## 1 Introduction  
To realize the fatigue limit design of mechanical structures, it is necessary to clearly understand the fatigue limit of materials through conducting fatigue tests. However, to accurately evaluate the fatigue limit, a large number of fatigue specimens and test cycles are required, and the process is costly and time-consuming. In general, the fatigue properties of materials are strongly correlated with tensile strength and hardness [1]. Especially in the context of steel materials, it is empirically known that there is a strong correlation between fatigue limit and tensile strength or hardness [2, 3, 4]. [PERSON] et al. [5] considered the effects of micro-surface defects, micro-surface crack shape and size, and Vickers hardness on the rotating bending and tensile-compressive fatigue limit and proposed an estimation formula by the square root \(\sqrt{area}\) of the projected area of the surface defect in the direction of maximum principal stress. Furthermore, it is known that fatigue limits are related to grain size in low-strength steels [1, 6]. These studies on estimating fatigue limits were primarily conducted utilizing the ordinary fatigue test procedure, with servo-hydraulic or electric motors serving as the driving force, up to a loading frequency of approximately 100 Hz. Since the late 1990s, the fatigue fracture of metallic materials in the long-life region where the number of stress cycles exceeds \(10^{7}\) times has been reported, and the ultra-high cycle fatigue properties of metallic materials have attracted attention [7]. However,it takes a long time to conduct fatigue tests up to the ultra-high cycle region at normal loading frequency. From the viewpoint of reducing the test time, ultrasonic fatigue tests using ultrasonic technology have been conducted, and many studies have been conducted in recent years [8; 9; 10; 11; 12; 13; 14; 15; 16]. Nevertheless, it has been pointed out that the fatigue properties of some metallic materials are affected by the loading frequency [9; 14]. In the 2000s, [PERSON] et al. [11] reported that the fatigue limit of body-centered cubic pure tantalum in ultrasonic fatigue tests was much higher than that in conventional servo-hydraulic tests. [PERSON] et al. [10; 17], [PERSON] [16], and [PERSON] et al. [18] conducted fatigue tests at various loading rates to clarify the effect of the loading rate on the fatigue properties. As a result, it was found that the fatigue properties of low-carbon steel are significantly affected by the loading frequency. Against this backdrop, [PERSON] [19] investigated the relationship between fatigue limit and loading frequency from a reaction kinetics perspective. They proposed an analytical model based on the strain dependence of yield strength, which can accurately explain the trend of experimental results with different loading frequencies. [PERSON] et al. [20] proposed a constitutive equation for mild steel that accounts for the effects of strain rate and temperature. [PERSON] et al. [21] developed a theoretical interpretation of the relation between the strain rate and yield strength in low-to-medium-strength steels. [PERSON] et al. [22] proposed a physical model of the loading frequency dependence of the fatigue limit of low-carbon steel based on the dislocation motion under cyclic loading. They reported that the fatigue limit of low- and medium-carbon steels, dependent on loading frequency, can be estimated by grouping data by axial-loading fatigue levels based on carbon content. In this study, rotating bending fatigue tests were conducted on low- and medium-carbon steels, and the estimation of the rotating bending fatigue limit in terms of the rate process theory proposed by [PERSON] et al. was attempted using a model [23]. As described above, the loading-frequency dependence of the fatigue limit of carbon steels with ferrite microstructures has been studied mainly for the axial-loading fatigue limit, which can be tested up to a high-frequency band.