Fatigue crack growth modeling of pipeline steels in high pressure gaseous hydrogen

International Journal of Fatigue 62 (2014) 249–257

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Fatigue crack growth modeling of pipeline steels in high pressure gaseous hydrogen q Robert L. Amaro, Elizabeth S. Drexler, Andrew J. Slifka ⇑ Applied Chemicals and Materials Division, Materials Measurement Laboratory, National Institute of Standards and Technology, 325 Broadway, Boulder, CO 80305, USA

a r t i c l e

i n f o

Article history: Received 1 November 2012 Received in revised form 16 April 2013 Accepted 10 October 2013 Available online 21 October 2013 Keywords: Hydrogen-assisted fatigue crack growth High-pressure hydrogen Fatigue crack growth modeling

a b s t r a c t Hydrogen will likely play a key role in a future clean energy economy. However, fundamental understanding of the deleterious effects of hydrogen on the fatigue and fracture properties of pipeline steels is lacking. Furthermore, engineering tools for design and lifetime prediction of pipeline steels in gaseous hydrogen are yet to be developed and implemented into national codes. A constitutive model that couples deformation and hydrogen-diffusion, supporting a phenomenological predictive model for fatigue crack growth, is presented for pipeline steels in high-pressure gaseous hydrogen. The semi-empirical model is predicated upon the hypothesis that one of two mechanisms dominates the fatigue crack growth response, depending upon the applied load and the material’s hydrogen concentration. The model correlates test results well, and illustrates how the deformation mechanisms contribute to fatigue crack propagation in pipeline steels in environments similar to those found in-service. Published by Elsevier Ltd.

1. Introduction Hydrogen has been envisioned as a key component to a future sustainable energy infrastructure. General Motors, Ford, Toyota, Honda and Daimler have all indicated that they will have a hydrogen fuel cell vehicle in production by 2015. However, the current gaseous hydrogen transportation infrastructure in the United States is nowhere near sufficient to support nation-wide sales of hydrogen fuel cell vehicles. Current gaseous hydrogen transportation schemes include hydrogen-specific tube trailers via highway or rail, and pipelines. Hydrogen transport via tube trailer is highly inefficient and insufficient to meet the envisioned demand. Hydrogen transport via steel pipelines is the most cost efficient method to transport hydrogen across the nation [1]. Given that hydrogen has deleterious effects upon steel deformation and fatigue response [2], a full understanding of the constitutive behavior of pipeline steels exposed to gaseous hydrogen is necessary prior to the introduction of hydrogen in pipelines. Furthermore, predictive models that account for the pipeline constitutive behavior and fatigue response are required. Finally, national codes and standards that incorporate the state of the art understanding of hydrogen’s effect upon pipeline steels, including predictive measures for deformation and fatigue, must be created. This work describes a simple constitutive model, coupled with an existing fatigue crack

q

Contribution of NIST, an agency of the US government; not subject to copyright.

⇑ Corresponding author. Tel.: +1 303 497 3744; fax: +1 303 497 5030. E-mail address: [email protected] (A.J. Slifka). 0142-1123/$ - see front matter Published by Elsevier Ltd. http://dx.doi.org/10.1016/j.ijfatigue.2013.10.013

growth rate (FCGR) predictive model, which has been implemented in MATLAB.1 The coupled constitutive-FCG model has been calibrated to two API pipeline steels [3]. Calibration was performed by use of monotonic tensile test results in high pressure gaseous hydrogen as well as FCG tests performed on compact tension (CT) specimens in high pressure gaseous hydrogen. Model predictive efficacy to CT specimen tests is provided. The model is then implemented in conjunction with a stress intensity factor solution in closed-form to predict FCGR in pipes having internal thumbnail shaped cracks. Model predictive capabilities for pipe geometry are discussed in reference to literature results on pressure vessels of similar geometry.

2. Materials Monotonic tension and constant load FCG tests were performed on API-5L X52 and X100 pipeline steels. The nominal chemical composition of the materials tested is provided in Table 1 and optical micrographs of the microstructures are provided in Fig. 1. The X52 microstructure had a characteristic length scale of approximately 10 lm and was composed of ferrite and pearlite. The X100 microstructure had a characteristic length scale on the 1 Commercial equipment, instruments, or materials are identified only in order to adequately specify certain procedures. In no case does such identification imply recommendation or endorsement by the National Institute of Standards and Technology, nor does it imply that the products identified are necessarily the best available for the purpose.

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Table 1 Nominal chemical composition (wt%) for alloys tested.

X100 X52

C

Mn

Si

S

P

Ni

Cr

Mo

Nb + V + Ti

Fe

0.064 0.060

1.870 0.087

0.099 0.120

<0.001 0.006

0.009 0.011

0.470 0.020

0.023 0.030

0.230 –

0.036 0.036

Bal Bal

Fig. 1. Optical micrographs of (a) X52 and (b) X100 microstructures [4].

order of 1 lm and was composed primarily of bainite and acicular ferrite [4]. 3. Experimental procedure Round tensile specimens were machined from late 1990s vintage X52 and X100 pipeline steels per ASTM E8/E8M [5]. Monotonic displacement-controlled tensile tests were performed on an MTS 100 kN servo hydraulic load frame equipped with control software from the manufacturer. All tests were performed in a stainless steel pressure vessel (regardless of environment) rated to 138 MPa internal pressure. Strain was measured by use of an extensometer having 25.4 mm gage length, 38 mm range and a resolution of 1 lm. Force was measured by use of a load cell of proving ring design. The load cell was located inside of the pressure vessel in series with the test specimen and hydraulic actuator. The load cell has a capacity of 10 kip (44 kN) and a resolution of 5 lbf (0.02 kN). Hydrogen used during tensile testing was generated by an on-site electrolyzer. Test results of hydrogen purity are provided in [4]. Monotonic tensile test results are provided in [4]. Load-controlled fatigue crack growth experiments were performed on the same API-5L pipeline steels discussed above. Compact tension specimens in the transverse-longitudinal orientation were tested on a closed-loop 22 kip (100 kN) servo-hydraulic load frame. Tests performed in high-pressure hydrogen (250 psi (1.72 MPa), 1000 psi (6.89 MPa), 3000 psi (20.68 MPa) and 7000 psi (48.26 MPa)) were conducted in a 20 ksi (138 MPa) pressure vessel. Test control was handled by the MTS Multi-Purpose Testware software with built-in load control functionality. All tests were performed per ASTM E647-08 [6]. Feedback for the force loop was provided by a 10 kip (44 kN) proving ring located inside the pressure vessel. The proving ring had a 5 lbf (0.02 kN) resolution. Crack mouth opening displacement was monitored via a clip gage having a 3 mm range and 0.001 mm resolution. Hydrogen gas was either electrolyzed from ultrapure water or provided in ‘‘commercially pure’’ form (99.9999% H2) from an outside vendor. Hydrogen purity was tested periodically by an outside vendor. Hydrogen pressure was maintained during testing via a program written in-house using LabVIEW. All test results reported here were conducted at a load ratio of Rf = 0.5 and a frequency of 1 Hz. The X100 specimens tested were all machined from the same section of pipe, i.e., the same heat, as was the case for the X52 specimens.

4. Test results Monotonic test results of API-5L X52 and X100 pipeline steels in high pressure gaseous hydrogen are shown as a function of hydrogen pressure in Fig. 2 [4]. The modeling here requires the use of true stress–strain relationships. As such the engineering stress– strain data from [4], and shown in Fig. 2, was converted to true stress–strain data up to the point of necking. The true stress–strain response up to 20% strain is shown in Fig. 3. In general, the presence of hydrogen considerably decreases the ductility of the material, as can be seen by the differences in elongation at failure for the specimens tested in air versus hydrogen (Fig. 2). Furthermore, hydrogen appears to have a minimal impact on the strain at which necking occurs. Fatigue crack growth test results in air and high pressure hydrogen are provided in Fig. 4. The FCGR results indicate that the hydrogen-assisted (HA) FCG response approaches that of the air response, at combinations of low hydrogen pressures and low DK. That is, the results indicate that at low hydrogen pressure and sufficiently low DK, the HA FCG would match the FCG response in air. Suresh and Ritchie [7] characterized the transition point at which the HA FCG deviates from that of air as K Tmax , or the transition cycle maximum stress intensity factor. Furthermore, the results indicate that the HA FCG increases with increasing hydrogen test pressure for values of DK below about 20 MPa m1/2. Above this value, the HA FCG trends tend to converge, regardless of hydrogen pressure. Finally, there exists a transition point in the HA FCG data at approximately DKtr  13.0 MPa m1/2, corresponding to a fatigue crack growth per cycle of da/dNtr  3104 mm/cycle. The HA FCG occurring below this transition (da/dN < da/dNtr) will be referred to as ‘‘transient’’ HA FCG. Hydrogen-assisted FCG occurring above the transition (da/dN > da/dNtr) will be referred to as ‘‘steady state’’ HA FCG, for the remainder of this work. The experimental results indicate that da/dNtr occurs at

da=dN tr ¼ 2  106 DK 2

ð1Þ

for both materials tested. The results of Eq. (1) are identical to the second-order estimate of the Irwin plastic zone size for X100 and are proportional to the plastic zone size for X52. It is hypothesized that the transient HA FCG results from crack extension per cycle on the order of (or smaller than) the region of highest hydrostatic

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251

Fig. 2. Engineering stress–strain monotonic response in two environments: (a) API-5L X100 steel and (b) API-5L X52 steel. From [4]. Tests performed in gaseous hydrogen shown as dotted lines.

Fig. 3. True stress–strain monotonic response up to the point of necking in two environments: (a) API-5L X100 steel and (b) API-5L X52 steel. Tests performed in gaseous hydrogen shown as dotted lines.

Δ

(a)

(b)

Fig. 4. FCGR of (a) API-5L X52 steel and (b) API-5L X100 steel as a function of hydrogen test pressure.

stress in front of the crack tip. As this location corresponds to the location of highest hydrogen concentration, the hypothesis assumes that the FCG is increased due to the effect of the high hydrogen concentration in this region. Once the crack extension (per cycle) extends some distance beyond the stress-assisted maximum hydrogen concentration, the rate of the FCGR decreases. The decrease in the rate of FCGR is due to the crack extending through material having steady state hydrogen concentration, which is considerably less than the maximum stress-assisted hydrogen concentration near the crack tip.

4.1. Constitutive modeling In order to characterize the material response to monotonic tensile loading, the true stress–strain experimental data from [4] was fit to a Ramberg–Osgood (R–O) stress–strain relationship [8]. The R–O equation models the elastic and inelastic deformation response of a strain hardening material. The R–O equation





e r r n ¼ þa ; r0 e0 r0

ð2Þ

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incorporates the stress and strain at the onset of inelastic deformation, r0 and e0, respectively. The 0.2% offset yield stress is used for r0, and the strain at yield is determined using Hooke’s law. The relationship also employs the dimensionless constant, a, and the hardening exponent, n. While a hardening exponent value approaching infinity indicates an elastic-perfectly plastic deformation response, a lower value of the exponent is associated with a hardening response in the inelastic regime. Values for the parameters n and a (in Eq. (2)) were determined for the true stress–strain response of the X52 and X100 materials in [4] and provided in Table 2 as a function of hydrogen test pressure. Table 2 also provides the 0.2% offset yield stress, r0, the experimentally determined modulus of elasticity, E, and the calculated strain at yielding, e0. There does not appear to be a statistically significant correlation between the hydrogen test pressure and the hardening exponent n, the parameter a, and the modulus of elasticity E. The Prandtl–Reuss constitutive relationship [9,10] is used to predict the three-dimensional deformation response of the pipeline steel. The Prandtl–Reuss relationship is given by

e_ ¼

r_ kk dij 9K

þ

S_ ij _ ij ; þ kS 2G

ð3Þ

where e_ is the total change in strain per time increment, the first through third terms to the right of the equal sign are the elastic volume change per time increment, the elastic shape change per time increment and the inelastic shape change per time increment, respectively. In this notation the ‘‘dot’’ over a parameter indicates the time rate-of-change of that parameter (d/dt). Parameters in Eq. (3) are defined as follows: r_ kk is the change in hydrostatic stress, dij is the Kronecker delta, K is the bulk modulus, Sij is the deviatoric stress defined by Sij ¼ rij  rkk dij =3; S_ ij , is the time increment in deviatoric stress, G is the shear modulus, and k_ is the time increment of yield surface evolution,

3e_ p k_ ¼ :  2r

ð4Þ

 in Eq. (4) are the time increment in effective The parameters e_ and r strain (defined in Eq. (5)) and the effective stress for each time increment (defined in Eq. (6)), respectively.

rffiffiffiffiffiffiffiffiffiffiffiffiffi 2 p p e_ ¼ e_ e_ 3 ij ij

r ¼

ð5Þ

rffiffiffiffiffiffiffiffiffiffiffiffiffi 3 Sij Sij : 2

ð6Þ

The term e_ pij is the current time increment in inelastic strain. The Ramberg–Osgood stress–strain relationship effectively defines the material’s yield surface evolution and therefore can be rearranged to be used as k_ in Eq. (3). To do so, one first recognizes that Eq. (2) can be explicitly decomposed into elastic and inelastic contribu-

Table 2 Ramberg–Osgood parameters for X52 and X100 steels tested. Material X100 Longitudinal

X100 Transverse X52 Longitudinal X52 Transverse

H2 pressure (Mpa)

r0

e0 –

n –

a

(Mpa)

–

E (Gpa)

AIR 5.5 13.8 27.6 68.9 AIR 13.8 AIR 13.8 AIR 13.8

693.01 700.37 700.90 708.86 714.01 804.47 810.23 442.21 422.11 432.71 440.62

0.0032 0.0032 0.0032 0.0031 0.0033 0.0035 0.0035 0.0021 0.0021 0.0020 0.0020

13.48 13.39 13.78 13.56 14.34 17.18 15.33 11.74 9.40 12.21 12.34

0.92 1.01 1.11 1.03 0.97 2.97 3.52 3.10 3.41 2.36 2.86

214.14 219.17 218.89 229.61 215.74 229.58 230.52 212.42 198.76 215.86 219.51

tions of strain, in which case the inelastic portion of the Ramberg– Osgood relationship is defined as



ep ¼ e0 a



r n : r0

ð7Þ

Replacing the inelastic strain and the stress with their ‘‘effective’’  , respectively, and differentiating ep with recounterparts, ep and r  spect to dr yields the instantaneous inelastic modulus

 Þn1 dep ae0 nðr ¼ : n  dr r0

ð8Þ

Solving for dep yields

dep ¼

 ae0 nðr  Þn1 dr

rn0

;

ð9Þ

which can then substituted into Eq. (4) to yield

 Þn1 3ae0 nðr k_ ¼ :  rn0 2r

ð10Þ

Finally, the Hutchinson, Rice, Rosengren (HRR) model [11,12] is used to determine the stress field in the fatigue process zone (FPZ). The HRR model predicts the stress distribution in front of a sharp crack of a strain-hardening material in polar coordinates (r and h). It is given by

rij¼ r0



EJ ar20 In r

1=nþ1

r~ ij ðn; hÞ;

ð11Þ

where r0 is the material yield strength, E the modulus of elasticity, J the value of the J-contour integral, a the dimensionless constant from the R–O equation, r the distance in front of the crack tip, n ~ ij a dimenthe work hardening exponent from the R–O equation, r sionless function of n and h, and In is an integration constant that ~ ij . The J-integral may be replaced by K 2I /E depends upon n and r for a globally linear elastic material [13], where K is the stress intensity factor. The crack propagation plane is assumed to occur at an angle of h = 0° and the distance, r, is taken as a piece-wise function defined by

8 da=dNN1 6 Eq:1 > < Eq:1; r ¼ da=dN N1 ; da=dNN1 > Eq:1 > :

ð12Þ

where da=dNN1 is the crack extension of the previous cycle. This formulation for the distance, r, assumes a critical distance damage approach to the HA crack extension, similar to that presented by [14]. It is assumed in this work that the stress-assisted hydrogen concentration sets up the critical distance for HA damage accumulation. For crack extension less than the transition crack extension, da/dNtr, the critical distance is taken as being proportional to the second-order estimate of the Irwin plastic zone size. In this case the crack extension occurs within the plastic zone and its associated zone of maximum hydrogen concentration. When crack extension exceeds da/dNtr, the critical distance occurs in a region of material beyond the plastic zone, and therefore within a region of diminished hydrogen concentration. The use of Eq. (12) ensures that the critical distance for HA damage accumulation is consistent with experimental results as well as avoiding the singularity condition presented by the HRR solution at the crack tip. The final form of the constitutive model (Eq. (3) in conjunction with Eqs. (11) and (12)) effectively accounts for the total deformation response of the API steels tested by modeling them as Ramberg–Osgood hardening materials within a constitutive model incorporating a Von Mises associated flow rule. The constitutive model was subsequently implemented with isotropic hardening in MATLAB.

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The constitutive model calibration was performed by determining the R–O parameters for each material. These values were then implemented in the constitutive relation. Results of the R–O monotonic tensile calibration are provided in Table 2. Model predictions to the monotonic tensile data provided in [4] (and shown in Fig. 3) for air and a representative hydrogen pressure are shown in Fig. 5. The model predicts the monotonic uniaxial tensile results well as a function of hydrogen test pressure. In order to validate the model calibration in compression, the model was used to predict the experimental results from [15]. Comparison of the experimental data to the predicted response is provided in Fig. 6. The constitutive model performs well when predicting both uniaxial tension and compression of X100 in air. The constitutive model was then used to predict the material response from a fully reversed load incursion. Though experimental data to calibrate the predicted response is lacking, the hysteresis loop provided by the constitutive model is reasonable, as shown in Fig. 7.

253

Fig. 6. Constitutive model prediction of literature [15] X100 tension and compression test results in air.

4.2. Fatigue crack growth modeling A phenomenological model to predict FCG of API steels in air and hydrogen was presented in [16]. The model is defined as

da da da ¼ þ dðPH  PHth Þ ; dNtotal dNfatigue dNH

ð13Þ

where the first term is given by the Paris relationship, dN da ¼ ADK b , fatigue d is the Dirac delta operator, P Hth is a threshold hydrogen pressure below which HA FCG does not occur, taken to be PHth ¼ 0:02 MPa da [17], and dN is the relationship for HA FCG given by H

da ¼ dNH

"

da dNPH

1



da þ dNDK

1 #1

:

ð14Þ

The first term in Eq. (14) predicts the transient HA FCG and is given by

  d1 da Q þ Vrh ¼ a1DK B1 Pm1 exp ; H dNPH RT

ð15Þ

where a1, B1, m1 and d1 are constants, PH is the hydrogen pressure, Q is the activation energy equal to Q = 27.1 kJ/(mol-K) [18], V = 2.0  106 m3/mol [19–22] is the partial molar volume of hydrogen in the metal, rh is the hydrostatic stress determined from the Prandtl–Reuss constitutive relationship. Note that   Q þVrh Pm1 is the stress-assisted hydrogen concentration withH exp RT in the material. The second term in Eq. (14) predicts the steady-state HA FCG and is defined as

Fig. 7. Constitutive model prediction for fully reversed loading (±800 MPa) at 6.89 MPa hydrogen pressure.

  d2 da Q þ Vrh ¼ a2DK B2 Pm2 ; H exp dNDK RT

ð16Þ

where a2, B2, m2 and d2 are constants, and all other parameters are defined above. The formulation of Eq. (14) treats the transient and steady-state HA FCG as if occurring in series, in which case the HA FCG is predicted to result from the interaction of these two deformation mechanisms. The HA FCG model was implemented in

Fig. 5. Constitutive model prediction and experimental results for X100 tested in: (a) air and (b) 69 MPa hydrogen (both in longitudinal orientation).

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MATLAB. The FCG model is coupled to the constitutive model so that at each time (or load) step the deformation state of the material at the critical damage location is updated. The Full model calibration for API-5L X100 steel exposed to high pressure gaseous hydrogen was provided in [16]. It has been shown however, that changing hydrogen pressure has little effect upon the steady state HA FCG regime. As such, a simplified version of the model has been proposed [3] in which parameter d2 is set to zero, thereby negating the effects of changing hydrogen pressure in Eq. (16). The simplified HA FCG model calibration for API-5L X52 and X100 steel exposed to high pressure gaseous hydrogen was provided in [3]. Model calibration results are provided in Table 3. Results of the hydrogen-specific constitutive model, coupled to the HA FCG model are provided in Fig. 8 for both materials and all hydrogen pressures tested. The HA FCG model, coupled to the constitutive model, performs well at predicting HA FCG of CT specimens for all conditions tested. 4.3. Model efficacy In order to validate the efficacy of the complete HA FCG model, the experimental and correlated FCG results are compared in Fig. 9. The model predicts 98% of the X100 data points and 97% of the X52 experimental data points within a factor of two. When used to predict the total number of cycles to test completion, the HA FCG model again predicts the results within a factor of two, as shown in Fig. 10. A parametric study was then conducted in order to illustrate the effects of changing the initial crack length (i.e. the smallest detectable flaw size in an operational pipe) as well as the hydrogen test pressure upon cycles to failure. The X100 constitutive and HA FCG model calibration was used in all subsequent analysis. The parametric study results are provided in Fig. 11. The study results indicate that decreasing the initial crack size from 5 mm to 1 mm increases the life of the CT specimen by tenfold. This result is not surprising, given the general understanding of the effects of the initial crack size upon overall fatigue crack growth. What should be noted, however, is that the current pipeline surface flaw detection capabilities are on the order of 1 mm. Furthermore, detection of 1 mm-sized surface flaw cracks is not guaranteed, as the current detection capabilities are not consistent. Fig. 11a) sheds light upon the importance of improving the flaw detection capabilities, by both decreasing the detectible flaw size and the repeatability in which a particular flaw size is detected. Fig. 11b) indicates that increasing the hydrogen pressure from approximately 7 MPa to approximately 20 MPa has only a minimal effect upon cycles to failure. However, decreasing the hydrogen pressure below about 4 MPa has a marked increase in life. The combined constitutive-HA FCG model was then implemented to predict FCG of pipeline geometries. Comparisons of model predictions are made to the test results presented in [23,24]. In this work ten 4130X grade steel pressure vessels (having diameters and thicknesses similar to what would be used in pipeline hydrogen transportation) were plumbed in parallel and pres-

Table 3 Fatigue crack growth model calibration for two API steels. API-5L X52 A

7.1  l09 Transient

a1 B1 m1 d1

1.2  l015 10.24 0.7 1

API-5L X100 b

2.9

A

Steady state a2 B2 m2 d2

3.4  107 2.9 N/A 0

9.9  l09 Transient

a1 B1 m1 d1

3  1013 7.96 0.7 1

b

2.83 Steady state

a2 B2 m2 d2

4  107 2.83 N/A 0

surized with gaseous hydrogen between 3 MPa and 44 MPa. Engineered flaws were electrical discharge machined (EDM) into the inner diameter of the pressure vessels at locations that ensured that the stress experienced by any given flaw was affected by neither neighboring flaws, nor the pressure vessel end caps. In this case the experimental results would be indicative of a pipe having an internal flaw experiencing similar hydrogen pressurization. The vast majority of the flaws were machined to have an aspect ratio of 2c/a = 3, where a is the length of the crack depth and 2c is the length of the crack intersecting the interior wall surface. Furthermore, all flaws had an inner root radius of 0.5 mm. In which case the flaws cannot strictly be modeled as cracks using Elastic–Plastic Fracture Mechanics (EPFM) or Linear-Elastic Fracture Mechanics (LEFM) because they are not sharp enough, though it is common practice to do so. Each pressure cycle occurred over five minutes, and cycling continued until a flaw grew to wall thickness in depth. Of the ten pressure vessels tested, four initiated and grew a crack from the EDM notch to wall thickness depth. These results speak to the statistical nature of real-world deformation and crack growth mechanisms. Initial modeling of the FCG and cycles to failure was performed by use of a crack driving force (DK) relationship from Anderson [13], which was also used by San Marchi et al. [24]. The relationship for a part through-crack on the interior of a pressurized pipe is given as

DK ¼ ðPmax  Pmin Þ

 rffiffiffiffiffiffi r pa F; t Q

ð17Þ

where Pmax and Pmin are the maximum and minimum internal pressure, respectively, r is the average radius of the vessel or pipe, t is the wall thickness, and a is the crack depth. The parameter Q is a crack-shape specific function defined as

Q ¼ 1 þ 1:464

a1:65 ; c

ð18Þ

where c is the half-length of the crack where it intersects the surface. The parameter F is a shape function defined as

F ¼ 1:12 þ 0:053n þ 0:0055n2 þ ð1 þ 0:02n þ 0:0191n2 Þ 

ð20  rtÞ2 ; 1400

ð19Þ

with n ¼ 2c=t. Utilizing Eq. (17), and making the necessary assumption that the EDM flaw can be characterized as a sharp crack, the model predicted 2100 cycles to through-wall thickness. The analysis was performed using an initial crack depth of 1 mm and maintaining a semi-circular shape throughout crack propagation. The experimental results, however, indicate that failure occurred in the vessels after between 8000 cycles and 15,000 cycles (with an average life of 12,900 cycles), depending upon the geometry of the EDM flaw. As detailed in San Marchi et al. [24], the difference between the predicted cycles to failure and the experimental cycles to failure may be due to the fact that the engineered flaw behaved as a stress riser, rather than a stress concentration, or a sharp crack. The difference between the actual cycles to failure and the predicted cycles to failure is likely the number of cycles required to initiate a sharp crack, or between 6000 cycles and 13,000 cycles. One will note, then, that the assumption that the EDM flaws behaved as a crack was necessary for the analysis, yet none-the-less strictly inappropriate. The analysis was then re-evaluated by varying the initial flaw size, while holding all other test parameters constant. Results indicate that an initial flaw size of 1 lm, or on the order of the microstructural length scale of the model calibration material, resulted in failure after 8000 cycles. Interestingly, if one uses the stress intensity solution proposed by Underwood [25], the predicted number of cycles to failure is drastically increased. The stress intensity solution is given by

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255

Fig. 8. Model prediction superimposed upon experimental data for (a) API-5L X52 steel and (b) API-5L X100 steel as a function of test environment. Test data shown as open symbols to aid in correlation trend visibility.

Fig. 9. Actual FCG versus correlated FCG. X52 left, X100 right. Bounds shown with dashed blue lines represent 2X predictive capability. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

Fig. 10. Comparison of experimental to model prediction for cycles to failure of CT specimens tested in gaseous hydrogen. Bounds shown with dashed blue lines represent 2X predictive capability. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

"

DK ¼ a 1:12

ðW 2 þ 1Þ ðW 2  1Þ

# þ 1:13 DP

pffiffiffiffiffiffi pa;

ð20Þ

where DP is the internal pressure range, W is the wall thickness ratio (W = router/rinner), a is the shape factor given in Table 4 and all other parameters have been defined above. Utilizing the stress intensity factor in Eq. (20) and an initial flaw size of 1 mm, the model predicts 10,500 cycles to failure for an aspect ratio of a/2c = 0.3 and 23,200 cycles to failure for an aspect ratio of a/2c = 0.5. In Rice and Levy [26], the original work describing

the shape factor used by Underwood and provided in Table 4, the stress intensity solution as well as the shape factor are described to be applicable for geometries encompassing those analyzed here. Furthermore, Rice and Levy predict considerable ‘‘load shedding’’ as the crack propagates through the wall thickness. It is this load shedding that is accounted for by the reduction in the shape factor (provided in Table 4) as a function of increased crack growth and the considerable increase in predicted life relative to the analysis using the solution of Anderson [13]. Finally, the stress intensity solution of Anderson is strictly only applicable for crack growth up to 80% of the wall thickness. Using the Anderson stress intensity solution, an aspect ratio of a/2c = 0.5, and an initial flaw size of 1 mm, the model predicts 2050 cycles for the crack to grow to 80% of the wall thickness. The stress intensity solution of Underwood is strictly only applicable for crack growth up to 60% of the wall thickness. Using the Underwood stress intensity solution, an aspect ratio of a/2c = 0.5, and an initial flaw size of 1 mm, the model predicts 11,000 cycles for the crack to grow to 60% of the wall thickness. Given that both stress-intensity solutions are based upon the assumption that the flaw is a ‘‘sharp crack’’, neither of the stress-intensity solutions are strictly applicable to the EDM flaws analyzed here prior to crack initiation. Rather, perhaps a large scale plasticity analysis should be performed incorporating a stress concentration factor to bridge the gap between EDM flaw geometry and EPFM/LEFM validity. Given the lack of models for crack initiation, however, this analysis is not possible at this time. Therefore, engineers must chose a stress-intensity solution from the literature and apply it cautiously. The authors have no opinion on the validity of either stress-intensity solution used here. Rather,

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Fig. 11. Results of the parametric study of the effect of (a) initial crack size (at a hydrogen pressure of 6.89 MPa) and (b) changing hydrogen pressure (at an initial crack size of 6 mm).

Table 4 Shape factor for stress intensity solution of Underwood [25].

5. Conclusions

Shape factor a a/t

Straight-front a/2c = 0

a/2c = 0.0l

a/2c = 0.1

a/2c = 0.3

Semicircular a/2c = 0.5

0 0.2 0.4 0.6

1 1 1 1

1 0.97 0.92 0.81

0.93 0.77 0.59 0.38

0.74 0.54 0.37 0.22

0.59 0.44 0.29 0.18

the purpose of this study is to apply two vastly different solutions to predict cycles to failure, and illuminate the issue involved with selecting appropriate stress-intensity solutions. Finally, a parametric study was performed by use of the Anderson stress intensity solution. Pipe geometry in this analysis is identical to the analysis above and detailed in [23,24] and the internal hydrogen pressure was varied between 10 MPa and 43.5 MPa. The load ratio was maintained constant at 0.07, identical to the San Marchi work. Results, in stress-life format, as a function of initial crack size are provided in Fig. 12. The results indicate that for an internal design pressure of 40 MPa and design life of 9000 cycles, the smallest detectable flaw size must be on the order of 0.1 mm. However, with the current flaw detection capabilities of 1 mm, and a design life of 9000 cycles, the maximum operating pressure could not exceed 30 MPa.

Fig. 12. Study results for pipe containing internally pressurized hydrogen. Results delineated by initial flaw size.

This paper presents a coupled constitutive-HA FCG model for pipeline steels exposed to high pressure gaseous hydrogen. Calibration of the constitutive model requires material-specific monotonic tensile tests to be performed in high-pressure gaseous hydrogen. Calibration of the HA FCG model requires FCG tests to be performed in high pressure gaseous hydrogen. However, as was shown elsewhere [3] model calibration can be performed with a minimal number of FCG tests. The coupled model has been implemented in MATLAB. The model implementation for CT specimens predicts FCG and cycles to final crack size within a factor of 2X. The model implementation for pressurized pipes having an internal flaw requires a closed-form stress intensity solution of unknown form, given that the two stress intensity solutions implemented above provided vastly different predictions. However, the model in its current form provides excellent insight to the effects of initial crack size and hydrogen operating pressure. References [1] Öney F, Vezirolu TN, Dülger Z. Evaluation of pipeline transportation of hydrogen and natural gas mixtures. Int J Hydrogen Energy 1994;19(10):813–22. [2] Somerday BP. Technical reference on hydrogen compatibility of materials. Plain carbon ferritic steels: C–MN alloys (code 1100). In: San Marchi C, Somerday BP, editors. Sandia National Laboratories, Livermore CA. [3] Amaro RL, Drexler ES, Rustagi N, Nanninga NE, Levy YS, Slifka AJ. Fatigue crack growth of pipeline steels in gaseous hydrogen-predictive model calibrated to API-5L X52. In: Sofronis P, Somerday BP, editors. 2012 International hydrogen conference. Jackson Lake, WY, USA; 2012. [4] Nanninga NE, Levy YS, Drexler ES, Condon RT, Stevenson AE, Slifka AJ. Comparison of hydrogen embrittlement in three pipeline steels in high pressure gaseous hydrogen environments. Corros Sci 2012;59:1–9. [5] ASTM Standard E8/E8M-09. Standard test method for tension testing of metallic materials. West Conshohocken, PA; 2009. [6] ASTM Standard E 647-08. Test method for measurement of fatigue crack growth rates. ASTM International; 2008. p. 45. [7] Suresh S, Ritchie RO. Mechanistic dissimilarities between environmentally influenced fatigue-crack propagation at near-threshold and higher growth rates in lower strength steels. Metal Sci 1982;16(11):529–38. [8] Ramberg W, Osgood WR. Description of stress–strain curve by three parameters. Washington (DC): National Advisory Committee for Aeronautics; 1943. p. Note 902. [9] Prandtl L. Spannungsverteilung im Plastischen Koerpern. In: Proceedings of the 1st international congress on applied mechanics; 1924. [10] Reuss A. Berücksichtigung der elastischen Formänderung in der Plastizitätstheorie. ZAMM – J Appl Math Mech/Zeitschrift für Angewandte Mathematik und Mechanik 1930;10(3):266–74. [11] Hutchinson JW. Singular behaviour at the end of a tensile crack in a hardening material. J Mech Phys Solids 1968;16(1):13–31. [12] Rice JR, Rosengren GF. Plane strain deformation near a crack tip in a power-law hardening material. J Mech Phys Solids 1968;16(1):1–12. [13] Anderson TL. Fracture mechanics fundamentals and applications. 3rd ed. Boca Raton (FL): CRC Press; 2005. 621.

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