## 1 Introduction  
With recent changes in climate, railroad vehicles are now operating in a wider range of conditions than in the past, including deep cold and heatwaves. Under such circumstances, the dynamic characteristics of the vehicle may be changed, and unexpected failures may occur [1,2]. For these reasons, interest in the mechanical properties of components at low and high temperatures is increasing. The axle is one of the key components affecting the structural safety of rolling stock. The failure of an axle can lead to the derailment of a vehicle, fatal accidents, property loss, etc. When flying ice cubes by train gust or gravel on the track impact the axle, a defect may occur on the surface of the axle [3]. Cracks may grow from the defect and lead to the axle breaking. So, in addition to fatigue strength, resistance to shock is one of the required standards of axles [4,5]. The fatigue strength and the impact resistance of a material depend on temperature, but the Korean standard [5] does not specify the temperature during tests. In general, railway vehicles are designed to have long lifetimes, from 20 to more than 40 years. Their components need to have long durability. During the lifetime of a railway vehicle, which is typically \(3\times 10^{6}\) km, an axle installed on the vehicle experiences very high cycles of fatigue loading (1.0-1.2 giga-cycles). However, the giga-cycle fatigue properties of axles have not been studied much because of the restrictions of testing time and costs. In the past, conducting fatigue tests in the giga-cycle range was not easy. Even when a fatigue test is performed at 100 Hz without rest, 115 days are needed to apply \(10^{9}\) loading cycles. Therefore, components under fatigue loading have been designed by using an S-N (stress-number of cycles to failure) curve, with an endurance limit of \(2\times 10^{6}\) cycles. However, according to the results of very-high-cycle fatigue tests, the fatigue strength of some materials decreases as the loading cycles increase above \(2\times 10^{6}\) cycles. In other words, fatigue limits were not observed for those materials. Under these circumstances, studies on fatigue characteristics at very high cycles are required. High-cycle fatigue testing technologies were widespread in the early 2000s [6; 7; 8] with the advent of new technologies such as ultrasonic resonance fatigue testing. Ultrasonic fatigue specimens have a cylindrical or rectangular cross section. Specim dimensions can be calculated by solving the elastic wave equations [6; 7]. In general, a 20 kHz ultrasonic transducer is widely used. Most tests are performed under tension-compression loading. [PERSON] et al. [9], [PERSON] [10], and [PERSON] et al. [11] presented test machines for torsional ultrasonic fatigue tests. Recently, fatigue specimens have been designed and studied under simultaneous tensile and shear stress [12; 13]. [PERSON] et al. [14] compared the fatigue behavior of two spring wires for automobile use (CrV and CrSi) under a giga-cycle regime. The CrV steel showed constant fatigue strength over \(10^{6}\) cycles, which means that this material has a fatigue limit. However, the CrSi had decreasing fatigue strength with loading cycles. In the case of fatigue life beyond \(10^{7}\) cycles, fatigue cracks occurred from the defects in the interior of the specimens. According to their calculated results, the crack initiation life was more than 99% of the total fatigue life. In their other research [15] on ultra-high-strength steels, they found that high-cycle fatigue cracks initiated from subsurface inclusions of sulfides. [PERSON] et al. [16] presented similar results with high-strength steels SUJ2 (high-carbon-chromium bearing steel) and SNCM439 (nickel chromium molybdenum steel). The fatigue strengths decreased over \(10^{7}\) cycles. In the short life regime, fatigue cracks started from the surface.