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Structural Integrity
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Procedia Structural Integrity 2 (2016) 1643-1651
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21st European Conference on Fracture, ECF21, 20-24 June 2016, Catania, Italy
Application of Coupled Damage and Beremin Model to Ductile-Brittle Transition Temperature Region Considering Constraint
Effect
Kiminobu Hojoa*, Naoki Ogawab, Takatoshi Hirotab, Kentaro Yoshimotob, Yasuto Nagoshia, and Shinichi Kawabataf
"Mitsubishi Heavy Industries, Ltd., 1-1, 1-chome, Hyogo-ku, Kobe, 652-8585 Japan bMitsubishi Heavy Industrins, Ltd., 1-1, 2-chome, Shinhama, Arai-cho, Takasago, 67J-8686 Japan c Ryoyu System Enyineering Co., Ltd., 1-6, 5-chome Knmatu-dAri, Hyogo-ku, Kobe, 652-0865 Japan
Abstract
For nuclear safety, fracture evaluation of reactor pressure vessels (RPV) under neutron irradiation is key issue. Fracture toughness from a CT specimen is used as a material constant for fracture evaluation, but it is well known that it has a targe constraint, which causes lower toughness than that of flowed structures, such as a eePV with a surface flow. In ductile to brittle toinsition temperature SDBTT) seiion ferric steel which is mdterial of RPV has a large scatter and it becomes important to know the accurate scatter of an irradiated matefial becaune of less maTgin o°" IRPV's integrity efter u long term operation. In this paper to establish a more precise fracture evaluation mathod in DBTT region for an ^ad^iat^i^ ItPV wgit]!:"^ a postulated surface flaw, a coupled model of damage mechanics for ductilf fracture and Beremin model for cleavage fracture was applied for correction of che effect o° a small ductile growth on the stress-strain lfield. To confirm the validity of the method, as the Arse trial, fracture testf using CT specimens were performed in several temperature conditions. The temperature dependence of the parameters of Beremin models were wvestigated as well.
Copy^t © 2(fl6 The AuUbMfe fu^ltlisli^c^ t>;ff liliievfst" B.V. Tjnig is an open i^ccctsss article under the CC BY-NC-ND license
(http://creativecommons.org/licenses/by-nc-nd/4.0/).
Peer-review under re spornibility of tde Stientific Committee ofECF21 .
Keywords: ductile to britttf traiaiitioa tsnpsratrrs (DBTT); Bfrfmia moCft; Camagf mechanics, Curson moCft; Roriiftifr moCft; CT specimen; Wfibrtt itrfii
* Corresponding author. Til.: +81-78-672-2249; fax: +81-78-685-2399. E-mail address: kiminobu_hojo@mhi.co.jp
Copyright © 2016 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license
(http://creativecommons.org/licenses/by-nc-nd/4.0/).
Peer review under responsibility of the Scientific Committee of ECF21.
10.1016/j.prostr.2016.06.208
1. Introduction
Irradiation embrittlement of RPV is an important issue because of its direct impact on plant restart or plant life extension. In Japan, there is an industrial code for the RPV's integrity evaluation in the PTS event considering neutron irradiation embrittlement. A revision of the JAEAC 4206 (2016) by introducing the latest knowledge such as the stress intensity factor considering the effect of the cladding, and the fracture toughness curve based on the master curve concept reduces the non-ductile fracture margin. The code is based on the conventional fracture mechanics approach using fracture toughness from CT specimen. It is well known that actual structures have larger toughness than that of CT specimens because of the constraint effect. For brittle fracture, Weibull stress proposed by Beremin (1983) and Mudry (1986) to consider the constraint effect, and the applicability of the local approach with a parameter of Weibull stress has been investigated by many researches like Minami et al. (1992), Ruggieri et al. (1993), Gao et al. (1998), and Wiesner et al. (1996). In the local approach, the parameters m and ou are regarded as material constants. Because the local approach is based on the weakest link theory, no ductile crack growth is assumed. On the other hand, for ductile fracture, damage mechanics like GTN model or Rousselier model has been applied to different constraint models. Le Delliou et al. (2014) determined the parameters of Rousselier model using a notched tensile specimen and simulated the ductile fracture behaviors of a CT specimen and a flawed large scaled pipe subjected to four point bending. When the PTS event of RPV occurs in the DBTT region of low alloy steel, a small ductile crack growth of 1 mm order or less is generated and the prediction accuracy by Beremin model decreases. To resolve this problem, Eriprit et al. (1996) developed the coupled model with Rousselier model and applied to SENB specimens. The similar approach was taken by Samal et al. (2008) and Gehrlicher et al. (2014). They applied the coupled model of Beremin and non-local Rousselier model to estimate KJc of low alloy steel using Weibull parameters at -100°C and Rousselier's parameters at room temperature. Most of the researches have been performed employing the fracture data of the material test specimens like CT or SENB specimen and a few papers mention the investigation results using a cylindrical or flat plate specimen with a surface flaw (Minami et al. (2006), Corre et al. (2006)).
From the background above, as the first step, the authors tried to apply the coupled Beremin and Damage models to estimate KJc values of CT specimen in DBTT region, and its applicability was discussed.
Nomenclature
DBTT ductile-brittle transition temperature
FEA finite element analysis
GTN Gurson-Tvergaard-Needleman
MC master curve method
NT notched tensile round bar
PTS pressurized thermal shock
RPV reactor pressure vessel
SENB single edge notched bend
D parameter of Rousselier model
m shape parameter of Weibull distribution
f void volume fraction
fc critical void volume fraction
fF void volume fraction at final failure
qb q2, and q3 parameters of GTN model
Ф yield function
01 maximum principal stress
o0 yield stress
oeq von Mises equivalent stress
ok parameter of Rousselier model
om hydro static stress
ou scale parameter of Weibull distribution
ctw Weibull stress p notch radius
2. Application model
The following items were focused on this research including a basic aspect.
• Equivalence of the parameters of GTN model from NT and CT specimens
• Temperature dependence of Weibull parameters
• Difference between the predicted fracture behaviors of GTN model and Rousselier model
• Difference between the predicted fracture behaviors of Beremin model and the coupled model
For investigation of these items, Beremin model for cleavage fracture, GTN and Rousselier models for ductile fracture were chosen. Evaluation equation of each model is shown below.
(1) Beremin model
Pf = 1 - exp
v v ' y
— f (fo )mdV v Jv„ v 1
V Vo V
m and au are the shape parameter and the scale parameter of Weibull distribution and theoretically Weibull stress at the onset of cleavage fracture should show Weibull distribution according to these parameters. Integral was performed in the region where von Mises stress exceeded yield stress. In DBTT region both of cleavage fractures with or without ductile crack growth can happen. Weibull parameter m is assumed as the material constants that are independent of temperature in the case of cleavage fracture without ductile crack growth in the previous papers (Bernauer et al. (1999), Samal et al. (2008)). In this paper m's independence of temperature was confirmed. Also some papers determine Weibull parameters from NT specimens and other papers determine those from CT specimens. The authors investigated its equivalence of the parameter's sets from both types of specimens as well.
(2) GTN model
GTN model simulates the void's behavior of initiation, growth and coalescence by expression of the yield function.
\ 2 / \
3 q2°m .( + q3f2)= o (3)
+ 2 fq1 cosh
L + k (f - fc) 1/q - L
Lf - fc
for for
f ± fc f > fc
When the secondary void nucleation is considered, the terms relating to the initial void and the secondary void are divided. The parameters of GTN model are determined by NT specimen or CT specimen depending on the researchers. Equivalence of the parameters from both specimens is discussed in this paper.
(3) Ro/ssiliir model
Rousselier model is often applied to evaluate ductile crack growth as another model of damage mechanics. An advantage over GTN model is that the model has less parameters as shown in the following yield function. ak and D are the parameters of Rousselier model.
ia V ( „ \
iq 1-f
+&kDf exp
(i - f >
-CT0 = o (6)
On the other hand this model has disadvantage of not considering of the secondary void's nucleation. In the paper the difference between the predicted behaviors by GTN model and Rousselier model is investigated.
(4) Co/plid model
As later described, GTN model and Beremin model are coupled to evaluate cleavage fracture after small ductile crack growth. The prediction results from the coupled model and Beremin model are compared. By using the coupling model, KJcs of flat plate specimens can be predicted from the fracture test of CT specimens without a ductile crack growth (at temperature -i25°C). Weibull parameters of m and au are assumed constant in different temperature and different constraint conditions if cleavage fracture is assured.
3. Experiment
In order to determine the parameters of each model and investigate the items mentioned above, the fracture tests were performed using three kinds of specimens at temperature -i25°C, -95°C, or -50 °C.
3.1. Material ond test condition
The specimens were made of low alloy pressure vessel steel ASTM A533 Grade B Class i (SQV2A (JIS G 3120)). Chemical compositions are shown in Table i. Table 2 gives a test matrix to determine the stress-strain (S-S) curves necessary to perform the FE analyses and the parameters of Bermin model or GTN model. The specimens used were round tensile bar, notched round tensile bar (NT), and 1/2TCT specimens. Geometries of the specimen are shown in each figure. Test data from the reference (Yoshimoto (2013)) were utilized to complement the data of this research. Tensile and fracture toughness tests were performed in accordance with JIS Z2241:1998 and ASTM E1921-10, respectively.
Table 1. Chemical compositions of A533B (weight %).
C Si Mn P S Ni Cr Mo Cu V Co B Al 0.19 0.26 1.38 0.007 0.008 0.62 0.15 0.48 0.09 0.01 0.023 0.0009 0.029
Table 2. Test matrix for investigations of the S-S curves and parameters of Beremin and GTN models.
Type of specimen Geometry Temperature (°C)
-125 -95 -50
Round tensile bar Fig. 1 (a) 2(*) 2(*) 1
NT (p=0.1mm) - - 1
NT (p=0.25mm) Fig. 1 (b) - - 1
NT (p=0.5mm) 1 - 4
1/2TCT Fig. 1 (c) 15(*) 15(*) 7
(*): Yoshimoto et al. (2013))
(a) Tensile round bar 90
R10 R10
(b) NT specimen
p= 0.1, 0.25, 0.5mm Details of A
W=25.4
B=12.7
(c) 1/2TCT specimen
Fig. 1 Specimen geometry. (unit: mm)
3.2. Test results
Material properties from the tensile tests using the round bar specimens are shown in Table 3. Figure 2 shows the true stress-true strain curve at each temperature for input of the FEA. Table 4 shows the test results of the notched tensile round bar specimens. No ductile crack initiation was observed at -125°C and cleavage fracture occurred at the bottom of the notch. For all of the specimens at -50°C, ductile crack growth from the notch was observed and the amount of the crack growth increases along with the notch radius.
Table 3. Mechanical properties.
Temperature (°C) 0.2% proof stress (MPa) Ultimate strength (MPa) Elongation (%) Reduction of Area (%)
-125 689 788 24.3 67.0
-95 621 748 24.1 65.6
-50 539 695 25.4 69.0
True strain (%)
Fig. 2. True stress- true strain curves for input of FEA.
Table 4. Fracture test results of notched round bar specimens.
Temperature (°C) Failure load Displacement Diameter Ductile crack
(mm) load (kN) (kN) at failure (mm) after test (mm) Area (%) Initiation location Aa (mm)
-125 0.5 108 108 0.64 9.55 9.0 No ductile crack
0.1 93 93 0.70 9.52 9.4 Notch 0.30
0.25 94 90 0.90 9.23 14.3 Notch 0.35
-50 0.5 93 90 1.12 8.97 19.2 Notch 0.44
0.5 94 92 1.09 9.10 17.5 Notch 0.29
0.5 94 90 1.19 8.97 19.7 Notch 0.27
0.5 94 91 1.13 9.05 18.3 Notch 0.40
Table 5 shows the results of the fracture toughness tests of 1/2TCT specimens at -50°C. The obtained data do not satisfy the validity condition of ASTM E1921-10 which relates to the limitation of KIc and the amount of ductile crack growth. No ductile crack initiation was observed at -125°C and very small ductile crack growth less than 0.1mm occurred at -95°C.
Table 5. Fracture toughness test results of 1/2TCT specimens.
Temperature (°C) Fracture toughness Jc (kJ/m2) Klc (MPaVm) Klc (1TCT) (MPaVm) Kjc(limit) (MPaVm) Validity Ductile crack growth Aa (mm)
688 392 333 226 invalid 1.01
612 370 314 226 invalid 0.86
514 339 288 226 invalid 0.62
-50 313 264 225 222 invalid 0.40
595 364 310 219 invalid 0.71
232 228 195 222 invalid 0.20
220 222 190 221 invalid 0.15
4. Analysis and discussion
4.1. Damage mechanics model
At first the difference of the predicted fracture behaviors from the numerical models was investigated using NT specimen. The focused numerical models are GTN model, Rousselier model and the computational cell model (Tvergaard (1982), Shih et al. (1995)) using GTN model. The FE code used was Abaqus (Ver.6.12-3). The S-S curves that were used are as shown in Fig. 2. The focused test data were those of NT specimen with p=0.5mm at -50°C. The minimum mesh size of both specimens was 0.03mm from the critical CTOD of fracture toughness test of 1/2TCT specimen. The parameters of the models were determined by the optimized method (Watanabe et al. (2014)) and the load-clip gauge displacement curve. Figure 3 shows the comparison between the stress contour of von Mises of the cell model and normal model using GTN model. All elements of the normal model have the constitution law of GTN model. From the figure the stress of the cell model concentrates on the notch, but that of the normal model shows an irregular distribution, which predicts that the fracture will initiate outside the cracked section. This cannot be actually observed. Figure 3 also shows the stress contour by Rousselier model, which resembles that of the cell model of GTN model.
Secondly the similar investigation was carried out to the fracture test result of 1/2TCT specimen at -50°C. The FE codes used, the size of mesh and the optimized method for determination of the parameters are the same as the NT specimen. The load-load line displacement curve of the fracture test was used for parameter fitting. Only the cell model was applied for the case of GTN model based on the knowledge of the result of NT specimen. Figure 4 shows the comparisons between the ductile crack growth distribution along the crack front from the fracture tests and the FEA by GTN and Rousselier model. GTN model simulated the actual behavior more appropriately than Rousselier model, especially around the side grooves. Further investigation was made by GTN model. The
parameters of GTN models from NT and CT specimens seem similar, but the parameter set from CT specimen could not predict the load-clip gage displacement relation of NT specimen.
(a) Normal model of GTN model (b) Cell model of GTN model (c) Rousselier model
Fig. 3. Stress contours of normal and Cell model of GTN model and Rousselier model
-0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 0.4 0.5 Thickness direction x/t
-0.5 -0.4 -0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 0.4 0.5 Thickness direction x/t
- CT-44 CT-45 CT-46 -CT-51 -CT-52 -CT-53 -CT-54 . CT-44 (FEA) CT-45 (FEA) CT-46 (FEA) CT-51 (FEA) CT-52 (FEA) CT-53 (FEA) CT-54 (FEA)
(a) GTN model (b) Rousselier model
Fig. 4. Comparison between ductile crack growth distribution along the crack front from the fracture test and FEA.
When the larger displacement was applied to the FE model of CT specimen, some elements were crashed and the calculation was not converged. A larger sized mesh model with 0.1mm was produced to avoid this kind of behavior and the parameters were determined by the same procedure as mentioned above. Table 6 shows the parameters of GTN model. q1, q2, and q3 were fixed to 1.5, 1, and 2.25, respectively for the case of the 0.1mm mesh model.
Table 6 Parameters of GTN model from CT specimens (-50°C)
fF fc £n Sn fN fo
0.196 0.001 0.424 0.108 0.037 1.0 X10-10
4.2. Coupled model
From the test results of CT specimens at -125°C, Weibull parameters m and au were determined as 35 and 2005MPa, respectively. On the assumption that m is constant regardless of temperature, Weibull distribution of Weibull stress were calculated at -95°C and -50°C using Beremin model and the coupled model with GTN model.
Figure 5 shows the comparison between three kinds of Weibull distribution of aWc; (1) aWc by Beremin model with its own m, (2) aWc by Beremin model with m =35 (from the tests at -125°C) , and (3) aWc by the coupled model with m=35. The results of -95°C are the nearly same among three cases. This means that m at -125 °C is applicable
to the condition of -95°C. No ductile crack growth by the coupled model at -125°C (not shown in Fig. 5) and -95°C occurred. At -50°C the distribution by (1) is very steep («/=87). Figure 6 is crw-Kj relationof Beremin model and the coupled model at each temperature. K, relates to loading or applied displacement. crw of Beremin model saturates earlier than lower temperature. Saturation of crw is caused by saturation of the stress adjacent to the crack tip even when loading or displacement increases. At -125°C the specimens broke at ATJc=57-131 MP;n/m. which corresponds to crw =1700-1900 MPa. This range is less than the saturation value of crw =2300 MPa. This is true of the condition of -95°C. On the other hand, at -50°C, K}c is in 225-430 MPaVm, and <rw is in the saturated region for the larger K}c region. As a result, the gradient of Weibull distribution of crw becomes very steep in this region. For the smaller side of K,c. the gradient of Weibull distribution in the case (1) looks similar. It may suggest the parameter m is independent of temperature, at least in the range of -125 °C to -50°C.
The coupled model makes the v\\—KSc curve increase. The curves of the coupled model deviate at A"lc=260-280 MPaVnu which corresponds to crw=2200MPa (-125°C), 2100MPa(-95°C), and 1900 MPa (-50°C), just before saturation of crw These do not affect o\Vc at -125°C and -95°C, but affect largely at -50°C. As a result the coupled model correlates Weibull distribution of crWc only at -50°C. However the correlation may be too much because the slope of the Weibull distribution in Fig.5 is too gentle. Figure 7 shows the relation between Kk and temperature. Beremin and the coupled model can predict the fracture tests at -125 and -95°C, but the upper bound of K]c at -50 °C is not precisely predicted. Further investigation is needed.
97.5 95 90
4«? D
A Beremin Model(-125°C) □ Beremin Model(-95°C) ■ Beremin Model(-95°C, m=35) X Coupled Model(-95°C, m=35) O Beremin Model(-50°C) • Beremin Model(-50°C, m=35) + Coupled Model(-50°C, m=35)
— -1
7.65 In CTW
2500 -
3 1500
Fig. 5. Failure probability-<rw from Beremin model and Coupled model
-CT, Beremin Model(-125°C) -CT, Coupled Model(-125°C) CT, Beremin Model(-95°C)
■ CT, Coupled Model(-95°C) CT, Beremin Model(-50°C)
■ CT, Coupled Model(-50°C)
■CT, Beremin Model(-50°C, m=87)
400 600 /O(MPaVm)
Fig. 6. <rw -Kj relation from Beremin and Coupled model
ro 0. 300
£ 200
0 Experiments
1 CT, Beremin Model(-125°C) ■ CT, Beremin Model(-95°C)
• CT, Beremin Model(-50°C) X CT, Coupled Model(-125°C)
* CT, Coupled Model(-95°C) + CT, Coupled Model(-50°C)
(1033)
Temparature (°C)
Fig. 7 Relation between Klc and temperature by the simulation models and fracture toughness test
5. Conclusion
In order to consider the constraint effect on DBTT region, fracture tests using NT and CT specimens were performed and several models which can handle this effect were applied to predict their fracture behaviours. As for damage mechanics model, GTN model predicted more precisely ductile crack growth of CT specimen with side grooves than Rousselier model. The parameter set of GTN model from one type of specimens could not simulate the fracture behaviour of the other's. This means that the parameters dependence on the specimen type may exist. The coupled model could correlate owc-£jc curve in DBTT, but its prediction accuracy has to be improved. The authors have a plan to perform fracture toughness tests using flat plates with a surface flaw and the evaluation method will be verified.
Acknowledgements
Prof. Fumiyoshi Minami kindly gave very important and discerning advice to the authors. The authors sincerely acknowledge the scientific and technical advice of Prof. Minami for the research work.
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