Creep constraint and fracture parameter C∗ for axial semi-elliptical surface cracks with high aspect ratio in pressurized pipes

Accepted Manuscript Creep constraint and fracture parameter C∗ for axial semi-elliptical surface cracks with high aspect ratio in pressurized pipes X.M. Tan, G.Z. Wang, S.T. Tu, F.Z. Xuan PII: DOI: Reference:

S0013-7944(18)30310-2 https://doi.org/10.1016/j.engfracmech.2018.06.005 EFM 6027

To appear in:

Engineering Fracture Mechanics

Received Date: Revised Date: Accepted Date:

22 March 2018 23 May 2018 7 June 2018

Please cite this article as: Tan, X.M., Wang, G.Z., Tu, S.T., Xuan, F.Z., Creep constraint and fracture parameter C∗ for axial semi-elliptical surface cracks with high aspect ratio in pressurized pipes, Engineering Fracture Mechanics (2018), doi: https://doi.org/10.1016/j.engfracmech.2018.06.005

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Creep constraint and fracture parameter C* for axial semi-elliptical surface cracks with high aspect ratio in pressurized pipes X.M. 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 Abstract The unified creep constraint parameter Ac and fracture parameter C* for axial semi-elliptical surface cracks with high aspect ratio in pressurized pipes have been investigated by three-dimensional finite element analyses. The results show that the constraint level near surface part is lower than that at the deeper part of crack front. With increasing crack aspect ratio a/c and decreasing crack depth a/t, the constraint levels decrease. The parameter Ac solution for high aspect ratio cracks has been obtained. The crack aspect ratio a/c has significant effect on the C* distributions along the crack fronts. For high aspect ratio cracks, the maximum C* along the crack fronts occurs near surface part. With increasing crack aspect ratio a/c from 0.2 to 2, the creep crack initiation location will change from the deepest part of the crack front to the near surface part. For high aspect ratio cracks, if the calculations of C* and Ac at the deepest point is used to predict creep crack initiation life, non-conservative results may be produced. More benefits from incorporating constraint effects may be obtained for creep life assessments of pipes with high aspect ratio cracks due to their lower constraint. Keywords: Creep constraint, Fracture parameter C*, High aspect ratio, Crack, Pipe

Corresponding author. Tel.: +86 21 64252681; fax: +86 21 64252681. E-mail address: [email protected] (Guozhen Wang). 1

Nomenclature a A1, A2 Ac

.

crack depth constants in 2RN creep model unified characterization parameter of in-plane and out-of-plane creep constraint area surrounded by equivalent creep strain isoline unified characterization parameter of in-plane and out-of-plane constraint area surrounded by equivalent plastic strain isoline at fracture measured in a standard test specimen thickness crack length C* integral analogous to the J integral inner diameter of pipes Young’s modulus stress intensity factor creep stress intensity factor stress exponents in 2RN creep model internal pressure constraint parameter creep constraint parameter inner radius of pipes load-independent creep constraint parameter out-of-plane constraint parameter creep time or pipe thickness stress redistribution time specimen width shape function in stress intensity factor K solutions creep strain rate

.

creep strain rate at normalized stress

ACEEQ Ap Aref B 2c C* D E K Kcr n1, n2 p Q R Ri R* Tz t tred W Y



0 c

equivalent creep strain

p

equivalent plastic strain or true plastic strain

0

normalizing stress



axial tension stress

 Φ Abbreviations 3D

Poisson’s ratio angular parameter characterizing crack front position three-dimensional 2

2RN CCG CCI C(T) CEEQ FEM M(T) SEN(T)

2-regime Norton creep crack growth creep crack initiation compact tension equivalent creep strain in ABAQUS code finite element method middle tension single-edge notched tension

1. Introduction The main failure modes of high-temperature components (such as pressurized pipes and vessels) are creep crack initiation (CCI) and creep crack growth (CCG). A lot of experimental and theoretical studies have shown that specimen or component geometries, sizes and crack sizes can affect creep crack-tip constraint, which can subsequently produce effects on the CCI time [1-3] and CCG rate [4-14] of materials. For a given C* value, the CCI time reduces and CCG rate increases with increasing crack-tip constraint [1-15]. The effect of creep constraint on CCG rate also was regarded as "structural brittleness" [10]. The standard plane strain C(T) specimen with high constraint is usually used to measure CCI time and CCG rate of materials [16]. However, the defects produced in manufacturing and service in actual components usually are shallower surface cracks with lower crack-tip constraint. Thus, the use of the CCI time and CCG rate data from the standard C(T) specimen in creep life assessments of actual components will produce overly conservative results[17]. To reduce the excessive conservatism, the creep constraint effect should be considered in creep life assessments by using appropriate constraint parameters. Over the past decade, the creep crack-tip constraint effect has been investigated, and some constraint parameters have been proposed, such as the parameters Q [18, 19], R [20-23], R* [24, 25], Tz [26, 27] and Q* (modified Q parameter) [4,9,28]. Shlyannikov et al. [29] proposed a creep stress intensity factor Kcr to quantify the 3

crack growth resistance and geometry constraint effect on CCG resistance. These parameters mainly are obtained by crack-tip stress field analyses. It has been shown that the parameters Q, R, R* and Q* mainly can characterize in-plane constraint, and the Tz can characterize out-of-plane constraint [30, 31]. The parameter R* solutions have been obtained for pressurized pipes in the previous studies [32, 33]. Based on the two parameter C*-R* concept with considering constraint effect, the CCG life has been assessed for axially cracked pipes [34]. In recent work of Xu et al. [35], the parameter Q* was used to analyze the creep constraint of pressurized pipes with axial surface cracks. However, in actual components, both in-plane and out-of-plane constraints exist simultaneously. To describe their interaction and the overall constraint level, it requires a unified creep constraint parameter which can capture both in-plane and out-of-plane creep constraints. In recent studies of authors [36, 37], a unified constraint parameter Ac based on crack-tip equivalent creep strain has been proposed, and its capability for characterizing both in-plane and out-of-plane creep constraints has been analyzed and verified. In order to establish creep life assessment method incorporating constraint effect based on the C*-Ac two-parameter concept, in a recent study of authors [38], the parameter Ac solutions have been investigated and obtained for pressurized pipes with axial and circumferential surface cracks. The creep constraint parameter and fracture mechanics parameter C* are two important parameters in creep life assessments of cracked components considering constraint effect. In the past studies and analyses for these two parameters in cracked pipes, it mainly focuses on the semi-elliptical surface cracks with lower aspect ratio (a/c≤1, a is crack depth and c is one half of crack length) [32-35, 38-40]. However, deep and short cracks with high aspect ratio (a/c>1) may also appear in actual 4

components due to pitting corrosion, fatigue and so on [41, 42]. Aspect ratio a/c could change the distribution of constraint parameters and fracture mechanics parameters (such as stress intensity factor K, J-integral and creep fracture parameter C*) along crack fronts, thereby probably changing the failure mode, crack initiation location and service life of components. In elastic-plastic fracture mechanics, the shape function Y and stress intensity factor K solutions for semi-elliptical cracks in finite plates and pipes have been widely investigated. The solutions of Y and K proposed by Newman and Raju [43-45] are widely used in various industries. In a recent work by Bocher, et al.[46], these solutions have been reviewed and expanded. But these solutions are mainly for low aspect ratio semi-elliptical surface cracks (a/c≤1). In recent several years, the stress intensity factor K solutions have been investigated for high aspect ratio semi-elliptical surface cracks (a/c > 1) in pipes and cylindrical vessels [41,42,47,48]. However, the creep constraint and fracture parameter C* for high aspect ratio cracks in pipes and cylindrical vessels still have not been investigated and understood. In this work, the unified creep constraint parameter Ac and fracture parameter C* for axial semi-elliptical surface cracks with high aspect ratio (1.2≤a/c≤2) in pressurized pipes were investigated by three-dimensional finite element analyses, and the parameter Ac solution was also obtained. 2. Procedures of finite element analysis 2.1 Material In this work, a Cr-Mo-V steel (Chinese 25Cr2NiMo1V steel) was used as pipe material. The two-region Norton (2RN) creep model has been developed for this steel at 566℃, and is expressed in Eq. (1) [49]. Mechanical properties and the different 5

Norton model parameters of (A1, n1) and (A2, n2) are listed in Table 1. The true stress-strain curve for the Cr-Mo-V steel at 566℃ is depicted in Fig.1 [49]. 2RN model

(1)

2.2 Finite element models The non-through axial surface cracks in pipes are usually simplified as axial semi-elliptical cracks, as shown in Fig. 2. The a, 2c, Ri and t in Fig. 2 are the crack depth, crack length, pipe inner radius and wall thickness, respectively. Φ is the angular parameter taking the center of the ellipse as its coordinate origin, and the 2Φ/π = 0 denotes the surface point and the 2Φ/π = 1 indicates the deepest point along the crack front. The length of all pipeline models employed in this work is selected as 30t for avoiding boundary effects [40]. The three-dimensional (3D) FE analyses were carried out by using ABAQUS code [50]. A cracked pipe with wall thickness t = 38mm and Ri/t = 10 was modeled. In order to investigate the effects of crack sizes with high aspect ratio a/c on the creep constraint and C*, the axial surface cracks with a/t = 0.1, 0.2, 0.4, 0.6 and 0.8, and a/c = 1.2, 1.4, 1.6, 1.8 and 2.0 were modeled. A typical 3D FE model and local meshes in the vicinity of the crack tip for a pipe crack (a/t =0.2, a/c =1.2) are shown in Fig. 3. The FE models and mesh details for other pipe cracks are similar to those in Fig. 3. Considering symmetry of the geometry, only one-fourth of the 3D cracked pipe was modeled. The symmetry boundary condition is applied on the un-cracked ligament and the cross section near the crack, while the cross section away from the crack is fixed by an encastre constraint. A fine mesh configuration is used around the crack-tip 6

for all FEM models and coarse meshes are used in other zones. The eight node brick elements with full integration (C3D8) and the elastic-plastic-power law creep material model were used. The material creep parameters in Table 1 and the true stress-strain curve in Fig.1 were used. The focused ring meshes with minimum size of 40μm were used around the crack tip zone according to the mesh sensitivity analysis. An internal pressure p = 15MPa was applied on both inside face and inner crack faces of the pipes. Axial tension stress caused by the internal pressure was applied in the pipe end. 2.3 Calculations of unified creep constraint parameter Ac and fracture parameter C* A unified creep constraint parameter Ac based on crack-tip equivalent creep strain has been defined in the previous work of authors [30, 36]： Ac 

ACEEQ Aref

at

t tred

1

(2)

where ACEEQ is the area surrounded by the equivalent creep strain (  c ) isoline ahead of the crack tip in a specimen or component, Aref is the reference area surrounded by the  c isoline in a standard reference specimen, t is creep time, and tred is stress redistribution time. The C(T) specimen (a/W = 0.5) with high constraint in plane strain is usually chosen as the standard reference specimen [36, 37]. The ACEEQ and Aref are calculated at the same  c isoline, the same creep time t/tred =1 (steady-state creep) and C* level by FEM analyses [36, 37]. The area Aref of the high constraint standard C(T) specimen is usually smaller than the area ACEEQ of the under evaluated specimen or component with lower constraint. Therefore, the Ac value is generally larger than 1, and the constraint level decreases with increasing Ac. It has been shown that Ac is independent on the choice of  c isolines and C* level [30, 36]. The 7

parameter Ac can effectively characterize both in-plane and out-of-plane creep constraints in various specimen geometries and pipes [30,36,37,51]. The correlation between the parameter Ac and CCG rate has also been obtained for 316H steel [51] and Cr-Mo-V steel [37], which may be used in creep life assessments for cracked components with any constraint levels [34]. In this paper, the parameter Ac has been used to characterize creep constraint of pipe cracks with high aspect ratio. The C(T) specimen with W = 20mm and a/W = 0.5 in plane strain was chosen as the standard reference specimen, and the ACEEQ and Aref surrounded by  c =0.05 isoline in Eq. (2) were calculated for the pipe cracks with different sizes and high aspect ratios at the same creep time t/tred =1 and load level C*, respectively. Then, the distributions of the unified creep constraint parameter Ac and fracture mechanics parameter C* along the crack fronts were calculated. 3. Results and discussion 3.1 Creep constraint parameter Ac for axial semi-elliptical surface cracks with high aspect ratio In the previous work of authors [30, 36-38], it has been shown that the parameter Ac based on crack-tip equivalent creep strain can effectively characterize both in-plane and out-of-plane creep constraints in test specimens with different geometries, crack sizes and loading modes, and it also can characterize constraints in pressurized pipes with lower aspect ratio cracks. In this work, the parameter Ac is adopted to analyze crack-tip constraint of pipe cracks with high aspect ratio. Fig. 4 shows distributions of the constraint parameter Ac along the crack fronts 8

for the cracks with high aspect ratio from a/c = 1.2 to 2 at different a/t. It can be seen that the parameter Ac decreases with increasing 2Φ/π from 0 to 0.2 and then slightly decreases with increasing the 2Φ/π from 0.2 to 1. This implies that the constraint level near free surface (2Φ/π = 0 to 0.2) is lower than that at the deeper part of crack front (2Φ/π = 0.2 to 1). For a given crack depth a/t, with increasing crack aspect ratio a/c from 1.2 to 2 (decreasing crack length 2c), the parameter Ac increases and constraint level decreases. Fig. 5 shows the distributions of constraint parameter Ac along the crack fronts for the cracks with different crack depths a/t at fixed a/c = 1.2 and 1.8, respectively. It shows that with increasing crack depth a/t, the Ac values decrease and constraint levels increase. The constraint of the cracks with lower aspect ratio a/c = 1.2 is higher than that of the cracks with higher a/c = 1.8. Compared the distributions of Ac along the crack fronts for the cracks with high aspect ratio (a/c = 1.2-2) in Figs.4 and 5 with those with low aspect ratio (a/c = 0.2-1) in the previous study [38], it can be seen that the distributions of Ac are similar, but the Ac values of cracks with high aspect ratio are higher than those of cracks with low aspect ratio. For the same a/t, with increasing crack aspect ratio a/c from 0.2 to 2 (decreasing crack length 2c), the Ac values continuously increase and constraint levels decrease. For validating the load-independence of Ac for high aspect ratio cracks, the Ac was calculated at different internal pressures p = 10, 15 and 20MPa. Fig. 6 shows the Ac values at the surface and deepest point of cracks with a/c =1.2 and different a/t = 9

0.1, 0.4 and 0.8 at different internal pressures p. It shows that with increasing a/t, the Ac decreases and the curves at different p are basically coincident. Fig. 7 shows the Ac values at the surface and deepest point of cracks with the same a/t = 0.2 and different a/c = 1.2, 1.6 and 2.0 at different internal pressures p. It indicates that with increasing a/c, the Ac increases and the curves at different p are also basically coincident. The results in Figs.6 and 7 imply that the constraint parameter Ac of high aspect ratio cracks is load-independent. This result is the same as that of low aspect ratio cracks in the previous study [38]. Thus, it can be concluded that for a wide of crack aspect ratio a/c from 0.2 to 2, the parameter Ac is load-independent. The solutions of Ac for a wide range of crack size and aspect ratio at a certain internal pressure load can be used for other internal pressure loads. This may bring convenience for the engineering application of Ac in creep life assessments of high-temperature components under different loads. 3.2 The parameter Ac solutions for axial cracks with high aspect ratio When the parameter Ac is used in creep life assessments of high-temperature pressurized pipes with high aspect ratio cracks, it is essential to estimate the Ac values for these cracks. In this work, the parameter Ac have been calculated for axially cracked pipes with high aspect ratio cracks by using 3D FE analyses. The parameter Ac values along the crack front for different crack sizes (a/t and a/c) are shown in Table 2. For applying parameter Ac conveniently in creep life assessments, the parameter Ac solutions in Table 2 can be fitted into empirical equations. The Ac values at the 10

surface point and deepest point are usually used in the creep crack growth analysis of cracked pipes. The average constraint parameter Acavg can also be used to characterize the constraint level of 3D semi-elliptical surface cracks in specimens and cracked pipes. Thus, the empirical equations of the Ac1 values at the surface point (2Φ/π=0), the Ac2 values at the deepest point (2Φ/π=1) and the average Acavg values along the crack front have been fitted. The parameter Ac1 solution for axial cracks with high aspect ratio (1.2 ≤ a/c ≤ 2) in pressurized pipes with Ri/t =10 is shown as follows:

a a Ac1  f A ( , ) t c

at the surface crack front， 2Φ/π  0 (3)

a a a a Ac1  A0  A1 ( )  A2 ( ) 2  A3 ( )3  A4 ( ) 4 t t t t

(4)

where

a a a a A0  98.209  276.748( )  292.293( ) 2  133.377( )3  22.241( ) 4 c c c c

a a a a A1  631.031  1931.527( )  2139.950( ) 2  1013.324( )3  173.555( ) 4 c c c c a a 2 a 3 a 4 A2  102.764  1325.397( )  2289.806( )  1362.440( )  267.429( ) c c c c a a a a A3  2700.806  5481.590( )  3793.977( ) 2  1023.580( ) 3  80.556( ) 4 c c c c a a a a A4  2606.050  5990.009( )  5004.620( ) 2  1815.275( )3  242.787( ) 4 c c c c The parameter Ac2 solution for axial cracks with high aspect ratio (1.2 ≤ a/c ≤ 2) in pressurized pipes with Ri/t =10 is expressed as follows:

a a Ac 2  f B ( , ) at the deepest crack front, 2Φ/π  1 t c a a a a Ac 2  B0  B1 ( )  B2 ( ) 2  B3 ( )3  B4 ( ) 4 t t t t 11

(5)

(6)

where

a a a a B0  75.680  198.114( )  187.972( ) 2  78.861( )3  12.325( ) 4 c c c c

a a a a B1  1477.361  3821.200( )  3660.684( ) 2  1542.853( ) 3  241.614( ) 4 c c c c a a 2 a 3 a B2  6714.491  17461.597( )  16810.250( )  7110.123( )  1115.849( ) 4 c c c c a a a a B3  11025.786  28754.807( )  27748.707( ) 2  11759.155( )3  1848.157( ) 4 c c c c a a a a B4  6018.989  15720.570( )  15189.942( ) 2  6444.096( )3  1013.733( ) 4 c c c c The parameter Acavg solution for axial cracks with high aspect ratio (1.2 ≤ a/c ≤ 2) in pressurized pipes with Ri/t = 10 is expressed as follows:

a a Acavg  f C ( , ) for average constraint parameter t c a a a a Acavg  C0  C1 ( )  C2 ( ) 2  C3 ( )3  C4 ( ) 4 t t t t

(7) (8)

where

a a a a C0  236.742  611.613( )  589.508( ) 2  249.028( ) 3  38.950( ) 4 c c c c a a 2 a 3 a C1  2584.667  6692.178( )  6416.507( )  2699.572( )  420.668( ) 4 c c c c a a 2 a 3 a C2  9105.536  23559.454( )  22577.701( )  9498.956( )  1480.930( ) 4 c c c c a a a a C3  13311.378  34490.617( )  33112.953( ) 2  13960.238( ) 3  2181.360( ) 4 c c c c a a a a C4  6824.909  17722.305( )  17056.413( ) 2  7209.696( ) 3  1129.561( ) 4 c c c c To validate the empirical equations, the comparison of the parameter Ac calculation results from the above empirical equations and the FEM results were given in Fig.8 for the high aspect ratio cracks with a/c from 1.2 to 2.0. It can be seen that calculation results of the Ac1 value (2Φ/π = 0) and the Ac2 value (2Φ/π = 1) from 12

empirical equations agree well with those from FEM for many different a/c values from 1.2 to 2.0. This proves the correctness of the empirical equations for calculating Ac values. These empirical equations can be used to calculate the constraint parameter Ac for axial cracks with high aspect ratio in pipes with Ri/t =10. The studies in the previous work [38] have shown that the radius-thickness ratio Ri/t has small effect on the parameter Ac. With increasing Ri/t from 5 to 20, the Ac value slightly decreases. Therefore, the Ac values for cracks in pipes with different Ri/t may be estimated by using the Ac values of cracks in pipe with Ri/t = 10. 3.3 Fracture parameter C* for axial semi-elliptical surface cracks with high aspect ratio In this study, the parameter C* has been calculated by FE analyses for axial semi-elliptical surface cracks with high aspect ratio a/c from 1.2 to 2. For comparison, the parameter C* for the cracks with low aspect ratio a/c from 0.2 to 1.0 was also calculated by FEM using the similar model in Fig.3. Figs.9 (a)-(e) give distributions of parameter C* along the crack front for axial cracks with a wide range of crack sizes of a/t = 0.1-0.8 and a/c = 0.2- 2. It can be seen that for a fixed pipe geometry Ri/t = 10 and internal pressure p = 15MPa, the C* level along the crack front increases with increasing crack depth a/t and decreasing crack aspect ratio a/c (increasing crack length 2c). For the cracks with high aspect ratio a/c from 1.2 to 2, the C* value initially increases and then decreases with increasing 2Φ/π from the surface point (2Φ/π = 0) to the deepest point (2Φ/π = 1), and the maximum C* value occurs near 13

surface point with a 2Φ/π value of about 0.2. For most cracks with high aspect ratio a/c from 1.4 to 2, the C* value at the deepest point is lower than that at surface point. For these cracks, if the C* values at the deepest point are used in CCI life assessments, non-conservative results may be produced. Thus, it is recommended that for the cracks with high aspect ratio a/c, the CCI life can be assessed at the location with maximum C* value at 2Φ/π = 0.2 along the crack front. For the cracks with low aspect ratio a/c from 0.2 to 0.6 in Figs.9 (a)-(e), the maximum C* occurs at the deepest part along the crack front, and the lower C* occurs near surface point with 2Φ/π from 0 to 0.2. The C* distributions of the cracks with a/c = 0.8 and 1 are similar to those of the cracks with high aspect ratio. The maximum C* occurs near surface point with a 2Φ/π value of about 0.2, but the C* at the deepest point is higher than that at the surface point. In general, for a given a/t, the crack aspect ratio a/c has significant effect on the C* distributions along the crack fronts. With increasing a/c from 0.2 to 2, the C* at the deepest point decreases significantly, but it reduces less at surface point. The distributions of parameter C* along the crack front of axial cracks in Fig.9 is similar to those of stress intensity factor K in the literature [41, 42]. It has been shown that for the high aspect ratio cracks, the maximum stress intensity factor K occurs near the surface point of the crack front, while for the low aspect ratio cracks, it occurs at the deepest point [41, 42]. This may suggest that the calculation results of parameter C* in Fig.9 is reasonable. From the energy point of view, the point with the maximum K of a high aspect ratio surface crack have more strain energy stored in the 14

corresponding domain, and the stress field of the point affects the strain energy [41]. The C* characterizes the creep crack-tip stress intensity at steady-state creep. Thus, the different C* values at different points along a semi-elliptical crack front may be explained by the crack-tip stress distributions. The crack-tip opening stress distributions at three points （2Φ/π = 0，2Φ/π = 0.2 and 2Φ/π = 1）along the crack fronts at t/tred = 1 have been calculated by FEM for two typical cracks. One is high aspect ratio crack with a/t = 0.4 and a/c = 1.6, and another is low aspect ratio crack with a/t = 0.4 and a/c = 0.4. The results calculated for the two cracks are shown in Fig.10. Fig.10(a) shows that for the high aspect ratio crack, the crack-tip opening stress is the highest at the point of 2Φ/π = 0.2 near free surface, and it is lower at the surface point (2Φ/π = 0) and the deepest point (2Φ/π = 1). The highest crack-tip stress corresponds to the highest C*, and the lower crack-tip stresses correspond to lower C* for the high aspect ratio crack with a/t = 0.4 and a/c = 1.6 in Fig.9. This implies that the highest C* at 2Φ/π = 0.2 near free surface is caused by the highest crack-tip stress when the high aspect ratio crack is loading. Fig.10 (b) shows that for the low aspect ratio crack, the crack-tip opening stress is the highest at the deepest point (2Φ/π = 1), and it is the lowest at the surface point (2Φ/π = 0). This corresponds to the C* values for the low aspect ratio crack with a/t = 0.4 and a/c = 0.4 in Fig.9. These results indicate that the C* value at different points along a semi-elliptical crack front depends on the crack-tip stress level at the point. The creep crack initiation is generally determined by both C* and constraint parameter Ac, and it will occur at the locations with higher C* and higher constraint 15

(lower Ac) along the crack fronts [52]. However, Figs.4, 5 and 9 show that the two locations with higher C* and higher constraint (lower Ac) for cracks with high aspect * * ratio does not coincide. A composite parameter C (Cref  Ac ) which consider the

combined effect of crack driving force C* and constraint Ac has been defined for predicting creep crack initiation locations [52]. The highest value of the composite parameter along the crack front indicates initiation location where there exist higher C* and higher constraint (lower Ac). If the normalized parameter C*ref is taken to be 1E-6MPam/h, the distributions of the composite parameter along the crack fronts can be calculated for cracks with high aspect ratio, and the results are typically shown in Fig.11 for cracks with a/t = 0.2 and 0.6. For comparison, the results for cracks with low aspect ratio (a/c = 0.2-1.0) were also calculated, and were included in Fig.11. The Ac values of low aspect ratio cracks were taken from the previous study [38]. It can be seen from Fig. 11 that the distributions of the composite parameter along the crack fronts are similar to those of C*, which shows that the creep crack initiation location is mainly determined by C* distributions. The highest value of the composite parameter for the high aspect ratio cracks occurs near surface point with 2/ =0.2. This implies that the creep crack will initiate at a location near surface point for the high aspect ratio cracks. It is also interesting to note that for a given a/t, with increasing the crack aspect ratio a/c from the lowest value of 0.2 to the highest value of 2, the crack initiation location (the highest composite parameter value) will change from the deepest part of the crack front (around 2/ =1) to the near surface part (around 2/ = 0.2). For the cracks with higher aspect ratio a/c from 0.8 to 2, the 16

crack initiation location will always occur near surface part with a 2/ value of about 0.2. This has been validated by the numerical simulation results of CCI based on creep ductility exhaustion model for the cracks with a/c = 0.8 and 1 in the previous work [52]. These results suggest that if the calculations of C* and Ac at the deepest point is used to predict CCI life for the cracks with higher aspect ratio a/c ≥ 0.8, the non-conservative results may be produced. It is recommended for these cracks that the C* and Ac at a location near surface point with 2/ = 0.2 are calculated for the CCI life assessments. In addition, in present creep crack growth shape and life analyses, two C* values at the deepest point and surface point are usually calculated and the crack growth shape is assumed remaining to be semi-elliptical [40]. Fig.11 shows that for high aspect ratio cracks, the CCG rate near surface part will be the fastest, and the semi-elliptical crack shape cannot be remained. The crack growth shape affects the distributions of C* and Ac, which will influence subsequent CCG rate. It also affects crack growth area in LBB (leak-before-break) analysis. Thus, for high aspect ratio cracks, natural crack growth shape may need to be calculated for accurate life assessments and LBB analysis. 3.4 The constraint-dependent CCI time and CCG rate of pipe cracks with high aspect ratio In the previous work [53]，the correlation equation between the CCI time and constraint parameter Ac for the Cr-Mo-V steel at 566℃ used in this work has been obtained, as shown in Eq.(9):

ti / t0  0.63 Acavg

1.36

(9) 17

where ti is CCI time in a cracked specimen or component, Acavg is average Ac value along the crack front, and t0 is the CCI time in a standard plane strain C(T) specimen with deep crack (a/W=0.5) . The constraint-dependent CCG rate equation for the Cr-Mo-V steel at 566℃ used in this work also has been obtained in the previous work [36], as shown in Eq.(10): .

.

f ( Ac )  a/ a0  1.312 Acavg where

1.623

(10)

is the CCG rate of a cracked specimen or component, the Acavg is average

Ac value along the crack front, and the

is the CCG rate of a standard plane strain

C(T) specimen. Based on the parameter Acavg solution in Eqs.(7) and (8) and Eqs.(9) and (10), the constraint-dependent CCI time ratio and CCG rate ratio for the cracks with high aspect ratio (1.2 ≤ a/c ≤ 2) can be calculated, and the results are shown in Fig.12. Fig.12 shows that for all pipe cracks, the CCI time ratio is larger than 1 and the CCG rate ratio is less than 1. This implies that the CCI time of pipe cracks with high aspect ratio is longer than that of the standard C(T) specimen, and their CCG rate is slower than that of the standard C(T) specimen. If the CCI time and CCG rate data of the standard C(T) specimen are used in life assessments of the cracked pipes, overly conservative results will be produced. Fig.12 also shows that for a given a/t, with increasing a/c, the CCI time ratio increases and the CCG rate ratio decreases. For a given a/c, with increasing a/t, the CCI time ratio decreases and the CCG rate ratio increases. For shallower and shorter cracks, the CCI time is about 1.55-2.15 times that of the standard C(T) specimen, and the CCG rate is about 0.3-0.4 times that of the 18

standard C(T) specimen. Compared with the pipe cracks with low aspect ratio (0.2 ≤ a/c ≤ 1) [38], the high aspect ratio cracks (1.2 ≤ a/c ≤ 2) have more low constraint levels (Figs.4 and 5). Thus, more benefits from incorporating constraint effects can be obtained for creep life assessments of pipes with high aspect ratio cracks.

4. Conclusion The unified creep constraint parameter Ac and fracture parameter C* for axial semi-elliptical surface cracks with high aspect ratio in pressurized pipes have been investigated by FE analyses. The main results obtained are as follows: (1) For the pipe cracks with high aspect ratio (1.2≤ a/c ≤2), the constraint level near surface part (2Φ/π = 0 to 0.2) is lower than that at the deeper part of crack front (2Φ/π = 0.2 to 1). For a given crack depth a/t, with increasing crack aspect ratio a/c, the parameter Ac increases and constraint level decreases. With increasing crack depth a/t, the Ac values decrease and constraint levels increase. The constraint parameter Ac of high aspect ratio cracks is load-independent. (2) Based on FE analyses, the parameter Ac solution for axial cracks with high aspect ratio (1.2≤ a/c ≤2) in pressurized pipes with Ri/t =10 has been obtained. It may be used to estimate the parameter Ac for high aspect ratio pipe cracks in creep life assessments incorporating constraint effect. (3) The crack aspect ratio a/c has significant effect on the C* distributions along the crack fronts. For high aspect ratio crack (1.2≤ a/c ≤2), the maximum C* along the crack fronts occurs near surface part. The C* values of most cracks (1.4≤ a/c ≤2) at the deepest point are lower than that at surface point. For low aspect ratio

19

cracks (0.2≤ a/c ≤0.6), the maximum C* occurs at the deepest part along the crack front, and the lower C* occur near surface part. * * (4) The compound parameter C (Cref  Ac ) composed of C* and Ac indicates that

for the high aspect ratio cracks, the creep crack initiation will occur near surface part. For a given a/t, with increasing the crack aspect ratio a/c from lower value of 0.2 to higher value of 2, the crack initiation location will change from the deepest part of the crack front (around 2/ =1) to the near surface part (around 2/ = 0.2). If the calculations of C* and Ac at the deepest point is used to predict CCI life for higher aspect ratio cracks (a/c ≥ 0.8), the non-conservative results may be produced. It is recommended for these cracks that the C* and Ac at a location near surface point with 2 / = 0.2 are calculated in CCI life assessments, and also natural crack growth shape may need to be calculated for accurate life assessments and LBB analysis. (5) The constraint level of high aspect ratio cracks is considerably lower than that of standard C(T) specimen, and it is also lower than that of low aspect ratio cracks. Thus, more benefits from incorporating constraint effects may be obtained for creep life assessments of pipes with high aspect ratio cracks. Acknowledgements This work was financially supported by the Projects of the National Natural Science Foundation of China (51575184 and 51375165) and the Fundamental Research Funds for the Central Universities (222201718005).

20

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26

Figure and Table Captions Fig. 1 True stress-strain curve of the Cr-Mo-V steel at 566℃ [49] Fig. 2 The geometry and dimension of axially cracked pipes with semi-elliptical surface crack Fig. 3 Typical finite element model of a pipe crack with high aspect ratio Fig. 4 The distributions of parameter Ac along crack fronts for high aspect ratio axial cracks, (a)a/t=0.1, (b)a/t=0.2, (c)a/t=0.4, (d)a/t=0.6, (e)a/t=0.8 Fig. 5 Distributions of Ac along the crack fronts for axial cracks in pipes with Ri/t = 10, (a) a/c = 1.2, (b) a/c = 1.8 Fig. 6 The parameter Ac for pipe cracks with a/c = 1.2 and different a/t at different internal pressures, (a) the surface point, (b) the deepest point Fig. 7 The parameter Ac for pipe cracks with a/t = 0.2 and different a/c at different internal pressures, (a) the surface point, (b) the deepest point Fig. 8 Comparison of parameter Ac calculated from empirical equations and FEM, (a, b) the surface point (2Φ/π = 0), (c, d) the deepest point (2Φ/π = 1) Fig. 9 Distributions of C* values for axial cracks, (a) a/t = 0.1, (b) a/t = 0.2, (c) a/t = 0.4, (d) a/t = 0.6, (e) a/t = 0.8 Fig. 10 The crack-tip opening stress distributions along the crack front for a typical high aspect ratio crack (a/t = 0.4, a/c = 1.6) (a) and a typical low aspect ratio crack (a/t = 0.4, a/c = 0.4). Fig. 11 Distributions of the composite parameter C*/(C*ref＊Ac) along the crack fronts, (a) a/t =0.2, (b) a/t =0.6 27

Fig. 12 The constraint-dependent CCI time ratio (a) and CCG rate ratio (b) for the cracks with high aspect ratio (1.2 ≤ a/c ≤ 2) in pipe with Ri/t = 10 Table 1 Mechanical properties and the 2RN creep model parameters of the Cr-Mo-V steel at 566℃ [49] Table 2 Values of unified constraint parameter Ac for axially cracked pipes with Ri /t =10 and a/c from 1.2 to 2

28

Fig. 1 True stress-strain curve of the Cr-Mo-V steel at 566℃[49]

29

Fig. 2 The geometry and dimension of axially cracked pipes with semi-elliptical surface crack

30

Fig. 3 Typical finite element model of a pipe crack with high aspect ratio

31

3.6

3.6

a/t=0.1

a/c=1.2 a/c=1.4 a/c=1.6 a/c=1.8

3.2 2.8

a/t=0.2

a/c=1.2 a/c=1.4 a/c=1.6 a/c=1.8 a/c=2

3.2 2.8

Ac

Ac

a/c=2 2.4 2.0

2.0

1.6

1.6 0.0

0.2

3.6

0.4

0.6

0.8

1.0

0.0

(b) a/t=0.6

Ac

2.8 2.4 2.0

1.6

1.6 0.6

0.8

1.2

1.0

0.0

0.2

0.4

0.6

0.8

1.0

2

2 (c)

(d) 3.6

a/t=0.8

a/c=1.2 a/c=1.4 a/c=1.6 a/c=1.8 a/c=2

3.2 2.8

Ac

1.0

a/c=1.2 a/c=1.4 a/c=1.6 a/c=1.8 a/c=2

3.2

2.0

0.4

0.8

3.6 a/c=1.2 a/c=1.4 a/c=1.6 a/c=1.8 a/c=2

2.4

0.2

0.6

(a) a/t=0.4

0.0

0.4

2

2.8

1.2

0.2

2

3.2

Ac

2.4

2.4 2.0 1.6 1.2 0.0

0.2

0.4

0.6

0.8

1.0

2 (e) Fig. 4 The distributions of parameter Ac along crack fronts for high aspect ratio axial cracks, (a)a/t=0.1, (b)a/t=0.2, (c)a/t=0.4, (d)a/t=0.6, (e)a/t=0.8 32

3.6

3.2

a/t=0.1 a/t=0.2 a/t=0.4 a/t=0.6 a/t=0.8

a/c=1.2 2.8

3.2 2.8

Ac

Ac

2.4 2.0

2.4

1.6

2.0

1.2

1.6 0.0

0.2

0.4

0.6

0.8

a/t=0.1 a/t=0.2 a/t=0.4 a/t=0.6 a/t=0.8

a/c=1.8

0.0

1.0

0.2

0.4

0.6

0.8

1.0

2

2

(a) (b) Fig. 5 Distributions of Ac along the crack fronts for axial cracks in pipes with Ri/t = 10, (a) a/c = 1.2, (b) a/c = 1.8

33

2.4

2.8

a/c=1.2 2/ =0

2.2

2.4

2.0

2.2

1.8

2.0

1.6

Ac

Ac

2.6

1.8 1.6

1.2

0.0

0.2

0.4

0.6

1.4 1.2

p=10MPa p=15MPa p=20MPa

1.4

a/c=1.2 2

p=10MPa p=15MPa p=20MPa

1.0 0.8 0.0

0.8

a/t

0.2

0.4

0.6

0.8

a/t

(a) (b) Fig. 6 The parameter Ac for pipe cracks with a/c = 1.2 and different a/t at different internal pressures, (a) the surface point, (b) the deepest point

34

2.6

3.0

a/t=0.2 2 2.4

2.6

2.2

2.4

2.0

Ac

Ac

a/t=0.2 2/ =0 2.8

1.8

2.2

p=10MPa p=15MPa p=20MPa

2.0 1.8

1.2

1.4

1.6

1.8

p=10MPa p=15MPa p=20MPa

1.6 1.4

2.0

a/c

1.2

1.4

1.6

1.8

2.0

a/c

(a) (b) Fig. 7 The parameter Ac for pipe cracks with a/t = 0.2 and different a/c at different internal pressures, (a) the surface point, (b) the deepest point

35

3.6

2 =0

3.2

2.4

2.4 2.0

1.6

1.6 0.4

a/t

0.6

0.2

0.8

0.4

a/t

(a)

3.2

0.8

3.2

2.4

21

FEM (a/c=1.3) FEM (a/c=1.5) FEM (a/c=1.7) FEM (a/c=1.9) Equations

2.8 2.4

Ac2

Ac2

2.8

0.6

(b) FEM (a/c=1.2) FEM (a/c=1.4) FEM (a/c=1.6) FEM (a/c=1.8) FEM (a/c=2.0) Equations

2 =1

FEM (a/c=1.3) FEM (a/c=1.5) FEM (a/c=1.7) FEM (a/c=1.9) Equations

2.8

2.0

0.2

2 =0

3.2

Ac1

2.8

Ac1

3.6

FEM (a/c=1.2) FEM (a/c=1.4) FEM (a/c=1.6) FEM (a/c=1.8) FEM (a/c=2.0) Equations

2.0 2.0

1.6

1.6 1.2

1.2 0.2

0.4

a/t

0.6

0.8

0.2

0.4

a/t

0.6

0.8

(c) (d) Fig. 8 Comparison of parameter Ac calculated from empirical equations and FEM, (a, b) the surface point (2Φ/π = 0), (c, d) the deepest point (2Φ/π = 1)

36

a/c=0.2 a/c=0.4 a/c=0.6 a/c=0.8 a/c=1.0 a/c=1.2 a/c=1.4 a/c=1.6 a/c=1.8 a/c=2.0

1E-7

0.0

0.2

0.4

0.6

0.8

1E-6

1E-7

1.0

0.0

0.2

(a)

0.4

0.6

1.0

0.8

Ri/t=10 p=15MPa a/t=0.6

a/c=0.2 a/c=0.4 a/c=0.6 a/c=0.8 a/c=1.0 a/c=1.2 a/c=1.4 a/c=1.6 a/c=1.8 a/c=2

1E-5

1E-6

0.0

1.0

0.2

0.4

0.6

0.8

1.0

2/

2/ (c)

(d) 1E-4

C*,MPam/h

0.8

C*,MPam/h

C*,MPam/h

a/c=0.2 a/c=0.4 a/c=0.6 a/c=0.8 a/c=1.0 a/c=1.2 a/c=1.4 a/c=1.6 a/c=1.8 a/c=2.0

1E-6

0.2

0.6

(b)

Ri/t=10 p=15MPa a/t=0.4

0.0

0.4

2

2/ 1E-5

a/c=0.2 a/c=0.4 a/c=0.6 a/c=0.8 a/c=1.0 a/c=1.2 a/c=1.4 a/c=1.6 a/c=1.8 a/c=2.0

Ri/t=10 p=15MPa a/t=0.2

C*,MPam/h

Ri/t=10 p=15MPa a/t=0.1

C*,MPam/h

1E-6

a/c=0.2 a/c=0.4 a/c=0.6 a/c=0.8 a/c=1.0 a/c=1.2 a/c=1.4 a/c=1.6 a/c=1.8 a/c=2

Ri/t=10 p=15MPa a/t=0.8

1E-5

1E-6 0.0

0.2

0.4

0.6

0.8

1.0

2/ (e) Fig. 9 Distributions of C* values for axial cracks, (a) a/t = 0.1, (b) a/t = 0.2, (c) a/t = 0.4, (d) a/t = 0.6, (e) a/t = 0.8 37

1.4 1.2

1.6

2 2 2

a/t=0.4 a/c=1.6 t/tred =1

1.4 1.2





1.0 0.8

1.0 0.8

0.6 0.4

2 2 2

a/t=0.4 a/c=0.4 t/tred =1

0.6

0.000 0.005 0.010 0.015 0.020 0.025 0.030

0.000 0.005 0.010 0.015 0.020 0.025 0.030

r/a

r/a

(a) (b) Fig. 10 The crack-tip opening stress distributions along the crack front for a typical high aspect ratio crack (a/t = 0.4, a/c = 1.6) (a) and a typical low aspect ratio crack (a/t = 0.4, a/c = 0.4).

C*/(C*refAc)

p=15MPa

0.1

0.0

0.2

0.4

0.6

0.8

a/c=0.2 a/c=0.4 a/c=0.6 a/c=0.8 a/c=1.0 a/c=1.2 a/c=1.4 a/c=1.6 a/c=1.8 a/c=2.0

Ri/t =10 t =38mm a/t =0.6 10 C*/(C*refAc)

a/c=0.2 a/c=0.4 a/c=0.6 a/c=0.8 a/c=1.0 a/c=1.2 a/c=1.4 a/c=1.6 a/c=1.8 a/c=2.0

1 Ri/t =10 t =38mm a/t =0.2

p=15MPa

1

0.0

1.0

0.2

0.4

0.6

0.8

1.0

2

2/

(a) (b) Fig. 11 Distributions of the composite parameter C*/(C*ref＊Ac) along the crack fronts, (a) a/t =0.2, (b) a/t =0.6

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2.6 2.4 2.2 2.0

0.7 0.6

1.6 1.4

a/c=1.7 a/c=1.8 a/c=1.9 a/c=2.0

0.4 0.3

1.2 1.0 0.0

a/c=1.2 a/c=1.3 a/c=1.4 a/c=1.5 a/c=1.6

0.5

1.8

. . a/a0

ti/t0

0.8

a/c =1.2 a/c =1.4 a/c =1.6 a/c =1.8 a/c =2.0

0.2 0.2

0.4

0.6

0.8

0.0

a/t

0.2

0.4

0.6

0.8

a/t

(a) (b) Fig. 12 The constraint-dependent CCI time ratio (a) and CCG rate ratio (b) for the cracks with high aspect ratio (1.2 ≤ a/c ≤ 2) in pipe with Ri/t = 10

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Table 1 Mechanical properties and the 2RN creep model parameters of the Cr-Mo-V steel at 566℃[49] Stress A(MPa-nh-1) n Material Elastic Yield modulus(GPa) stress(MPa) σ≤250MPa A1=7.26×10-26 n1=8.75 160 383 Cr-Mo-V σ>250MPa A2=3.53×10-36 n2=13.08

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Table 2 Values of unified constraint parameter Ac for axially cracked pipes with Ri /t =10 and a/c from 1.2 to 2 a/t 0.1

0.2

0.4

0.6

0.8

2Φ/π

a/c 1.2 1.4 1.6 1.8 2 1.2 1.4 1.6 1.8 2 1.2 1.4 1.6 1.8 2 1.2 1.4 1.6 1.8 2 1.2 1.4 1.6 1.8 2

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

2.383 2.581 2.701 2.702 2.839 2.225 2.341 2.478 2.657 2.837 2.033 2.178 2.302 2.510 2.700 1.836 1.986 2.151 2.341 2.578 1.723 1.871 1.976 2.177 2.472

2.134 2.269 2.489 2.440 2.681 2.095 2.208 2.347 2.436 2.672 1.925 1.986 2.203 2.401 2.509 1.725 1.830 1.947 2.211 2.398 1.610 1.749 1.810 2.026 2.325

2.024 2.210 2.382 2.324 2.496 1.963 2.128 2.274 2.321 2.475 1.843 1.952 2.079 2.244 2.403 1.695 1.748 1.855 2.125 2.289 1.561 1.676 1.755 1.987 2.251

1.974 2.192 2.303 2.346 2.469 1.955 2.115 2.169 2.268 2.425 1.820 1.937 1.988 2.176 2.372 1.691 1.737 1.833 2.102 2.268 1.541 1.644 1.740 1.952 2.228

2.008 2.120 2.246 2.326 2.438 1.958 2.113 2.160 2.242 2.435 1.806 1.930 1.963 2.134 2.310 1.662 1.739 1.826 2.059 2.242 1.534 1.620 1.735 1.944 2.204

1.966 2.150 2.266 2.317 2.433 1.945 2.098 2.128 2.282 2.386 1.793 1.894 1.955 2.106 2.305 1.655 1.704 1.817 2.056 2.253 1.524 1.605 1.728 1.937 2.184

2.036 2.116 2.229 2.266 2.478 1.923 2.086 2.113 2.260 2.351 1.786 1.882 1.933 2.095 2.338 1.633 1.699 1.807 2.040 2.231 1.512 1.588 1.714 1.931 2.170

2.029 2.093 2.254 2.295 2.399 1.901 2.034 2.099 2.255 2.331 1.785 1.843 1.927 2.032 2.323 1.626 1.690 1.795 1.988 2.217 1.495 1.576 1.698 1.912 2.160

1.976 2.001 2.229 2.292 2.383 1.875 1.989 2.051 2.187 2.357 1.772 1.832 1.916 2.080 2.311 1.572 1.680 1.788 1.963 2.207 1.488 1.564 1.684 1.903 2.151

1.969 2.024 2.217 2.289 2.378 1.903 1.973 2.066 2.137 2.353 1.770 1.823 1.903 2.067 2.268 1.612 1.664 1.775 1.952 2.206 1.477 1.558 1.676 1.892 2.143

1.961 2.085 2.207 2.287 2.373 1.885 1.956 2.088 2.191 2.347 1.764 1.800 1.892 2.061 2.253 1.599 1.651 1.759 1.948 2.195 1.469 1.548 1.667 1.872 2.130

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Highlights  Creep constraint and fracture parameter C* of high aspect ratio cracks are studied.  Constraint levels decrease with increasing aspect ratio of pipe cracks.  Creep constraint parameter Ac solution for high aspect ratio cracks is obtained.  Maximum C* along crack front of high aspect ratio cracks occurs near surface part.  Crack initiation location along crack front changes with increasing aspect ratio.

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