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Dynamic fracture analysis of buried steel gas pipeline using cohesive model
Soil Dynamics and Earthquake Engineering 128 (2020) 105881
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Soil Dynamics and Earthquake Engineering journal homepage: http://www.elsevier.com/locate/soildyn
Dynamic fracture analysis of buried steel gas pipeline using cohesive model Xiaohua Zhu a, *, Zilong Deng a, Weiji Liu a a
School of Mechatronic Engineering, Southwest Petroleum University, Chengdu, 610500, China
A R T I C L E I N F O
A B S T R A C T
Keywords: Finite element method Crack propagation Cohesive model Dynamic fracture Crack-tip-opening angle
Pipeline play an irreplaceable role in long-distance oil and gas transportation. However, the initiation and propagation of cracks associated with defects may lead to pipeline fracture or explosion. In this paper, the nu merical model of dynamic fracture in full-size natural gas buried pipeline are established based on cohesive model. The dynamic crack propagation of X80 pipeline steel is simulated with and without backfill soil. These parameters, like the crack-tip-opening angle (CTOA), are investigated quantitatively in the simulation in this paper. The results show that a threshold value of the crack propagation rate and CTOA does exist. With considering the backfill soil, the crack propagation rate decreases dramatically, and the curve of crack propa gation distance will appear many horizontal trend stage against time. The lower the internal pressure, the more obvious the horizontal trend stage will be. The soil layer stress caused by pipeline crack propagation is mainly near the pipeline. Comparing the buried pipeline with penetrating cracks and surface cracks, it is found that surface cracks are more dangerous. The study of the dynamic fracture mechanism of full-scale pipeline provides necessary supplement for experiment.
1. Introduction Pipeline is a most convenient and economical carrier to the highpressure oil and gas and long-distance place. In recent years, high strength steel has been used in the many countries. X80 pipeline steel is considered to be a relatively economical high-grade pipeline steel, which is more popular than X65 and other pipeline steels. The pipeline in the course of service, due to defects in the bonding surface, grain interface and inclusions of materials during manufacture, and the effects of various alternating stresses, such as formation stress and fluctuation of internal pressure of pipelines, cause damage accumulation continu ously and micro-cracks may gradually develop into macro-cracks, which will eventually lead to damage, and structural dysfunction, even the risks of fracture, explosion [1]. Traditional pipeline design idea is to use Charpy V-notched Impact Test (CVN) and Drop Weight Tear Test (DWTT). In recent years, many scholars have explored reasonable parameters and simulation methods to describe and predict pipeline cracks, studied the propagation mech anism of pipeline cracks. Many scholar carried out full-scale blasting test and DWTT (Igi [2], Tronskar [3], Chaouadi [4]). The results show that the relationship between impact toughness of drop weight and crack propagation energy was analyzed and compared with the prediction of empirical formula, the effect of loading rate on ductile fracture behavior
was analyzed, and the ductile crack resistance under dynamic loading rate is significantly higher than that under quasi-static loading rate. Through experiments, the factors affecting pipeline fracture propagation velocity are analyzed, combine with the velocity curve of decompression wave and the formula of pipeline fracture velocity. Acoustic emission is used to study the propagation of pipeline cracks (Yusof [5]). Battelle Two-Curve (BTC) method is used to calculate crack arrest toughness, fracture resistance curves and pressure reduction curves, and deter mined the principle and application scope of BTC method (Huo [6], Keim [7], Scheider [8]). Bogatov [9] evaluated the crack resistance of steel based on plastic diagram. Rudland [10] proposed a method for calculating dynamic J-R curves based on indentation drop weight tearing test. Wu [11] studied the propagation of surface cracked pipes with different geometric sizes in creep stratum by simulation. Some scholars have applied the numerical simulation to the pipeline fracture and achieved good results. Shim [12] and Uddin [13] presents the development of a dynamic ductile crack growth model to simulate an axially running crack in a pipe by cohesive zone model. The tuned SRDD model is used by Oikonomidis [14] for the simulation of axial crack propagation and arrest in X100 natural gas pipelines. Nordhagen [15] considers a predictive numerical modelling approach for fracture-propagation control in CO2-transport pipelines. A coupled flu id–structure interaction model is developed by Aursand [16] and finite
* Corresponding author. E-mail address: [email protected] (X. Zhu). https://doi.org/10.1016/j.soildyn.2019.105881 Received 10 June 2019; Received in revised form 29 September 2019; Accepted 29 September 2019 Available online 22 October 2019 0267-7261/© 2019 Elsevier Ltd. All rights reserved.
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Fig. 1. Axial stress analysis of pipeline.
difference method is used by Nakai [17] to simulate the ductile crack propagation in full-scale pipes which was able to capture the global deformation as well as the crack speed, which are considered soil backfill effect. Crack-tip-opening angle (CTOA), formed between two separated fracture surfaces, is an engineering fracture parameter. There is no theoretical solution for crack driving force based on CTOA for fracture specimens or cracked pipes [18]. Shibanuma [19] and Hashemi [20] carried out crack propagation test on X80 pipeline steel and explored the measurement method of crack tip opening angle. Therefore, the appli cation of CTOA concept relies on finite element calculation and requires powerful models. Rybicki et al. [21] and O’Donoghue [22] simulated the steady crack growth of thin-walled specimens by finite element method, and studied the criterion of crack tip opening angle. Dunbar [23] et al. used cohesive model to simulate ductile crack growth. The ductile crack growth characteristics of different steel DWTT specimens were compared with CTOA. Parmar [24] studied the effect of changing the boundary layer on the crack growth resistance curve and CTOA. The results show that CTOA can be used as a criterion for crack propagation. The study of Amara [25] also shows that the critical value of CTOA can be used as a criterion of fracture resistance for characterizing stable tearing in thin metallic materials. Zhuang [26] et al.and YOU [27] et al. developed an iteration algorithm for FEM simulation of ductile induced deceleration by using energy balance equation. Combining with the drop weight tearing test of two samples for determining heat dissipation rate, the crack tip opening angle criterion in FEM simulation and the critical value in DWTT for two samples were compared. According to the aforementioned literature, most pipelines are buried in the ground, less researches have paid attention to the crack propagation of buried pipelines, which is more appropriate with the practical situation. In present study, a finite element model based on cohesive model is presented to simulate the dynamic fracture of pipeline cracks. Firstly, this paper establishes a pipeline model without soil layer, secondly, it establishes a model of buried pipeline which distinguishes backfill from unexcavated soil layer to studies the crack propagation of pipeline. The quantitative relationship between crack propagation rate and pipeline internal pressure is predicted, and the key parameters (such as CTOA, crack tip stress-strain field, etc.) are studied quantitatively. The dynamic fracture mechanism of full-scale pipeline is revealed, which provides necessary supplement for the experiment. The calcula tion results are accurate, which lays a foundation for further research on the fracture problems in engineering structures, especially in the treat ment of pipeline cracks.
2. Establishment of pipeline model 2.1. Force analysis of pipeline The stress on the pipe wall can be divided into axial stress, circum ferential stress and radial stress according to the direction. In the thinwalled pipe such as natural gas pipeline, the axial stress and circum ferential stress play a major role. As shown in Fig. 1, the analysis of the axial stress is carried out. Assuming that the internal pressure of the pipeline is P and the diameter of the pipeline is D, Intercepting any section of the pipeline, assuming that the end face is closed, the pressure of natural gas on the closed end is π ðD=2Þ2 P. In practice, the pipeline is not closed, so it is understood that the gas produces an axial resultant force on the wall of the pipeline, any section of the pipeline is the stress balance state. That is, the axial resultant force of gas on the pipe wall and the circumfer ential stress on the pipe wall are equal, so:
πðD=2Þ2 P ¼ πDtσx σx ¼
PD 4t
(1) (2)
As shown in Fig. 2, it is an analytical diagram of the circumferential stress. The component force in the y direction of the resultant force which produces the internal gas on the wall of the pipeline is balanced with the stress in the two sections of the pipeline. Take integral infini tesimal dα for pipe wall, Z π D PL sin αdα ¼ 2Ltσ y (3) 2 0
σy ¼
PD 2t
(4)
It can be found that the σ y is greater than the axial stress σx and has a double relationship. Therefore, it is necessary to control the circumfer ential stress of the pipeline in design. Since there is little influence on the axial force when the soil layer is added, the circumferential force is analyzed. It is assuming that the pipeline is buried in an infinite elastic body and that each elastomer maintains complete contact on the contact surface, that is, it neither separates from each other nor slides with each other. The polar coor dinate system as shown in Fig. 3 is established.
Fig. 2. Circumferential stress analysis of pipeline. 2
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Fig. 6. Traction-separation failure curve of linear softening.
cross sections of the crack. The stress is a function of the crack displacement value, that is, the tension-displacement relationship. Cohesive element is widely used in the initiation and propagation of cracks in most materials. For example, it is used for fatigue fracture, overload failure and phase interface separation of metal materials, composite matrix cracking, etc. For the three-dimensional cohesive element, it is divided into pentahedron and hexahedron. The upper and lower sections are the main load-bearing surfaces, while the surrounding surfaces are not loaded. As shown in Fig. 5, the surface of the red shadow line is the main load-bearing surface. It is supposed that there is a Fracture Process Zone at the crack tip. In three-dimensional FPZ, there are normal (tn ) and two shear directions (ts and tt ) traction forces on the crack surface. Fig. 6 shows the tractionseparation failure curve (TSL: Traction-separation laws) of normal traction (tn ) with the change of crack opening displacement (δn ). The traction-separation curves of shear traction with crack opening displacement similar to those of this kind. Before the crack initiates, it is assumed that the material is in the elastic stage, which is represented by a linear elastic rising stage. After cracking, with the monotonous decrease of the corresponding separation function, called tension or stress softening stage. The linear softening shown in Fig. 5 is also widely used, in addition, bilinear and exponential softening are also widely used. In order to express the combined action of the whole damage and various mechanisms in materials, the damage evolution of cohesive element gives a damage index D, which is a function of equivalent displacement δm . The effective relative displacement δm combines the effects of δn , δs and δt , describe the mixed displacement after the inter face being loaded. qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi δm ¼ 〈δn 〉2 þ δ2s þ δ2t (7)
Fig. 3. Stress analysis of pipeline in soil layer.
Fig. 4. Cohesive zone at crack tip.
Fig. 5. The load-bearing diagram of cohesion model.
σρ ¼
P
σϕ ¼
P
2
½1 þ ð1
2μÞn� Dρ2
ð1
nÞ
½1 þ ð1
2μ
2 Þn� Dd2
ð1
nÞ
½1 þ ð1
2μÞn� Dρ2 þ ð1
nÞ
½1 þ ð1
2μÞn� Dd2
2 2
ð1
(5)
where, 〈〉 is Macaulay bracket and � δn ; δn � 0 〈δn 〉 ¼ 0; δn < 0
(6)
nÞ
ð1þμÞ where, n ¼ EEð1þ , E is the elastic modulus of pipeline, μ is the Poisson’s μ0 Þ 0
Taking the linear softening law (or linear damage evolution in Fig. 6) as an example, the damage index can be calculated by
ratio of pipeline, E is the elastic modulus of soil, μ is the Poisson’s ratio of soil, D is the outer diameter of pipeline, d is the inner diameter of pipeline, P is the internal pressure of pipeline. 0
(8)
0
D¼
2.2. Finite element model of pipeline
δmf ðδm;max δm;max δmf
δm0 Þ � δm0
(9)
where δm0 and δmf are effective relative displacements at damage initi ation and complete failure, respectively. δm;max is the maximum effective relative displacement attained during the loading history. Initially, D ¼ 0, when the cohesive element satisfies the condition of damage generation, the value of D monotonically increases from 0 to 1 upon further loading after the initiation of damage, and then, D ¼ 1, it is complete failure. When the natural gas pipeline is under pressure, the main stress is circumferential stress, which is perpendicular to the surface of the cohesive model. Therefore, it is feasible and reliable to simulate the crack propagation of the natural gas pipeline in this way. The section
Cohesive Model is a computational model widely used in fracture mechanics. Dugdale and Barenblatt proposed the concept of cohesive model to explain the theoretical infinite stress of the crack tip. Cohesive Model avoids the stress singularity of crack tip in linear elastic fracture mechanics, and calculates the stress and fracture energy in crack pro cess. As shown in Fig. 4, the cohesive model assumes that a small area at the crack tip is a cohesive zone, which is composed of a crack and two cross sections. There is an interaction force between molecules or atoms on both sides of the crack, which is called cohesive force. The value of cohesive force σ is related to the separation displacement δ of the two 3
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Fig. 7. Pipeline numerical model and meshing.
Fig. 8. The FEA simulation of DWTT.
cracking of the upper and lower continuous elements is simulated by setting a cohesive element in the middle of the continuous element. The stress and fracture energy of cohesive element is changed according to the predefined tension-displacement relationship. Based on the background of the second West-to-East Gas Trans mission Line in China, X80 high-grade steel pipe is selected. The diam eter of the pipeline is 1219 mm, the wall thickness is 18.4–25.6 mm, the total length of the pipeline is 5 m, and the maximum design pressure is 14 MPa. Fracture toughness coefficient is KIC ¼ 115MPa⋅m1=2 , The elastic modulus is 203 GPa, Poisson’s ratio is 0.3, and material density is 7800 kg/m3, The corresponding hardening property is shown in Fig. 7 (c). It is difficult to grasp the specific trend of internal pressure change in
the process of natural gas leakage in buried pipelines. Therefore, the internal pressure of natural gas is set to be constant in process of crack initiation and stability. Symmetrical constraints are applied at both ends of the pipeline. It indicates that the pipeline extends toward both ends. According to Hashemi [20] and Shim [12] experimental research and the example of pipeline rupture, pipeline cracks tend to propagate along the axis, so the direction of crack propagation along the axis path is established, and the crack propagation path is set up. The established pipeline model is shown in Fig. 7. Because the crack propagation path on the pipeline is the focus of analysis, the local mesh is refined by the transition of coarse and fine meshes, which ensures the accuracy of calculation and saves calculation time. 4
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2.4. Model validation with X100 pipeline steel In order to verify the rationality of the model, we have chosen two literature sources. The simulations reproduce as closely as possible conditions. The burst tests of X100 pipes with 914 mm outer diameter were reported [30]. Oikonomidis [14]. simulated the pipe burst of an X100 pipe. The same conditions were chosen. The model was a 10 m long pipe with a 250 mm long axial through thickness crack introduced at one end. Fig. 9 show that the model predicted the crack arrest time, the crack velocity and the crack arrest length to within 10%. 3. Pipeline model result analysis In order to verify the reliability of the numerical model, empirical formulas are used for fitting. Battelle Research Institute of the United States fitted the formula of fracture propagation velocity through a se ries of full-scale ductile fracture tests of pipelines, which is authoritative and valuable [31]. � σ f Pd V ¼ 0:275 pffiffiffi R Pa
2.3. Determining model parameters
�16 1 =
Fig. 9. The experimental and the model profiles of crack length and time for a X100 pipe with 914 mm outer diameter and 13 mm wall thickness.
(10)
where, Vis pipeline fracture propagation velocity, m=s; R is resistance of steel to fracture propagation, KJ=m2 ; σ f is flow stress, MPa; Pd is pipeline crack tip pressure; Pa is pipeline pressure at crack initiation, MPa. Substituting material and size parameters into the semi-empirical formulas, the relationship between crack propagation rate and inter nal pressure represented by semi-empirical formulas is obtained, as shown in the black curve in Fig. 10(b). Fig. 10(a) shows the curve of the crack propagation distance with time in the finite element simulation process. According to the curve, the steady-state velocity of the crack propagation is obtained, which is in fitting well with the results obtained by semi-empirical formula. It shows that the numerical simulation has certain reliability. Both empirical formulas and numerical simulations show that there is an upper limit for the crack propagation rate. Even if the internal pressure increases, the crack propagation rate will not increase infinitely. When other parameters are fixed (such as crack length, inner diameter, etc.), there exists a critical internal
Based on the Drop Weight Tear Test (DWTT) experiments and hun dreds of simulations, we adopt the following parameters of X80 by Shi and Wang [28]:K ¼ 80GPa=mm,δm0 ¼ 0:0225mm, δmf ¼ 4δm0 . The corresponding hardening property of X80 pipeline steel is specified by Fig. 8(c) which determines the dependence of Mises yielding stress on equivalent plastic strain. [29] The fracture parameters chosen in such way achieves good agreement with tests not only for fracture speed but also for the loading force-displacement curves. The fracture parameters can also be applied to pipeline fracture. From Fig. 8(d), because of the dynamic simulation, the data expe rienced a slight fluctuating trend, but the overall trend is fitted. The parameters of CMZ are used to simulate the fracture of X80 steel.
Fig. 10. Relation between crack distance and propagation velocity without soil layer constraints. 5
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A common formula is to determine the distance between the two sides of the crack at a specific distance from the back of the instantaneous crack tip. CTOA ¼ 2arctanðδ = 2xÞ
(11)
where, δ is the opening displacement of the crack tip at a certain distance x behind the crack tip. Kazuki [16] conduct a full gas burst test to clarify the CTOA history during ductile cracking in a high-pressure pipeline. In addition to the CTOA based on the COD 2 mm and COD 10 mm behind the crack tip was evaluated. The values of both CTOA(2 mm) and CTOA(10 mm) clearly remained nearly constant during crack propagation. Thus, the average values were 13.2� for CTOA(2 mm) and 13.9� for CTOA(10 mm). The difference of CTOA between the CTOA(2 mm) and CTOA(10 mm) is 0.7� .In order to facilitate statistics and ensure the accuracy of data, the x value selected in this paper is 0.005 m. As shown in Fig. 12(a), the abscissa is the crack propagation dis tance, and the ordinate is the crack tip opening angle corresponding to the internal pressure. It can be seen that the crack tip opening angle is a higher value at the beginning of the crack propagation, because in the initial stage of crack propagation, natural gas in the pipeline leaks from the crack, the pressure is concentrated, and the crack tip opening angle decreases to a relatively stable value just after the initial velocity of crack propagation being obtained. Different pressures correspond to different steady-state velocities, and different pressures correspond to different steady-state crack tip opening angles.
Fig. 11. Definition schematic of CTOA.
pressure for crack propagation. When this pressure is reached, the crack begins to propagate. This critical internal pressure can simulate the crack several times. Compared with the experimental results from Shibanuma et al. [19], the simulation results are reliable, which can better illustrate the reli ability of using this method to simulate the propagation of pipeline cracks. The crack tip opening angle means the magnitude of crack driving force. The crack tip opening angle (CTOA) has been applied to the evaluation of crack arrest toughness of high toughness pipeline steel. It can combine the local micro-mechanism of crack propagation with the stress and plastic flow depending on the geometric shape. It is more truly reflecting the deformation of crack tip and stress distribution. The definition of CTOA is related to plastic deformation energy, as shown in Fig. 11. The current definition is based on experimental measurements.
Fig. 12. Changes of crack tip opening angle and crack cloud figure of the stress without soil layer.
Fig. 13. Pipeline model and meshing. 6
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Fig. 14. Curve of crack length with time and crack propagation rate under soil layer.
4. Pipeline model under soil layer 4.1. Pipeline-soil model of prefabricated penetrating crack Soil layer is an important factor affecting the mechanical properties of pipelines. Most scholars did not consider the actual situation of pipeline surrounding soil when studying pipeline fracture behavior, and reasonably divided backfill and soil layer. In order to simulate buried pipelines more realistically and increase the applicability of the model, a soil layer and pipeline model is established, as shown in Fig. 13 Supposing that the backfill and the soil layer are loess, Mohr-Coulomb elastic-plastic constitutive model is used to describe the non-linear characteristics of rock and soil. Its elastic modulus 14.4 MPa, Poisson’s ratio is 0.2, cohesive force is 18 KPa, internal friction angle is 30� , dilation angle is 16� , density is 1400 kg/m3, the gravity load is applied to the soil, and the acceleration of gravity is 9.8 m/s2. In order to distin guish backfills, cohesive elements with zero thickness are added be tween backfills, as shown in Fig. 13 (b), so that they can crack and separate. The friction coefficient between pipe wall and soil is 0.3. In order to make backfill soil contact with pipeline better, the mesh of backfill soil is refined. Similarly, pressure loads are applied to the inside of the pipe to simulate the fluid action in the pipe. Through finite element simulation, the curve of crack propagation distance of pipeline with time is obtained. As shown in Fig. 14(a), the length of crack propagation is considered to be the distance exceeding the length of prefabricated crack, and the slope of the curve is expressed as the instantaneous velocity of crack propagation at each moment. With the decrease of pressure, there are many horizontal trend stages in the curve, that is, the crack propagation rate slows down, and then returns to a relatively stable rate. The lower the pressure is, the longer the horizontal trend stage time will be. Under the same pressure, compared with the case without soil layer, the horizontal trend stage is not obvious. According to Fig. 14(a), the steady-state crack propagation rate is obtained. As shown in Fig. 14(b), the crack propagation rate decreases obviously due to the restraint of the soil layer and the consumption of part of the energy of the soil layer. With the increase of the internal pressure, the crack propagation rate increases gradually. When the in ternal pressure is large, the crack propagation rate increase slightly. Fig. 15 shows the curve of crack tip opening angle with crack length under different internal pressures. Compared with the case without soil
Fig. 15. Curve of crack tip opening angle with crack propagation distance.
layer, the crack tip opening angle is also a higher value at the beginning of crack propagation, and then tends to be stable. The difference is that the crack tip opening angle is larger than that in the case without soil layer, and there will be more significant fluctuations in the later stage than that in the case without soil layer. Crack tip opening angle indicates the crack driving force. In the initial stage of crack propagation, a large driving force is required for crack propagation. When the initial velocity of crack propagation is obtained, the driving force required for crack propagation decreases and tends to be stable. However, due to the in fluence of soil layer, the soil layer deformation will consume part of the driving force for crack propagation. As a result, the crack tip opening angle will decrease, and then the effect of internal pressure will continue to drive the crack propagation, and the crack tip opening angle will increase. Fig. 16 is a cloud figure of formation damage and pipeline stress. The cohesive elements in backfill soil shown in Fig. 16(a) are broken and mainly concentrated in the soil layer near pipeline. The red 7
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Fig. 16. Soil layer damage cloud diagram and stress cloud diagram.
Fig. 17. The propagation of cracks under soil layer and pictures taken at the scene.
part between elements in Fig. 16(b) is the damage of cohesive elements. Therefore, the leakage of pipeline crack propagation causes soil collapse and explosion accidents which mainly attribute to the disturbance of soil caused by gas inside the pipeline. The deformation of pipeline canno
t cause large-scale collapse and explosion. Fig. 17(a) shows a cloud diagram of crack propagation in pipelines under soil layer. Similar to Fig. 12(b), the stress concentration area is mainly near the crack tip, but in the case of covering by soil layer, the 8
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Fig. 18. Curves of surface crack propagation length and propagation rate under rate.
Fig. 19. Cloud diagram of surface crack propagation under soil layer.
stress concentration area is wider. Therefore, when considering pipeline crack arrest, relevant crack arrest measures in the near area where cracks are found can better achieve the effect of crack arrest. Fig. 17(b) is pipeline crack propagation pictures taken at the scene by Bogatov [9]. It can be inferred that the numerical simulations can well simulate the process from initiation to stable expansion of pipeline, and can provide reference for the subsequent anti-cracking measures.
along the wall thickness direction of the pipeline when the surface crack propagates. When the crack propagates to a certain threshold, the crack begins to propagate in the axial direction. At this time, the initial rate has been obtained, so the horizontal trend stage is not obvious when the crack propagates. As shown in Fig. 18(b), surface crack propagation rate is faster than penetrating crack propagation rate, because surface crack propagation along the wall thickness direction has obtained a certain initial rate, so under the same internal pressure, surface crack propagation rate is faster than penetrating crack propagation rate. From the cloud diagram of Fig. 19, it can be seen that the surface crack of the pipeline propagates along the wall thickness of the pipeline, and when it reaches a certain threshold, the crack begins to propagate in the axial direction.
4.2. Pipeline-soil model of prefabricated surface crack According to the analysis in section 2.1, the most dangerous point in the pipeline is found out. The penetrating crack in the pipeline is replaced by surface crack. The wall thickness of the pipeline model is 18.4 mm, the crack is internal crack, and the crack depth is 9 mm. This defective pipeline of internal surface crack is simulated and analyzed under soil layer. The curve of crack propagation length with time is obtained as shown in Fig. 18(a). The horizontal trend stage is not obvious when the surface crack propagates, because it first propagates
5. Conclusion In this paper, the fracture propagation process of full-scale pipeline is simulated using cohesive model, with considering the influence of strata 9
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on crack propagation. The crack propagation velocity and propagation angle are quantitatively analyzed, and the following conclusions can be drawn.
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Numerical model for unstable ductile crack propagation and arrest in pipelines using finite difference method. Eng Fract Mech 2016;162:179–92. https://doi.org/10.1016/j.engfracmech.2016.03.048. [18] Salvini P, Fonzo A, Mannucci G. Identification of ctoa and fracture process parameters by drop weight test and finite element simulation. Eng Fract Mech 2003;70(3–4):553–66. https://doi.org/10.1016/S0013-7944(02)00137-6. [19] Shibanuma Kazuki, Hosoe Takahiro, Yamaguchi Hikaru, Tsukamoto Masatoshi, Suzuki Katsuyuki, Aihara Shuji. Crack tip opening angle during unstable ductile crack propagation of a high-pressure gas pipeline. Eng Fract Mech 2018;204: 434–53. https://doi.org/10.1016/j.engfracmech.2018.10.020. [20] Hashemi SH, Howard IC, Yates JR, Andrews RM, Edwards AM. A single specimen CTOA test method for evaluating the crack tip opening angle in gas pipeline steels. In: International Pipeline Conference IPC2004-0610; 2004. p. 1703–9. https://doi. org/10.1115/IPC2004-0610. [21] Rybicki EF, Kanninen MF. A finite element calculation of stress intensity factors by a modified crack closure integral. Eng Fract Mech 1977;9(4):931–8. https://doi. org/10.1016/0013-7944(77)90013-3. [22] O’Donoghue PE, Kanninen MF, Leung CP, Demofonti G, Venzi S. The development and validation of a dynamic fracture propagation model for gas transmission pipelines. Int J Press Vessel Pip 1997;70(1):11–25. https://doi. org/10.1016/S0308-0161(96)00012-9. [23] Dunbar A, Wang X, Tyson W, Xu S. Simulation of ductile crack propagation and determination of ctoas in pipeline steels using cohesive zone modelling. Fatigue Fract Eng Mater Struct 2014;37(6):592–602. https://doi.org/10.1111/ffe.12143. [24] Parmar S, Wang X, Tyson B, Xu S. Simulation of ductile fracture in pipeline steels under varying constraint conditions using cohesive zone modeling. In: International Journal of Pressure Vessels & Piping V06BT06A043; 2018. https:// doi.org/10.1115/PVP2015-45873. [25] Amara MB, Pluvinage G, Capelle J, Azari Z. Crack tip opening angle as a fracture resistance parameter to describe ductile crack extension and arrest in steel pipes under service pressure. Phys Mesomech 2015;18(4):355–69. https://doi.org/10. 1134/S1029959915040086. [26] Zhuang Z, You XC. The toughness-induced crack deceleration mechanism on ultrahigh pressure steel gas pipelines. In: Asme pressure vessels & piping conference PVP2003-2003; 2003. p. 13–21. https://doi.org/10.1115/PVP2003-2003. [27] You XC, Zhuang Z, Huo CY, Zhuang CJ, Feng Y. Crack arrest in rupturing steel gas pipelines. Int J Fract 2003;123(1/2):1–14. https://doi.org/10.1023/b:frac .0000005791.79914.8. [28] Shi Y, Wang YJ. Shape effects of the traction–separation law on the global response of the dynamic fracture for pipeline steels. Acta Mech 2017. https://doi.org/10. 1007/s00707-017-1913-5.
(1) The cohesive model can well simulate the dynamic crack prop agation process in pipelines. In the case of pipelines without soil layer, the simulation results and empirical formulas show that there is an upper limit for the crack propagation rate, that is, the internal pressure increases continuously, and the crack propa gation rate will not increase infinitely. In the design of crack arresting tools, it is necessary to control the circumferential stress on the pipeline. (2) Whether or not there is soil layer, in the initial stage of pipeline crack propagation, the crack tip opening angle is a high value, and then it decreases to a relatively stable value. The crack tip opening angle increases and greater fluctuates under soil layer. The formation deformation caused by pipeline cracking mainly concentrates near the pipeline. (3) Using cohesive element to simulate the backfill of pipeline can more accurately describe the crack propagation phenomenon of pipeline. The crack propagation rate will appear many horizontal trend stage. The lower the internal pressure is, the more obvious the horizontal trend stage is, the crack propagation rate will decrease significantly, and the crack tip opening angle will in crease and the fluctuation will be more obvious. (4) Internal surface cracks are more dangerous than penetrating cracks. The surface cracks of pipelines propagate along the wall thickness of pipelines. When they reach a certain threshold, the cracks begin to propagate in the axial direction. Therefore, in the detection of pipeline defects, we should also pay attention to the impact of surface cracks. Declaration of competing interestCOI We declare that we do not have any commercial or associative in terest that represents a conflict of interest in connection with the work submitted. Acknowledgments This study is supported by the China Postdoctoral Science Founda tion (2018M633403), Applied Basic Research of Sichuan Province (Free Exploration-2019YJ0520), National Natural Science Foundation of China (51674214 and 51709231), Youth Science and Technology Innovation Research Team of Sichuan Province (2017TD0014). Such supports are greatly appreciated by the authors. Appendix A. Supplementary data Supplementary data to this article can be found online at https://doi. org/10.1016/j.soildyn.2019.105881. References [1] Bussiba A, Darcis PP, Mccolskey JD, Mccowan CN, Kohn G, Smith R, et al. Fatigue crack growth rates in six pipeline steels. In: International pipeline conference; 2006. https://doi.org/10.1115/IPC2006-10320. [2] Igi S, Guan C, Rothwell B, Hiraide T. Investigation of crack propagation characteristics using instrumented charpy and DWT tests for full-scale burst tested 1219 mm OD grade 550 (X80) linepipe. In: International pipeline conference; 2016. https://doi.org/10.1115/IPC2016-64240. [3] Tronskar JP, Mannan MA, Lai MO. Measurement of fracture initiation toughness and crack resistance in instrumented charpy impact testing. Eng Fract Mech 2002; 69(3):321–38. https://doi.org/10.1016/S0013-7944(01)00077-7. [4] Chaouadi R, Puzzolante JL. Loading rate effect on ductile crack resistance of steels using precracked charpy specimens. Int J Press Vessel Pip 2008;85(11):752–61. https://doi.org/10.1016/j.ijpvp.2008.08.004.
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[29] Ren ZJ, Ru CQ. Numerical investigation of speed dependent dynamic fracture toughness of line pipe steels. Eng Fract Mech 2013;99:214–22. https://doi.org/10 .1016/j.engfracmech.2012.12.016. [30] Andrews RM, Millwood N, Batte A, Lowesmith B. The fracture arrest behavior of 914 mm diameter X100 grade steel pipelines. In: ASME Proceedings of the 5th international pipeline conference; 2004. p. 1693–701.
[31] Maxey W, Kiefner J, Eiber R, Duffy A. Ductile fracture initiation, propagation, and arrest in cylindrical vessels. In: Fracture toughness: part II: ASTM international; 1972. https://doi.org/10.1520/STP38819S.
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Soil Dynamics and Earthquake Engineering journal homepage: http://www.elsevier.com/locate/soildyn
Dynamic fracture analysis of buried steel gas pipeline using cohesive model Xiaohua Zhu a, *, Zilong Deng a, Weiji Liu a a
School of Mechatronic Engineering, Southwest Petroleum University, Chengdu, 610500, China
A R T I C L E I N F O
A B S T R A C T
Keywords: Finite element method Crack propagation Cohesive model Dynamic fracture Crack-tip-opening angle
Pipeline play an irreplaceable role in long-distance oil and gas transportation. However, the initiation and propagation of cracks associated with defects may lead to pipeline fracture or explosion. In this paper, the nu merical model of dynamic fracture in full-size natural gas buried pipeline are established based on cohesive model. The dynamic crack propagation of X80 pipeline steel is simulated with and without backfill soil. These parameters, like the crack-tip-opening angle (CTOA), are investigated quantitatively in the simulation in this paper. The results show that a threshold value of the crack propagation rate and CTOA does exist. With considering the backfill soil, the crack propagation rate decreases dramatically, and the curve of crack propa gation distance will appear many horizontal trend stage against time. The lower the internal pressure, the more obvious the horizontal trend stage will be. The soil layer stress caused by pipeline crack propagation is mainly near the pipeline. Comparing the buried pipeline with penetrating cracks and surface cracks, it is found that surface cracks are more dangerous. The study of the dynamic fracture mechanism of full-scale pipeline provides necessary supplement for experiment.
1. Introduction Pipeline is a most convenient and economical carrier to the highpressure oil and gas and long-distance place. In recent years, high strength steel has been used in the many countries. X80 pipeline steel is considered to be a relatively economical high-grade pipeline steel, which is more popular than X65 and other pipeline steels. The pipeline in the course of service, due to defects in the bonding surface, grain interface and inclusions of materials during manufacture, and the effects of various alternating stresses, such as formation stress and fluctuation of internal pressure of pipelines, cause damage accumulation continu ously and micro-cracks may gradually develop into macro-cracks, which will eventually lead to damage, and structural dysfunction, even the risks of fracture, explosion [1]. Traditional pipeline design idea is to use Charpy V-notched Impact Test (CVN) and Drop Weight Tear Test (DWTT). In recent years, many scholars have explored reasonable parameters and simulation methods to describe and predict pipeline cracks, studied the propagation mech anism of pipeline cracks. Many scholar carried out full-scale blasting test and DWTT (Igi [2], Tronskar [3], Chaouadi [4]). The results show that the relationship between impact toughness of drop weight and crack propagation energy was analyzed and compared with the prediction of empirical formula, the effect of loading rate on ductile fracture behavior
was analyzed, and the ductile crack resistance under dynamic loading rate is significantly higher than that under quasi-static loading rate. Through experiments, the factors affecting pipeline fracture propagation velocity are analyzed, combine with the velocity curve of decompression wave and the formula of pipeline fracture velocity. Acoustic emission is used to study the propagation of pipeline cracks (Yusof [5]). Battelle Two-Curve (BTC) method is used to calculate crack arrest toughness, fracture resistance curves and pressure reduction curves, and deter mined the principle and application scope of BTC method (Huo [6], Keim [7], Scheider [8]). Bogatov [9] evaluated the crack resistance of steel based on plastic diagram. Rudland [10] proposed a method for calculating dynamic J-R curves based on indentation drop weight tearing test. Wu [11] studied the propagation of surface cracked pipes with different geometric sizes in creep stratum by simulation. Some scholars have applied the numerical simulation to the pipeline fracture and achieved good results. Shim [12] and Uddin [13] presents the development of a dynamic ductile crack growth model to simulate an axially running crack in a pipe by cohesive zone model. The tuned SRDD model is used by Oikonomidis [14] for the simulation of axial crack propagation and arrest in X100 natural gas pipelines. Nordhagen [15] considers a predictive numerical modelling approach for fracture-propagation control in CO2-transport pipelines. A coupled flu id–structure interaction model is developed by Aursand [16] and finite
* Corresponding author. E-mail address: [email protected] (X. Zhu). https://doi.org/10.1016/j.soildyn.2019.105881 Received 10 June 2019; Received in revised form 29 September 2019; Accepted 29 September 2019 Available online 22 October 2019 0267-7261/© 2019 Elsevier Ltd. All rights reserved.
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Fig. 1. Axial stress analysis of pipeline.
difference method is used by Nakai [17] to simulate the ductile crack propagation in full-scale pipes which was able to capture the global deformation as well as the crack speed, which are considered soil backfill effect. Crack-tip-opening angle (CTOA), formed between two separated fracture surfaces, is an engineering fracture parameter. There is no theoretical solution for crack driving force based on CTOA for fracture specimens or cracked pipes [18]. Shibanuma [19] and Hashemi [20] carried out crack propagation test on X80 pipeline steel and explored the measurement method of crack tip opening angle. Therefore, the appli cation of CTOA concept relies on finite element calculation and requires powerful models. Rybicki et al. [21] and O’Donoghue [22] simulated the steady crack growth of thin-walled specimens by finite element method, and studied the criterion of crack tip opening angle. Dunbar [23] et al. used cohesive model to simulate ductile crack growth. The ductile crack growth characteristics of different steel DWTT specimens were compared with CTOA. Parmar [24] studied the effect of changing the boundary layer on the crack growth resistance curve and CTOA. The results show that CTOA can be used as a criterion for crack propagation. The study of Amara [25] also shows that the critical value of CTOA can be used as a criterion of fracture resistance for characterizing stable tearing in thin metallic materials. Zhuang [26] et al.and YOU [27] et al. developed an iteration algorithm for FEM simulation of ductile induced deceleration by using energy balance equation. Combining with the drop weight tearing test of two samples for determining heat dissipation rate, the crack tip opening angle criterion in FEM simulation and the critical value in DWTT for two samples were compared. According to the aforementioned literature, most pipelines are buried in the ground, less researches have paid attention to the crack propagation of buried pipelines, which is more appropriate with the practical situation. In present study, a finite element model based on cohesive model is presented to simulate the dynamic fracture of pipeline cracks. Firstly, this paper establishes a pipeline model without soil layer, secondly, it establishes a model of buried pipeline which distinguishes backfill from unexcavated soil layer to studies the crack propagation of pipeline. The quantitative relationship between crack propagation rate and pipeline internal pressure is predicted, and the key parameters (such as CTOA, crack tip stress-strain field, etc.) are studied quantitatively. The dynamic fracture mechanism of full-scale pipeline is revealed, which provides necessary supplement for the experiment. The calcula tion results are accurate, which lays a foundation for further research on the fracture problems in engineering structures, especially in the treat ment of pipeline cracks.
2. Establishment of pipeline model 2.1. Force analysis of pipeline The stress on the pipe wall can be divided into axial stress, circum ferential stress and radial stress according to the direction. In the thinwalled pipe such as natural gas pipeline, the axial stress and circum ferential stress play a major role. As shown in Fig. 1, the analysis of the axial stress is carried out. Assuming that the internal pressure of the pipeline is P and the diameter of the pipeline is D, Intercepting any section of the pipeline, assuming that the end face is closed, the pressure of natural gas on the closed end is π ðD=2Þ2 P. In practice, the pipeline is not closed, so it is understood that the gas produces an axial resultant force on the wall of the pipeline, any section of the pipeline is the stress balance state. That is, the axial resultant force of gas on the pipe wall and the circumfer ential stress on the pipe wall are equal, so:
πðD=2Þ2 P ¼ πDtσx σx ¼
PD 4t
(1) (2)
As shown in Fig. 2, it is an analytical diagram of the circumferential stress. The component force in the y direction of the resultant force which produces the internal gas on the wall of the pipeline is balanced with the stress in the two sections of the pipeline. Take integral infini tesimal dα for pipe wall, Z π D PL sin αdα ¼ 2Ltσ y (3) 2 0
σy ¼
PD 2t
(4)
It can be found that the σ y is greater than the axial stress σx and has a double relationship. Therefore, it is necessary to control the circumfer ential stress of the pipeline in design. Since there is little influence on the axial force when the soil layer is added, the circumferential force is analyzed. It is assuming that the pipeline is buried in an infinite elastic body and that each elastomer maintains complete contact on the contact surface, that is, it neither separates from each other nor slides with each other. The polar coor dinate system as shown in Fig. 3 is established.
Fig. 2. Circumferential stress analysis of pipeline. 2
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Fig. 6. Traction-separation failure curve of linear softening.
cross sections of the crack. The stress is a function of the crack displacement value, that is, the tension-displacement relationship. Cohesive element is widely used in the initiation and propagation of cracks in most materials. For example, it is used for fatigue fracture, overload failure and phase interface separation of metal materials, composite matrix cracking, etc. For the three-dimensional cohesive element, it is divided into pentahedron and hexahedron. The upper and lower sections are the main load-bearing surfaces, while the surrounding surfaces are not loaded. As shown in Fig. 5, the surface of the red shadow line is the main load-bearing surface. It is supposed that there is a Fracture Process Zone at the crack tip. In three-dimensional FPZ, there are normal (tn ) and two shear directions (ts and tt ) traction forces on the crack surface. Fig. 6 shows the tractionseparation failure curve (TSL: Traction-separation laws) of normal traction (tn ) with the change of crack opening displacement (δn ). The traction-separation curves of shear traction with crack opening displacement similar to those of this kind. Before the crack initiates, it is assumed that the material is in the elastic stage, which is represented by a linear elastic rising stage. After cracking, with the monotonous decrease of the corresponding separation function, called tension or stress softening stage. The linear softening shown in Fig. 5 is also widely used, in addition, bilinear and exponential softening are also widely used. In order to express the combined action of the whole damage and various mechanisms in materials, the damage evolution of cohesive element gives a damage index D, which is a function of equivalent displacement δm . The effective relative displacement δm combines the effects of δn , δs and δt , describe the mixed displacement after the inter face being loaded. qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi δm ¼ 〈δn 〉2 þ δ2s þ δ2t (7)
Fig. 3. Stress analysis of pipeline in soil layer.
Fig. 4. Cohesive zone at crack tip.
Fig. 5. The load-bearing diagram of cohesion model.
σρ ¼
P
σϕ ¼
P
2
½1 þ ð1
2μÞn� Dρ2
ð1
nÞ
½1 þ ð1
2μ
2 Þn� Dd2
ð1
nÞ
½1 þ ð1
2μÞn� Dρ2 þ ð1
nÞ
½1 þ ð1
2μÞn� Dd2
2 2
ð1
(5)
where, 〈〉 is Macaulay bracket and � δn ; δn � 0 〈δn 〉 ¼ 0; δn < 0
(6)
nÞ
ð1þμÞ where, n ¼ EEð1þ , E is the elastic modulus of pipeline, μ is the Poisson’s μ0 Þ 0
Taking the linear softening law (or linear damage evolution in Fig. 6) as an example, the damage index can be calculated by
ratio of pipeline, E is the elastic modulus of soil, μ is the Poisson’s ratio of soil, D is the outer diameter of pipeline, d is the inner diameter of pipeline, P is the internal pressure of pipeline. 0
(8)
0
D¼
2.2. Finite element model of pipeline
δmf ðδm;max δm;max δmf
δm0 Þ � δm0
(9)
where δm0 and δmf are effective relative displacements at damage initi ation and complete failure, respectively. δm;max is the maximum effective relative displacement attained during the loading history. Initially, D ¼ 0, when the cohesive element satisfies the condition of damage generation, the value of D monotonically increases from 0 to 1 upon further loading after the initiation of damage, and then, D ¼ 1, it is complete failure. When the natural gas pipeline is under pressure, the main stress is circumferential stress, which is perpendicular to the surface of the cohesive model. Therefore, it is feasible and reliable to simulate the crack propagation of the natural gas pipeline in this way. The section
Cohesive Model is a computational model widely used in fracture mechanics. Dugdale and Barenblatt proposed the concept of cohesive model to explain the theoretical infinite stress of the crack tip. Cohesive Model avoids the stress singularity of crack tip in linear elastic fracture mechanics, and calculates the stress and fracture energy in crack pro cess. As shown in Fig. 4, the cohesive model assumes that a small area at the crack tip is a cohesive zone, which is composed of a crack and two cross sections. There is an interaction force between molecules or atoms on both sides of the crack, which is called cohesive force. The value of cohesive force σ is related to the separation displacement δ of the two 3
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Fig. 7. Pipeline numerical model and meshing.
Fig. 8. The FEA simulation of DWTT.
cracking of the upper and lower continuous elements is simulated by setting a cohesive element in the middle of the continuous element. The stress and fracture energy of cohesive element is changed according to the predefined tension-displacement relationship. Based on the background of the second West-to-East Gas Trans mission Line in China, X80 high-grade steel pipe is selected. The diam eter of the pipeline is 1219 mm, the wall thickness is 18.4–25.6 mm, the total length of the pipeline is 5 m, and the maximum design pressure is 14 MPa. Fracture toughness coefficient is KIC ¼ 115MPa⋅m1=2 , The elastic modulus is 203 GPa, Poisson’s ratio is 0.3, and material density is 7800 kg/m3, The corresponding hardening property is shown in Fig. 7 (c). It is difficult to grasp the specific trend of internal pressure change in
the process of natural gas leakage in buried pipelines. Therefore, the internal pressure of natural gas is set to be constant in process of crack initiation and stability. Symmetrical constraints are applied at both ends of the pipeline. It indicates that the pipeline extends toward both ends. According to Hashemi [20] and Shim [12] experimental research and the example of pipeline rupture, pipeline cracks tend to propagate along the axis, so the direction of crack propagation along the axis path is established, and the crack propagation path is set up. The established pipeline model is shown in Fig. 7. Because the crack propagation path on the pipeline is the focus of analysis, the local mesh is refined by the transition of coarse and fine meshes, which ensures the accuracy of calculation and saves calculation time. 4
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2.4. Model validation with X100 pipeline steel In order to verify the rationality of the model, we have chosen two literature sources. The simulations reproduce as closely as possible conditions. The burst tests of X100 pipes with 914 mm outer diameter were reported [30]. Oikonomidis [14]. simulated the pipe burst of an X100 pipe. The same conditions were chosen. The model was a 10 m long pipe with a 250 mm long axial through thickness crack introduced at one end. Fig. 9 show that the model predicted the crack arrest time, the crack velocity and the crack arrest length to within 10%. 3. Pipeline model result analysis In order to verify the reliability of the numerical model, empirical formulas are used for fitting. Battelle Research Institute of the United States fitted the formula of fracture propagation velocity through a se ries of full-scale ductile fracture tests of pipelines, which is authoritative and valuable [31]. � σ f Pd V ¼ 0:275 pffiffiffi R Pa
2.3. Determining model parameters
�16 1 =
Fig. 9. The experimental and the model profiles of crack length and time for a X100 pipe with 914 mm outer diameter and 13 mm wall thickness.
(10)
where, Vis pipeline fracture propagation velocity, m=s; R is resistance of steel to fracture propagation, KJ=m2 ; σ f is flow stress, MPa; Pd is pipeline crack tip pressure; Pa is pipeline pressure at crack initiation, MPa. Substituting material and size parameters into the semi-empirical formulas, the relationship between crack propagation rate and inter nal pressure represented by semi-empirical formulas is obtained, as shown in the black curve in Fig. 10(b). Fig. 10(a) shows the curve of the crack propagation distance with time in the finite element simulation process. According to the curve, the steady-state velocity of the crack propagation is obtained, which is in fitting well with the results obtained by semi-empirical formula. It shows that the numerical simulation has certain reliability. Both empirical formulas and numerical simulations show that there is an upper limit for the crack propagation rate. Even if the internal pressure increases, the crack propagation rate will not increase infinitely. When other parameters are fixed (such as crack length, inner diameter, etc.), there exists a critical internal
Based on the Drop Weight Tear Test (DWTT) experiments and hun dreds of simulations, we adopt the following parameters of X80 by Shi and Wang [28]:K ¼ 80GPa=mm,δm0 ¼ 0:0225mm, δmf ¼ 4δm0 . The corresponding hardening property of X80 pipeline steel is specified by Fig. 8(c) which determines the dependence of Mises yielding stress on equivalent plastic strain. [29] The fracture parameters chosen in such way achieves good agreement with tests not only for fracture speed but also for the loading force-displacement curves. The fracture parameters can also be applied to pipeline fracture. From Fig. 8(d), because of the dynamic simulation, the data expe rienced a slight fluctuating trend, but the overall trend is fitted. The parameters of CMZ are used to simulate the fracture of X80 steel.
Fig. 10. Relation between crack distance and propagation velocity without soil layer constraints. 5
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A common formula is to determine the distance between the two sides of the crack at a specific distance from the back of the instantaneous crack tip. CTOA ¼ 2arctanðδ = 2xÞ
(11)
where, δ is the opening displacement of the crack tip at a certain distance x behind the crack tip. Kazuki [16] conduct a full gas burst test to clarify the CTOA history during ductile cracking in a high-pressure pipeline. In addition to the CTOA based on the COD 2 mm and COD 10 mm behind the crack tip was evaluated. The values of both CTOA(2 mm) and CTOA(10 mm) clearly remained nearly constant during crack propagation. Thus, the average values were 13.2� for CTOA(2 mm) and 13.9� for CTOA(10 mm). The difference of CTOA between the CTOA(2 mm) and CTOA(10 mm) is 0.7� .In order to facilitate statistics and ensure the accuracy of data, the x value selected in this paper is 0.005 m. As shown in Fig. 12(a), the abscissa is the crack propagation dis tance, and the ordinate is the crack tip opening angle corresponding to the internal pressure. It can be seen that the crack tip opening angle is a higher value at the beginning of the crack propagation, because in the initial stage of crack propagation, natural gas in the pipeline leaks from the crack, the pressure is concentrated, and the crack tip opening angle decreases to a relatively stable value just after the initial velocity of crack propagation being obtained. Different pressures correspond to different steady-state velocities, and different pressures correspond to different steady-state crack tip opening angles.
Fig. 11. Definition schematic of CTOA.
pressure for crack propagation. When this pressure is reached, the crack begins to propagate. This critical internal pressure can simulate the crack several times. Compared with the experimental results from Shibanuma et al. [19], the simulation results are reliable, which can better illustrate the reli ability of using this method to simulate the propagation of pipeline cracks. The crack tip opening angle means the magnitude of crack driving force. The crack tip opening angle (CTOA) has been applied to the evaluation of crack arrest toughness of high toughness pipeline steel. It can combine the local micro-mechanism of crack propagation with the stress and plastic flow depending on the geometric shape. It is more truly reflecting the deformation of crack tip and stress distribution. The definition of CTOA is related to plastic deformation energy, as shown in Fig. 11. The current definition is based on experimental measurements.
Fig. 12. Changes of crack tip opening angle and crack cloud figure of the stress without soil layer.
Fig. 13. Pipeline model and meshing. 6
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Fig. 14. Curve of crack length with time and crack propagation rate under soil layer.
4. Pipeline model under soil layer 4.1. Pipeline-soil model of prefabricated penetrating crack Soil layer is an important factor affecting the mechanical properties of pipelines. Most scholars did not consider the actual situation of pipeline surrounding soil when studying pipeline fracture behavior, and reasonably divided backfill and soil layer. In order to simulate buried pipelines more realistically and increase the applicability of the model, a soil layer and pipeline model is established, as shown in Fig. 13 Supposing that the backfill and the soil layer are loess, Mohr-Coulomb elastic-plastic constitutive model is used to describe the non-linear characteristics of rock and soil. Its elastic modulus 14.4 MPa, Poisson’s ratio is 0.2, cohesive force is 18 KPa, internal friction angle is 30� , dilation angle is 16� , density is 1400 kg/m3, the gravity load is applied to the soil, and the acceleration of gravity is 9.8 m/s2. In order to distin guish backfills, cohesive elements with zero thickness are added be tween backfills, as shown in Fig. 13 (b), so that they can crack and separate. The friction coefficient between pipe wall and soil is 0.3. In order to make backfill soil contact with pipeline better, the mesh of backfill soil is refined. Similarly, pressure loads are applied to the inside of the pipe to simulate the fluid action in the pipe. Through finite element simulation, the curve of crack propagation distance of pipeline with time is obtained. As shown in Fig. 14(a), the length of crack propagation is considered to be the distance exceeding the length of prefabricated crack, and the slope of the curve is expressed as the instantaneous velocity of crack propagation at each moment. With the decrease of pressure, there are many horizontal trend stages in the curve, that is, the crack propagation rate slows down, and then returns to a relatively stable rate. The lower the pressure is, the longer the horizontal trend stage time will be. Under the same pressure, compared with the case without soil layer, the horizontal trend stage is not obvious. According to Fig. 14(a), the steady-state crack propagation rate is obtained. As shown in Fig. 14(b), the crack propagation rate decreases obviously due to the restraint of the soil layer and the consumption of part of the energy of the soil layer. With the increase of the internal pressure, the crack propagation rate increases gradually. When the in ternal pressure is large, the crack propagation rate increase slightly. Fig. 15 shows the curve of crack tip opening angle with crack length under different internal pressures. Compared with the case without soil
Fig. 15. Curve of crack tip opening angle with crack propagation distance.
layer, the crack tip opening angle is also a higher value at the beginning of crack propagation, and then tends to be stable. The difference is that the crack tip opening angle is larger than that in the case without soil layer, and there will be more significant fluctuations in the later stage than that in the case without soil layer. Crack tip opening angle indicates the crack driving force. In the initial stage of crack propagation, a large driving force is required for crack propagation. When the initial velocity of crack propagation is obtained, the driving force required for crack propagation decreases and tends to be stable. However, due to the in fluence of soil layer, the soil layer deformation will consume part of the driving force for crack propagation. As a result, the crack tip opening angle will decrease, and then the effect of internal pressure will continue to drive the crack propagation, and the crack tip opening angle will increase. Fig. 16 is a cloud figure of formation damage and pipeline stress. The cohesive elements in backfill soil shown in Fig. 16(a) are broken and mainly concentrated in the soil layer near pipeline. The red 7
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Fig. 16. Soil layer damage cloud diagram and stress cloud diagram.
Fig. 17. The propagation of cracks under soil layer and pictures taken at the scene.
part between elements in Fig. 16(b) is the damage of cohesive elements. Therefore, the leakage of pipeline crack propagation causes soil collapse and explosion accidents which mainly attribute to the disturbance of soil caused by gas inside the pipeline. The deformation of pipeline canno
t cause large-scale collapse and explosion. Fig. 17(a) shows a cloud diagram of crack propagation in pipelines under soil layer. Similar to Fig. 12(b), the stress concentration area is mainly near the crack tip, but in the case of covering by soil layer, the 8
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Fig. 18. Curves of surface crack propagation length and propagation rate under rate.
Fig. 19. Cloud diagram of surface crack propagation under soil layer.
stress concentration area is wider. Therefore, when considering pipeline crack arrest, relevant crack arrest measures in the near area where cracks are found can better achieve the effect of crack arrest. Fig. 17(b) is pipeline crack propagation pictures taken at the scene by Bogatov [9]. It can be inferred that the numerical simulations can well simulate the process from initiation to stable expansion of pipeline, and can provide reference for the subsequent anti-cracking measures.
along the wall thickness direction of the pipeline when the surface crack propagates. When the crack propagates to a certain threshold, the crack begins to propagate in the axial direction. At this time, the initial rate has been obtained, so the horizontal trend stage is not obvious when the crack propagates. As shown in Fig. 18(b), surface crack propagation rate is faster than penetrating crack propagation rate, because surface crack propagation along the wall thickness direction has obtained a certain initial rate, so under the same internal pressure, surface crack propagation rate is faster than penetrating crack propagation rate. From the cloud diagram of Fig. 19, it can be seen that the surface crack of the pipeline propagates along the wall thickness of the pipeline, and when it reaches a certain threshold, the crack begins to propagate in the axial direction.
4.2. Pipeline-soil model of prefabricated surface crack According to the analysis in section 2.1, the most dangerous point in the pipeline is found out. The penetrating crack in the pipeline is replaced by surface crack. The wall thickness of the pipeline model is 18.4 mm, the crack is internal crack, and the crack depth is 9 mm. This defective pipeline of internal surface crack is simulated and analyzed under soil layer. The curve of crack propagation length with time is obtained as shown in Fig. 18(a). The horizontal trend stage is not obvious when the surface crack propagates, because it first propagates
5. Conclusion In this paper, the fracture propagation process of full-scale pipeline is simulated using cohesive model, with considering the influence of strata 9
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on crack propagation. The crack propagation velocity and propagation angle are quantitatively analyzed, and the following conclusions can be drawn.
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(1) The cohesive model can well simulate the dynamic crack prop agation process in pipelines. In the case of pipelines without soil layer, the simulation results and empirical formulas show that there is an upper limit for the crack propagation rate, that is, the internal pressure increases continuously, and the crack propa gation rate will not increase infinitely. In the design of crack arresting tools, it is necessary to control the circumferential stress on the pipeline. (2) Whether or not there is soil layer, in the initial stage of pipeline crack propagation, the crack tip opening angle is a high value, and then it decreases to a relatively stable value. The crack tip opening angle increases and greater fluctuates under soil layer. The formation deformation caused by pipeline cracking mainly concentrates near the pipeline. (3) Using cohesive element to simulate the backfill of pipeline can more accurately describe the crack propagation phenomenon of pipeline. The crack propagation rate will appear many horizontal trend stage. The lower the internal pressure is, the more obvious the horizontal trend stage is, the crack propagation rate will decrease significantly, and the crack tip opening angle will in crease and the fluctuation will be more obvious. (4) Internal surface cracks are more dangerous than penetrating cracks. The surface cracks of pipelines propagate along the wall thickness of pipelines. When they reach a certain threshold, the cracks begin to propagate in the axial direction. Therefore, in the detection of pipeline defects, we should also pay attention to the impact of surface cracks. Declaration of competing interestCOI We declare that we do not have any commercial or associative in terest that represents a conflict of interest in connection with the work submitted. Acknowledgments This study is supported by the China Postdoctoral Science Founda tion (2018M633403), Applied Basic Research of Sichuan Province (Free Exploration-2019YJ0520), National Natural Science Foundation of China (51674214 and 51709231), Youth Science and Technology Innovation Research Team of Sichuan Province (2017TD0014). Such supports are greatly appreciated by the authors. Appendix A. Supplementary data Supplementary data to this article can be found online at https://doi. org/10.1016/j.soildyn.2019.105881. References [1] Bussiba A, Darcis PP, Mccolskey JD, Mccowan CN, Kohn G, Smith R, et al. Fatigue crack growth rates in six pipeline steels. In: International pipeline conference; 2006. https://doi.org/10.1115/IPC2006-10320. [2] Igi S, Guan C, Rothwell B, Hiraide T. Investigation of crack propagation characteristics using instrumented charpy and DWT tests for full-scale burst tested 1219 mm OD grade 550 (X80) linepipe. In: International pipeline conference; 2016. https://doi.org/10.1115/IPC2016-64240. [3] Tronskar JP, Mannan MA, Lai MO. Measurement of fracture initiation toughness and crack resistance in instrumented charpy impact testing. Eng Fract Mech 2002; 69(3):321–38. https://doi.org/10.1016/S0013-7944(01)00077-7. [4] Chaouadi R, Puzzolante JL. Loading rate effect on ductile crack resistance of steels using precracked charpy specimens. Int J Press Vessel Pip 2008;85(11):752–61. https://doi.org/10.1016/j.ijpvp.2008.08.004.
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