Research Article | Open Access
Applicability of the Modified Ritchie-Knott-Rice Failure Criterion to Examine the Feasibility of Miniaturized Charpy Type SE(B) Specimens
This paper examined whether the modified Ritchie-Knott-Rice (RKR) failure criterion can be applied to examine the feasibility of miniaturized Charpy type SE(B) specimens of thickness-to-width ratio . The modified RKR failure criterion considered in this paper is the () criterion which predicts the onset of cleavage fracture when the midplane crack-opening stress measured at a distance equal to four times the crack-tip opening displacement, denoted as , exceeds a critical stress . Specimens with values of 25, 10, 3, and 2 mm (denoted as 25t, 10t, 3t, and 2t specimens, resp.) manufactured with 0.55% carbon steel were tested at 20°C. The results showed that the modified RKR criterion could appropriately predict the occurrence of cleavage fracture accompanied by negligibly small stable crack extension (denoted as fracture) naturally for the 25t and 10t specimens. The modified RKR criterion could also predict that fracture does not occur for the 2t specimen. The obtained from specimens for the 25t and 10t specimens exhibited only a small difference, indicating that the obtained from the 10t specimens can be used to predict the that will be obtained with the 25t specimens.
Test specimen size effects on the cleavage fracture toughness of a material in the ductile-to-brittle transition temperature (DBTT) region are important when assessing aging steel structures and reactor pressure vessels. Large scatter in has also been identified. A practical way to maintain conservatism in used in a structural integrity assessment is to specify the test specimen type and thickness, as found in the IAEA recommendations for monitoring the degradation of irradiated nuclear reactor pressure vessels (RPVs) . To understand and become able to convert obtained with different specimen types and thicknesses, many studies have been performed in the past [2–10]. The cause of the test specimen size effect on has usually been categorized into the planar size effect on (i.e., different planar specimen configurations, including crack depth) and the test specimen thickness (TST) effect on . The former has been assumed to result from constraint loss, and the latter results from statistical weakest link (SWL) size effects . Here, this constraint loss is the loss in one-to-one correspondence between and the crack-opening stress distribution (hereinafter denoted as ’s inability to characterize the crack-tip stress). Anderson et al.  gave theoretical background to the empirical TST effect on , described as , by assuming the SWL model. However, the fact that has a nonnegligible lower bound value for large TST seemed to indicate that assuming the TST effect on the effect is thoroughly due to the SWL size effect not being perfect.
Thus, the authors have been working to explain the TST effect on based on ’s inability to characterize the crack-tip stress field [11–18]. The first work was to analyze and test whether there is “out-of-plane” stress difference between SE(B) specimens with identical planar specimen configuration but with different thicknesses (hereinafter denoted as nonproportional specimens) . This was because the formerly mentioned theoretical work of Anderson et al. implicitly assumed the nonproportional specimens. SE(B) specimens of width mm and four thickness-to-width ratios of 0.25, 0.5, 1, and 1.5 were considered. Fracture toughness tests results using 0.55% C steel JIS S55C showed that the empirical relationship holds for = 0.25~1 and that is bounded for and 1.5. The difference in , especially the bounding nature for large TST, clearly correlated with the difference in at the fracture load. The idea to explain the TST effect by using the difference in was supported by many researchers, such as [19–25].
Our second work was to become able to transfer ’s between different TSTs. According to Chen et al. “… it is necessary to distinguish the concepts of the minimum toughness or the lower boundary of toughness values from that of the scatter band of toughness. The former is a definite parameter determined by the specimen geometry and yielding properties, and the latter is statistical behavior determined by the distribution of the weakest constituent” ; it was thought that the minimum toughness of a material, observed for a specific specimen and temperature, can be transferred by running an elastic-plastic finite element analysis (EP-FEA) with a given stress-strain relationship and a failure criterion. For this failure criterion, one of the modified Ritchie-Knott-Rice failure criterion, that is, the criterion , which predicts the onset of cleavage fracture when the crack-opening stress , measured at a distance from the crack-tip equal to four times the crack-tip opening displacement (CTOD) (hereinafter denoted as ), exceeds a critical value (Figure 1), was considered. The criterion successfully explained the TST effect on observed for S55C steel nonproportional SE(B) specimens [13, 17] and RPV ASTM A533 Grade B class 1 steel (A533B) proportional SE(B) specimens (whose ratio is held constant, though is changed) . The tests were conducted in the DBTT region. The specimen thickness ranged from 6.25 to 37.5 mm for S55C steel and from 8 to 254 mm for A533B steel. Although the for the specimens exhibited a variation of 30.5 to 198 N/mm for S55C steel and 7 to 106 N/mm for A533B steel, the critical values for exhibited variations of only 5% and 4%, respectively. Another finding was that the stress level of is maintained as for increasing load. This finding suggested that the minimum that satisfied corresponds to the minimum for a specific specimen , which seemed to be consistent with the formerly mentioned opinion of Chen et al. .
This paper is an extension of our previous works to examine whether the failure criterion can be applied to examine the feasibility of miniaturized Charpy type SE(B) specimens, that is, (1) whether the criterion could predict the occurrence of cleavage fracture accompanied by negligibly small stable crack extension (hereinafter denoted as fracture) and (2) whether is identical to that observed in full sized specimens in the case where fracture occurred. SE(B) specimens of (Charpy type) with of 25 (full size), 10, 3, and 2 (miniaturized sizes) mm were examined (hereinafter denoted as 25t, 10t, 3t, and 2t specimens, resp.). Additionally, 0.55% carbon steel JIS S55C at 20°C, which is in the DBTT region, was selected for material and test temperature. The results showed that the criterion could appropriately predict a fracture naturally for the 25t and 10t specimens. The criterion could also predict that fracture does not occur for the 2t specimen. The obtained from specimens for the 25t and 10t specimens exhibited only a small difference, indicating that the obtained from the 10t specimens can be used to predict the that will be obtained with the 25t specimens.
2. Outline of This Work and Material Selection
The outline of this work is summarized in Figure 2. First, fracture toughness tests with the 25t specimens were conducted and fracture toughness for this size was obtained. The critical stress was obtained by running EP-FEA. Then, EP-FEA for miniaturized-size specimens of 10t, 3t, and 2t were run, and the failure criterion was applied to predict whether these specimens will experience fracture. For specimens for which fracture was expected, was predicted from the of the 25t specimen. Finally, fracture toughness tests were conducted to confirm the predictions.
Considering that the nominal tensile strength to the nominal yield stress ratio for EURO RPVs and Japanese RPVs is equal to 1.3, 0.55% C steel JIS S55C, whose room temperature is known to show a higher value of 1.8 was selected for examination and tested at 20°C. The material was quenched at 850°C and tempered at 650°C. Chemical contents were C: 0.55%, Si: 0.17%, Mn: 0.61%, P: 0.015%, S: 0.004%, Cu: 0.13%, Ni: 0.07%, and Cr: 0.08%, respectively. Charpy impact test results and a true stress-true strain curve are shown in Figure 3. These tests were conducted in accordance with JIS Z2242  and Z2241 , respectively. Averaged nominal tensile properties were yield strength MPa, tensile strength MPa, and elongation of 23.7%.
3. Fracture Toughness Tests and EP-FEA for 25t SE(B) Specimen to Obtain the Critical Stress of Failure Criterion
3.1. Fracture Toughness Tests for 25t SE(B) Specimen
The fracture toughness tests for the 25t SE(B) specimen were conducted in accordance with ASTM E1921 . The dimensions of the SE(B) specimen are shown in Figure 4. The length and the support span of the specimen were designed to satisfy and and were fabricated as and , where width mm.
Fatigue precrack was inserted using loads corresponding to and 19 MPam1/2 for the 1st and final stages, respectively, which satisfied the requirement from the standard of and 20 MPam1/2. For each discrete step, the reduction in for any of these steps was 15%, which satisfied the suggestion from the standard that the reduction in for any of these steps be no greater than 20%. A maximum force and a minimum force with a ratio of were applied at a loading frequency of 10 Hz.
During the fracture toughness test, the loading rate was controlled to be in a specified range from 0.1 to 2.0 MPam1/2/s to comply with the standard, resulting in a range from 1.21 to 1.71 MPam1/2/s. Test temperature requirements are to hold the temperature constant at °C for longer than minutes, where the specimen thickness is 25 mm, resulting in °C for 45 minutes. Six test results satisfying the ASTM E1921 requirements were considered for examination. The results clearly showed fracture.
Fracture toughness data are summarized in Table 1. The variable in the table is the stress intensity factor calculated from the fracture load and the measured crack depth-to-width ratio . in the table is the fracture toughness in terms of stress intensity factor, calculated as , where GPa is Young’s modulus and is Poisson’s ratio used for this conversion. and in the table denote the mean and standard deviation of each parameter. is a requirement of the standard placed on the ligament size, where is the crack length. Although not listed, negligible stable crack extension was measured using SEM observations.
In Table 1, the standard deviation of was 0.00, meaning that the potential scatter due to crack depth difference was minimized. The mean of was 99.8 MPam1/2. Therefore, the master curve reference temperature in ASTM E1921 was approximately equal to the test temperature of 20°C. The standard deviation to average ratio of % for these data was sufficiently small compared with the ASTM E1921 prediction of % = 45%. The minimum was 80.6, which satisfied the ASTM E1921 requirement of .
3.2. EP-FEA for 25t SE(B) Specimen
The FE model used for the elastic-plastic analysis of the SE(B) specimen is shown in Figure 5. In this study, the FE models were fundamentally generated based on the FE model described in the work by Gao and Dodds Jr. , so that total element number of 31,990 and node number of 139,816 do not change for specimen size. Using symmetry conditions, one-quarter of an SE(B) specimen containing a straight crack was analyzed, with appropriate constraints imposed on the symmetry planes, as illustrated in Figure 5. An initial blunted notch with a radius of = 4.305 μm was inserted at the crack-tip and “spiderweb” radius of mm was selected to fully cover the high stress region for the 25t specimen. For all cases, 20-node isoparametric three-dimensional solid elements with reduced (2 × 2 × 2) Gauss integration were employed. A load line displacement was applied for each EP-FEA. In the EP-FEA, the applied load was measured as the total reaction force on the supported nodes. The simulated by the EP-FEA, denoted as , was evaluated using a load-versus-crack-mouth opening displacement diagram (- diagram), in accordance with ASTM E1921 . CTOD was measured at node A in Figure 5 .
(a) Global mesh
(b) Detail of crack-tip mesh
The mechanical properties of the test specimens are a Young’s modulus of 206 GPa and a Poisson’s ratio of 0.3. The FEA material behavior was assumed to be governed by the incremental theory of plasticity, the isotropic hardening rule, and the Prandtl-Reuss flow rule. Two total true stress-strain curves, shown in Figure 3(b), were averaged and used in the EP-FEA. WARP3D  was used as the FEA solver.
First, to validate the EP-FEA results, the EP-FEA diagram for the 25t specimen was compared with the experimental results in Figure 6. The open rectangles represent the EP-FEA results.
As shown in Figure 6, the path of the EP-FEA diagram showed good agreement with the experimental results. Therefore, the EP-FEA result was considered to be valid and proceeded to obtain the critical stress .
The relationship between (i.e., crack-opening stress measured at a distance from the crack-tip equal to 4 on -axis at the specimen midplane) and for each load step was summarized in Figure 7. The stars in the figure indicate the fracture toughness values (’s) obtained from the experiments.
From this figure, it is revealed that fracture always occurred after reached 3.82; thus, this value was determined as the critical value of the criterion. If the criterion is applicable to miniaturized specimens, fracture is predicted when for these miniaturized specimens reaches the identical critical value of 3.82.
4. Applicability of the Failure Criterion to Examine the Feasibility of Miniaturized Charpy Type SE(B) Specimens
4.1. Selection of Miniaturized Specimen Size for Examination
Wallin et al.  noted that the minimum size of a Charpy type SE(B) specimen that satisfies the requirement of for a RPV steel and tested at the temperature (master curve reference temperature) ±50°C is a thickness of 5 and a width of 5 mm (hereinafter denoted as the 5t specimen). Considering that was approximately equal to 20°C for S55C steel, the minimum size for the S55C steel to observe fracture was expected to be 5t. However, some smaller specimens, such as 3.3t, have been examined in the past [34, 35]. Because we are examining the feasibility of miniature specimens with to exhibit fracture, smaller specimens were also considered.
By using = 33.5~61.1 N/mm of 25t specimen and MPa and assuming crack length and , the range of for 10t was estimated to be in the range of 20.4~37.2, that is, close to 30. This is different from the abovementioned Wallin et al.’s prediction. Thus, to be conservative, the 10t specimen was selected as the specimen for which fracture is expected. For the specimens below , the 3t specimen was selected as a candidate of fracture from the past experience of the 3.3t specimen (though there might be an opinion that the material and test temperature were different). The 2t specimen was selected as a possible candidate for which fracture should not be observed.
The configurations of the 10t, 3t, and 2t SE(B) specimens are shown in Figure 8. The support span size was set as equal to .
4.2. Prediction of Cleavage Fracture and for Miniaturized Specimens by Applying the Criterion
EP-FEA was run for the 10t, 3t, and 2t specimens shown in Figure 8 to predict whether cleavage fracture occurs. Details of EP-FEA are fundamentally identical to those for the 25t specimen, except the near crack-tip generated meshes; was in a range of 1.84 to 3.98 μm and was in a range of 0.46 to 2.3 mm. The relationships between and are shown in Figure 9. The value 3.82 in the figure indicates the critical value obtained from the 25t specimens.
From Figure 9(a), fracture was predicted for the 10t specimen because reached the critical value and the stress level was maintained for increasing load. The constancy of for increasing load assures that even if a significant microcrack is not located at the original crack-tip, cleavage fracture might occur after small stable crack extension and a significant microcrack is encountered (SWL model). Thus, the minimum for the 10t specimen was predicted as 58 N/mm because exceeded the critical value when reached 58 N/mm.
In contrast, for the 3t and 2t specimens showed a maximum for increasing load; the stress level could not be maintained. Because is measured at a variable location of and because 4 increases with load, Figures 9(b) and 9(c) indicate that the region of high stress level does not continue to grow after the load shows a maximum for . Thus, qualitatively, large stable crack extension was expected. This was the first experience for the criterion and out of scope of what was considered within the criterion. However, because large stable crack extension was expected, fracture was not predicted for the 3t and 2t specimens.
4.3. Fracture Toughness Tests for the Miniaturized Specimen
Fracture toughness testing for the miniaturized specimen was conducted in accordance with ASTM E1921 . The exception was the ligament size requirement of and the quantity of the tests. The dimensions of the miniaturized SE(B) specimens are shown in Figure 8. The 3t and 2t specimens were cut from the 10t specimen after the fatigue crack was inserted.
Fatigue precrack was inserted using loads corresponding to MPam1/2 for both the 1st and final stage, which satisfied the requirement from the standard of and 20 MPam1/2. For each discrete step, the reduction in for any of these steps was 13–15%, which satisfied the suggestion from the standard that the reduction in for any of these steps be no greater than 20%. The ratio of the maximum force and the minimum force , or , was used, and the loading frequency was 30 Hz.
In the fracture toughness test, the loading rate was controlled to be within a specified range from 0.1 to 2.0 MPam1/2/s, which resulted in an actual range between 0.12 and 1.4 MPam1/2. In the following, the test results for the 10t, 3t, and 2t SE(B) specimens are described.
4.3.1. Test Results for the 10t SE(B) Specimens
The P- diagrams of the 10t SE(B) specimens are compared with , which is the calculated from the elastic compliance given in ASTM E1820 .
As shown in Figure 10, the two experimental diagrams showed good reproducibility. Moreover, the linear slope of the experimental diagrams exhibited good agreement with the . Therefore, the experimental results of the 10t SE(B) specimens were considered to be valid and capable of being compared with the numerical predictions, despite the fact that only two tests were conducted.
The diagrams for both tests showed a sudden increase in after the maximum load was reached, indicating that the clip gauge dropped off. This suggested that an unstable fracture occurred.
The fracture surface of the 10t specimens clearly showed cleavage fracture after very small stable crack extension , satisfying the ASTM E1921 requirement to be less than . The values of the measured are listed in Table 2. Thus, the so-called fracture was observed, as predicted with the criterion.
Fracture toughness test results are summarized in Table 2. The average of N/mm was 1.7 times larger than that of the 25t SE(B) specimen. This magnification agreed with the empirical thickness relationship describing , that is, (10/25)−1/2 = 1.6. was close to but smaller than 30, as predicted. Although only two data points were available, the standard deviation to the average ratio of was 7.75% for the current data, which was sufficiently small compared with the ASTM E1921 prediction of % = 47%. Thus, from a fracture toughness standpoint, the experimental results for the 10t SE(B) specimens were considered to be valid and capable of being compared with the numerical predictions, despite the fact that only two data points were obtained.
Considering the fact that the used in the EP-FEA was 0.50, which was different from the experimental result of 0.54, EP-FEA with was rerun, resulting in negligible small difference in the predicted minimum of 58 N/mm. Comparing the test results of and 68.1 N/mm with the updated prediction, it was concluded that the criterion could properly and consequently predict the minimum of the 10t SE(B) specimen.
In summary, it was concluded that the criterion could properly predict the occurrence of fracture and the minimum of the 10t SE(B) mm specimen, despite the fact that the for the specimen did not satisfy ASTM E1921’s requirement of .
4.3.2. Test Results for the 3t and 2t SE(B) Specimens
Fracture surfaces and diagrams of the 2t and 3t specimens are summarized in Figures 11 and 12, respectively. Note that mm does not represent the value at final fracture, because the maximum measurement capacity of the clip gauge was this value. As expected, the 2t specimens in Figure 11 showed that large stable crack extension preceded before cleavage fracture, obviously not satisfying the ASTM E1921 requirement of . Thus, the criterion properly predicted the nonoccurrence of fracture for the 2t SE(B) specimen.
diagrams for the 3t specimens (Figure 12(b)) were different from those of the 2t specimens; that is, specimen id 1 showed “pop-in” phenomena. From the fracture surface of specimen id 1 in Figure 12(a), this pop-in was correlated with one of the two significant cleavage fracture areas on the fracture surface. Because ASTM E1921 states that “all pop-in crack initiation values for cracks that advance by a cleavage-driven mechanism are to be regarded as eligible data,” specimen id 1 showed fracture. This might have some relationship with the fact that exceeded for the 3t SE(B) specimen, but further study in the future is necessary. The fracture surface of specimen id 2 showed an obviously large stable crack extension. In summary, one of the two 3t specimens did not show fracture. Thus, it was concluded that the criterion properly predicted the nonoccurrence of fracture for the 3t SE(B) specimen.
In summary, it was concluded that the criterion could appropriately predict invalid fracture for the 2t and 3t SE(B) specimens.
This paper examined whether the failure criterion was applicable to the miniaturized SE(B) specimens. Past experience of application of the criterion to small SE(B) was for and mm for A533B steel . The guideline of the minimum miniaturized Chary type SE(B) specimen satisfying the ASTM E1921 requirements (e.g., and stable crack extension ; described as fracture in this paper) was known to be 5t in size . Considering the restriction for the material S55C and these backgrounds, 2t-, 3t-, 10t-, and 25t-sized Charpy type SE(B) were examined. Focus was placed on (1) whether the criterion could predict the occurrence of fracture and (2) whether is identical to that observed in full-sized specimens in the case where fracture occurred.
Regarding point (1), the results showed that the criterion could appropriately predict a fracture naturally for specimens for the 25t and 10t specimens. The criterion could also predict that fracture does not occur for the 3t and 2t specimens if the necessary condition, that is, “the stress level of is maintained as for increasing load,” is explicitly considered. This necessary condition was not stated in the original proposal of the criterion  but seems to be automatically satisfied for ordinary-sized specimens whose . Because is measured at a variable location of and because 4 increases with load to reflect the stable crack extension, cases such as Figures 9(b) and 9(c) indicate that the region of high stress level ahead of the crack cannot be maintained after the load shows a maximum for . Thus, qualitatively, large stable crack extension was expected.
Regarding point (2), obtained from the 10t specimen exhibited only a small difference with that of 25t. The minimum that could satisfy for the 10t specimen conservatively predicted experimental . The process can definitely be applied to the reverse case; obtained from the 10t mm specimen can be used to predict the obtained from the 25t specimens.
In summary, the results from this work appear to demonstrate the applicability of the modified Ritchie-Knott-Rice (i.e., ) failure criterion for examining the feasibility of miniaturized Charpy type SE(B) specimens. It is expected that if a fracture is predicted by the criterion and the implicit necessary condition (i.e., “the stress level of is maintained as for increasing load”) is satisfied for a miniaturized SE(B) specimen for a specific material and temperature, it is possible to transfer obtained using this miniaturized specimen to the expected for a 25t SE(B) specimen.
There might be an opinion that in the experiments cleavage fracture occurs after microductile (or stable) crack extension, whereas in the FE-modeling a stationary crack without crack extension is treated. Strictly speaking, the current FE-modeling reflects some crack extension due to crack-tip blunting, which has been understood to correspond to the stretch-zone. Though the stable crack extension is very large compared with this stretch-zone, the conclusions of the current study are considered as valid, because the effect of the crack-length on the critical stress is small. The criterion was originally proposed to transfer ’s obtained with specimens of different crack length  and was the critical value that is independent of the crack length.
There might be another opinion that the condition “stress level of is maintained for increasing load” is questionable. However, it has long been known that crack-tip stress distribution reaches a steady state above some load level if distance from the crack-tip is normalized as () . Usually, has a linear relationship with in this load range; thus, is expected to be constant and independent of . Considering the fact that fracture was always observed to occur after reached for A533B steel  and for S55C steel in this work and in , the minimum that can satisfy seems to correspond to the minimum observed with the specimen configuration and the material at a specific temperature. This finding is consistent with the opinion of Chen et al. that “the minimum fracture toughness is a definite parameter determined by the specimen geometry and yielding properties”  and will be examined in more detail in the future.
This paper examined whether the failure criterion, which is one of the modified RKR criteria, was applicable to the miniaturized Charpy type SE(B) specimens, that is, (1) whether the criterion could predict the occurrence of cleavage fracture accompanied by negligibly small stable crack extension (denoted as fracture in this paper) and (2) whether is identical to that observed in full-sized specimens in the case where fracture occurred. 25t, 10t, 3t, and 2t Charpy type SE(B) specimens manufactured with S55C steel were tested at 20°C. The results showed that the criterion could appropriately distinguish the occurrence of fracture, by explicitly considering the necessary condition (i.e., “the stress level of is maintained as for increasing load”) with the original criterion (i.e., “ fracture occurs when exceeds a critical value ”). The obtained for the 25t and 10t specimens exhibited only a small difference, indicating that obtained from the 10t mm specimens can be used to predict the that will be obtained with the 25t specimens.
|:||obtained from FEA|
|:||SIF corresponding to the fracture load|
|:||Cleavage fracture toughness =|
|:||Parameter that gives information regarding the initial ligament size to fracture process zone size:|
|:||Crack-mouth opening displacement (CMOD)|
|:||Stable crack extension|
|:||Initial ligament size:|
|:||Crack-tip opening displacement (CTOD)|
|:||True and nominal tensile strength|
|:||True and nominal yield stress|
|:||Critical crack-opening stress|
|:||measured at a distance from the crack-tip equal to four times at the specimen midplane|
|25t, 10t, 3t, and 2t specimens:||SE(B) specimens of , with values of 25, 10, 3, and 2 mm|
|fracture:||Specimens which resulted with cleavage fracture which satisfied ASTM E1921 requirement of <|
|ASTM:||American Society for Testing and Materials|
|A533B:||ASTM A533 Grade B class 1 steel|
|CTOD:||Crack-tip opening displacement|
|DBTT:||Ductile-to-brittle transition temperature|
|EP-FEA:||Elastic-plastic finite element analysis|
|IAEA:||International Atomic Energy Agency|
|JIS:||Japanese Industrial Standards|
|RPV:||Reactor pressure vessel|
|S55C:||JIS 0.55% carbon steel|
|SE(B):||Single-edge notched bend bar|
|SWL:||Statistical weakest link|
|TST:||Test specimen thickness.|
The authors declare that they have no competing interests.
Part of this work was supported by Chubu electric power’s research based on the selected proposals. This support and Dr. Hideki Yuya’s advice were greatly appreciated. Discussion with Dr. Hiroaki Kurishita, former Professor at Tohoku University was also helpful. Testing of miniaturized SE(B) specimens was conducted as a joint research work between the authors and Kobe Material Testing Laboratory Co., Ltd. Mr. Kazuto Nakadate, Mr. Katsuya Taguchi, Ms. Yuriko Inoue, Mr. Kazukiyo Takahashi, Mr. Katsuki Yoshioka, Mr. Yoshihiro Saeki, and Mr. Satoshi Yuki contributed to the testing and are appreciated.
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