A unified creep interaction factor to characterize multiple cracks interaction at elevated temperatures

Engineering Fracture Mechanics 223 (2020) 106786

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A unified creep interaction factor to characterize multiple cracks interaction at elevated temperatures

T

⁎

Lianyong Xua,b, Lei Zhaoa,b, , Zhifang Gaoc, Yongdian Hana,b, Hongyang Jinga,b a b c

School of Materials Science and Engineering, Tianjin University, Tianjin 300072, China Tianjin Key Laboratory of Advanced Joining Technology, Tianjin 300072, China School of Materials Science and Engineering, Tianjin University of Technology, Tianjin, China

A R T IC LE I N F O

ABS TRA CT

Keywords: Multiple surface cracks Unified creep interaction factor Equivalent creep strain Combination criteria

Multiple cracks are inevitable to simultaneously occur in components, the growth and coalescence of which greatly reduced loading capacity of defected components at elevated temperatures. Considering the driving force of crack growth determined by the equivalent creep strain ahead of crack front, this paper proposed a unified creep interaction factor based on the ratio of the equivalent creep strain area between multiple cracks and single crack at the same creep strain isoline. This proposed interaction factor successfully demonstrated the interaction among coplanar cracks, eliminated the dependence of position along the crack front and retain the development trend of the maximum creep interaction. The proposed unified factor was greatly influenced by crack depth and creep hardening exponent, besides crack distance. Moreover, through the analyses of the variation of the crack growth rate in the cases of multiple cracks or single crack, the combination rule of multiple cracks could be determined by the value of the interaction level and thus the criteria for combination of interacting multiple cracks was proposed.

1. Introduction The development of fracture mechanics approaches and structural integrity codes is stimulated by the crucial needs to assure and evaluate the reliability of pressure vessels and piping components with defects. These defects usually initiated and propagated during the production or installation stage of engineering components or were caused by the continuous operation with high temperature, cyclic temperature or load across the thickness and microstructure deteriorations [1,2]. Specially, more than one defects simultaneously exist in welded components or steam pipe lines, nuclear pipe lines, off shore pipe lines, aircrafts, and pressure vessels. Many theoretical evidences and experimental data have revealed that multiple cracks could change the coalescence of interacting cracks and accelerate the crack growth rate in comparison with single crack growing individually and independently [3,4]. The interacting multiple cracks would change crack growth directions, crack displacement modes and in turn potentially reduce the damage capacity of defected components [5–7]. Hence, developing a comprehensive understanding of the effect induced by multiple cracks on the fracture behavior of materials plays a crucial role in the structural integrity. In the cases of two or more cracks in close proximity, the possibility of the interacting cracks on the enhanced crack growth rate must be considered for defected components during the safety or reliability assessment. Now, the guidelines for conducting the failure assessment for the defected components with more than one cracks have been established in the current engineering design codes

⁎

Corresponding author. E-mail address: [email protected] (L. Zhao).

https://doi.org/10.1016/j.engfracmech.2019.106786 Received 22 August 2019; Received in revised form 14 November 2019; Accepted 17 November 2019 Available online 19 November 2019 0013-7944/ © 2019 Elsevier Ltd. All rights reserved.

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[8–10], which also recommend the flaw characterization approaches of multiple coplanar cracks. These codes are only based on the adjacent distance between multiple cracks. When the relative spacing between the adjacent cracks is smaller than the specific value, these cracks are assumed to be interacting cracks and then should be recharacterized as a new planar crack, the dimensions of which would envelope interacting cracks on the projected plane. Experimental evidences and numerical efforts examining the influence of double cracks in cracked plate models had revealed that the crack interaction was related to crack orientations, crack type, crack sizes and crack locations [11–15]. Tan and Chen [16] and Chang [17] proposed a single semi-elliptical crack with a fracture area equivalent to the sum of the fracture areas of adjacent multiple cracks to recharacterize the interacting cracks. Xuan et al. [18] modified the crack interaction criterion in the creep regimes based on the numerical computations of the references stress for defected components. Dai et al. [19] analyzed the influences of different creep exponents and mismatching factors on the creep interacting effect using the estimation of the C* integral with modified reference stress. Xu et al. and Zhao et al. [7,20–22] used a new big crack multiplied with the creep intensity factor of double cracks to replace the interacting double cracks. Hence, more and more researches were conducted to quantitatively describe the interaction level of multiple cracks using the fracture parameter and establish an appropriate combination rule on the basis of the value of the fracture parameter. The crack driving forces quantified by K, J-integral, δCTOD, T-stress and C*-integral were computed from numerical analyses to measure the crack interaction phenomena [12,14,18,23–25]. Moreover, these literatures reported that the combination rules in engineering design codes were conservative if the multiple flaws were evaluated as to whether they were combined or not on the basis of the relative distance of double cracks. Especially, the crack driving forces quantified by K, J-integral, δCTOD, T-stress and C*-integral were significantly affected by the position along the crack front. The level of the crack driving forces dominates whether multiple cracks are combined or not and the established combination rule may overestimate the critical distance for cracks interaction. In order to describe the whole level of multiple cracks interacting phenomena, a new interaction factor is required, which could eliminate the dependence of the crack position on the interaction factors. Seah and Qian [26] proposed a crack interaction factor that was determined by the integrated opening stress over a finite zone near the crack tip, to characterize the variated opening stress of the crack in the cases of multiple cracks. In addition, the constraint level of multiple cracks could be represented by this new interaction factor. Because the interacting multiple cracks would affect the stress distribution and then cause the crack initiation and growth behavior different from the single crack. This can be attributed to the variation of constraint along the crack front, either. In pure creep regime, two-parameter fracture mechanics has been introduced, such as C* − Q [27–29], C* − Q* [30], C* − A2 or C(t) − A2(t) [31–33], C* − R [34], C* − R* [35], Ac [36] and C* − Tz [37], where the first parameter C* integral is employed to characterize the mechanical field in front of crack tips and the secondary parameters are introduced to represent the crack-tip constraint level in creep regimes. Dai et al. [32,33] had proposed a higher order asymptotic analysis with separable variable form to represent stress filed for crack tip under mixed mode creep conditions and then modified time dependent failure assessment diagram for well demonstrating the effect of constraint on the creep crack growth behavior based on the C* − A2 or C(t) − A2(t) theory. Mostafavi et al. [38,39] had proposed a unified constraint parameter as a function of the plastic region size at fracture between the cracked sample and a standard testing sample. This is defined as follows

φ=

As Assy

(1)

where As is the plastic region at fracture and Assy is the reference plastic region at the fracture for a standard specimen. On the basis of this, Yang et al. [40,41] modified and proposed a new parameter Ap to eliminate the limitations for higher fracture toughness material, as:

Ap =

APEEQ Aref

(2)

where APEEQ is employed to represent the area surrounded by the equivalent plastic strain (εp) isoline ahead of a crack tip and Aref is the reference area surrounded by the εp isoline in a standard contact tension (CT) specimen with high constraint. Analogous to Ap in the elastic-plastic regimes, Ma et al. [42–44] established a unified parameter Ac in the creep regimes, which was as function of the equivalent creep strain area ACEEQ ahead of the crack tip in any specimen or components normalized by the equivalent creep strain area Aref in a CT specimen. This is computed as follows:

Ac =

ACEEQ at t Aref

tred = 1

(3)

where tred denotes the redistribution time. Because finite element software only provides the calculation of Ct values. As the creep time increases more than tred, the Ct value changes slightly and then this value is assumed to be C* values. It was recommended that for comparison Ac was computed at the same equivalent creep strain εc isoline, the same creep time t/tred = 1 and C* level. Aref means the equivalent creep strain area ACEEQ ahead of crack tip in a standard CT specimen. In the creep regimes, a majority experimental and theoretical evidences [30,35,45–49] had revealed that the creep crack propagation rate was greatly affected by the constraint level ahead of the crack-tip. The unified constraint parameter normalized by the standard CT specimen with high constraint can be correlated with the crack growth behavior. Similarly to unified constraint parameter Ac at elevated temperatures, a unified creep interaction factor is proposed to demonstrate the intensity caused by multiple cracks in the creep regimes, which is computed as a ratio of ACEEQ in the case of multiple cracks and ACEEQ in the presence of single crack at the same equivalent creep strain εc isoline to quantify the variation of crack driving forces. 2

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Fig. 1. Schematic of the plate with double semi-elliptical surface cracks (a) the whole model and (b) details of double semi-elliptical surface cracks.

This is illustrated detailed in Section 3. In order to understand the application of the unified proposed creep interaction factor in the creep regimes, the comprehensive numerical analyses of coplanar double cracks in flat plate model exposed to remote tension load were conducted. Subsequently, the criteria of the unified proposed creep interaction factor for recharacterizing the interacting multiple cracks was studied. 2. Finite element model and numerical procedures 2.1. Specimen geometries In current standards, in the cases of more than one defects, the combination rule for multiple cracks is only applied to adjacent cracks. In addition, the highest crack intensity factor quantified by the ratio of the crack driving force between multiple cracks and single crack occurs in the coplanar cracks with the same sizes [26]. Hence, a finite thickness plate with two identical coplanar surface cracks with a semi-elliptical shape is used to investigate the interaction behavior, as shown in Fig. 1. The cracks are assumed to be located in the center plane of the plate (see Fig. 1a). a and c respectively represents the half-length and depth of the crack. s demonstrates the spacing between adjacent two cracks. t is the thickness of the plate. The semi-elliptical cracks have a length 2c of 20 mm. The plate length 2W and the plate height L equal to 10 times the crack length 2c, which is sufficiently large to remove the boundary effects on the stress and strain field ahead of crack tip [14,50]. The other dimensions are all calculated according to the designed ratios. To investigate the influence of the crack dimensions on the interaction factor, various dimensionless parameters are set: crack shape ratios denoted by a/c (0.2, 0.4, 0.6, 0.8 and 1.0), crack depth ratios represented by a/t (0.2, 0.4, 0.6, 0.8) and relative crack distance ratios denoted by s/c (0.5, 1, 2, 3). 2.2. Finite element model Considering the loading and geometry symmetry, a quarter of the cracked plate was modeled in ABAQUS software, as detailed shown in Fig. 2a, which utilized eight-node 3-D reduced elements (C3D8R) in a range of 28676–58076 and nodes in a range of 32012–62284. The remote tensile load was applied to the end of the cracked plate. The symmetry boundary conditions were exerted on the un-crack ligament and the center plane of the cracked plate. The mesh near the crack front was refined with the finest element 3

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Fig. 2. Illustrations of the employed FE model and mesh (a) the whole model (b) the refine mesh of the crack front region.

size of 0.050 mm due to that the steep gradients of stress and strain fields were located in this zone, as illustrated in Fig. 3b. The areas surrounded by the equivalent creep strain isoline ahead of the crack front were computed to demonstrate the intensity factor of multiple cracks in the creep regimes. The adjacent regions were meshed with a gradually coarse mesh for reducing computational time. The rest regions are meshed with element sizes ranged from 0.050 mm to 1 mm at the centerplane of the plate. As the employed finest mesh size was smaller than 0.050 mm, the computed stress and strain filed from fine element (FE) method, Ct and C* values near the crack front changed slightly with the element sizes. The Ct integral was computed using domain integral approach. Five different rings were used to compute the Ct integral and the average values of 2nd to 5th contours were chosen to determine the C*values along the crack front, where the Ct integral converged to within 1% variation among adjacent rings of elements.

2.3. Material properties In this paper, a heat resistant ASME SA335 P92 steel is chosen. Considering high creep resistance and good oxidation resistance at the temperature over 600 °C, ASME SA335 P92 steel has been widely used to manufacture the main steam pipes, reheaters and headers with large thickness in Ultra Super Critical (USC) Power Plants. For computing the Ct and C* along the crack front, a power law creep model is adopted to compute the accumulation of the creep deformation as follows:

εċ = Aσ n

(4)

where εċ denotes the creep strain rate; A is the creep constant in power law creep model; n is the creep hardening exponent. A and n are dependent on the material and temperature. The variation of the temperature would affect the creep deformation, so the interaction behavior of multiple cracks would be influenced, either. In order to obtain a wide range of useful results, the materials properties employed in this paper are listed in Table 1. It can be noted that considering the influence of the temperature, the mechanical and creep properties of P92 steel at different temperatures are provided. It is assumed that the temperature only affect the creep hardening exponent of P92 steel [47].

Fig. 3. Dependence of creep interaction factor on the position along the crack front (a) different relative distance between the two cracks and (b) different crack depths. 4

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Table 1 Mechanical and creep properties of P92 steel at different temperatures. Temperature (°C)

E (MPa)

v

B (MPa-nh−1)

n

εf

550 600 650 700 750

160,000 140,000 125,000 75,000 6000

0.3 0.3 0.3 0.3 0.3

3.5085E-32 3.5085E-32 3.5085E-32 3.5085E-32 3.5085E-32

5 8 12.72 16 20

0.2 0.2 0.2 0.2 0.2

2.4. Creep damage model and creep crack growth simulation The crack growth behavior of a cracked structure under tensile loading can be analyzed through the damage model. In this section, the creep crack growth behaviors in the presence of multiple cracks with different crack configurations are simulated and compared with single crack growing independently. The same FE models as Fig. 2(a) and (b) are employed to conduct the crack growth behavior simulations. The creep damage model related to the creep ductility exhaustion is used, where the accumulation damage during the creep process at any time could be computed from the time integral of the creep damage accumulation rate. The creep damage accumulation rate ω̇ is correlated with the function of the equivalent creep strain rate εċ and multiaxial creep ductility ε ∗f . This is defined as:

ω=

ε̇ ̇ = ∫ ∗c dt ∫ ωdt ε

(5)

f

The creep ductility in a material is influenced by the variation of the stress state during the creep deformation, which is usually correlated with the stress triaxiality (σm σe : σm the hydrostatic stress and σe the equivalent Mises stress) [51]. Many researchers, such as Cock and Ashby [52], Rice and Tracey [53], Spindler [54] and Nikbin [55] all proposed many models to describe the influence of the triaxial stress state on the reduction of the creep ductility. Previous studies had revealed that the material deteriorations in P92 steel or welded joint is caused by the nucleation, grow and aggregation of creep voids along the grain boundaries, which dominates the creep damage accumulation [56]. Thus, the Cocks and Ashby model based on the void growth theory with a power law creep deformation model is employed, which is computed from this expression [52]:

ε ∗f =

εf sinh[2(n − 0.5) 3(n + 0.5)] sinh[2σm (n − 0.5) σe (n + 0.5)]

(6)

where εf denotes the uniaxial failure strain and is determined from the conventional uniaxial creep tests. This damage model has been extensively and successfully employed in the simulations of the creep crack growth behavior and creep crack growth properties and the simulated results matched well with the experimental data to a large extent [57,58]. The virtual creep crack growth is simulated using the element removal approach. When the computed time increases, the accumulated creep damage ω increases from 0 to 1. As the damage ω exceeds to the set critical damage value (0.999), the belonging element is considered to be failed. The corresponding Elastic modulus of this element is dropped to a very small value of 1 MPa, which in turn leads to the loading capacity of this element to be dropped to a very small value for simulating the failed elements. Then, the crack front propagates to the next adjacent element and an update crack front forms. This failure simulation approach is implemented into user subroutines CREEP and USDFLD of ABAQUS. 3. Definition of the unified creep interaction factor For characterizing the interaction intensity caused by double cracks, the interaction value is computed by the magnification of the stress intensity factor, which is defined as [12,23]:

γElastic =

KDouble K Single

(7)

KDouble and K Single are the stress intensity factor in the cases of two cracks and one crack, respectively. For comparison, it is recommended that they should be computed at the same location in specimens exerted with the equivalent remote tensile loads. In the creep regimes, the parameter C* is extensively employed to characterize the stress and strain field ahead of crack tip [30,35,46,59]. The C* values are as a function of the stress intensity factor K, the reference stress σref and the corresponding creep ̇ c in cracked components [9]: deformation rate εref 2

K ⎞ ̇ c ⎛⎜ C ∗ = σref εref ⎟ σ ⎝ ref ⎠

(8)

Analogous to the stress intensity factor employed in the elastic regimes (see Eq. (7)), the interaction factor in the creep regimes can be computed by [14,18]: 5

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γCreep = ∗ CDouble

∗ CDouble ∗ CSingle

(9)

∗ CSingle

and represent the integral C* values in the presence of two cracks and one crack. Noting that they should be where calculated at the same location and same loading condition in specimens. In previous studies, it has been suggested that the creep interaction factor represented by γCreep is heavily affected by the position on the crack front, as shown in Fig. 3. The position along the crack front is identified by the parameter θ as shown in Fig. 1(b). For normalization, the crack position is represented by 2θ/π in the range of 0 to 2. Shi et al. [14] and Xuan et al. [25] also reported similar phenomenon. The highest value of the creep interaction factor is prone to occur near the inner surface point (2θ/π = 0). This means that employing the maximum creep interaction factor as a combination rule would overestimate the crack growth tendency in the case of two cracks and provide a much conservative prediction. For eliminating this, it is required to employ a unified interaction factor to represent the whole stress and strain filed ahead of crack tip. Yang et al. [40,41] and Ma et al. [42–44] respectively correlated the constraint level with the deformation area of cracked components. Particularly, Ma et al. [42–44] had successfully employed the parameter Ac to correlate the change of the crack growth behavior for various specimen geometries, pressurized pipes at elevated temperatures. They [42–44] also revealed that the creep crack propagation rate was a function of the unified constraint parameter.

ȧ = ȧ0 f (Ac )

(10)

where ȧ denotes the creep crack propagation rate; ȧ0 is the creep crack propagation rate obtained from the CT specimen, as well known with high constraint level; f (Ac ) is employed to demonstrate the role of the unified constraint level Ac on the crack growth behavior. This demonstrates that the creep crack growth rate is related to the C* values and also is affected by the level of the creep deformation ahead of crack tip. In the cases of two cracks, the crack growth rate is accelerated, which is controlled by the increase of the C* accompanied by the increase of the creep strain along the crack front. The variation of the creep deformation region at the same creep strain isoline could be used to represent the intensity induced by multiple cracks. Hence, analogous to the definition of the unified constraint parameter Ac as a function of the normalized equivalent creep strain region ahead of crack tip [42–44], a unified A creep interaction factor γCreep for multiple cracks is proposed and is herein defined as: (ACEEQ )double A γCreep

=

Aref (ACEEQ )single Aref

=

(ACEEQ )double (ACEEQ )single

at t

tred = 1 (11)

A γCreep

where denotes the intensity factor induced by multiple cracks; (ACEEQ )double and (ACEEQ )single means the area of the equivalent creep strain along the crack front at the same creep strain isoline for double cracks and single cracks subjected to same remote tensile load, respectively. Moreover, the difference of the constraint along the crack front induced by creep deformation or crack types could A could reflect the role of the be considered through the new proposed creep interaction factor. In contrast, the employment of γCreep multiple cracks interacting level on the whole and then could better represent the crack growth tendency in the cases of two cracks. Because at a same specific creep strain isoline, there is only a certain value of the creep deformation area along the crack tip. So, for A one case of multiple cracks, there is only one certain value of unified factor γCreep along the crack tip, unlike the conventional creep interaction factor distribution in Fig. 3. In this way, the position dependence can be eliminated using the unified creep interaction factor. The profiles of the equivalent creep strain εc isoline in the cases of two cracks and single crack are compared in Fig. 4. The red region means the equivalent creep strain area at the specific εc isoline. The area of these creep strain regions can be calculated by the image processing software. It can be clearly noted that the shapes of the equivalent creep strain region in two cases are similar. The area of the equivalent creep strain region in the presence of multiple cracks at the same creep strain εc isoline is larger than the area of single crack, which reflects the interaction behavior of multiple cracks. Moreover, the deformation area near the inner surface of crack is greatly enlarged, which represents the stimulated interaction. This also causes the change of the interaction factor and the

Fig. 4. Illustration of the variation of equivalent creep strain area in the presence of multiple cracks and single cracks at different εc isoline (a) εc = 0.004 and (b) εc = 0.006. 6

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Fig. 5. Dependence of creep interaction factor on the chosen εc isoline values.

dependence on the position of the crack front (see Fig. 3). In addition, with the changing of the isoline, the equivalent creep strain A area increases both in single crack and multiple cracks. The variation of γCreep against the chosen value of εc isoline is shown in Fig. 5. A The value of γCreep changes slightly with the value of εc isoline increasing. Because the area of the equivalent strain for double cracks A and single crack proportionally enhances as stated in Fig. 4b. Hence, the value of γCreep seems independent of the chosen value of εc isoline. This had been stated in literatures about the unified constraint parameter related to the deformation area [42,43,59]. For comparison, a same value of creep strain isoline is used in this paper.

4. Application of the unified creep interaction factor 4.1. Influence of crack distance ratio A against The relative crack distance ratio between the two flaws significantly affects interaction intensity. The variation of the γCreep the crack distance ratio is plotted in Fig. 6. When the two cracks are gradually away from each other, the new proposed creep interaction factor exhibits a non-linear decrease trend, which is similar with the reduction of the γCreep reported in literatures [7,14,20]. As the cracks approach other, the creep interaction intensity would steeply increase. This is due to that as the multiples crack meet each other, the mutual interaction among the two cracks is strong and significantly improves the stress value ahead of the crack front. This can be reflected by the improvement of the equivalent creep strain area ahead of the crack front when the spacing between two cracks decreases (see Fig. 7). The red region representing the area of the equivalent creep strain εc isoline increases as the cracks gradually meet to each other. In turn, this accelerates the accumulation of creep damage and then stimulates the rates of crack growth. This is reason for the current standards employing the crack distance as the combination rule. Moreover, as the relative A distance increases, γCreep gradually decreases. Particularly, if the relative distance between the two cracks is twice higher than crack length, the reduction rate of the creep interaction factor is reduced. When the cracks are away from each other, the mutual interaction of multiple cracks has slight effect on the stress field. This is certified in Fig. 7, as the relative crack distance exceeds 2c, the equivalent creep strain region at the same creep strain isoline changes

Fig. 6. Influence of crack distance ratio on the unified creep interaction factor (a) a/t = 0.4 (shallow crack) and (b) a/t = 0.8 (deep crack). 7

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Fig. 7. Variation of the equivalent creep strain area against the crack distance for multiple cracks with a/t = 0.2 and a/c = 1.0 (a) s/c = 0.5, (b) s/ c = 1.0, (c) s/c = 2.0 and (d) s/c = 3.0.

slightly. However, this reduction trend is influenced by crack depths. When a deep crack occurs (see Fig. 6b), there is still a high A reduction rate of γCreep . The variation of the creep interaction intensity phenomenon for multiple cracks is well represented by the new A proposed creep interaction γCreep .

A and crack shape ratio 4.2. Relation between γCreep A with the crack shape ratio. When the crack changes from a semi-ellipse (a/c = 0.2) to a Fig. 8 shows the development of γCreep A semi-circular (a/c = 1.0), there exists a increasing trend of γCreep . For the case of shallow crack depth, the creep interaction factor A changes slightly [7]. When the crack depth ratio becomes higher than 0.2, the increase trend of γCreep against the crack shape ratio becomes more obviously. Specially, when the crack shape ratio increases than 0.8, the creep interaction factor changes slightly, even a slight reduction. This means that the influence of the crack shape ratio on the creep interaction factor is diminished as a/c is higher than 0.8. The increase trend is influenced by the crack depth. When investigating the intensity factor for cracked components under tension or complex loading conditions in the creep regimes, Si et al. [14] and Xu et al. [21] found that the interaction effect with the crack shape ratio is different at three locations of the crack front (inner surface point 2θ/π = 0, deepest point 2θ/π = 1 and outer surface point 2θ/π = 2). For the locations at the inner surface point or deepest point, the creep interaction factor γCreep firstly increases to the maximum value and then as the crack shape changes there is reduction tendency. In contrast, for the location at the outer surface point, the creep interaction factor γCreep only owns an increase development, which is different from the other two positions. However, using the proposed interaction factor correlated with the equivalent creep strain area ahead of crack tip not only eliminates the dependence of the position on the crack front, but also retains the development trend of the maximum intensity factor.

8

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Fig. 8. Relationship between crack depth ratio against creep interaction factor at different crack distance ratios (a) s/c = 0.5 and (b) s/c = 1.0. A 4.3. Relation between γCreep and crack depth ratio A A . It is worth noting that γCreep increases Fig. 9 depicts the influence of the crack depth against the proposed interaction factor γCreep A with increasing the crack depth. This is same with γCreep observed in literatures. The highest of the creep interaction factor γCreep occurs at the plates with the deepest crack. The increase trend is independent of the crack distance. Moreover, the increase rate of the creep A interaction factor γCreep is accelerated as the crack depth ratio is higher than 0.6. The interacting intensity denoted by the proposed interaction factor of the plate with a/t = 0.8 is three times more than that of a shallow crack when the s/c is 0.5. When the cracks are far away from each other, the discrepancy is only about 1.6 times.

Fig. 9. Influence of crack depth ratio on the creep interaction factor at different crack distance ratios (a) s/c = 0.5, (b) s/c = 1.0, (d) s/c = 2.0 and (d) s/c = 3.0. 9

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Fig. 10. (a) Relationship between creep interaction factor and crack distance ratio (b) Relationship between creep interaction factor and creep hardening exponent. A 4.4. Relation between γCreep and creep hardening exponent n

At high temperature, the creep deformation is controlled by the material type and the temperature, which can be demonstrated by the variation of the creep hardening exponent n. Five different creep hardening exponents are examined in Table 1. The role of the A creep hardening exponent n on the interaction factor γCreep is illustrated in Fig. 10. Under different creep hardening exponents, the A A variation of γCreep with the crack distance ratio is similar, which means that γCreep could well demonstrate the influence of the creep A at the nearest crack distance hardening exponent on the interaction intensity of multiple cracks. The creep interaction factor γCreep increases from 1.69 to 4.49, approximately 2.65 times, when the creep hardening exponent increases from 5 to 20. Because the creep deformation is significantly relied on the creep hardening exponent. A high n produces large creep deformation as the external load is A same. The larger creep deformation causes a stronger interaction of multiple cracks. When γCreep is plotted with the creep hardening exponent in Fig. 10b, it is worth noting that the creep interaction factor would increase non-linearly with the creep hardening exponent [25]. Whereas the increasing rate of the creep interaction factor is small, as the creep hardening exponent is less than 12. Especially, as the crack distance exceeds twice of the crack length, the creep interaction factor nearly keeps constant [7,18]. The increase of the creep interaction factor accelerates when the creep hardening exponent exceeds this transition point.

5. Discussions As described, the new proposed creep interaction factor related to the normalized equivalent creep strain area ahead of crack tip would well reflect the intensity phenomena caused by multiple cracks, eliminate the dependence of the interaction factor difference on the crack front position and retain the development trend of the most dangerous crack interactions. Moreover, it can be noted that besides the relative spacing between the two cracks greatly affects the creep interaction factor, leads to the coalescence of cracks and accelerates the crack growth behavior, the crack depth and the creep hardening exponent could obviously stimulate the interaction factor. Moreover, the influence of the crack shape ratio is weaker than other factors. Particularly, there exists non-linear relationships A between γCreep and these factors. For the engineering issues, a closed empirical solution is required to evaluate the interaction intensity in the cases of more than one cracks at elevated temperatures and provide recommends for structural integrity of cracked components. Hence, the proposed creep interaction factor in the presence of more than one cracks is recommend to be computed by the following empirical equation [3]:

a A3 s A4 a A5 A γCreep = A1 + A2 ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ n A6 ⎝ c ⎠ ⎝c⎠ ⎝ t ⎠

(12)

Table 2 Coefficients of empirical equation for estimating the unified creep interaction factor. Coefficients

Values

A1 A2 A3 A4 A5 A6

1.187 ± 0.076 0.0196 ± 1.47E-02 0.371 ± 8.99E-02 −0.745 ± 8.12E-02 2.860 ± 0.382 2.101 ± 0.307

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where A1–A6 are coefficients and could be calculated using non-linear fitting approach based on the FE results under different factors. For assuring 95% confidence, the value of coefficients for empirical equations are listed in Table 2. The calculations evaluated by the proposed closed-form solution match well with the corresponding FE calculations, where the highest discrepancy is less than 15%. This means that the creep interaction level for multiple cracks in practical application could be well demonstrated by the proposed empirical solution related to the normalized equivalent creep strain area. The current standards mainly use the crack distance between multiple cracks as the combination criteria. This is established by the analyses of the variation of the stress intensity factor for two cracks in plates shown in Fig. 1. In the elastic regimes, the driving force is controlled by the stress intensity factor ahead of crack tip and the interaction factor diminishes to approximately 1 as the A relative distance exceeds 2c. However, the creep interaction factor γCreep still owns a high value as the relative spacing between the two cracks becomes higher than 2c (see Fig. 4). Moreover, the intensity at high temperature is greatly affected by crack depth and material properties, especially in the cases of deep crack or high creep hardening exponent. The creep intensity factor nearly equals to 1 as the short crack exists, in contrast, it increases steeply as the crack approaches to the depth of the cracked flat plate. Even if the multiple crack interaction in the creep regimes was higher than the crack interaction in the elastic regimes at the same crack dimensions and loading conditions, Xuan et al. [25] had indicated that a higher value of the creep interaction factor would be permissible in comparison with the value specified in current linear-elastic fracture conditions. Xu et al. [22] reported that a more than 50% increment in crack growth rates under fatigue condition could be obtained as a 10% improvement of the stress intensity factor occurred. But, for the cases of the creep deformation, it required a 30% higher creep interaction factor to obtain the same improved rate of the creep crack growth. The virtual crack growth behavior in the presences of the two cracks in the flat plate model is conducted to reveal the relationship between the proposed creep interaction factor and the accelerated crack growth behavior. Considering that the crack growth rate of multiple cracks would be higher than the crack growth rate of single crack when subjecting to the same load and the same crack dimensions, a magnified ratio of the crack growth rate is defined to quantify this magnification effect:

β=

(da dt )double (da dt )single

(13)

where (da dt )double means the steady creep crack growth rate of the two cracks and (da dt )single represents the steady creep crack growth rate for a single crack at the same condition. Fig. 11 compares with the simulated crack growth contour and the experimental crack morphology. The simulated crack growth profiles for plates with double cracks are in a good agreement of experimental observation. The crack growth profile is similar and the highest crack growth length occurs near the inner surface point where exhibits the highest equivalent creep deformation (see Fig. 4). The variation of the creep interaction factor is plotted with the magnified ratio in Fig. 12. Different scatter factors at 1.5 and 2 are plotted to demonstrate the different confidence bands. The correlation between the magnification ratio and the new proposed creep interaction factor is limited in the scatter band of 1.5. This reflects a high confidence. There is a linear relationship between the proposed creep interaction factor and the magnified crack growth behavior in the log-log curve. As the creep interaction factor is approximately 4, the crack growth rate of multiple cracks is 10 times more than that of single crack at the same conditions, matching well with [25]. When the creep interaction factor reduces to approximately 1.2, the improvement of the crack growth rate is less than 150%. When the creep interaction factor approaches to 1, the accelerated creep crack growth rates of the two cracks are limited, approximately 1.1. This phenomenon means that the interaction on the crack growth behavior is slight. Traditionally, the multiple crack treatment rules in current standards have been constructed in terms of the acceptable increment of stress intensity factor due to the interaction. In the elastic regimes, the crack growth is mainly caused by the fatigue, the driving force of which is mainly determined by the stress intensity factor. The combination for multiple cracks is assessed by the increasing of the stress intensity factor, which could introduce an approximately 50% improvement of the crack growth rate. If the same rule is employed in the creep regimes, a limitation of the creep interaction factor could be obtained as shown in Fig. 12. The specific values of the unified creep interaction factor would be evaluated according to Eq. (12). The limitation value of the unified creep interaction factor is about 1.25, similar to reported Xu et al. [22] and Xuan et al. [25] literature. This means that as the unified creep interaction factor of multiple cracks in engineering components is higher than 1.25, the stimulated crack growth rate of multiple cracks could not be ignored. Then,

Fig. 11. Simulated crack growth profile in the presence of double cracks and experimental observations. 11

Engineering Fracture Mechanics 223 (2020) 106786

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Fig. 12. Relation between magnified creep crack growth ratio and new interaction factor.

the multiple cracks should be considered by the combination of the multiple cracks, rather than in the form of single crack. In contrast, as the interaction intensity is smaller than 1.25, the influence of the interacting cracks on the crack growth behavior is relatively small. Two cracks still can be evaluated as a single crack, respectively. 6. Conclusions (1) A unified creep interaction factor based on the equivalent creep isoline ahead of crack tip was proposed to eliminate the dependence the conventional creep interaction factor on the position along the crack tip and retain the development tendency of the maximum interaction intensity. (2) The variation of the unified creep interaction factor with crack configurations and material properties were investigated. The unified creep interaction behavior was greatly relied on the crack distance, crack depth and creep hardening exponent. As the relative crack distance met each other or the crack depth approached to the plate thickness or the creep hardening exponents become higher than 12, the increasing of the creep interaction factor was accelerated. In contrast, the crack shape ratio had slight effect on the creep interaction factor. (3) The variation of the creep crack growth rate in the presences of two cracks and single crack was conducted. There existed a linear relationship between the creep interaction factor and the magnified ratio of the crack growth rate in the presence of two cracks in log-log curve. (4) A criteria of creep interaction factor for recharacerizing the interacting multiple cracks was established on the basis of the analyses of the crack growth behavior. As the creep interaction factors exceeded the limit value 1.25, the cracks should be combined; otherwise, they should be considered as a single crack individually. Even, the creep interaction factor for crack components with different crack configurations could be estimated using an empirical equations considering the role of crack depth, crack distance, crack shape and material properties. Declaration of Competing Interest The authors declared that they have no conflicts of interest to this work. Acknowledgement This research work has been financially supported by National Key R&D Program of China (Grant No. 2018YFB0704000) . References: [1] Ab Razak N, Davies CM, Nikbin KM. Testing and assessment of cracking in P91 steels under creep-fatigue loading conditions. Eng Fail Anal 2018;84:320–30. [2] Chen G, Wang GZ, Zhang JW, Xuan FZ, Tu ST. Effects of initial crack positions and load levels on creep failure behavior in P92 steel welded joint. Eng Fail Anal 2015;47(Part A):56–66. [3] Judt PO, Ricoeur A. Crack growth simulation of multiple cracks systems applying remote contour interaction integrals. Theor Appl Fract Mech 2015;75:78–88. [4] Azizi S, Bagheri R. Mixed mode transient analysis of functionally graded piezoelectric plane weakened by multiple cracks. Theor Appl Fract Mech 2019;101:127–40. [5] Kamaya M. A crack growth evaluation method for interacting multiple cracks. JSME Int J 2003;46:15–23. [6] Xu L, Zhang X, Zhao L, Han Y, Jing H. Characterization of creep crack-tip constraint levels for pressurized pipelines with axial surface cracks. Adv Eng Softw 2017;134:63–74. [7] Zhao L, Guo W, Xu L, Han Y, Jing H. Evaluation of the multiple embedded cracks interaction effect in creep regime by creep damage method. Adv Eng Softw 2019;128:125–35.

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