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Effect of crack depth on the shift of the ductile–brittle transition curve of steels
Engineering Fracture Mechanics 74 (2007) 2437–2448 www.elsevier.com/locate/engfracmech
Effect of crack depth on the shift of the ductile–brittle transition curve of steels Zhong-An Chen a, Zhen Zeng a, Yuh J. Chao
b,*
a
b
Jiangsu University, Jiangsu, Zhenjiang, China Department of Mechanical Engineering, University of South Carolina, Columbia, SC 29208, USA Received 13 May 2006; received in revised form 6 November 2006; accepted 7 November 2006 Available online 17 January 2007
Abstract Nuclear reactor pressure vessel (RPV) steels degrade due to neutron irradiation during normal operation. As a result, the ductile–brittle transition curve of the steel shifts to higher temperature which decreases operation margins in both the temperature and pressure. The loss of these margins however can be offset somewhat by appealing to arguments based on constraint of potential/postulated shallow cracks. In this paper, it is demonstrated that the fracture toughness values for shallow flaws are higher than those determined from standard deep cracked test specimens based on constraint consideration. The J–A2 three-term solution is used to characterize the crack-tip stress field where J represents the level of loading and A2 quantifies the level of constraint. Based on the RKR cleavage model, procedures to quantify the temperature shift between specimens with different constraint levels are developed. The experimental data by Sherry et al. [Sherry AH, Lidbury DPG, Beardsmore DW. Validation of constraint based structural integrity assessment methods. Final report, Report No. AEAT/RJCB/RD01329400/R003, AEA Technology, UK, 2001] for the A533B RPV steel are used to demonstrate the procedure and it is shown that the ductile–brittle transition curve shifts to lower temperature from high constraint to low constraint specimens. 2006 Elsevier Ltd. All rights reserved. Keywords: Nuclear reactor pressure vessel; Constraint effect; J–A2; Cleavage fracture; Ductile–brittle transition curve
1. Introduction Steels in general are ductile at high temperature and brittle at low temperature. This material behavior is normally presented as the ductile–brittle transition curve and plotted as fracture energy or toughness versus temperature. A typical ductile–brittle transition curve is shown in Fig. 1. Under normal operating conditions, pressure vessels are designed to operate at temperature above the onset of upper shelf toughness (OUST) in Fig. 1 in order to avoid brittle fracture. In RPV the criteria for safety include consideration of the effect of neutron irradiation over time on the ferritic steels which makes the steel *
Corresponding author. Tel.: +1 803 777 5869; fax: +1 803 777 0106. E-mail address: [email protected] (Y.J. Chao).
0013-7944/$ - see front matter 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.engfracmech.2006.11.010
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Toughness
Upper shelf (ductile)
Lower Constraint
Irradiation
Lower shelf (brittle)
Temperature Fig. 1. Ductile–brittle transition curve of steels and the effects of irradiation and constraint on the shift of the curve.
brittle and shifts the material’s ductile–brittle transition curve to a higher temperature range (the dashed curve in Fig. 1). The shift decreases operation margins in both the temperature and pressure and is inevitable in the aging process of the RPV. It has been a common industry practice to use standard fracture specimens with high constraint (i.e. deep cracked specimens) to determine the ductile–brittle transition curve of RPV steels. For instance, all ASTM testing standards specify specimens with a/W around 0.5, where a is the crack length or depth and W is the width of the specimen, which has high constraint. On the other hand, low constraint cracks are often used by codes for structural integrity assessment. For instance, ASME Pressure Vessel Code Section III, Appendix G for protection against non-ductile failure adopts a postulated reference flaw size of a/W = 0.25. And, in Appendix A of Section XI, a basic allowable flaw size of a/W = 0.025 was suggested as acceptable in non-destructive examination of piping and vessels. It has been demonstrated in many places that low constraint shallow flaws, such as a/W = 0.025 and 0.25, exhibit higher fracture toughness than those from high constraint specimens [2–5]. Applying the same argument to different temperature covering the entire life and operating conditions of RPV, it could result in a shift of the ductile–brittle transition curve to lower temperature, as shown in Fig. 1. The loss of temperature and pressure margins arising from irradiation effects during the lifetime of RPV can therefore be offset by the argument based on constraint as applied to shallow flaws in RPV. A review of this topic on the effect of constraint and dose attenuation in determining the operating margin between OUST and normal steady state service temperature can be found in Dolby et al. [6]. For a stationary crack in an elastic–plastic material, effect of constraint on crack-tip fields has been investigated extensively for different specimen geometry and loading configurations. The three most commonly used methods to quantify crack-tip constraints are (1) the J–T approach by Betegon and Hancock [7], (2) the J–Q approach by O’Dowd and Shih [8,9], and (3) the J–A2 approach by Yang et al. [10] and Chao et al. [11]. The J–T approach has limited use in elastic–plastic fracture because T-stress only exists in linear elastic field. The J–Q approach is only good for small applied loads because the parameter Q becomes distance-dependent under large applied loads [12]. The J–A2 approach, however, is based on a rigorous asymptotic mathematical solution. Moreover, A2 is nearly independent of its position near the crack-tip [13] and has been successfully used to quantify the constraint effect on fracture toughness including the J–R curve for different geometry and loading configurations [14]. A salient feature of using A2 to quantify the constraint is that A2 is independent of loading and only a function of specimen geometry and material under fully plastic conditions [14]. As K and T-stress for small scale yielding (SSY) or very little plasticity, J–A2 is best used for materials under large scale yielding (LSY) at fracture when constraint effect is included in the interpretation of test data. The J–A2 approach therefore holds notable promise for two-parameter fracture testing as outlined by Chao and Zhu [15] and Zhu and Chao [16]. Applying the constraint to the shift of the ductile–brittle transition curve has been studied by various researchers. Gao and Dodds [17,18] used the scaling technique combined with Weibull statistics to investigate cleavage fracture. T-stress was used as the constraint parameter by Gao et al. [19] to predict the shift between specimens exhibiting different in-plane constraint levels. Bezensek and Hancock [2,20] investigated the temperature shift based on the argument of loss of crack-tip constraint using T-stress.
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The objective of the present paper is to demonstrate the shift of the ductile–brittle curve due to difference in constraint in deep and shallow flaws. The J–A2 methodology is used and A2 is used to quantify the constraint. Procedures to transfer (or shift) the ductile–brittle transition curve between specimens of different in-plane constraint levels are developed which enables the determination of the transition curve of non-standard specimens (or flawed structures) from standard three-point bending specimens. The experimental data on A533B RPV steel by Sherry et al. [1] are used to demonstrate and validate the procedure. 2. Theoretical consideration Our attention is focused on mode-I cracks in elastic–plastic materials under plane strain conditions. The material behavior described by the Ramberg–Osgood power-law stress–strain relation in uni-axial tension can be written as n e r r ¼ þa e 0 r0 r0
ð1Þ
where e is the uni-axial strain, r is the uni-axial stress, r0 is a reference stress, e0 = r0/E is a reference strain with E the Young’s modulus, a is a material constant and n is the strain hardening exponent. For actual elastic–plastic solids, r0 and e0 may be taken to be the yield stress and the yield strain respectively. Using J2 deformation theory of plasticity, the uni-axial stress–strain relation (1) can be generalized to the multi-axial state as n1 eij rij rkk 3 re S ij ¼ ð1 þ mÞ v dij þ a ð2Þ 2 r0 e0 r0 r0 r0 where m is the Poisson’s ratio, dij is the Kronecker delta, Sij is the deviatoric stress and re is the Von Mises pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi effective stress defined as re ¼ 3S ij S ij =2. 2.1. J–A2 three-term solution for the crack-tip stress Making use of rigorous asymptotic mathematical analysis, Yang et al. [10] and Chao et al. [11] developed the J–A2 three-term solution to characterize the crack-tip fields. The crack-tip stress field in polar coordinate system (r, h) can be written as r S 2 S 3 rij r S 1 ð1Þ ð2Þ ð3Þ 2 r ~ij ðhÞ þ A2 ~ij ðhÞ þ A2 ~ij ðhÞ ¼ A1 ð3Þ r r r L L L r0 ðkÞ
~ij ðhÞ ðk ¼ 1; 2; 3Þ, the stress power exponents Sk (S1 < S2 < S3) and the dimenwhere the angular functions r sionless integration constant In are dependent only on the hardening exponent n and independent of other material constants (i.e. e0, r0) and applied loads. L is a characteristic length parameter which can be chosen as the crack length a, the specimen width W, the thickness B or unity. The first term in Eq. (3) is the HRR solution and the parameters A1 and S1 are given by S 1 J 1 ð4Þ A1 ¼ ; S1 ¼ ae0 r0 I n L nþ1 ðkÞ
~ij , Sk and In have been calculated and tabulated by Chao and Plane strain mode-I dimensionless functions r Zhang [21] and S3 = 2S2 S1 for n P 3. The parameter A2 is related to the specimen geometry and loading and can be determined using a point matching technique or a simple weight average technique [22]. The three-term mathematical solution, Eq. (3), is the first three terms from the series solution for a crack in a Ramberg–Osgood power-law material. Although it contains three terms, there are only two amplitudes J and A2. When A2 = 0, the three-term asymptotic solution Eq. (3) reduces to the leading-term HRR singularity field. Physically, therefore, the amplitude J can be treated as the loading parameter and A2 can be used as the parameter to quantify the constraint level existing at the crack-tip. Therefore, any cracked specimen or
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structure member holds a specific constraint level represented by a value of A2. After all, the fundamental reason which contributes to the observed constraint effect is the higher order terms that make the crack-tip stress deviate from the classical HRR J-fields at a finite distance from the crack tip. 2.2. The Ritchie–Knott–Rice (RKR) model Once the crack-tip stress fields are determined from either numerical finite element analysis or Eq. (3) provided that the loading level J and the constraint level A2 are known, a fracture criterion is needed in determining the fracture event. The RKR model is adopted in this work which states that ‘‘cleavage fracture is controlled by a critical local fracture stress attained over a characteristic micro-structural distance ahead of the crack-tip’’. The fracture stress has been studied extensively and is considered to be independent of temperature as long as transgranular cleavage is the dominant failure mechanism [23–25], specimen size [25–27] and loading rate [28]. The critical distance rc in the RKR model can be considered as the size of the fracture process zone and is widely argued to be of the order of two to three grains dimension [29,31]. The RKR model can therefore connect the macroscopic fracture toughness with the local steel microstructure through the critical local fracture stress rf. Applying the J–Q stress fields, the RKR model has been used by O’Dowd and Shih [30] to examine constraint effects on fracture. In this current work, it is assumed that the same cleavage mechanisms, and therefore the RKR model, operate throughout the transition temperature range. We further assume that the same critical local fracture stress rf and critical distance rc in the RKR model apply to both high constraint and low constraint specimen geometry allowing a connection to be established between the two. 3. Scaling of the ductile–brittle transition curve based on constraint variation Consider two specimens, having different constraint level A2, operating at the same temperature. Applying Eq. (3) to the opening stress (h = 0 degree) and utilizing the RKR model, one has r S1
r S 3 c ð3Þ ~ij ð0Þ rf ¼ r0 A1 þ A2 þ r L L L S 2 S 3 rc S 1 ð1Þ ð2Þ ð3Þ rc 2 r c ~ij ð0Þ þ A2 ~ij ð0Þ þ A2 ~ij ð0Þ ¼ r0 A1 r r r L L L c
r S 2 c
ð1Þ ~ij ð0Þ r
ð2Þ ~ij ð0Þ r
A22
ð5Þ
where, A1 , A2 are for one constraint specimen (e.g. low constraint), and A1, A2 are for another constraint specimen (e.g. high constraint). Since the two specimens are at the same temperature, material properties are identical. With (4), Eq. (5) then yields J C ¼
S1 k 1 JC k
ð6Þ
where r S1
r S 3 c ð3Þ ~ij ð0Þ k¼ þ A2 þ r L L L r S 2 r S 3 rc S 1 ð1Þ c c ð2Þ ð3Þ ~ij ð0Þ þ A2 ~ij ð0Þ þ A2 ~ r r r ð0Þ k ¼ ij 2 L L L c
ð1Þ ~ij ð0Þ r
r S 2 c
ð2Þ ~ij ð0Þ r
A22
ð7Þ ð8Þ
Therefore, knowing the toughness JC for one specimen of constraint A2 and the critical distance rc, one can then predict the toughness of another specimen of constraint A2 . No rf is needed in Eq. (6). Here the change of fracture toughness is due to the difference in constraint level A2 of the two specimens at the same temperature
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assuming the RKR model is operative. It results in a vertical scaling of the ductile–brittle transition curve shown in Fig. 1. Alternatively, if both the critical stress and critical distance are known and the specimens obey a fracture criterion characterized by the RKR model, the toughness can then be predicted using the first line of Eq. (5) by S1 kr0 1 JC ¼ e0 r0 I n La ð9Þ rf ð2þS1 Þ
Note the constraint A2 is embedded in k, JC is proportional to r0 1 and r0 is strongly dependent on temperature. If one further assumes that the material holds the same strain hardening, which leads to identical In, L, ð1Þ ð2Þ ð3Þ ~hh ð0Þ, r ~hh ð0Þ, r ~hh ð0Þ, S1 and S2 at different temperature, Eq. (9) can then be applied to predict the ductile– a, r brittle transition curve for any specimen or structure with constraint A2 over the temperature range. This will be demonstrated later when the test data are discussed. 4. Scaling of the ductile–brittle transition curve based on variation in yield stress For two specimens at two different temperatures if the toughness, hardening for the material, the critical local fracture stress rf and the critical distance rc in the RKR model are assumed to be the same at the two temperatures, using Eq. (5) the relation between the yield stresses (at different temperature) of the two specimens becomes r0
2S 1þ1 k 1 ¼ r0 k
ð10Þ
Since the yield stress r0 is the only term in Eq. (10) that depends on temperature, the ductile–brittle transition curve in Fig. 1 can be scaled horizontally from one specimen of constraint A2 to another specimen of constraint A2 provided that the temperature dependence of r0 is known. Note that Eq. (10) does not contain the fracture stress rf although the RKR model is implied. 5. Test data and numerical procedure 5.1. Material and test data The procedure discussed in Sections 3 and 4 is applied to experimental data obtained by Sherry et al. [1] on an A533B pressure vessel steel. The material constant a is 1.0 and the hardening exponent, n, is set at 12 [1], ð1Þ ~hh ð0Þ ¼ 2:5542, and e0 = r0/E. Corresponding to n = 12, the dimensionless parameters are found to be r ð2Þ ð3Þ ~hh ð0Þ ¼ 0:3067, r ~hh ð0Þ ¼ 7:0021, S1 = 0.07692, S2 = 0.06653, S3 = 0.20999, and In = 4.44111 from Chao r and Zhang [21]. Poisson’s ratio is m = 0.3. The dependence of yield stress with temperature in the range 170 C 6 T 6 40 C can be described by r0 ðT Þ ¼ 0:003T 2 1:137T þ 454 ðMPaÞ
ð11Þ
where T is temperature in C. Eq. (11) was from fitting to measured data reported by Sherry et al. [1] and was used by Bezensek and Hancock [2,20]. The dependence of Young’s modulus on temperature is weak for this material in the temperature range of interest. A constant Young’s modulus E of 215 GPa was therefore used in the analysis which is the average in the temperature range 170 C 6 T 6 40 C [1]. Fracture tests were performed and toughness data were reported by Sherry et al. [1] on A533B pressure vessel steel conforming to ASTM E813-87. Three-point bend specimens were used which included one group of shallow cracked (B · W = 50 · 50 mm, a = 3.75 mm) specimens and another group of deep cracked (B · W = 50 · 67 mm, a = 33.5 mm) specimens. Tests were performed throughout the ductile–brittle
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temperature range. Fig. 2 shows theffi test data where the toughness is expressed in term of the critical stress pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi intensity factor K J ¼ J C E=ð1 v2 Þ. In Fig. 2, the data were fitted with two solid lines as K J ¼ 24:8e0:033ðT þ125Þ þ 65 – for the deep crack a=w ¼ 0:5 data K J ¼ 24:8e0:033ðT þ170Þ þ 60 – for the shallow crack a=w ¼ 0:075 data
ð12Þ
The data show a significant increase in fracture toughness from deep cracked specimens to shallow cracked specimens at the same temperature. In Section 6, we will extract necessary parameters from the deep cracked (high constraint) specimens and then predict the toughness data for the shallow cracked (low constraint) specimens. 5.2. Finite element modeling Plane strain elastic–plastic finite element analysis (FEA) was performed using ABAQUS (version 6.4-1) to calculate the crack-tip stress fields and to determine fracture parameters for the test specimens. Due to symmetry, only one half of each specimen is modeled. Typical FEA mesh is illustrated in Fig. 3a for the deep crack and in Fig. 3b for the shallow crack. A fine mesh with the smallest element size of 0.0047 mm is focused on the crack-tip with increasingly coarse mesh generated elsewhere. The FEA mesh consists of 1161 eight-node plane 600
K J (MPavm)
500
× a/ W = 0.5 a/ W = 0.075
400 300 200 100 0 -160
-140
-120
-100
-80
-60
-40
Temperature (ºC) Fig. 2. Fracture toughness as a function of temperature for A533B steel [1] with fitted curves by Eq. (12).
Fig. 3. Finite element mesh for three-point bend specimens, (a) deep crack a/W = 0.5, W = 67 mm and (b) shallow crack a/W = 0.075, W = 50 mm.
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strain isoparametric elements with reduced integration and 3648 nodes. The local mesh encircling the crack-tip contains 30 rings of elements with 24 elements in each ring. The A2 parameter is extracted from the crack opening stress when the loading of the specimen reaches the fracture toughness. To obtain a distance-independent constraint parameter A2, we adopted the simple weight average technique. It is assumed that the resultant force due to the crack opening stress rhh(r, 0) on the remaining ligament in the region of 1 < r/(J/r0) < 5 from finite element calculation is the same as that from the threeterm solution Eq. (3). Details can be found in Chao et al. [22]. The characteristic length L was arbitrarily set at 1 mm in all calculations. 6. Results Fig. 4 shows the comparison of the crack-tip opening stress from FEA and from the J–A2 three-term solution at J = 64.3 N/mm for a/W = 0.5 and J = 405.3 N/mm for a/W = 0.075 at T = 100 C. The good match between the FEA and the J–A2 three-term solution indicates that the J–A2 three term solution can indeed characterize the crack-tip stress fields in the region of interest. In Chao et al. [11] and Chao and Zhu [14] a detailed discussion on the constraint parameter A2 was provided. It was shown that for pure power law or fully plastic materials A2 is independent of applied J, a, r0, and e0 and is a function of the hardening exponent and geometry only or A2 jfully plastic ¼ f ðn; geometryÞ
ð13Þ
In FEA calculations of three-point bending specimens, the loading was applied until J reached the JC on the fitted curves in Fig. 2 (shown in Fig. 2 by KJ) at a given temperature. Fig. 5 shows the values of A2 obtained from the FEA solution for the two specimen geometries. It is shown that A2 is approximately equal to 0.21 for the deep cracked (high constraint) specimen and 0.34 for the shallow (low constraint) specimen in the transition temperature range. The nearly constant A2 value for a given specimen geometry over a wide range of temperature indicates that the specimen indeed has reached the state of large scale yielding at fracture and is consistent with Eq. (13). To apply the RKR model, a critical fracture stress rf and a critical distance rc are required. Ritchie et al. [29] have investigated the fracture mechanisms of A533B steel and suggested 2–4 austenitic grains of size 0.025 mm for rc and 1830 MPa for rf. Since the test data are available for A533B as shown in Fig. 2, one can alternatively determine these two constants using the fracture toughness of high constraint specimens pffiffiffiffi at lower shelf. Using the fitted toughness value at 135 C of Fig. 2, i.e. KJ = 78.7 MPa m corresponding 6
FEA a/W = 0.5 Δ FEA a/W = 0.075 — J -A 2 a/W = 0.5 — J -A 2 a /W = 0.075
5
σ θ /σ 0
4 3 2 1 0
0
1
2
r
3
4
5
J σ
Fig. 4. Comparison of the opening stress directly ahead of the crack tip at JC = 64.3 N/mm for a/W = 0.5 and JC = 405.3 N/mm for a/W = 0.075 at T = 100 C.
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a /W = 0.5 a /W = 0.075
-A 2
0.5 0.4 0.3 0.2 0.1 0 -165 -150 -135 -120 -105 -90 -75 -60
-45
-30
Temperature (ºC) Fig. 5. A2 at fracture as a function of temperature showing nearly constant value for the two specimen geometries indicating large scale yielding (LSY).
to Jc = 26.2 N/mm for the deep crack specimen, and choosing rc = 0.12 mm, FEA reveals rf = 1955 MPa. Note that rc = 0.12 mm was chosen here to be consistent with those determined by Ritchie et al. [29], Ortner and Hippsley [32] and Bezensek and Hancock [2,20]. Bezensek and Hancock [2,20] have obtained rf = 2300 MPa at rc = 0.12 mm using the same specimen. The difference in rf is believed to come from the FEA modeling. A modified boundary layer formulation with an assumed T-stress applied on the outside boundary was used by Bezensek and Hancock [2,20] while the full specimen model shown in Fig. 3 is used in the current analysis. The two critical values in the RKR model rf = 1955 MPa at rc = 0.12 mm are used next for predicting the entire ductile–brittle transition curves for specimens with various constraint. Using Eq. (6), the transition curve for the deep cracked specimen group (a/W = 0.5, A2 = 0.21) in Fig. 2 is scaled upward to the curve for the shallow cracked specimen group ða=W ¼ 0:075; A2 ¼ 0:34Þ using rc = 0.12 mm and is shown in Fig. 6. The predicted curve for the shallow cracked group compares moderately well with the test data. The shift results in an overall move of the transition curve to lower temperature. Following the discussion in Section 3, if one assumes that the material has the same strain hardening at different temperature, Eq. (9) can be used to predict the fracture toughness for any specimen having constraint A2. Combining with Eq. (11), Eq. (9) can then be extended to determine the entire ductile–brittle transition curve over the transition temperature range. The two curves in Fig. 7 are obtained following this procedure
600
KJ (MPa m)
500
× a/W = 0.5 a/W = 0.075
400 300 200 100 0 -160
-140
-120 -100 -80 Temperature (ºC)
-60
-40
Fig. 6. The predicted ductile–brittle transition curve for a/W = 0.075 group of specimens from the fitted curve for a/W = 0.5 based on rc = 0.12 mm and Eq. (6).
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600 500
× a/W = 0.5
KJ (MPa √m)
a/W = 0.075
400 300 200 100 0 -160
-140
-120
-100
-80
-60
-40
Temperature (ºC)
Fig. 7. The predicted ductile–brittle transition curves based on rf = 1955 MPa at rc = 0.12 mm and Eq. (9).
600 × a/W = 0.5 500
a/W = 0.075
KJ (MPa √m)
400 300 200 100 0 -160
-140
-120
-100
-80
-60
-40
Temperature (ºC) Fig. 8. Ductile–brittle transition curve for a/W = 0.075 group of specimens predicted from the a/W = 0.5 fitted curve and Eq. (10) with rc = 0.12 mm.
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi using A2 = 0.21 (for a/W = 0.5) and A2 = 0.34 (for a/W = 0.075), K J ¼ J C E=ð1 v2 Þ, rf = 1955 MPa and rc = 0.12 mm. The predicted curves compare very well with the experimental data. Using Eq. (10) and rc = 0.12 mm the fitted transition curve for the deep cracked specimen group (i.e. A2 = 0.21) in Fig. 2 is shifted to predict the transition curve for the shallow cracked specimen group ðA2 ¼ 0:34Þ. The results are shown in Fig. 8 where again the predicted curve for the shallow cracked specimen group is reasonable. The shift again results in a temperature shift (or scaling) of the curve to lower temperature. 7. Concluding remarks Assuming that the mechanism of local cleavage and the hardening behavior of the steel are independent of temperature and given the fact that yield stress is strongly temperature dependent, the procedure developed in this article allows the prediction of fracture toughness of a group of specimens with certain constraint from the fracture toughness of another group of specimens with a different constraint. The J–A2 three-term solution is used to characterize the crack-tip stress fields and the constraint level of test specimen is quantified by A2. It is demonstrated that combined with the RKR model as the fracture criterion all three procedures, i.e. Eqs. (6), (9) and (10), shift (or scale) the ductile–brittle transition curve to lower temperature as it transfers
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the curve from high constraint specimen (e.g. a/W = 0.5) to low constraint specimen (e.g. a/W = 0.075). The predicted curves from the three procedures compared well with the test data from A533B RPV steel. Fig. 9 schematically shows the ductile–brittle transition curves as parameterized by the constraint A2. Practically, doing a set of tests on deep cracked specimens such as the ASTM specimens the ductile–brittle transition of shallow cracked specimens can then be predicted. More importantly the ductile–brittle transition curve of any structure component can be obtained as well provided that the constraint parameter A2 of the structure component can be determined. This paper provides a theoretical basis for this shift using constraint as an argument. Similar shift from deep cracked three-point-bend or CT type of specimens to shallow surface flaws as specified by the ASME Pressure Vessel Code warrants further studies. The shift would allow the increase of the operation margins in either temperature or pressure in RPV as attention is placed upon shallow flaws. In addition, the procedure is useful in transferability of test data from laboratory to large scale structure. Note that the same set of test data shown in Fig. 2 were used by Sherry et al. [1] and Bezensek and Hancock [2,20] in studying the transition curve for A533B RPV steels. Qualitatively similar shift of the ductile–brittle transition curve was obtained relative to the current work. When dealing with constraint, Sherry et al. [1] adopted the elastic T-stress and the Q-parameter while Bezensek and Hancock [2,20] used the elastic T-stress. As discussed in Section 1, T-stress only exists in linear elastic field which is certainly not the case for A533B steel even near the lower shelf region, e.g. at fracture LSY condition has prevailed especially for shallow cracks. Forcing the linear elastic theory for predicting fracture under LSY conditions therefore requires extra justifications and assumptions. The authors would like to point out that the T-stress is indeed a rigorous parameter in predicting fracture event including the constraint effect for brittle materials when plasticity is negligible at the moment of fracture [35,36]. The Q-stress is a summation of the second and the third terms in Eq. (3). It contains no details in distance r from the crack tip, angular function as well as material properties. It therefore suffers the distance and load dependence as has been experienced by many researchers independently. The J–A2 approach on the other hand is a derivative of the rigorous mathematical solution for non-linear power-law materials. It is best for quantifying fracture in LSY such as the A533B steels in the transition region. In summary, the purpose of any fracture mechanics theory is to provide a mathematical model to describe the phenomenological fracture event in materials. Models with fewer assumptions and can better represent the material behavior are to prevail. The current paper provides an alternative using the A2 parameter for quantifying the constraint level of deep and shallow specimens for A533B steels in the transition region. Rigorous analysis, assumptions and procedures are outlined. As a final note, it is generally believed that statistical model such as the Weibull should be used in interpreting the test data for cleavage fracture. While any statistical model is indispensable particularly for accommodating the large scatter of cleavage fracture data, one of the fundamental physical reasons of the shift or Structure specimen #2
or
Fracture Toughness
A2*** Lower constraint
A2**
Structure or specimen #1
A2 *
ASTM CT
A2*** < A2** < A2* Temperature Fig. 9. Schematic showing the ductile–brittle transition curves as parameterized by the constraint A2.
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scaling of the ductile–brittle fracture toughness transition curve is due to the constraint or size effect, either from the thickness or the in-plane specimen size and crack depth. The current paper is to demonstrate the latter. Furthermore, purely statistical model without considering the constraint effect, e.g. Master Curve based on weakest link [33,34], may not predict the crack depth effect on the lower shelf as observed however in the current data and most recently by Balart and Knott [37]. Acknowledgement The work is sponsored by Jiangsu University Fund (1281190001). The support is greatly appreciated. References [1] Sherry AH, Lidbury DPG, Beardsmore DW. Validation of constraint based structural integrity assessment methods. Final report, Report No. AEAT/RJCB/RD01329400/R003, AEA Technology, UK, 2001. [2] Bezensek B, Hancock JW. Increased temperature margins due to constraint loss. ASME J Pres Ves Technol 2005;127:173–8. [3] Chao YJ, Lam PS. Effects of crack depth, specimen size, and out-of-plane stress on the fracture toughness of reactor vessel steels. ASME J Pres Ves Technol 1996;118:415–23. [4] Kim Y, Chao YJ, Zhu X. Effect of specimen size and crack depth on 3D crack-front constraint for SENB specimens. Int J Solids Struct 2003;40:6267–84. [5] Sorem WA, Dodds RH, Rolfe ST. An analytical and experimental comparison of rectangular and square crack-tip opening displacement fracture specimens on an A36. Nonlinear fracture mechanics. Elastic–plastic fracture, ASTM STP 995, vol. II. Philadelphia: American Society for Testing and Materials; 1989. p. 470–94. [6] Dolby RE, Wiesner CS, Ainsworth RA, Burdekin FM, Hancock J, Milne I, et al. Review of a procedure for performing constraint and attenuation-corrected fracture mechanics safety case calculations for magnox reactor steel pressure vessels. Int J Pres Ves Pip 2005;82:496–508. [7] Betegon C, Hancock JW. Two parameter characterization of elastic–plastic crack-tip fields. ASME J Appl Mech 1991;58:104–10. [8] O’Dowd NP, Shih CF. Family of crack-tip fields characterized by a triaxiality parameter – I. Structure of fields. J Mech Phys Solids 1991;39:989–1015. [9] O’Dowd NP, Shih CF. Family of crack-tip fields characterized by a triaxiality parameter – II. Fracture applications. J Mech Phys Solids 1992;40:939–63. [10] Yang S, Chao YJ, Sutton MA. Higher order asymptotic crack fields in a power-law hardening material. Engng Fract Mech 1993;45:1–20. [11] Chao YJ, Yang S, Sutton MA. On the parameters: theory and practice. J Mech Phys Solids 1994;42:629–47. [12] Sharma SM, Aravas N, Zelman MG. Two-parameter characterization of crack tip fields in edged-cracked geometries: plastic and creep solutions. Fracture mechanics. ASME STP 1220, vol. 25. Philadelphia: American Society for Testing and Materials; 1995. p. 309–27. [13] Nikishkov GP, Bruckner-Foit A, Munz D. Calculation of the second fracture parameter for finite cracked bodies using a three-term elastic–plastic asymptotic expansion. Engng Fract Mech 1995;52:685–701. [14] Chao YJ, Zhu XK. Constraint-modified J–R curves and its application to ductile crack growth. Int J Fract 2000;106:135–60. [15] Chao YJ, Zhu XK. J–A2 characterization of crack-tip fields: extent of J–A2 dominance and size requirements. Int J Fract 1998;89:285–307. [16] Zhu X, Chao YJ. Specimen size requirements for two-parameter fracture toughness testing. Int J Fract 2005;135:115–36. [17] Gao XS, Dodds Jr RH. Constraint effects on the ductile-to-brittle transition temperature of ferritic steels: a Weibull stress model. Int J Fract 2000;102:43–69. [18] Gao XS, Dodds Jr RH. An engineering approach to assess constraint effects on cleavage fracture toughness. Engng Fract Mech 2000;68:263–83. [19] Gao X, Dodds RH, Tregoning RI, Joyce JA. Prediction of the T0 shift between specimens of different constraints using the T-stress based T-functions. Int J Fract 2000;104:L3–8. [20] Bezensek B, Hancock JW. Increased temperature margins due to constraint loss. In: Lam PS, et al., editor. Proc int conf ASME 2003 pressure vessels and piping, PVP-462, Cleveland, OH, USA, July 2003. p. 49–56. [21] Chao YJ, Zhang L. Tables of plane strain crack tip fields: HRR and higher order terms. ME-Report, 97-1, Department of Mechanical Engineering, University of South Carolina, 1997. [22] Chao YJ, Zhu XK, Lam PS, Louthan MR, Lyer NC. Application of the two-parameter J–A2 description to ductile crack growth. In: Halford GR, Gallagher JP, editors. ASTM STP 1389. American Society for Testing and Materials; 2000. p. 165–82. [23] Chen JH, Zhu L, Ma H. On the scattering of the local fracture stress. Acta Metall Mater 1990;38:2527–35. [24] Cury DA. Comparison between two models of cleavage fracture. Metal Sci 1980;14(2):78–80. [25] Wang GZ, Liu GH, Chen JH. Effects of precracked specimen geometry on local cleavage fracture stress rf of low alloy steel. Int J Fract 2001;112:183–96.
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[26] Bowen P, Knott JF. Size effect on the microscopic cleavage fracture stress rf in martensitic microstructures. Metall Trans A 1986;17:231–41. [27] Wang GZ, Wang JG, Chen JH. Effects of geometry of notched specimens on the local cleavage fracture stress rf of C–Mn steel. Engng Fract Mech 2003;70:2499–512. [28] Wang GZ, Wang YJ, Chen JH. Effects of loading rate on the local cleavage fracture stress rf in notched specimens. Engng Fract Mech 2005;72:675–89. [29] Ritchie RO, Knott JF, Rice JR. On the relationship between critical tensile stress and fracture toughness in mild steel. J Mech Phys Solids 1973;21:395–410. [30] O’Dowd NP, Shih CF. Two-parameter fracture mechanics: theory and application. In: Landes JD, McCabe DE, Boulet JAM, editors. Fracture mechanics: twenty-fourth symposium. ASTM STP 1207. Philadelphia: American Society of Testing Materials; 1994. [31] Ritchie RO, Server WL, Wulleart RA. Critical fracture stress and fracture strain models for prediction of lower and upper shelf toughness in nuclear pressure vessel steel. Metall Trans 1979;10A:1557–70. [32] Ortner SR, Hippsley CA. Two component description of ductile to brittle transition in ferritic steel. Mater Sci Technol 1996;12:1035–42. [33] Wallin K. The size effect in KIC results. Engng Fract Mech 1985;22(1):149–63. [34] Standard test method for determination of reference temperature, T0, for ferritic steels in the transition range, ASTM E1921-05, Annual Book of ASTM Standards, vol. 03.01, 2004. [35] Chao YJ, Zhang X. Constraint effect in brittle fracture. In: Robert Piascik, editor. ASTM STP 1296, Fatigue and fracture mechanics, vol. 27, 1997. p. 41–60. [36] Chao YJ, Liu S, Broviak B. Brittle fracture: constraint effect and crack curving under mode I conditions. Exp Mech 2001;41(3):232–41. [37] Balart MJ, Knott JF. Effects of geometry and flow properties on the fracture toughness of a C–Mn reactor pressure vessel steel in the lower shelf region. Int J Pres Ves Pip 2006;83:205–15.
Effect of crack depth on the shift of the ductile–brittle transition curve of steels Zhong-An Chen a, Zhen Zeng a, Yuh J. Chao
b,*
a
b
Jiangsu University, Jiangsu, Zhenjiang, China Department of Mechanical Engineering, University of South Carolina, Columbia, SC 29208, USA Received 13 May 2006; received in revised form 6 November 2006; accepted 7 November 2006 Available online 17 January 2007
Abstract Nuclear reactor pressure vessel (RPV) steels degrade due to neutron irradiation during normal operation. As a result, the ductile–brittle transition curve of the steel shifts to higher temperature which decreases operation margins in both the temperature and pressure. The loss of these margins however can be offset somewhat by appealing to arguments based on constraint of potential/postulated shallow cracks. In this paper, it is demonstrated that the fracture toughness values for shallow flaws are higher than those determined from standard deep cracked test specimens based on constraint consideration. The J–A2 three-term solution is used to characterize the crack-tip stress field where J represents the level of loading and A2 quantifies the level of constraint. Based on the RKR cleavage model, procedures to quantify the temperature shift between specimens with different constraint levels are developed. The experimental data by Sherry et al. [Sherry AH, Lidbury DPG, Beardsmore DW. Validation of constraint based structural integrity assessment methods. Final report, Report No. AEAT/RJCB/RD01329400/R003, AEA Technology, UK, 2001] for the A533B RPV steel are used to demonstrate the procedure and it is shown that the ductile–brittle transition curve shifts to lower temperature from high constraint to low constraint specimens. 2006 Elsevier Ltd. All rights reserved. Keywords: Nuclear reactor pressure vessel; Constraint effect; J–A2; Cleavage fracture; Ductile–brittle transition curve
1. Introduction Steels in general are ductile at high temperature and brittle at low temperature. This material behavior is normally presented as the ductile–brittle transition curve and plotted as fracture energy or toughness versus temperature. A typical ductile–brittle transition curve is shown in Fig. 1. Under normal operating conditions, pressure vessels are designed to operate at temperature above the onset of upper shelf toughness (OUST) in Fig. 1 in order to avoid brittle fracture. In RPV the criteria for safety include consideration of the effect of neutron irradiation over time on the ferritic steels which makes the steel *
Corresponding author. Tel.: +1 803 777 5869; fax: +1 803 777 0106. E-mail address: [email protected] (Y.J. Chao).
0013-7944/$ - see front matter 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.engfracmech.2006.11.010
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Toughness
Upper shelf (ductile)
Lower Constraint
Irradiation
Lower shelf (brittle)
Temperature Fig. 1. Ductile–brittle transition curve of steels and the effects of irradiation and constraint on the shift of the curve.
brittle and shifts the material’s ductile–brittle transition curve to a higher temperature range (the dashed curve in Fig. 1). The shift decreases operation margins in both the temperature and pressure and is inevitable in the aging process of the RPV. It has been a common industry practice to use standard fracture specimens with high constraint (i.e. deep cracked specimens) to determine the ductile–brittle transition curve of RPV steels. For instance, all ASTM testing standards specify specimens with a/W around 0.5, where a is the crack length or depth and W is the width of the specimen, which has high constraint. On the other hand, low constraint cracks are often used by codes for structural integrity assessment. For instance, ASME Pressure Vessel Code Section III, Appendix G for protection against non-ductile failure adopts a postulated reference flaw size of a/W = 0.25. And, in Appendix A of Section XI, a basic allowable flaw size of a/W = 0.025 was suggested as acceptable in non-destructive examination of piping and vessels. It has been demonstrated in many places that low constraint shallow flaws, such as a/W = 0.025 and 0.25, exhibit higher fracture toughness than those from high constraint specimens [2–5]. Applying the same argument to different temperature covering the entire life and operating conditions of RPV, it could result in a shift of the ductile–brittle transition curve to lower temperature, as shown in Fig. 1. The loss of temperature and pressure margins arising from irradiation effects during the lifetime of RPV can therefore be offset by the argument based on constraint as applied to shallow flaws in RPV. A review of this topic on the effect of constraint and dose attenuation in determining the operating margin between OUST and normal steady state service temperature can be found in Dolby et al. [6]. For a stationary crack in an elastic–plastic material, effect of constraint on crack-tip fields has been investigated extensively for different specimen geometry and loading configurations. The three most commonly used methods to quantify crack-tip constraints are (1) the J–T approach by Betegon and Hancock [7], (2) the J–Q approach by O’Dowd and Shih [8,9], and (3) the J–A2 approach by Yang et al. [10] and Chao et al. [11]. The J–T approach has limited use in elastic–plastic fracture because T-stress only exists in linear elastic field. The J–Q approach is only good for small applied loads because the parameter Q becomes distance-dependent under large applied loads [12]. The J–A2 approach, however, is based on a rigorous asymptotic mathematical solution. Moreover, A2 is nearly independent of its position near the crack-tip [13] and has been successfully used to quantify the constraint effect on fracture toughness including the J–R curve for different geometry and loading configurations [14]. A salient feature of using A2 to quantify the constraint is that A2 is independent of loading and only a function of specimen geometry and material under fully plastic conditions [14]. As K and T-stress for small scale yielding (SSY) or very little plasticity, J–A2 is best used for materials under large scale yielding (LSY) at fracture when constraint effect is included in the interpretation of test data. The J–A2 approach therefore holds notable promise for two-parameter fracture testing as outlined by Chao and Zhu [15] and Zhu and Chao [16]. Applying the constraint to the shift of the ductile–brittle transition curve has been studied by various researchers. Gao and Dodds [17,18] used the scaling technique combined with Weibull statistics to investigate cleavage fracture. T-stress was used as the constraint parameter by Gao et al. [19] to predict the shift between specimens exhibiting different in-plane constraint levels. Bezensek and Hancock [2,20] investigated the temperature shift based on the argument of loss of crack-tip constraint using T-stress.
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The objective of the present paper is to demonstrate the shift of the ductile–brittle curve due to difference in constraint in deep and shallow flaws. The J–A2 methodology is used and A2 is used to quantify the constraint. Procedures to transfer (or shift) the ductile–brittle transition curve between specimens of different in-plane constraint levels are developed which enables the determination of the transition curve of non-standard specimens (or flawed structures) from standard three-point bending specimens. The experimental data on A533B RPV steel by Sherry et al. [1] are used to demonstrate and validate the procedure. 2. Theoretical consideration Our attention is focused on mode-I cracks in elastic–plastic materials under plane strain conditions. The material behavior described by the Ramberg–Osgood power-law stress–strain relation in uni-axial tension can be written as n e r r ¼ þa e 0 r0 r0
ð1Þ
where e is the uni-axial strain, r is the uni-axial stress, r0 is a reference stress, e0 = r0/E is a reference strain with E the Young’s modulus, a is a material constant and n is the strain hardening exponent. For actual elastic–plastic solids, r0 and e0 may be taken to be the yield stress and the yield strain respectively. Using J2 deformation theory of plasticity, the uni-axial stress–strain relation (1) can be generalized to the multi-axial state as n1 eij rij rkk 3 re S ij ¼ ð1 þ mÞ v dij þ a ð2Þ 2 r0 e0 r0 r0 r0 where m is the Poisson’s ratio, dij is the Kronecker delta, Sij is the deviatoric stress and re is the Von Mises pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi effective stress defined as re ¼ 3S ij S ij =2. 2.1. J–A2 three-term solution for the crack-tip stress Making use of rigorous asymptotic mathematical analysis, Yang et al. [10] and Chao et al. [11] developed the J–A2 three-term solution to characterize the crack-tip fields. The crack-tip stress field in polar coordinate system (r, h) can be written as r S 2 S 3 rij r S 1 ð1Þ ð2Þ ð3Þ 2 r ~ij ðhÞ þ A2 ~ij ðhÞ þ A2 ~ij ðhÞ ¼ A1 ð3Þ r r r L L L r0 ðkÞ
~ij ðhÞ ðk ¼ 1; 2; 3Þ, the stress power exponents Sk (S1 < S2 < S3) and the dimenwhere the angular functions r sionless integration constant In are dependent only on the hardening exponent n and independent of other material constants (i.e. e0, r0) and applied loads. L is a characteristic length parameter which can be chosen as the crack length a, the specimen width W, the thickness B or unity. The first term in Eq. (3) is the HRR solution and the parameters A1 and S1 are given by S 1 J 1 ð4Þ A1 ¼ ; S1 ¼ ae0 r0 I n L nþ1 ðkÞ
~ij , Sk and In have been calculated and tabulated by Chao and Plane strain mode-I dimensionless functions r Zhang [21] and S3 = 2S2 S1 for n P 3. The parameter A2 is related to the specimen geometry and loading and can be determined using a point matching technique or a simple weight average technique [22]. The three-term mathematical solution, Eq. (3), is the first three terms from the series solution for a crack in a Ramberg–Osgood power-law material. Although it contains three terms, there are only two amplitudes J and A2. When A2 = 0, the three-term asymptotic solution Eq. (3) reduces to the leading-term HRR singularity field. Physically, therefore, the amplitude J can be treated as the loading parameter and A2 can be used as the parameter to quantify the constraint level existing at the crack-tip. Therefore, any cracked specimen or
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structure member holds a specific constraint level represented by a value of A2. After all, the fundamental reason which contributes to the observed constraint effect is the higher order terms that make the crack-tip stress deviate from the classical HRR J-fields at a finite distance from the crack tip. 2.2. The Ritchie–Knott–Rice (RKR) model Once the crack-tip stress fields are determined from either numerical finite element analysis or Eq. (3) provided that the loading level J and the constraint level A2 are known, a fracture criterion is needed in determining the fracture event. The RKR model is adopted in this work which states that ‘‘cleavage fracture is controlled by a critical local fracture stress attained over a characteristic micro-structural distance ahead of the crack-tip’’. The fracture stress has been studied extensively and is considered to be independent of temperature as long as transgranular cleavage is the dominant failure mechanism [23–25], specimen size [25–27] and loading rate [28]. The critical distance rc in the RKR model can be considered as the size of the fracture process zone and is widely argued to be of the order of two to three grains dimension [29,31]. The RKR model can therefore connect the macroscopic fracture toughness with the local steel microstructure through the critical local fracture stress rf. Applying the J–Q stress fields, the RKR model has been used by O’Dowd and Shih [30] to examine constraint effects on fracture. In this current work, it is assumed that the same cleavage mechanisms, and therefore the RKR model, operate throughout the transition temperature range. We further assume that the same critical local fracture stress rf and critical distance rc in the RKR model apply to both high constraint and low constraint specimen geometry allowing a connection to be established between the two. 3. Scaling of the ductile–brittle transition curve based on constraint variation Consider two specimens, having different constraint level A2, operating at the same temperature. Applying Eq. (3) to the opening stress (h = 0 degree) and utilizing the RKR model, one has r S1
r S 3 c ð3Þ ~ij ð0Þ rf ¼ r0 A1 þ A2 þ r L L L S 2 S 3 rc S 1 ð1Þ ð2Þ ð3Þ rc 2 r c ~ij ð0Þ þ A2 ~ij ð0Þ þ A2 ~ij ð0Þ ¼ r0 A1 r r r L L L c
r S 2 c
ð1Þ ~ij ð0Þ r
ð2Þ ~ij ð0Þ r
A22
ð5Þ
where, A1 , A2 are for one constraint specimen (e.g. low constraint), and A1, A2 are for another constraint specimen (e.g. high constraint). Since the two specimens are at the same temperature, material properties are identical. With (4), Eq. (5) then yields J C ¼
S1 k 1 JC k
ð6Þ
where r S1
r S 3 c ð3Þ ~ij ð0Þ k¼ þ A2 þ r L L L r S 2 r S 3 rc S 1 ð1Þ c c ð2Þ ð3Þ ~ij ð0Þ þ A2 ~ij ð0Þ þ A2 ~ r r r ð0Þ k ¼ ij 2 L L L c
ð1Þ ~ij ð0Þ r
r S 2 c
ð2Þ ~ij ð0Þ r
A22
ð7Þ ð8Þ
Therefore, knowing the toughness JC for one specimen of constraint A2 and the critical distance rc, one can then predict the toughness of another specimen of constraint A2 . No rf is needed in Eq. (6). Here the change of fracture toughness is due to the difference in constraint level A2 of the two specimens at the same temperature
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assuming the RKR model is operative. It results in a vertical scaling of the ductile–brittle transition curve shown in Fig. 1. Alternatively, if both the critical stress and critical distance are known and the specimens obey a fracture criterion characterized by the RKR model, the toughness can then be predicted using the first line of Eq. (5) by S1 kr0 1 JC ¼ e0 r0 I n La ð9Þ rf ð2þS1 Þ
Note the constraint A2 is embedded in k, JC is proportional to r0 1 and r0 is strongly dependent on temperature. If one further assumes that the material holds the same strain hardening, which leads to identical In, L, ð1Þ ð2Þ ð3Þ ~hh ð0Þ, r ~hh ð0Þ, r ~hh ð0Þ, S1 and S2 at different temperature, Eq. (9) can then be applied to predict the ductile– a, r brittle transition curve for any specimen or structure with constraint A2 over the temperature range. This will be demonstrated later when the test data are discussed. 4. Scaling of the ductile–brittle transition curve based on variation in yield stress For two specimens at two different temperatures if the toughness, hardening for the material, the critical local fracture stress rf and the critical distance rc in the RKR model are assumed to be the same at the two temperatures, using Eq. (5) the relation between the yield stresses (at different temperature) of the two specimens becomes r0
2S 1þ1 k 1 ¼ r0 k
ð10Þ
Since the yield stress r0 is the only term in Eq. (10) that depends on temperature, the ductile–brittle transition curve in Fig. 1 can be scaled horizontally from one specimen of constraint A2 to another specimen of constraint A2 provided that the temperature dependence of r0 is known. Note that Eq. (10) does not contain the fracture stress rf although the RKR model is implied. 5. Test data and numerical procedure 5.1. Material and test data The procedure discussed in Sections 3 and 4 is applied to experimental data obtained by Sherry et al. [1] on an A533B pressure vessel steel. The material constant a is 1.0 and the hardening exponent, n, is set at 12 [1], ð1Þ ~hh ð0Þ ¼ 2:5542, and e0 = r0/E. Corresponding to n = 12, the dimensionless parameters are found to be r ð2Þ ð3Þ ~hh ð0Þ ¼ 0:3067, r ~hh ð0Þ ¼ 7:0021, S1 = 0.07692, S2 = 0.06653, S3 = 0.20999, and In = 4.44111 from Chao r and Zhang [21]. Poisson’s ratio is m = 0.3. The dependence of yield stress with temperature in the range 170 C 6 T 6 40 C can be described by r0 ðT Þ ¼ 0:003T 2 1:137T þ 454 ðMPaÞ
ð11Þ
where T is temperature in C. Eq. (11) was from fitting to measured data reported by Sherry et al. [1] and was used by Bezensek and Hancock [2,20]. The dependence of Young’s modulus on temperature is weak for this material in the temperature range of interest. A constant Young’s modulus E of 215 GPa was therefore used in the analysis which is the average in the temperature range 170 C 6 T 6 40 C [1]. Fracture tests were performed and toughness data were reported by Sherry et al. [1] on A533B pressure vessel steel conforming to ASTM E813-87. Three-point bend specimens were used which included one group of shallow cracked (B · W = 50 · 50 mm, a = 3.75 mm) specimens and another group of deep cracked (B · W = 50 · 67 mm, a = 33.5 mm) specimens. Tests were performed throughout the ductile–brittle
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temperature range. Fig. 2 shows theffi test data where the toughness is expressed in term of the critical stress pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi intensity factor K J ¼ J C E=ð1 v2 Þ. In Fig. 2, the data were fitted with two solid lines as K J ¼ 24:8e0:033ðT þ125Þ þ 65 – for the deep crack a=w ¼ 0:5 data K J ¼ 24:8e0:033ðT þ170Þ þ 60 – for the shallow crack a=w ¼ 0:075 data
ð12Þ
The data show a significant increase in fracture toughness from deep cracked specimens to shallow cracked specimens at the same temperature. In Section 6, we will extract necessary parameters from the deep cracked (high constraint) specimens and then predict the toughness data for the shallow cracked (low constraint) specimens. 5.2. Finite element modeling Plane strain elastic–plastic finite element analysis (FEA) was performed using ABAQUS (version 6.4-1) to calculate the crack-tip stress fields and to determine fracture parameters for the test specimens. Due to symmetry, only one half of each specimen is modeled. Typical FEA mesh is illustrated in Fig. 3a for the deep crack and in Fig. 3b for the shallow crack. A fine mesh with the smallest element size of 0.0047 mm is focused on the crack-tip with increasingly coarse mesh generated elsewhere. The FEA mesh consists of 1161 eight-node plane 600
K J (MPavm)
500
× a/ W = 0.5 a/ W = 0.075
400 300 200 100 0 -160
-140
-120
-100
-80
-60
-40
Temperature (ºC) Fig. 2. Fracture toughness as a function of temperature for A533B steel [1] with fitted curves by Eq. (12).
Fig. 3. Finite element mesh for three-point bend specimens, (a) deep crack a/W = 0.5, W = 67 mm and (b) shallow crack a/W = 0.075, W = 50 mm.
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strain isoparametric elements with reduced integration and 3648 nodes. The local mesh encircling the crack-tip contains 30 rings of elements with 24 elements in each ring. The A2 parameter is extracted from the crack opening stress when the loading of the specimen reaches the fracture toughness. To obtain a distance-independent constraint parameter A2, we adopted the simple weight average technique. It is assumed that the resultant force due to the crack opening stress rhh(r, 0) on the remaining ligament in the region of 1 < r/(J/r0) < 5 from finite element calculation is the same as that from the threeterm solution Eq. (3). Details can be found in Chao et al. [22]. The characteristic length L was arbitrarily set at 1 mm in all calculations. 6. Results Fig. 4 shows the comparison of the crack-tip opening stress from FEA and from the J–A2 three-term solution at J = 64.3 N/mm for a/W = 0.5 and J = 405.3 N/mm for a/W = 0.075 at T = 100 C. The good match between the FEA and the J–A2 three-term solution indicates that the J–A2 three term solution can indeed characterize the crack-tip stress fields in the region of interest. In Chao et al. [11] and Chao and Zhu [14] a detailed discussion on the constraint parameter A2 was provided. It was shown that for pure power law or fully plastic materials A2 is independent of applied J, a, r0, and e0 and is a function of the hardening exponent and geometry only or A2 jfully plastic ¼ f ðn; geometryÞ
ð13Þ
In FEA calculations of three-point bending specimens, the loading was applied until J reached the JC on the fitted curves in Fig. 2 (shown in Fig. 2 by KJ) at a given temperature. Fig. 5 shows the values of A2 obtained from the FEA solution for the two specimen geometries. It is shown that A2 is approximately equal to 0.21 for the deep cracked (high constraint) specimen and 0.34 for the shallow (low constraint) specimen in the transition temperature range. The nearly constant A2 value for a given specimen geometry over a wide range of temperature indicates that the specimen indeed has reached the state of large scale yielding at fracture and is consistent with Eq. (13). To apply the RKR model, a critical fracture stress rf and a critical distance rc are required. Ritchie et al. [29] have investigated the fracture mechanisms of A533B steel and suggested 2–4 austenitic grains of size 0.025 mm for rc and 1830 MPa for rf. Since the test data are available for A533B as shown in Fig. 2, one can alternatively determine these two constants using the fracture toughness of high constraint specimens pffiffiffiffi at lower shelf. Using the fitted toughness value at 135 C of Fig. 2, i.e. KJ = 78.7 MPa m corresponding 6
FEA a/W = 0.5 Δ FEA a/W = 0.075 — J -A 2 a/W = 0.5 — J -A 2 a /W = 0.075
5
σ θ /σ 0
4 3 2 1 0
0
1
2
r
3
4
5
J σ
Fig. 4. Comparison of the opening stress directly ahead of the crack tip at JC = 64.3 N/mm for a/W = 0.5 and JC = 405.3 N/mm for a/W = 0.075 at T = 100 C.
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a /W = 0.5 a /W = 0.075
-A 2
0.5 0.4 0.3 0.2 0.1 0 -165 -150 -135 -120 -105 -90 -75 -60
-45
-30
Temperature (ºC) Fig. 5. A2 at fracture as a function of temperature showing nearly constant value for the two specimen geometries indicating large scale yielding (LSY).
to Jc = 26.2 N/mm for the deep crack specimen, and choosing rc = 0.12 mm, FEA reveals rf = 1955 MPa. Note that rc = 0.12 mm was chosen here to be consistent with those determined by Ritchie et al. [29], Ortner and Hippsley [32] and Bezensek and Hancock [2,20]. Bezensek and Hancock [2,20] have obtained rf = 2300 MPa at rc = 0.12 mm using the same specimen. The difference in rf is believed to come from the FEA modeling. A modified boundary layer formulation with an assumed T-stress applied on the outside boundary was used by Bezensek and Hancock [2,20] while the full specimen model shown in Fig. 3 is used in the current analysis. The two critical values in the RKR model rf = 1955 MPa at rc = 0.12 mm are used next for predicting the entire ductile–brittle transition curves for specimens with various constraint. Using Eq. (6), the transition curve for the deep cracked specimen group (a/W = 0.5, A2 = 0.21) in Fig. 2 is scaled upward to the curve for the shallow cracked specimen group ða=W ¼ 0:075; A2 ¼ 0:34Þ using rc = 0.12 mm and is shown in Fig. 6. The predicted curve for the shallow cracked group compares moderately well with the test data. The shift results in an overall move of the transition curve to lower temperature. Following the discussion in Section 3, if one assumes that the material has the same strain hardening at different temperature, Eq. (9) can be used to predict the fracture toughness for any specimen having constraint A2. Combining with Eq. (11), Eq. (9) can then be extended to determine the entire ductile–brittle transition curve over the transition temperature range. The two curves in Fig. 7 are obtained following this procedure
600
KJ (MPa m)
500
× a/W = 0.5 a/W = 0.075
400 300 200 100 0 -160
-140
-120 -100 -80 Temperature (ºC)
-60
-40
Fig. 6. The predicted ductile–brittle transition curve for a/W = 0.075 group of specimens from the fitted curve for a/W = 0.5 based on rc = 0.12 mm and Eq. (6).
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600 500
× a/W = 0.5
KJ (MPa √m)
a/W = 0.075
400 300 200 100 0 -160
-140
-120
-100
-80
-60
-40
Temperature (ºC)
Fig. 7. The predicted ductile–brittle transition curves based on rf = 1955 MPa at rc = 0.12 mm and Eq. (9).
600 × a/W = 0.5 500
a/W = 0.075
KJ (MPa √m)
400 300 200 100 0 -160
-140
-120
-100
-80
-60
-40
Temperature (ºC) Fig. 8. Ductile–brittle transition curve for a/W = 0.075 group of specimens predicted from the a/W = 0.5 fitted curve and Eq. (10) with rc = 0.12 mm.
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi using A2 = 0.21 (for a/W = 0.5) and A2 = 0.34 (for a/W = 0.075), K J ¼ J C E=ð1 v2 Þ, rf = 1955 MPa and rc = 0.12 mm. The predicted curves compare very well with the experimental data. Using Eq. (10) and rc = 0.12 mm the fitted transition curve for the deep cracked specimen group (i.e. A2 = 0.21) in Fig. 2 is shifted to predict the transition curve for the shallow cracked specimen group ðA2 ¼ 0:34Þ. The results are shown in Fig. 8 where again the predicted curve for the shallow cracked specimen group is reasonable. The shift again results in a temperature shift (or scaling) of the curve to lower temperature. 7. Concluding remarks Assuming that the mechanism of local cleavage and the hardening behavior of the steel are independent of temperature and given the fact that yield stress is strongly temperature dependent, the procedure developed in this article allows the prediction of fracture toughness of a group of specimens with certain constraint from the fracture toughness of another group of specimens with a different constraint. The J–A2 three-term solution is used to characterize the crack-tip stress fields and the constraint level of test specimen is quantified by A2. It is demonstrated that combined with the RKR model as the fracture criterion all three procedures, i.e. Eqs. (6), (9) and (10), shift (or scale) the ductile–brittle transition curve to lower temperature as it transfers
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the curve from high constraint specimen (e.g. a/W = 0.5) to low constraint specimen (e.g. a/W = 0.075). The predicted curves from the three procedures compared well with the test data from A533B RPV steel. Fig. 9 schematically shows the ductile–brittle transition curves as parameterized by the constraint A2. Practically, doing a set of tests on deep cracked specimens such as the ASTM specimens the ductile–brittle transition of shallow cracked specimens can then be predicted. More importantly the ductile–brittle transition curve of any structure component can be obtained as well provided that the constraint parameter A2 of the structure component can be determined. This paper provides a theoretical basis for this shift using constraint as an argument. Similar shift from deep cracked three-point-bend or CT type of specimens to shallow surface flaws as specified by the ASME Pressure Vessel Code warrants further studies. The shift would allow the increase of the operation margins in either temperature or pressure in RPV as attention is placed upon shallow flaws. In addition, the procedure is useful in transferability of test data from laboratory to large scale structure. Note that the same set of test data shown in Fig. 2 were used by Sherry et al. [1] and Bezensek and Hancock [2,20] in studying the transition curve for A533B RPV steels. Qualitatively similar shift of the ductile–brittle transition curve was obtained relative to the current work. When dealing with constraint, Sherry et al. [1] adopted the elastic T-stress and the Q-parameter while Bezensek and Hancock [2,20] used the elastic T-stress. As discussed in Section 1, T-stress only exists in linear elastic field which is certainly not the case for A533B steel even near the lower shelf region, e.g. at fracture LSY condition has prevailed especially for shallow cracks. Forcing the linear elastic theory for predicting fracture under LSY conditions therefore requires extra justifications and assumptions. The authors would like to point out that the T-stress is indeed a rigorous parameter in predicting fracture event including the constraint effect for brittle materials when plasticity is negligible at the moment of fracture [35,36]. The Q-stress is a summation of the second and the third terms in Eq. (3). It contains no details in distance r from the crack tip, angular function as well as material properties. It therefore suffers the distance and load dependence as has been experienced by many researchers independently. The J–A2 approach on the other hand is a derivative of the rigorous mathematical solution for non-linear power-law materials. It is best for quantifying fracture in LSY such as the A533B steels in the transition region. In summary, the purpose of any fracture mechanics theory is to provide a mathematical model to describe the phenomenological fracture event in materials. Models with fewer assumptions and can better represent the material behavior are to prevail. The current paper provides an alternative using the A2 parameter for quantifying the constraint level of deep and shallow specimens for A533B steels in the transition region. Rigorous analysis, assumptions and procedures are outlined. As a final note, it is generally believed that statistical model such as the Weibull should be used in interpreting the test data for cleavage fracture. While any statistical model is indispensable particularly for accommodating the large scatter of cleavage fracture data, one of the fundamental physical reasons of the shift or Structure specimen #2
or
Fracture Toughness
A2*** Lower constraint
A2**
Structure or specimen #1
A2 *
ASTM CT
A2*** < A2** < A2* Temperature Fig. 9. Schematic showing the ductile–brittle transition curves as parameterized by the constraint A2.
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