## 1 Introduction
This is the principal reason that much research has been applied to describe the peculiar behaviour of the fatigue resistance of resistance spot welding, which is employed in a wide range of industrial applications [31; 32; 33]. The analysis of static and fatigue behaviour of laser-butt welded joints made of titanium grade 2 and grade 5 was conducted by [PERSON] et al. [34]. They claimed that fatigue failures in the titanium grade 2 and grade 5 samples occur inside the base materials but not at the weld location. In terms of static characteristics, titanium grade 5 was found to be superior to titanium grade 2. There was a significant difference between grades 2 and 5, with 780,182 cycles being the maximum fatigue cycle number to failure for grade 2, and this value was 980,511 cycles for grade 5. The research conducted by [PERSON] et al. [35] further advanced the understanding of titanium alloy welds, specifically through electron beam welding. Their findings indicated that the fatigue strength of joints could exceed the yield strength of the base metal, particularly when post-weld treatments were applied. The identification of fatigue crack initiation points and propagation characteristics within the weld zones provides essential insights into the behaviour of welded titanium joints under varying stress conditions, highlighting the intricate relationship between microstructural features and fatigue performance. [PERSON] et al. [36] evaluated the fatigue life of Ti-6 Al-4V laser welded joints. The study's findings aimed to identify the damage mode of the components through thermographic tests. Thermographic analyses showed that specimens failed in various ways; for example, fatigue fractures started at the horizontal plate's radius of curvature and advanced longitudinally until they broke completely. Additionally, cracks spread laterally along the radius of curvature of the vertical plate, which causes the welded region to experience the highest strain and temperature rise. Based on their findings, [PERSON] et al. [37] concluded that all tensile and fatigue samples failed in the welding zone when testing the tensile and fatigue strength of TC17 titanium alloy laser joints. Moreover, when the laser heat input increased, the tensile strength declined, while the fatigue strength improved at first and then dropped and increased again. [PERSON] et al. [38] specifically examined the fatigue behaviour of similar linear friction welded (LFW) joints of Ti-6 Al-2 Sn-4 Zr-2 Mo-0.1 Si. Their investigation emphasized that, under monotonic loading, failure occurred at the oscillating workpiece, 10 mm away from the welding centre zone (WCZ) in the base metal (BM). Both the BM and LFW cross-weld samples showed fatigue strengths of 625 MPa and 600 MPa after 105 cycles, respectively. All of the samples' fatigue crack initiation sites were 1.5 mm from the WCZ, close to the thermo-mechanically affected zone (TMAZ), which is in line with the early strain activity and likely related to residual stresses. [PERSON] et al. [39] proposed the fatigue parameter (\(\lambda\)) for both the imperfection size and depth based on the fatigue life of TC4 titanium alloy welded joints. This paper emphasised that dislocation-slip defects and the absence of fusing, commonly known as typical crack causes, exclusively cause surface failures, while pores alone determine internal failures. The size and depth of defects have both positive and negative effects on fatigue life, leading to substantial dispersion. A fatigue parameter (\(\lambda\)) based on these factors may better reflect the distribution law of the fatigue life. The fatigue life evaluation of notched samples under size impacts was suggested by [PERSON] et al. [40] in a probabilistic framework. In this model, the multiaxial stress state at the notch tip is characterised by the Smith-Watson-Topper damage factor. By taking the location of flaws and their sensitivity to size into account, a calibrated weakest-link theory (WLT)-based probabilistic fatigue life prediction model was suggested [41].