A creep crack growth life assessment method for pressurized pipes based on a two-parameter approach

Engineering Fracture Mechanics 220 (2019) 106676

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A creep crack growth life assessment method for pressurized pipes based on a two-parameter approach

T

⁎

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

A R T IC LE I N F O

ABS TRA CT

Keywords: Constraint Creep crack growth life Two-parameter C*-Ac Pipe Crack

The unified creep constraint parameter Ac based on crack tip equivalent creep strain can incorporate both in-plane and out-of-plane constraints in specimens and components. To improve the accuracy of creep crack growth (CCG) life assessment for cracked components, it needs to develop new life assessment procedures to consider the unified constraint. In this work, the methodology of the CCG life assessment based on two-parameter C*-Ac has been studied and given for pressurized pipes with a wide range of initial crack sizes. The assessment results of CCG life and crack size/shape evolution based on the two-parameter C*-Ac approach are quantitatively analyzed and compared with those based on the conventional single-parameter C* approach. The effects of calculation methods of C* on the CCG life assessments are also analyzed. The results show that the application of the two-parameter approach can reduce conservatism and predict reasonable crack growth size and shape. The conventional CCG life assessment based on the single-parameter C* and the reference stress C* calculation for shallower and shorter surface cracks with lower constraint in cracked pipes may produce very high overall conservatism degree. Therefore, in the CCG life assessments for these cracks, it is highly suggested that the constraint effect is considered and the finite element solution of C* is used.

1. Introduction For high-temperature structures containing defects, the creep crack growth (CCG) is the main failure mode, and accurate predictions and assessments of CCG life are very important for design and ensuring structural integrity. It has been well-known that the crack-tip constraint effect caused by specimen or component geometries, crack sizes and loading modes can influence CCG rate of materials [1–13]. The loss of crack-tip constraint leads to lower CCG rate. In the current standard for measuring CCG rate of materials [14], the plane strain C(T) specimen is generally used to ensure high crack-tip constraint and measure upper bound CCG rate data of materials. However, the defects produced in actual components (such as pressurized pipes and vessels) usually are shallower surface cracks with lower crack-tip constraint. If the upper bound CCG rate data from the plane strain C(T) specimen is used to predict the CCG life of the lower constraint cracks in components, over-conservative results may be produced. This excessive conservatism may lead to unnecessary maintenance or replacement of in-service high-temperature components. For reducing conservatism and improving accuracy, the methodology of CCG life assessment with considering constraint effect should be investigated and developed. The development and application of appropriate creep constraint parameters are the keys to achieving the constraint dependent CCG life assessments of engineering structures. Some creep constraint parameters for mode I cracks have been proposed, such as Q [15,16], Q* [4,17], A2 [18,19], R [20–23], R* [24–26] and Ac [27–30]. The effects of constraint on creep crack initiation (CCI) time

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Corresponding author. E-mail address: [email protected] (G.Z. Wang).

https://doi.org/10.1016/j.engfracmech.2019.106676 Received 2 August 2019; Received in revised form 11 September 2019; Accepted 13 September 2019 Available online 14 September 2019 0013-7944/ © 2019 Elsevier Ltd. All rights reserved.

Engineering Fracture Mechanics 220 (2019) 106676

X.M. Tan, et al.

Nomenclature a a0 ȧ

ȧ0 Δa Ac ACEEQ Aref A 2c 2c0 Δc C* Cd∗ Cs∗ E f1, f2, f3 Gi K n p Q

Q*

R Ri Ro R* t tf, C*-Ac

crack depth initial crack depth creep crack growth rate of a cracked specimen or component creep crack growth rate of a standard plane strain C(T) specimen crack depth increment unified characterization parameter of in-plane and out-of-plane creep constraint area surrounded by equivalent creep strain isoline area surrounded by equivalent creep strain isoline in a standard specimen the Norton’s coefficient in power-law creep strain rate expression crack length initial crack length crack length increment creep fracture mechanics parameter C* value at the deepest point along crack front C* value at the surface point along crack front Young’s modulus conservatism factors in creep crack growth life analysis influence coefficients stress intensity factor power-law creep stress exponent internal pressure constraint parameter under elastic-plastic condition or an elliptical integral defining the shape of the ellipse a load-independent creep constraint parameter

tf,

C*

tf,

FEM

tf,

RS

tred Δt εe εc σref ν Φ

based on Q creep constraint parameter inner radius of pipes outer radius of pipes load-independent creep constraint parameter creep time or pipe thickness creep crack growth life from two-parameter C*-Ac assessment creep crack growth life from single-parameter C* assessment creep crack growth life predicted by FEM solution of C* creep crack growth life predicted by reference stress calculation of C* stress redistribution time time increment elastic strain equivalent creep strain reference stress Poisson’s ratio angular parameter characterizing crack front position

Abbreviations 3D CCG CCI C(T) CEEQ FEM RS

three-dimensional creep crack growth creep crack initiation compact tension equivalent creep strain in ABAQUS code finite element method reference stress

and growth rate in specimens or components also have been analyzed by using the two-parameter C*-Q [15,16], C*-Q* [4,17,31,32], C*-R* [24,33] and C*-Ac [29,34,35]. It has been shown that with increasing crack-tip constraint, the creep crack initiation time becomes short and CCG rate increases. In recent studies, the CCI time in pressurized pipes has been predicted by using the twoparameter C*-Q* [31,32], C*-R* [35] and C*-Ac [35] approaches. The CCG life in pressurized pipes also has been predicted by using the two-parameter C*-R* approach [33]. The constraint parameters Q, Q*, A2, R and R* are defined based on crack-tip stress field analysis, and they can mainly characterize in-plane constraint in specimens or components [18,26,27]. In recent work, higher order asymptotic analyses of crack tip fields of pure mode II and mixed mode I/II cracks in power-law creeping solids have been conducted by Dai et al. [36,37]. The two-order asymptotic solutions developed by them may potentially provide basis for the development of two-parameter C*-A2 approach for pure mode II and mixed I/II creep cracks [36,37]. In actual engineering structures, the in-plane and out-of-plane constraints coexist due to complex structure geometries, different crack sizes and shape and loading modes. It needs to develop a unified creep constraint parameter which can characterize both inplane and out-of-plane constraints. It has been shown that the constraint parameter Ac defined based on crack-tip equivalent creep strain can characterize both in-plane and out-of-plane constraints [27–30]. The parameter Ac is load-independent, and it may accurately and conveniently characterize overall constraint level in structures at different load levels. Because of this advantage of the parameter Ac, it can be expected to use in constraint dependent CCG life assessments of engineering structures. In order to realize the application of Ac in pressurized pipes, the finite element solutions of Ac for pressurized pipes with a wide range of axial and circumferential surface crack sizes have been investigated and obtained in the previous work of authors [30,38,39]. The effects of creep properties of materials on the parameter Ac have also been investigated for cracked pipes, and the estimating method of Ac for materials with different creep properties has been given [40]. However, the methodology of the CCG life assessment based on twoparameter C*-Ac for pressurized pipes has not been investigated and established. In this work, the CCG life assessment methodology based on two-parameter C*-Ac for pressurized pipes with a wide range of initial crack sizes was studied. The assessment results of CCG life and crack size/shape evolution based on the two-parameter C*-Ac approach were quantitatively analyzed and compared with those based on the conventional single-parameter C* approach. The effects of calculation methods of C* on the assessment result also were analyzed.

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2. Materials and geometry of cracked pipes with semi-elliptical surface cracks The pipe material used in this work was a Cr-Mo-V steel (25Cr2NiMo1V steel), and its elastic-plastic and creep properties parameters at 566 °C are listed in Table 1 [1]. The cracked pipes with axial internal surface cracks were used in CCG life assessments based on two-parameter C*-Ac. The geometry and dimension of the cracked pipes with semi-elliptical surface cracks are shown in Fig. 1. The a, 2c, Ri and t are the crack depth, crack length, pipe inner radius and wall thickness, respectively. The Φ is the angular parameter for characterizing the crack front location, and the 2Φ/π = 0 denotes the surface point and the 2Φ/π = 1 indicates the deepest point along the crack front. The pipe dimensions are taken as t = 38 mm and radius-thickness ratio Ri/t = 10. The initial crack sizes (a0/t, a0/c0) for the CCG life analyses include a0/t = 0.1, 0.2, 0.4, 0.6, 0.7 and a0/c0 = 0.2, 0.4, 0.6, 0.8, 1. For the material, geometry and crack sizes of the pipes used in this work, the constraint parameter Ac solutions have been obtained by extensive finite element analyses in the previous work of authors [30,38].

3. Procedure of creep crack growth life assessment based on two-parameter C*-Ac for cracked pipes 3.1. Unified creep constraint parameter Ac and its application For incorporating both in-plane and out-of-plane creep constraints in specimens or components, a unified creep constraint parameter Ac based on crack-tip equivalent creep strain has been proposed in the work of Ma et al. [27,28], as shown in Eq. (1):

Ac =

ACEEQ t at =1 Aref tred

(1)

where ACEEQ is the area surrounded by the equivalent creep strain (εc ) isoline ahead of the crack tip in a specimen or component, Aref is a reference area surrounded by the εc isoline in a standard plane-strain reference specimen with high constraint (such as C(T) specimen), t is creep time, and tred is stress redistribution time. The ACEEQ and Aref are calculated at the same εc isoline, the same creep time t/tred = 1 (steady-state creep) and C* level by FEM analyses [27,28]. The tred is also calculated by FEM. The Ac value calculated by Eq. (1) is independent on the choice of εc isolines and C* level [27,28]. The increase of Ac value implies the decrease of crack-tip constraint level (constraint loss), which leads to the decrease of CCG rate. The Ac > 1 means that the constraint level of a cracked specimen or component and CCG rate are lower than those of the standard reference specimen. The unified constraint parameter Ac has been successfully used in in-plane and out-of-plane constraint analyses for specimens and cracked pipes with different geometries [12,13,27–30]. To apply the parameter Ac in creep life assessment for cracked pipes, the Ac solutions for pressurized pipes with different crack sizes, crack orientations and locations have been investigated and obtained in the previous studies of authors [30,38,39]. The effects of creep properties of materials on the parameter Ac of pressurized pipes have also been studied [40]. In this work, the methodology of creep crack growth life assessment based on two-parameter C*-Ac approach for pressurized pipes will be investigated. In CCG life assessments considering constraint effect for cracked components based on two-parameter C*-Ac approach, the CCG rate along the crack fronts will be controlled by the two parameters C* and Ac. Referring to the previous studies [29,30,33,35], the general life calculation procedure may be as follows. (1) (2) (3) (4)

To establish constraint-dependent CCG rate formula for component material at service temperature. To calculate constraint parameter Ac for cracked component. To calculate fracture parameter C* for cracked component at service load. To calculate CCG rate and crack growth increment during a short time period by substituting the Ac and C* values into the constraint-dependent CCG rate formula of component material. (5) To calculate Ac and C* for a new growth crack after step (4), and repeat step (4) to calculate the next new crack size. (6) To repeat steps (4) and (5), the CCG life (the total crack growth time) can be obtained by accumulating the short time at each step until the crack size reaches a critical size for component failure. In order to understand and achieve the CCG life assessment by using the procedure above for cracked components, the cracked pipe as a typical case is studied. The constraint-dependent CCG rate formulas of the pipe material (Cr-Mo-V steel) and the calculation methods of the parameters Ac and C* for the cracked pipe will be described in the followings. Table 1 The material properties of Cr-Mo-V steel at 566 °C [1]. Young’s modulus, E (GPa)

Poisson’s ratio, ν

Strain hardening coefficient, α

Yield stress, σ0 (MPa)

Strain hardening exponent, n

Norton’s coefficient, A (MPa−n h−1)

Norton’s exponent, n

160

0.3

1

383

10.4

7.26E−26

8.75

3

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Fig. 1. The geometry and dimension of the axially cracked pipes with semi-elliptical surface crack.

3.2. The constraint-dependent CCG rate formulas of the pipe material Based on the parameter Ac, the constraint-dependent CCG rate formulas of the pipe material (Cr-Mo-V steel) at 566 °C have been investigated and established in the previous work [28]. It has been found that the da/dt-C* curves of the C(T) specimens with different in-plane and out-of-plane constraints in a wide range of C* can be divided into three regions, including low C* region, transition C* region and high C* region. In these three regions, the da/dt-C* correlation lines have different slopes, which reflect different CCG behavior of the steel due to the stress dependent creep deformation and fracture mechanism [28]. In low and transition C* regions, the constraint has significant effect on CCG rate, thus the correlations between CCG rate ratio ȧ/ ȧ0 and parameter Ac were established in these two regions, as shown in Fig. 2 [28]. The ȧ0 is CCG rate of a standard plane-strain C(T) reference specimen. Fig. 2 shows that a monotonic correlation line between the ȧ/ ȧ0 and Ac on log–log scale can be formed for all C(T) specimens with different in-plane and out-of-plane constraints, and the correlation line is independent of C* (the data at two typical C* values all are located on one correlation line). This implies that the parameter Ac can characterize both in-plane and out-of-plane constraints. The formulas between ȧ/ ȧ0 and Ac for the low C* region (3 × 10−7 MPa.m/h ≤ C* < 3 × 10−5 MPa.m/h) and transition C* region (3 × 10−5 MPa.m/h ≤ C* < 3 × 10−4 MPa.m/h) have been fitted from Fig. 2, as shown in Eqs. (2) and (3), respectively [28]:

f (Ac ) =

ȧ = 1.312Ac−1.623 , ȧ = 0.793C ∗0.611 for 3 × 10−7 ≤ C ∗ < 3 × 105 MPa. m/h a ̇0

(2)

f (Ac ) =

ȧ = 1.212Ac−1.008 , ȧ = 0.157C ∗0.484 for 3 × 10−5 ≤ C ∗ < 3 × 10 4 MPa. m/h a ̇0

(3)

3.3. The FEM solutions of parameter Ac for cracked pipes For the analyzed geometry and dimension of pipes with axial semi-elliptical internal surface cracks (Fig. 1) in this work, the FEM solutions of parameter Ac in the previous work of authors [30] can be used. Fig. 3 shows the FEM results of parameter Ac at the surface point (2Ф/π = 0) and at the deepest point (2Ф/π = 1) in Ref. [30]. It can be seen that for a fixed crack depth a/t, the parameter Ac increases with increasing aspect ratio a/c (decreasing crack length 2c). For a fixed aspect ratio a/c, the parameter Ac decreases with increasing crack depth a/t. For a given crack size, the Ac1 at surface point (2Ф/π = 0) is higher than the Ac2 at the deepest point (2Ф/π = 1), which means that the constraint level at surface point is lower than that at the deepest point.

Fig. 2. The correlations between ȧ/ ȧ0 and Ac for low C* region (3 × 10−7 ≤ C* < 3 × 10−5 MPa.m/h) (a) and transition C* region (3 × 10−5 ≤ C* < 3 × 10−4 MPa.m/h) (b) for all specimens with different in-plane and out-of-plane constraint levels [26]. 4

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3.0 2.8 2.6

1.8

2.2

1.6 1.4

2.0

1.2

1.8 1.6

a/c = 0.2 a/c = 0.4 a/c = 0.6 a/c = 0.8 a/c = 1.0

Axial crack, 2Φ/π =1

2.0

Ac2

2.4

Ac1

2.2

a/c = 0.2 a/c = 0.4 a/c = 0.6 a/c = 0.8 a/c = 1.0

Axial crack, 2Φ/π =0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

1.0

0.8

0.1

0.2

a/t

0.3

0.4

0.5

0.6

0.7

0.8

a/t

(a)

(b)

Fig. 3. The constraint parameters Ac1 at the surface point (a) and Ac2 at the deepest point (b) for different axial crack sizes in pressurized pipes with Ri/t = 10 [28].

The empirical equations for the parameter Ac1 at the surface point (2Φ/π = 0) and the Ac2 at the deepest point (2Φ/π = 1) along the crack front for axial cracks in cracked pipes with Ri/t = 10 are expressed in Eqs. (4) and (5), respectively [30]: At the surface point, 2Φ/π = 0:

a a 2 a 3 a 4 Ac1 = D0 + D1 ⎛ ⎞ + D2 ⎛ ⎞ + D3 ⎛ ⎞ + D4 ⎛ ⎞ t t t ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝t ⎠

(4)

where

a a 2 a 3 a 4 D0 = 3.967 − 9.839 ⎛ ⎞ + 25.63 ⎛ ⎞ − 25.312 ⎛ ⎞ + 7.955 ⎛ ⎞ ⎝c⎠ ⎝c⎠ ⎝c⎠ ⎝c⎠ a a 2 a 3 a 4 D1 = −3.258 − 17.494 ⎛ ⎞ + 172.302 ⎛ ⎞ − 340.165 ⎛ ⎞ + 187.597 ⎛ ⎞ ⎝c⎠ ⎝c⎠ ⎝c⎠ ⎝c⎠ a a 2 a 3 a 4 D2 = 20.892 − 59.094 ⎛ ⎞ − 348.245 ⎛ ⎞ + 991.925 ⎛ ⎞ − 606.336 ⎛ ⎞ c c c ⎝ ⎠ ⎝c⎠ ⎝ ⎠ ⎝ ⎠ a a 2 a 3 a 4 D3 = −44.889 + 216.347 ⎛ ⎞ + 227.516 ⎛ ⎞ − 1214.877 ⎛ ⎞ + 818.84 ⎛ ⎞ ⎝c⎠ ⎝c⎠ ⎝c⎠ ⎝c⎠ a a 2 a 3 a 4 D4 = 28.851 − 163.222 ⎛ ⎞ + 5.439 ⎛ ⎞ + 508.369 ⎛ ⎞ − 381.479 ⎛ ⎞ ⎝c⎠ ⎝c⎠ ⎝c⎠ ⎝c⎠ At the deepest point, 2Φ/π = 1:

a a 2 a 3 a 4 Ac2 = E0 + E1 ⎛ ⎞ + E2 ⎛ ⎞ + E3 ⎛ ⎞ + E4 ⎛ ⎞ t t t ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝t ⎠

(5)

where

a a 2 a 3 a 4 E0 = 11.317 − 62.061 ⎛ ⎞ + 149.311 ⎛ ⎞ − 154.121 ⎛ ⎞ + 57.713 ⎛ ⎞ ⎝c⎠ ⎝c⎠ ⎝c⎠ ⎝c⎠ a a 2 a 3 a 4 E1 = −148.482 + 951.42 ⎛ ⎞ − 2267.028 ⎛ ⎞ + 2331.717 ⎛ ⎞ − 871.333 ⎛ ⎞ ⎝c⎠ ⎝c⎠ ⎝c⎠ ⎝c⎠ a a 2 a 3 a 4 E2 = 624.821 − 4057.259 ⎛ ⎞ + 9756.559 ⎛ ⎞ − 10128.38 ⎛ ⎞ + 3817.706 ⎛ ⎞ c c c ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝c⎠ a a 2 a 3 a 4 E3 = −1006.52 + 6584.61 ⎛ ⎞ − 15954.23 ⎛ ⎞ + 16693.939 ⎛ ⎞ − 6339.06 ⎛ ⎞ ⎝c⎠ ⎝c⎠ ⎝c⎠ ⎝c⎠ a a 2 a 3 a 4 E4 = 552.792 − 3637.626 ⎛ ⎞ + 8858.129 ⎛ ⎞ − 9318.639 ⎛ ⎞ + 3556.083 ⎛ ⎞ ⎝c⎠ ⎝c⎠ ⎝c⎠ ⎝c⎠ The Eqs. (4) and (5) will be used to calculate the constraint parameters Ac1 and Ac2 for the axial semi-elliptical internal surface cracks in the cracked pipes. 5

Engineering Fracture Mechanics 220 (2019) 106676

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3.4. Calculation of fracture parameter C* for cracked pipes 3.4.1. The FEM solutions of fracture parameter C* for cracked pipes For the cracked pipes with axial semi-elliptical internal surface cracks, the fracture parameter C* has been obtained by 3D FEM analyses in the work of Wen et al. [41]. The parameter C* solution is expressed as follows:

R 2 + Ri2 n + 1 a a R C ∗ = A·t·Hc ( , , i , ϕ, n )·(Pi· O2 ) t c t RO − Ri2

(6)

where a, 2c, Ri, Ro, t, Φ and Pi are the crack depth, crack length, pipe inner radius, pipe outer radius, wall thickness, crack angular parameter and internal pressure load, respectively. The A and n are the Norton’s coefficient and creep exponent of materials, respectively. Hc describes a function which affects parameter C* (influence function). It can be seen from Eq. (6) that pipe geometries and sizes, crack sizes, pressure load and creep properties of materials have influence on parameter C*, indicating that the parameter C* is a function of a/t, a/c, Ri/t, Ro, Φ, Pi, A and n. The values of the influence function Hc for different a/t, a/c, Ri/t, Φ and n have been tabulated in Ref. [41], and they have been fitted as empirical formulas. These FEM solutions of fracture parameter C* will be used in the calculation of C* for the cracked pipes. 3.4.2. Reference stress calculation of fracture parameter C* for cracked pipes The parameter C* is usually estimated by the reference stress (RS) method [42,43], as shown in Eq. (7) for Norton creep material: n−1 2 C ∗ = Aσref K

(7)

where K is stress intensity factor and σref is reference stress. The reference stress σref for the semi-elliptical axial surface crack in pressurized pipes in the R6 [44] is expressed as Eq. (8):

σref =

p R a R + ( i )In( o ) Ri M Ri + a Ri + a 1.61c 2

M = (1 +

Ri a

)1/2

(8)

where Ri, Ro and p are the pipe inner radius, pipe outer radius and internal pressure load, respectively. The a is the crack depth and c is the half crack length. The stress intensity factor K for axial internal surface cracks in pressurized pipes has been obtained in the work of Raju and Newman [45], as shown in Eqs. (9) and (10):

K=

pRi t

Fi =

R2 t a a a ( 2 o 2 ) ⎡2G0 − 2( ) G1 + 3( )2G2 − 4( )3G3⎤ ⎢ ⎥ Ri Ro − Ri ⎣ Ri Ri Ri ⎦

πa a a t Fi ( , , , ϕ) Q c t Ri

(9)

(10)

where Gi (i = 0, 1, 2, 3) is the influence coefficients and Q is a parameter for defining the shape of the ellipse. The Q can be determined by using Eq. (11) [46]:

Q = 1 + 1.464(a/ c )1.65 for a/ c ⩽ 1 Q = [1 + 1.464(c / a)1.65](a/ c )2 for a/ c > 1

RS

(11)

RS

FEM a/c = 0.2

a/c = 0.2

2Φ/π =0

a/c = 0.4

1E-3

FEM

a/c = 0.6 a/c = 0.8

a/c = 1.0

C*, MPa⋅m/h

C*, MPa⋅m/h

a/c = 0.8

1E-4

2Φ/π =1

a/c = 0.4

1E-3

a/c = 0.6

1E-5 1E-6

1E-4

a/c = 1.0

1E-5 1E-6 1E-7

1E-7 0.1

0.2

0.3

0.4

0.5

0.6

0.1

0.7

0.2

0.3

0.4

a/t

a/t

(a)

(b)

0.5

0.6

0.7

Fig. 4. Comparison of C* calculated by FEM solution and reference stress (RS) method for different crack sizes at the surface point (a) and at the deepest point (b) in cracked pipes with Ri/t = 10 at pressure load p = 15 MPa. 6

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In the work of Liu [47], an F factor which is equal to Fi / Q in Eq. (9) was defined, and the F factor solution has been given. In this work, the solution of F factor for semi-elliptical surface cracks with a/c ≤ 1 at the deepest point (2Φ/π = 1) and at the surface point (2Φ/π = 0) for axially cracked pipes with Ri/t = 10 was used to calculate the stress intensity factor K in Eq. (9). When the σref and K are calculated by using Eqs. (8) and (9), respectively, the parameter C* can be calculated by using Eq. (7). Fig. 4 shows a comparison of C* calculated by the FEM solution and reference stress (RS) method for different crack sizes with 0.1 ≤ a/t ≤ 0.7 and 0.2 ≤ a/c ≤ 1 in the cracked pipes with a pressure load p = 15 MPa. It can be seen that with increasing crack depth a/t and decreasing aspect ratio a/c (increasing crack length 2c), the C* values increase. For a given crack size, the C* values calculated by reference stress method are higher than those calculated by the FEM solution. These results are consistent with those in the work of Wen et al. [41] and Liu et al. [33]. The different C* values calculated by the FEM and RS methods will affect CCG life assessment results. This will be analyzed later.

3.5. Calculation method of creep crack growth life for cracked pipes In this work, the constraint-dependent CCG rate formulas in Section 3.2, the FEM solutions of parameter Ac in Section 3.3 and the calculation methods of fracture parameter C* (FEM and RS methods) in Section 3.4 were used in the calculations of creep crack growth life for the axially cracked pipes. A step-by-step procedure was adopted in the creep crack growth analyses, and the crack shape is assumed to be semi-elliptical during creep crack growth. Only the CCG life was analyzed, and it is assumed that there is no crack initiation time. Firstly, the parameters Ac and C* at the surface point and at the deepest point in each step need to be calculated. Secondly, the crack depth increment Δa in each step is set as 0.02 t (the pipe wall thickness t = 38 mm). Based on Eqs. (2) and (3), the corresponding time increment Δt for the crack depth increment Δa at the deepest point in each step for single-parameter C* and two-parameter C*-Ac assessments can be calculated, as shown in Eqs. (12) and (13), respectively: Δa

⎧ 0.793C ∗0.611 ⎪ d Δt = Δa ⎨ ⎪ 0.157Cd∗0.484 ⎩

for 3E − 7 MPa m/h ⩽ C ∗ < 3E−5 MPa m/h for 3E − 5 MPa m/h ⩽ C ∗ ⩽ 3E−4 MPa m/h

Δa

⎧ −1.623 ) ⎪ (0.793Cd∗0.611)·(1.312Acd Δt = Δa ⎨ −1.008 ) ⎪ (0.157Cd∗0.484)·(1.221Acd ⎩

(12)

for 3E − 7 MPa m/h ⩽ C ∗ < 3E−5 MPa m/h for 3E − 5 MPa m/h ⩽ C ∗ ⩽ 3E−4 MPa m/h

(13)

where Cd∗ and Acd represent the C ∗ and Ac values at the deepest point. Then the crack length increment Δc at the surface point for single-parameter C* and two-parameter C*-Ac assessments can be obtained by Eqs. (14) and (15), respectively, as follows:

0.793Cs∗0.611·Δt Δc = ⎧ ⎨ 0.157Cs∗0.484·Δt ⎩

for 3E − 7 MPa m/h ⩽ C ∗ < 3E−5 MPa m/h for 3E − 5 MPa m/h ⩽ C ∗ ⩽ 3E−4 MPa m/h

(1.312Acs−1.623 ·0.793Cs∗0.611)·Δt for 3E − 7 MPa m/h ⩽ C ∗ < 3E−5 MPa m/h Δc = ⎧ ⎨ (1.221Acs−1.008 ·0.157Cs∗0.484 )·Δt for 3E − 5 MPa m/h ⩽ C ∗ ⩽ 3E−4 MPa m/h ⎩

(14)

(15)

where Cs∗ and Acs represent the C ∗ and Ac values at the surface point, respectively. In the end, a new crack size in each step can be calculated based on the crack depth increment Δa and the crack length increment Δc by Eqs. (16) and (17), as follows:

ai + 1 = ai + Δa ci + 1 = ci + Δc

i = 0, 1, 2. ..

(16)

i = 0, 1, 2. ..

(17)

The analyses will be terminated when the crack depth ai+1 reaches 0.8 time of the pipe thickness (ai+1 = 0.8 t), and the sum of time accumulated by the time increment Δt in each step is the CCG life tf. According to the BS7910 standard [48], it is necessary to verify the steady creep conditions at each step for the CCG life assessment. Thus, the elastic strain εe and the accumulated creep strain εc at the reference stress σref for a short time period Δt from uniaxial creep data are calculated by Eqs. (18) and (19), respectively. If εc ≥ εe, the stress redistribution is completed and the CCG rate can be calculated by Eqs. (2) and (3). If εc < εe, the stress redistribution is incomplete and the CCG rate calculated from Eqs. (2) and (3) should be multiplied by a factor of 2 [48].

εe = σref / E

(18)

εc = A (σref )nΔt

(19) 7

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4. Results and discussion 4.1. Creep crack growth assessment based on FEM solutions of C* and Ac In this section, the creep crack growth life assessment for pressurized pipes are conducted based on FEM solutions of C* and Ac. The CCG life predicted by two-parameter C*-Ac is compared with that predicted by single-parameter C*. In order to analyze the effect of initial crack sizes on CCG life assessment, the initial crack sizes are set as different a0/t (a0/t = 0.1, 0.2, 0.4, 0.6, 0.7) and a0/c0 (a0/ c0 = 0.2, 0.4, 0.6, 0.8, 1). To investigate the effect of internal pressure load on CCG life, two internal pressure loads p = 7.5 MPa and 15 MPa are applied in the analyses, respectively. According to the calculation method in Section 3.5, the CCG life and crack profile are calculated by using two-parameter C*-Ac and single-parameter C*, respectively. The CCG life calculation results for typical cracks with a0/t = 0.1 and 0.4 and different a0/c0 are shown in Fig. 5. It can be found that the CCG life predicted by two-parameter C*-Ac is longer than that predicted by singleparameter C* for both two pressure loads, which indicates the conservatism of single-parameter C* assessment. For a given initial crack depth a0/t, the CCG life tf increases with increasing initial aspect ratio a0/c0. For a given initial aspect ratio a0/c0, the CCG life tf decreases with increasing initial crack depth a0/t. For the initial crack with larger depth (a0/t = 0.4), the a0/c0 has more significant effect on the CCG life. These results are caused by the effects of crack sizes on the parameters Ac (Fig. 3) and C* (Fig. 4). With increasing a0/t and decreasing a0/c0, the constraint and C* increase which accelerate the CCG rate. Higher pressure load can cause higher C*, which leads to faster CCG rate and shorter CCG life. A conservatism factor f1 can be defined as the ratio of the CCG life from two-parameter C*-Ac (tf, C*-Ac) to that from singleparameter C* (tf, C*), as shown in Eq. (20):

f1 =

t f , C ∗−Ac tf , C∗

(20)

The f1 depicts the conservatism degree of CCG life prediction based on single-parameter C* compared with that based on twoparameter C*-Ac. The factor f1 values for a wide range of initial crack sizes (a0/t = 0.1, 0.2, 0.4, 0.6, 0.7 and a0/c0 = 0.2, 0.4, 0.6, 0.8, 1) at two pressure loads (p = 7.5 and 15 MPa) are calculated, and the results are shown in Fig. 6. It can be seen that the factor f1 increases with decreasing initial crack depth a0/t and increasing initial aspect ratio a0/c0. For the cracks with a0/c0 ≥ 0.6, the factor f1 is not sensitive to a0/c0. For the shallower (a0/t ≤ 0.4) and shorter (a0/c0 ≥ 0.6) cracks, the value of the factor f1 is in a range of 1.6–2.1. This implies that for the shallower and shorter surface cracks in pressurized pipes, the CCG life predicted by single-parameter C* may be overly conservative. The defects in pipes are usually shallower and shorter surface cracks. Therefore, it is suggested that the CCG life of cracked pipes may require to be assessed based on the two-parameter C*-Ac for reducing the conservatism in traditional assessment using single-parameter C*. It is believed that if the CCG rate of materials is more sensitive to crack-tip constraint, more high accuracy benefit will be gained by using the CCG life assessments considering constraint effect. Fig. 6 also shows that the internal pressure load almost has no influence on the factor f1. This may come from the load-independence of the constraint parameter Ac. The load-independence of the factor f1 may bring benefit for creep life analysis of cracked components. The service load in actual components may change under different working conditions. The factor f1 obtained at one load may be used to estimate the f1 at another load. The creep crack size and shape evolutions during crack growth process determine the final crack size and crack growth area, which influence the limit load, fracture behavior and LBB (leak-before-break) analysis of a cracked structure. When the constraint is considered, the crack size and shape evolutions are controlled by two parameters C* and Ac. This may be different from the case controlled by single-parameter C*, and will be analyzed in the followings. 106

a0/t=0.1 a0/t=0.4 C*-FEM

a0/t=0.1 a0/t=0.4 C*-FEM

p=7.5MPa, C* p=7.5MPa, C*-Ac

108

105

tf , h

tf , h

p=15MPa, C* p=15MPa, C*-Ac

107

106 104

0.2

0.4

0.6

0.8

0.2

1.0

0.4

0.6

a0/c0

a0/c0

(a)

(b)

0.8

1.0

Fig. 5. The CCG life predicted from FEM solutions of C* and Ac for initial semi-elliptical surface cracks with a0/t = 0.1 and a0/t = 0.4 in pressurized pipes at internal pressure loads p = 15 MPa (a) and p = 7.5 MPa (b). 8

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p=7.5MPa p=15MPa

Conservatism factor, f 1

2.4 2.2

p=7.5MPa p=15MPa

a0/t =0.1

a0/t =0.2

a0/t =0.4

a0/t =0.6

a0/t =0.7

2.0 1.8 1.6 1.4 1.2 1.0 0.8

0.2

0.4

0.6

0.8

1.0

a0/c0 Fig. 6. The conservatism factor f1 for semi-elliptical surface cracks with various initial sizes in pressurized pipes at two pressure loads (p = 7.5 and 15 MPa).

The creep crack shape evolutions for two typical initial surface cracks based on two-parameter C*-Ac and single-parameter C* calculations are shown in Figs. 7 and 8, respectively. The final crack growth depth increment Δa and half crack length increment Δc are marked in the two figures. Comparing Fig. 7 with Fig. 8, it can be found that for the same initial crack size a0/t and a0/c0, when the final crack depth increment Δa is the same (this is determined by the assumption that the final failure occurs at crack depth a/ t = 0.8), the final half crack length increment Δc calculated by the two-parameter C*-Ac is smaller than that calculated by the singleparameter C*. This implies that the crack growth length and area predicted from two-parameter C*-Ac is smaller than that from the single-parameter C*. This is caused by the lower constraint at the surface point along the crack front (Fig. 3). Fig. 7 shows that for a

Fig. 7. Predicted creep crack shape evolution based on two-parameter C*-Ac for initial cracks with a0/t = 0.1, a0/c0 = 1 (a) and a0/t = 0.2, a0/ c0 = 0.4 (b) at pressure load p = 15 MPa. 9

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Fig. 8. Predicted creep crack shape evolution based on single-parameter C* for initial cracks with a0/t = 0.1, a0/c0 = 1 (a) and a0/t = 0.2, a0/ c0 = 0.4 (b) at pressure load p = 15 MPa.

given initial crack size a0/t and a0/c0, the final half crack length increment Δc from two-parameter C*-Ac is smaller than the final crack depth increment Δa. This result is basically consistent with the experimental and numerical creep crack growth behavior of a semi-elliptical surface crack in the literature [49,50], which suggests that the two-parameter C*-Ac approach may reasonably predict crack growth size and shape. While Fig. 8 shows that in the case without considering constraint effect, the increment Δc from the single-parameter C* is close to Δa. This means that the final crack growth length may be overestimated by the traditional singleparameter C* approach. To further understand the creep crack size evolutions during crack growth process, Fig. 9 gives the change of crack depth increment Δa and half crack length increment Δc with time calculated from the two-parameter C*-Ac (Fig. 9(b) and (d)) and singleparameter C* (Fig. 9(a) and (c)) for the typical initial crack sizes in Fig. 5. Fig. 9 shows that the Δa and Δc and their growing rate increases with time due to the increase of C* and constraint level during crack growth. For a given initial crack depth a0/t, the final crack depth increment Δa is the same for different a0/c0 due to the assumption that the final failure occurs at crack depth a/t = 0.8. The time corresponding to the final crack depth increment Δa is the CCG life tf predicted in Fig. 5. Fig. 9 indicates that at the same pressure load p = 15 MPa, the life tf and final crack length increment Δc depends on the initial crack sizes (a0/t and a0/c0) and life calculation methods (two-parameter or single-parameter). For the same initial crack sizes, the growing rate of Δc from two-parameter calculation is smaller than those from single-parameter, which leads to shorter final crack length increment Δc and smaller crack growth area in two-parameter calculations, as typical shown in Fig. 7. For the same calculation method and initial crack depth a0/t, with decreasing a0/c0 (increasing initial crack length), the growing rate of Δc increases and the final crack length increment Δc becomes short. To further understand the crack shape evolutions during crack growth process, Fig. 10 shows the variation of crack aspect ratio a/ c with crack depth a/t calculated from the two-parameter C*-Ac (Fig. 10(b) and (d)) and single-parameter C* (Fig. 10(a) and (c)) for the typical initial crack sizes in Figs. 5 and 9. Fig. 10 shows the crack shape evolution curves (a/c-a/t curves) from the two-parameter and single-parameter calculations are similar. But for a given initial crack size, the aspect ratio a/c predicted by two-parameter C*-Ac is slightly larger than that predicted by single-parameter C* due to the smaller increment Δc of the two-parameter assessment. For the shallower initial cracks with a0/t = 0.1, the aspect ratio a/c is approaching to an asymptotic value as the crack grows for different initial aspect ratio a0/c0. This result is basically consistent with that in Ref. [51] for CCG analysis of elliptic surface cracks in pressure 10

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Δa

Δc

25

a0/c0=0.4

20

a0/c0=0.8 a0/c0=1.0

10 5 a0/t =0.1, C*

0

50000

100000

a0/c0=0.2 a0/c0=0.6 a0/c0=0.8

20

a0/c0=1.0

15 10 5 0

150000

Δc

a0/c0=0.4

25

a0/c0=0.6

15

0

Δa

30

a0/c0=0.2

Δa or Δc, mm

Δa or Δc, mm

30

a0/t =0.1, C*-Ac

0

100000

t, h

(b) 20

a0/t =0.4, C*

a0/t =0.4, C*-Ac

15

10

Δa

Δc

a0/c0=0.2 a0/c0=0.4

5

a0/c0=0.6

Δa or Δc, mm

Δa or Δc, mm

15

10

Δa

a0/c0=1.0

0

10000

20000

30000

40000

50000

Δc a0/c0=0.2 a0/c0=0.4

5

a0/c0=0.6

a0/c0=0.8

0

300000

t,h

(a) 20

200000

a0/c0=0.8

0

a0/c0=1.0

0

20000

40000

t, h

t, h

(c)

(d)

60000

80000

Fig. 9. The variations of crack depth increment Δa and half crack length increment Δc with time during creep crack growth based on singleparameter C* and two-parameter C*-Ac calculations, (a), (b) a0/t = 0.1 and (c), (d) a0/t = 0.4 at pressure load p = 15 MPa.

vessels. For the deeper initial cracks with a0/t = 0.4, the aspect ratio a/c increases with the increase of crack depth a/t (except for the crack with a0/c0 = 1). For the cracks with higher a0/c0 (a0/c0 ≥ 0.6), after a/t is > 0.6, the a/c almost does not change. In general, the results in Figs. 5–10 show that compared with the traditional single-parameter C* assessment of CCG life for cracked pipes, the assessment based on two-parameter C*-Ac approach can reduce conservatism degree and produce smaller crack growth length and area. The crack growth length can affect limit load and crack-tip fracture parameter; hence influence the failure assessment results of cracked components. In addition, the crack growth area is related to the leakage rate calculation in LBB (leakbefore-break) analysis of cracked pipes. The reasonable calculation results of CCG life, crack growth length and area based on twoparameter approach with considering constraint effect may improve accuracy of life design and integrity assessments for cracked components. 4.2. Effect of reference stress calculation of C* on creep crack growth life in cracked pipes The results in Figs. 5–10 are calculated and obtained by the FEM solutions of C* and Ac. Fig. 4 shows that the C* from the reference stress (RS) calculation is higher than that of the FEM calculation. Thus, the RS calculation of C* can affect the CCG life assessment results of both two-parameter and single-parameter approaches. Fig. 11 shows the CCG life tf calculated from the RS and FEM calculations of C*. It can be seen that for different initial crack sizes, the CCG life from the RS calculation of C* is significantly shorter than that from the FEM calculation of C* for both two-parameter and single-parameter approaches. This implies that the RS calculation of C* will produce conservative life predication results. A conservatism factor f2 can be defined as the ratio of the CCG life tf, FEM predicted by FEM solution of C* and the CCG life tf, RS predicted by RS calculation of C*, as shown in Eq. (21):

f2 =

t f , FEM t f , RS

(21)

Fig. 12 shows the conservatism factor f2 for a wide range of initial crack sizes based on single-parameter C* and two-parameter C*-Ac assessments at two internal pressure loads (p = 7.5 and 15 MPa). It can be found that the single-parameter and two-parameter 11

Engineering Fracture Mechanics 220 (2019) 106676

1.2 1.1 1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1

a0/c0=0.2 a0/c0=0.4

a/c

a/c

X.M. Tan, et al.

a0/c0=0.6 a0/c0=0.8

a0/t =0.1, C*

0.1

0.2

0.3

0.4

0.5

a0/c0=1.0

0.6

0.7

0.8

1.2 1.1 1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1

a 0/c 0=0.2 a 0/c 0=0.4 a 0/c 0=0.6

0.1

0.2

0.3

0.4

a0/c0=0.2 a0/c0=0.4

a0/t =0.4, C*

a0/c0=0.6 a0/c0=0.8 a0/c0=1.0

0.4

0.5

0.6

0.5

a 0/c 0=1.0

0.6

0.7

0.8

a/t (b)

a/c

a/c

a/t (a) 1.6 1.5 1.4 1.3 1.2 1.1 1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1

a 0/c 0=0.8

a 0/t =0.1, C*-A c

0.7

0.8

1.6 1.5 1.4 1.3 1.2 1.1 1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1

a0/c0=0.2 a0/c0=0.4

a0/t =0.4, C*-A c

a0/c0=0.6 a0/c0=0.8 a0/c0=1.0

0.4

0.5

0.6

a/t

a/t

(c)

(d)

0.7

0.8

Fig. 10. Variation of crack aspect ratio a/c with crack depth a/t during crack growth process calculated from the two-parameter C*-Ac ((b) and (d)) and single-parameter C* ((a) and (c)) for different initiation crack sizes at pressure load p = 15 MPa.

C* (RS) C*-Ac (RS)

C* (RS) C*-Ac (RS)

106

C* (FEM) C*-Ac (FEM)

105

tf /h

tf /h

C* (FEM) C*-Ac (FEM)

105

104

a0/t = 0.1, p =15MPa 0.2

0.4

0.6

0.8

1.0

a0/t =0.4, p =15MPa 0.2

0.4

0.6

a0/c0

a0/c0

(a)

(b)

0.8

1.0

Fig. 11. The CCG life calculated from RS and FEM calculations of C* for the two-parameter C*-Ac and single-parameter C* approaches at pressure load p = 15 MPa. (a) a0/t = 0.1, (b) a0/t = 0.4.

approaches have similar factor f2 due to the load-independence of Ac. For all initial crack sizes and two pressure loads, the f2 value is larger than 1, which suggests that the RS calculation of C* can produce conservative life assessment results. The factor f2 is mainly related to initial crack sizes and internal pressure loads. For the same initial crack size, the factor f2 at lower load p = 7.5 MPa is higher than that at higher p = 15 MPa. This may be caused by the overestimation of C* and underestimation of the CCG life tf, RS of the RS method at lower load. The effect of the initial crack size on the factor f2 is related to load level. Under the lower load 12

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single-parameter C* two-parameter C*-Ac

single-parameter C* two-parameter C*-Ac a0/t=0.1 a0/t=0.2

p = 15MPa

3.5

a0/t=0.4

3.0

a0/t=0.7

Conservatism factor, f2

Conservatism factor, f2

4.0

a0/t=0.6

2.5 2.0 1.5 1.0

0.2

0.4

0.6

0.8

1.0

6.0 5.5 5.0 4.5 4.0 3.5 3.0 2.5 2.0 1.5 1.0

a0/t=0.1 a0/t=0.2

p = 7.5MPa

a0/t=0.4 a0/t=0.6 a0/t=0.7

0.2

0.4

0.6

a0/c0

a0/c0

(a)

(b)

0.8

1.0

Fig. 12. The conservatism factor f2 for a wide range of initial crack sizes in pressurized pipes based on single-parameter C* and two-parameter C*-Ac assessments at two internal pressure loads: (a) p = 15 MPa and (b) p = 7.5 MPa.

1.6 1.5 1.4 1.3 1.2 1.1 1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1

a0/c0 =0.2

a0/c0 =0.2 a0/c0 =0.4

a0/t =0.4, C*(RS)

a0/c0 =0.6 a0/c0 =0.8 a0/c0 =1.0

a/c

a/c

p = 7.5 MPa, for the initial cracks with a0/c0 > 0.5, with decreasing initial crack depth a0/t, the factor f2 increases; for the cracks with a0/c0 < 0.5, with decreasing a0/t from 0.7 to 0.2, the factor f2 decreases. Under the higher load p = 15 MPa, with decreasing initial crack depth a0/t, overall the factor f2 increases and the a0/c0 has small effect on f2 (except for the increase of f2 of the cracks with a0/t > 0.6 and a0/c0 < 0.5). The effect of initial crack sizes on factor f2 mainly comes from the difference of C* calculated using RS and FEM methods for various crack sizes, as shown in Fig. 4. The overall trends in Fig. 12 show that for shallower and some deeper and longer initial cracks, the f2 value is higher and the RS calculation of C* will produce over-conservative life assessment. For improving the accuracy of life assessment, it is suggested that the FEM solution of C* is used. To examine the effect of the RS calculation of C* on the crack size and shape evolution during crack growth process, the variation of crack aspect ratio a/c with crack depth a/t has been calculated by using both two-parameter and single-parameter approaches for cracks with a0/t = 0.4 and different a0/c0, as shown in Fig. 13. It can be seen that the a/c-a/t curves calculated by RS calculation of C* in Fig. 13 are similar to those calculated by FEM C* solution in Fig. 10(c) and (d). This result illustrates that different C* calculation methods only affect the C* value and CCG rate, and almost has no effect on the crack size and shape evolution during crack growth process. This may bring convenience to the analysis of the crack size and shape evolution. To examine the effect of the RS calculation of C* on the conservatism factor f1 in Eq. (20), the f1 values have been calculated by using the RS C* for various initial crack sizes, as shown in Fig. 14. It can be found that the curves in Fig. 14 are similar to those in Fig. 6. The only difference is that the pressure load has little effect on the factor f1 in Fig. 14. The comparison between Fig. 14 and Fig. 6 illustrates that different C* calculation methods almost have no effect on the factor f1. This may be convenient for the conservatism analysis in CCG life assessments with and without considering constraint effect.

0.4

0.5

0.6

0.7

0.8

1.6 1.5 1.4 1.3 1.2 1.1 1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1

a0/c0 =0.4 a0/t =0.4, C*-A c(RS )

a0/c0 =0.6 a0/c0 =0.8 a0/c0 =1.0

0.4

0.5

0.6

a/t

a/t

(a)

(b)

0.7

0.8

Fig. 13. The variation of crack aspect ratio a/c with crack depth a/t calculated by using single-parameter (a) and two-parameter (b) for cracks with a0/t = 0.4 and different a0/c0 at pressure load p = 15 MPa. 13

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p=7.5MPa p=15MPa

Conservatism factor, f 1

2.4 2.2

p=7.5MPa p=15MPa

a0/t =0.1

a0/t =0.2

a0/t =0.4

a0/t =0.6

a0/t =0.7

2.0 1.8 1.6 1.4 1.2 1.0 0.8

C*-RS 0.2

0.4

0.6

0.8

1.0

a0/c0 Fig. 14. The conservatism f1 values calculated by using the RS C* for various initial crack sizes in pressurized pipes at two pressure loads p = 7.5 and 15 MPa.

4.3. Overall conservatism degree analysis in conventional CCG life assessments According to the analyses above, in the conventional CCG life assessment procedure [43,48], no account of constraint effect and the RS calculation of C* may lead to greater conservatism which is expressed by the conservatism factors f1 and f2, respectively. The analyses in Section 4.1 show that the CCG life assessments based on two-parameter approach and FEM calculations of C* and Ac produce the longest CCG life which may be reasonable and accurate in principle. This life can be denoted as tf, C*-Ac(FEM). The analyses in Section 4.2 indicate that the conventional CCG life assessments based on single-parameter approach and the RS calculation of C* produce the shortest CCG life. This life can be denoted as tf, C*(RS). To analyze overall conservatism degree in the conventional CCG life assessments without considering constraint effect and with RS calculation of C*, a conservatism factor f3 can be defined as:

f3 =

t f , C ∗−Ac (FEM ) t f , C ∗ (RS )

(22)

The factor f3 reflects overall conservatism degree caused by the conservatism factors f1 and f2. The factor f3 have been calculated for a wide range of initial crack sizes at two internal pressure loads (p = 7.5 and 15 MPa), as shown in Fig. 15. It can be found that the factor f3 values depend on initial crack sizes and pressure loads. For shallower and shorter initial surface cracks, the factor f3 values are very high. For the same initial crack sizes, the factor f3 value at lower load is higher than that at higher load. To further clearly show the effects of constraint and the RS calculation of C* on the CCG life, the CCG life data together with the conservatism factors f1, f2 and f3 for some typical initial shallower cracks (0.1 ≤ a0/t ≤ 0.4, 0.2 ≤ a0/c0 ≤ 1) at pressure load p = 15 MPa are listed in Table 2. The data in Table 2 show that the shallowest cracks with a0/t = 0.1 and different a0/c0 have the highest values of f1, f2 and f3. This shows that for these cracks, the CCG life assessment without considering constraint effect and with the RS calculation of C* will produce very high overall conservatism degree (f3 = 4.08–4.77). For example, for the crack with a0/

a0/t=0.1

p = 15MPa

Composite conservatism factor, f 3

Composite conservatism factor, f 3

8

a0/t=0.2

7

a0/t=0.4 a0/t=0.6

6

a0/t=0.7

5 4 3 2 1

0.2

0.4

0.6

0.8

1.0

8

a0/t=0.1

a0/t=0.2

a0/t=0.6

a0/t=0.7

a0/t=0.4

7 6 5 4 3 2 1

p = 7.5MPa

0.2

0.4

0.6

a0/c0

a0/c0

(a)

(b)

0.8

1.0

Fig. 15. The conservatism factor f3 for semi-elliptical surface cracks with various initial sizes in pressurized pipes with different pressure loads: (a) p = 15 MPa and (b) p = 7.5 MPa. 14

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Table 2 The CCG life data from two-parameter C*-Ac and single-parameter C* with FEM solution of C* for shallower cracks in pressurized pipes with a pressure load p = 15 MPa. Initial crack sizes

a0/t0 = 0.1, a0/t0 = 0.1, a0/t0 = 0.1, a0/t0 = 0.1, a0/t0 = 0.1, a0/t0 = 0.2, a0/t0 = 0.2, a0/t0 = 0.2, a0/t0 = 0.2, a0/t0 = 0.2, a0/t0 = 0.4, a0/t0 = 0.4, a0/t0 = 0.4, a0/t0 = 0.4, a0/t0 = 0.4,

a0/c0 = 0.2 a0/c0 = 0.4 a0/c0 = 0.6 a0/c0 = 0.8 a0/c0 = 1.0 a0/c0 = 0.2 a0/c0 = 0.4 a0/c0 = 0.6 a0/c0 = 0.8 a0/c0 = 1.0 a0/c0 = 0.2 a0/c0 = 0.4 a0/c0 = 0.6 a0/c0 = 0.8 a0/c0 = 1.0

Creep life tf (year) Single-parameter C*

Two-parameter C*-Ac

12.55 12.81 14.53 15.86 17.29 4.55 6.91 8.55 9.74 10.74 1.68 2.74 3.77 4.61 5.36

21.59 24.79 28.36 32.23 36.04 6.49 12.08 15.55 18.63 20.15 2.10 4.00 6.06 7.53 8.60

f1

f2

f3

1.72 1.94 1.95 2.03 2.08 1.43 1.75 1.82 1.91 1.88 1.25 1.46 1.61 1.63 1.60

2.59 2.14 2.12 2.27 2.29 1.96 1.87 1.93 2.04 1.93 2.09 1.95 1.86 1.84 1.73

4.53 4.08 4.18 4.46 4.77 2.79 3.32 3.52 3.76 3.77 2.41 2.53 3.05 3.03 2.96

t = 0.1 and a0/c0 = 1, the CCG life with considering constraint is 36.04 years, and that without considering constraint is only 17.29 years. If the C* from RS calculation is used in the single-parameter assessment, the CCG life will be only 7.55 years (17.29/ f2 = 17.29/2.29 = 7.55). This implies that if the conventional CCG life assessment method is used, the life will be significantly underestimated by 4.77 time (f3 = 4.77). Table 2 also shows that the factor f1 is more sensitive to a0/t than to a0/c0. This suggests that for initial shallower cracks, the effect of constraint on life is more significant and it should be given more consideration. The initial crack size has less effect on f2, and for different crack sizes, the average value of f2 is about 2. This shows that the RS calculation of C* will underestimate the CCG life by about 2 times. The high-temperature components usually work at lower load and have shallower and shorter surface cracks. The conventional CCG life assessments based on the single-parameter and the RS calculation of C* may produce over-conservative results. Therefore, in the CCG life assessments for high-temperature components, it is highly suggested that the constraint effect is considered and the FEM solution of C* is used. In this work, the CCG life assessment procedure using the two-parameter C*-Ac approach has been investigated and given for the cracked pipes. In principle, this procedure may be used for other cracked components if the FEM solutions of the parameters Ac and C* are available and the constraint-dependent CCG rate of materials can be established. This may need to be further studied. 4.4. Discussion on validation of two-parameter C*-Ac approach The two-parameter C*-Ac approach for CCG life assessment of pressurized pipes may need to be validated by testing the cracked pipes. The CCG tests of large-scale pipes (such as the pressurized pipes in this work) at high temperature are very difficult and almost no studies can be found in the literature. In addition, for the validation of two-parameter C*-Ac approach with considering constraint effect, the cracked pipes with shallower cracks and lower load level C* (with significant constraint effect) should be tested. Under these conditions, the creep crack initiation (CCI) and CCG will take very long time. Such very long-term CCI and CCG test for largescale pipes at high temperature is a very big challenge and almost impossible. Fortunately, with the developments of computing technology and creep damage model, the FEM simulation based on creep damage model may provide a virtual test for the long-term CCI and CCG behavior in large-scale cracked components. In a previous study of authors [28], the CCG rates in C(T) specimens of the Cr-Mo-V steel used in this work were simulated over a wide range of C* by FEM based on creep ductility exhaustion model. The simulated CCG rates agree well with experimental data, thus the simulation method has been validated. In the CCG simulation based on creep ductility exhaustion model, the crack growth is controlled by local equivalent creep strain rate and triaxial stress at the crack tip, and the effects of crack-tip constraint (stress state) on CCG rate have been incorporated. In a recent study of authors [35], the CCI time and location of axial semi-ellipse surface cracks with different sizes (different constraint levels) in cracked pipes of the Cr-Mo-V steel have been comparatively predicted by the FEM simulation based on creep ductility exhaustion model and the two-parameter C*Ac approach. The CCI time and location predicted by the FEM simulation are basically consistent with those predicted by the twoparameter C*-Ac approach. The prediction capability of the two-parameter C*-Ac approach to a certain extent may have been validated by this result. Because the CCI time is usually defined for a small creep crack extension of 0.2 mm or 0.5 mm [35], the CCG rate in cracked pipes may also be reasonably predicted by the FEM simulation. To further validate the CCG prediction capability of the two-parameter C*-Ac approach for cracked pipes, we are doing work on the CCG simulation in cracked pipes with different initial crack sizes. It needs a long time to calculate for longer creep crack growth in large-scale pressurized pipes. In addition, the two-parameter C*-Ac approach may also be validated by a comparison between the CCG prediction data and experimental data of laboratory specimens with different geometries (different constraint levels). This comparison has been done for different specimen geometries of 316H steel in one of our studies [52]. The predicted CCG rate by using the two-parameter C*-Ac for 15

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each specimen agrees well with the experimental data. This result may also provide validation for the two-parameter approach. More validation work may need to be done in further work for cracked components and specimens and different high-temperature materials. 5. Conclusions In this work, the methodology of creep crack growth life assessment based on two-parameter C*-Ac approach for pressurized pipes with a wide range of initial crack sizes have been studied and given. The assessment results of CCG life and crack size/shape evolution based on the two-parameter C*-Ac approach have been quantitatively analyzed and compared with those based on the conventional single-parameter C* approach. The effect of the calculation method of C* on the CCG life assessments also have been analyzed. The main results obtained are as follows. (1) Compared with the two-parameter C*-Ac assessments, the CCG life predicted by single-parameter C* approach may be overly conservative for shallower and shorter surface cracks in pressurized pipes. (2) The analyses of crack size and shape evolution show that the crack growth length and area predicted from two-parameter C*-Ac approach are smaller than those from single-parameter C* approach due to the lower constraint at the surface point along the crack front. The conventional single-parameter C* approach may overestimate the crack growth length and area. (3) Compared with the FEM calculation of C*, the reference stress (RS) calculation of C* will produce conservative life predication results. The conservatism degree is related to initial crack sizes and internal pressure loads. For shallower cracks and lower loads, the conservatism degree is higher. The C* calculation method almost has no effect on the crack size and shape evolution during crack growth process. (4) The conventional CCG life assessments based on the single-parameter C* approach and the RS calculation of C* for shallower and shorter surface cracks with lower constraint in cracked pipes may produce very high overall conservatism degree. Therefore, in the CCG life assessments for these cracks, it is highly suggested that the constraint effect is considered and the FEM solution of C* is used. (5) More validation work for the two-parameter C*-Ac approach may need to be done in further work for cracked components and specimens and different high-temperature materials. Declaration of Competing Interest The authors declared that there is no conflict of interest. Acknowledgements This work was financially supported by the Projects of the National Key Research Program (2018YFC080880), the Projects of the National Natural Science Foundation of China (51575184 and 51975212) and the Fundamental Research Funds for the Central Universities (50321071918005). 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