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Fracture toughness transition curve estimation from a notched round bar specimen using the local approach method
Nuclear Engineering and Design 174 (1997) 247 – 258
Fracture toughness transition curve estimation from a notched round bar specimen using the local approach method K. Hojo a,*, I. Muroya a, A. Bru¨ckner-Foit b a
Takasago Research and De6elopment Center, Mitsubishi Hea6y Industries Limited, 2 -1 -1 Shinhama, Arai-cho, Takasago 676, Japan b IMF-II, Forschungszentrum Karlsruhe, Postfach 3640, D-76021 Karlsruhe, Germany Received 4 March 1996; received in revised form 11 October 1996; accepted 30 April 1997
Abstract The local approach method for the brittle or transition region describes the fracture probability of specimens or structures using Weibull statistics. Many papers have discussed the characteristics of the Weibull parameter using notched tensile specimens and the applicability to fracture toughness scatter evaluation using CT specimens. However few papers have made clear whether the Weibull parameter of the Weibull stress is a material property or not. In this paper the distribution of Weibull stress in the brittle fracture region using notched round bar specimens and CT specimens were investigated and it was confirmed that both distributions agreed well. Furthermore the estimation method for the fracture toughness transition curve including its scatter from notched round bar tensile tests was proposed based on the relation between the Weibull stress and the Wallin’s fracture toughness transition curve. As a result, the estimated fracture toughness curve in the brittle and lower transition region from the notched round bar specimens coincided with the measured fracture toughness curve from CT specimens. This method will be applicable to fracture toughness curve estimation under plane strain conditions even if there is no possibility of obtaining thick enough CT specimens from a structure because of geometry or some other restrictions. © 1997 Elsevier Science S.A.
1. Introduction Recent structural integrity assessment uses fracture mechanics parameters based on continuum mechanics. For example, the stress intensity factor K is used for brittle fracture and the experimental method to obtain the fracture toughness KIC has been standardized by ASTM. However the conventional fracture toughness tests specify the min* Corresponding author. Tel.: + 81 794 456716; fax: + 81 794 456795.
imum thickness to meet the plane strain condition and this may cause difficulty in obtaining a test specimen with the required thickness from a structure depending on the fracture toughness level. Furthermore it is well known that the fracture toughness scatters from the lower shelf to the transition region of the fracture toughness transition curve in ferritic steel. This requires many specimens to evaluate ferritic steel’s fracture behavior. The local approach to fracture has been introduced in order to quantitatively evaluate the scat-
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ter behavior of fracture toughness in the lower shelf region. In this method the Weibull stress was introduced as a parameter of Weibull statistics by Pineau (1981), Beremin (1983) and Mudry (1987) and the applicability has been investigated mainly using notched tensile specimens. Also recently the relation between conventional fracture mechanics parameter K and the Weibull parameter of the local approach was studied by Minami et al. (1992) and Tagawa et al. (1991, 1993). However there is no paper that confirms that the Weibull parameters from CT specimens and notched tensile specimens are the same values or describes the temperature dependency of the Weibull parameter in the lower transition region. This paper discusses the applicability to the fracture evaluation of CT specimens and notched tensile specimens and proposes an estimation method for the fracture transition curve from notched tensile specimens using the local approach.
2. Evaluation method
2.1. Local approach When the weakest link model can be applied to cleavage fracture of steel, failure probability P tot f of a structure is expressed by Eq. (2) using the Weibull stress sW defined by Eq. (1). sW $
& 1 V0
V pl
sm 1 dVpl
1/m
n
P tot f =1− exp −
sW su
(1)
m
(2)
where s1 is the maximum principal stress, V0 is the reference volume (1 mm3 was chosen for normalization), Vpl is the plastic zone volume, and m and su are the Weibull parameters. The Weibull parameters are considered as material constants. In order to obtain the Weibull stress and Weibull parameters, the detailed stress information of the plastic zone of the analyzed structure or specimen at fracture is required. Therefore the local approach needs numerical simulation using the finite element method (FEM). Fig. 1 shows
the Weibull statistics analysis flow chart with a CT specimen as an example. The procedure is as follows. (1) KICs scatter is obtained from the fracture toughness test using CT specimens. (2) After sorting and numbering the KIC values in ascending order, the failure probability Pf is estimated by (i− 0.5)/N where i is the rank number and N is the sample size. (3) The finite element model with fine mesh division in the vicinity of the crack tip is prepared. (4) and (5) The relation between KIC and stress distribution is found after the finite element elastic plastic analysis using the model in (3). (6) m0 and su0 are given as the initial Weibull parameters for the iterative calculation. (7) From (4) and (5) the stress distribution at fracture is estimated and the Weibull stress shown in the equation in Fig. 1 is calculated. (8) From the relation between KIC and fracture probability in (2) and the result of (7), the relation between Weibull stress and failure probability can be plotted on a Weibull statistics graph. Weibull shape parameter m1 and scale parameter su1 are obtained from this graph. (9) and (10) If m1 is not equal to m0, input values are changed to m1 and su1, and goes back to (7). (9) and (11) If m1 is equal to m0, the iteration has converged.
2.2. Fracture toughness transition cur6e of the lower shelf-transition temperature region In the lower shelf-transition temperature region of ferritic steel, fracture toughness KIC increases as temperature rises. Wallin (1993) proposed the temperature transition curve of ferritic steel including the KIC scattering curve after investigation of several kinds of ferritic steel. According to Wallin (1993) if the shift temperature T0 can be defined, it is possible to describe the temperature dependency of the steels’ fracture toughness with a single curve. The expression is given by the following equation:
Pf = 1−exp −
B KIC − Kmin B0 K0 − Kmin
n 4
(3)
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249
Fig. 1. Weibull parameter analysis flow.
where K0 is the temperature dependent scale parameter corresponding to 63.2% fracture probability, B is the thickness of the tested specimen, KIC is fracture toughness from the specimen with thickness B, and B0 is the reference thickness. Kmin is a
lower limiting KI value below which cleavage fracture is impossible and 20 MPa m is assumed. For ferritic steel, the temperature dependency of fracture toughness is described by Eq. (4) introducing shift temperature T0 (K).
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K0(MPa m) = 31 + 77 exp[0.019(T −T0)]
(4)
According to the local approach, in the case of cleavage fracture, the failure probability is expressed by Eq. (2). This equation is equivalent to Eq. (3) in the case of the fracture toughness test, therefore the following relation should exist:
sW su
m
KIC −Kmin K0 −Kmin
=
4
(5)
Under small scale yielding conditions, the stress field in the vicinity of the crack tip is approximated by Eq. (6) s1
K
(6)
g(u)
2pr
where r is the distance from the crack tip and g(u) is the function with the angle from the crack surface. In order to normalize the coordinate system, using u= rs 2y/K 2, Eq. (6) is rewritten as follows, s1 =
g(u)
2pu
sy
(7)
Similarly, normalizing small plastic zone area dApl with K and sy, dApl is equal to K 4/s 4y · dupl where dupl is dimensionless small plastic zone area. Assuming that the stress distribution is uniform in the thickness direction, Eq. (1) can be written as sW =
& 1 V0
u pl
g(u)
2pu
sy
K 4IC du s 4y pl
m
From Eq. (8) and Eq. (5), su =
K (K0 −K ) s BCu (KIC −K ) V 4 IC
4
m−4 y min 4 min 0
where Cu =
&& u pl
g(u)
2pu
n
1/m
n
(8)
1/m
(9)
m
u du duu
(10)
For simplification, neglecting Kmin, su =
−4 K 40s m y BCu V0
n
It can also be deduced from Eq. (11), K0 =
V0s m u −4 BCus m y
1/4
(12)
Therefore, if su, m and sy are found from the tensile tests, from Eq. (12) K0 is obtained. Consequently the shift temperature T0 can be defined by Eq. (4), finally from Eq. (3) the fracture toughness transition curve and its scatter from the lower shelf-transition region can be estimated.
3. Material test and numerical analysis In this section, for investigation of applicability of the local approach and Wallin’s fracture toughness transition curve to ferritic steel’s fracture behavior, the material tests and numerical simulation analysis using FEM were carried out.
3.1. Test conditions 3.1.1. Round bar tensile test Using the round bar tensile specimen shown in Fig. 2, the stress–strain curve was measured. The material is A533B reactor pressure vessel steel. The chemical composition is shown in Table 1. The number of test temperature points was 6 from − 150°C to 200°C. The speed of testing at room temperature and below was 9.81–29.4 MPa and that over room temperature was 0.1–0.5% min − 1. 3.1.2. Fracture toughness test Fracture toughness was obtained using 1/ 2TCT, 1TCT and 2TCT specimens with sidegrooves as shown in Fig. 2b. The material is the same as that for the round bar tensile specimen. The test temperatures were − 150°C and − 100°C. KIC was determined by ASTM E399-90. If KIC was not valid, JIC based on ASTM E813-89 was converted to the KIC value.
1/m
(11)
Hence the relation between Weibull parameters su, m and fracture toughness parameter K0 can be expressed by Eq. (11).
3.1.3. Notched tensile bar test Tensile tests were conducted using the notched tensile bar specimen shown in Fig. 2c. The material is the same as mentioned above, and twelve specimens were used at − 150°C. To compare the
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Fig. 3. Engineering stress – strain curve (temperature − 150°C).
3.2.2. Fracture toughness test The test results at − 150°C and − 100°C are shown in Tables 2 and 3, respectively. In this Table, to increase the test number for Weibull statistics, other test results using the same material are added. Fracture toughness from 1/2TCT and 2TCT specimens were converted to that from the 1TCT specimen by thickness correction. The equations of thickness correction were as follows: Fig. 2. (a) Round tensile bar specimen; (b) CT specimen; (c) notched tensile bar specimen.
result of FE analysis, four strain gauges were pasted at a distance of 20 mm from the notched section of each specimen.
3.2. Test results 3.2.1. Round bar tensile test An example of the stress strain curves is shown in Fig. 3. Fig. 4 shows the temperature dependency of the yield stress by exponential type curve fitting.
K(1TCT)= K1
B1 25
1/4
(13)
where K(1TCT) is the corrected KI value for the 1TCT specimen; K1 is the experimental KI value of the specimen used (MPa); and B1 is the nominal thickness of the specimen used (mm) From Table 2 all test results at −150°C show no ductile crack growth and relatively small scatter of KIC. On the other hand as shown in Table 3, at −100°C very small ductile crack growth can be seen from many specimen’s fracture surfaces and the scatter of KIC increases in comparison with − 150°C.
Table 1 Chemical composition of A533B C
Si
Mn
P
S
Ni
Cr
Mo
Cu
N
0.21
0.22
1.48
0.007
0.008
0.62
0.09
0.48
0.06
0.01
Weight %.
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3.3. Analysis 3.3.1. FE analysis
Fig. 4. Temperature dependency of yield stress.
3.2.3. Notched tensile bar test Table 4 shows the fracture load of each specimen. As the result of fracture surface observation, it is confirmed that the river patterns flow from the center of the fracture surface to the outer surface and there is no ductile crack growth. This means the onset of cleavage fracture in the center of the fracture surface.
3.3.1.1. Analytical condition. For the numerical simulation FE code ABAQUS 4.9-1 was used. The element type was an eight node plane strain element(CPE8H) for CT specimens and an eight node axial symmetric element (CAX8) for notched tensile bar specimens. Since cleavage fracture is assumed for the local approach the analysis temperature of the CT specimen was chosen to be − 150°C. The average value of the stress–strain curve in Fig. 3 was used for FE analysis and the material constants are as follows: Young’s modulus E=2.19× 105 MPa; Poisson’s ratio n= 0.3; and yield stress sy = 7.11× 102 MPa. Fig. 5a and b show FE mesh divisions and boundary conditions of the specimens. In order to calculate the detailed stress distribution at the stress concentration zone, the minimum element size is 0.05 mm for the CT specimen and 0.1 mm for the notched tensile bar specimen.
Table 2 Result of fracture toughness tests (temperature −150°C) Specimen No.
Specimen type
Initial crack length (mm)
Maximum load (kN)
Stable crack growth Da (mm)
KIC (MPa · m)
Note
CT01 CT02 CT03 CT04 CT05 CT06 CT07 CT08 CT09 CT10 CT11 CT12 CT13 CT14 CT15 CT16
ITCT ITCT ITCT ITCT ITCT ITCT ITCT ITCT ITCT ITCT ITCT 2TCT 2TCT 1/2TCT 1/2TCT 1/2TCT
28.6 28.2 28.6 28.4 28.6 28.6 28.6 28.6 28.5 28.0 28.3 55.6 55.0 14.4 14.2 14.2
19.7 22.7 19.0 20.7 20.5 16.0 18.5 20.9 18.4 14.5 18.1 38.2 67.6 7.7 4.4 5.0
0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00
48.3 54.1 46.6 50.1 50.2 39.2 45.1 51.2 44.9 36.5 44.1 44.1 66.5 45.1 25.1 25.1
A A A A A A A A A A A A B B A A
Note: A, valid KIC in ASTM; B, converted KIC from J.
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Table 3 Result of fracture toughness tests (temperature −100°C) Specimen No.
Specimen type
Initial crack length (mm)
Maximum load (kN)
Stable crack growth Da (mm)
KIC (MPa · m)
Note
CT21 CT22 CT23 CT24 CT25 CT26 CT27 CT28 CT29 CT30 CT31 CT32 CT33 CT34 CT35
ITCT ITCT ITCT ITCT ITCT ITCT ITCT ITCT ITCT ITCT ITCT 2TCT 2TCT 1/2TCT 1/2TCT
28.5 28.6 28.6 28.6 28.6 28.6 28.4 28.6 28.5 28.5 28.4 55.8 55.2 14.2 14.1
31.2 42.2 37.9 22.5 40.8 44.0 28.4 45.0 43.9 43.6 24.4 38.2 147.7 10.5 9.3
0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00
80.2 119.6 103.8 58.9 114.1 130.5 71.0 133.8 125.1 65.2 55.8 56.1 128.1 56.1 128.1
A A A A A A A A A A A A B B B
Note: A, valid KIC in ASTM; B, converted KIC from J.
3.3.2. Results 3.3.2.1. CT specimen. As an example of the stress distribution, von Mises’s equivalent stress and the principal stress contours are shown in Fig. 6a and b respectively. Fig. 7 shows the stress distribution vertical to the crack surface near the crack tip of the CT specimen. From these figures, the stress distribution is very smooth and the precise stress results seem to be obtained. Table 4 Result of notched tensile bar tests (temperature −150°C) Specimen No.
Maximum load (kN)
N1BIM-1 N1BIM-2 N1BIM-3 N1BIM-4 N1BIM-5 N1BIM-6 N1BIM-7 N1BIM-8 N1BIM-9 N1BIM-10 N1BIM-11 N1BIM-12
98.5 97.6 96.6 99.1 97.1 88.8 90.6 93.6 98.4 93.9 97.0 101.6
3.3.2.2. Notched tensile bar test. From the FE analysis result it is shown that the maximum principal stress is large around the central axis, and it is estimated that the onset of fracture is the model center. This agrees well with the observation of the fracture surface as mentioned in Section 3.2. Fig. 8 shows the comparison of the fracture strain at a smoothed cross section of analysis and experimental results. Both coincide well and the FE analysis of the notched tensile bar specimen also gives a good result. 3.3.3. Weibull parameter analysis The Weibull stress and Weibull parameters were calculated using the post files of the FE code. The average value and 90% confidence interval of the Weibull parameters of the CT and those of the notched tensile bar specimens are shown in Table 5. From this Table the Weibull parameters of both specimens agree fairly well. Fig. 9 shows fracture probability for two kinds of specimens. As a result there is no remarkable difference between the parameters of both specimens, and these results give one of the proofs that the Weibull parameters su and m are material constants independent of the specimen geometry.
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254
Fig. 5. (a) Mesh division of the CT specimen; (b) mesh division of the notched tensile bar specimen.
4. Discussion
4.1. Applicability of Wallin’s fracture toughness transition cur6e In order to show the applicability of Wallin’s fracture toughness transition curve, the parameter T0 of Eq. (4) is determined based on the experimental data in Table 2. T0 can be derived from
the relation K0 = 1.1Kave and Eq. (4), where Kave is the average value of fracture toughness. The factor 1.1 of the relation of K0 and Kave can be derived from Eq. (3). If the variable x obeys the Weibull distribution, the failure probability F(x) is expressed by the following equation:
F(x)= 1− exp −
x b
m
(14)
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Fig. 6. (a) von Mises equivalent stress contour (CT specimen, temperature −150°C); (b) principal stress contour (CT specimen, temperature − 150°C).
K. Hojo et al. / Nuclear Engineering and Design 174 (1997) 247–258
256
Table 5 Weibull parameters CT specimen
Fig. 7. Stress distribution near the crack tip vertical to the crack surface (CT specimen, temperature −150°C).
where b is the scale factor and m is the shape factor. In this case, the expectation value, that is the average value E(x) is expressed by the following equation: E(x)= b
1 1 != bG +1 m m
1 Kave =Kmin + (K0 −Kmin)G +1 m
Shape factor m (90% 17.08 (12.17, confidence interval) 24.11)
17.43 (11.74, 26.25)
Shape factor su (MPa) (90% confidence interval)
1891.1 (1837.1, 1948.2)
1954.3 (1904.7, 2006.1)
interval in comparison with the experimental data. From this figure, although there are only two kinds of data at −150°C and − 100°C, most data are within the 90% confidence interval of Wallin’s prediction curve. This suggests Eqs. (3) and (4) are applicable to estimate the fracture behavior at the lower shelf-transition temperature region.
(15)
Comaring Eq. (3) with Eq. (14) and using m= 4 in Eq. (15), the following equation is obtained.
Notched tensile bar specimen
(16)
Here if Kmin is neglected, K0 =1.1 Kave can be derived. Fig. 10 shows the estimated fracture toughness transition curve using resolved T0 value and upper and lower bounds defined by the 90% confidence
Fig. 8. Comparison of the measured strain and FE results for the notched tensile bar specimens on fracture.
4.2. Temperature dependence of the Weibull parameter su According to Tagawa et al. (1991), the Weibull parameter su has been considered temperature independent. However there has been no data logically explained. This will be discussed here. According to Eq. (11), su is a function of K0 and sy. When the relations between parameters K0, sy and temperature are obtained, the temperature dependency of su can be estimated. Since
Fig. 9. Comparison of fracture probability of the CT specimens and the notched tensile bar specimens.
K. Hojo et al. / Nuclear Engineering and Design 174 (1997) 247–258
257
Fig. 10. Wallin’s fracture transition curve and experimental results.
Fig. 12. Comparison of the fracture toughness transition curve of the CT specimens and the notched tensile bar specimens.
both relations have been obtained from the CT specimens and the round bar tensile specimens in the above sections, it is possible to estimate the variation of su with temperature using Eq. (11) (Fig. 11). In this figure, su directly defined from the experimental results obtained with the notched tensile bars is also plotted for comparison. This figure shows su is almost constant from −150°C to − 100°C. Therefore it seems justified to assume that the Weibull parameters are independent of temperature in the lower shelf region. However, the behavior in the transition region is not clear, because ductile crack growth may introduce additional fracture mechanics.
4.3. Correlation of fracture beha6ior of the CT specimens and the notched tensile bar specimens Fig. 12 shows the fracture toughness curve from the CT specimen using Eqs. (3) and (4) and that from the notched tensile bar specimen using Eqs. (12), (3) and (4). Both curves agree well and there is a possibility that the fracture toughness curve from the lower shelf-transition region can be estimated from the notched tensile bar test at one temperature condition.
5. Conclusion In this paper it has been shown that for cleavage fracture the Weibull parameters are material constants independent of the specimen shape. Furthermore by relating the Weibull parameters and conventional fracture mechanics parameter K, the procedure that estimates the fracture toughness transition curve from the notched tensile bar test was proposed and its applicability was shown by fracture toughness tests and tensile tests.
References
Fig. 11. Temperature dependency curve for Weibull parameter of su.
Beremin, F.M., 1983. Metall. Trans. 14A, 2277 – 2287. Minami, F., Bru¨ckner-Foit et al, A., 1992. Estimation procedure for the Weibull parameters used in the local approach. Int. J. Fract. 54, 197 – 210.
258
K. Hojo et al. / Nuclear Engineering and Design 174 (1997) 247–258
Mudry, F., 1987. A local approach to cleavage fracture. Nucl. Eng. Des. 105, 65–76. Pineau, A., 1981. In: Francois, D. et al., (Eds), Proc. 5th Int. Conf. on Fracture, Cannes, vol. 2, Pergamon, Oxford, pp. 533-577. T.Tagawaetal.,1991.PredictionofCleavageFractureToughness by Statistical Local Approach, CAMP-ISIJ, Vol.4, p. 1887 (in Japanese).
Tagawa, T., Miyata, T., Aihara, S., Okamoto, K., 1993. Analysis of embrittlement in weld heat-affected zone due to formation of martensitic islands by local criterion approach. Iron Steel 79 (10) 55-61 (in Japanese). Wallin, K., 1993. Irradiation damage effects on the fracture toughness transition curve shape for reactor pressure vessel steels. Int. J. Pressure Vessels Piping 55, 61– 79.
.
.
Fracture toughness transition curve estimation from a notched round bar specimen using the local approach method K. Hojo a,*, I. Muroya a, A. Bru¨ckner-Foit b a
Takasago Research and De6elopment Center, Mitsubishi Hea6y Industries Limited, 2 -1 -1 Shinhama, Arai-cho, Takasago 676, Japan b IMF-II, Forschungszentrum Karlsruhe, Postfach 3640, D-76021 Karlsruhe, Germany Received 4 March 1996; received in revised form 11 October 1996; accepted 30 April 1997
Abstract The local approach method for the brittle or transition region describes the fracture probability of specimens or structures using Weibull statistics. Many papers have discussed the characteristics of the Weibull parameter using notched tensile specimens and the applicability to fracture toughness scatter evaluation using CT specimens. However few papers have made clear whether the Weibull parameter of the Weibull stress is a material property or not. In this paper the distribution of Weibull stress in the brittle fracture region using notched round bar specimens and CT specimens were investigated and it was confirmed that both distributions agreed well. Furthermore the estimation method for the fracture toughness transition curve including its scatter from notched round bar tensile tests was proposed based on the relation between the Weibull stress and the Wallin’s fracture toughness transition curve. As a result, the estimated fracture toughness curve in the brittle and lower transition region from the notched round bar specimens coincided with the measured fracture toughness curve from CT specimens. This method will be applicable to fracture toughness curve estimation under plane strain conditions even if there is no possibility of obtaining thick enough CT specimens from a structure because of geometry or some other restrictions. © 1997 Elsevier Science S.A.
1. Introduction Recent structural integrity assessment uses fracture mechanics parameters based on continuum mechanics. For example, the stress intensity factor K is used for brittle fracture and the experimental method to obtain the fracture toughness KIC has been standardized by ASTM. However the conventional fracture toughness tests specify the min* Corresponding author. Tel.: + 81 794 456716; fax: + 81 794 456795.
imum thickness to meet the plane strain condition and this may cause difficulty in obtaining a test specimen with the required thickness from a structure depending on the fracture toughness level. Furthermore it is well known that the fracture toughness scatters from the lower shelf to the transition region of the fracture toughness transition curve in ferritic steel. This requires many specimens to evaluate ferritic steel’s fracture behavior. The local approach to fracture has been introduced in order to quantitatively evaluate the scat-
0029-5493/97/$17.00 © 1997 Elsevier Science S.A. All rights reserved. PII S 0 0 2 9 - 5 4 9 3 ( 9 7 ) 0 0 1 2 5 - 8
K. Hojo et al. / Nuclear Engineering and Design 174 (1997) 247–258
248
ter behavior of fracture toughness in the lower shelf region. In this method the Weibull stress was introduced as a parameter of Weibull statistics by Pineau (1981), Beremin (1983) and Mudry (1987) and the applicability has been investigated mainly using notched tensile specimens. Also recently the relation between conventional fracture mechanics parameter K and the Weibull parameter of the local approach was studied by Minami et al. (1992) and Tagawa et al. (1991, 1993). However there is no paper that confirms that the Weibull parameters from CT specimens and notched tensile specimens are the same values or describes the temperature dependency of the Weibull parameter in the lower transition region. This paper discusses the applicability to the fracture evaluation of CT specimens and notched tensile specimens and proposes an estimation method for the fracture transition curve from notched tensile specimens using the local approach.
2. Evaluation method
2.1. Local approach When the weakest link model can be applied to cleavage fracture of steel, failure probability P tot f of a structure is expressed by Eq. (2) using the Weibull stress sW defined by Eq. (1). sW $
& 1 V0
V pl
sm 1 dVpl
1/m
n
P tot f =1− exp −
sW su
(1)
m
(2)
where s1 is the maximum principal stress, V0 is the reference volume (1 mm3 was chosen for normalization), Vpl is the plastic zone volume, and m and su are the Weibull parameters. The Weibull parameters are considered as material constants. In order to obtain the Weibull stress and Weibull parameters, the detailed stress information of the plastic zone of the analyzed structure or specimen at fracture is required. Therefore the local approach needs numerical simulation using the finite element method (FEM). Fig. 1 shows
the Weibull statistics analysis flow chart with a CT specimen as an example. The procedure is as follows. (1) KICs scatter is obtained from the fracture toughness test using CT specimens. (2) After sorting and numbering the KIC values in ascending order, the failure probability Pf is estimated by (i− 0.5)/N where i is the rank number and N is the sample size. (3) The finite element model with fine mesh division in the vicinity of the crack tip is prepared. (4) and (5) The relation between KIC and stress distribution is found after the finite element elastic plastic analysis using the model in (3). (6) m0 and su0 are given as the initial Weibull parameters for the iterative calculation. (7) From (4) and (5) the stress distribution at fracture is estimated and the Weibull stress shown in the equation in Fig. 1 is calculated. (8) From the relation between KIC and fracture probability in (2) and the result of (7), the relation between Weibull stress and failure probability can be plotted on a Weibull statistics graph. Weibull shape parameter m1 and scale parameter su1 are obtained from this graph. (9) and (10) If m1 is not equal to m0, input values are changed to m1 and su1, and goes back to (7). (9) and (11) If m1 is equal to m0, the iteration has converged.
2.2. Fracture toughness transition cur6e of the lower shelf-transition temperature region In the lower shelf-transition temperature region of ferritic steel, fracture toughness KIC increases as temperature rises. Wallin (1993) proposed the temperature transition curve of ferritic steel including the KIC scattering curve after investigation of several kinds of ferritic steel. According to Wallin (1993) if the shift temperature T0 can be defined, it is possible to describe the temperature dependency of the steels’ fracture toughness with a single curve. The expression is given by the following equation:
Pf = 1−exp −
B KIC − Kmin B0 K0 − Kmin
n 4
(3)
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249
Fig. 1. Weibull parameter analysis flow.
where K0 is the temperature dependent scale parameter corresponding to 63.2% fracture probability, B is the thickness of the tested specimen, KIC is fracture toughness from the specimen with thickness B, and B0 is the reference thickness. Kmin is a
lower limiting KI value below which cleavage fracture is impossible and 20 MPa m is assumed. For ferritic steel, the temperature dependency of fracture toughness is described by Eq. (4) introducing shift temperature T0 (K).
K. Hojo et al. / Nuclear Engineering and Design 174 (1997) 247–258
250
K0(MPa m) = 31 + 77 exp[0.019(T −T0)]
(4)
According to the local approach, in the case of cleavage fracture, the failure probability is expressed by Eq. (2). This equation is equivalent to Eq. (3) in the case of the fracture toughness test, therefore the following relation should exist:
sW su
m
KIC −Kmin K0 −Kmin
=
4
(5)
Under small scale yielding conditions, the stress field in the vicinity of the crack tip is approximated by Eq. (6) s1
K
(6)
g(u)
2pr
where r is the distance from the crack tip and g(u) is the function with the angle from the crack surface. In order to normalize the coordinate system, using u= rs 2y/K 2, Eq. (6) is rewritten as follows, s1 =
g(u)
2pu
sy
(7)
Similarly, normalizing small plastic zone area dApl with K and sy, dApl is equal to K 4/s 4y · dupl where dupl is dimensionless small plastic zone area. Assuming that the stress distribution is uniform in the thickness direction, Eq. (1) can be written as sW =
& 1 V0
u pl
g(u)
2pu
sy
K 4IC du s 4y pl
m
From Eq. (8) and Eq. (5), su =
K (K0 −K ) s BCu (KIC −K ) V 4 IC
4
m−4 y min 4 min 0
where Cu =
&& u pl
g(u)
2pu
n
1/m
n
(8)
1/m
(9)
m
u du duu
(10)
For simplification, neglecting Kmin, su =
−4 K 40s m y BCu V0
n
It can also be deduced from Eq. (11), K0 =
V0s m u −4 BCus m y
1/4
(12)
Therefore, if su, m and sy are found from the tensile tests, from Eq. (12) K0 is obtained. Consequently the shift temperature T0 can be defined by Eq. (4), finally from Eq. (3) the fracture toughness transition curve and its scatter from the lower shelf-transition region can be estimated.
3. Material test and numerical analysis In this section, for investigation of applicability of the local approach and Wallin’s fracture toughness transition curve to ferritic steel’s fracture behavior, the material tests and numerical simulation analysis using FEM were carried out.
3.1. Test conditions 3.1.1. Round bar tensile test Using the round bar tensile specimen shown in Fig. 2, the stress–strain curve was measured. The material is A533B reactor pressure vessel steel. The chemical composition is shown in Table 1. The number of test temperature points was 6 from − 150°C to 200°C. The speed of testing at room temperature and below was 9.81–29.4 MPa and that over room temperature was 0.1–0.5% min − 1. 3.1.2. Fracture toughness test Fracture toughness was obtained using 1/ 2TCT, 1TCT and 2TCT specimens with sidegrooves as shown in Fig. 2b. The material is the same as that for the round bar tensile specimen. The test temperatures were − 150°C and − 100°C. KIC was determined by ASTM E399-90. If KIC was not valid, JIC based on ASTM E813-89 was converted to the KIC value.
1/m
(11)
Hence the relation between Weibull parameters su, m and fracture toughness parameter K0 can be expressed by Eq. (11).
3.1.3. Notched tensile bar test Tensile tests were conducted using the notched tensile bar specimen shown in Fig. 2c. The material is the same as mentioned above, and twelve specimens were used at − 150°C. To compare the
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Fig. 3. Engineering stress – strain curve (temperature − 150°C).
3.2.2. Fracture toughness test The test results at − 150°C and − 100°C are shown in Tables 2 and 3, respectively. In this Table, to increase the test number for Weibull statistics, other test results using the same material are added. Fracture toughness from 1/2TCT and 2TCT specimens were converted to that from the 1TCT specimen by thickness correction. The equations of thickness correction were as follows: Fig. 2. (a) Round tensile bar specimen; (b) CT specimen; (c) notched tensile bar specimen.
result of FE analysis, four strain gauges were pasted at a distance of 20 mm from the notched section of each specimen.
3.2. Test results 3.2.1. Round bar tensile test An example of the stress strain curves is shown in Fig. 3. Fig. 4 shows the temperature dependency of the yield stress by exponential type curve fitting.
K(1TCT)= K1
B1 25
1/4
(13)
where K(1TCT) is the corrected KI value for the 1TCT specimen; K1 is the experimental KI value of the specimen used (MPa); and B1 is the nominal thickness of the specimen used (mm) From Table 2 all test results at −150°C show no ductile crack growth and relatively small scatter of KIC. On the other hand as shown in Table 3, at −100°C very small ductile crack growth can be seen from many specimen’s fracture surfaces and the scatter of KIC increases in comparison with − 150°C.
Table 1 Chemical composition of A533B C
Si
Mn
P
S
Ni
Cr
Mo
Cu
N
0.21
0.22
1.48
0.007
0.008
0.62
0.09
0.48
0.06
0.01
Weight %.
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3.3. Analysis 3.3.1. FE analysis
Fig. 4. Temperature dependency of yield stress.
3.2.3. Notched tensile bar test Table 4 shows the fracture load of each specimen. As the result of fracture surface observation, it is confirmed that the river patterns flow from the center of the fracture surface to the outer surface and there is no ductile crack growth. This means the onset of cleavage fracture in the center of the fracture surface.
3.3.1.1. Analytical condition. For the numerical simulation FE code ABAQUS 4.9-1 was used. The element type was an eight node plane strain element(CPE8H) for CT specimens and an eight node axial symmetric element (CAX8) for notched tensile bar specimens. Since cleavage fracture is assumed for the local approach the analysis temperature of the CT specimen was chosen to be − 150°C. The average value of the stress–strain curve in Fig. 3 was used for FE analysis and the material constants are as follows: Young’s modulus E=2.19× 105 MPa; Poisson’s ratio n= 0.3; and yield stress sy = 7.11× 102 MPa. Fig. 5a and b show FE mesh divisions and boundary conditions of the specimens. In order to calculate the detailed stress distribution at the stress concentration zone, the minimum element size is 0.05 mm for the CT specimen and 0.1 mm for the notched tensile bar specimen.
Table 2 Result of fracture toughness tests (temperature −150°C) Specimen No.
Specimen type
Initial crack length (mm)
Maximum load (kN)
Stable crack growth Da (mm)
KIC (MPa · m)
Note
CT01 CT02 CT03 CT04 CT05 CT06 CT07 CT08 CT09 CT10 CT11 CT12 CT13 CT14 CT15 CT16
ITCT ITCT ITCT ITCT ITCT ITCT ITCT ITCT ITCT ITCT ITCT 2TCT 2TCT 1/2TCT 1/2TCT 1/2TCT
28.6 28.2 28.6 28.4 28.6 28.6 28.6 28.6 28.5 28.0 28.3 55.6 55.0 14.4 14.2 14.2
19.7 22.7 19.0 20.7 20.5 16.0 18.5 20.9 18.4 14.5 18.1 38.2 67.6 7.7 4.4 5.0
0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00
48.3 54.1 46.6 50.1 50.2 39.2 45.1 51.2 44.9 36.5 44.1 44.1 66.5 45.1 25.1 25.1
A A A A A A A A A A A A B B A A
Note: A, valid KIC in ASTM; B, converted KIC from J.
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Table 3 Result of fracture toughness tests (temperature −100°C) Specimen No.
Specimen type
Initial crack length (mm)
Maximum load (kN)
Stable crack growth Da (mm)
KIC (MPa · m)
Note
CT21 CT22 CT23 CT24 CT25 CT26 CT27 CT28 CT29 CT30 CT31 CT32 CT33 CT34 CT35
ITCT ITCT ITCT ITCT ITCT ITCT ITCT ITCT ITCT ITCT ITCT 2TCT 2TCT 1/2TCT 1/2TCT
28.5 28.6 28.6 28.6 28.6 28.6 28.4 28.6 28.5 28.5 28.4 55.8 55.2 14.2 14.1
31.2 42.2 37.9 22.5 40.8 44.0 28.4 45.0 43.9 43.6 24.4 38.2 147.7 10.5 9.3
0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00
80.2 119.6 103.8 58.9 114.1 130.5 71.0 133.8 125.1 65.2 55.8 56.1 128.1 56.1 128.1
A A A A A A A A A A A A B B B
Note: A, valid KIC in ASTM; B, converted KIC from J.
3.3.2. Results 3.3.2.1. CT specimen. As an example of the stress distribution, von Mises’s equivalent stress and the principal stress contours are shown in Fig. 6a and b respectively. Fig. 7 shows the stress distribution vertical to the crack surface near the crack tip of the CT specimen. From these figures, the stress distribution is very smooth and the precise stress results seem to be obtained. Table 4 Result of notched tensile bar tests (temperature −150°C) Specimen No.
Maximum load (kN)
N1BIM-1 N1BIM-2 N1BIM-3 N1BIM-4 N1BIM-5 N1BIM-6 N1BIM-7 N1BIM-8 N1BIM-9 N1BIM-10 N1BIM-11 N1BIM-12
98.5 97.6 96.6 99.1 97.1 88.8 90.6 93.6 98.4 93.9 97.0 101.6
3.3.2.2. Notched tensile bar test. From the FE analysis result it is shown that the maximum principal stress is large around the central axis, and it is estimated that the onset of fracture is the model center. This agrees well with the observation of the fracture surface as mentioned in Section 3.2. Fig. 8 shows the comparison of the fracture strain at a smoothed cross section of analysis and experimental results. Both coincide well and the FE analysis of the notched tensile bar specimen also gives a good result. 3.3.3. Weibull parameter analysis The Weibull stress and Weibull parameters were calculated using the post files of the FE code. The average value and 90% confidence interval of the Weibull parameters of the CT and those of the notched tensile bar specimens are shown in Table 5. From this Table the Weibull parameters of both specimens agree fairly well. Fig. 9 shows fracture probability for two kinds of specimens. As a result there is no remarkable difference between the parameters of both specimens, and these results give one of the proofs that the Weibull parameters su and m are material constants independent of the specimen geometry.
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Fig. 5. (a) Mesh division of the CT specimen; (b) mesh division of the notched tensile bar specimen.
4. Discussion
4.1. Applicability of Wallin’s fracture toughness transition cur6e In order to show the applicability of Wallin’s fracture toughness transition curve, the parameter T0 of Eq. (4) is determined based on the experimental data in Table 2. T0 can be derived from
the relation K0 = 1.1Kave and Eq. (4), where Kave is the average value of fracture toughness. The factor 1.1 of the relation of K0 and Kave can be derived from Eq. (3). If the variable x obeys the Weibull distribution, the failure probability F(x) is expressed by the following equation:
F(x)= 1− exp −
x b
m
(14)
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Fig. 6. (a) von Mises equivalent stress contour (CT specimen, temperature −150°C); (b) principal stress contour (CT specimen, temperature − 150°C).
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Table 5 Weibull parameters CT specimen
Fig. 7. Stress distribution near the crack tip vertical to the crack surface (CT specimen, temperature −150°C).
where b is the scale factor and m is the shape factor. In this case, the expectation value, that is the average value E(x) is expressed by the following equation: E(x)= b
1 1 != bG +1 m m
1 Kave =Kmin + (K0 −Kmin)G +1 m
Shape factor m (90% 17.08 (12.17, confidence interval) 24.11)
17.43 (11.74, 26.25)
Shape factor su (MPa) (90% confidence interval)
1891.1 (1837.1, 1948.2)
1954.3 (1904.7, 2006.1)
interval in comparison with the experimental data. From this figure, although there are only two kinds of data at −150°C and − 100°C, most data are within the 90% confidence interval of Wallin’s prediction curve. This suggests Eqs. (3) and (4) are applicable to estimate the fracture behavior at the lower shelf-transition temperature region.
(15)
Comaring Eq. (3) with Eq. (14) and using m= 4 in Eq. (15), the following equation is obtained.
Notched tensile bar specimen
(16)
Here if Kmin is neglected, K0 =1.1 Kave can be derived. Fig. 10 shows the estimated fracture toughness transition curve using resolved T0 value and upper and lower bounds defined by the 90% confidence
Fig. 8. Comparison of the measured strain and FE results for the notched tensile bar specimens on fracture.
4.2. Temperature dependence of the Weibull parameter su According to Tagawa et al. (1991), the Weibull parameter su has been considered temperature independent. However there has been no data logically explained. This will be discussed here. According to Eq. (11), su is a function of K0 and sy. When the relations between parameters K0, sy and temperature are obtained, the temperature dependency of su can be estimated. Since
Fig. 9. Comparison of fracture probability of the CT specimens and the notched tensile bar specimens.
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Fig. 10. Wallin’s fracture transition curve and experimental results.
Fig. 12. Comparison of the fracture toughness transition curve of the CT specimens and the notched tensile bar specimens.
both relations have been obtained from the CT specimens and the round bar tensile specimens in the above sections, it is possible to estimate the variation of su with temperature using Eq. (11) (Fig. 11). In this figure, su directly defined from the experimental results obtained with the notched tensile bars is also plotted for comparison. This figure shows su is almost constant from −150°C to − 100°C. Therefore it seems justified to assume that the Weibull parameters are independent of temperature in the lower shelf region. However, the behavior in the transition region is not clear, because ductile crack growth may introduce additional fracture mechanics.
4.3. Correlation of fracture beha6ior of the CT specimens and the notched tensile bar specimens Fig. 12 shows the fracture toughness curve from the CT specimen using Eqs. (3) and (4) and that from the notched tensile bar specimen using Eqs. (12), (3) and (4). Both curves agree well and there is a possibility that the fracture toughness curve from the lower shelf-transition region can be estimated from the notched tensile bar test at one temperature condition.
5. Conclusion In this paper it has been shown that for cleavage fracture the Weibull parameters are material constants independent of the specimen shape. Furthermore by relating the Weibull parameters and conventional fracture mechanics parameter K, the procedure that estimates the fracture toughness transition curve from the notched tensile bar test was proposed and its applicability was shown by fracture toughness tests and tensile tests.
References
Fig. 11. Temperature dependency curve for Weibull parameter of su.
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