Microsoft Word - numero_60_art_24_3470.docx H. Djeloud et alii, Frattura ed Integrità Strutturale, 60 (2022) 346-362; DOI: 10.3221/IGF-ESIS.60.24 346 Investigation fatigue crack initiation and propagation cruciform welded joints by extended finite element method (XFEM) and implementation SED approach Djeloud Hamza, Moussaoui Mustafa Laboratory of Development in Mechanics and Materials (LDMM) - University of Djelfa, (17000) Algeria. hamzadjaloud@gmail.com, http://orcid.org/0000-0002-0726-3466 moussaoui_must@yahoo.fr Kellai Ahmed Research Center in Industrial Technologies, CRTI, P.O. Box 64, Cheraga, 16014 Algiers, Algeria. a.kellai@crti.dz Hachi Dahmane Laboratory of Development in Mechanics and Materials (LDMM) - University of Djelfa, (17000) Algeria. hachi_dahmane@yahoo.fr Filippo Berto Department of Mechanical and Industrial Engineering, NTNU – Norwegian University of Science and Technology, Trondheim, Norway. filippo.berto@ntnu.no, http://orcid.org/0000-0002-4207-0109 Benattou Bouchouicha Laboratory of Materials and Reactive Systems (LMSR), Department of Mechanical Engineering, University of Sidi-Bel-Abbes, Bp 89, cité Ben M'hidisidi- Bel-Abbes 22000-Algeria. benattou.bouchouicha@gmail.com, https://orcid.org/0000-0002-6051-5108 Hachi Brahim Elkhalil Laboratory of Development in Mechanics and Materials (LDMM) - University of Djelfa, (17000) Algeria. br_khalil@yahoo.fr, http://orcid.org/0000-0002-6672-746X ABSTRACT. This study has used the strain energy density (SED) approach to evaluate the stress intensity factor (SIF) of cracked cruciform welded joints in Hardox 450 steel. A microstructural analysis was made of Hardox 450 steel which is composed of refined and tempered low carbon martensite. The Citation: Djeloud, H., Moussaoui, M., Kellai, A., Hachi, D., Berto, F., Bouchouicha, B., Hachi, B. E., Investigation fatigue crack https://youtu.be/Ecus9L7GERs H. Djeloud et alii, Frattura ed Integrità Strutturale, 60 (2022) 346-362; DOI: 10.3221/IGF-ESIS.60.24 347 obtained results of simulation will be compared with those provided by J-integral method for different enriched zones and contours based on the extended finite element method (XFEM) coupled with the level set technique (LST). Crack initiation and propagation under cyclic loading have been adopted for the modeling of cruciform welded joints. KEYWORDS. Strain energy density approach; XFEM; Stress intensity factor; crack initiation and propagation; Hardox 450. initiation and propagation cruciform welded joints by extended Finite Element Method (XFEM) and implementation SED approach, Frattura ed Integrità Strutturale, 60 (2022) 346-362. Received: 13.02.2022 Accepted: 22.02.2022 Online first: 02.03.2022 Published: 01.04.2022 Copyright: © 2022 This is an open access article under the terms of the CC-BY 4.0, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited. INTRODUCTION elding is an efficient and long lasting joining procedure. The various welding types are used in almost all industries. However, the welding operation, usually creates different types of defects, like cracks, porosity, elemental segregation, and brittle phases. These defects significantly decrease the fatigue life [1,2]. XFEM has been successfully applied to solve many welding-related problems. For instance, Kai uses XFEM to create a repaired welding model of welded joints of P91 steel plates with particular cracks[3]. Chen et al. build a numerical model by XFEM and experimental investigation of crack growth in T-joints to better understand the mechanisms of crack growth in welded joints [4]. Wang et al. also used XFEM for numerical simulation of the fatigue crack growth and suggested developing simple solutions for practical prediction of kM factors [5]. XFEM will be utilized to make a model of a semi-elliptical weld toe crack in a fillet weld for different sizes. The obtained kM parameter is represented in the form of curves as a function of crack dimensions. Pang makes use of the volumetric approach to evaluate the SIF in a pipe made P264GH steel under internal pressure by adopting XFEM [6]. He chose P264GH due to its weldability and ductility properties that help make this material appropriate for piping [7]. Taheri used XFEM to evaluate the effects of welding residual stresses on crack growth rate[8]. Kraedegh et al. in [9] examined fatigue crack growth in T joints under three-point bending that were simulated numerically by XFEM. Chatziioannou et al. manufactured X-joint specimens, S420 steel and used them [10]. A comparative study between repaired and unrepaired cruciform welded. A new correlation was proposed to assess the SIF of repaired cruciform welded joints based on the reduction and the correction factors of un-repaired cruciform welded joints [11]. They are vulnerable to high cyclic loading, and rigorous numerical models are used to simulate the experimental Can provide accurate predictions [12–15]. There are various approaches that can be used for assessment of the SIF of welded joints, peak stress, volumetric approach, and average strain energy density[16–18]. We implemented the last one in the XFEM couple with level set technique because it is faster and less resource consuming. In this study, we use a code developed by Hachi and his team that was used to solve problems involving cracks in case of static and dynamic load and crack growth prediction, homogenization in 2D and 3D, etc [19–21]. Sih developed the SED theory for the purpose of solving fracture mechanics problems [22–24]. He identified the global and local strain energy density, as well as the fracture behavior factors. Lazzarin and co-workers were the first to formalize and publish a series of very crucial papers using a synthesis based on the ASED calculated in the control volume are around the crack tip or U-notched or V-notched, See references [25–29]. many researchers utilize the strain energy density criterion investigated experimentally and numerically in case of brittle materials [30][31]. Lazzarin studied ductile materials and in- plane tensile loading mode I. However, mode II is generally negligible in applications[32]. To estimate the NSIFs directly from local stress distributions, we need very fine meshes. Note that refined meshes are not required when the goal of the finite element analysis is to estimate the average value of the local strain energy density on a control volume surrounding the V-notches or crack tip [33]. The SED approach has been successfully used by Aliha et al. in mixed mode (I/II). Aliha also studied sharp notched disc bend specimens under mixed mode (I/III) loading [34][35], Campagnolo et al. characterized different control volumes and exposed them to different combinations of static loading types in order to evaluate the resistance of different materials, [36–38]. Furthermore, In the case of cracks subjected to mixed mode (I/II) loading, the relationship between the averaged strain energy density SED technique and the peak stress method has been explored [39]. Just a few number of studies used SED approach to analyse cruciform welding joint with predefined cracks. That’s why we were interested in this study in using this approach in XFEM to evaluate SIF value and compare it with another SIF value W H. Djeloud et alii, Frattura ed Integrità Strutturale, 60 (2022) 346-362; DOI: 10.3221/IGF-ESIS.60.24 348 obtained using J-integral. The paper is organized in the following way. Section 2 describes the materials and models used in this study. Section 3 microscopic study of the state for different welding regions weld metal (WM), heat affected zone (HAZ), basse metal (BM) Section 4.1 provides the theoretical background of the ASED approach. In subsection 4.2 we show the theoretical background XFEM, and in subsection 4.3 we present the theory of Level-set technique. Section 5 describes the method adopted to get results in two phases, the first part of static load, and the second part in fatigue load. Section 6 summarizes the main conclusions of the paper. MATERIALS AND MODELS igh-strength steel is used to make cruciform welded joints made of high-strength martensitic abrasion resistant Hardox 450 steel. Chemical composition of the base metal, weld metal and mechanical properties are indicated in Tab. 1, Tab. 2, and Tab. 3 respectively. All data from Tabs. 1, was determined using a Thermo Fisher Scientific instrument. For the manufacturing of steel structures like steel bridges, offshore structures, etc. Welding is nowadays considered an efficient metal joining process. The type of fillet welded cruciform joint is commonly used in the construction of long spanned bridges with improved design and higher weld quality. However, the existence of geometrical discontinuities such as cracks and porosity in addition to metallurgical nonuniformities, which lead to crack initiations from different positions that can be difficult to detect [40]. Figure 1: Geometry and dimension of cruciform welded joints involves a cracks specimen (mm scale). C Si Mn Cr Ni Mo 0.21 0.7 1.6 0.25 0.25 0.25 Table 1: Chemical composition Hardox 450. C Si Mn 0.11 0.8 1.5 Table 2: Chemical composition of the electrode. E(GPa) t (MPa) ICK (MPa(mm)0.5) A% v 210 1660 3067 7 0.29 Table 3: Mechanical properties of Hardox 450. H J-integral contour Control volume Enriched zone Crack Weld toe 50 5 H. Djeloud et alii, Frattura ed Integrità Strutturale, 60 (2022) 346-362; DOI: 10.3221/IGF-ESIS.60.24 349 NUMERICAL MODELLING he local strain energy density SED approach is one of such modern methods, currently used in the evaluation of fatigue. The main idea of the SED method is to fully surround the crack tip or notch tip with a size control volume to calculate the strain energy for each finite element that can be achieved through the Eqn. 1.                     2 2 2 21 2 2 1 ) 2 i xx yy zz xx yy xx zz yy zz xyW v v E (1) where   0zz under plane-stress and     zz xx yyv under plane-strain [41]. The total average elastic energy included in the area of control volume according to the SED approach is determined by Eqn. 2.    1 1 i i e i i W W A (2) The next stage in the SED calculation is to determine an ideal control volume radius cR , which varies depending on the material. There are already different investigations about the control volume radius it can be calculated from the Eqn. 3 . Generally for steel  0.2  0.4cR [42]. For a sharp V-notch, the critical volume becomes a circular sector of radius cR centered at the notch tip Fig. 2a.              2 1 5 8 4 Ic c t v v K R (3) Figure 2: Control volume (a) pointed V-notch (b) crack. Where v is the Poisson's ratio IcK critical stress intensity factor mode I and t conventional ultimate tensile strength. When utilizing the SED approach, it is critical to adjust the finite elements in the control volume to ensure that there are enough finite elements to approximate the actual value[43–45] in the case of using XFEM, the crack tip enrichment and the crack enrichment compensate for the density of the mesh. The analytical evaluation for the total elastic ASED over the control volume is based on the leading order terms of William’s solution and is evaluated as shown in the following equation [33].          1) 2 1 1 2(1 I c e K W E R (4) E is the Young’s modulus which is given for different materials and 1 are the eigenvalues of the Williams' stress field solution for the N-SIF K1 for modes I. The eigenvalues 1 can be derived from the case of crack 1 =0.5 [39]. The values for this are already listed in the literature for different important 2α [40]. 1e correction factors which depend on the stress- T 𝑅 𝑅 (a) (b) 2𝛼 2𝛼 0 H. Djeloud et alii, Frattura ed Integrità Strutturale, 60 (2022) 346-362; DOI: 10.3221/IGF-ESIS.60.24 350 strain field, Poisson’s ratio and the notch opening angle 2α. Wither calculation can be made for 1e from the following empirical equations [46].         26 41 5.373.10 2 6.151.10 2 0.133e (5) XFEM coupled to the LST The displacement field is described by the following finite element approximation equation.      i iu x N x u (6) The XFEM is used to represent the discontinuities independent of the mesh. The discontinuities can be modeled by enriching all discontinuous elements using enrichment functions that satisfy the discontinuous behavior and adding additional nodal degrees of freedom, mention here Belytschko and Moes the first to formalize and publish a series of very important papers using XFEM [47–50]. In general, the approximation of the field of displacement in the XFEM takes the following form. 1 ( ) ( ) ( ) ( ) e f a i i ii S ai n n u x N x u H x a x b                  (7) fn is the set of nodes are which contains the crack tip (represented by yellow squares on Fig. 3), en is the set of nodes entirely cut by the crack (represented by blue circles in Fig. 3). The iu are the classical degrees of freedom. The ia are the degrees of freedom linked to the discontinuity and the aib are the degrees of freedom linked to the singularities. Figure 3: Enrichment strategy in XFEM. The onset and crack growth are characterized using the Paris law [51], which relates the change in SIF to crack growth rates. The stress intensity factor range can be evaluated by proposed by [52].   2 2I IIK K K (8) Once the crack is defined as a level set segment, the model of XFEM evaluate the IK and IIK through this, the increment of the crack is deduced by Eqn. 9.      6 * * 10 m K da C dN (9) H. Djeloud et alii, Frattura ed Integrità Strutturale, 60 (2022) 346-362; DOI: 10.3221/IGF-ESIS.60.24 351 Figure 4: Flowchart of our calculation code. Insertion of the geometry properties of the discontinuities structure Insert material properties Reading mesh data files Localization by Level set Construction of stiffness matrices Insertion of boundary conditions System resolution 𝐾𝑈 𝐹 Insert material properties Determinations of the displacement field Calculation of deformations and stresses field If the crack exists Calculate stresses and principal vectors Calculation of the J-integral and propagation If σ 𝜎 Crack creation End Meshing with Gmsh Insertion of material properties Assembly by vectorization technique No Yes No Yes H. Djeloud et alii, Frattura ed Integrità Strutturale, 60 (2022) 346-362; DOI: 10.3221/IGF-ESIS.60.24 352 The maximum circumferential stress criterion is used in this study to determine the direction of the crack growth, which is given by Eqn. 10. The direction of crack growth is a consequence of the mixed-mode stress intensity factors, as can be shown in Eqn. 2, and the crack will propagate in the direction where  I is a maximum [53].             2 1 2 8 4 I I II II K K arctangent K K (10) where KI and KII are respectively maximum stress intensity factor of mode I and II during cyclic loading. Level-set technique The level set method is a numerical technique used for analysing and tracking moving interfaces. The interface can be evolved by representing it with is contours of a level set function  . Without the knowledge of the exact location [54, 55] of the interface, it can be moved implicitly by updating the level set function  . At all times, the interface is represented as its zero level the Eqn. 11 represented the analytical form.                  1 2 1 2   1 p p c cx x y yx a a (11) Presentation of the developed code The modeling by XFEM coupled with LST for the evaluation of the basic parameters in fracture mechanics presented in the previous section (Numerical modeling) was programmed according to the flowchart proposed in Fig. 4. ANALYSIS OF METALLURGICAL TRANSFORMATION he Hardox 450 steel is characterised by a microstructure of quenching, contains lamellar crystals, which are likely due to γ → α shear transformation, and fragmented α phase crystals with weak misorientations, consist of the martensite with a slate-like morphology with areas of tempered martensite Fig. 5 [56,57]. Figure 5: Optical micrograph of base metal BM. According to the cooling rate and the amount of carbon, the microstructure consists of the refined and tempered low- carbonlath-type martensite with fine acicular ferrite, widmanstatten ferrite and pearlite were formed Fig. 6 [64]. T H. Djeloud et alii, Frattura ed Integrità Strutturale, 60 (2022) 346-362; DOI: 10.3221/IGF-ESIS.60.24 353 Figure 6: Optical micrograph of heat affected zone HAZ. The microstructure of the weld metal is dendritic, consists of a fine-grained non-equilibrium (acicular) ferrite, and pearlite structure with troostite precipitates, nucleating mainly at the solidification grain boundaries which results from the lower cooling rate Fig. 7 [58]. Figure 7: Optical micrograph of weld metal zone WM. RESULTS AND DISCUSSIONS fter examining a lot of studies, it is clear that few researchers used the SED approach to assess SIF in the case of the cracked component and mention here [59–62]. As stated in the paper F. Berto ‘This approach cannot be applied to notch with zero opening angle (cracks) subjected to mixed mode loading’, this is confirmed in this study, by comparing the results with one of the most effective methods J-integral. A H. Djeloud et alii, Frattura ed Integrità Strutturale, 60 (2022) 346-362; DOI: 10.3221/IGF-ESIS.60.24 354 Effect of the static loading a) Variation in crack length In this section, cruciform welded joints are made from Hardox 450 modeling by XFEM in 2D plane strain meshing by the triangular element. All the dimensions and boundary conditions are shown in Fig. 1 and Fig. 5. The welding fatigue crack initiation point is difficult to predict accurately because it usually occurs in the vicinity of the weld toe. This has been found in many studies [63,64]. The length of the cracks evolves from 2 mm and increases with step of 1 mm to reach a maximum length of 8 mm. Here, an interest is focused on two study stages: the first stage focuses on the evaluation of the SIF values for several enriched zones of the diameters 3 eh , 4 eh and 5 eh ( eh is element size) while keeping the contour of J-integral constant with a diameter equal to 4 eh , and in the second step keeping the enriched zone constant with a diameter equal to 4 eh and the variation will be carried on the diameter of J-integral contours with the following values 3 eh , 4 eh and 5 eh . The obtained results are compared with the values given by the SED approach. All results are summarized in Figs. 9-14. Figure 8: Mesh distribution and boundary condition. 2 3 4 5 6 7 8 0.0 5.0x103 1.0x104 1.5x104 2.0x104 2.5x104 3.0x104 3.5x104 S tr e s s i n te n s it y f a c to r K I [ (M P a )( m m )0 .5 ] Crack length [mm] Enriched zone (diameter=5he) Enriched zone (diameter=4he) Enriched zone (diameter=3he) Enriched element SED approach Figure 9: Crack length versus SIF calculated by SED and J-integral with different enriched zone sizes IK . H. Djeloud et alii, Frattura ed Integrità Strutturale, 60 (2022) 346-362; DOI: 10.3221/IGF-ESIS.60.24 355 2 3 4 5 6 7 8 0.0 5.0x103 1.0x104 1.5x104 2.0x104 2.5x104 3.0x104 3.5x104 S tr e s s i n te n s it y f a c to r K I [ (M P a )( m m )0 .5 ] Crack length [mm] J-integral contour (diameter=5he) J-integral contour (diameter=4he) J-integral contour (diameter=3he) Enriched element SED approach Figure 10: Crack length versus SIF calculated by SED and J-integral with different contour sizes IK . 2 3 4 5 6 7 8 -2.0x103 -1.5x103 -1.0x103 -5.0x102 0.0 S tr e s s i n te n s it y f a c to r K II [ M P a (m m )0 .5 ] Crack length [mm] Enriched zone (diameters=5he) Enriched zone (diameters=4he) Enriched zone (diameters=3he) Enriched element Figure 11: Crack length versus SIF calculated by SED and J-integral with different enriched zone sizes IIK . H. Djeloud et alii, Frattura ed Integrità Strutturale, 60 (2022) 346-362; DOI: 10.3221/IGF-ESIS.60.24 356 2 3 4 5 6 7 8 -2.0x103 -1.5x103 -1.0x103 -5.0x102 0.0 S tr e s s i n te n s it y f a c to r K II [ M P a (m m )0 .5 ] Crack length [mm] J-integral contour (diameter=5he) J-integral contour (diameter=4he) J-integral contour (diameter=3he) Enriched element Figure 12: Crack length versus SIF calculated by J-integral with different contour sizes IIK . b) Variation in crack orientation The same as the previous example by changing the location of the crack angle   10, 20, 30, 40, 50, 60, 70, 80, 90 Fig. 13 to simulate the possibility of defect to the weld root. Changing the contour of the J-Integral was dispensed because there was no effect on the results /I IIK K . 10 20 30 40 50 60 70 80 90 0 1x103 2x103 3x103 4x103 5x103 6x103 S tr e s s i n te n s it y f a c to r K I [ M P a (m m )0 .5 ] Crack orientation [deg] Enriched zone (diameter=5he) Enriched zone (diameter=4he) Enriched zone (diameter=3he) Enriched element Figure 13: Crack orientation versus SIF calculated J-integral with different enriched zone sizes IK . H. Djeloud et alii, Frattura ed Integrità Strutturale, 60 (2022) 346-362; DOI: 10.3221/IGF-ESIS.60.24 357 10 20 30 40 50 60 70 80 90 0.0 2.0x102 4.0x102 6.0x102 8.0x102 1.0x103 1.2x103 S tr e s s i n te n s it y f a c to r K II [ M P a (m m )0 .5 ] Crack orientation [deg] Enriched zone (diameter=5he) Enriched zone (diameter=4he) Enriched zone (diameter=3he) Enriched element Figure 14: Crack orientation versus SIF calculated J-integral with different enriched zone sizes IIK . Simulation of fatigue crack initiation and propagation a) Crack initiation Perform crack initiation and propagation simulation was carried out by XFEM under cyclic loading (all the dimensions, boundary conditions and mesh distribution like the previous example).  610  max Pa .Fig. 15 represents the loading changes depending on time. When the crack grows the level set defined by Eqn. (2) is used to make control volume and trace the crack tip and extract the SED value every cycle, through the Eqn. (4) is calculated can calculate SIF value represented in the Fig. 16. After applying load, the element with the maximum principal stress is checked, if the value is greater than ultimate tensile stress these elements contain crack and is oriented towards in the direction of second principal vector and its length at the beginning 2 eh . Figure 15: Change the load in terms of time. b) Crack propagation with different orientation of crack Perform predefined crack simulation with different orientations      60 , 70 , 80 , 90 see Fig. 17 was carried out by XFEM under cyclic loading (all the dimensions, boundary conditions, and mesh distribution like the previous example). The results obtained are shown in Fig. 18. 𝜎 Load Time H. Djeloud et alii, Frattura ed Integrità Strutturale, 60 (2022) 346-362; DOI: 10.3221/IGF-ESIS.60.24 358 0 10000 20000 30000 40000 50000 60000 2.0x103 4.0x103 6.0x103 8.0x103 1.0x104 1.2x104 S tr e s s i n te n s it y f a c to r [M P a (m m )0 .5 ] Number of cyclic KI J-integral KI SED approach Figure 16: Stress intensity factor IK versus number of cyclic. Figure 17: Cruciform welded joints with different crack orientation. CONCLUSIONS he fatigue performance of cruciform welded joints made Hardox 450 steel has been studied in this paper, the comparison between the SIF by SED and SIF by J-integral was carried out. The following conclusions can be proposed based on the results: - The SED approach was used to successfully calculate the mode I, SIF of crack-containing cruciform welded joints using XFEM. T Crack Weld toe 𝜃 9 0 H. Djeloud et alii, Frattura ed Integrità Strutturale, 60 (2022) 346-362; DOI: 10.3221/IGF-ESIS.60.24 359 30000 60000 90000 120000 150000 2.0x106 4.0x106 6.0x106 8.0x106 1.0x107 1.2x107 1.4x107 1.6x107 1.8x107 S tr e s s i n te n s it y f a c to r [M P a (m m )0 .5 ] Number of cyclic KI J-integral =90 KI J-integral =80 KI J-integral =70 KI J-integral =60 Figure 18: Stress intensity factor IK versus number of cyclic with different orientation. - When using XFEM, the SED approach ensures a rapid and simple assessment of the SIF in the case of the cracked component in mode I. - Has been utilized the level set formulation is used to model the crack and update the crack tip at each cyclic also the control volume is an LST in the form of a circle whose center is a crack tip. - Despite the complexity of programming the J-integration method, it is the most widely used and stable method for calculating the SIF of cracked components. - 5 eh is considered the radius of the optimal enriched zone. - The use of XFEM ensures that eW and SIF values are accurately extracted without a refine mesh. - In the cruciform welded joints the crack initiation in the weld toe is caused by stress concentration and vicinity to the heat affected area. 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