New unloading compliance correlation for ... - Wiley Online Library

doi: 10.1111/ffe.12160

New unloading compliance correlation for throughwall circumferentially cracked pipe to measure crack growth during fracture tests J. CHATTOPADHYAY, W. VENKATRAMANA and B. K. DUTTA Reactor Safety Division, Bhabha Atomic Research Centre, Mumbai 400085, India Received Date: 30 August 2013; Accepted Date: 11 January 2014; Published Online: 12 February 2014

A B S T R A C T Crack growth is generally measured during fracture experiment of specimen or component.

The unloading compliance technique is commonly used for this purpose because of its simplicity. It infers the crack length from unloading compliance of cracked component. The pre-requisite of this technique is the availability of an equation that correlates crack length with unloading compliance. While such correlations are available for compact tension and three-point bend specimens, it is not available for big components such as pipe or pipe bend. Development of such a correlation for throughwall circumferentially cracked (TCC) straight pipe under bending, therefore, forms the objective of the present study. However, the challenge to develop such correlation for TCC pipe is that the equation should contain not only crack length as a function but also the current deformation or load level as a parameter. This is attributed to the fact that the circular cross section of the pipe ovalizes during deformation leading to change of bending stiffness of the cracked body. New compliance correlations have been proposed for TCC pipe under bending load considering these complexities. Elastic-perfectly plastic material behaviour has been assumed to characterize the material stress–strain response. However, it has been shown that error due to this approximation with respect to the actual stress–strain behaviour is negligible if one chooses flow stress equal to average of yield and ultimate strength. The proposed correlations are expressed in terms of normalized parameters to make them independent of specific values of geometric dimensions such as radius, thickness and span length of four-point bending loading system. Effectiveness of this normalization has also been verified by carrying out a sensitivity study. Keywords NOMENCLATURE

C C0 CMOD CT E I L M PL R t TCC TPB UTS YP Z δ δe 0 EI λ0 ¼ CπR 2 θ σf

crack; crack growth measurement; fracture test; pipe; unloading compliance.

= compliance = initial elastic compliance = crack mouth opening displacement calculated at mid-length, mid-thickness = compact tension = Young’s modulus = area moment of inertia of pipe cross section = inner span length of four-point bending load = bending moment = plastic limit load = radius of pipe cross section = wall thickness of pipe cross section = throughwall circumferentially cracked = three-point bend = ultimate tensile stress = yield point = outer span length of four-point bending load = crack mouth opening displacement = elastic crack mouth opening displacement = normalized compliance = semi-circumferential crack angle in a pipe = flow stress in elastic-perfectly plastic material tensile property idealization

Correspondence: J. Chattopadhyay. E-mail: [email protected]; [email protected]

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© 2014 Wiley Publishing Ltd. Fatigue Fract Engng Mater Struct, 2014, 37, 928–936

NEW UNLOADING COMPLIANCE CORRELATION OF CIRCUMFERENTIALLY CRACKED PIPE

INTRODUCTION

Fracture experiments are conducted on small specimens and full-scale components for various fracture toughness parameters such as KIC, JIC and J–R curve.1,2 These parameters are required in the detailed fracture mechanics analysis of various real-life components.3 They are derived from load versus load line displacement and load versus crack growth data, measured during fracture experiments. While load and load line displacement are measured directly from the in-built devices in the modern servo-hydraulic machines, crack growth measurement is a challenge in any fracture experiment. Various methods, for example, unloading compliance, potential drop4 and image processing techniques2 are commonly employed to measure crack growth. The unloading compliance technique is the most commonly used and also the simplest method to measure crack growth during fracture test of small specimens, for example, CT and TPB.5,6 This technique allows crack growth to be inferred from the load and displacement transducers that are part of any fracture mechanics test. A specimen can be partially unloaded at various points during the test in order to measure the elastic compliance, which can be related to the current crack length. As the crack grows, the specimen becomes more compliant (less stiff). Figure 1 shows the pictorial representation of the aforementioned technique. The pre-requisite of this technique is the availability of an equation that correlates crack length with unloading compliance. In the absence of such equations for full-scale components, for example, cracked pipes and elbows, compliance technique is rarely used to measure crack growth during fracture experiments of such components. It may be mentioned that fracture experiments of piping components with various crack configurations are carried out worldwide to study various fracture mechanics issues1–3 in the context of application of leak-before-break concept to primary heat transport system piping in pressurized heavy-water reactor and pressurized water reactor.

Fig. 1 The unloading compliance technique to measure crack growth during fracture experiment.

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TCC straight pipe under four-point bending load is a common component for fracture studies. Straight pipes are critical components of primary heat transport system piping of nuclear reactors. Conventionally, compliance correlations are derived by generating compliance (reciprocal of stiffness of load–deflection curve) versus crack length by performing small displacement linear elastic finite element (FE) analysis. However, it does not account for the large geometric deformation that may occur during the loading of the specimen. The unloading compliance may be influenced by the increasing/decreasing stiffness of the specimen because of the change in basic geometry during deformation. Therefore, it is of interest to study the effect of deformation on the unloading compliance. An earlier study by Chattopadhyay et al.7 had shown that the deformation has practically no effect on the compliance of the TPB specimens. In other words, unloading compliance of TPB specimen depends exclusively on current crack length (a/w). This behaviour was attributed to the fact that deformation of TPB specimens does not affect the area moment of inertia (i.e. bending stiffness). However, the same is not true for pipe. The same study7 showed that deformation of TCC pipe significantly affects the unloading compliance. In other words, the unloading compliance of TCC pipe depends not only on the current crack length but also on the deformation level.

PRESENT WORK

As mentioned earlier, an equation correlating crack length and unloading compliance is the pre-requisite for using the compliance technique to measure crack growth during fracture experiments. While these types of correlations are available for small specimens, for example, CT and TPB,5,6 no such correlation is available for TCC pipe. Development of such a correlation, therefore, forms the objective of the present study. However, the challenge to develop such correlation for TCC pipe is that the equation should contain not only crack length as a function but also the current deformation or load level as a parameter. Consequently, nonlinear FE analysis considering both geometric and material nonlinearity has been carried out for straight pipes of different R/t ratio with throughwall circumferential cracks of various sizes. The analyses have been performed up to load with large-scale plastic deformation. First, an equation correlating compliance and crack angle of TCC pipe has been developed from initial elastic solution of load versus CMOD. Subsequently, the equations have been modified to include the effect of load/deformation.


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FINITE ELEMENT ANALYSIS

The FE method is used to develop the compliance correlations. Straight pipe with throughwall circumferential crack under four-point bending load (Fig. 2) is considered for analysis. Each pipe is loaded a little beyond the theoretical plastic collapse load with periodic unloading. Theoretical plastic collapse load is calculated using the following equation: PL ¼

16R2 tσ f ZL

cos

θ 0:5 sinðθÞ ; 2

(1)

where R is the mean radius of the pipe cross section, t is the pipe wall thickness, θ is the semi-circumferential crack angle, σf is the material flow stress, and Z and L are the outer and inner span of the four-point bending load, respectively. The load versus CMOD has been generated for the entire load range with periodic unloading. The CMOD is calculated at the middle of crack length (where it is maximum) and at mid-thickness. The amount of unloading is 15% of collapse load. Stiffness is evaluated from the slope of the load–CMOD curve by linear curve fitting of the middle portion of the unloading path leaving aside the top and bottom 1% of collapse load. Compliance is evaluated by taking reciprocal of this stiffness evaluated at different stages of loading including the initial elastic portion. Compliance is defined as follows: C¼

δ P ðZ LÞ ;M ¼ ; M 4

(2)

where M is the moment at the cracked cross section of the pipe, δ is the total CMOD, P is the total applied load and C is the compliance. The initial elastic compliance has been normalized as follows: λ0 ¼

C 0 EI ; πR2

(3)

where λ0 is the normalized initial elastic compliance (C0), E is the Young’s modulus of the pipe material, I is the area

moment of inertia of the pipe cross section and R is the mean pipe radius. Compliance values at higher load levels (C) are normalized with respect to the initial elastic value (C/C0). Applied moment is also normalized with respect to the theoretical collapse moment as follows: m¼

M P L ðZ LÞ ; ; ML ¼ ML 4

(4)

where m is the normalized applied moment and ML is the theoretical plastic collapse moment of pipe containing a throughwall circumferential crack. The rationale of normalizing the compliance and load level is to make the equations independent of specific geometric dimensions of pipe. Thus, normalized compliances (λ) have been generated for various values of normalized load levels (m) and crack angles (θ/π). Geometric details The geometric details are as follows: mean radius (R) = 50 mm; R/t = 5, 10, 15 and 20; 2θ = 30°, 60°, 90°, 120°, 150° and 180°; inner span (L) = 1480 mm; and outer span (Z) = 4000 mm. Material properties Elastic-perfectly plastic material model is used in this study. The other material properties chosen are as follows: Young’s modulus (E), 203 GPa; yield stress, 288 MPa; and Poisson’s ratio, 0.3. However, it has been shown later that choice of these specific values does not affect the compliance correlations significantly. Finite element model Twenty-noded solid elements are used to model the straight pipes. Because of symmetry, only one fourth of the pipe is modelled. The total number of elements varied between 832 and 468 with 1–2 element(s) across the thickness of the pipe, and the number of nodes varied

Fig. 2 Pipe with throughwall circumferential crack under four-point bending load.



between 4917 and 3499. Figure 3 shows a typical FE mesh of straight pipe. Two mesh convergence studies have been carried out: one for R/t = 10, 2θ = 90°, and another for R/t = 20, 2θ = 180°, to check the adequacy of the FE mesh employed. For R/t = 10, 2θ = 90°, three mesh sizes are tried: coarse mesh (as used in the study), fine spider web-type mesh near crack tip with one element across the thickness and same spider web-type mesh with two elements across the thickness. For R/t = 20, 2θ = 180°, two mesh sizes are tried: coarse mesh (as used in the study) and fine spider web-type mesh near crack tip with one element across the thickness. It is found that the unloading compliance values remain almost unchanged with finer mesh (average difference of 0.14% and 0.5% for these two cases, respectively). This proves the adequacy of the FE mesh employed. It is quite expected as load–deflection and unloading compliance are gross structural behaviours that do not show much mesh dependence.

RESULTS AND DISCUSSION

Figure 4 shows the typical moment versus CMOD data for R/t = 20. From these data, initial elastic compliance (C0) has been calculated by linear regression of the initial 5% data. It is then normalized using Eq. (3). Zahoor8 proposed the elastic CMOD (δe) of throughwall circumferentially cracked pipe subjected to pure bending moment as follows:

δe ¼ f 2 :πR2

M EI

(5)

( " 1:5 2:94 )# θ θ θ : 1 þ A 6:071 f2 ¼4 þ 24:15 π π π (6)

0:25 R 0:25 A ¼ 0:125 t 0:25 R 3 ¼ 0:4 t π Ro 4 Ri 4 I¼ : 4

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R for 5≤ ≤10 t R ≤20 for 10≤ t (7)

(8)

By rearranging the aforementioned equations, one can write the following: λ0 ¼

C 0 EI ¼ f 2: πR2

(9)

Figure 5 shows the comparison of normalized initial 0 EI elastic compliance, λ0 ¼ CπR 2 , between the present values calculated from the FE results and the Zahoor’s prediction as per Eqs (6)–(9). A good matching may be observed. This validates the general accuracy of the present data. Subsequently, unloading compliance values have been calculated and normalized with respect to initial elastic compliance λ C at various load levels expressed in normalized λ0 ¼ C 0 form as per Eq. (4). Figure 6a–d shows the variation of normalized compliance λλ0 ¼ CC0 with normalized load level m ¼ MML for various crack angles for R/t = 5, 10, 15 and 20. The line λλ0 ¼ CC0 ¼ 1 indicates the condition when there is no geometric hardening/softening; in other words, it indicates that deformation does not affect unloading compliance. It is observed that deformation effect is most prominent for thinner pipes and it gradually diminishes as the relative thickness (t/R) increases. It is also observed that for a given relative thickness (t/R), the effect increases with crack angles and load/deformation. Figure 6a–d also

Fig. 3 Typical finite element mesh.


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Fig. 4 Moment versus CMOD data for various crack angles for R/t = 20.

Fig. 5 Comparison of initial elastic compliance with Zahoor’s8 equations.

shows that except for relatively thicker pipe and smaller crack angles, there is mostly geometric hardening, which leads to decrease of compliance with increased load or deformation. Geometric hardening increases with decrease in relative thickness (t/R) and increase in crack angles and load/deformation. Geometric softening is observed for R/t = 5 for crack angles of 2θ = 30°–90° and for R/t = 10 for crack angles of 2θ = 30°–60°. However, the degree of geometric softening is almost insignificant compared to geometric hardening. The degree of geometric hardening is most prominent for thinner pipe (R/t = 20) in the order of 10% for 2θ = 180° for normalized load level of m ¼ MML ¼ 1. This geometric hardening is attributed to

Fig. 6 Variation of normalized compliance (λ/λ0 = C/C0) with normalized load level (m = M/ML) for various crack angles for (a) R/t = 5, (b) R/ t = 10, (c) R/t = 15 and (d) R/t = 20.



ovalization of the circular cross section of the pipe during deformation. Figure 7 shows the cracked cross section of a pipe with R/t = 20 and 2θ = 180° after deformation at close to plastic collapse moment. The deformation is shown 10 times magnified for clarity. It is seen that circular cross section of the pipe ovalizes during deformation in such a way that area moment of inertia of the cross section is increased leading to increased bending stiffness and hence lower compliance.

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(Eqs (11)–(14)) of normalized compliance (C/C0) have been proposed for R/t = 5, 10, 15 and 20. They are as follows: For R/t = 5 " 1:2335 1:234 # C θ θ (11) ¼ 1 0:301024 0:3012 C0 π π

0:0339m0:736439 0:249899m1:5864 For R/t = 10

" 1:0719 1:07516 # C θ θ ¼ 1 þ 0:000696284: 0:0007: (12) C0 π π

PROPOSED COMPLIANCE EQUATIONS

The present FE results and Zahoor’s equations (Eqs (5)–(9)) of C0 match quite closely (Fig. 5), which validates the present numerical model. However, Zahoor8 expressed CMOD as a function of moment and crack length, where as for use of unloading compliance method to measure crack growth during fracture tests, crack length needs to be expressed as a function of compliance values. Therefore, a new correlation expressing crack length (θ/π) as a func tion of normalized initial elastic compliance λ0 ¼ CπRo EI2 is first proposed. Equation (10) shows the correlation: θ ¼ 1:68471 2:52469:λo 0:760581 π R0:136538 2:39055:λo 0:772705 t

(10)

Figure 8 shows the comparison of FE data points and predictions of the proposed equation (10). The average error is around 1.42%. Subsequently, equations correlating the ratio of compliance at large deformation level and the initial elastic compliance λλ0 ¼ CC0 with normalized crack angle (θ/π) and load level (m = M/ML) are proposed for various R/t values. It is carried out through regression analyses on the FE data. Four different equations

Fig. 7 Ovalization of circular cross section of a pipe (R/t = 20 and 2θ = 180°) at the crack plane.

0:0248:m0:727444 þ 19:9127:m1:6218

For R/t = 15 " 3:97111 4:0437 # C θ θ (13) ¼ 1 0:8238: 0:849025: C0 π π

0:1758:m0:99484 þ 0:0452771:m1:5565 For R/t = 20 " 0:8804 0:886615 # C θ θ (14) ¼ 1 þ 1:28195: 1:3015: C0 π π

0:876637:m1:0408 þ 0:8358:m1:07393

EFFECT OF STRAIN HARDENING ON THE COMPLIANCE

The FE analyses to generate the compliance correlations are carried out assuming elastic-perfectly plastic material behaviour to simplify the material characterization. However, actual materials show strain hardening behaviour. Therefore, a study has been undertaken to check the effect of material strain hardening on compliance values in comparison to postulated elastic-perfectly plastic material behaviour. Two cases have been taken for studies with R/t = 5 and 20, and 2θ = 90° to represent two extreme cases of R/t within the range considered for the present investigation. To represent a realistic material property, true stress–true strain data of one nuclear grade piping carbon steel SA333Gr6 material are taken. Figure 9 shows the stress–strain diagram of the material. For each geometric case, four sets of results are shown with different material parameters, namely (i) actual true stress–true strain data, (ii) elastic-perfectly plastic at YP = 288 MPa, (iii) elasticperfectly plastic at UTS = 420 MPa and (iv) elasticperfectly plastic at flow stress = 354 MPa (average of YP and UTS). Figure 10a and b shows the comparison. It


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Fig. 8 Fitting of FE data to propose initial elastic compliance for various crack angles and R/t: comparison of FE data with predictions of proposed equations.

Fig. 9 Stress–strain diagram of SA333Gr6 material.

may be seen from the figure that if actual true stress–true strain material characterization is idealized as elastic-perfectly plastic material behaviour at flow stress equal to average of yield and ultimate strength, compliance values are very close to each other. It proves that elastic-perfectly plastic material idealization does not affect significantly the compliance values provided one chooses perfect plasticity at appropriate flow stress. However, further studies may be needed to investigate this aspect in more detail.

INDEPENDENCE OF NORMALIZED COMPLIANCE WITH RESPECT TO CHANGES OF GEOMETRIC PARAMETERS

To make the compliance value independent of geometric parameters, for example, radius and thickness, it has been normalized as λ ¼ CEI . A study has been undertaken to πR2

investigate the effectiveness of the normalization of compliance. Two cases of mean radius (R) of 50 and 100 mm are taken for studies with R/t = 5 and 20, and 2θ = 90° to represent two extreme cases of R/t within the range considered for the present investigation. Figure 11 shows that the difference is within acceptable limit. Also, a study has been undertaken to investigate the effect of change of inner and outer span lengths of four-point bending loading on the normalized compliance (C/C0). One typical case of R/t = 20 and 2θ = 90° has been taken for study. In addition to the standard outer and inner span lengths of 4000 and 1480 mm, analysis for another set of outer and inner span lengths of 2000 and 740 mm, respectively, has been carried out and compared with the standard set results. Figure 12 shows the comparison, which shows that the difference is within acceptable limit. These exercises prove that the normalization of the compliance parameters makes the correlations independent of specific geometric dimensions.

HOW TO USE THESE EQUATIONS

The following procedure should be followed to use the previously proposed equations to measure crack growth during fracture tests of TCC pipe: 1 Get unloading compliance (C) value at a particular load level (m = M/ML) from test data. 2 Use flow stress, σf equal to average of yield stress and UTS to calculate M ′L . 3 Get the initial elastic compliance (C0) value corresponding to m using the proposed (C/C0) Eqs (11)–(14) for a particular value of R/t (linear interpolation for intermediate R/t may be adopted). 4 Get the value of (θ/π) from the C0 or λ0 value using the proposed Eq (10).



Fig. 10 Effect of strain hardening on compliance values for 2θ = 90°: (a) R/t = 5 and (b) R/t = 20.

Fig. 11 Effect of change of radius and thickness on the normalized compliance values.


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Fig. 12 Effect of change of outer and inner span lengths of four-point bending loading on the normalized compliance values for R/t = 20 and 2θ = 90°.

CONCLUSION

REFERENCES

New compliance correlations have been proposed for throughwall circumferentially cracked straight pipe under bending load for measurement of crack growth during fracture experiments. Unlike the conventional compliance correlations, the presently proposed ones account for the ovalization of original circular cross section of pipe during deformation. Elastic-perfectly plastic material behaviour has been assumed to characterize the material stress–strain response. However, it has been shown that error due to this approximation with respect to the actual stress–strain behaviour is negligible if one chooses flow stress equal to average of yield and ultimate strength. The proposed correlations are expressed in terms of normalized parameters to make them independent of specific values of geometric dimensions such as radius, thickness and span length of four-point bending loading system.

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Acknowledgements The work was carried out as a part of research programme pursued in Reactor Safety Division, Bhabha Atomic Research Centre (RSD, BARC). CONFLICT OF INTEREST

There are no conflicts of interest.
