Hybrid approach for calculation of J-R curve using R6

Engineering Fracture Mechanics 215 (2019) 16–35

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Hybrid approach for calculation of J-R curve using R6 MK. Sahua,b, , J. Chattopadhyaya,b, BK. Duttab ⁎

a b

T

Reactor Safety Division, Hall-7, Bhabha Atomic Research Centre, Trombay, Mumbai 400075, India Homi Bhabha National Institute, Training School Complex, Anushaktinagar, Mumbai 400094, India

ARTICLE INFO

ABSTRACT

Keywords: Fracture toughness R6 J-R curve Cracked pipe Cracked elbow

The fracture property, J-R curve is an essential material property for assessing the cracked structures for unstable ductile tearing. For obtaining this material property, fracture test is conducted and resultant test data namely load, load line displacement (LLD) and relevant crack extension data are post processed to obtain J-R curve. However, in this method, geometry functions and are essential which are available for only limited geometries. To avoid this limitation, recently two approaches based on R6 failure assessment diagram were proposed. First was the load based R6 approach where load vs. crack growth data is converted to fracture property J-R curve using R6 Failure Assessment Diagram (FAD). This method was further modified to displacement based R6 approach where LLD vs. crack extension data is converted to J-R curve using R6 FAD. The predicted J-R curves were in good agreement with conventionally calculated values especially for displacement based approach. However, when these load and displacement based approaches are attempted to be used in case of throughwall cracked elbows, the predicted J-R curve deviated significantly with respect to those obtained using conventional method. On further investigation, it is found that the assumption involved in the displacement based approach is violated for cracked elbows because of severe ovalization of cross section. Hence, a modified approach is proposed in this work where both load and displacement parameter are used for calculation of J-R curves and named as hybrid approach. Further, this approach is adopted for the six through wall cracked pipes which are already investigated by load and displacement based R6 approaches for J-R curves. The predicted J-R curves are found to be in good agreement for all the investigated pipes and elbows. Hence, a simpler yet more accurate hybrid approach based on R6 method is proposed in this paper for post processing the test data to obtain, the material J-R curve.

1. Introduction The fracture property J-R curve is an essential material property for assessment of the unstable ductile tearing of structures made of ductile material. This property is an important parameter for Leak Before Break (LBB) qualification of pressurized piping systems of PHWR and PWR type nuclear reactors. For obtaining material J-R curve, fracture test is usually conducted on the pre-cracked test specimen which gives experimental results like load, load line displacement (LLD) with relevant crack growth data. Conventionally, these experimental data are post processed to obtain material J-R curve using certain geometry functions and as per ASTM code E1820 [1]. These functions are available for very limited geometries only. Hence, the calculation of J-R curve for the geometries without these functions, is not possible using the conventional methodology. It is also established that the J-R curve depends

⁎

Corresponding author at: Reactor Safety Division, Hall-7, Bhabha Atomic Research Centre, Trombay, Mumbai 400075, India. E-mail address: [email protected] (M. Sahu).

https://doi.org/10.1016/j.engfracmech.2019.04.031 Received 19 February 2019; Received in revised form 15 April 2019; Accepted 23 April 2019 Available online 25 April 2019 0013-7944/ © 2019 Elsevier Ltd. All rights reserved.

Engineering Fracture Mechanics 215 (2019) 16–35

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Nomenclature

bend radius of an elbow mean radius of cross section wall thickness weakening factor due to crack on limit load of uncracked elbow a coefficient in Ramberg – Osgood stress-strain equation elastic strain e reference strain ref yield strain y half crack angle in circumferentially cracked elbow poisson’s ratio bending stress for uncracked pipe cross section b reference stress ref yield strength of material y ultimate strength of material u B d = tan 1 L angular dimension of an elbow d angular deformation of a cracked elbow applied load line displacement applied displacement corresponding to limit load L of load vs. load line displacement data k a factor for getting reference strain by multiplication with normalized angular deformation d / d a function used to multiply the area under load vs. load line deflection curve to get plastic J-integral a function used to get plastic J-integral under crack growth situation tR = 2b elbow factor

Rb Rm t X

crack size initial crack size crack growth height between loading point and mid cross section of elbow in the test configuration Do outer diameter of pipe/elbow E young’s modulus 2) E ' = E/(1 effective Young’s modulus H = B2 + L2 a length parameter of elbow test configuration J total J-integral (summation of elastic and plastic) Je elasticJ -integral Jmat material fracture toughness in plastically deformed regime KI stress intensity factor

a a0 a B

Kmat = Jmat E ' material fracture toughness Kr =KI /Kmat normalized crack driving force V moment arm length in the test configuration of an elbow Lr = P /PL normalized remote loading M0 limit moment for defect free elbow Mb bending moment ML limit bending moment n Ramberg-Osgood strain hardening exponent parameter P applied load P0.2 load corresponding to crack growth a=0.2 mm PL limit load

Rm

significantly on the test specimen geometries and loading configurations. Hence, the calculation of J-R curve of the exact practical case is important to avoid over-conservatism or under-conservatism in the design. To bypass the need of theses geometry functions and , Sahu et al. [2] proposed a simpler load based R6 method to post process the experimental load vs. crack growth data to obtain material J-R curve using R6 Failure Assessment Diagram (FAD). This was named as load based R6 approach. This procedure has been applied for calculation of J-R curves for total six pipes of two sizes, namely, 8 in. and 16 in. nominal diameters with different crack sizes. For these pipes, using fracture test data of load, LLD and relevant crack growth, J-R curves were calculated and compared with reported J-R curves obtained using conventional load-LLD integration approach using and functions by Chattopadhyay et al. [3]. The predicted J-R curves were found to be in good agreement with conventional values for three of total six investigated pipes. However, remaining three cases were showing significant deviation from conventional values. This observation inspired for the need of improvement in the proposed load based R6 approach. Based on limited literatures available [4,5], it is established that for fracture assessment of any cracked component under large plastic deformation, the applied displacement is better parameter instead of applied load. Thus, an improved displacement based R6 approach for calculation of J-R curve was proposed by Sahu et al. [6] instead of load based approach. In displacement based approach, experimental LLD vs. crack growth data was post processed to obtain J-R curve. Displacement based predictions were found to be in better agreement with conventionally calculated J-R curves than the predictions of load based approach. For more validation of the displacement based R6 approach, it is realized that this method should be tested with other geometries and/or loading configurations. As a part of comprehensive Component Integrity Test Program in Bhabha Atomic Research Centre (BARC), fracture tests were carried out on a large number of cracked pipes and elbows of various diametrical sizes having different crack sizes and configurations by Chattopadhyay et al. [3,7]. Out of all these tests, test data of five elbows having two different diameters, namely, 8 in. and 16 in. with different sizes of throughwall cracks subjected to in-plane bending moment were taken up for comparison of J-R curves for validation of the newly proposed displacement based R6 approach. While comparing the J-R curves of these five elbows evaluated by conventional approach using and factors as reported by Chattopadhyay et al. [7] with those obtained using the proposed displacement based R6 approach, significant variation was observed. This necessitated further improvement in displacement based R6 approach to predict J-R curve from test data. The present paper is an effort in that direction. In this paper, the displacement based R6 approach is further modified to displacement – load (hybrid) based R6 method. In this proposed method, both experimental load and LLD with crack extension data are required for calculation of J-R curve. Hence, this method is named as ‘displacement-load’ or ‘hybrid’ R6 approach. The estimated J-R curves using hybrid approach are compared with

17

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conventionally calculated values already reported for these elbows. The results are found to be in good agreement with conventionally calculated J-R curves for all the elbows. This hybrid approach is further extended to the pipes, already investigated using load based and displacement based approaches. These estimated J-R curves are also found to be in very good agreement with conventional J-R curves for all the pipes as well. Thus, proposed hybrid method is found to be more accurate yet retaining the simplicity of the earlier proposed load/displacement based R6 approach to evaluate J-R curve from test data. The following sections describes the development of the proposed hybrid R6 approach. 2. R6 failure assessment diagram (FAD) to obtain J-R curve R6 Failure Assessment Diagram (FAD) is a widely used simple method for failure assessment of any cracked component. In this methodology, two failure modes, namely, brittle fracture and plastic collapse is included and in between ductile tearing is mapped as explained in R6 procedure revision 4 [8], as shown in Fig. 1. British Energy researchers, Dowling and Townley [9] and Harrison [10] addressed the significant interaction of fracture and plastic collapse and proposed a two-criteria failure assessment diagram (FAD), R6. A Failure Assessment Line (FAL) was proposed

where normalized crack tip loading, Kr = KI / KJ was proposed as a function of normalized remote loading Lr = P /PL . Here KJ = JE ' 2) where E ' = E for plane stress while E/(1 for plane strain condition.P and PL are applied load and limit load respectively for the assessed structure. Hence, failure assessment line is simply the variation of Je / J with normalized loading Lr as shown in Fig. 1. In this paper, option-1 failure assessment line of R6 FAD, is used for investigation which is a unique line and expressed by a simple exponential mathematical expression as shown in Eq. (1).

Kr =

Je / J = f1 (Lr ) = (1 + 0.5Lr 2)

1/2 [0.3

+ 0.7exp ( 0.6Lr 6)]

(1)

For assessment, mathematical solutions of elastic stress intensity factor, KI and limit load PL are required which are available for wider range of geometries than cases covered by η and γ functions. At fracture loading point, fracture toughness, Jmat can be estimated using Eq. (1), which can be written mathematically,

J = Jmat ( a) = Je (a 0 + a)[f (Lr )]

(2)

2

Using Eq. (2), fracture toughness, Jmat can be calculated for applied normalised loading Lr using FAD assessment line. Fig. 2 shows an assessment point 1 corresponding to any arbitrary loading. It is well established that the assessment path with increasing load P will be linear as shown in Fig. 2. On reaching the failure assessment line the ductile crack initiation occurs. Beyond the crack initiation, in stable crack growth regime, the loading path will follow the failure assessment line as explained by Milne [11]. For calculation of Jmat for this loading point, the function f (Lr ) has to be calculated corresponding to point 1, which is shown as point 1′ in failure assessment line. It should be noted that the FAL is shown to be extended infinitely with Lr and not curtailed at usual maximum value of Lr because here FAL is being used for calculation of Jmat and not for any failure assessment to account for collapse load. In order to apply FAD methods, it is necessary to evaluate the stress intensity factor, KI and limit load, PL . This methodology is being extended in this paper to the throughwall cracked elbows under bending moment. Hence, the related mathematical expressions are explained in further details. 3. Closed form expressions The utilized mathematical expressions of stress intensity factor, KI and limit load PL are given in the following sub-sections:

Fig. 1. Schematic representation of different modes of failure in R6. 18

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M. Sahu, et al.

Fig. 2. Schematic Illustration of the FAD methodology [6].

3.1. Stress intensity factor for elbows In order to apply FAD methods, it is necessary to evaluate the stress intensity factor, KI . The following solution for circumferentially through-wall cracked elbows under in-plane bending moment proposed by Chattopadhyay [12], was used:

KI = Fb

b

where the bending stress, b

(3)

a b,

is defined in terms of the bending moment Mb as (4)

= Mb/( Rm2 t )

where the geometry factor, Fb is given as:

Fb = ( 3.4628 + 4.446

0.1366)

+ ( 2.2524 + 1.1102

+ ( 52.429 + 52.445

0.1267)

2.6137

0.1848)

+ (0.8634 + 1.7283

0.4587

0.0695)

(t / Rm)

0.5119

(5)

where = tRb /Rm2 with t , Rb and Rm are wall thickness, mean bend radius and mean radius of an elbow cross section respectively. Other parameter / is normalized crack size with 2 being the angular size of the through wall crack. These expressions are applied for calculation of KI for all the elbows tested under both opening and closing mode of loadings. It should be noted that Eq. (5) have been developed for closing mode of moment. Because of non-availability of any mathematical expression of KI for elbows under opening mode of moment that time, Eq. (5) is used by Chattopadhyay et al. [7] for determination of linear stress intensity factor KI and finally for calculation of J-R curves for all elbows tested under both opening and closing mode of loadings. Hence, this same expression of KI is used in this investigation for determination of J-R curves for all the five elbows. 3.2. Limit load for elbows The limit load expressions for circumferentially cracked elbows under in-plane bending moment are proposed by Chattopadhyay et al. [13] for both opening and closing mode of loadings. The basic form of limit moment equations for both mode of loadings is represented by the following expression: (6)

ML = M0 X

where M0 is the limit moment for defect free elbow and X is the weakening factor due to existing flaw, which will obviously follow the condition of X 1. 3.2.1. Opening mode The limit moment for defect-free elbows, M0 , for opening mode of moment is proposed as

M0 = 4Rm2 t y (1.048

1/3

(7)

0.0617)

The weakening factor X due to existing crack is given as: 19

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M. Sahu, et al.

X = 1.127

1.8108

(8)

3.2.2. Closing mode The limit moment for defect free elbow under closing mode is proposed as:

M0 = 4Rm 2t y·1.075

(9)

2/3

For all the investigated elbows, the ratio, Rm / t 5. Thus, calculations of J-R curves using conventional approach are calculated by Chattopadhyay et al. [7] with the assumption of Rm / t = 5. Weakening factor, X for Rm / t = 5 is proposed as:

X = 1.1194 for

45o

0.7236

2.0806

2

(10)

150o .

2 It can be observed later that for all the elbow cases investigated, the existing crack sizes are within the validity limit. It should be noted that many researchers have proposed different closed form expressions of limit load PL with slight variations. However, the explained expressions of PL from Eqs. (6) to (10) have been used for getting and functions which were used for prediction of conventional J-R curves for the elbows. Hence, this particular set of closed form expressions of limit load have been chosen for calculation of J-R curves. 4. Fracture test results: Elbows The testing of these elbows has been performed under the component integrity test program by Chattopadhyay et al. [14] at SERC (Structural Engineering Research Centre), Chennai. The test specimens selected for the investigation consist of 90 degree elbows having two sizes; nominal bore 200 mm (8 in.) and 400 mm (16 in.), with circumferential throughwall crack, either at intrados or extrados. Notches have been machined on the elbow by milling process. Before carrying out the fracture tests, each elbow was fatigue pre-cracked through remote loading by around 3–10 mm on each side of the crack to have sharp crack tips. These tests have been carried out by applying in-plane bending moment. The elbows which were cracked at the extrados were tested under closing mode and those cracked at the intrados were tested under opening mode as shown in Fig. 3(a) and (b) respectively. Straight pipes were welded to each side of an elbow and to flanges, bolted to circular plates, for connection to the loading. Fig. 3(c) is a schematic representation of an elbow test set up. Total five elbows with circumferential through-wall cracks, either at the intrados or extrados have been picked up for the present study. They were tested by applying monotonic and quasi-static in-plane bending moment. Relevant data for FAD analyses for the elbows tested under closing mode of bending moment with extrados throughwall cracks has been given in Table 1. Similar information is given in Table 2 for elbows tested under opening mode of bending moment. Crack initiation loads for all the elbows have also been interpolated from the load vs. crack growth data, corresponding to the crack growth of 0.2 mm, termed asP0.2 . These experimental data of cracked pipes and elbows have been used for further investigation by using FE analyses and available closed

Fig. 3. Loading configuration of an elbow, under in plane bending moment: (a) crack at extrados – closing mode, (b) crack at intrados – opening mode; (c) test set up. 20

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Table 1 Elbows having extrados throughwall crack: tested under closing mode of moment. Designation

ELTWEX16-4 ELTWEX16-5

Rb (mean bend radius)

Do (Outer Diameter.)

t (avg. wall thickness)

(mm) 609 609

(mm) 406 406

(mm) 35.7 37.6

2 (crack angle)

P0.2 (crack initiati-on load)

V (moment arm length)

(degree) 94.1 124.0

kN 1004.2 741.1

(mm) 840 840

Rm/ t

2 (crack angle)

P0.2 (crack initiation load)

V (moment arm length)

5.32 5.07 5.01

(degree) 94.96 95.89 122.79

kN 114.3 639.5 607.4

(mm) 826 840 840

Rm/ t

5.19 4.90

Table 2 Elbows having intrados throughwall crack: tested under opening mode of moment. Designation

ELTWIN8-1 ELTWIN16-1 ELTWIN16-2

Rb (mean bend radius)

Do (Outer Diameter)

t (avg. wall thickness)

(mm) 207 609 609

(mm) 219 406 406

(mm) 18.80 36.43 36.85

Table 3 Comparison of experimental and predicted crack initiation loads and plastic collapse moments [14,15]. Crack Initiation Load

ELTWEX16-4 ELTWEX16-5 ELTWIN8-1 ELTWIN16-1 ELTWIN16-2

% Error1= 2

% Error =

Plastic Collapse Moment

ExperimentalP0.2

FEA estimation PFE

% Error1

Experimental MLexp

Using Eqs. (6)–(9) MLeqn

% Error2

kN 1004.2 741.1 114.3 639.5 607.4

kN 989 770 104 621 474

% 1.5 −3.9 9.0 2.9 21.9

kN-m 985.6 792.3 857.0 699.1 98.4

kN-m 962.1 819.4 847.1 678.8 101.1

% 2.4 −3.4 1.2 2.9 −2.7

(P 0.2 PFE ) × P0.2 exp eqn (ML ML ) exp ML

100

× 100

form expressions of limit load. The predicted crack initiation loads [14,15] and limit loads [14,16] are in good agreement with experimental values as shown in Table 3. Using Gurson based continuum damage model, GTN model [17,18], one elbow has been simulated using micromechanical properties of the material by Dutta et al. [19]. The computed results of load vs. crack growth curve and load vs. LLD curve are found to be in very good agreement with corresponding fracture test results. These earlier investigations validate the experimental results obtained from the fracture tests. The relevant dimensions, outer diameter, D0 , mean bend radius, Rb , wall thickness, t , circumferential crack size, 2 and moment arm length V are shown in the Fig. 3. These parameters including ratio Rm / t and crack initiation loads P0.2 are tabulated in Tables 1 and 2. R6 based methodology will be applied to calculate fracture toughness, Jmat using the experimental load and/or LLD for any arbitrary loading point with relevant crack extension value. 4.1. Material tensile properties Tensile stress-strain data obtained for the piping material carbon steel SA333Gr6 is already reported for both 8 in. and 16 in. piping material by Sahu et al. [6] during displacement based approach for pipes. It should be noted that the piping materials are identical for same sizes of straight pipes and elbows as explained by Chattopadhyay et al. [14]. The relevant material property is reproduced here for ready reference. Material properties were evaluated for both these two piping materials separately and expressed corresponding to pipe diameter. The specimens were prepared with their loading axis parallel to the length of pipe as shown in Fig. 4(a). As shown in Fig. 4(b), round tensile specimens of 4 mm diameter and 30 mm gauge length following ASTM standard E8 were fabricated from both the piping material SA333Gr6 [20]. All tests were carried out at a strain rate of 1.2 × 10−4 sec−1. 21

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M. Sahu, et al.

Fig. 4. (a) Orientation of specimen in pipe block. (b) Sketch of round tensile specimens [20].

For getting material tensile properties, tests were performed according to ASTM standard E08 [21]. Results are presented in Table 4.Tensile stress strain data obtained for the 8 in. and 16 in. piping materials by Chattopadhyay et al. [3]. These material properties are used for respective sizes of elbows in this work. For displacement based R6 approach, tensile stress –strain data is fitted in average way to remove the Luder regime. This Ramberg-Osgood (RO) fitted curve is quantified by two parameters: Coefficient and strain hardening exponent n as shown in Table 4. The RO fitting will depend on the strain range chosen. For smoothing the tensile data in the Luders regime, the true stress and true strain is fitted from yield stress point to ultimate stress point. For curve fitting, the strain ranges are from yield strain y = y / E to ultimate strain u , which is corresponding to ultimate stress. The values of y and u are 0.00142 and 0.183, and 0.00154 and 0.199 respectively for the 8 in. and 16 in. piping materials respectively The detailed approach of calculation of these parameters is explained by Sahu et al. [6]. These parameters will be used for estimation of reference stress ref from reference strain ref with Ramberg-Osgood equation as shown in Eq. (11). ref

=

y

ref y

+

ref

n

(11)

y

This approach removes the Luder regime from stress strain data and ensures the smooth increase of reference stress consequent material fracture toughness Jmat with applied reference strain ref .

ref

and

5. Application of R6 for prediction of J-R curves for elbows Load and displacement based R6 approaches were already proposed and employed for prediction of J-R curves of cracked pipes by Sahu et al. in [2] and [6] respectively. The results were found to be in good agreement with conventionally calculated J-R curves especially for the displacement based approach. This approach is further adopted for one of the elbows, ELTWEX16-4 which is fabricated with extrados throughwall crack and tested under closing mode of bending moment. The details are already given in Table 1 and loading configuration is schematically shown in Fig. 3. 5.1. Displacement based approach For application of displacement based approach, nominal loading Lr is calculated using applied LLD instead of experimental load which is the established methodology in the R6 approach in failure assessment also. For using LLD, elastic beam theory is applied and a mathematical relation between reference strain ref and LLD was proposed for a cracked pipe [6]. Similarly, for elbows, curved beam theory is going to be used for getting the similar mathematical relation to calculate reference strain using LLD. 5.1.1. Curved beam theory for elbow for displacement based R6 approach Using the angular rotation d , curved beam theory is applied as shown in Fig. 5(a), for the deformed curved beam and elastic strain e at distance y from neutral axis is calculated as,

Table 4 Properties of SA 333 Grade 6 steel for two different pipe/elbow diameters. D0

E (Young’s modulus)

(mm) 219 406

(GPa) 203 203

(Poisson’s ratio) 0.3 0.3

y (yield

(MPa) 288 312

stress)

u (ultimate

(MPa) 420 459

22

stress)

(coefficient of RO eqn.) 6.71 6.27

n (hardening exponent of RO eqn.) 5.30 5.27

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Fig. 5. (a) Curved beam theory. (b) Schematic representation of deformed elbow.

e

=

d k d

(12)

where d is the total angle created by the curved beam at centre as shown in the Fig. 5(a). Parameter k is a non-dimensional parameter of y and eccentricity e . It should be noted that unlike a pipe, for an elbow centroid and neutral axis are not same. Hence, for an elbow an additional parameter e other than y , is needed for getting strain distribution on cross section. This parameter e is the distance between centroid and neutral axis which is a non-dimensional function of bend radius, Rb , mean radius Rm and thickness t for an elbow as explained by Timoshenko [22]. For application of this theory, it is assumed that all the deformation/rotation in elbow due to LLD is concentrated at cracked section only. This assumption is quite reasonable because all the elbows under investigation, are cracked throughwall with reasonably large size. Hence, all the deformation is mainly concentrated at cracked section and, deformations in other parts may be neglected for practical purpose. The deformed elbow at cracked section is shown schematically in Fig. 5(b). The angular rotation d is shown as,

d =

2

cosd H

(13)

+ where is applied LLD, H = and d = tan as shown in Fig. 5(b). Here V and B are moment arm length and vertical distance between cracked section and loading point as shown in the figure. Putting value of d from Eqs. (13) in (12) and renaming elastic strain e to reference strain ref because this strain will represent the material stress strain data for converting to reference stress ref . Thus ref in terms of applied LLD, V2

ref

=

2

cosd d H

B2

1 (B / V )

k

where k is now the function of LLD,

(14) and reference strain

ref

for a cracked elbow. The parameters 23

and

ref

are global parameters

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M. Sahu, et al.

Fig. 6. Experimental load vs. crack growth data for elbow ELTWEX16-4 [7,14].

representing the geometrical parameters like bend radius, Rb , mean radius Rm thickness t including additional parameters crack size 2 , for a cracked elbow. The parameter k is still not known. For determination of k , ref is calculated by using load and displacement data corresponding to crack initiation point. Reference strain at crack initiation ref (i) in terms of crack initiation load is derived by Sahu et al. [6] as following. ref (i)

= (1 + )

y

Using Eq. (14), ref (i)

ref (i )

=

i

2

ref

cosd Hd

y

= (1 + )

Pi PL ( y )

y

(15)

at crack initiation point can be written as,

k

(16)

Now using Eqs. (15) and (16), k can be written in terms of crack initiation parameters,

k=

y (1

+ )

Pi 2 Hd PL ( y ) i cosd

(17)

Using Eq. (17) with d = /4 , k parameter can be calculated for crack initiation point with parameters Pi and i , where subscript ‘i ’ represents the crack initiation point. Once, k is known for an elbow, it is assumed to be constant for an elbow and it is used with further loading points LLD= for determination of reference strain using Eq. (16). Rest of the methodology is similar as explained by Sahu et al. [6] for pipes for determination of reference stress, ref and bending stress, b which are used for calculation of nominal loading, Lr and stress intensity factor, KI , respectively. Finally Jmat is calculated using Eq. (2). 6. Calculation for elbow ELTWEX16-4 First, R6 based approaches are employed here for the elbow ELTWEX16-4. Fracture test data, namely, load vs. crack growth and LLD vs. crack growth are shown in Figs. 6 and 7 respectively for this elbow. It should be noted that only an arbitrary elbow ELTWEX16-4 is chosen for the explanation of the R6 calculation, to make it simpler. 6.1. Load based calculation For employing the load based R6 approach, experimental load vs. crack extension data is required. In load based R6 approach, reference stress ref = P / PL is directly calculated using the applied load as explained by Sahu et al. [2] for pipes. In this approach, each point of this experimental data of Fig. 6 can be converted to a point in the fracture property J-R curve. Reference stress is calculated using the experimental load P using the established load approach i.e. Lr = ref / y = P / PL (a, y ) . Similarly, the bending stress b is also calculated by converting experimental load to corresponding bending moment as shown in Eq. (4). Finally using this bending stress b , stress intensity factor KI is evaluated using the Eq. (3). The calculated J-R curve using this fracture data is shown in Fig. 8. It may be noted that the load-based calculation is quite lower than the conventional J-R curve. The reason of this deviation is that the load is not an appropriate parameter for this case to quantify the instantaneous plastic deformation level and consequent reference stress ref and reference strain ref near crack tip. Reference stress ref is a parameter which is an important factor to evaluate the Jmat in forms of f (Lr = ref / y ) as shown in Eq. (2). 24

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Fig. 7. Experimental load line displacement (LLD) vs. crack growth data for elbow ELTWEX16-4 [7,14].

6.2. Displacement based calculation For using displacement based approach, the input is experimental LLD vs. crack extension data which is shown for elbow ELTWEX16-4 in Fig. 7. In this approach, each point of this experimental curve is converted to the corresponding point of fracture property J-R curve as explained in displacement based approach by Sahu et al. [6]. First, using the curved beam theory, LLD is converted to reference strain ref and then corresponding reference stress ref is obtained from the fitted material stress – strain data. This ref will now represent the fracture property J-R curve which is always rising with more and more loading. If this parameter ref would have been predicted by load as in case of load approach, it may not have represented properly the energy absorbed during more and more applied LLD which is quantified as Jmat . Another parameter which is very important for determination of Jmat is elastic J -integral, Je , which is determined from ref . The methodology of this calculation is explained further in detail. It should be noted that employed mathematical expression of elastic SIF (Stress Intensity Factor), KI for an elbow have been proposed for uncracked bending stress b . Here, this bending stress is related to global bending moment M as M = Rm2 t b which is corresponding to similar pipe section. Hence, this reference stress ref is calculated by balancing the moment for uncracked section of a pipe in terms of bending stress b and for cracked section of an elbow in terms of ref . For a closing moment case of an cracked elbow, using Eqs. (6), (9) and (10), the moment in terms of reference stress ref can be written as shown in Eq. (18).

M = 4Rm2t

ref

1.075

2/3

1.1194

0.7236

2.0806

2

(18)

This is equalized with bending moment of similar pipe section as explained earlier. The bending stress the reference stress ref as shown in Eq. (19). b

=

4

1.075

2/3

1.1194

0.7236

2.0806

b

can be written in terms of

2 ref

(19)

It should be noted that the mathematical expression of limit moment ML is different for opening and closing moment. Hence, using relevant Eqs. (6), (7) and (8) the mathematical expression can be easily derived for calculating bending stress b from reference stress ref for opening mode as shown in Eqn. (20). b

=

4

(1.048

1/3

0.0617) 1.127

1.8108

ref

(20)

For calculation of bending stress b from reference stress ref , Eqs. (19) and (20) are used for cracked elbows tested under closing mode and opening mode, respectively. Thus, bending stress b is evaluated from applied LLD which is finally used for calculation ofJe . Thus, in this approach only displacement vs. crack extension data is required for prediction of fracture property J-R curve and experimental load data is not required. For elbow ELTWEX16-4, the experimental LLD vs. crack extension data as shown in Fig. 7, is converted to fracture property i.e. Jmat vs. a plot as shown in Fig. 8. These predicted J-R curve using displacement based R6 approaches is compared with conventional J-R curve as shown in Fig. 8. It can be observed that predicted J-R curves by load based approach is under predicting the fracture property J-R curve while displacement based approach is over predicting. This deviation of J-R curve obtained from displacement based R6 approach with respect to the conventional one suggests a need for a probable modification in this proposed displacement based R6 approach to calculate J-R curve. 25

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Fig. 8. J-R curves calculated by different methods for elbow ELTWEX16-4.

6.3. Reason of deviation of displacement based R6 approach As mentioned in the previous section, Sahu et al. [6] observed that predictions of J-R curves by displacement based R6 approach are in very good agreement with conventional values for all the pipes. However, here the predictions by this approach for the elbow ELTEEX16-4 is showing significant deviation from conventional J-R curve. Thus, an attempt is made to investigate the possible reason for this deviation. It should be noted that the uncracked bending stress b is calculated using Eq. (19), which is based on the assumption that the cracked cross section will remain circular during loading. However, it is observed by Chattopadhyay et al. [16] that if an elbow having a circumferential crack greater than some threshold size 2 th is subjected to closing mode of bending moment, its load carrying capacity is weakened significantly due to ovalization of cross section under significant plastic deformation, as shown schematically in Fig. 9. Thus, the inherent assumption of circular cross section during deformation is not valid for elbows. Applied LLD is used for calculation of reference strain ref , and further, this ref is used for estimation of reference stress ref , which is finally used for calculation of bending stress b . Hence, the calculation is based on the assumption that the crack section is circular while it has actually turned to oval shape because of loading configuration and significant plastic deformation. In reality, this shape change has caused significant reduction in the load carrying capacity of the cross section. Thus, the prediction of the load from LLD based on circular cross section is an erroneous approach. This approach will give significant over prediction of the bending stress/load because of the present ovalization in closing mode of the loading. This over prediction of b will lead to higher estimate of KI or Je , which will eventually lead to significant over prediction of material fracture toughness Jmat as shown in Fig. 8. 6.4. Displacement-load (hybrid) based approach It is observed that using the LLD for calculation of displacement based approach is further modified.

b

and further KI is not an appropriate approach in case of an elbow. Thus,

a) Load based approach for J-elastic Instead of LLD, directly experimental measured load P should be used for more accurate prediction of bending stress b and consequentKI using Eqs. (3) and (4) respectively. Thus, load based approach is adopted for calculation of stress intensity factorKI and consequent J-elastic, Je = KI2 /E ' . b) Displacement based approach for normalized loading ‘Lr ’ It should be noted that displacement is used for calculation of reference strain and reference stress and finally for calculating theLr . So Lr and f1 (Lr ) are predicted by displacement based approach. The Eq. (2), Jmat ( a) = Je (a 0 + a)[f (Lr )] 2 is used for calculation of fracture property Jmat . First term Je is calculated by load based approach using experimental loadP , while second term f (Lr ) is evaluated by displacement based approach using experimental LLD. Thus, this methodology is termed as, “hybrid approach” because it is using both the experimental results of load and displacement with relevant crack extension data. Comparison of hybrid approach based calculation with other approaches are shown in Fig. 10. It can be observed that prediction by hybrid approach is in better agreement with conventional J-R curve than load based and displacement based approaches for the elbow ELTWEX16-4. The work flow diagram of this methodology is shown schematically in Fig. 11. The displacement approach and load approach are depicted in left and right side respectively, in mid part of the flow diagram. 26

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Fig. 9. Schematic representation of weakening of an elbow under closing mode of moment [16].

Fig. 10. J-R curve by hybrid approach is in agreement with conventional one for elbow ELTWEX16-4.

7. Results 7.1. Different parameters in the steps of Jmat – estimation Relevant experimental parameters crack initiation load Pi and corresponding LLD, i are given in Table 5 for all the elbows. Different important parameters e.g. non-dimensional parameter ‘k ’, normalized reference strain ref (i) / y , normalized reference stress, Lr (i) and J-elasticJe (i) evaluated in the steps of determination of J-R curve are also shown in the table. It should be noted that the parameters like normalized reference stress Lr (i) and J-elastic Je (i) are identical for load based and displacement based approaches because the loading level is within elastic regime. In the elastic regime, reference stress is in elastic limit i.e. P PL or ref y. Variation of different important parameters namely normalized reference strain ref (i) / y , normalized reference stress, Lr (i) , bending stress b , and J-elasticJe (i) with crack extension a , are shown in Fig. 12 for elbow ELTWEX16-4. The parameters are initially in agreement near crack initiation point because of elastic loading regime where load based and displacement based approaches yield identical results. However, on higher crack growth range, the significant differences in the parametric values are quite evident because of large scale plastic deformation. 7.2. J-R Curves for remaining elbows Hybrid approach is also employed for calculation of J-R curves for remaining four elbows. Experimental fracture data of load and 27

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Fig. 11. Work flow diagram for the hybrid approach.

Table 5 Different parameters at crack initiation point during J-R curve calculation for the elbows. Elbow Designation

Pi (KN)

ELTWEX16-4 ELTWEX16-5 ELTWIN16-1 ELTWIN16-2 ELTWIN8-1

1004.16 741.10 639.48 607.43 114.26

i (mm)

22.81 21.00 19.26 24.80 27.81

‘k ’ 1.05 0.98 0.77 0.71 0.88

28

ref (i) / y

6.43 5.52 4.62 5.47 6.66

Lr (i) = 0.88 0.76 0.64 0.75 0.94

ref (i) / y

Je (i) (N/mm) 173.58 249.58 75.44 165.81 28.69

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Fig. 12. Variation of different parameters with crack extension Δa for elbow ELTWEX16-4. (a) Normalized reference strain, εref/εy. (b) Normalized reference stress, Lr = σref/σy. (c) Bending stress, σb for uncracked section. (d) J-elastic, Je calculated by different methods.

Fig. 13. Experimental load and LLD curves with corresponding crack growth for elbow ELTWEX16-5.

LLD curves with relevant crack extension are shown in Fig. 13 for the elbow ELTWEX16-5. The calculated J-R curves using this fracture data by different approaches are shown in Fig. 14. Similarly, for remaining 16 in. elbows with intrados cracks, these curves are shown in Figs. 15 and 16 for elbow ELTWIN16-1, and, in Figs. 17 and 18 for elbow ELTWIN16-2. For 8 in. elbow ELTWIN8-2, the experimental load and LLD vs. crack extension data are plotted in Fig. 19 and calculated J-R curves using this experimental data by the different proposed approaches are shown in Fig. 20. It can be observed that the predictions of J-R curves using hybrid approach are in better agreement with conventional values compared to both load and displacement based R6 approaches.

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Fig. 14. J-R curves calculated by different methods for elbow ELTWEX16-5.

Fig. 15. Experimental load and LLD curves with corresponding crack growth for elbow ELTWIN16-1.

7.3. Hybrid approach to straight pipes under bending moment 7.3.1. Experimental details of the pipes under bending moment Relevant geometries and loading configurations are already discussed by Sahu et al. [2,6] for cracked pipes tested under four point bending moment. Closed form expression of stress intensity factor, KI for circumferentially through-wall cracked pipes under in-plane bending moment were proposed in R6 [8,23]. Closed form expression of limit moment, ML , is also available in R6 [8,24] for straight pipes under bending moment. These expressions have been already explained by Sahu et al. [2] in details. The load and displacement based R6 approaches have already been implemented for total six cracked straight pipes by Sahu et al. [2] and Sahu et al. [6] respectively. The experimental load and LLD data with relevant crack extension values are reported in those papers. In this paper, hybrid approach is implemented for these pipes and J-R curves predicted by hybrid approach is compared with already reported load based and displacement based J-R curves. 7.3.2. J-R Curves calculated using hybrid approach It is already observed by Sahu et al. [6] that the predictions of J-R curves by displacement cased approach are in good agreement with conventional values for all six pipes. However, for elbows instead of displacement based approach, prediction by hybrid approach is in good agreement with conventional values. Thus, this hybrid approach is extended to the pipes for prediction of J-R curves. These J-R curves are compared with J-R curves predicted by other approaches and shown in Fig. 21 for all pipes. It can be observed that J-R curve predictions by hybrid approach are very close to displacement based approach and in good agreement with conventional J-R curves for all the pipes. This can be explained by the fact that degree of ovalization of circular cross section of pipe during deformation is minimal. Hence, unlike elbow where degree of ovalization is significant, hybrid and displacement based 30

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Fig. 16. J-R curves calculated by different methods for elbow ELTWIN16-1.

Fig. 17. Experimental load and LLD curves with corresponding crack growth for elbow ELTWIN16-2.

Fig. 18. J-R curves calculated by different methods for elbow ELTWIN16-2.

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Fig. 19. Experimental load and LLD curves with corresponding crack growth for elbow ELTWIN8-2.

approaches yield almost identical J-R curve for pipes. 8. Comparison of different approaches for calculation of J-R curve In conventional approach of calculation of J-R curve from experimental fracture test data, load and load line displacement (LLD) with related crack extension are required. Load and LLD curve are integrated using relevant and functions. The values of these functions depend on the instantaneous geometry, so it has to be updated for changed crack size due to crack growth. Hence, this approach requires both experimental load and LLD data as shown in part A of Fig. 22, with related crack extension. This is fundamentally appropriate approach because it is based on the energy consumed by the cracked component during the crack extension. In the usual R6 approach, for assessment of failure, the normalized parameters, like crack driving force, Kr = KI / Kmat and remote loading, Lr = P /PL are evaluated by the applied load only. As shown in Eq. (2), Jmat ( a) = Je [f (Lr )] 2 , material fracture toughness, Jmat is multiplication of two variables J elastic, Je and functional value of Lr , [f (Lr )] 2 . Thus, the value of fracture toughness Jmat at this crack extension point a is dependent on these two parameters. In conventional R6 methodology, both these parameters are evaluated by using the instantaneous value of experimental load P with crack extension a as shown in part B of Fig. 22. Thus, this approach is termed as load based R6 approach. In displacement based approach, both these parameters Je and Lr are evaluated by the instantaneous experimental LLD value at this crack extension value a as shown in part C of Fig. 22. The extent of plastic deformation is better quantified by the applied displacement rather than applied load as done in load based R6 approach. Hence, this approach is appropriate in prediction of material fracture toughness in the regime of significant plastic deformation near crack tip. However, the limitation of this approach is that in this approach bending stress b is also calculated by applied displacement. In case of significant plastic deformation and/or geometric hardening or softening, this approach is not able to correctly predict bending stress b . On using this parameter to calculate Je , it results in to erroneous prediction of fracture toughnessJmat .

Fig. 20. J-R curves calculated by different methods for elbow ELTWIN8-2.

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Fig. 21. Comparative plot of J-R curves calculated by hybrid- R6 approach for pipes. Figures (a)–(f) respectively for pipes SPBMTWC8-1, SPBMTWC8-2, SPBMTWC8-3, SPBMTWC16-1, SPBMTWC16-2 and SPBMTWC16-3.

This limitation of displacement based approach is corrected by calculating the elastic J (Je ) by using the experimental load P like load based approach. Therefore, to have the best predictions, displacement based approach is used for calculating the normalized loading Lr , and load based approach is adopted for calculating Je as shown in part D of Fig. 22. Thus, this approach is termed as hybrid approach, which improves the prediction of J-R curve. 9. Conclusion The conventional approach for post processing the test data to obtain fracture property J-R curve requires the geometry functions and which are available for limited geometries. Hence post processing of fracture test data to obtain J-R curve is not possible for a crack geometry without these functions. In this investigation, a simpler yet accurate approach is proposed using R6 failure assessment diagram for that purpose. This paper can be concluded with the following points: 33

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Fig. 22. Conventional and different R6 –approaches for calculation of J-R curve using experimental load and displacement data.

• Load based R6 approach is not appropriate to predict the J-R curve for both pipes and elbows. The reason is that the load is not an appropriate parameter to quantify the fracture property J -integral in the significantly plastically deformed regime. • Displacement based approach is appropriate for pipes but nor for elbows. The reason is that the significant ovalization violates the basic assumptions involved in the calculation of material fracture property, namely the J-R curve. • The displacement based approach is modified to hybrid approach in this paper for post processing the test data of load, LLD and crack growth to obtain the material J-R curve. • The predicted J-R curves using hybrid (load-displacement) R6 approach, are found to be in good agreement with conventionally • • • • •

calculated J-R curves for all the pipes and elbows. Hence, hybrid approach is recommended as a final methodology for prediction of J-R curves from fracture test data. This method requires closed form expressions of stress intensity factor KI and limit load PL which are available for wider range of geometries than and functions. On the way, an innovative approach is developed and utilized to calculate the reference stress and reference strain for the deeply cracked elbows using curved beam theory. The utility of R6 is extended from just failure assessment to calculation of fracture toughness Jmat as well. In this paper, elastic beam theory approach is utilized for proposing a methodology to calculate references strain ref and Jmat using experimental LLD. It is recommended to validate this methodology by extensive finite element analyses. This work will further substantiate the proposed methodology. This methodology may be adopted for other materials and geometries in future investigations.

Appendix A. Supplementary material Supplementary data to this article can be found online at https://doi.org/10.1016/j.engfracmech.2019.04.031. References [1] “ASTM E1820-99a: Annual book ASTM standards,” in Standard test method for measurement of fracture toughness., vol. vol. 03.01, 1999. [2] Sahu MK, Chattopadhyay J, Dutta BK. Application of R6 failure assessment method to obtain fracture toughness. Theor Appl Fract Mech 2015;81:67–75. [3] Chattopadhyay J, Dutta BK, Kushwaha HS. Experimental and analytical study of three point bend specimens and throughwall cicumferentially cracked straight pipes. Int J Press Vessels Pip 2000;77:455–71. [4] Jayadevan KR, Ostby E, Thaulow C. Fracture response of pipelines subjected to large plastic deformation under tension. Int J Press Vessels Pip 2004;81(9):771–83. [5] Otsby E, Hellesvik AO. Large-scale experimental investigation of the effect of biaxial loading on the deformation capacity of pipes with defects. Int J Press Vessels Pip 2008;85(11):814–24. [6] Sahu MK, Chattopadhyay J, Dutta BK. Displacement based calculation of fracture toughness for cracked pipes using R6 method. Theor Appl Fract Mech 2018;93:211–21. [7] Chattopadhyay J, Pavankumar TV, Dutta BK, Kushwaha HS. J-R curves from through-wall cracked elbow subjected to in-plane bending moment. J Press Vessel Technol, Trans ASME 2003;125:36–45. [8] R6, “Assessment of the integrity of structures containing defects, Revision 4, including subsequent updates,” Gloucester, UK, 2013. [9] Dowling AR, Townley AC. The effects of defects on structural failure: a two-criterion approach. Int J Press Vessels Pip 1975;3:77–107. [10] R. P. Harrison, K. Loosemore and I. Milne, “Assessment of the Integrity of Structures Containing Defects,” London, 1977. [11] Milne I. Failure analysis in the presence of ductile crack growth. Mat Sci Eng 1979;39:65–79. [12] Chattopadhyay J, Dutta BK, Kushwaha HS, Mahajan SC, Kakodkar A. A database to evaluate stress intensity factors of elbows with throughwall flaws under

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combined internal pressure and bending moment. Int J Press Vessels Pip 1994;60:71–83. [13] Chattopadhyay J, Tomar AKS, Dutta BK, Kushwaha HS. Closed form collapse moment equation of throughwall circumferentially cracked elbows subjected to inplane bending moment. ASME J Press Vessel Technol 2004;126:307–17. [14] Chattopadhyay J, Pavan Kumar T, Dutta BK, Kushwaha HS. Fracture experiments on throughwall cracked elbows under in-plane bending moment: test results and theoretical/numerical analyses. Engng Fract Mech 2005;72:1461–97. [15] Chattopadhyay J, Tomar AKS, Dutta BK, Kushwaha HS. Elastic-plastic J and COD estimation schemes for throughwall circumferentially cracked elbow under inplane closing moment. Engng Fract Mech 2005;72:2186–217. [16] Chattopadhyay J, Tomar AKS. New plastic collapse moment equations of defect-free and throughwall circumferentially cracked elbows subjected to internal pressure and in-plane bending moment. Engng Fract Mech 2006;73:829–54. [17] Tvergaard V, Needleman A. Analysis of the cup- cone fracture in a round tensile bar. Acta Metall 1984;32:157–69. [18] Gurson A. Continuum theory of ductile rupture by void nucleation and gorwth. Part-I-Yield criteria and flow rules for porous ductile media. 1977. p. 2–15. [19] Dutta BK, Guin S, Sahu MK, Samal MK. A phenomenological form of the q2 parameter in the Gurson model. Int J Press Vessels Pip 2008;85(4):199–210. [20] Singh PK, Chattopadhyay J, Kushwaha HS, Tarafder S, Ranaganath VR. Tensile and fracture properties evaluation of PHT system piping material of PHWR. Int J Press Vessels Pip 1998;75:271–80. [21] “ASTM E8-91: Annual book of ASTM standards,” in Test methods for tension testing of metallic materials, vol. 03.01, 1991. [22] Timoshenko S. Strength of Materials. New Delhi: CBS Publishers; 1986. [23] Zahoor A. Closed form expressions for fracture mechanics analysis of cracked pipes. ASME, J Press Vessel Technol 1985;107:203–5. [24] Jones MR, Eshelby JM. Limit load solutions for circumferentially cracked cylinders under internal pressure and combined tension and bending. Nuclear Electric Report TD/SID/REP/0032. 1990.

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