## 1 Introduction  
To describe the relationship between the load amplitude and fatigue life expressed in terms of S-N curves, generally, a large number of specimens and fatigue tests are required. For stress-controlled fatigue tests, a power law relationship, according to [PERSON] [1], is frequently used to describe S-N curves in the high-cycle fatigue (HCF) regime mathematically, according to Equation (1). The equation consists of the stress amplitude \(\sigma_{a}\), the number of cycles to failure \(N_{f}\), the fatigue strength coefficient \(\sigma^{\prime}_{f}\), and the fatigue strength exponent \(b\). \[\sigma_{a}=\sigma^{\prime}_{f}\ \left(2\ N_{f}\right)^{b} \tag{1}\]Usually, fatigue tests in the low-cycle fatigue (LCF) regime are performed while strain-controlled and can be mathematically described by the relationship according to [PERSON] and Manson [2; 3], which is given by Equation (2), using the plastic strain amplitude \(\epsilon_{a,p}\), the fatigue ductility coefficient \(e_{f}^{\prime}\), and the fatigue ductility exponent \(c\). \[\epsilon_{a,p}=e_{f}^{\prime}\ \Big{(}2\ N_{f}\Big{)}^{c} \tag{2}\]  
The fatigue behaviors in the LCF, HCF, and VHCF (very-high-cycle fatigue) regimes are characterized by significantly differing damage mechanisms [4]. This is especially so for the failure mechanisms in the VHCF regime, which particularly depend on the specific material properties and heat treatment conditions. They are, therefore, crucial as they influence the way cracks initiate and propagate, which is the reason why specific failure mechanisms must be differentiated. [PERSON] distinguishes between type I and II materials. While type I materials tend to be ductile metallic materials without internal defects, type II materials have crack-like defects such as non-metallic inclusions or pores [5; 6]. In the HCF and VHCF regimes, type I materials like SAE 1045 (C45E, 1.1191) steel tend to show a microstructural degradation in terms of persistent slip bands (PSB), which reach through grains to the material surface. PSBs are formed when local acting stresses are above a critical threshold value [7; 8]. Especially in the VHCF regime, where very high-cycle numbers occur, the stress amplitudes are often below this threshold. Although PSBs typically play a significant role in damage mechanisms, the continuous cyclic loading leads to damage accumulation even at low-stress amplitudes. This accumulation can ultimately result in premature failure of the material. Investigations referring to the influence of frequency on fatigue behavior have already been carried out by [PERSON] in the early 1920s, which showed a significant increase in fatigue life at higher frequencies. Within this research, it was assumed that loads that are sufficient to cause fatigue and fracture for lower frequencies do not have enough time to activate this effect at high test frequencies, and therefore, higher load amplitudes are needed [9]. Studies by [PERSON] and [PERSON] regarding the frequency effect on cyclic plastic deformation, damage localization, dislocation movement, and fatigue crack growth in pure metals with body-centered cubic (bcc) and face-centered cubic (fcc) structures were performed in the early 1980s [10]. [PERSON] [11] provided a more detailed review, demonstrating that a frequency effect is differently pronounced depending on the lattice structure of the materials. Pure fcc metals tend to be insensitive regarding an increase in test frequency since the critical shear stress is low and largely unaffected by the strain rate; thus, the slip systems are still active even under high test frequencies. As a result, increased test frequencies have minimal impact on their behavior. In contrast, for pure bcc metals, the high activation energy required for dislocation movement and a greater critical shear stress cause slip systems to become inactive at high frequencies, thus making the frequency effect more pronounced [12; 13].