Service life of corrosion pitted pipes subject to fatigue loading and hydrogen embrittlement

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Service life of corrosion pitted pipes subject to fatigue loading and hydrogen embrittlement Ali Rajabipour*, Robert E. Melchers Centre for Infrastructure Performance and Reliability, The University of Newcastle, NSW 2308, Australia

article info

abstract

Article history:

The structural service life of brittle material pipes with exterior corrosion pits is likely to

Received 10 January 2018

depend on crack initiation and crack development and this may be influenced by pressure

Received in revised form

loading fluctuations and the possibility of material hydrogen embrittlement. Recently

9 March 2018

developed methods are used to estimate the cracking pattern, the failure state of the crack

Accepted 11 March 2018

development from external pits and the rate of Hydrogen-Assisted Cracking under fluc-

Available online xxx

tuating loadings. The effect of hydrogen from the surrounding environment on the

Keywords:

initiated from surface pits is estimated as a function of pipe age. A realistic example is

Service life prediction

presented and the results discussed.

cracking rate is formulated using a generalized form of Paris' law. The depth of cracks

Pitting corrosion

© 2018 Hydrogen Energy Publications LLC. Published by Elsevier Ltd. All rights reserved.

Fatigue loading Hydrogen embrittlement

Introduction Water supply pressure pipes typically are subject to daily operating pressure variations that are essentially low frequency loading [1]. Conventionally the resulting crack development is considered in the context either of Stress Corrosion Cracking [2,3] or of corrosion fatigue [1,4e7]. Where hydrogen is involved, the crack growth rate usually is considered to increase with decreasing loading frequency since hydrogen has more time to diffuse into and within the metal under lower frequencies [4,7e11]. Many numerical studies [12e16] and experimental reports [17e20] show that cracks along pipes often commence at the sites of corrosion pits [21,22]. Stress concentrations at corrosion pits magnify the effects of hoop stress resulting from pressure fluctuations in a pipe. Thus, compared with the intact surface of a pipe, pitted sites receive a wider range of

stresses. This intensifies the fatigue loading at pits. Cracking from corrosion pits can be modelled using the approach proposed in Refs. [23,24]. This situation is magnified if there is also the presence of hydrogen. Under certain exposure conditions hydrogen can reach into pits, gouges and cracks in the microstructure on the external surface of pipes [25e27]. Also, in aqueous soil environments hydrogen can be generated electrochemically on the external surface of ferrous pipes [28]. Hydrogen embrittlement can reduce the structural strength [29e31] and the ductility [32e34] of metals and thus increase the rate of fatigue cracking. As such, rate of fatigue cracking increases with hydrogen pressure in pipelines [35e38]. When the crack is assumed to be of a simple, predefined general shape, the approach described in Ref. [1] can be used for modelling crack propagation from pits under hydrogen conditions (herein termed Hydrogen-Assisted Fatigue Cracking, HA-FC). However, more realistically, the shape of a crack starting from a pit is likely to be influenced by the

* Corresponding author. E-mail address: [email protected] (A. Rajabipour). https://doi.org/10.1016/j.ijhydene.2018.03.063 0360-3199/© 2018 Hydrogen Energy Publications LLC. Published by Elsevier Ltd. All rights reserved. Please cite this article in press as: Rajabipour A, Melchers RE, Service life of corrosion pitted pipes subject to fatigue loading and hydrogen embrittlement, International Journal of Hydrogen Energy (2018), https://doi.org/10.1016/j.ijhydene.2018.03.063

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stress field in the neighbourhood of the crack front and this is likely to be influenced by pit shape. Ideally, therefore, both the crack front and the pit shape should be incorporated in modelling the development of the crack. For this a computational model coupling hydrogen diffusion and crack propagation is required. Computational models for representing Hydrogen-Assisted Fatigue Cracking (HA-FC) usually are classified in two general categories according to the effect hydrogen has on material fracture properties [39]. In the first category the failure of hydrogen affected region is attributed to hydrogen enhanced decohesion” (HEDE). In this approach a so-called “traction separation law” [40e43] is used to overcome the singularity of the crack tip. The second category includes methods in which failure is attributed to “hydrogen enhanced local plasticity” (HELP) [44,45]. Herein a novel methodology based on Linear Elastic Fracture Mechanics (LEFM) is used to estimate the service life of pipes affected by corrosion pitting and influenced by Hydrogen-Assisted Fatigue Cracking. The main concepts of the approach used are as follows. First, for a pipe under the influence of hydrogen the likely cracking patterns are estimated using the cracking pattern in the equivalent unaffected pipe. Second, the failure state of a pitted pipe is defined, based on the state of crack development from the pits. In the third step, the rate of cracking under HA-FC is estimated by reducing the material fracture toughness as a consequence of hydrogen diffusion from the crack front, using the method proposed in Ref. [46]. Finally, having available the rate of crack development from the previous step, the location of the crack front is estimated under a HA-FC mechanism and the time required to reach the failure state is estimated. Fatigue formulation herein has been adopted from authors previous works and are based on experimental tests by others. The theory on which service life prediction method is based on in this paper should be an incentive to conduct future long term experimental observations/tests to validate the method proposed in here.

Hydrogen diffusion under low frequency cyclic loading Like all diffusion processes, the diffusion of hydrogen is driven by the gradient of hydrogen concentration (potential) [43]. Several material factors influence the diffusion process. These include the diffusivity ‘constant’ D, the extent of local plasticity, the rate of plastic strain, the presence of sites that trap hydrogen and the intensity and form of the local stress field [47e50]. Herein these are taken into account by an equivalent (or apparent) diffusivity, Dapp . For most water pipes under normal operational conditions the pressure fluctuations cause only a limited amount of fluctuation in the range of stress intensity ðDKÞ at the tip of any crack that may have formed. This means it is plausible to assume Dapp changes only slightly with DK and hence it is reasonable to assume a constant value of Dapp in calculations. In the following only one-dimensional diffusion is considered. This is based on reports that diffusion in the direction normal to the crack plane does not influence the distribution of hydrogen concentration ahead the crack

and is therefore of minor influence in governing diffusion [43,51,52]. Based on these assumptions, the diffusion equation can be simplified to: vu v2 u ¼ Dapp 2 vt vx

(1)

in which: u: Relative concentration of hydrogen in the metal, defined 
concentration of hydrogen in the metal as concentration of hydrogen in the envoronment t: Elapsed time, starting from the formation of a new crack tip Dapp : Apparent diffusivity of hydrogen in the metal x: Distance in the direction of the crack direction, from the crack tip to where the hydrogen concentration is being considered. For simplicity, let the relative concentration of hydrogen at the tip of the crack, where the metal is in direct contact with hydrogen, be assumed to be unity, irrespective of the hydrogen pressure in the surrounding environment. This means that the boundary conditions are assumed to move with the crack as it grows. Thus, to solve Equation (1) under moving boundary conditions the origin of the coordination system is reset to be at the crack tip for each incremental computational step. In consequence, the boundary conditions for each increment of crack growth are: un ð0; tÞ ¼ 1

(2)

lim un ðx; tÞ ¼ 0

(3)

x/∞

in which un is the relative concentration of hydrogen in the metal after the occurrence of the nth computational increment. As noted, the solution of Equation (1) can be considered as a moving boundary condition problem, with the initial condition for the next increment being the solution of the previous increment. Recalling that the origin is set at the crack tip, the initial condition for nth increment is: un ðx; 0Þ ¼ un1 ðx þ xn ; tn Þ

(4)

Let it now be assumed that the rate of cracking in fatigue is slow and that the frequency of applied loading is low. As a result, there is more time for hydrogen to diffuse through the metal. It is therefore reasonable to assume that the hydrogen concentration at the crack tip is similar to that in the surrounding environment. Further, this is likely to be the case at every stage in the development of the crack. Some notion of the hydrogen concentration ahead the crack tip can be obtained by considering just the first increment (i.e. the time at which the crack just starts to grow). In this first increment of crack growth, hydrogen is unlikely to have diffused fully into the metal so the initial condition, Equation (4), is zero. The hydrogen concentration in this step of development is obtained as a simple closed form solution of the diffusion problem [53]:

Please cite this article in press as: Rajabipour A, Melchers RE, Service life of corrosion pitted pipes subject to fatigue loading and hydrogen embrittlement, International Journal of Hydrogen Energy (2018), https://doi.org/10.1016/j.ijhydene.2018.03.063

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u1 ðxÞ ¼ 1  erf

x pffiffiffiffiffiffiffiffiffiffiffiffi 2 Dapp t

 (5)

For illustration consider now the case of a water pipe for which the period of internal pressure fluctuations is 12 h for diurnal pressure variations [5]. Also, let the period of maximum pressure be assumed 8 h. Under this condition the solution to Equation (5) is shown in Fig. 1 for a hydrogen 13

ðm =sÞ (from Ref. [54]). diffusion diffusivity of ¼ 9  10 For the specific case of cast iron pipes under fatigue loading the crack increment in one cycle has been observed to be of the order of 105 mm [5]. This lies at the very left-hand side of Fig. 1 and for the present example it shows that during an increment of crack tip growth the hydrogen concentration ahead the crack tip is almost the same as that in the surrounding environment. For the next step in the growth of the crack, that is, crack tip growth in the second load cycle, the initial condition is not zero (cf. Equation (4)). This means that the hydrogen concentration will be somewhat greater than that shown in Fig. 1. It follows that it is plausible to assume that under normal cyclic loading periods, and under the normal range of pressure fluctuations, the crack tip moves in a hydrogen concentration which is approximately uniform and closely equal to the hydrogen concentration in the surrounding environment. This allows the application of the method described in Ref. [46] for estimating the rate of Hydrogen-Assisted Fatigue Cracking. This method is outlined in the next section and used to estimate the development of crack growth to a defined failure criterion. 2

Service life prediction In this section the service life of a pitted pipe is estimated under several different scenarios. Fatigue cracking, both in the absence and in the presence of hydrogen, is modelled using the formulation in Ref. [46] for the rate of cracking in a HA-FC mechanism, as follows: daH DK2þa ¼ 2 a F  ½lðDKÞa dN sy KIc

(6)

in which:

lðDKÞ ¼

1 1  mðDKÞ

8 lmin > > > < lmax  lmin ðDK  DK0 Þ lðDKÞ ¼ lmin þ > DK1  DK0 > > : lmax

DK  DK0 DK0 < DK  DK1

(7)

DK1 < DK

in which lmin ¼

1 1 and lmax ¼ 1  mmin 1  mmax

in which: daH dN :

Crack growth rate under hydrogen assisted fatigue DK: Range of stress intensity factor in a loading cycle sy : Yield stress of the virgin metal (not embrittled) KIC : Fracture toughness of the virgin metal (i.e. not embrittled) m: Reduction factor for fracture toughness reduction caused by hydrogen concentration a and F are constant values for the problem herein.

Let the depth of the crack be used as an indicator of damage, as described in Ref. [55]. In particular, the critical stage of crack growth occurs when crack depth, growing towards the pipe centreline inwards from the exterior surface of the pipe, reaches the internal surface of the pipe. Let this be the failure criterion and be termed ‘full penetration cracking’ [55]. An example is as follows. Consider a cast iron pipe with 15 mm wall thickness and 500 mm external diameter. For this pipe several different shapes of pits can be considered and cracking patterns under monotonically increasing internal pressure obtained [55]. Also, eXtended Finite Element Method (XFEM) [56] can be employed to model the cracking process numerically. As in Ref. [55] let the contours of the crack fronts computed when hydrogen has not diffused into the metal be termed ‘homogenous front contours’. For the present example these can be estimated from typical material properties for cast iron pipelines as summarized in Table 1 [5,57,58]. Fig. 2 shows the homogeneous crack fronts when the pit depth ðDÞ and width ðWÞ are 6 (mm) and 12 (mm), respectively.

Estimating stress at the crack front The stress intensity factor at the crack front can be estimated using the ‘homogeneous front contours’ defined above, such as those shown in Fig. 2. When the actual crack front (including the effect of hydrogen) is at a homogeneous front contour (for example at HFC4 in Fig. 2) the crack commences

Table 1 e Material properties in the model. Parameter Fig. 1 e Relative hydrogen concentration ahead the crack under different loading periods. Hydrogen diffusivity ¼ 9 £ 10¡13ðm2 =sÞ.

Elastic modulus Poisson's ratio Yield stress Fracture toughness

Value 80 GPa 0.3 80 MPa 14 MPa m0.5

Please cite this article in press as: Rajabipour A, Melchers RE, Service life of corrosion pitted pipes subject to fatigue loading and hydrogen embrittlement, International Journal of Hydrogen Energy (2018), https://doi.org/10.1016/j.ijhydene.2018.03.063

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3 1 HF C

2

PIT

HFC

1

A homogeneous front contour

3 H

FC

4

C HF

3.77MPa

5 6

C HF

HF C

2.60MPa 2.89MPa 3.18MPa

4.35MPa 4.65MPa 4.84MPa

Fig. 2 e Homogeneous front contours and the associated internal pressures when W ¼ 12 mm and D ¼ 6 mm. HFC: Homogeneous Front Contour.

Crack front at HFC1 Crack front at HFC3 Crack front at HFC6

K = K IC

4.84

3.77

2.60

K , Stress I ntensi ty Factor at the crack front (MPa.m )

to propagate when the stress intensity factor reaches its critical value. Thus when the crack front is at HFC4 and the pressure is 4.35 MPa, the stress intensity factor K is equal to KIC . Also, when the pressure is zero K is equal to zero. To estimate the stress at the stress front let it be assumed that the so-called Small Scale Yield (SSY) condition is valid [59]. It allows the use of plasticity immediately local to the crack tip while elsewhere the metal has brittle or closely brittle

behaviour. Under the assumption of Small Scale Yielding it is feasible to apply Linear Elastic Fracture Mechanics (LEFM) for computational purposes [60e62]. This means that using the SSY assumption results in a linear relation between pressure and K. It follows that for every development of crack front (HFC1, HFC2, etc. in Fig. 2) and given the pipe pressure, the value of K at the crack front can be calculated. Fig. 3 shows the value of the stress intensity factor at the crack front under different pipe internal pressure values and for different stages of crack development. In the analysis the elastic modulus is required, but this is known not change by more than 2% as a result of hydrogen embrittlement [43,63,64]. Thus it is reasonable to assume that the linear relation between K and pipe internal pressure (Fig. 3) remains almost unchanged even when under hydrogen diffusion conditions.

Hydrogen-Assisted Fatigue Cracking To show how cracking under hydrogen assisted fatigue can be computed using the formulation presented above, it will be convenient to consider the example of two pit shapes for which homogeneous front contours are estimated. Consider two pits, A and B, with size parameters W and D as defined in Fig. 2 with values shown in Table 2. In both cases the pipe wall thickness ðSÞ is 15 mm and pipes diameter is 500 mm. Following earlier work [46], consider first the application of Paris' law for fatigue cracking and assume the values of Paris' law presented in Ref. [5] are valid for water pipe cast iron in the present example. This gives Paris' law as Equation (8):

Pressure (MPa) Fig. 3 e Stress intensity factor ðKÞ at the crack front versus pipe internal pressure. The linear relation between pressure and K depends on level of crack development (HFC1, HFC2, etc. in Fig. 2).

Table 2 e Shape parameters of the considered pits. Pit name A B

WðmmÞ in Fig. 2

D ðmmÞ in Fig. 2

12 27

6 13.5

Please cite this article in press as: Rajabipour A, Melchers RE, Service life of corrosion pitted pipes subject to fatigue loading and hydrogen embrittlement, International Journal of Hydrogen Energy (2018), https://doi.org/10.1016/j.ijhydene.2018.03.063

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  da ¼ 1:3 1014  DK10 dN

(8)

It is now possible to employ Equation (6) to give the rate of cracking as: 8
  daH 1 ¼ 1:3 1014  DK10  1m dN

(9)

The next step is to use Equation (9) to estimate the development (growth) of the crack in each cycle of loading and, when the crack increment has been estimated for a cycle, to update the crack front. For this computation a program written in Mathematica 10 was used. The inputs are the contours of the homogeneous fronts, the range of pressure fluctuation ðDPÞ and the constants in Equation (9). The program calculates the range of stress intensity factor ðDKÞ at the crack front respective to ðDPÞ and the location of the crack front. The calculated DK is then used to estimate the crack development in a cycle. Results for pits A and B were obtained as presented in Figs 4e7. These Figures also show the sensitivity of crack depth development to the range of pressure change, DP and to the reduction factor for fracture toughness due to hydrogen diffusion, m.

Combined effects of Hydrogen-Assisted Fatigue Cracking and pressure surge As noted in the Introduction, it is possible for pressure surges (water hammer) to influence the development of cracks. This is now considered. Let it be assumed that the influence of strain rate on material properties can be ignored [65,66]. This allows it to be assumed that a pressure surge acts as an incrementally increasing internal pressure. With this assumption and in the special case in which hydrogen embrittlement is not present, it follows that the development of the crack front under a specific pressure surge can be obtained directly using the homogeneous front contours. For pits A and B (as defined above) and for a pressure of 1 MPa at the time of the surge Fig. 8 shows the increase in crack depth with increasing pressure surge. It is emphasised that hydrogen diffusion and the resultant metal embrittlement are not considered in Fig. 8. These aspects are considered further below. For convenience let the function CDS ðPÞ define the relationship between pressure surge P and crack depth (for example, as shown by the functions in Fig. 8). Also, let function CDF ðtÞ defines the relationship between time t and crack depth under a fatigue mechanism (such as shown by the functions in Figs. 4e7). Assume now that at time ts a pressure

μ=0.3

15.0

10

μ=0.2

14.8

μ=0.4 14.6

Crack depth mm

Crack depth mm

9

μ=0.3

8

μ=0.2 7

μ=0.1

14.4 14.2

μ=0.0

14.0 13.8

μ=0.0

13.6

6

13.4 0

50

100

150

200

250

300

0

50

100

Fig. 4 e Crack depth versus time for pit A. DP ¼ 1 MPa and m varies.

150

200

250

300

Time Year

Time Year

Fig. 6 e Crack depth versus time for pit B. DP ¼ 1 MPa and m varies.

15.0

=0.8 MPa

14.8

=0.7MPa

Crack depth mm

14.6 14.4 14.2 14.0

=0.6 MPa

13.8

=0.5 MPa

13.6 13.4

0

50

100

150

200

250

300

Time Year

Fig. 5 e Crack depth versus time for pit A. m ¼ 0.3 and DP varies. DP values are in MPa.

Fig. 7 e Crack depth versus time for pit B. m ¼ 0.3 and DP varies. DP values are in MPa.

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Crack depth

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*

CDF(t) Fig. 8 e Crack depth versus pressure surge for pits A and B. Working pressure at the time of pressure surge is 1 MPa.

surge of magnitude Ps occurs. Under this scenario if CDF ðts Þ  CDS ðPs Þ, the pressure surge is not sufficiently large to develop the crack front. Thus the crack depth is influenced only by the fatigue mechanism: CDðtÞ ¼ CDF ðts Þ

(10)

in which CDðtÞ is the crack depth at time t. However, if CDF ðts Þ < CDS ðPs Þ, the pressure surge is large enough to develop the crack front and in this scenario the crack depth is influenced both by the fatigue mechanism and by the pressure surge. In this case the rate of fatigue cracking will change immediately after the pressure surge ends. This change is due to sudden development of the crack front causing DK to decrease immediately after the pressure surge. It should be evident that the rate of fatigue cracking will become equal to the rate corresponding to the crack having reached a depth of CDS ðPs Þ. This is illustrated schematically in Fig. 9. The curve CDF ðts Þ shows crack depth as a function of time when there is no occurrence of a pressure surge. However, if there is a pressure surge, such as at time ts there is a sudden increase in crack depth as shown. The new curved is marked CDðtÞ. The crack growth due to the pressure surge at ts (the jump at ts ) is equivalent to the crack growth in CDF ðtÞ (without pressure surge) when time reaches to t* . t* is the time needed for the crack depth to reach CDS ðPs Þ on the CDF ðtÞ diagram. Therefore, mathematically t* is the value of the inverse function of CDF ðtÞ when the crack depth is CDS ðPs Þ. In summary, the total crack depth as a function of time under fatigue and under pressure surge will be: 8 <



CDF ðtÞ   t  ts CDS þ CDF t* þ t  ts t > ts : CDF ðts Þ  CDS ðPs Þ ¼ > CDðtÞ ¼ CDF ðtÞ CDF ðts Þ < CDS ðPs Þ ¼ > CDðtÞ ¼

(11) in which: t* ¼ CD1 F ðCDS ðPs ÞÞ (See Fig. 9). Ps : Pressure surge at time ts CDF ðtÞ : Crack depth as a function of time due to fatigue cracking

CD(t)

ts

t*

Time

Fig. 9 e Schematic diagram of crack depth versus time when pressure surge has happened at ts and CDF ðts Þ  CDS ðPs Þ.

CDS ðPÞ : Crack depth as a function of pressure magnitude due to pressure surge CDðtÞ : Crack depth influenced by fatigue cracking and pressure surge as a function of time Let attention now be turned to the more complex case involving the combined effect of fatigue and pressure surge under the specific condition that hydrogen is available and diffuses into the metal. As before, hydrogen concentration decreases with increasing distance from the crack tip. Since fracture toughness depends on hydrogen concentration, it increases with distance from the crack tip. It follows that when hydrogen diffusion occurs, the homogeneous front contours that are obtained under constant fracture toughness cannot be used directly to find the new crack front under a pressure surge. However, similar to the case of hydrogen assisted cracking (HAC) [55] it is reasonable to assume that the crack front tends to become a homogeneous front after a pressure surge. The reason for this can be outlined as follows. Firstly it proposed that K varies along the crack front when the crack front matches none of the homogeneous front contours. Then it is argued that the further a point on the crack front is located from the homogeneous front contours, the higher is the range of DK it will receive. As a result, that point will grow faster than other points on that crack front. Finally, it is concluded that any deviation from the homogeneous front contour is a transient phenomenon and that eventually the crack front tends to follow the succession of homogeneous front contours. Therefore, when hydrogen is available and right after the occurrence of a pressure surge, the crack

Please cite this article in press as: Rajabipour A, Melchers RE, Service life of corrosion pitted pipes subject to fatigue loading and hydrogen embrittlement, International Journal of Hydrogen Energy (2018), https://doi.org/10.1016/j.ijhydene.2018.03.063

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KIC;H ¼ KIC ð1  m  uÞ

1 .0

Relative hydrogen concentration

front deviates from the homogeneous front contours but with the passage of time and under fatigue cracking conditions, the crack front will tend to follow the homogeneous front contours. To determine the hydrogen concentration likely to exist ahead of the crack tip use can be made of the method described in Ref. [46]. It assumes that fracture toughness KIC,H under hydrogen conditions is related to the normal fracture toughness KIC and hydrogen concentration through:

Aer 50 cycles 0 .8

0 .6

0 .4

Aer 1 cycle

0 .2

(12) 0 .0 0 .0

KIC;H : Fracture toughness of the uniformly embrittled metal KIC : Fracture toughness of the virgin metal (i.e. not embrittled) u : Hydrogen concentration in the metal m : Reduction factor for fracture toughness reduction caused by hydrogen concentration Also, the method described in Ref. [46] to estimate the hydrogen concentration ahead the crack tip assumes that fatigue loading period (“B” in Fig. 10) is very short compared to normal working pressure, “A” and the periods of high pressure, “C”. Assuming this is valid also for the present case, and considering one dimensional hydrogen diffusion, the hydrogen concentration ahead of the crack tip is:

un ðx;tÞ¼ erfc

! rffiffiffiffiffiffiffiffi ! rffiffiffiffiffiffiffiffiffi Z∞ DðxzÞ2 DðxþzÞ2 1 D x þ un1 ðxþxn ;tÞ e 4t e 4t dz 4Dt 4Pt 0

(13) in which: un : Relative concentration metal after n loading


of hydrogen in cycles, defined 

the as

Number of hydrogen particles in a unit volume of the metal Number of hydrogen particles in a unit volume of the envoronment

t: Elapsed time, starting from formation of a new crack tip D : Apparent diffusivity of hydrogen in the metal x: Distance in the direction of the crack, from the crack tip to the point where the hydrogen concentration is being considered

Fig. 10 e Schematic illustration of pressure fluctuation in a pipe. Regions “A” and “B” show the normal working pressure and maximum daily pressure, respectively.

0 .5

1 .0

1 .5

2 .0

D is ta n ce a h e a d th e cra ck fro n t m m

Fig. 11 e Hydrogen concentration ahead the crack front in pit “A” after 365 cycles. Hydrogen diffusivity ¼ 9 £ 10¡13 ðm2 =sÞ. Period of loading ¼ one day.

Fig. 11 shows, for Pit A, the relative hydrogen concentrations ahead the crack front for cycles ranging from 1 to 50 as functions of the distance from the crack front, as obtained using Equation (13). Hydrogen concentration ahead the crack front will cause a reduction in the metal fracture toughness. This can be estimated using Equation (12). The trends in Fig. 11 show that under pressure surges considerable development of cracking occurs. This suggests that the material toughness along the cracking path is not homogeneous during the cracking process under a pressure surge. For example, when the crack development is about 1.2 mm the change in toughness is about 40%. This can be interpreted as showing that under pressure surge conditions the crack front does not follow the homogeneous front contours. It follows that some means is needed to estimate the shape of the crack front. This can be done using numerical modelling that allows for the changes of material toughness. As explained above, following the conclusion of the pressure surge the crack front tends to become a homogeneous front. Fig. 12 shows the effect of one pressure surge on crack depth development form pit “A”. A pressure surge of 1 (MPa) is considered at fifth, 50th, 100th and 150th year of pipe service life.

Discussion Although all the results presented so far are for a specific pit shape, this is not a limitation on the procedure. The analyses can be used also for other pit geometries. Further, they also can be used for estimating the development of crack shape where the corrosion pits themselves change shape with time, noting that it is well known that corrosion pits grow in size and shape with time. This means that the diagrams of crack depth versus time presented above will be different and should change as the pit shape changes with time. One practical limitation is that the temporal development of pit shape is not well understood and a considerable amount of research is being conducted in this area [67e69]. Irrespective of this, the present results do permit estimation of crack depth as a function of time once information is available about pit shapes.

Please cite this article in press as: Rajabipour A, Melchers RE, Service life of corrosion pitted pipes subject to fatigue loading and hydrogen embrittlement, International Journal of Hydrogen Energy (2018), https://doi.org/10.1016/j.ijhydene.2018.03.063

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10

7

=150 Year

=100 Year

8

=50 Year

=5 Year

Crack depth mm

9

6 0

50

100

150

200

250

300

Time Year

Fig. 12 e Crack depth in pit “A” under combined effects of Hydrogen-Assisted Fatigue Cracking and pressure surge. D P ¼ 1:0MPa, m ¼ 0:35, Pressure surge: PS ¼ 1:0MPa, Normal working pressure ¼ 0:8MPa. Time at which pressure surge happens ðtS Þ varies.

Fracture toughness of cast iron as used in water trunk mains of different ages has been measured to be in the range of 11e15 (MPa m0.5) [5,70]. Also, fluctuations in normal operating pressure for water mains have been measured in the range of 0.75 MPae1.0 MPa, as reported in diagram number 14 in Ref. [5]. When these values are used with Figs. 4e7 it is found that under practical pressure fluctuations it is very unlikely that a crack will propagate any considerable distance under fatigue cracking unless the cracking process is assisted by hydrogen. Even when a pit reaches 90% of the wall thickness of a cast iron water pipe, failure is unlikely under normal operating pressure fluctuations. A generally similar conclusion was reported in Refs. [71e73]. The parameter m in (12) depends on hydrogen pressure and material metallurgical properties. In Ref. [52] this parameter was obtained at about 0.2 for X-100 steel for samples pre-charged with hydrogen. It is known that m increases with hydrogen pressure and with the range of stress intensity factor [46]. However, for water pipes in water supply systems it can be expected that the range of pressure fluctuation and thus the range of the stress intensity factor at the crack front do not change very much. This is likely because water pipe pressure depends on water consumption and this usually does not deviate very from the average pressure range. The pressure fluctuations usual in water supply pipes are moderate and rarely exceed 1.0 MPa [5]. This is much lower than the level of pressure required to develop pipe wall cracking (at least 2.6 MPa in Fig. 2). From this it may be inferred that DK at the crack front seldom exceed one third of KIC (compare with KIC ¼ 14 in Fig. 2). In turn this means that for the examples considered in section 3.1 above, DK is unlikely to exceed 5 MPa m0.5. These conclusions are based on a value of m about 0.2 as found for X100 steel. For other pipeline steels such as X60, X70 and X80 m has a minimum value, even under high hydrogen pressures, of about 0.5 [46] and thus the effect of hydrogen is potentially more severe. In contrast, for cast iron pipes the

possible effect of hydrogen scarcely has been considered [54] and a reasonable range of m can not be suggested at the present time. In addition to the magnitude of pressure surges, the history of such surges also can affect crack development from pits. This is illustrated in Fig. 12. It shows that the earlier a pressure surge occurs, the more effective it is in developing a crack of a given size. Unfortunately there is little information about the history of pressure surge for most of the water systems. Current operational experience is not necessarily a good guide since pressure control for many water systems has much improved in recent years [74]. It is possible that older water pipes have experienced high levels of pressure surge in their early service lives. Also, pressure surges do not occur in all pipes and depends on the pipe location relative to the pump stations, valves etc. This is supported by field data that show higher numbers of failures in particular regions of water networks [75,76]. Hydrogen assisted cracking is a time dependent process, as is evident from Equation (13). This is the case even for constant DK since hydrogen diffusion itself is time dependent. Fig. 11 shows that hydrogen concentration increases with the number of loading cycles and that only a relative small number of cycles, typically no more than 50 in the examples considered herein, is required for the hydrogen concentration to reach steady conditions. As noted above, for pressure surges less than about 1.0 MPa, crack growth usually is not sufficient to cause significant change in the hydrogen concentration ahead the crack front. For example, Fig. 11 indicates that for event under a crack growth of 0.5 (mm), relative hydrogen concentration at the end of cracking path will be quite close to one. It follows that for engineering applications it is plausible to assume that critical pressures surges occur in the steady state condition. As a result, the relative hydrogen concentration is almost unity and the material toughness can be given as is KIC  ð1  mÞ. This assumption is conservative but reduces considerably the computational time, thereby making the calculations straightforward for engineering application. Considering now the effect of internal pressure in pipes, it is clear from Fig. 2 that higher pressure is required for a crack to continue to grow as it develops. It follows that under the same amount of internal pressure, the stress range DK required at the crack front increases with crack development. As a result, rate of crack growth versus time has a negative derivative and the diagrams of crack depth versus time are all concave down (Figs. 4e7). In some cases, when crack initiates, the homogeneous front contours are almost vertical. This indicates a sharp increase in the growth of crack depth at the early stages of crack development. It is easily verified that this occurs when the crack initiates from the bottom of a pit. Usually this is associated with wide pits (W D ¼ 2 for example). An important result from these analyses is that internal pressures less than or equal to the normal working pressure may not assist cracking throughout the life of pipe (under a Hydrogen Induces Cracking mechanism). This can be seen in Fig. 13. It shows that under a static loading conditions the development of a crack from a pit depends on the internal

Please cite this article in press as: Rajabipour A, Melchers RE, Service life of corrosion pitted pipes subject to fatigue loading and hydrogen embrittlement, International Journal of Hydrogen Energy (2018), https://doi.org/10.1016/j.ijhydene.2018.03.063

i n t e r n a t i o n a l j o u r n a l o f h y d r o g e n e n e r g y x x x ( 2 0 1 8 ) 1 e1 1

Fig. 13 e Solid line: Cracked area vs pressure, W/D ¼ 0.5, D/ (pipe thickness) ¼ 0.6. Dashed line: Schematic diagram if SCC is in process.

pressure gradual being increased. Provided the pressure is increased the crack will continue to grow and it will stop if the pressure ceases to increase. Under SCC conditions, a similar scenario holds but the pressure values needed for cracking are lower. This is because the SCC threshold stress intensity factor is lower than KIC [77e79]. It follows that it is possible that development of a crack under SCC stops after a while and that fatigue cracking remains the only cracking process.

Conclusion 1. Within the range of the frequency of fatigue loading and of hydrogen diffusivity normally expected for pipes it is shown that crack propagation is by the crack front proceeding through a metal that can be considered to be spatially almost uniformly embrittled. 2. The range of daily pressure fluctuations normally expected for water pipes is insufficient to cause the pipe to fail through fatigue cracks breaching the pipe wall. Pipe wall failure is more likely to occur if the metal is embrittled by hydrogen and the probability of failure is increased with increased hydrogen concentration and with increased number of daily cycles. 3. The influence of pressure surge on the development of cracks decreases with time and the later in time a pressure surge occurs the less effective it is in causing pipe wall failure.

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