reorientation would not occur at temperatures less than 400°C when the cladding stress is below the 90 MPa limit.94 At temperatures ≧400°C, hydride reorientation may occur at stresses below the 90 MPa stress limit, but more experimental data are needed.94 On the other hand, the PCMI transient stresses during in-reactor power ramps are in the range of 200–600 MPa. The high PCMI stresses are sufficiently high to cause hydride reorientation in the cladding tubes. A variety of hydride microstructures have been reported, including hydride blisters, hydride rims, sunburst hydrides and radial hydrides.16,19 ^– 21,26– 36 Once formed during power ramps, the hydride microstructures may remain in the cladding tubes and probably persist after the fuel rods are removed from the reactors. Thus, there are considerable uncertainties on the initial cladding conditions (crack distribution, hydride orientation and morphology) before the fuel rods are emplaced in a repository.

OPEN IN VIEWER

Figure 22.

Critical tensile stresses for hydride reorientation as a function of temperature compared against experimental data compiled by Chung,25 cold worked and stress relieved (CWSR) Zircaloy-4 data compiled by Ferry and Poinssot,95 and stress relief annealed (SRA) Zircaloy-4 by Chu et al. 96 and Daum et al. 97 The stress limit defined in the NRC Interim Staff Guidance-II Revision 3 (ISG-II Rev. 3)94 is shown as the dotted line

Potential repository environment

For fuel rods in a potential waste repository environment, the cladding stress was computed to be in the range of 60–100 MPa (8·7–14·5 ksi), based on consideration of the internal rod pressure due to fission gas release and a uniform cladding diameter.48,98 This estimate of the cladding stress is likely a lower bound because non-uniform cladding diameter changes induced by PCMI and stress concentration at the ridges was not taken into account in the previous stress calculation.

Computing the local hoop stress at the ridges in cladding tubes in a potential repository requires knowledge of (1) the anticipated temperature gradient between the pellet fragments and cladding, (2) swelling or the presence of voids in the fuels61 and (3) bonding or welding between the fuel fragments and cladding wall.36,61 The latter two phenomena could prevent the fuel pellets from separating from the cladding, thereby allowing a high local stress to be maintained at the ridges after a power ramp.59 One possible estimate of the upper bound hoop stress in cladding tubes after service may be the relaxed stable hoop stress in the cladding after an in-reactor power ramp, because the temperature gradient, the extent of fuel expansion, and the contact between fuel and cladding are at the maximum levels. The relaxed PCMI stress values range from 200–400 MPa (29–58 ksi) based on the computational results of Brochard et al.,53,54 Suzuki and Uetsuka58 and Schäffler et al. 57 The actual unrelaxed PCMI stresses may be less than 200–400 MPa (29–58 ksi), depending on the stress relaxation processes and the bonding between pellet and cladding. Over a period of 10⁴–10⁶ years, cladding tubes in a potential repository might encounter one or more seismic events and impact loading from rockfalls that can cause transient loading on the cladding in successive bursts. The transient cladding stress during a seismic event or a rockfall is expected to lead to a high stress intensity factor at an existing crack in the cladding in a very short time. The general characteristics of the K profiles for these transient stress fields would be similar, but with different amplitudes, durations and stress relaxation times. For impact loading, subsequent DHC growth may be feasible under the stress transient results in a fracture of the surface oxide layer, hydride blister or rim, or radial hydrides to create or extend a crack so that K>K _IH. Under this circumstance, DHC may occur after a rock fall event; the DHC growth kinetics would then be dictated by the duration of the transient stress, the crack lengths, the cladding temperature, the cooling rate and the properties of the degraded cladding.

Internal gas pressure

The propensity to DHC in fuel rod cladding tubes due to internal pressure and local transient stresses induced by PCMI is considered in this section. In particular, the plot of critical stress versus crack length shown in Fig. 7 for Zr–2·5Nb has been extended to zircaloy fuel cladding because comparable experimental data for Zircaloy-2 or Zircaloy-4 are incomplete and are insufficient for statistical analysis. As shown in Fig. 6, experimental results on Zircaloy-2 tubing by Huang and Mills38 indicated that the K _IH value for Zircaloy-2 tubing ranged from 5·2 to 8·4 MPa m^1/2 (4·7–7·6ksi in.^1/2), compared to 5·5–9 MPa m^1/2 (5–8·2 ksi in.^1/2) for Zr–2·5Nb materials with closely matched crystallographic textures. The activation energy for DHC in Zircaloy-2 was determined to be 65·3 kJ mol⁻¹ (15·60kcal mol⁻¹),38 which is essentially identical to the experimental value of 65·5 kJ mol⁻¹ (15·65 kcal mol⁻¹) for Zr–2·5Nb.12 Mills and Huang38 indicated that the DHC growth rates in Zircaloy-2 tubes were comparable to those observed in Zr–2·5Nb tubes with similar crystallographic textures. Because the DHC properties are similar, extending Fig. 7 for Zr–2·5Nb to Zircaloy-2, therefore, requires only the assumption that the fracture strengths for hydrides in Zr–2·5Nb are comparable to those for fracture of hydrides in Zircaloy-2. On this basis, the calculated curves based on large crack K _IH thresholds of 5·36 and 12 MPa m^1/2 (4·9 and 10·9 ksi in.^1/2) are applicable to both Zircaloy-2 and Zr–2·5Nb, because the actual experimental data of these two alloys lie within these two limits. There is a general consensus that the cladding stress level due to internal fission gas pressure [60–100 MPa (8·7–14·5 ksi)] results in a stress intensity level that is on the order of 0·5–2 MPa m^1/2 (0·46–1·82 ksi in.^1/2), which is lower than the K _IH of 5 MPa m^1/2 (4·6 ksi in.^1/2) required for the onset of DHC growth. Hence, DHC by internal gas pressure alone is considered unlikely, as indicated earlier, when the initial crack sizes are estimated on the basis of the size of manufacturing defects in the cladding. Figure 23 compares the cladding stresses against the critical stresses for the onset of DHC for semicircular cracks with large crack thresholds of 5·38 MPa m⁻¹ (4·9 ksi in.^1/2) and 12 MPa m⁻¹ (10·9 ksi in.^1/2) and the corresponding small crack threshold stresses. The nominal stresses in the cladding due to the internal gas pressure are well below the critical threshold stresses required to cause DHC in either Zr–2·5Nb or Zircaloy-2.

OPEN IN VIEWER

Figure 23.

A comparison of cladding stresses under various loading conditions and crack lengths against the large crack and small crack threshold stresses for the onset of DHC in Zr alloy cladding tubes made of Zircaloy-2 or pressure tubes made of Zr–2·5Nb (1 MPa = 0·145 ksi; 1 MPa m^1/2 = 0·91 ksi in.^1/2; 1 μm = 0·0394 mil)

Transient stress

The transient stress during power ramps can be high and sufficient to cause DHC. The transient PCMI stresses during power ramp are compared with the threshold stress in Figure 23. The comparison indicates that the PCMI transient stress exceeds the threshold stress for DHC for crack depths greater than 80–120 μm (3·2–4·7 mil). Applying the transient hoop stresses computed by Brochard et al.,53,54 Suzuki and Uetsuka58 and Schäffler et al. 57 to DHC indicated that the extent of DHC growth critically depends on the time duration at which the local stress intensity exceeds the K _IH. The DHC growth calculations were performed using a semicircular crack assumption. The cladding tube was assumed to contain a 100 μm (3·9 mil) thick oxide layer with a 150 μm (5·9 mil) hydride blister layer formed beneath the oxide layer. The initial crack depth was taken to be half of the oxide thickness. The oxide layer failed when K> and the crack depth was set to the oxide thickness. Similarly, the hydride layer failed when K> and the crack depth was increased by the layer thickness of the hydride blister. DHC growth continued if K>K_IH under the stress transient or ceased to occur if K≤K _IH. Figure 24 shows the DHC growth results for the three stress transients at 318°C (604°F). For DHC growth under a transient stress with a short duration of 7 min (dashed line, Fig. 24), the amount of DHC growth is ≈108 μm (4·3 mil) and crack growth ceases after K drops below K _IH (Type II transient) resulting in equal values of crack length (a) and crack depth (b). For a transient stress duration of 130 min (solid line, Fig. 24), the amount of DHC growth is about 0·1776 mm (6·9 mil) and the crack depth (b) exceeded the wall thickness, leading to leakage but not failure, because the crack length (a) is still below the critical crack length for a K _IC of 70 MPa m^1/2 (63·7 ksi in.^1/2). In comparison, DHC growth proceeds throughout the entire wall thickness, and the crack length exceeds the critical crack length for K _IC = 70 MPa m^1/2 (63·7 ksi in.^1/2) when the transient stress duration is 330 min (dot dashed line). In this case, K>K _IH for the entire duration of the stress transient (Type I) and K≧K _IC at time ≧200 min. It is evident that DHC growth during in-reactor power ramp is an intermittent process that depends on the induced stress or K amplitude, the duration of the transient, the cladding temperature and the frequency of ramping. DHC growth occurs as long as the induced K level exceeds K _IH. The crack growth rate is fastest at about 318–320°C (604–608°F) for Zr–2·5Nb. Thus the amounts of DHC crack extension are expected to increase with increasing cumulative times of high stress transients or the number of power ramps experienced by the fuel rod.

OPEN IN VIEWER

Figure 24.

DHC growth resulting from PCMI stress transients during power ramps. Crack extension was computed for Zr–2·5Nb cladding on the basis of PCMI stress transients reported by Schäffler et al. 7 and Brochard et al. 53,54 (1 mm = 39·4 mil; 318°C = 604·4°F). Leakage occurs at t _l where the crack depth (b) ≧cladding wall thickness. Cladding fracture occurs at t _f where K≧K _IC

In an overview paper, Chung25 summarised several instances involving failure of high burnup cladding. Under various usage conditions, high burnup cladding is subjected to PCMI stresses not only at high temperatures [380–420°C (716–788°F)], but also during subsequent slow cooling to a lower temperature [e.g. 200°C (392°F)]. The transient local stresses can induce the precipitation of radial hydrides in significant amounts in some local regions, sometimes leading to cleavage fracture and RHA-DHC. Inner wall cracks were found in a high burnup Zircaloy-4 PWR fuel during pre-test heating of a simulated RIA-like pulse test. The length of the inner wall cracks ranged from 20–480 μm (0·79–18·9 mil), and many were partially covered with hydrides.25 Based on the metallographic and fractographic characteristics of the inner wall cracks, Chung25 suspected that these cracks might have been caused by RHA-DHC. This is because numerous local regions were found to contain radial hydrides that were in tight contact with the pellet–cladding bonding layer where high transient tensile stresses could be induced by PCMI. Brittle cracks consistent with RHA-DHC were reported at the outer and inner walls of Zircaloy-2 BWR fuel cladding tested under a power ramp with a hold time for tubes with and without a primary defect.20,22,36 In non-lined BWR failed rods with Zircaloy-2 cladding, Lin et al. 20 observed a long [287 μm (11·3 mil)] outside wall crack with radial hydrides at the crack tip. In addition, radial hydrides were observed to emanate from the outer and the inner cladding walls. The lengths of the radial hydrides that emanated from an outer wall were on the order of 300–390 μm (11·8–15·3 mil), compared to 90–240 μm (3·5–9·4 mil) from an inner wall. Cheng et al. 21 reported hydride cracks of various lengths [50–200 μm (2–7·9 mil)] initiated from the outer and inner walls in both Zr lined and non-lined Zircaloy-2 cladding tubes. Localised oxide layer on the order of 70–150 μm (2·8–5·9 mil) in thickness was observed.22 A hydride rim over 80 μm (3·1 mil) thick was observed on the outer surface. Radial hydrides that emanated from the hydride rim ranged from 160 to 340 μm (6·3–13·4 mil) in length. Non-lined Zircaloy-2 was also prone to the formation sunburst hydrides below the hydride rim. Hayashi et al. 36 also reported that brittle cracks initiated from radial hydrides formed at the outer walls of Zircaloy-2 cladding tubes of high burnup fuel rods. The crack size ranged from 40 to 330 μm (1·6–13 mil) in the through-thickness direction. Some of the observed cracks propagated entirely through the wall thickness and became axial cracks. The length of the axial cracks was 700 μm and higher. Hayashi et al. 36 identified RHA-DHC as the cladding failure mechanism by showing the presence of radial hydrides at the crack tip. They also reported that the oxide thickness ranged from 30 to 100 μm (1·2–3·9 mil) and the length of the radial hydrides was about 70 μm (2·8 mil).36

The transient nature of the PCMI stresses dictates that DHC growth during power ramp does not lead to immediate cladding failure or the development of a leak in the cladding. Instead, the damage process appears to involve the accumulation of small increments of DHC growth over a long time or after numerous power ramping events. The power ramp transient stresses produce two effects that can affect the integrity of cladding tubes under subsequent repository conditions: (1) the generation of local residual stresses at ridges in cladding under normal operating conditions in reactors and repositories and (2) a distribution of defect or crack sizes that evolve with time and are considerably larger than the initial manufacturing defect sizes. Both the local stress and the crack size distribution can significantly affect the propensity to DHC growth in cladding tubes in a repository environment, illustrated by the box in Fig. 23. The relaxed PCMI stresses computed by Brochard et al. 53,54 and Suzuki and Uetsuka58 are about 100–300 MPa (14·5–43·5 ksi), and the corresponding crack depths are 50–250 μm (1·97–9·84 mil). The larger crack depth is based on a 120 μm (4·72 mil) oxide crack and a 130 μm (5·12 mil) hydride crack formed beneath the oxide layer by DHC or fracture of a hydride blister during power ramps. These oxide thicknesses and hydride blister sizes have been estimated on the basis of the experimental observations by Cheng et al. 21 on Zircaloy-2 that showed the localised oxide layer thickness was about 70–150 μm (2·8–5·9 mil). A hydride rim over 80 μm (3·1 mil) thick formed on the outside surface with radial hydrides 160–340 μm (6·3–13·4 mil) in length that emanated from the hydride rim.22 For a crack depth of 250 μm (9·8 mil), the relaxed PCMI stresses are comparable to those required to start DHC. In contrast, DHC does not occur at stresses below 200 MPa for K _IH = 5·36 MPa m^1/2 (4·88 ksi in.^1/2). For zirconium alloys with low oxide thickness and hydrogen pickup (e.g. M5), the initial crack depth may be substantially less than 250 μm (9·8 mil); DHC would not be feasible at a cladding stress of 200 MPa (29 ksi).

Type I transient stress during a cool-down

Without the local stresses at the ridges, the cladding hoop stresses from the internal gas pressure alone may not be sufficiently high to cause DHC in cladding tubes under dry storage and repository conditions. To produce substantial DHC, the local stress intensity factor at cladding must exceed K _IH for a sufficient time period without being relaxed below this critical value. Of the two transient stress types, the one modelled by Schäffler et al. 57 and another described for reactor ramp-up by Suzuki and Uetsuka58 are capable of producing substantial DHC growth because of an extended duration where K>K _IH. For cladding tubes under potential repository conditions, the amount of crack extension generated by this type of local stress distribution also depends on the cladding temperature and cooling rate. For DHC under a cool-down from a given temperature [e.g. 320°C (608°F)], the crack extension per unit temperature change dT can be derived from equation (11) and is given by (16) and (17) where is the cooling rate. The importance of the cooling rate on the DHC growth response is illustrated in Fig. 25, which plots da/dT as a function of temperature for three different cooling rates for Zr–2·5Nb with V _o = 0·14 m s⁻¹ (0·459 ft s⁻¹), R = 8·31J mol⁻¹ K⁻¹ (1·98 cal mol⁻¹ K⁻¹) and Q _DHC = 65·5kJ mol⁻¹ (15·65 kcal mol⁻¹) for T<318°C (604°F) where TSSP is controlled by diffusion kinetics. For T>318°C (604°F) where hydrogen solubility controls TSSP,  = 8·83×10⁻⁴² m s⁻¹ (2·897×10⁻⁴¹ ft s⁻¹) and Q _h = 389 kJ mol⁻¹ (92·93 kcal mol⁻¹). Figure 25 shows that the amount of DHC growth per temperature change is the highest at about 318°C (604°F) and decreases with both increasing and decreasing temperatures for all three cooling rates. DHC growth is essentially nonexistent when the temperature is above 360°C (680°F) because hydrogen is in solid solution and hydrides cannot be formed at the crack tip. DHC growth is very limited when the temperature drops below 150°C (302°F) because of slow diffusion kinetics. A high cooling rate reduces the DHC growth because there is less time for the crack to propagate during the prescribed cooling rate. Conversely, a slow cooling rate allows more time for DHC to occur. The cumulative crack extension by DHC during the cooldown is given by the area under the curves shown in Fig. 25 and is obtained by integrating or summing equations (16) and (17) over the temperature range from T ₁ to T ₃. T ₁ and T ₃ are the initial, intermediate and final temperatures in the cool-down cycle leading to (18) The results for cooling from T ₃ = 400°C (752°F) to T ₁ = 0°C (32°F) with T ₂ = 318°C (604°F) are summarised in Table 3.

OPEN IN VIEWER

Figure 25.

DHC growth rate per unit temperature change as a function of cladding temperature for three cooling rates [1 mm °C⁻¹ = 0·022 in. °F⁻¹; T (°F) = 1·8T (°C)+32; 1°C s⁻¹ = 1·8°F s⁻¹]

OPEN IN VIEWER

Table 3.

DHC crack extensions and time of growth as function of cooling rates

Cooling rate/°C s−1 (°F s−1)	Δa/mm (mil) (T 3→T 2)	Δa/mm (mil) (T 2→T 1)	Δa/mm (mil) (T 3→T 1)	Time/h
0·1 (0·18)	0·0109 (0·43)	0·0889 (13·5)	0·0998 (3·93)	1·11
0·01 (0·018)	0·1807 (7·11)	0·8888 (35·0)	1·0695 (42·1)	11·11
0·001 (0·0018)	1·8075 (71·2)	8·8876 (350·0)	10·695 (421)	111·11

Table 3 shows that the amount of DHC extension is about 0·1 mm (3·94 mil) for a cooling rate of 0·1°C s⁻¹ (0·18°F s⁻¹, and the time spent in DHC growth is 1·1 h. When the cooling rate is reduced to 0·01°C s⁻¹ (0·018°F s⁻¹), the time for DHC growth increases to 11·11 h, and the amount of DHC extension increases to 1·07 mm (42·1 mil). Further decrease in the cooling rate to 0·001°C s⁻¹ (0·0018°F s⁻¹) increases the DHC time to 111 h and the amount of DHC extension to 10·7 mm (0·42 in.).

Type II transient stress

For Type II transient stress, the peak stress intensity factor exceeds the K _IH value and results in DHC growth for a short time period during a power ramp. After the power ramp is completed, the equilibrium stress is stabilised at a stress intensity factor K that drops below the K _IH and terminates DHC growth. Under this circumstance, DHC remains dormant until either the local stress is increased or the K _IH value is equivalently reduced so that K>K _IH. When K>K _IH, DHC resumes and the crack growth process can be analysed using the procedure described for treating Type I transient stress. When K<K _IH, DHC ceases and the crack length remains constant until another transient stress cycle causes the K value to exceed K _IH again. This is shown in Fig. 24, which presents the results of computations based on the Type I PCMI stress transients reported by Brochard et al. 53 Consequently, the cumulative DHC extension depends on the number of transient stress cycles, the time period within which K>K _IH, the cladding temperature, and the cooling rate. The time t _h required for the diffusion of hydrogen to form hydrides at a crack tip can be estimated from the expression (19) with (20) where l _h is the distance that hydrogen must diffuse to reach the crack tip to form hydrides, R is the universal gas constant (R = 8·31 J mol⁻¹ K⁻¹), T is absolute temperature (K), and D _h is the diffusion coefficient of hydrogen in zirconium (m² s⁻¹).4,98 Using equations (19) and (20), the times required for diffusion distances of 20, 100 and 1000 μm (0·79, 3·94 and 39·37 mil) are computed, and the results are shown as a function of temperature in Fig. 26. For all three diffusion distances, the times for diffusion of hydrogen to the crack tip are small compared to the stress transient in power ramp, which is on the order of 10 min or higher. Thus, there are sufficient times for the diffusion of hydrogen to the crack tip to form hydrides.

OPEN IN VIEWER

Figure 26.

Times required for the diffusion of hydrogen atoms from various distances to form hydrides at the crack tip [T (°F) = 1·8T (°C)+32; 1 μm = 0·0394 mil]

The high transient stresses associated with power ramps can potentially cause hydride reorientation (Fig. 22), and leading to crack initiation in cladding by fracturing a hydride blister and a number of radial hydrides. Once a crack of sufficient depth is formed, the stress due to the internal gas pressure might then be large enough to cause subsequent DHC growth. If not, DHC growth would cease and remain dormant until the crack is reactivated by a subsequent transient stress cycle. At 318·5°C (605·3°F), the time to leakage and future DHC are 24 and 198 min respectively. They are in accordance with the experimental observation (9–149 min) reported as power ramp test of segmental rods.36

DHC in zirconium alloy cladding tubes

Extensive review of literature data2,10,11 and FEM computations53 ^– 59 indicate that DHC in zirconium alloy cladding requires local transient stresses induced by PCMI during power ramping. Without these transient PCMI stresses, the cladding stresses due to internal gas pressure alone are not sufficiently high to produce a stress intensity factor at a manufacturing defect or crack to exceed the growth threshold K _IH for DHC. The relatively large PCMI stresses can cause hydride reorientation, leading to a variety of hydride microstructures that are prone to brittle crack formation, followed by DHC. The frequency of PCMI occurrence during power ramps is not known, and the corresponding hydride microstructures have not been fully characterised. Thus, characterisation of hydride morphology in the cladding tubes of high burnup fuel rods would indicate the frequency of the PCMI occurrence.

At least two types of transient stress profiles have been identified. Type I transient leads to continuous DHC growth until cladding failure, while Type II transient results in intermittent DHC growth and arrest. High burnup fuel rods, which may experience numerous power ramps, are likely to contain cracks that have been initiated by oxide cracking, blister fracture or corrosion and subsequently extended by DHC. A range of crack depths generated by DHC may be expected because the driving stresses are transient and vary with thermal and mechanical loading histories, hold times and the number of power ramps. Currently, very little information is available about the hydride size distribution and the crack depth distribution in cladding tubes after they are removed from service. Without an accurate assessment of crack depth and hydride blister size, there are great uncertainties on the cladding stress required to cause K>K _IH, the cladding resistance against DHC growth during the operation period before disposal.

The peak cladding temperature history for the centre rod in the waste package during the operation period before disposal is about 325°C (617°F).49 Because of the high DHC propagation rate near 318°C (604°F), the time to leakage is on the order of a few hours if K>K _IH in any cladding tubes. DHC may occur based on internal gas pressure alone if large hydride blisters or radial hydrides are present in the microstructure, the fracturing of which leads to K>K _IH. There is also a concern that slow cooling rates (10⁻⁶–10⁻⁷°C s⁻¹) may reduce the K _IH value by forming a long continuous hydride at the crack tip. Experimental K _IH data needed to address this issue currently do not exist.

Effects of temperature on DHC

The cladding temperature has a significant effect on DHC growth in zirconium cladding tubes. The worst DHC growth resistance in Zr–2·5Nb occurs at around 310–320°C (590–608°F), with an estimated time to leakage of 24 min and an estimated time to failure of about 3·3 h. For both cladding leakage or fracture, the time to failure increases with increasing temperature above 310–320°C (590–608°F) because of slower hydride formation kinetics, while the time to failure increases with decreasing temperatures below 310–320°C (590–608°F) because of slower hydrogen diffusion. As a result, power ramping at around 310–320°C (590–608°F) can cause cladding failure by DHC in as little as 24 min for leakage or 3·3 h for cladding fracture. Power ramping above 350°C (662°F) would prevent DHC initially, but results in DHC when the cladding temperature cools slowly from 350°C (662°F) to lower temperatures, while the transient stresses are still high. The time to leakage is about one-eighth of the time to fracture at a given temperature. The lower DHC growth resistance at 310–320°C (590–608°F) occurs as a result of a high crack growth rate, despite a higher fracture toughness compared to that at lower temperatures. At ambient temperature, the DHC growth is exceedingly slow, making the time to propagate a DHC through a cladding wall thickness of 0·56 mm (22·1 mil) about 10 years. Thus, a low cladding temperature can effectively suppress DHC in cladding tubes, regardless of the local stresses at the ridges.

Effect of burnup

The burnup level of fuel rods is one of the most important factors affecting the cladding performance and integrity. A high burnup can be detrimental to cladding integrity because the number of power ramps increases with increasing burnup, while the gap between pellet and cladding decreases. In high burnup fuel, the cladding tubes are likely to have experienced a greater number of transient stress cycles where high PCMI stresses can cause hydride reorientation, producing a variety of hydride microstructure and DHC growth compared to low burnup fuels. High burnup cladding tubes are also likely to contain thicker oxide or corrosion layers that can fracture during power ramping and serve as stress risers to initiate DHC at multiple sites. Power ramp tests conducted on segmented fuel rods36 revealed a change of cladding failure from one of pinhole perforation at low burnup (<45 GWd MTU⁻¹) to one of RHA-DHC at high burnup levels (53 and 63 GWd MTU⁻¹).

Effect of cooling rate

The propensity of DHC decreases with increasing cooling rates. Because DHC growth rates are higher at temperatures between 200 and 350°C (392–662°F), a high cooling rate means that less time is spent in this temperature range where the DHC propagates at high rates and consequently forms smaller crack extensions. A cooling rate exceeding 0·1°C s⁻¹ (0·18°F s⁻¹) gives a crack extension of <0·01 mm (0·39 mil), because the time spent in the high DHC growth regime is only about 1 h. The amount of DHC growth and time spent in the high DHC growth regime increase proportionately with decreasing cooling rates, as shown in Table 3. The slow cooling rates likely in a potential repository environment make the zirconium alloy cladding tubes susceptible to DHC growth, because there could be a significant amount of time spent at an elevated temperature where DHC growth kinetics are high.