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Load-independent creep constraint parameter and its application
Engineering Fracture Mechanics 116 (2014) 41–57
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Engineering Fracture Mechanics journal homepage: www.elsevier.com/locate/engfracmech
Load-independent creep constraint parameter and its application J.P. Tan, G.Z. Wang ⇑, S.T. Tu ⇑, F.Z. Xuan Key Laboratory of Pressure Systems and Safety, Ministry of Education, East China University of Science and Technology, Shanghai 200237, China
a r t i c l e
i n f o
Article history: Received 22 May 2013 Received in revised form 24 October 2013 Accepted 18 December 2013
Keywords: Creep constraint Load independence Finite element Creep crack growth Application
a b s t r a c t A load-independent creep constraint parameter R was proposed, and its load-independence was validated using finite element results in previous studies. A fixed distance r = 0.2 mm from a crack tip is chosen to define the R, and the R at steady-state creep can be used to evaluate constraint level with little conservatism for whole creep time. The R can be used for ranking constraint levels for different specimens or components, and for predicting constraint-dependent creep crack growth rates. The constraint-dependent creep crack growth rate equations of a Cr–Mo–V steel have been obtained, and it may be used in creep life assessments. Ó 2013 Elsevier Ltd. All rights reserved.
1. Introduction Many experimental and theoretical evidences have shown that crack-tip constraint state has great influence on the fracture behavior of materials, and the loss of constraint causes the increases in fracture toughness [1]. The quantification of constraint has been widely investigated within the elastic–plastic fracture mechanics frame, and led to the development of two parameter fracture mechanics, such as J–T, J–Q and J–A2 [2–5]. In these approaches, the first parameter J integral sets the size scale over which high stresses and strains develop, and the secondary parameters T [2], Q [3,4] and A2 [5] were introduced to quantify the crack-tip constraint. The Hutchinson–Rice–Rosengren (HRR) singular stress field or the small scale yielding (SSY) solution with T-stress = 0 is generally used as the reference field to study the crack-tip constraint [1,3,4,6,7]. Under creep conditions, some experimental and theoretical evidences have shown that constraint can affect creep crack growth (CCG) rate [8–13]. In a recent study, it has been found that there is a significant constraint effect on CCG rate in low C region [14], and the CCG rates increase with increasing out-of-plane constraint (specimen thickness). To achieve accurate structural integrity assessment for high temperature components, it is necessary to find a simple and accurate constraint parameter to quantify the creep crack-tip constraint level in specimens or components, and then the correlation of constraint-dependent CCG rates of specimens or components can be obtained. However, the studies for two-parameter characterization of creep crack-tip fields are very limited. The creep crack-tip stress and strain rate fields are usually described by the C–Q two-parameter and the Q is used to quantify the constraint [15–17]. The effect of in-plane constraint on CCG using Q parameter was examined [16]. Combined the C–Q two-parameter concept with the NSW model, Nikbin [13] investigated the effect of constraint on the CCG rate. Based on the C–Q two-parameter concept and finite element (FE) analysis, Bettinson et al. [17] examined the effect of specimen type and load level on the Q from short to long term creep conditions for elasticcreep materials. ⇑ Corresponding authors. Tel.: +86 021 64252681; fax: +86 021 64253513. E-mail addresses: [email protected] (G.Z. Wang), [email protected] (S.T. Tu). 0013-7944/$ - see front matter Ó 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.engfracmech.2013.12.015
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Nomenclature a a0 a_ a_ 0 A B Bn C C(t) D0 E In J L n q Q Q R R Rav g Rz0 r rc t tred W z
a dij
e0 e_ 0 h
r0 r22 rij
r~ ij t
crack length initial crack length creep crack growth rate creep crack growth rate from the standard specimen coefficient in the power-law creep stain rate expression specimen thickness net specimen thickness C integral analogous to the J integral C(t) integral material constants of CCG rate Young’s modulus dimensionless constant related to n J-integral characteristic length, usually is set as 1 m power-law creep stress exponent or power-law stain hardening exponent in Ramberg–Osgood relation material constant of CCG rate constraint parameter under elastic–plastic condition load-independent constraint parameter under elastic–plastic condition creep constraint parameter load-independent creep constraint parameter average value of R along 3D crack front R value at specimen center distance from a crack tip creep damage zone creep time creep redistribution time specimen width distance from specimen center along crack front strain hardening coefficient in Ramberg–Osgood relation dimensionless function of n, h yield strain creep strain rate at yield stress polar coordinate at the crack tip yielding stress opening stress deviatoric stress dimensionless stress function of n, h Poisson’s ratio
Abbreviations 3D three dimension CCG creep crack growth CT compact tension CT2 compact tension specimen with 2 mm thickness CT5 compact tension specimen with 5 mm thickness CT10 compact tension specimen with 10 mm thickness CT10-SG compact tension specimen with 10 mm thickness and side grooves CCT center-cracked tension FEM finite element method LSC large-scale creep LSY large-scale yield PE plane strain PS plane stress SENB single-edge notched bend SENT single-edge notched tension SENTDp single-edge notched tension with 0.05 W loading point offset SSC small-scale creep SSY small-scale yield
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In the previous work of authors [18], it was found that HRR field is not suitable to be used as a reference field for defining the constraint parameter under creep condition. The stress singularity near the crack tips does not exist due to the crack-tip blunting caused by the larger creep deformation and damage. Because the deep crack compact tension (CT) specimen (a/W is about 0.5) with high constraint in plane strain (PE) state are usually used for measurement of creep fracture toughness (which is the creep toughness corresponding to a given crack extension at a given time [19]) and CCG rate of materials, the crack-tip stress field calculated by the FEM in this specimen was considered as the reference field for defining the constraint parameter. A new creep constraint parameter R was defined as follows [18]:
R¼
r22 ðr22 ÞCT ; at h ¼ 0; r ¼ distance from a crack tip r0
ð1Þ
where r0 is the normalizing stress. R represents stress field difference Dr = r22(r22)CT between the under evaluated specimen or component and the CT specimen with a/W = 0.5 at the same C value. The definition of R is similar to the Q, but the reference field and distance r is different. The use of the physical distance r from a crack tip in Eq. (1) allows the direct measurement for high constraint zone size which is relative to the creep damage and fracture process zone ahead of a crack tip. Based on the constraint parameter R, the creep crack-tip constraint induced by the crack depths [18,20], specimen thicknesses [21] and loading configurations [22] has been investigated in detail, and the correlation of creep crack-tip constraint between axially cracked pipelines and test specimens in plane strain state has also been studied [23]. However, the constraint parameter R not only depends on specimen or component geometries, crack sizes and loading configurations, but also is a function of loading level (it is characterized by C), the distance form crack-tip r and creep time t/tred [20–22]. It was found that the creep constraint parameter R decreases with increasing loading level C. In the creep life assessment, the fracture parameter C (crack-tip load level) increases with crack growth, while the R cannot remain constant during the crack growth. In this case, it is difficult to calculate R for predicting constraint-dependent creep life. Therefore, it is necessary to find a load-independent creep constraint parameter. In this study, a load-independent creep constraint parameter R has been defined by modifying the constraint parameter R. The effects of distance from the crack tip r and creep time t/tred also were analyzed. Based on the FE calculation results of the parameter R and size of creep damage zone ahead of crack tips, a fixed distance r from a crack tip was used for defining the R. Finally, the application of the new constraint parameter R was discussed. 2. Load-independent creep constraint parameter R The J–Q theory is widely used in constraint analysis in the elastic–plastic fracture mechanics framework. However, the Q depends on structure geometry, crack size and applied load, and so cannot remain constant during the crack growth. Thus, it is improper to describe the constraint effect on the J–R curves in ductile crack growth [24,25]. To resolve this issue, Zhu et al. [25] proposed a modified crack-tip stress field of the J–Q theory with the HRR reference field, as follows,
rij J ¼ r0 ae0 r0 In L
1=ðnþ1Þ 1=ðnþ1Þ r r~ ij ðh; nÞ þ Q dij ; L
for r > J=r0 and jhj < p=2
ð2Þ
where e0 is the yield strain, r0 is the yield stress, a and n are material constant, J is the contour integral, In is an integration ~ ij ðh; nÞ are dimensionless stress function and (r, h) are the polar coordinates centered at constant, L is a characteristic length, r the crack tip. The Q is a modified constraint parameter and defined by
Q ¼
1=ðnþ1Þ
J
ae0 r0 In L
Q
for r ¼ 2J=r0 and h ¼ 0
ð3Þ
It has been revealed that the Q is a load-independent constraint parameter under large-scale yielding (LSY) or fully plastic deformation, and can be used to predict constraint-corrected J–R curves [25]. In the case of steady-state creep, power law creep is analogous to power law plasticity and the C parameter is analogous to the J integral [26]. Following the J–Q two-parameter field under the elastic–plastic condition, the creep crack-tip stress and strain rate fields are usually described by the C–Q two-parameter as follows [16,17],
C
rij =r0 ¼ _ e0 r0 In r
1=ðnþ1Þ
r~ ij ðh; nÞ þ Q dij
ð4Þ
where r and h is distance and angle from the crack tip, respectively, r0 is the yield stress, e_ 0 is the creep strain rate at the yield ~ ij ðh; nÞ are dimensionless funcstress, In is a parameter which depends on the creep exponent n and in-plane stress state, r tions of n, h and in-plane stress state and dij is the Kronecker delta. The HRR stress field is generally considered as the reference field. The Q quantifies the deviation of the stress from the HRR field, and can be obtained by FE calculation. However, as described in the Introduction, the HRR field is improper to be a reference field for defining the creep constraint parameter. Based on an actual reference stress field in the deep crack CT specimen (a/W = 0.5, a is the crack length and W is the specimen width) with high constraint, a creep constraint parameter R was defined [18], as shown by Eq. (1). Further studies [20–23] show that the parameter R depends on specimen or structure geometries, crack sizes, loading configurations, load level (C), distance from crack tips and creep time. So the R cannot remain constant during the creep crack
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growth due to the change in load level C. It is analogous to that the Q is improper to describe the constraint effect on the J–R curves in ductile crack growth. The R is improper to describe the constraint effect on CCG rate. To resolve this issue, a loadindependent creep constraint parameter under large-scale creep condition is required. The power law creep is analogous to power law plasticity and the C parameter is analogous to the J integral. Following the load-independent constraint parameter Q defined in Eq. (3) under LSY or fully plastic deformation, a load-independent creep constraint parameter R can be defined as follows,
R ¼
1=ðnþ1Þ C R; r0 e_ 0 L
for h ¼ 0 and r ¼ distance from crack tip
ð5Þ
where r0 is the yield stress, e_ 0 is the creep strain rate at the yield stress, n is material constant, C is the contour integral, L is a characteristic length (it is usually set to be 1 m [25]) and R is the constraint parameter defined in Eq. (1). Because the HRR field is not used in the definition of R, the parameter In is not used in Eq. (5). It should be noted that in the definition of the constraint parameter R in Eq. (1) and the parameter R in Eq. (5), the distance r from crack tip is not a constant value. This allows that the changes of parameters R and R with crack-tip distance r could be analyzed and discussed. For the use of the parameters R and R, a constant r value should be determined, which is analogous to the parameters Q and Q in Eq. (3). This will be discussed in Section 4, a fixed distance r = 0.2 mm will be used in the definition of R (Eq. (6)). 3. Validation of the parameter R To validate that the modified constraint parameter R is independent on the load level C, some FE calculation results of the R in various specimens and cracked pipes at different C from the previous papers of authors [20–23] were analyzed. The specimens include the CT specimens with different crack depths a/W under PE condition [20], the CT specimens with different thicknesses B and a/W = 0.5 (3D FE calculations [21]), the specimens with different loading configurations under PE condition [22], and the cracked pipes include axially cracked pipes with a/t = 0.35 under PE condition [23]. The constraint parameter R was calculated by using Eq. (5). The parameter R at various load levels (C) comes from the literature [20– 23]. The material used in the previous work [20–23] was the 2.25Cr1Mo steel. The power law creep material parameters n and A of the steel at 565 °C are 9.732 and 1.733 1026 MPan h1, respectively. The normalizing stress r0 was taken 1 as 330 MPa, and the corresponding e_ 0 ¼ 0:0561 h . A typical distribution of R ahead of crack tips under various C values is shown in Fig. 1 for the specimens with different loading configurations under PE condition [22]. It can be obviously found that with increasing the load level C, the negative R increases and constraint decreases. Fig. 2 shows the distributions of calculated parameter R ahead of crack tips at different C and steady-state creep of t/ tred = 1 (tred is stress redistribution time) for the specimens with different loading configurations (Fig. 2(a)) [22], for specimens with different loading modes and axially cracked pipe with a/t = 0.35 (Fig. 2(b)) [23], for CT specimens with different crack depths a/W (Fig. 2(c)) [20] and for CT specimens with different thicknesses B/W (W = 25 mm) at middle plane z/B = 0 ((Fig. 2(d)) [21]. It can be found that the R–r curves at different C are basically coincident for each type of specimens and cracked pipes. The little difference near the crack tips may be caused by larger creep deformation occurring at crack-tip region at higher C values. Fig. 2 also shows that the specimens with different loading configurations (CT, SENB, SENT, CCT) have different constraints (Fig. 2(a and b)), and with increasing crack depth a/W (Fig. 2(c)) and specimen thickness B/W (Fig. 2(d)) of CT specimens, the constraint R increases. Fig. 3 depicts the distributions of calculated parameter R ahead of crack tips at different C (load level) and transition creep time t/tred. It can be also found that the R–r curves at different C are basically coincident at transition creep time.
Fig. 1. The distributions of parameter R ahead of crack tips at different C (load level) and t/tred = 1 for specimens with different loading configurations [22].
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Fig. 2. The distributions of parameter R ahead of crack tips at different C (load level) and t/tred = 1, (a) for specimens with different loading configurations [22], (b) for specimens with different loading modes and axially cracked pipe with a/t = 0.35 [23], (c) for CT specimens with different crack depths a/W [20], (d) for CT specimens with different thicknesses B at middle plane z/B = 0 (3D) [21].
Therefore, it has been revealed that R is a load-independent constraint parameter at steady-state creep (large-scale creep) and transition creep (small-scale creep). The R–r curves only depend on specimen or structure geometries, crack sizes and loading configurations. Since the R is independent on C, it can be deduced that the R may be same during creep when the C(t) (for transition creep) reaches to C (for steady-state creep). Actually, this result can be seen from Fig. 5. Thus, the definition of the R in Eqs. (5) and (6) may be used for both transition creep and steady-state creep. 4. Effects of distance from a crack tip and creep time on R Although the constraint parameter R eliminates the effect of load level, the parameter still changes with the distance from a crack tip and creep time (Figs. 2 and 3). In the constraint analysis under elastic–plastic condition, the parameter Q is defined as the difference between the full-field solution and the HRR field [3,4]. The difference field within the sector |h| < p/2 and J/r0 < r < 5J/r0 corresponds effectively to a spatially uniform hydrostatic stress state of adjustable magnitude [1,6]. This two parameter J–Q solution is valid in the interest range of 1 < r/(J/r0) < 5 at small-scale yielding (SSY) and intermediate loading or contained yielding, and the parameter Q is approximately distance-independent due to the self-similar invariant nature of the stress fields (normalized by the initial yield stress or reference stress) with respect to the applied load [27]. In this case, a fixed distance of r/(J/r0) = 2 is chosen so that Q or Q is evaluated outside the finite-strain but still within the J–Q [4] or J–Q annulus [25], and a fixed Q or Q value can be obtained for engineering application. However, under largescale yielding (LSY) condition, the J–Q solution loses its effectiveness due to the loss of the self-similar invariant nature of the stress fields, and the Q is distance-dependent [27]. For convenient engineering application of the constraint parameter R, it is analogous to the parameter Q or Q, and a finite distance from a crack tip should be fixed to define a fixed R value for accurately describing creep crack-tip constraint in specimens or components. For PE results in Fig. 2, the R was essentially independent of the distance r from crack tips in a range of r = 0.1–0.4 mm. A recent study [14] of authors on creep fracture mechanism of a Cr–Mo–V steel in CT specimens with different thicknesses and loading levels shows that the size of creep damage zone rc with creep voids ahead of crack tips is in a range of about 0.18– 0.41 mm, as shown in Fig. 4. This zone encompasses the region over which creep damage accumulates locally at the crack tip.
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Fig. 3. The distributions of parameter R ahead of crack tips at different load level C, (a) for CT specimens with different crack depths a/W at t/tred = 0.03 [20]; (b) for CT specimens with different thicknesses B at middle plane z/B = 0 (3D) at t/tred = 0.001 [21].
The constraint evaluation in this damage zone and at the boundary near the damage zone is very important to determine the creep damage and fracture process [18]. Thus, a fixed distance r = 0.2 mm within the creep damage zone (its size is the critical distance rc for creep damage) may be chosen to define the R. In addition, Fig. 2 and the stress field results calculated by FEM in Refs. [18,20–22] show that the influence of blunting of the crack tip on the crack-tip stress fields are mainly in a region of r < 0.1 mm ahead of crack tips for the small-scale creep (SSC) and large-scale creep (LSC). The choice of a distance r = 0.2 mm for defining the parameter R can avoid the influence of crack-tip blunting on the stress fields, and also it is suitable to both the SSC and LSC conditions. Therefore, a fixed distance r = 0.2 mm can be chosen to define the R. Actually, an arbitrary distance choice in the range of 0.1–0.4 mm has no significant effect on the R value (Fig. 2). The creep constraint parameter R has been analyzed at steady-state creep in Fig. 2. The R should also be investigated at transition creep to obtain the influence of creep time t/tred on it. Prior to the attainment of widespread creep conditions, the crack tip stress and strain rate fields are usually characterized by the C(t) integral. Hence, considering transition creep and the fixed distance r = 0.2 mm, Eq. (5) can be amended as
R ¼
1=ðnþ1Þ CðtÞ R; r0 e_ 0 L
at h ¼ 0 and r ¼ 0:2 mm
ð6Þ
Fig. 5(a) and (b) are the typical changes of constraint parameter R with creep time t/tred in the CT specimens with different crack depths [20] and in the specimens with different loading configurations [23], respectively. The R was calculated by using Eq. (6). It can be seen that the patterns of R–t/tred curves are similar for different specimens. During the early stage of creep (0 < t/tred < 0.1), the R increases with creep time, and then it becomes independent on the t/tred. Because the period of 0 < t/tred < 0.1 is very short and R is low in this period, the parameter R under steady-state creep can be used to evaluate the constraint level with little conservatism during the whole creep time.
5. Application of constraint parameter R The load-independent constraint parameter R is valuable to accurate CCG life analyses for high temperature components and specimens. This will be shown in following two aspects.
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Fig. 4. The creep damage zones ahead of crack tips, (a) for CT10 specimen with lower C level; (b) for CT10 specimen with middle C level; (c) for CT2 specimen with lower C level [14].
5.1. Rank constraint level for different specimens and components According to the analyses above, the constraint parameter R in Eq. (6) is mainly dependent on specimen or component geometries, crack sizes and loading configurations. The constraint levels in various specimens or components with different crack sizes may be ranked by the parameter R. The R values were calculated by using Eq. (6) for some typical specimens and pipes of 2.25Cr1Mo steel in the literature [20–23], and are shown in Table 1. The PE in Table 1 means plane strain state. The
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Fig. 5. The distribution of constraint parameter R with creep time t/tred, (a) in the CT specimens with different crack depths [20], (b) in the specimens with different loading modes and in pipes [23].
Table 1 Constraint parameter R of some typical specimens and pipes of 2.25Cr1Mo steel. Type a/W B/W R
CT 0.1 PE 0.20
CT 0.2 PE 0.02
CT 0.35 PE 0.24
CT 0.5 PE 0.00
CT 0.7 PE 0.10
CT 0.5 0.08 1.12
CT 0.5 0.2 0.33
CT 0.5 0.4 0.03
CT 0.5 0.5 0.08
Type a/W B/W R
CT 0.5 1 0.09
SENB 0.35 PE 0.50
SENB 0.5 PE 0.30
SENT 0.5 PE 0.47
SENT 0.35 PE 0.77
SENTDp 0.35 PE 1.05
Pipe 0.35 PE 1.08
CCT 0.5 PE 1.58
CCT 0.35 PE 1.64
increase of negative R implies the decrease of constraint. The rank by parameter R is consistent with that by parameter R in the literature [20–23]. But the parameter R is independent on load level C, distance r from crack tip and creep time, and it may be more convenient in engineering applications. Based on the parameter R, a creep constraint handbook may be created to lookup the constraint level of different specimens and components. 5.2. Predict constraint-dependent CCG rate Under creep conditions, some experimental and theoretical evidences have shown that constraint can affect CCG rate [11]. For a given C value (creep fracture mechanics parameter), the model predications showed that the CCG rates in PE state were significantly greater than those in PS state [13,28–31]. The experimental results of Tabuchi et al. [32] have shown that there was an effect of specimen thickness on the CCG rate, and the specimen with larger thickness exhibits the higher CCG rate than the smaller one. Recently, a significant constraint effect on CCG rate has been found in low C region and the CCG rates increase with increasing specimen thickness [14]. It also has been found that at the same C value the CCG rates measured in the center-cracked tension (CCT) specimens are lower than those obtained from deep crack CT specimens for the
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austenitic stainless steels [8,33,34] and ferritic steels [10]. Zhao et al. [35] have found that the initiation CCG growth rates increase with increasing crack depth in CT specimens. These results imply that the CCG rates have obvious in-plane and outof-plane constraint effects induced by specimen geometries, crack sizes and loading modes. However, the deep crack and thick CT specimen data measured by using standard test procedure [36] are conventionally used in CCG assessments [37]. The use of such bounding data will clearly produce conservative assessment result for the creep life of components in plants. Therefore, there is a strong incentive to reduce excess conservatism in order to provide more realistic life estimate [11]. To incorporate the creep constraint effect in the life assessments, it is necessary to obtain quantitative relationship between creep constraint parameter and CCG rate. In a recent study of authors [14], the CCG tests were conducted using CT specimens with different thicknesses for a Cr– Mo–V steel at 566 °C. The effect and mechanism of out-of-plane constraint and C levels on the CCG rates were investigated by the microscopic observations of creep fracture modes. The results show that the effect of out-of-plane constraint on the CCG rates is related to the C levels, as shown in Fig. 6(a). In low C region, with increasing the out-of-plane constraint, there is a change in creep fracture mode from a ductile intergranular fracture to a brittle intergranular fracture, which leads to the increase in CCG rates. In middle C region, the ductile intergranular fracture mode does not change with specimen thickness, and the out-of-plane constraint almost has no effect on the CCG rates. In high C region, the mixed intergranular and transgranular ductile fracture mode occurred in all specimens, and the out-of-plane constraint only has slight effect on the CCG rates. In the low C range, there is a large difference in CCG rates for the CT specimens with different thicknesses, as shown in Fig. 6(b). Usually, the service load levels (C levels) of high temperature components is low, and the constraint effect on CCG rate may be obvious. To predict the constraint-dependent CCG rate, a quantitative relation between CCG rate and constraint parameter should be established, and the parameter R may be valuable for this purpose. 5.2.1. Creep constraint parameter R in CT specimens with different thicknesses To quantify the relation between CCG rate and constraint in Fig. 6(b), the constraint parameter R of CT specimens with different thicknesses (B = 10, 5 and 2 mm) in the previous experimental study [14] needs to be calculated by FEM. Threedimensional and two-dimensional (PS and PE) FE models were established for the cracked CT specimens. Only one quarter of the 3D CT geometry is modeled due to symmetry. The specimen thickness B is 10, 5 and 2 mm, and denoted as CT10, CT5
Fig. 6. The correlation between CCG rates and C for the CT specimens with different thicknesses [14], (a) at whole C range, (b) at low C range.
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Fig. 7. The 3D FE model of the typical CT10 specimen, (a) meshes of the whole model, (b) local meshes at the specimen bottom, (c) local meshes around the crack tip.
and CT2 specimen, respectively. The specimen width W is 20 mm and crack length a0 is 10 mm (a/W = 0.5). One specimen CT10 has side groove, and is denoted as CT10-SG. The typical 3D FE meshes for the cracked CT specimen are illustrated in Fig. 7. The symmetry boundary condition is applied on the un-cracked ligament and mid-plane (z/B = 0) of the specimen. The crack tip is initially sharp. The load is applied to the loading hole as a distributed load. The mesh sensitivity studies have been done for four smallest element sizes of 10, 20, 40 and 80 lm around the crack tips. The results show that when the smallest element size is less than 40 lm, the FE results of the C(t), C and stresses do not essentially change with the element sizes. So the smallest element size of 20 lm around the crack tips is used. The typical model for the cracked CT10 specimens in Fig. 7 contains 11696 eight-node linear 3D elements (C3D8H) and 14319 nodes. The 2D FE model contains 1595 four-node linear plan strain elements (CPE4H) or plan stress elements (CPS4H) and 1701 nodes. All analyses were carried out using ABAQUS codes [38] with large deformation. The material used in the experimental study [14] was a Cr–Mo–V steel. An elastic–plastic power law creep material model was used in FE calculations. The elastic–plastic behavior is represented by a Ramberg–Osgood law as follows
r r e¼ þa r0 E
n ð7Þ
where E is the Young’s modulus, r0 is a normalizing stress, n is the hardening exponent and a is the strain hardening coefficient. The creep strain rate e_ is given by:
e_ ¼ Arn ¼ e_ 0 ðr=r0 Þn
ð8Þ
where n is the creep stress exponent, A is a constant. The material parameters are listed in Table 2 for the Cr–Mo–V steel at 566 °C [14]. The contour integral C(t) and C were evaluated by using the in-built ABAQUS routines. Thirteen equivalent thickness planar layers were set near crack-tips in 3D models, as shown in Fig. 7(b). Twenty contours were set around the crack tip at each layer. The C is usually path-independent, and C(t) is not path-independent, but with increasing time the C(t) gradually becomes path-independent [20,22]. The C(t) was defined as the value from the tenth contour near the crack tip for the different thickness specimens with the same size of meshes and contours (The C(t) defined by other contours has similar value as the tenth contour). The redistribution time tred was determined as the time at which the value of C(t)/C is about 1.02 [20,22].
Table 2 Material parameters of the Cr–Mo–V steel at 566 °C [14]. Young’s modulus E (GPa)
Poisson’s ratio (m)
Reference stress r0 (MPa)
Strain hardening coefficient a
Strain hardening exponent n
Norton’s coefficient A (MPan h1)
Norton’s exponent n
160
0.3
383
0.002
10.4
7.26E26
8.75
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Fig. 8. The distribution of parameter R ahead of crack tips in the CT2 specimen at different load levels C and t/tred = 1 (a) and at different creep time t/tred (b).
The R was calculated by using Eq. (6) at middle plane (z/B = 0) of the 3D CT specimens. Figs. 8 and 9 show the distributions of R for two typical specimens of CT2 and CT10 at various C and creep time t/tred. It can be found from Figs. 8(a) and 9(a) that the R–r curves are almost the same for different C levels, except slight discrepancy at very near crack tips. This difference may be due to the different crack-tip blunting under various loading level. These results further show that the new parameter R is independent on the load level C. Figs. 8(b) and 9(b) show that the R ahead of the crack tips are different for the creep time t/tred < 0.1, and almost the same for t/tred > 0.1. It is further proved that the constraint parameter R is almost independent on creep time t/tred. Because the constraint parameter R is almost the same at various C levels and creep time t/tred, it is not necessary to compare the constraint level of specimens at the same C and the same time. Fig. 10(a) shows the distributions of constraint parameter R ahead of crack tips at middle plane (z/B = 0) in CT specimens with different thicknesses at steady-state creep of t/tred = 1. It can be seen that significant constraint effect has been induced by the specimen thickness. The crack-tip constraint parameter R increases with increasing the specimen thickness B, and the specimen order in terms of crack-tip constraint from high to low is CT10-SG, CT10, CT5 and CT2. With increasing the distance r, the negative R increases and the constraint decreases. From Fig. 10(a), the constraint parameter R in Eq. (6) at middle plane (z/B = 0) can be determined, as shown by Rz0 in Table 3. The middle plane in CT10 specimens and plane strain specimen (PE) have highest constraint and the Rz0 is around zero. The plane stress specimen (PS) has the lowest constraint, and the constraint of CT5 specimen is higher than that of CT2 specimen. Fig. 10(b) shows the distributions of constraint parameter R ahead of crack tips at free surface (z/B = 0.5) in CT specimens with different thicknesses at steady-state creep of t/tred = 1. It can be seen that the constraint at free surface is much lower than that at the middle plane (z/B = 0) in Fig. 10(a), which shows significant out-of-plane constraint effect. From Fig. 10(b), the constraint parameter R in Eq. (6) at free surface (z/B = 0.5) can be determined, as shown by Rz0:5 in Table 3. It shows that with increasing specimen thickness the constraint at free surface increases. Fig. 10(c) depicts the variation of the constraint parameter R with the specimen thickness z/B at the fixed distance r = 0.2 mm. It can be found that with increasing the z/B from zero (middle plane) to 0.5 (free surface) and decreasing specimen thickness, the parameter R (constraint) decreases. This implies that the out-of-plane creep constraint effect can be characterized by the parameter R. In the 3D constraint analysis, in addition to the constraint parameter evaluated at
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Fig. 9. The distribution of parameter R ahead of crack tips in the CT10 specimen at different load levels C and t/tred = 1 (a) and at different creep time t/tred (b).
mid-thickness of the specimen, an average value of constraint parameter along the crack front also is usually determined to characterize the average constraint level [15,39]. This average constraint level may incorporate both in-plane and out-ofplane constraint effects. For actual test specimens for measuring creep crack growth (CCG) rate, the average crack growth length is usually measured. For analyzing constraint effect on the average crack growth length, the use of the average value of parameter R along the crack front may be useful. So the average value of parameter R along the crack front was calculated, and denoted as Rav g ,
Rav g ¼
1 z
Z
0
z
RðzÞ dz
ð9Þ
where z is the distance from middle plane along specimen thickness. The calculated value of the parameter Rav g is listed in Table 3 for the CT specimens with different thicknesses. It can be seen that the Rav g values are lower than the Rz0 values, and higher than the Rz0:5 values. The constraint levels are ranked well by the Rav g , the specimen order in terms of constraint from high to low is CT(PE), CT10-SG, CT10, CT5, CT2 and CT(PS). 5.2.2. Relation between CCG rate and creep constraint R The CCG rates are usually expressed in the form [30],
a_ ¼ D0 C q
ð10Þ
where a_ is CCG rate and in mm/h, C is crack-tip fracture parameter and in MPa m/h. The D0 and q are material constants which are often measured experimentally. When the constraint effect is considered, the CCG rate can be expressed as a function of C and constraint parameter. If the creep constraint is characterized by parameter R, the CCG rate a_ can be described in the following form [16],
a_ ¼ a_ 0 f ðR Þ where a_ 0 is the CCG rate from the standard specimen with high constraint, and f(R) is a function of R.
ð11Þ
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Fig. 10. The distributions of constraint parameter R ahead of crack tips at middle plane (z/B = 0) (a), at free surface (z/B = 0.5) (b) in the 3D CT specimens at t/tred = 1, and the distribution of parameter R along specimen thickness z/B in the 3D CT specimens (c).
Table 3 Constraint parameter R of the CT specimens with various thicknesses. Type
CT
CT
CT
CT
CT
CT
a/W Stress state Rz0 Rav g
0.5 PE 0 0
0.5 B = 10, SG 0.061 0.207
0.5 B = 10 0.046 0.514
0.5 B=5 0.256 0.980
0.5 B=2 1.021 1.631
0.5 PS 2.185 2.185
Rz0:5
0
1.411
2.427
2.709
2.775
2.185
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Table 4 A summary of CCG rate equations in low C region for different specimens [14]. Specimen
CCG rate equation
C region
CT10-SG
C 5 1.1E3 MPa m/h
CT5
a_ ¼ 0:428C 0:512 a_ ¼ 0:157C 0:484 a_ ¼ 0:083C 0:478
CT2
a_ ¼ 0:044C 0:485
C 5 5.6E5 MPa m/h
CT10
C 5 1.1E3 MPa m/h C 5 2.8E4 MPa m/h
Fig. 11. The relationship between constraint parameter (1R) and CCG rates.
The CCG rate data in Fig. 6(b) at low C region can be fitted by Eq. (10), and the CCG rate equations can be obtained for the CT specimens with different thicknesses, as shown in Table 4. For a given lower value C1 = 3E5 MPa m/h, the CCG rates of the CT10-SG, CT10, CT5 and CT2 specimens can be determined by the equations in Table 4, and their values are 2.07E3 mm/ h, 1.02E3 mm/h, 5.72E4 mm/h and 2.82E4 mm/h, respectively. Combining with the constraint parameter R of the CT _ a_ 0 and (1R) ((1 Rav g ), (1 Rz0 ) and (1 Rz0:5 )) can be depicted, as shown in specimens in Table 3, relations between a= Fig. 11. It can be found that the curve between CCG rate with 1 Rav g shows linear relation on log–log scale. In contrast, the relationship between CCG rate and 1 Rz0 and 1 Rz0:5 are complex. Power law relation can be used to fit the _ a_ 0 (1R) curves in Fig. 11. The fitting results are as follows, a=
_ a_ 0 ¼ 2:987ð1 Rav g Þ2:367 f ðR Þ ¼ a= 1:723
_ a_ 0 ¼ 0:932ð1 Rz0 Þ f ðR Þ ¼ a=
_ a_ 0 ¼ 27:678ð1 f ðR Þ ¼ a=
ð12Þ
2:954 Rz0:5 Þ
where a_ 0 is the CCG rate for the standard CT10 specimen. Then, the relation of the CCG rates for the CT specimens with different thicknesses can be expressed as, C a_ ¼ C 1 ð1 R Þ 2 a_ 0 ;
a_ 0 ¼ 0:157C 0:484
ð13Þ
where the C1 and C2 are constants. Using Eq. (13) and constraint level shown in Table 3, the CCG rate can be predicted for the specimens with different thicknesses. The comparison between prediction and experimental results has shown in Fig. 12. It can be found that the prediction results using the average constraint parameter Rav g are more accurate than those using the middle-thickness constraint parameter Rz0 and the free surface constraint parameter Rz0:5 . Therefore, the 3D average constraint parameter Rav g is recommended to estimate the constraint level of specimens or components. For a specimen or component with a certain geometry and loading mode, as long as the constraint parameter R is calculated under an arbitrary loading level C using FE analysis, the constraint-dependent CCG rate can be predicted at different C levels using Eq. (13). This constraint-dependent CCG rate equation (Eq. (13)) may be used in accurate creep life assessments of specimens or components with various constraint levels. The life assessment methodology incorporating creep constraint effect by the parameter R needs to be further studied.
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55
Fig. 12. Comparison between prediction and experimental results of CCG rates, (a) using C Rav g , (b) using C Rz0 and (c) using C Rz0:5 .
6. Conclusions (1) The creep constraint parameter R depends on load level C, and it cannot remain constant during the CCG. So the R is improper to describe the constraint effect on CCG rate. To resolve this issue, a load-independent creep constraint parameter R has been proposed, and its load independence has been validated using FE results of parameter R in various specimens and pipes in previous studies. (2) The parameter R was essentially independent of the distance r from crack tips in a range from 0.1 to 0.4 mm, and the size of creep damage zone ahead of crack tips for evaluating constraint is in a range of about 0.18–0.41 mm. So a fixed distance r = 0.2 mm is chosen to define the R. (3) The change of the parameter R with creep time t/tred shows that during the early stage of creep (0 < t/tred < 0.1), the R increases with creep time, and then it becomes independent on the t/tred. The parameter R at steady-state creep can be used to evaluate the constraint level with little conservatism for the whole creep time.
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(4) Two aspects of application of the load-independent creep constraint parameter R have been identified. One is to rank constraint levels for different specimens or structures, and another is to predict constraint-dependent CCG rate. (5) Based on recent experimental results of CCG rate in CT specimens with different thicknesses (out-of-plane constraint) and FE calculations of the parameter R, the constraint-dependent CCG rate equation of a Cr–Mo–V steel has been obtained. It may be used in accurate creep life assessments of specimens or components with various constraint levels. The life assessment methodology incorporating creep constraint effect by the parameter R needs to be further studied.
Acknowledgments This work was financially supported by the Projects of the National Natural Science Foundation of China (51375165, and 51325504), and the National High Technology Research and Development Program of China (2012AA040103). References [1] Dodds RH, Shih JCF, Anderson TL. Continuum and micromechanics treatment of constraint in fracture. Int J Fract 1993;64:101–33. [2] Betegon C, Hancock JW. Two-parameter characterization of elastic–plastic crack tip fields. J Appl Mech 1991;58:104–10. [3] 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(8): 989–1015. [4] O’Dowd NP, Shih CF. Family of crack-tip fields characterized by a triaxiality parameter-II. Fracture applications. J Mech Phys Solids 1991;40: 939–63. [5] Chao YJ, Yang S, Sutton MA. On the fracture of solids characterized by one or two parameters: theory and practice. J Mech Phys Solids 1994;42: 629–47. [6] O’Dowd NP, Shih CF. Two-parameter fracture mechanics: theory and applications. Philadelphia: American Society for Testing and Materials; 1994. [7] O’Dowd NP. Applications of two parameter approaches in elastic–plastic fracture mechanics. Engng Fract Mech 1995;52(3):445–65. [8] Bettinson AD, O’Dowd NP, Nikbin KM, Webster GA. Experimental investigation of constraint effects on creep crack growth. In: 2002 ASME pressure vessels and piping conference, Vancouver, BC, Canada; 2002 [PVP2002-1117]. [9] Tabuchi M, Adachi T, Yokobori AT, Fuji A, Ha J, Yokobori T. Evaluation of creep crack growth properties using circular notched specimens. Int J Pres Ves Pip 2003;80:417–25. [10] Takahashi Y, Igari T, Kawashima F, Date S, Titoh NI, Noguchi Y, et al. High temperature crack growth behavior of high-chromium steels. In: 18th International conference on structural mechanics in reactor technology, Beijing, China; 2005. p. 1904–15. [11] Budden PJ, Dean DW. Constraint effects on creep crack growth. In: Eighth international conference on creep and fatigue at elevated temperatures. American Society of Mechanical Engineers, San Antonio, Texas, USA; 2007 [CREEP2007-26104]. [12] Yatomi M, Nikbin KM. Numerical prediction of constraint effect of creep crack growth. In: Proceeding of the ASME 2009 pressure vessels and piping disvision conference, Prague, Czech Republic; 2009 [PVP2009-78032]. [13] Nikbin KM. Justification for meso-scale modelling in quantifying constraint during creep crack growth. Mater Sci Eng A 2004;365:107–13. [14] Tan JP, Tu ST, Wang GZ, Xuan FZ. Effect and mechanism of out-of-plane constraint on creep crack growth behavior of a Cr–Mo–V steel. Engng Fract Mech 2013;99:324–34. [15] Shih CF, O’Dowd NP, Kirk MT. A framework for quantifying crack tip constraint. In constraint effects in fracture. Philadelphia: American Society for Testing and Materials; 1993. p. 2–20. [16] Budden PJ, Ainsworth RA. The effect of constraint on creep fracture assessments. Int J Fract 1997;87:139–49. [17] Bettinson AD, O’Dowd NP, Nikbin K, Webster GA. Two parameter characterization of crack tip fields under creep conditions. In: IUTAM Symposium on creep in structures, Netherlands; 2001. p. 95–104. [18] Wang GZ, Liu XL, Xuan FZ, Tu ST. Effect of constraint induced by crack depth on creep crack-tip stress field in CT specimens. Int J Solids Struct 2010;47:51–7. [19] Davies CM, O’Dowd NP, Dean DW, Nikbin KM, Ainsworth RA. Failure assessment diagram analysis of creep crack initiation in 316H stainless steel. Int J Pres Ves Pip 2003;80:541–51. [20] Sun PJ, Wang GZ, Xuan FZ, Tu ST, Wang ZD. Quantitative characterization of creep constraint induced by crack depths in compact tension specimens. Engng Fract Mech 2011;78:653–65. [21] Sun PJ, Wang GZ, Xuan FZ, Tu ST, Wang ZD. Three-dimensional numerical analyses of out-of-plane creep crack-tip constraint in compact tension specimens. Int J Pres Ves Pip 2012;96–97:78–89. [22] Wang GZ, Kui LB, Xuan FZ, Tu ST. Numerical investigation on the creep crack-tip constraint induced by loading configuration of specimens. Engng Fract Mech 2012;79:353–62. [23] Tan JP, Wang GZ, Xuan FZ, Tu ST. Correlation of creep crack-tip constraint between axially cracked pipelines and test specimens. Int J Pres Ves Pip 2012;98:16–25. [24] Faleskog J. Effects of local constraint along three-dimensional crack fronts – a numerical and experimental investigation. J Mech Phys Solids 1995;43:447–93. [25] Zhu XK, Jang SK, Chen YF. A modification of J–Q theory and its applications. Int J Fract 2001;111:47–52. [26] Landes JD, Begley JA. A fracture mechanics approach to creep crack growth. Mechanics of crack growth. ASTM STP 590. American Society for Testing and Materials; 1976. p. 128–48. [27] Zhu XK, Leis BN. Bending modified J–Q theory and crack-tip constraint quantification. Int J Fract 2006;141:115–34. [28] Nikbin KM, Smith DJ, Webster GA. Prediction of creep crack growth from uniaxial creep data. Proc R Soc London, Ser A 1984;396:183–97. [29] Nikbin KM, Smith DJ, Webster GA. An engineering approach to the prediction of creep crack growth. J Eng Mater Technol 1986;108:186–91. [30] Webster GA, Ainsworth RA. High temperature component life assessment. London: Chapman and Hall; 1994. [31] Yatomi M, O’Dowd NP, Nikbin KM, Webster GA. Theoretical and numerical modelling of creep crack growth in a carbon-manganese steel. Engng Fract Mech 2006;73:1158–75. [32] Tabuchi M, Kubo K, Yagi K. Effect of specimen size on creep crack growth rate using ultra-large CT specimens for 1Cr–Mo–V steel. Engng Fract Mech 1991;40:311–21. [33] Bettinson A, Nikbin K, O’Dowd NP, Webster GA. The influence of constraint on the creep crack growth in 316H stainless steel. In: Proceedings 5th international conference structural integrity assessment, Cambridge, UK; 2000. [34] Ozmat B, Argon AS, Parks DM. Growth modes of cracks in creeping Type 304 stainless steel. Mech Mater 1991;11:1–17. [35] Zhao L, Jing HY, Xu LY, Han YD, Xiu JJ. Evaluation of constraint effects on creep crack growth by experimental investigation and numerical simulation. Engng Fract Mech 2012;96:251–66.
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[36] ASTM E 1457. Standard test method for measurement of creep crack growth rates in metals, vol. 3(1). Annual Book of ASTM Standards; 2001. p. 945– 66. [37] Ainsworth RA. Assessment procedure for the high temperature response of structures. Nuclear electric procedure R5, Issue 3. British Energy Generation Ltd.; 2003. [38] Hibbitt, Karlsson, Sorensen I. ABAQUS v6.7. 2007. [39] Silva LAL, Cravero S, Ruggieri C. Correlation of fracture behavior in high pressure pipelines with axial flaws using constraint designed test specimens– Part II: 3-D effects on constraint. Engng Fract Mech 2006;73:2123–38.
Contents lists available at ScienceDirect
Engineering Fracture Mechanics journal homepage: www.elsevier.com/locate/engfracmech
Load-independent creep constraint parameter and its application J.P. Tan, G.Z. Wang ⇑, S.T. Tu ⇑, F.Z. Xuan Key Laboratory of Pressure Systems and Safety, Ministry of Education, East China University of Science and Technology, Shanghai 200237, China
a r t i c l e
i n f o
Article history: Received 22 May 2013 Received in revised form 24 October 2013 Accepted 18 December 2013
Keywords: Creep constraint Load independence Finite element Creep crack growth Application
a b s t r a c t A load-independent creep constraint parameter R was proposed, and its load-independence was validated using finite element results in previous studies. A fixed distance r = 0.2 mm from a crack tip is chosen to define the R, and the R at steady-state creep can be used to evaluate constraint level with little conservatism for whole creep time. The R can be used for ranking constraint levels for different specimens or components, and for predicting constraint-dependent creep crack growth rates. The constraint-dependent creep crack growth rate equations of a Cr–Mo–V steel have been obtained, and it may be used in creep life assessments. Ó 2013 Elsevier Ltd. All rights reserved.
1. Introduction Many experimental and theoretical evidences have shown that crack-tip constraint state has great influence on the fracture behavior of materials, and the loss of constraint causes the increases in fracture toughness [1]. The quantification of constraint has been widely investigated within the elastic–plastic fracture mechanics frame, and led to the development of two parameter fracture mechanics, such as J–T, J–Q and J–A2 [2–5]. In these approaches, the first parameter J integral sets the size scale over which high stresses and strains develop, and the secondary parameters T [2], Q [3,4] and A2 [5] were introduced to quantify the crack-tip constraint. The Hutchinson–Rice–Rosengren (HRR) singular stress field or the small scale yielding (SSY) solution with T-stress = 0 is generally used as the reference field to study the crack-tip constraint [1,3,4,6,7]. Under creep conditions, some experimental and theoretical evidences have shown that constraint can affect creep crack growth (CCG) rate [8–13]. In a recent study, it has been found that there is a significant constraint effect on CCG rate in low C region [14], and the CCG rates increase with increasing out-of-plane constraint (specimen thickness). To achieve accurate structural integrity assessment for high temperature components, it is necessary to find a simple and accurate constraint parameter to quantify the creep crack-tip constraint level in specimens or components, and then the correlation of constraint-dependent CCG rates of specimens or components can be obtained. However, the studies for two-parameter characterization of creep crack-tip fields are very limited. The creep crack-tip stress and strain rate fields are usually described by the C–Q two-parameter and the Q is used to quantify the constraint [15–17]. The effect of in-plane constraint on CCG using Q parameter was examined [16]. Combined the C–Q two-parameter concept with the NSW model, Nikbin [13] investigated the effect of constraint on the CCG rate. Based on the C–Q two-parameter concept and finite element (FE) analysis, Bettinson et al. [17] examined the effect of specimen type and load level on the Q from short to long term creep conditions for elasticcreep materials. ⇑ Corresponding authors. Tel.: +86 021 64252681; fax: +86 021 64253513. E-mail addresses: [email protected] (G.Z. Wang), [email protected] (S.T. Tu). 0013-7944/$ - see front matter Ó 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.engfracmech.2013.12.015
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J.P. Tan et al. / Engineering Fracture Mechanics 116 (2014) 41–57
Nomenclature a a0 a_ a_ 0 A B Bn C C(t) D0 E In J L n q Q Q R R Rav g Rz0 r rc t tred W z
a dij
e0 e_ 0 h
r0 r22 rij
r~ ij t
crack length initial crack length creep crack growth rate creep crack growth rate from the standard specimen coefficient in the power-law creep stain rate expression specimen thickness net specimen thickness C integral analogous to the J integral C(t) integral material constants of CCG rate Young’s modulus dimensionless constant related to n J-integral characteristic length, usually is set as 1 m power-law creep stress exponent or power-law stain hardening exponent in Ramberg–Osgood relation material constant of CCG rate constraint parameter under elastic–plastic condition load-independent constraint parameter under elastic–plastic condition creep constraint parameter load-independent creep constraint parameter average value of R along 3D crack front R value at specimen center distance from a crack tip creep damage zone creep time creep redistribution time specimen width distance from specimen center along crack front strain hardening coefficient in Ramberg–Osgood relation dimensionless function of n, h yield strain creep strain rate at yield stress polar coordinate at the crack tip yielding stress opening stress deviatoric stress dimensionless stress function of n, h Poisson’s ratio
Abbreviations 3D three dimension CCG creep crack growth CT compact tension CT2 compact tension specimen with 2 mm thickness CT5 compact tension specimen with 5 mm thickness CT10 compact tension specimen with 10 mm thickness CT10-SG compact tension specimen with 10 mm thickness and side grooves CCT center-cracked tension FEM finite element method LSC large-scale creep LSY large-scale yield PE plane strain PS plane stress SENB single-edge notched bend SENT single-edge notched tension SENTDp single-edge notched tension with 0.05 W loading point offset SSC small-scale creep SSY small-scale yield
J.P. Tan et al. / Engineering Fracture Mechanics 116 (2014) 41–57
43
In the previous work of authors [18], it was found that HRR field is not suitable to be used as a reference field for defining the constraint parameter under creep condition. The stress singularity near the crack tips does not exist due to the crack-tip blunting caused by the larger creep deformation and damage. Because the deep crack compact tension (CT) specimen (a/W is about 0.5) with high constraint in plane strain (PE) state are usually used for measurement of creep fracture toughness (which is the creep toughness corresponding to a given crack extension at a given time [19]) and CCG rate of materials, the crack-tip stress field calculated by the FEM in this specimen was considered as the reference field for defining the constraint parameter. A new creep constraint parameter R was defined as follows [18]:
R¼
r22 ðr22 ÞCT ; at h ¼ 0; r ¼ distance from a crack tip r0
ð1Þ
where r0 is the normalizing stress. R represents stress field difference Dr = r22(r22)CT between the under evaluated specimen or component and the CT specimen with a/W = 0.5 at the same C value. The definition of R is similar to the Q, but the reference field and distance r is different. The use of the physical distance r from a crack tip in Eq. (1) allows the direct measurement for high constraint zone size which is relative to the creep damage and fracture process zone ahead of a crack tip. Based on the constraint parameter R, the creep crack-tip constraint induced by the crack depths [18,20], specimen thicknesses [21] and loading configurations [22] has been investigated in detail, and the correlation of creep crack-tip constraint between axially cracked pipelines and test specimens in plane strain state has also been studied [23]. However, the constraint parameter R not only depends on specimen or component geometries, crack sizes and loading configurations, but also is a function of loading level (it is characterized by C), the distance form crack-tip r and creep time t/tred [20–22]. It was found that the creep constraint parameter R decreases with increasing loading level C. In the creep life assessment, the fracture parameter C (crack-tip load level) increases with crack growth, while the R cannot remain constant during the crack growth. In this case, it is difficult to calculate R for predicting constraint-dependent creep life. Therefore, it is necessary to find a load-independent creep constraint parameter. In this study, a load-independent creep constraint parameter R has been defined by modifying the constraint parameter R. The effects of distance from the crack tip r and creep time t/tred also were analyzed. Based on the FE calculation results of the parameter R and size of creep damage zone ahead of crack tips, a fixed distance r from a crack tip was used for defining the R. Finally, the application of the new constraint parameter R was discussed. 2. Load-independent creep constraint parameter R The J–Q theory is widely used in constraint analysis in the elastic–plastic fracture mechanics framework. However, the Q depends on structure geometry, crack size and applied load, and so cannot remain constant during the crack growth. Thus, it is improper to describe the constraint effect on the J–R curves in ductile crack growth [24,25]. To resolve this issue, Zhu et al. [25] proposed a modified crack-tip stress field of the J–Q theory with the HRR reference field, as follows,
rij J ¼ r0 ae0 r0 In L
1=ðnþ1Þ 1=ðnþ1Þ r r~ ij ðh; nÞ þ Q dij ; L
for r > J=r0 and jhj < p=2
ð2Þ
where e0 is the yield strain, r0 is the yield stress, a and n are material constant, J is the contour integral, In is an integration ~ ij ðh; nÞ are dimensionless stress function and (r, h) are the polar coordinates centered at constant, L is a characteristic length, r the crack tip. The Q is a modified constraint parameter and defined by
Q ¼
1=ðnþ1Þ
J
ae0 r0 In L
Q
for r ¼ 2J=r0 and h ¼ 0
ð3Þ
It has been revealed that the Q is a load-independent constraint parameter under large-scale yielding (LSY) or fully plastic deformation, and can be used to predict constraint-corrected J–R curves [25]. In the case of steady-state creep, power law creep is analogous to power law plasticity and the C parameter is analogous to the J integral [26]. Following the J–Q two-parameter field under the elastic–plastic condition, the creep crack-tip stress and strain rate fields are usually described by the C–Q two-parameter as follows [16,17],
C
rij =r0 ¼ _ e0 r0 In r
1=ðnþ1Þ
r~ ij ðh; nÞ þ Q dij
ð4Þ
where r and h is distance and angle from the crack tip, respectively, r0 is the yield stress, e_ 0 is the creep strain rate at the yield ~ ij ðh; nÞ are dimensionless funcstress, In is a parameter which depends on the creep exponent n and in-plane stress state, r tions of n, h and in-plane stress state and dij is the Kronecker delta. The HRR stress field is generally considered as the reference field. The Q quantifies the deviation of the stress from the HRR field, and can be obtained by FE calculation. However, as described in the Introduction, the HRR field is improper to be a reference field for defining the creep constraint parameter. Based on an actual reference stress field in the deep crack CT specimen (a/W = 0.5, a is the crack length and W is the specimen width) with high constraint, a creep constraint parameter R was defined [18], as shown by Eq. (1). Further studies [20–23] show that the parameter R depends on specimen or structure geometries, crack sizes, loading configurations, load level (C), distance from crack tips and creep time. So the R cannot remain constant during the creep crack
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growth due to the change in load level C. It is analogous to that the Q is improper to describe the constraint effect on the J–R curves in ductile crack growth. The R is improper to describe the constraint effect on CCG rate. To resolve this issue, a loadindependent creep constraint parameter under large-scale creep condition is required. The power law creep is analogous to power law plasticity and the C parameter is analogous to the J integral. Following the load-independent constraint parameter Q defined in Eq. (3) under LSY or fully plastic deformation, a load-independent creep constraint parameter R can be defined as follows,
R ¼
1=ðnþ1Þ C R; r0 e_ 0 L
for h ¼ 0 and r ¼ distance from crack tip
ð5Þ
where r0 is the yield stress, e_ 0 is the creep strain rate at the yield stress, n is material constant, C is the contour integral, L is a characteristic length (it is usually set to be 1 m [25]) and R is the constraint parameter defined in Eq. (1). Because the HRR field is not used in the definition of R, the parameter In is not used in Eq. (5). It should be noted that in the definition of the constraint parameter R in Eq. (1) and the parameter R in Eq. (5), the distance r from crack tip is not a constant value. This allows that the changes of parameters R and R with crack-tip distance r could be analyzed and discussed. For the use of the parameters R and R, a constant r value should be determined, which is analogous to the parameters Q and Q in Eq. (3). This will be discussed in Section 4, a fixed distance r = 0.2 mm will be used in the definition of R (Eq. (6)). 3. Validation of the parameter R To validate that the modified constraint parameter R is independent on the load level C, some FE calculation results of the R in various specimens and cracked pipes at different C from the previous papers of authors [20–23] were analyzed. The specimens include the CT specimens with different crack depths a/W under PE condition [20], the CT specimens with different thicknesses B and a/W = 0.5 (3D FE calculations [21]), the specimens with different loading configurations under PE condition [22], and the cracked pipes include axially cracked pipes with a/t = 0.35 under PE condition [23]. The constraint parameter R was calculated by using Eq. (5). The parameter R at various load levels (C) comes from the literature [20– 23]. The material used in the previous work [20–23] was the 2.25Cr1Mo steel. The power law creep material parameters n and A of the steel at 565 °C are 9.732 and 1.733 1026 MPan h1, respectively. The normalizing stress r0 was taken 1 as 330 MPa, and the corresponding e_ 0 ¼ 0:0561 h . A typical distribution of R ahead of crack tips under various C values is shown in Fig. 1 for the specimens with different loading configurations under PE condition [22]. It can be obviously found that with increasing the load level C, the negative R increases and constraint decreases. Fig. 2 shows the distributions of calculated parameter R ahead of crack tips at different C and steady-state creep of t/ tred = 1 (tred is stress redistribution time) for the specimens with different loading configurations (Fig. 2(a)) [22], for specimens with different loading modes and axially cracked pipe with a/t = 0.35 (Fig. 2(b)) [23], for CT specimens with different crack depths a/W (Fig. 2(c)) [20] and for CT specimens with different thicknesses B/W (W = 25 mm) at middle plane z/B = 0 ((Fig. 2(d)) [21]. It can be found that the R–r curves at different C are basically coincident for each type of specimens and cracked pipes. The little difference near the crack tips may be caused by larger creep deformation occurring at crack-tip region at higher C values. Fig. 2 also shows that the specimens with different loading configurations (CT, SENB, SENT, CCT) have different constraints (Fig. 2(a and b)), and with increasing crack depth a/W (Fig. 2(c)) and specimen thickness B/W (Fig. 2(d)) of CT specimens, the constraint R increases. Fig. 3 depicts the distributions of calculated parameter R ahead of crack tips at different C (load level) and transition creep time t/tred. It can be also found that the R–r curves at different C are basically coincident at transition creep time.
Fig. 1. The distributions of parameter R ahead of crack tips at different C (load level) and t/tred = 1 for specimens with different loading configurations [22].
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Fig. 2. The distributions of parameter R ahead of crack tips at different C (load level) and t/tred = 1, (a) for specimens with different loading configurations [22], (b) for specimens with different loading modes and axially cracked pipe with a/t = 0.35 [23], (c) for CT specimens with different crack depths a/W [20], (d) for CT specimens with different thicknesses B at middle plane z/B = 0 (3D) [21].
Therefore, it has been revealed that R is a load-independent constraint parameter at steady-state creep (large-scale creep) and transition creep (small-scale creep). The R–r curves only depend on specimen or structure geometries, crack sizes and loading configurations. Since the R is independent on C, it can be deduced that the R may be same during creep when the C(t) (for transition creep) reaches to C (for steady-state creep). Actually, this result can be seen from Fig. 5. Thus, the definition of the R in Eqs. (5) and (6) may be used for both transition creep and steady-state creep. 4. Effects of distance from a crack tip and creep time on R Although the constraint parameter R eliminates the effect of load level, the parameter still changes with the distance from a crack tip and creep time (Figs. 2 and 3). In the constraint analysis under elastic–plastic condition, the parameter Q is defined as the difference between the full-field solution and the HRR field [3,4]. The difference field within the sector |h| < p/2 and J/r0 < r < 5J/r0 corresponds effectively to a spatially uniform hydrostatic stress state of adjustable magnitude [1,6]. This two parameter J–Q solution is valid in the interest range of 1 < r/(J/r0) < 5 at small-scale yielding (SSY) and intermediate loading or contained yielding, and the parameter Q is approximately distance-independent due to the self-similar invariant nature of the stress fields (normalized by the initial yield stress or reference stress) with respect to the applied load [27]. In this case, a fixed distance of r/(J/r0) = 2 is chosen so that Q or Q is evaluated outside the finite-strain but still within the J–Q [4] or J–Q annulus [25], and a fixed Q or Q value can be obtained for engineering application. However, under largescale yielding (LSY) condition, the J–Q solution loses its effectiveness due to the loss of the self-similar invariant nature of the stress fields, and the Q is distance-dependent [27]. For convenient engineering application of the constraint parameter R, it is analogous to the parameter Q or Q, and a finite distance from a crack tip should be fixed to define a fixed R value for accurately describing creep crack-tip constraint in specimens or components. For PE results in Fig. 2, the R was essentially independent of the distance r from crack tips in a range of r = 0.1–0.4 mm. A recent study [14] of authors on creep fracture mechanism of a Cr–Mo–V steel in CT specimens with different thicknesses and loading levels shows that the size of creep damage zone rc with creep voids ahead of crack tips is in a range of about 0.18– 0.41 mm, as shown in Fig. 4. This zone encompasses the region over which creep damage accumulates locally at the crack tip.
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Fig. 3. The distributions of parameter R ahead of crack tips at different load level C, (a) for CT specimens with different crack depths a/W at t/tred = 0.03 [20]; (b) for CT specimens with different thicknesses B at middle plane z/B = 0 (3D) at t/tred = 0.001 [21].
The constraint evaluation in this damage zone and at the boundary near the damage zone is very important to determine the creep damage and fracture process [18]. Thus, a fixed distance r = 0.2 mm within the creep damage zone (its size is the critical distance rc for creep damage) may be chosen to define the R. In addition, Fig. 2 and the stress field results calculated by FEM in Refs. [18,20–22] show that the influence of blunting of the crack tip on the crack-tip stress fields are mainly in a region of r < 0.1 mm ahead of crack tips for the small-scale creep (SSC) and large-scale creep (LSC). The choice of a distance r = 0.2 mm for defining the parameter R can avoid the influence of crack-tip blunting on the stress fields, and also it is suitable to both the SSC and LSC conditions. Therefore, a fixed distance r = 0.2 mm can be chosen to define the R. Actually, an arbitrary distance choice in the range of 0.1–0.4 mm has no significant effect on the R value (Fig. 2). The creep constraint parameter R has been analyzed at steady-state creep in Fig. 2. The R should also be investigated at transition creep to obtain the influence of creep time t/tred on it. Prior to the attainment of widespread creep conditions, the crack tip stress and strain rate fields are usually characterized by the C(t) integral. Hence, considering transition creep and the fixed distance r = 0.2 mm, Eq. (5) can be amended as
R ¼
1=ðnþ1Þ CðtÞ R; r0 e_ 0 L
at h ¼ 0 and r ¼ 0:2 mm
ð6Þ
Fig. 5(a) and (b) are the typical changes of constraint parameter R with creep time t/tred in the CT specimens with different crack depths [20] and in the specimens with different loading configurations [23], respectively. The R was calculated by using Eq. (6). It can be seen that the patterns of R–t/tred curves are similar for different specimens. During the early stage of creep (0 < t/tred < 0.1), the R increases with creep time, and then it becomes independent on the t/tred. Because the period of 0 < t/tred < 0.1 is very short and R is low in this period, the parameter R under steady-state creep can be used to evaluate the constraint level with little conservatism during the whole creep time.
5. Application of constraint parameter R The load-independent constraint parameter R is valuable to accurate CCG life analyses for high temperature components and specimens. This will be shown in following two aspects.
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Fig. 4. The creep damage zones ahead of crack tips, (a) for CT10 specimen with lower C level; (b) for CT10 specimen with middle C level; (c) for CT2 specimen with lower C level [14].
5.1. Rank constraint level for different specimens and components According to the analyses above, the constraint parameter R in Eq. (6) is mainly dependent on specimen or component geometries, crack sizes and loading configurations. The constraint levels in various specimens or components with different crack sizes may be ranked by the parameter R. The R values were calculated by using Eq. (6) for some typical specimens and pipes of 2.25Cr1Mo steel in the literature [20–23], and are shown in Table 1. The PE in Table 1 means plane strain state. The
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Fig. 5. The distribution of constraint parameter R with creep time t/tred, (a) in the CT specimens with different crack depths [20], (b) in the specimens with different loading modes and in pipes [23].
Table 1 Constraint parameter R of some typical specimens and pipes of 2.25Cr1Mo steel. Type a/W B/W R
CT 0.1 PE 0.20
CT 0.2 PE 0.02
CT 0.35 PE 0.24
CT 0.5 PE 0.00
CT 0.7 PE 0.10
CT 0.5 0.08 1.12
CT 0.5 0.2 0.33
CT 0.5 0.4 0.03
CT 0.5 0.5 0.08
Type a/W B/W R
CT 0.5 1 0.09
SENB 0.35 PE 0.50
SENB 0.5 PE 0.30
SENT 0.5 PE 0.47
SENT 0.35 PE 0.77
SENTDp 0.35 PE 1.05
Pipe 0.35 PE 1.08
CCT 0.5 PE 1.58
CCT 0.35 PE 1.64
increase of negative R implies the decrease of constraint. The rank by parameter R is consistent with that by parameter R in the literature [20–23]. But the parameter R is independent on load level C, distance r from crack tip and creep time, and it may be more convenient in engineering applications. Based on the parameter R, a creep constraint handbook may be created to lookup the constraint level of different specimens and components. 5.2. Predict constraint-dependent CCG rate Under creep conditions, some experimental and theoretical evidences have shown that constraint can affect CCG rate [11]. For a given C value (creep fracture mechanics parameter), the model predications showed that the CCG rates in PE state were significantly greater than those in PS state [13,28–31]. The experimental results of Tabuchi et al. [32] have shown that there was an effect of specimen thickness on the CCG rate, and the specimen with larger thickness exhibits the higher CCG rate than the smaller one. Recently, a significant constraint effect on CCG rate has been found in low C region and the CCG rates increase with increasing specimen thickness [14]. It also has been found that at the same C value the CCG rates measured in the center-cracked tension (CCT) specimens are lower than those obtained from deep crack CT specimens for the
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austenitic stainless steels [8,33,34] and ferritic steels [10]. Zhao et al. [35] have found that the initiation CCG growth rates increase with increasing crack depth in CT specimens. These results imply that the CCG rates have obvious in-plane and outof-plane constraint effects induced by specimen geometries, crack sizes and loading modes. However, the deep crack and thick CT specimen data measured by using standard test procedure [36] are conventionally used in CCG assessments [37]. The use of such bounding data will clearly produce conservative assessment result for the creep life of components in plants. Therefore, there is a strong incentive to reduce excess conservatism in order to provide more realistic life estimate [11]. To incorporate the creep constraint effect in the life assessments, it is necessary to obtain quantitative relationship between creep constraint parameter and CCG rate. In a recent study of authors [14], the CCG tests were conducted using CT specimens with different thicknesses for a Cr– Mo–V steel at 566 °C. The effect and mechanism of out-of-plane constraint and C levels on the CCG rates were investigated by the microscopic observations of creep fracture modes. The results show that the effect of out-of-plane constraint on the CCG rates is related to the C levels, as shown in Fig. 6(a). In low C region, with increasing the out-of-plane constraint, there is a change in creep fracture mode from a ductile intergranular fracture to a brittle intergranular fracture, which leads to the increase in CCG rates. In middle C region, the ductile intergranular fracture mode does not change with specimen thickness, and the out-of-plane constraint almost has no effect on the CCG rates. In high C region, the mixed intergranular and transgranular ductile fracture mode occurred in all specimens, and the out-of-plane constraint only has slight effect on the CCG rates. In the low C range, there is a large difference in CCG rates for the CT specimens with different thicknesses, as shown in Fig. 6(b). Usually, the service load levels (C levels) of high temperature components is low, and the constraint effect on CCG rate may be obvious. To predict the constraint-dependent CCG rate, a quantitative relation between CCG rate and constraint parameter should be established, and the parameter R may be valuable for this purpose. 5.2.1. Creep constraint parameter R in CT specimens with different thicknesses To quantify the relation between CCG rate and constraint in Fig. 6(b), the constraint parameter R of CT specimens with different thicknesses (B = 10, 5 and 2 mm) in the previous experimental study [14] needs to be calculated by FEM. Threedimensional and two-dimensional (PS and PE) FE models were established for the cracked CT specimens. Only one quarter of the 3D CT geometry is modeled due to symmetry. The specimen thickness B is 10, 5 and 2 mm, and denoted as CT10, CT5
Fig. 6. The correlation between CCG rates and C for the CT specimens with different thicknesses [14], (a) at whole C range, (b) at low C range.
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Fig. 7. The 3D FE model of the typical CT10 specimen, (a) meshes of the whole model, (b) local meshes at the specimen bottom, (c) local meshes around the crack tip.
and CT2 specimen, respectively. The specimen width W is 20 mm and crack length a0 is 10 mm (a/W = 0.5). One specimen CT10 has side groove, and is denoted as CT10-SG. The typical 3D FE meshes for the cracked CT specimen are illustrated in Fig. 7. The symmetry boundary condition is applied on the un-cracked ligament and mid-plane (z/B = 0) of the specimen. The crack tip is initially sharp. The load is applied to the loading hole as a distributed load. The mesh sensitivity studies have been done for four smallest element sizes of 10, 20, 40 and 80 lm around the crack tips. The results show that when the smallest element size is less than 40 lm, the FE results of the C(t), C and stresses do not essentially change with the element sizes. So the smallest element size of 20 lm around the crack tips is used. The typical model for the cracked CT10 specimens in Fig. 7 contains 11696 eight-node linear 3D elements (C3D8H) and 14319 nodes. The 2D FE model contains 1595 four-node linear plan strain elements (CPE4H) or plan stress elements (CPS4H) and 1701 nodes. All analyses were carried out using ABAQUS codes [38] with large deformation. The material used in the experimental study [14] was a Cr–Mo–V steel. An elastic–plastic power law creep material model was used in FE calculations. The elastic–plastic behavior is represented by a Ramberg–Osgood law as follows
r r e¼ þa r0 E
n ð7Þ
where E is the Young’s modulus, r0 is a normalizing stress, n is the hardening exponent and a is the strain hardening coefficient. The creep strain rate e_ is given by:
e_ ¼ Arn ¼ e_ 0 ðr=r0 Þn
ð8Þ
where n is the creep stress exponent, A is a constant. The material parameters are listed in Table 2 for the Cr–Mo–V steel at 566 °C [14]. The contour integral C(t) and C were evaluated by using the in-built ABAQUS routines. Thirteen equivalent thickness planar layers were set near crack-tips in 3D models, as shown in Fig. 7(b). Twenty contours were set around the crack tip at each layer. The C is usually path-independent, and C(t) is not path-independent, but with increasing time the C(t) gradually becomes path-independent [20,22]. The C(t) was defined as the value from the tenth contour near the crack tip for the different thickness specimens with the same size of meshes and contours (The C(t) defined by other contours has similar value as the tenth contour). The redistribution time tred was determined as the time at which the value of C(t)/C is about 1.02 [20,22].
Table 2 Material parameters of the Cr–Mo–V steel at 566 °C [14]. Young’s modulus E (GPa)
Poisson’s ratio (m)
Reference stress r0 (MPa)
Strain hardening coefficient a
Strain hardening exponent n
Norton’s coefficient A (MPan h1)
Norton’s exponent n
160
0.3
383
0.002
10.4
7.26E26
8.75
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Fig. 8. The distribution of parameter R ahead of crack tips in the CT2 specimen at different load levels C and t/tred = 1 (a) and at different creep time t/tred (b).
The R was calculated by using Eq. (6) at middle plane (z/B = 0) of the 3D CT specimens. Figs. 8 and 9 show the distributions of R for two typical specimens of CT2 and CT10 at various C and creep time t/tred. It can be found from Figs. 8(a) and 9(a) that the R–r curves are almost the same for different C levels, except slight discrepancy at very near crack tips. This difference may be due to the different crack-tip blunting under various loading level. These results further show that the new parameter R is independent on the load level C. Figs. 8(b) and 9(b) show that the R ahead of the crack tips are different for the creep time t/tred < 0.1, and almost the same for t/tred > 0.1. It is further proved that the constraint parameter R is almost independent on creep time t/tred. Because the constraint parameter R is almost the same at various C levels and creep time t/tred, it is not necessary to compare the constraint level of specimens at the same C and the same time. Fig. 10(a) shows the distributions of constraint parameter R ahead of crack tips at middle plane (z/B = 0) in CT specimens with different thicknesses at steady-state creep of t/tred = 1. It can be seen that significant constraint effect has been induced by the specimen thickness. The crack-tip constraint parameter R increases with increasing the specimen thickness B, and the specimen order in terms of crack-tip constraint from high to low is CT10-SG, CT10, CT5 and CT2. With increasing the distance r, the negative R increases and the constraint decreases. From Fig. 10(a), the constraint parameter R in Eq. (6) at middle plane (z/B = 0) can be determined, as shown by Rz0 in Table 3. The middle plane in CT10 specimens and plane strain specimen (PE) have highest constraint and the Rz0 is around zero. The plane stress specimen (PS) has the lowest constraint, and the constraint of CT5 specimen is higher than that of CT2 specimen. Fig. 10(b) shows the distributions of constraint parameter R ahead of crack tips at free surface (z/B = 0.5) in CT specimens with different thicknesses at steady-state creep of t/tred = 1. It can be seen that the constraint at free surface is much lower than that at the middle plane (z/B = 0) in Fig. 10(a), which shows significant out-of-plane constraint effect. From Fig. 10(b), the constraint parameter R in Eq. (6) at free surface (z/B = 0.5) can be determined, as shown by Rz0:5 in Table 3. It shows that with increasing specimen thickness the constraint at free surface increases. Fig. 10(c) depicts the variation of the constraint parameter R with the specimen thickness z/B at the fixed distance r = 0.2 mm. It can be found that with increasing the z/B from zero (middle plane) to 0.5 (free surface) and decreasing specimen thickness, the parameter R (constraint) decreases. This implies that the out-of-plane creep constraint effect can be characterized by the parameter R. In the 3D constraint analysis, in addition to the constraint parameter evaluated at
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Fig. 9. The distribution of parameter R ahead of crack tips in the CT10 specimen at different load levels C and t/tred = 1 (a) and at different creep time t/tred (b).
mid-thickness of the specimen, an average value of constraint parameter along the crack front also is usually determined to characterize the average constraint level [15,39]. This average constraint level may incorporate both in-plane and out-ofplane constraint effects. For actual test specimens for measuring creep crack growth (CCG) rate, the average crack growth length is usually measured. For analyzing constraint effect on the average crack growth length, the use of the average value of parameter R along the crack front may be useful. So the average value of parameter R along the crack front was calculated, and denoted as Rav g ,
Rav g ¼
1 z
Z
0
z
RðzÞ dz
ð9Þ
where z is the distance from middle plane along specimen thickness. The calculated value of the parameter Rav g is listed in Table 3 for the CT specimens with different thicknesses. It can be seen that the Rav g values are lower than the Rz0 values, and higher than the Rz0:5 values. The constraint levels are ranked well by the Rav g , the specimen order in terms of constraint from high to low is CT(PE), CT10-SG, CT10, CT5, CT2 and CT(PS). 5.2.2. Relation between CCG rate and creep constraint R The CCG rates are usually expressed in the form [30],
a_ ¼ D0 C q
ð10Þ
where a_ is CCG rate and in mm/h, C is crack-tip fracture parameter and in MPa m/h. The D0 and q are material constants which are often measured experimentally. When the constraint effect is considered, the CCG rate can be expressed as a function of C and constraint parameter. If the creep constraint is characterized by parameter R, the CCG rate a_ can be described in the following form [16],
a_ ¼ a_ 0 f ðR Þ where a_ 0 is the CCG rate from the standard specimen with high constraint, and f(R) is a function of R.
ð11Þ
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Fig. 10. The distributions of constraint parameter R ahead of crack tips at middle plane (z/B = 0) (a), at free surface (z/B = 0.5) (b) in the 3D CT specimens at t/tred = 1, and the distribution of parameter R along specimen thickness z/B in the 3D CT specimens (c).
Table 3 Constraint parameter R of the CT specimens with various thicknesses. Type
CT
CT
CT
CT
CT
CT
a/W Stress state Rz0 Rav g
0.5 PE 0 0
0.5 B = 10, SG 0.061 0.207
0.5 B = 10 0.046 0.514
0.5 B=5 0.256 0.980
0.5 B=2 1.021 1.631
0.5 PS 2.185 2.185
Rz0:5
0
1.411
2.427
2.709
2.775
2.185
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Table 4 A summary of CCG rate equations in low C region for different specimens [14]. Specimen
CCG rate equation
C region
CT10-SG
C 5 1.1E3 MPa m/h
CT5
a_ ¼ 0:428C 0:512 a_ ¼ 0:157C 0:484 a_ ¼ 0:083C 0:478
CT2
a_ ¼ 0:044C 0:485
C 5 5.6E5 MPa m/h
CT10
C 5 1.1E3 MPa m/h C 5 2.8E4 MPa m/h
Fig. 11. The relationship between constraint parameter (1R) and CCG rates.
The CCG rate data in Fig. 6(b) at low C region can be fitted by Eq. (10), and the CCG rate equations can be obtained for the CT specimens with different thicknesses, as shown in Table 4. For a given lower value C1 = 3E5 MPa m/h, the CCG rates of the CT10-SG, CT10, CT5 and CT2 specimens can be determined by the equations in Table 4, and their values are 2.07E3 mm/ h, 1.02E3 mm/h, 5.72E4 mm/h and 2.82E4 mm/h, respectively. Combining with the constraint parameter R of the CT _ a_ 0 and (1R) ((1 Rav g ), (1 Rz0 ) and (1 Rz0:5 )) can be depicted, as shown in specimens in Table 3, relations between a= Fig. 11. It can be found that the curve between CCG rate with 1 Rav g shows linear relation on log–log scale. In contrast, the relationship between CCG rate and 1 Rz0 and 1 Rz0:5 are complex. Power law relation can be used to fit the _ a_ 0 (1R) curves in Fig. 11. The fitting results are as follows, a=
_ a_ 0 ¼ 2:987ð1 Rav g Þ2:367 f ðR Þ ¼ a= 1:723
_ a_ 0 ¼ 0:932ð1 Rz0 Þ f ðR Þ ¼ a=
_ a_ 0 ¼ 27:678ð1 f ðR Þ ¼ a=
ð12Þ
2:954 Rz0:5 Þ
where a_ 0 is the CCG rate for the standard CT10 specimen. Then, the relation of the CCG rates for the CT specimens with different thicknesses can be expressed as, C a_ ¼ C 1 ð1 R Þ 2 a_ 0 ;
a_ 0 ¼ 0:157C 0:484
ð13Þ
where the C1 and C2 are constants. Using Eq. (13) and constraint level shown in Table 3, the CCG rate can be predicted for the specimens with different thicknesses. The comparison between prediction and experimental results has shown in Fig. 12. It can be found that the prediction results using the average constraint parameter Rav g are more accurate than those using the middle-thickness constraint parameter Rz0 and the free surface constraint parameter Rz0:5 . Therefore, the 3D average constraint parameter Rav g is recommended to estimate the constraint level of specimens or components. For a specimen or component with a certain geometry and loading mode, as long as the constraint parameter R is calculated under an arbitrary loading level C using FE analysis, the constraint-dependent CCG rate can be predicted at different C levels using Eq. (13). This constraint-dependent CCG rate equation (Eq. (13)) may be used in accurate creep life assessments of specimens or components with various constraint levels. The life assessment methodology incorporating creep constraint effect by the parameter R needs to be further studied.
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Fig. 12. Comparison between prediction and experimental results of CCG rates, (a) using C Rav g , (b) using C Rz0 and (c) using C Rz0:5 .
6. Conclusions (1) The creep constraint parameter R depends on load level C, and it cannot remain constant during the CCG. So the R is improper to describe the constraint effect on CCG rate. To resolve this issue, a load-independent creep constraint parameter R has been proposed, and its load independence has been validated using FE results of parameter R in various specimens and pipes in previous studies. (2) The parameter R was essentially independent of the distance r from crack tips in a range from 0.1 to 0.4 mm, and the size of creep damage zone ahead of crack tips for evaluating constraint is in a range of about 0.18–0.41 mm. So a fixed distance r = 0.2 mm is chosen to define the R. (3) The change of the parameter R with creep time t/tred shows that during the early stage of creep (0 < t/tred < 0.1), the R increases with creep time, and then it becomes independent on the t/tred. The parameter R at steady-state creep can be used to evaluate the constraint level with little conservatism for the whole creep time.
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(4) Two aspects of application of the load-independent creep constraint parameter R have been identified. One is to rank constraint levels for different specimens or structures, and another is to predict constraint-dependent CCG rate. (5) Based on recent experimental results of CCG rate in CT specimens with different thicknesses (out-of-plane constraint) and FE calculations of the parameter R, the constraint-dependent CCG rate equation of a Cr–Mo–V steel has been obtained. It may be used in accurate creep life assessments of specimens or components with various constraint levels. The life assessment methodology incorporating creep constraint effect by the parameter R needs to be further studied.
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