Failure Analysis of Resistance Spot-Welded Structure Using XFEM: Lifetime Assessment

Abstract

Due to their effective and affordable joining capabilities, resistance spot-welded (RSW) structures are widely used in many industries, including the automotive, aerospace, and manufacturing sectors. Because spot-welded structures are frequently subjected to cyclic stress conditions while in service, fatigue failure is a serious concern. It is essential to comprehend and predict their fatigue behavior in order to guarantee the dependability and durability of the relevant engineering products. The analysis of fatigue failure in spot-welded structures is the main topic of this paper, along with the prediction of fatigue life (N_f) and identification of failure mechanisms. Also, the effects of parameters such as the amount of cyclic load applied, the load ratio, and size of the spot-welding on the N_f were investigated. To achieve this, the fatigue performance of spot-welded joints was simulated using the extended finite element method (XFEM). The XFEM method is particularly suited for capturing intricate crack patterns in spot-welded structures because it allows for the modeling of crack propagation without the need for remeshing. It was observed that when the cycling load was decreased by 20%, N_f increased by around 250%. On the other hand, the fatigue life of the structure, and, hence, the crack propagation rate, was significantly affected by the load ratio and diameter of the spot-welding. This paper presents the details of the novel approach to studying spot-weld fatigue characterization using XFEMs to simulate crack propagation.

1. Introduction

The resistance spot-welding process, from now named spot welding, is essential to the joining of sheet metals in industries such as the automotive, aviation, marine and energy sectors. The fatigue failure of spot-welded components was cited as one of the major concerns in the literature [1,2]. To achieve fatigue-safe designs, numerical methods and computer-aided engineering are crucial tools, while simulation reliability depends strongly on the modeling methodology.

The researchers have put effort into using a variety of modeling techniques to perform finite element analyses for the fatigue simulation of spot-welded structures. The umbrella type of spot-weld modeling technique has been used in a computer-aided optimization study by Zhang and Taylor [3]. Based on finite element simulations, Dincer et al. [4] assessed the performance of rigid-beam, elastic-beam, umbrella, and nine-point contact modeling methods in estimating the spot-weld low-cycle fatigue life. Wagere [5] presented analytical and numerical approaches for the fatigue life calculations of point-welded structures. The performance evaluation of solid and shell elements has been carried out in spot-weld fatigue life analyses by Duran [6] and Deng et. al. [7], where the comparison of FEM results with experimental data was presented in both studies. Duran conducted comparative research for several FEM modeling techniques for multi-axial high-cycle fatigue [8] and low-cycle fatigue [9] life assessment of spot-welded components. In a different study, Duran [10] used rigid and distributed coupling elements to determine the impact of load distribution modeling on the life calculation of weld regions. A recent study by Duran [11] shows the sensitivity of the solid element spot-weld modeling technique to nugget diameter. In finite element simulations of spot-welded structures, where rigid bars and solid elements are used to describe the weld nuggets, Donders et al. [12] investigated the effect of mesh size on the accuracy of the simulations. The study presented by El-Sayed et al. [13] examines the spot welds under direct shear and variable amplitude loading. El-Sayed et al.’s [13] simulations were carried out using the commercial Nastran program and the displacements were then exported to a custom fatigue life evaluation tool. Rahman et al. [14] provided durability analysis with rigid spider modeling of spot-welded sheets without providing any test data or correlation. Ertas and Sonmez [15] used elastic beam modeling to perform simulations of point-welded components while Coffin–Manson criterion was applied for the life assessment. Vadivel et al. [16] investigated spot-welded joints with HCF simulations using commercial Ansys software (https://www.ansys.com/), but their study lacked test data and correlation. The determination of residual stresses prior to and following the application of mechanical loadings of point-welded structures was the subject of a paper by Mohammari and Hemmati [17]. Draffourg et al. [18] created a spot-weld life prediction criterion based on mechanical damage and fracture mechanics. In his study [18], finite element studies were carried out to derive strain levels for fatigue predictions. Yang et al. [19] conducted point-weld life assessments using Sheppard, Rupp, and Swellam methods by exporting stress and strain data from finite element analyses using the elastic beam modeling method. Pote [20] gave details of FE-based fatigue life analysis of spot welds in which a wheel geometry was examined and compared to experimental data. Farrahi et al. [21] carried out hybrid multi-body dynamics and finite element analysis for the body-in-white simulations and spot-weld failures. Zigo et al. [22] investigated FE-based fatigue life estimation of spot-welded structures by using a PSD function obtained using Dirlik method, where the Smith–Watson–Topper parameter was employed for the damage and life calculations. Akbulut [23] examined hybrid finite element analysis—a life assessment method in which strain-based fatigue calculations were performed after FE simulations of a welded components. Nikolic et al. [24] carried out analysis of the fatigue life of spot-welded joints using linear elastic fracture mechanics, and two identical materials were shown to have a longer life than two dissimilar sheet metals with differing expansion coefficients. Kang et al. [25] analyzed weld failure for tensile–shear and coach–peel samples. Their study [25] used rigid spiders for modeling the weld nugget and FEM simulations carried out to determine the nodal forces and moments on a spot weld, which were then used for fatigue life calculations. Kang et al. [26] presented a correlation study for the results of point-weld fatigue life analyses with the test data generated by Auto/Steel partnership. Rupp et al. [27] performed analyses to assess the spot-weld life with a custom post-processor and they reported experimental data verification. Sohn et al. [28] carried out fatigue life estimation of point-welded structures using their custom tool named as ‘Expert System for Fatigue Design’. A study by Ghanbari et al. [29] examined the fatigue behavior of spot-welded joints in ferrite–martensite steel and hybrid steels. Several welding parameters were tested by authors [29], including welding time, electric current intensity, and compressive force. According to the results, electric current intensity significantly affected the fatigue life of the joints and the adhesive used in hybrid joints contributed to the increase in fatigue strength. An experimental study in [30] examined the fatigue behavior of spot-welded sheet metals. In a study by Ertas and Akbulut [30], modified tensile shear (MTS) samples were subjected to fatigue life tests with varying electrode forces. The results of the research revealed that the fatigue life and failure modes were influenced by the spot-generating schemes and electrode force effects. Through optical micrograph analysis, it was observed that fatigue failure was predominantly due to through-thickness cracking. In optical micrograph analysis, through-thickness cracking was detected to be the primary cause of fatigue failure. Shahani and Farrahi [31] carried out analysis for the fatigue behavior of friction stir spot welding (FSSW) in lap–shear samples of Al 6061-T6 alloy. The study identified shear fractures at elevated load levels and transverse crack growth at low load magnitudes as two fatigue failure modes. Using tensile–shear and cross-tension samples, Wang et al. [32] performed microstructural characterization, microhardness testing, and tensile and fatigue tests on spot-welded Q&P980 steel. Fatigue cracks were observed to initiate at the interface between the sheets, with fatigue failure modes varying between fracture along the circumference or width direction for tensile–shear test components and pullout or fracture along the width direction for cross-tension samples. Long and Khanna [33] investigated fatigue properties and failure characterization of high-strength spot-welded steels, including DP600 GI, TRIP600-bare, and HSLA340Y GI. DP600 GI exhibited the highest microhardness, followed by TRIP600-bare and HSLA340Y GI, and both DP600 GI and HSLA340Y GI experienced softening during high-load and low-cycle fatigue tests, with HSLA showing more prominent softening behavior. Fatigue properties for welded composites were evaluated by Tsang et al. [34] and Lockwood et al. [35], where experimental results were presented for nylon 6 and nylon 66 reinforced with glass fibers.

A few numerical studies were carried out to understand the underpinning deformation mechanisms of the spot-welded structures. Amiri et al. [36] introduced an integrated search system that evaluates the tensile strength and fatigue behavior of spot-welded joints using ultrasonic testing (UT) and machine-learning approaches. The authors employed resistance spot-welded (RSW) samples made of different low-carbon steel sheets and utilized artificial neural network (ANN) simulations to predict the static strength and fatigue life. The research successfully demonstrated the potential of using UT in combination with machine learning for the reliable and cost-effective assessment of spot-weld properties. Jia et al.’s study [37] concentrated on the thermodynamic entropy approach for evaluating the fatigue of structures. The proposed approach by Jia et al. [37] was applied to spot-welded joints and a high level of accuracy was reported through comparison with the traditional methods.

There are three basic modes of spot-weld failure—interfacial, partial interfacial, and pullout—which appear during static tensile–shear testing of spot-welded specimens. However, dynamic stress initiates a crack usually nearby the nugget and propagates until a sudden fracture occurs. Investigating the fatigue characteristics of spot-welded structures is crucial for ensuring the reliability and durability of welded components. Understanding the failure mechanisms and failure modes associated with fatigue in spot-welded joints provides valuable insights for developing strategies to improve fatigue resistance and extend the service life of spot-welded assemblies. However, an accurate prediction of fatigue life is a challenge, especially in terms of the position of crack initiation, its propagation in the course of cycling loading, and the failure cycle with their dependency on the different design parameters. The majority of studies in the literature evaluate stress and displacement data using finite element models, which are then used to the assessment of the fatigue life of spot-welded components using high-cycle and low-cycle fatigue theories. Literature review reveals the lack of spot-weld fatigue analysis involving the crack initiation and its propagation, which is addressed in the current numerical study. In this paper, an extended finite element (XFEM) model for the crack propagation in the spot-welded structure is developed. The XFEM method allows for the simulation of crack growth in spot-welded structures without requiring the mesh to be modified, which makes it particularly effective in capturing intricate crack formations. Fatigue life values of spot-welded joints using the developed XFEM method have been correlated with the experimental data for three different load cases. The effect of various model parameters including the amount of the applied load, the load ratio, and the nugget diameter on the fatigue life assessment of spot-welded components were evaluated through advanced XFEM analyses.

2. XFEM Modelling

A numerical analysis was conducted to evaluate the fatigue life of a spot-welded structure. The analysis employed the direct cyclic approach in Abaqus/Standard, incorporating the extended finite element method (XFEM) and linear elastic fracture mechanics (LEFM) to simulate crack propagation. The Paris Law, which relates the fracture energy release rate to crack growth, was utilized to model the initiation and progression of the crack. The maximum energy release rate (G_pl) and the lower limit (G_thresh) were defined as G_pl = 0.85 $G_{eq, c}$ and G_thresh = 0.01 $G_{eq, c}$ [38], respectively, where $G_{eq, c}$ represents the critical equivalent strain energy release rate. The calculation of $G_{eq, c}$ was based on the mode-mix criterion using the linear power law model.

The calculation is performed according to the mode-mix criterion, utilizing the linear power law model [39].

$$\frac{G_{eq}}{G_{eq, c}}={\left(\frac{G_1}{G_{1c}}\right)}^{{\alpha }_1}+{\left(\frac{G_2}{G_{2c}}\right)}^{{\alpha }_2}+{\left(\frac{G_3}{G_{1c}}\right)}^{{\alpha }_3}$$

(1)

In this model, the critical energy release rates in the opening (Mode I), first shear (Mode II), and second shear (Mode III) modes are denoted as $G_{1c}$, $G_{2c}$, and $G_{3c}$, respectively. The coefficients ${\alpha }_1$, ${\alpha }_2,$ and ${\alpha }_3$ are assumed to have a value of 1.0 in the analysis.

During the low-cycle fatigue analysis, the difference in energy release rates between the minimum and maximum loads, referred to as ΔG, serves as a measure of the fatigue crack in the structure. The growth rate of the crack per cycle was determined using the Paris Law, expressed in Equation (2), where ‘a’ represents the crack length, N is the cycle number, and $c_3$ and $c_4$ are material constants [38].

$$\frac{da}{dN}={c_3\left(∆G\right)}^{c_4}$$

(2)

In Abaqus, the crack propagation rate was defined in terms of the energy release rate rather than the stress intensity factor, resulting in slightly different definitions for $c_3$ and $c_4$ compared to the classical Paris Law constants C and m ($da/dN=C{∆K}^m$). The relationship between these constants is given as the following:

$$c_3=C{E}^{c_4},$$

(3)

$$c_4=\frac{m}{2}.$$

(4)

It is important to satisfy the inequality given in Equation (5) in the computations to accurately represent the onset and growth of the crack.

G_pl > G_max> G_thresh

(5)

After crack initiation, the procedure proceeded as follows: The software incrementally advanced the crack length (a_N) from the current cycle to a_N+ΔN by releasing one element at the interface after each cycle N. The number of cycles needed to cause failure in each interface element at the crack tip was determined as ΔN_j, where j represents the node corresponding to the crack tip. This calculation relied on the material constants c₃ and c₄, along with the determined node spacing at the interface elements near the crack tips as given in Equation (6).

Δa_Nj = a_N+ΔN − a_N

(6)

At the end of each stable loading cycle, the analysis aimed to release at least one interface element. In this regard, the element that required the least number of cycles for propagation was identified, denoted as ΔN_min = min (ΔN_j), indicating the number of cycles needed for the crack to propagate over its element length, Δa_Nmin = min (Δa_Nj). Consequently, the element with the fewest required cycles was determined to be ready for release, resulting in zero stiffness and constraint. During the subsequent cycle, a new relative fracture energy release rate was calculated for the interface elements at the crack tip due to the release of the interface element, as stated in [38].

The spot-welded structure’s three-dimensional XFEM model is depicted in Figure 1. XFEM was defined only in the overlap regions of the adherends as the potential sites for the onset and propagation of the cracks. Elsewhere, the standard FE calculations were performed. To confirm its accuracy, the model’s performance was compared to the experimental findings presented in [19]. The dimensions of the XFEM model correspond to those of the samples utilized in the referenced experimental study. The spot welding was modelled using the tie constraint module available in Abaqus where the interfaces of both adherends on the welding side were defined as master and slave surfaces.

Regarding the boundary conditions, the model underwent loading from one end while being fixed at the other end.

Table 1. Details of the cases (F_min, F_max, R, and d) studied in the numerical analysis.

	Nugget Diameter, d (mm)	Fmin (kN)	Fmax (kN)	R
Case 1	4.5	2000	4000	0.5
Case 2	4.5	1800	3600	0.5
Case 3	4.5	1600	3200	0.5
Case 4	4.5	0	4000	0.0
Case 5	4.5	1000	4000	0.25
Case 6	4.5	3000	4000	0.75
Case 7	3.5	2000	4000	0.5
Case 8	4.0	2000	4000	0.5
Case 9	5.0	2000	4000	0.5
Case 10	5.5	2000	4000	0.5

provides details of the minimum (F_min) and maximum (F_max) forces applied, along with the load ratio (R), during the fatigue loading as well as the nugget diameter. Ten cases were studied with R ranging from 0.0 to 0.75, d ranging from 3.5 mm to 5.5 mm, F_min ranging from 0 N to 3000 N, and F_max ranging from 3200 N to 4000 N. A direct cyclic analysis with 25 Fourier terms was conducted. Prior to the cyclic loading, a static step was performed where F_max was applied to induce crack formation at the stress concentration site. The mesh discretization employed eight-noded plane stress quadrilateral elements (C3D8R) with varying element sizes ranging from 0.00025 mm to 0.001 mm. A finer mesh was utilized in the welding region to ensure accuracy. Convergence of the mesh was achieved by assessing the maximum von Mises stress in the element adjacent to the welding region under maximum static loading, with a change of less than 5.0% observed when compared to a mesh with element sizes twice as small [39,40]. The material constants required for the numerical model can be found in

Table 2. The material parameters for austenitic steel used in the simulations.

Source	Property	Austenitic Steel
[11]	E (GPa)	193
	υ	0.3
	σmax (MPa)	608
	Sy (MPa)	400
[41]	G1c (N/m)	6500
	G2c (N/m)	6500
	G3c (N/m)	6500
[This paper]	c3	7.5 × 10−8
	c4	1.75

.

3. Results and Discussion

3.1. Influence of the Amount of Load F

In the developed XFEM model, firstly the static loading with a value of F_max was applied to initiate a crack in the spot-welded structure. As seen in Figure 2a, a crack with a length of 0.0005 mm appeared on the thinner adherend of the structure next to the spot-welding point, in compliance with the observations in the literature [8,11]. When cycling loading is applied this crack expands. Figure 2b,c demonstrates how the crack expands along the width of the spot-welded structure. The validation in the numerical model was performed using the experimental results from [19]. Figure 3a compares the obtained values for Case 1, where the cyclic loading with minimum and maximum values of 2000 N and 4000 N were applied to the spot-welded structure. It was observed that N_f equaled to 179,251 and 212,620 cycles from the experiment and XFEM simulations. We believe that they match each other reasonably well. In the experimental study in [19], the authors did not mention how N_f was defined. In the simulations, the N_f was taken to be the number of cycles when the crack length reached to the size of the nugget diameter, 0.45 mm, as can be seen in Figure 3a. This matches well with the experimentally observed crack pattern in [42].

To see the effect of F on the fatigue performance of spot-welded structure, two extra loading cases, Cases 2 and 3 (see Table 1), were also considered, where the F_max was changed into 3600 N and 3200 N, respectively, by keeping the load ratio R constant. From the simulations, N_f was obtained to be 287,120 and 582,180 cycles, respectively, where they matched well with those obtained experimentally with 290,223 and 485,628 cycles, respectively (see Figure 4). The obtained crack patterns are presented in Figure 3b,c, where they are similar to each other and also not very different from the one with F_max = 4000 N. When the F_max was reduced by 10% and 20% (from 4000 N into 3600 N and 3200 N), the N_f increased by 73% and 244%. Overall, it was observed that the applied cycling load influenced the fatigue life significantly.

The crack growth rate is of paramount importance in assessing the structural integrity and durability of materials and components. It provides critical information about the rate at which cracks propagate under cyclic loading conditions, which directly influences the remaining fatigue life and the potential for catastrophic failure.

Table 3. Numerically obtained crack propagation rates (da/dN) for different cases studied.

	Nugget Diameter (mm) (Fmin/Fmax (kN))	da/dN (mm/Cycle)
Case 1	4.5 (2000/4000)	1.88 × 10−8
Case 2	4.5 (1800/3600)	1.39 × 10−8
Case 3	4.5 (1600/3200)	0.69 × 10−8
Case 4	4.5 (0/4000)	17.2 × 10−8
Case 5	4.5 (1000/4000)	9.13 × 10−8
Case 6	4.5 (3000/4000)	0.66 × 10−8
Case 7	3.5 (2000/4000)	6.33 × 10−8
Case 8	4.0 (2000/4000)	4.59 × 10−8
Case 9	5.0 (2000/4000)	0.621 × 10−8
Case 10	5.5 (2000/4000)	-

compares the obtained average crack expansion rates (da/dN), i.e., the change in crack length for the number of loading cycles (up to N_f), for the cases investigated. They were 1.88 × 10⁻⁸, 1.39 × 10⁻⁸, and 0.69 × 10⁻⁸ mm/cycle for Cases 1–3, respectively. It was observed that when F_max was decreased by 20% (4000 N to 3200 N), da/dN decreased by approx. 63%.

3.2. Influence of the Load Ratio R

The load ratio is one of the important parameters that has a significant impact on the fatigue performance of a structure. In this section, its effect on the fatigue characteristics of the spot-welded structure was analyzed, where the load ratio R was varied between 0.0 and 0.75. Figure 5 shows the obtained N_f values. It was observed that they were 23,185; 43,778; 212,620; and 600,580 cycles when R equaled 0.0, 0.25, 0.50, and 0.75, respectively. It was noticed that when the R got smaller, the fatigue life decreased; the change in N_f was very dramatic with values of 85% and 50.0% when R decreased from 0.75 to 0.5 and later to 0.25, respectively. No significant change was observed when R reduced to 0.0 from 0.25. The smaller value of R reduces the fatigue life of a structure sensibly. This was due to increased stress levels during the cyclic loading, leading to accelerated fatigue crack growth and overall durability. This can be seen from the comparison of da/dN values presented in Table 3. Its value for R = 0.0 increased by more than 25 times when compared to that of R = 0.75.

3.3. Influence of Nugget Size

Lastly, the influence of diameter of the spot weld on the fatigue life of the spot-welded structure was analyzed. For this purpose, this size was changed from 3.5 mm to 5.5 mm with an increment of 0.5 mm. Figure 6 presents the obtained N_f values, where they are 47,369; 76,128; 212,620; 724,440; and 10⁶ cycles, respectively. For the last one, the service life of the welded structure was around 1.73 × 10⁶ cycles, but in this study, a fatigue life greater than 10⁶ cycles was considered as the run-out. It should be mentioned that N_f represents the number of cycles when the crack length reaches to the nugget diameter. For instance, for d = 3.5 mm, the crack length reached to 3.5 mm at 47,369 cycles. When the nugget diameter was increased from 3.5 mm to 4.0 mm, the change in N_f was not so significant (approx. 60% increase). However, a dramatic increase in N_f was observed for larger d values. For instance, the N_f increased by 2.8 and 3.40 times, when d increased from 4.0 mm to 4.5 mm and from 4.5 mm to 5.0 mm, respectively. Here, we concluded that the size of the spot weld affected the fatigue performance of the structure but at different levels for different size ranges of the spot weld. When the average crack expansion rates were compared for different d values (see Table 3), the crack propagated at a rate of 6.33 × 10⁻⁸ mm/cycle for d = 3.5 mm, more than 10 times faster than the structure with d = 5.0 mm. For smaller nugget size, the crack expanded very fast and, hence, the fatigue life of the welded structure was much shorter. This is in line with the finding in [8], where the smaller nugget diameter led to shorter fatigue life on the basis of notch effect, where higher strains and stresses evolved around the spot-welding point.

4. Conclusions

This numerical study analyzed the fatigue failure of the spot-welded structure for its different influential parameters such as the amount of cyclic load, the load ratio, and size of the spot welding. The XFEM modelling approach was used to simulate the crack propagation in the welded-structure under cyclic tensile loading. The model was validated using the experimental results from the literature. The following conclusions were drawn:

• The crack in the structure was nucleated around the spot-welding point and later it expanded along the width of the spot-welded structure when the cycling loading was continually applied.
• The applied cycling load influenced the fatigue life substantially. When the F_max was increased by 25%, the N_f decreased more than 2.7 times.
• For smaller load ratios, the fatigue life of a structure decreased notably, where, in parallel, the crack propagation rate increased. For instance, this increase was more than 25 times when R equaled 0.0 compared to that of 0.75 for the simulated spot-welded austenitic steel plates.
• The fatigue life of the welded structure was significantly reduced as the crack propagated rapidly when a smaller nugget size was used.
• XFEM analysis provide valuable insights into the fatigue life estimation and failure mechanisms of spot-welded structures, aiding in the design and optimization of such components for enhanced durability and reliability.
• The correlated XFEM methodology introduced in this study is new to literature and can be applied for any kind of material at the design phase of spot-welded structures.
• The resistance spot-welding parameters affected the heat-affected zone and, thus, induced residual stresses, which may impact fatigue life estimations in the nugget region. The assessment of fatigue life including the simulations of heat-affected zone and resistance spot-weld parameters should be considered in a future work.