Computation of the stress intensity factor KI for external longitudinal semi-elliptic cracks in the pipelines by FEM and XFEM methods

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, semi-analytical, 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 X-FEM method. Moustabchir [2] calculated the Stress Intensity Factor KI, mode I, in the pipes structures containing an axial semi-elliptic 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 semi-elliptic 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 semi-elliptic 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 X-FEM method to evaluate the rupture of the pipes. Martin [8] used the X-FEM 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 semi-elliptic cracks. Shim [9] calculated the Stress Intensity Factor KI in mode I using the X-FEM 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 semi-elliptical crack in the pipe and elbow using the X-FEM method.

In this work, were computed the Stress Intensity Factor KI numerically by the classical finite element (FEM) and extended (X-FEM) method for an axial semi-elliptic 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.

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 Charpy-V impact test Berer [3]. The cracked engineering components are examined in terms of KI by various analytical or semi-elliptical 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.

The strategy applied in this work to calculate the Stress Intensity Factor are presented as follows:

In the Abaqus/Standard code calculation the evaluation of the J integral is done in an automatic way by applying the integral method over the energy domain. It is based on the formulation described by Eq. (1) developed by Shih et al. [14, 15] where A* is the area of the surface between the contours \( \Gamma _{0} \) and \( \Gamma _{1} \) (as presented in Fig. 1). The parameter (q) is a smoothed function that can be choice as q = 1 on \( \Gamma _{0} \) and q = 0 on \( \Gamma _{1} \). The contour \( \Gamma _{0} \) is reduced in practice at the crack tip. The outline \( \Gamma _{1} \) coincides with the edges of the elements.

Further discretization is presented on the Eq. (2) given by formula:with: np: number of Gauss points, \( {\text{w}}_{\text{p}} \): Integration weight, \( \upxi_{\text{k}} \): Coordinates of elements in local landmarks.

There, the integral J was used to calculate the factor KI for the mechanical case with linear elastic rupture. The integral J is equivalent to the rate of the energy restitution G under a single mode for the linear elastic problems. The factor KI were calculated from the integral J according to the following equation Rice [16]:with \( {\text{E}}^{\prime } = {\text{E}} \) plane stress, \( {\text{E}}^{\prime } = \frac{\text{E}}{{1 - {\upnu }^{2} }} \) plane strain.

Equation (3) was introduced in the code “ABAQUS” and “ANSYS” to compute the KI Stress Intensity Factor numerically from the integral J.

To extract the numerical values of displacements were obtained through numerical integration Eq. (4). Some difficulty appears in classical FEM when is encountered a discontinuous domain, however it was solved much easy, for a cracking problem, applying the method (X-FEM). Digital integration presents two major difficulties: the discontinuity along the crack and the singularity at the bottom of the crack. Therefore, these difficulties require special treatment because of the presence of a discontinuity in integration [17].

The integration of these enriched elements were obtained by dividing them into several tetrahedra located above and below the surface, as is highlighted in Fig. 2. For the element comprising the crack tip, it is first necessary to divide it into two types of elements by a vertical line passing through the crack tip, then the numerical integration will be completed in each type of element, as illustrated in figures Fig. 3b–d. For the cracked element, the numerical integration will be completed in two subdomains, as shown in Fig. 3a. The integration of enriched elements generally requires a higher order Gauss quadrature.

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.

The Extended Finite Element Method (X-FEM) 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 X-FEM was made by Belytschkol and Black [23]. Sukumar et al. [24] was the first to extend the XFEM method to model three-dimensional cracks. Stolarska and all [25] coupled the level set method and the X-FEM method to predict crack propagation. Finally, Belytschko et al. [26] developed a new X-FEM 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 X-FEM method to simulate the behavior of fracture mechanics in the case of stationary or dynamic cracks.

According to Belytschko and Moes [33], the displacement field may be described by the finite element approximation using the following equation:where: Ni (x): Interpolation functions of standard finite elements. H(x): The Heaviside enrichment function presents the ‘Jump’ of the displacement of the lips. a_i: Additional degrees of freedom associated with the Heaviside function. F_j (x): Enrichment functions with singularity near the crack front. b _i ^j : The additional degrees of freedom associated with singular functions.

The function H (x) is presented by the following equation:

The function H (x) thus takes the values of + 1 or − 1 according to the side of the crack on which one is placed.

The enrichment function, presented as sketch in Fig. 4, shows the singularity of the vicinity of the crack front Fj (x). It can be presented by the following equation:where (r, θ) is the polar coordinate system with its origin at the crack tip

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).

Figure 10 shows a plot of Von Mises distribution stresses for the ratio (a/c = 1, a/t = 0.8), where the maximum stress reaches the value 151.1 MPa in both methods. There is a small disturbance of the stresses at along the crack front for the FEM method, that may be is due to the presence of singularity at the point of crack.

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.

Figure 11 presents the comparison between the XFEM method, the FEM method and the Raju and Newman method Raju [18] along the crack front for the case (a/c = 1). The variation of KI factor was obtained as a function generated by the crack angle where the maximum value lies in the angle φ = π/2, and represent the deepest point of the crack. The approximation highlights good agreement between the numerical results calculated by the two methods (FEM and XFEM) and the analytical results found by Raju [18].

Figures 12 and 13 shows the variation of the KI factor around the crack front for the case (a/c = 0.4 and a/c = 0.2). The results acquired show good agreement between the numerical and analytical results, and the maximum value of the KI factor was obtained for an angle φ = π/2.

In Table 3, were provided a comparison between the two numerical methods (FEM and XFEM) and the analytical method of Raju and Newman [18], in terms of errors and KI results values. They proves that the modelling strategy implemented using the XFEM method does not exceed an error of 1.81%, however, were detected a larger error when is used the FEM method, error that is in turn of 6.81%. This difference is due to the power of the XFEM method, once modelling cracked structures. The robustness is derived from the enrichment functions that they are added to the formulation of the classical FEM method.

In Fig. 14 presents the variation of the KI factor as a function of the depth of the crack in the tube. It is noted that when the crack depth increases the value of the KI factor also increases until it reaches the critical value 280 MPa.mm^0.5 for the case (a/c = 0.2).

Besides, when the value of the ratio a/c decreases the values ​​of the KI also increases until it reaches critical values.

The outcomes of this study can be summarized as: