## 1 Introduction  
With the development of modern industry, the total life cycle of mechanical components, such as aircraft engines, exceeds already up to 10\({}^{9}\)-10\({}^{10}\) during the whole lifetime. Very high cycle fatigue (VHCF) is defined as fatigue after 10\({}^{7}\) cycles [1]. Most traditional fatigue test machines that work at several hundred hertz are not suitable for experimental investigation of VHCF range due to the limitation of time consumption. [PERSON] introduced ultrasonic resonance fatigue testing at 20 kHz [2]. Afterwards, the ultrasonic methodology was used for exploring fatigue lives [3] and mechanisms [4] at very high cycle domain and for investigating threshold crack propagation behavior [5], which is complicated and time consuming by conventional test machines. However, relative to the standardized low-frequency tests methods, high-frequency fatigue results are influenced by some factors, such as loading mode and frequency, specimen geometry and size, material type and heat treatment, etc. [PERSON] and [PERSON] [1] analyzed frequency sensitivity for different types of materials. They concluded that frequency effect was much more significant between 0.001 and 10 Hz than between 10 and 20,000 Hz. The insignificant influence of loading frequency was also reported in ferrous-type alloy [6] and nonferrous metals [7; 8]. [PERSON], [PERSON], et al. [9; 10; 11; 12] evaluated the size effect analytically in the VHCF regime and validated on the full-scale specimen. It is important to monitor and record test parameters such as frequency, vibration amplitude, surface temperature of specimen, etc. during the ultrasonic fatigue tests for further analysis. High-temperature superalloy Inconel 718 is widely used in aeronautic applications, due to its high resistance to corrosion, oxidation, thermal creep deformation, and high mechanical strength at elevated temperatures up to 700 \({}^{\circ}\)C [13]. Rotary speed of aeroengine is from tens of thousands to hundreds of thousands of revolutions per minute (rpm), depending on the type and dimension of the engines. Low-amplitude but high-frequency vibration induced by the disturbance could lead to accidental rupture in the aeroengine. [PERSON] et al. [14; 15; 16; 17] analyzed the high cycle fatigue (HCF) behavior of Inconel 718 in as-built and double-aging conditions at 10 Hz. [PERSON] et al. [18] compared the HCF performance of solution and aged material using a high-frequency machine at 130 Hz. However, few results have been presented on the VHCF behavior and crack mechanism of Inconel 718 using ultrasonic loading at 20 kHz to date. [PERSON]'s team [19; 20; 21] proved that fatigue strength increases with decrease in grain size. Moreover, the endurance limit under ultrasonic loading is higher than that tested under rotary bending (RB). [PERSON]'s team [22; 23] found that failure cycles for a given loading level increased with an increase in loading frequency. In other words, higher frequency results in an upper shift of the stress vs. number of cycles (**S-N**) curves. There is a transition from fatigue crack multisite initiations at the surface to single initiation in the subsurface with decrease in loading levels. [PERSON] et al. [24] presented that the improvement in fatigue life is mainly attributed to the induced compressive residual stress in the subsurface. [PERSON] [25; 26] found that Inconel 718 can still fracture after exceeding 10\({}^{8}\) cycles. Both low and high cycle fatigue cracks are initiated at large carbides whose length is greater than 50 \(\upmu\)m. [PERSON] et al. [27] indicated that competition between surface and interior crack initiation causes the inflection of **S-N** curve. Manufacturing defects in the microstructure may act as original fatigue micro-cracks. [PERSON] et al. [28; 29] found that the refinement in grain structure and higher \(\Sigma\)3 twin boundary density is associated with substantial reductions in lifetime. [PERSON] et al.