## 4 Numerical Procedure and Results  
The FAD has most recently been utilised for pipeline integrity [42], evaluating a cracked pipe using the FAD construction built on the curve of stress-strain. In addition, in [41], they modelled a pipe with an external defect to build the FAD, and the modelling was done by the CFEM using CASTEM software. We can also find similar research works that have been interested in cracked pipelines such as [7; 8; 9], where they have focused on cases where plastic deformation is important. They have been interested in circumferential cracks, and the classical finite element model is used to study the CTOD failure parameter. Thus, the FAD and XFEM were used to evaluate the liquefied petroleum gas cylinder (API 579) [25]. In this study, we have combined the volumetric technique (VM) based on the XFEM and FAD for studying the pipe steel P264 GH. In order to determine the parameters of the VM, we have to calculate the elastoplastic circumferential stress distribution around the defect \(\sigma_{yy}\). The geometry of our study was numerically modelled with XFEM in a three-dimensional space of the solid type established with ABAQUS/CAE with the standard/explicit scheme, for the purpose of establishing the curve \(\sigma_{yy}\) of a steel pipe exposed to inner pressure. The cylinder configuration was simulated as a 3D deformable solid structure, and to reduce the calculation time, we have turned our problem into a symmetry problem, which is why, in the transversal and axis direction, we applied symmetric boundary conditions, so just a quarter model could be simulated. The XFEM model design with enforced limit conditions is introduced in Figure 4. where symmetric boundary conditions on the transversal and axial direction are illustrated in Figure 4. The mesh is shown in Figure 4, with 338,148 nodes and 322,000 elements. In the XFEM approach, the geometry is independently meshed using an eight-node linear brick element with reduced integration and hourglass control (C3D8R) in the Abaqus element library. The crack initial is semi-elliptical, and it is axial because, in the axial direction, the defect becomes critical. Regarding the loads, we applied a pressure of 60 bar for studying the elastoplastic behaviour because the plastic compartment appears when the pressure exceeds 40 bar; this choice is based on the experimental study of [43]. To make a good calculation, the dimensions of the elements through the thickness of the wall (i.e., the path of propagation) is 0.1 mm, although the rest of the structure was meshed with coarser elements. For this study, crack propagation modelling is carried out by combining the XFEM method and cohesive segments approach [44], where this technique has been most recently employed [45; 46]. This method was based on the traction separation law. The use of this methodology lets us avoid the singular shape function, and just the Heaviside function was considered [47]. The failure mechanism includes two parameters: The damage initiation and damage  
Figure 4: Finite element model and boundary conditions: (**a,b**) Symmetric boundary in the X, Y, and Z direction, and (**c**) mesh of the quart model using extended finite element method (XFEM). evolution criterion. The structural behaviour is assumed linear-elastic up to the initiation criterion being reached, and this parameter is based on stress or strain, where both are available in Abaqus software. Then, material degradation depends on the damage evolution, where two kinds of parameters define the evolution of damage: The first parameter consists of specifying the effective displacement at total failure, and the second involves specifying the energy dissipated because of the failure. In this study, mode I is the mode of rupture under consideration; it is the opening of the crack where the tensile stress will be normal to the crack plane.