Computation of the stress intensity factor KI for external longitudinal semielliptic cracks in the pipelines by FEM and XFEM methods
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Abstract
Evaluation of structural integrity of a cracked structure has become an important matter in the industrial field since couples of decades. However, damage process occurred in a structural component is not yet fixed. The objective of this research was to compute the stress intensity factor KI, in mode I, using in the linear elastic domain, by the finite element method and the extended finite element method. The defect studied in this survey has a form of a longitudinal semielliptic crack, located on the outer surface of the tube. A summary of the paper contains a numerical convergence for each method in terms of accuracy and limitations. The proposed methodology and outcomes released from this study act as novel design tool for the industrial engineers when is required to generate a robust solution for product development working in critical conditions.
Keywords
Stress intensity factor (SIF) Finite element method (FEM) Extended finite element method (XFEM) Semielliptic crack J integral1 Introduction
In the field of oil industry, the pipelines are the most used means of transporting petroleum materials such as gas, oil and hydrogen. The pipes installed above ground are of great issue. During their operation, they can generate several unexpected damages that cause significant material and human damage as well as environmental damage, especially if they are in the oceanic environment. To reduce these accidents the costs of maintenance can be very high, triggering a major concern for companies operating in this field. The research carried out by Lam [1] on the statistics of the accidents of the pipes, shows that the majority of the breaks are due to: pits of corrosion, the impact of a bullet lost if one is in a hunting zone (for piping install above ground), or at the impact of a bucket excavating machinery. These defects are in the form of a crack or a scratch; they affect the pipeline resistance and cause a sudden break that can be catastrophic. For that reason, researchers are interested in evaluating the structural integrity of pressurized pipelines.
These defects are generally treated by the fracture mechanics approach, which provide accurate details of the distribution of stresses and deformations near the defect zone, but also can indicate an estimative lifetime of the resistance of these structures according to the critical size of defects.
The Stress Intensity Factor KI is a key parameter used in the fracture mechanics field. Researchers as Moustabchir [2], Berer [3], Zareei [4] use this parameter to predict the initiation and propagation of cracks in the pipes. KI were calculated using several methods such as analytical, semianalytical, and numerical methods.
Modern FEM simulations permits to reduce considerable the cost of product design process and to estimate in real time potential harmful situation when a crack nucleate and propagate. Besides, applying numerical simulation allows investigate the structural response of a components and predicting its live during working condition [5]. It permits to highlights improvement of design by eliminating the physical/real constrain (incapacity to test large/long products (chain of pipe), visualize their behavior in harsh environments, etc.),however they are much easy solved when are transferred to virtual constraint guidance (VCG) [6].
The literature state many researchers who deal with the problem of crack in the pipes using the FEM methods, in contrast a very rare works was noted for the XFEM method. Moustabchir [2] calculated the Stress Intensity Factor KI, mode I, in the pipes structures containing an axial semielliptic crack using the Finite Element Method (FEM). Berer [3] studied the effect of the opening of a crack, in mode I, in cracked cylinders when compressive loading generates the KI factor.
Zareei [4] calculated the KI factor for an internal circumferential semielliptic crack in a pipe subjected to any arbitrary load. Study that was based on the finite element analysis in three dimensions. Sahu [7] calculated the KI factor for a semielliptic crack located at the inner surface of the tube for different ratios, ratios that depends on the defect geometry (a/t) and (a/c). The finite element method (FEM) embedded was generated on the ANSYS software.
The literature indicates great potential to use the XFEM method to evaluate the rupture of the pipes. Martin [8] used the XFEM method to evaluate crack propagation in SUBSEA equipment.
This research investigates the mechanism of a defect that occurs in a pipeline and propagate at external surface in a longitudinal manner as a semielliptic cracks. Shim [9] calculated the Stress Intensity Factor KI in mode I using the XFEM for various types of plate and pipe cracks. Sharma et al. [10] and Sharma et al. [11] evaluate the stress intensity factors (SIF) of an axial/circumferential semielliptical crack in the pipe and elbow using the XFEM method.
In this work, were computed the Stress Intensity Factor KI numerically by the classical finite element (FEM) and extended (XFEM) method for an axial semielliptic crack located in the outer surface of the pipe. The key objective is to highlight the power of each method using a robust convergence, strategy that provides critical values of the KI parameters in the different positions of the crack.
2 Stress intensity factor KI
The Stress Intensity Factor KI represent the most important parameter in the linear elastic fracture mechanics. It allows to predict whether the crack is stable or not in respect to the toughness of the KIC material. Determination of the KIC is usually performed using the CharpyV impact test Berer [3]. The cracked engineering components are examined in terms of KI by various analytical or semielliptical methods. These methods are based on displacement extrapolation and/or energetic approach such as integral J/integral interaction. Methods detailed in the reference Qian [12]. Zhu [13] showed that the energy approach provides good result in comparison with analytical results, reason why the computation of factor KI by the energetic approach is independent of the mesh near the crack.
 1.
Computation of the KI factor by the integral contour method:
 2.
Integration technique (Gauss) in the XFEM method:
 3.Calculation of the KI factor by the Raju and Newman method Raju [18]:$$ {\text{K}}_{{\text{I}}} = \frac{{{\text{PR}}_{i}^{2} }}{{{\text{R}}_{{\text{e}}}^{2}  {\text{R}}_{{\text{i}}}^{2} }}\sqrt {\frac{{\uppi {\text{a}}}}{{\text{Q}}}} \left[ {2{\text{G}}_{{\text{0}}} + 2{\text{G}}_{{\text{1}}} \left( {\frac{{\text{a}}}{{{\text{R}}_{{\text{e}}} }}} \right) + 3{\text{G}}_{{\text{2}}} \left( {\frac{{\text{a}}}{{{\text{R}}_{{\text{e}}} }}} \right)^{2} + {\text{4G}}_{3} \left( {\frac{{\text{a}}}{{{\text{R}}_{{\text{e}}} }}} \right)^{3} } \right] $$(5)
With \( {\text{Q}} = 1.0 + 1.464\left( {\frac{{\text{a}}}{{\text{c}}}} \right)^{1.65} \) For \( {\text{a}}/{\text{c}} \le 1 \)
G0, G1, G2, G3, are functions dependent on geometry Raju [18].
Ri and Re are the inner radius and the outer radius.
P: Pressure applied to the tube.
3 The principle of the XFEM method
The Extended Finite Element Method (XFEM) has emerged as a powerful numerical procedure for analyzing crack propagation problems. The approach of XFEM was introduced on 1974 by Benzley [19] who proposed the idea of enrichment near the crack front using asymptotic solutions for static failure problems. Atluri et al. [20] and Nash Gifford et al. [21] subsequently developed this method and obtained highly accurate results for stationary cracks.
A few years later, Melenek and Babuska [22] developed the fundamental unit partition method for the finite element method (PUFEM). The first real upgrade “development” effort of XFEM was made by Belytschkol and Black [23]. Sukumar et al. [24] was the first to extend the XFEM method to model threedimensional cracks. Stolarska and all [25] coupled the level set method and the XFEM method to predict crack propagation. Finally, Belytschko et al. [26] developed a new XFEM formulation for the arbitrary propagation of cracks in hulls.
Yet, several researchers Moes [27], Chahine [28], Budyn [29], Gupta [30], Wang [31], Hou [32] used the XFEM method to simulate the behavior of fracture mechanics in the case of stationary or dynamic cracks.
The function H (x) thus takes the values of + 1 or − 1 according to the side of the crack on which one is placed.
4 Numerical simulation with ABAQUS software
4.1 The geometry of the problem
Geometric characteristics of the tube
Inner radius  Ri = 100 mm 
Outer radius  Re = 110 mm 
Thickness  t = 10 mm 
Length  L = 200 mm 
Mechanical properties of P265GH material Moustabchir [35]
Elasticity module  E = 207,000 MPa 
Poisson coefficient  ν = 0.3 
The limit of elasticity  Re = 340 MPa 
Limit of rupture  R_{M} = 440 MPa 
Elongation  A = 35% 
Critical stress intensity factor  K_{IC} = 94.99 MPa.m^{0.5} 
4.2 Numerical modeling
In this present study, were extracted the Stress Intensity Factor values of KI in mode I, simulating the behavior in the linear elastic domain. The ABAQUS 6.14 software may offers two different ways to evaluate the full contour. The first is based on the classical finite element method (FEM), which usually requires the user to define explicitly the crack front. In the ABAQUS 6.14 software, the special command (* CONTOUR INTEGRAL) were accessed that is dedicated to calculate the integral J in the crack front. This command uses a predefined model based on the discretized formula of J (Eq. 2).
In the second method of XFEM, the data required for the contour integral is determined automatically by level set for a specified distance of the functions related to the nodes connected to an element Zhu [36].
This study treats a longitudinal crack/cracks in pipes because they are more critical than circumferential cracks. A longitudinal crack/cracks have been studied carefully in this work Khoramishad [37].
The crack configuration is described by some nondimensional parameters, namely the relative wall thickness (t/R), the relative crack depth (a/t) and the aspect ratio of the crack (a/c). The geometry of the pipe and the position of the longitudinal crack on the pipes and their effects on the socalled Stress Intensity Factor were studied at different positions along the crack front, a/t: 0.2, 0.5, 0.8 and a/c: 0.2, 0.4, 1 with t/R = 0.1
4.3 Meshing, loadings and boundary conditions
The mesh step of the geometry studied is a very important phase that determines the precision of the results and the computation time. A satisfactory mesh is generated for a mesh when it makes possible to have precise results in an optimal computation time.
The new XFEM method permit to eliminate the limitations of the classical finite element method (FEM). This method makes possible to study the problems of crack propagation without remeshing. In addition, the second advantage of this method is that the mesh of the structure is independent of the crack geometry, this is feasible with new enrichment functions that allow dealing with the problem of the singularity at the point of the crack, and the discontinuity of the displacement.
5 Results and discussions
The numerical results obtained using the ABAQUS software were compared in terms of KI factor between the two FEM and XFEM methods. In the numerical simulation was considered the energetic approach derived from Eq. (3).
On the other hand, in the XFEM method, the constraints are more stable and well presented. Here we can note here, the advantage of the XFEM method which permits to overcome the problem of the singularity at the point of the crack tip.
The error of the KI values for the critical default
KI (MPa mm^{0.5}) (angle φ = π/2)  

Default size  a/c = 0.2, a/t = 0.8  a/c = 0.4, a/t = 0.8  a/c = 1, a/t = 0.8 
Analytical  270.99  169.25  89.65 
FEM method  290.8  171.5  92.1 
XFEM method  271.9  170.33  88.02 
FEM method error (%)  6.81  1.31  2.66 
XFEM method error (%)  0.33  0.63  1.81 
Besides, when the value of the ratio a/c decreases the values of the KI also increases until it reaches critical values.

The problem of stress singularity on the crack tip is treated better by the XFEM method, compared to the classical method (FEM), FEM approach that require to use a very fine and very regular mesh around the point of the crack that may have great influence on the results of the stresses; however, this is not the case for the XFEM method where the mesh is independent of the geometry of the crack. The treatment of the singularity problem is evaluated using the enrichment functions.

The value of the Stress Intensity Factor KI is maximum at the 90° angle of the crack front. This represents the deep point of the crack near the same time of the inner surface of the tube where the maximum value of the pressure is applied.

The numerical and analytical results present a good agreement, having only a difference of 1.81% by using XFEM method and excending 6.81% in FEM method. In addition, the results by the XFEM method are closer to the analytical results of Raju [16]. The global agreement observed gives confidence for the use of the XFEM method for the determination of KI values.
6 Conclusion
The two numerical methods (FEM and XFEM) used for calculation of the Stress Intensity Factor KI, in mode I, in the elastic linear domain prove a robust tool for assessment of structural components. It was validated well against the analytical results existing in the literature. The results also show that the position of the longitudinal crack on the pipe has a significant influence on the stress intensity factor KI; and the XFEM method permits to overcome the stress singularity problem at the point of the crack tip. This strategy gives confidence in the use of the XFEM method to deal with cracking problems in complex structures.
The success of this methods can be extended, such as this work using the XFEM method permits to create robust platforms to study the problem of fatigue in cracked pipes, that can be easy coupled to the Paris law. And having the final objective to obtain KI, in dynamic case, while applying the XFEM method.
Notes
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