1 Introduction

Aluminum and titanium alloys as well as nickel-based superalloys are commonly used materials in the aeronautical and aerospace industry. In this industry sector, high-strength aluminum alloys, such as AA2024-T3 are widely used in lightweight structural applications in the aircraft’s fuselage or lower wing section in the form of sheets or plates (Ref 1). Titanium alloys and nickel-based superalloys, such as Ti-6Al-4V and Inconel 718 are superior engineering materials and find great application in the aerospace field. Ti-6Al-4V with its low density, high strength, and good corrosion resistance is used in different sections of aircraft including fuselage, landing gear, static and rotating engine components as well as window frames (Ref 2, 3). Inconel 718 is mostly used in high-temperature sections of aircraft engines due to the combination of high strength, high corrosion resistance, and good high-temperature behavior (Ref 4).

The aerospace industry is one of the fields where fatigue is design critical, and fatigue damage may also occur after up to 1010 load cycles, referred to as very high cycle fatigue (VHCF) (Ref 5). Although, numerous experiments were performed on the fatigue behavior of aluminum and titanium as well as nickel-based superalloys, mostly focusing on low (LCF) or high cycle fatigue (HCF) with 104 or 107 load cycles to failure, respectively, with 10-100 Hz of testing frequency (Ref 6,7,8). Few research articles describe the fatigue behavior of the materials over 107 or 108 load cycles for a very high cycle fatigue regime (Ref 4, 5, 8, 9). Where VHCF is usually performed with a higher value of testing frequency 1-20 kHz reaching 108 or 109 load cycles to failure to decrease the testing time (Ref 10,11,12). However, the fatigue damage may depend on the testing frequency within HCF and VHCF regimes, which also needs to be considered during the design process of aeronautical structures (Ref 4, 11,12,13,14).

Engler-Pinto et al. (Ref 12) studied the frequency and environment effect on fatigue behavior of different cast aluminum alloys tested up to 107 and 109 load cycles to failure at 75 Hz and 20 kHz testing frequencies, respectively, both in air and water. It was observed that some aluminum alloys are very sensitive to the testing frequencies as well as to the environment. Mayer et al. in (Ref 15) investigated AA2024-T351 aluminum alloy under HCF as well as VHCF with 8-10 Hz and 20 kHz frequencies, respectively. It was concluded that the aluminum alloy did not show any frequency effect or dependency and the absence of the fatigue limit up to 1010 load cycles was confirmed on the AA2024-T351. In addition, the fracture behavior of aluminum alloy did not show any dependency on the testing frequency (Ref 15, 16).

Morrissey et al. (Ref 14) described the frequency effects (70, 400, and 1800 Hz) on Ti-6Al-4V for the HCF regime. It was found that the fatigue strength of the titanium alloy increases with an increase in the testing frequency. Yoshinaka et al. (Ref 17) paid attention to the fatigue behavior of Ti-6Al-4V alloy subjected to different surrounding environments such as air and close to vacuum conditions: 100, 10−2, and 10−4 Pa. All tests were performed with a stress ratio of 0.1 and a testing frequency of 60 Hz. From the results, it was concluded that there was no noticeable difference compared to fatigue behavior tested in air. However, with a decrease in pressure, for 10−2 and 10−4 Pa, the decrease in the crack propagation rate, as well as the increase in crack initiation time, were observed. Three different failure mechanisms were identified for the titanium alloy with a surface, near the surface as well as interior crack initiations (Ref 5, 17).

For the Inconel 718, Yang et al. (Ref 18) observed the frequency influence and compared the material behavior at 52.5 Hz and 20 kHz of testing frequency. It was noticed that crack initiation behavior for the two testing frequencies was different: for low-frequency experiments, the cracks in the material were initiated from multiple positions, however, for the high-frequency test a single crack initiation position was found. Some of the literature shows multiple surface crack initiations without any frequency dependency (Ref 19, 20). Although, a limited amount of publications exists on HCF and VHCF of aluminum alloys, titanium alloys, and nickel-based superalloys most studies were performed either at a low testing frequency bellow 90 Hz for HCF or at a very high frequency, which is in most cases equal to 20 kHz for VHCF.

In the present study, the impact of 20, 90, and 1000 Hz testing frequencies is investigated on AA2024-T3, Ti-6Al-4V, and Inconel 718 materials in HCF regime at ambient temperature and normal atmosphere. The quantitative analysis of frequency sensitivity study is performed and compared up to 107 cycles to failure for all the materials. In addition, VHCF behavior is evaluated for 1000 Hz of testing frequency. Threshold stress intensity factor (SIF) ranges are determined as well as fatigue fracture surfaces are studied for 20, 90, and 1000 Hz testing frequencies and with respect to different materials.

2 Materials and Experimental Methods

2.1 Microstructure and Fracture Surface Analysis

AA2024 aluminum alloy was provided in the form of extruded cylindrical bars in the T3 temper condition. The bars had a nominal diameter of 12 mm and had typical characteristic mechanical properties, including a yield strength of 360 MPa, a tensile strength of 475 MPa, and a fracture strain value of 0.2. Inconel 718 was obtained in the form of hot-rolled bars with a diameter of 12.4 mm. The material was subjected to an annealing process at a temperature of 980°C for a duration of one hour, followed by precipitation hardening. The resulting material exhibited characteristic mechanical properties of a yield strength of 485 MPa, ultimate tensile strength of 950 MPa, and fracture strain value of 0.3. The Ti-6Al-4V alloy was supplied as cylindrical bars of 12 mm in diameter, forged at 950°C for one hour, annealed at 710°C for two hours, and then cooled down in the air. The characteristic mechanical properties are as follows: 885, 1050, and 0.17 MPa for yield strength, ultimate tensile strength, and fracture strain, respectively. The microstructure depicted in Fig. 1, was evaluated from the lateral surface, which is parallel to the cross-section area plane enabling a determination of the grain size of each material. Kroll's etchant was used to etch the Ti-6Al-4V and AA2024 specimens for a duration of 10 to 15 seconds, while Inconel 718 was subjected to an etchant solution containing 15% HCl, 10% HNO3, and 10% acetic acid for 60 seconds. The microstructural investigations were carried out within the cross-section area depicted with respect to the specimen diameter (D1) in Fig. 2. The chemical composition of all three materials is similar to the reported ones for AA2024-T3, Ti-6Al-4V, and Inconel 718 alloys (Table 1) (Ref 21,22,23).

Fig. 1
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Microstructure of AA2024-T3 (A), Ti-6Al-4V (B) and Inconel 718 (C) alloys

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Fig. 2
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Round specimen geometry for tensile and fatigue tests

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Table 1 Chemical composition of AA2024, Ti-6Al-4V and Inconel 718 alloys [wt. %]
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The fatigue fracture surfaces of the respective materials tested at different frequencies were investigated microfractographically using light microscope (Keyence Digital Microscope VHX-7000) in combination with a scanning electron microscope (SEM, Jeol JSM-6490LV) in order to identify crack initiation sites and typical fracture characteristics. The following SEM parameters were used: 25 keV high voltage, 71 µA emission current, 0.013 nA beam current, 30 µm aperture and 20-22 mm working distance.

2.2 Mechanical Testing and Data Analysis

Mechanical properties of the alloys such as tensile yield strength, ultimate tensile strength and fracture strain were determined. At least two samples were taken to carry out quasi static uniaxial tensile experiments for AA2024-T3, Ti-6Al-4V, and Inconel 718. Round test specimens were designed and manufactured according to ASTM standard E466 (Fig. 2) with the gauge diameter (D1) equal to 5, 3.5, and 5 mm for AA2024, Ti-6Al-4V, and Inconel 718, respectively. The radius of curvature (R1) was equal to 40 mm for all the materials. The surface of the specimens was polished to Rz 3.4 µm. The tensile experiments were performed using an electronic universal testing machine (Zwick Roell Group, Germany). The crosshead speed was equal to 0.5 mm/s for all materials. The strain was measured using a video extensometer videoXtens 1-270 (Zwick Roell, Germany).

Specimen geometry for fatigue analysis was determined based on ASTM standard E466 (Fig. 2). AA2024-T3 specimens were cut out and manufactured with 4.5 and 5 mm in diameter within the gauge for 20 and 90 Hz frequencies experiments respectively, as well as 4 mm for 1000 Hz testing frequency. Ti-6Al-4V specimens were produced with diameters of 6, 5, and 4 mm for testing frequencies of 20, 90, and 1000 Hz, respectively. However, despite the high yield strength of the material, the maximum cyclic stress level caused the specimens to break within the clamping region. In response, the diameter of the specimens was reduced further to 3.5 mm for all the testing frequencies in order to reduce stress concentration on the clamping region. Inconel 718 specimens were designed to be 6 and 5 mm in the gauge diameter for 20 and 90 Hz, respectively, as well as 4 mm for 1000 Hz. The current investigation has neglected the influence of specimen size, based on previous research findings indicating that the effect can be insignificant considering the specimen size differences from the current study or falls within the scope of error for the alloys (Ref 24, 25). The gauge length L0 was equal to 15 mm for experiments at 20 and 90 Hz of testing frequency as well as 10 mm for 1000 Hz for all the specimens. The gauge length reduction was performed to avoid the high-frequency testing machine overload. In total, three sets of 9 to 15 specimens for each frequency and material were manufactured.

Fatigue experiments were performed in a force control mode with a sine-wave form. The stress ratio for all the frequencies was equal to 0.1. 20 Hz testing frequency tests were conducted using a Schenck servo hydraulic testing machine (PC160N) with a 25 kN load cell. The limits for the 20 Hz frequency experiments were 107 load cycles as well as force drop by 80 %. 90 and 1000 Hz testing frequencies were utilized using resonant testing machines Testronic and Gigaforte (RUMUL®, Russenberger Prüfmaschinen AG, Neuhausen am Rheinfall, Switzerland) with 25 kN and 50 kN load cells, respectively. For the 90 and 1000 Hz testing frequencies the limits were set up to be 107 and 109 load cycles or delta frequency drop by 2 % for HCF and VHCF, respectively. A cooling system with compressed air was utilized for 1000 Hz frequency fatigue testing for Inconel 718 as well as for Ti-6Al-4V due to high strain rates and rapid temperature increase during the test. The fatigue testing was conducted under controlled environmental conditions using a K-type thermocouple, which was integrated with RUMUL® software. The temperature was maintained at an average of 24 ± 4 °C throughout the entire testing period.

Fatigue results were represented using stress to number of cycles to failure (S-N) curves with respect to testing frequency. The region on S-N curves from 104 to 2 × 106 load cycles to failure was chosen for Basquin’s equation:

$${\mathrm{ln}}{N}_{f}={\mathrm{ln}}C-k{\sigma }_{\mathrm{max}}$$
(1)

where C and k are Basquin’s constants. Nf is the number of cycles to failure and σmax is the maximum stress level applied. The result is being plotted as a mean S-N curve with survival probability of 50 %. Basquin’s endurance limit was determined at 2 × 106 load cycles to failure evaluating from 4 to 10 tested samples for all the materials as well as testing frequencies.

Weibull analysis (also called lifetime distribution) is a method to evaluate the reliability characteristics of the material under different loading conditions (Ref 26,27,28,29). In order to observe the effect of different testing frequencies on the material, six specimens were fabricated for the analysis of each testing frequency and material. To stay within the HCF regime the maximum testing stress level was determined to be 300 for AA2024-T3, 860 for Ti-6Al-4V, and 650 MPa for Inconel 718. To quantify the effect of the testing frequency, scale and shape parameters were determined using the following equation:

$${\mathrm{ln}}\left({\mathrm{ln}}\frac{1}{1-F}\right)=m{\mathrm{ln}}{N}_{f}-m{\mathrm{ln}}{\sigma }_{0}$$
(2)

where F is the probability of failure, m is the Weibull modulus or shape parameter and σ0 is the scale parameter.

The fatigue crack initiation model is described by the following governing equation to quantify the effect of testing frequency and determine the values of the threshold SIF range, ΔKth [MPa m1/2] (Ref 30, 31):

$${(\sigma }_{\mathrm{max}}-{\sigma }_{f})\cdot {N}_{i}^{\alpha }={\left(\frac{\Delta {K}_{th}^{2}\cdot {\mu }^{2}}{c\cdot d\cdot \vartheta \cdot {\sigma }_{y}^{2}}\right)}^\frac{1}{2}$$
(3)

where Ni is the number of load cycles to crack initiation, σf is the fatigue limit, σy is the yield strength of the material, µ is the shear modulus, d is the grain size, ϑ is Poisson’s ratio and α as well as c are the fatigue strength constants.

3 Results and Discussions

3.1 Fatigue Testing

The investigation of the frequency influence is of high importance due to strain hardening effect and time-dependent chemical processes such as oxidation, which affects the fatigue behavior of sensitive aluminum alloys, as discussed and reported in (Ref 11, 12). A similar conclusion was made and reported by Mayer et al. in (Ref 15) performing fatigue tests with 8-10 Hz and 20 kHz. Based on fatigue test results of the current study for AA2024-T3 represented as a function of testing frequencies for approximately 20, 90, and for 1000 Hz (Fig. 3A), no noticeable frequency effect on the behavior of AA2024-T3 was observed. The HCF strength (at 107 load cycles to failure) of AA2024 was determined to be equal to 230, 225 MPa for 20, and 90 Hz of testing frequencies. Whereas, in the case of VHCF strength value (at 109 load cycles to failure) it is equal to 155 MPa for 1000 Hz of testing frequency. It can also be seen in Fig. 3(A) that the AA2024-T3 alloy does not show any endurance limit of up to 109 load cycles. Predicted Basquin’s endurance limits for the AA2024 alloy with a probability of failure of 50 % (Fig. 3B) are equal to 206, 221, and 205 MPa for 20, 90, and 1000 Hz testing frequencies. Due to the absence of an endurance limit in AA2024-T3, the predicted values of the Basquin's endurance limit as well as experimental values are not matching. In general, the fatigue behavior of the aluminum alloys well correlates with the previously published results (Ref 7, 10,11,12).

Fig. 3
figure 3

S-N curves (A, C, E) and mean S-N curve within the Basquin’s region with the survival probability of 50 % (B, D, F) with 20 Hz, 90 Hz as well as 1000 Hz testing frequencies of AA2024, Ti-6Al-4V, and Inconel 718. Three types of identified crack initiation behavior of the titanium alloy (C): internal (◊), near the surface (○), and surface crack sites (rest)

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Fatigue behavior of Ti-6Al-4V alloy exhibited relatively significant frequency sensitivity. With an increase in the testing frequency the fatigue strength of the material increases, as can be seen in Fig. 3(C) and (D). The mean S-N curve within the Basquin’s region with a survival probability of 50 % is depicted in Fig. 3(D) and it shows an increase in the fatigue strength from 20 over 90 to 1000 Hz. The HCF strengths for 20 and 90 Hz of testing frequencies are also different; they are equal to 820 MPa as well as 845 MPa, respectively. The VHCF strength is equal to 700 MPa at 109 load cycles to failure. Predicted values of the Basquin’s endurance limit at 2 × 106 load cycles to failure very well correlate with the HCF strength for the Ti-6Al-4V alloy and are equal to 830, 847, and 855 MPa for 20, 90 and 1000 Hz of testing frequencies, respectively. Several factors may contribute to the frequency dependency of titanium alloy, such as its strain hardening mechanism, creep, and environmental impact. The strain hardening mechanism might result from the first cycles of the fatigue testing at 1000 Hz especially at higher loads (Ref 13, 14). It is clearly seen in Fig. 3(C) that between 104 and 106 the fatigue strength increases in the case of 1000 Hz and the maximum testing stress levels are ranging between 840 and 950 MPa, which is close to the yield strength of the alloy of 898 MPa (Table 3). After the maximum stress level drops, the behavior of the Ti-6Al-4V coincides with the results from 20 and 90 Hz of testing frequencies. In (Ref 13) the strain-hardening effect of the titanium alloy was also identified in testing at 20 kHz. The strain hardening mechanism can be related to the crystal structure of the alloy (Ref 14). The Ti-6Al-4V alloy has a two-phase microstructure consisting of alpha (α) and beta (β) phases. The alpha phase has a hexagonal close-packed (HCP) crystal structure, while the beta phase has a body-centered cubic (BCC) crystal structure. HCP materials only have three primary slip systems, which means that dislocations have limited mobility during deformation. The BCC crystal structure have five primary slip systems, however, it also has 48 secondary slip systems that can become active at higher stresses or strains. In contrast, a face-centered cubic (FCC) crystal structure, which is present in AA2024-T3 and Inconel 718 alloys, contains only twelve primary slip systems. Therefore, in the case of the titanium alloy Ti-6Al-4V the interface between the alpha (α) HCP and beta (β) BCC phases can create stresses that might potentially influence the fatigue behavior of the material at higher testing frequency and at the same time increase the strain-hardening rate. It has also been reported that Ti-6Al-4V alloy can exhibit room temperature creep even under stress levels lower than its yield stress (Ref 32, 33). By increasing the loading frequency to 1000 Hz, the duration of each loading cycle is reduced, leading to a decrease in the effect of creep. In addition to the strain hardening and relatively low creep resistance, Peters et al. conducted research on the behavior of Ti-6Al-4V alloy under various environmental conditions, including air and vacuum, which demonstrated that the fatigue strength of the titanium alloy was considerably lower when tested in the air than when tested in a vacuum (Ref 34). In the case of the present study, during low frequency testing at 20 Hz, the material was exposed to the air environment for an extended duration, resulting in faster material degradation due to a greater amount of time for oxidation in comparison to 1000 Hz. Therefore, the combination of the strain-hardening mechanism, creep and the duration of exposure to the surrounding environment can play a critical role in the fatigue behavior of the alloy.

Frequency dependency was not so pronounced or might be within the error range in the case of nickel-based superalloy, Inconel 718. It was observed that higher frequency produces minor effect on the fatigue strength of the material and incorporates an upper shift in the S-N curve, which is shown in Fig. 3(E) and (F). The HCF strength for Inconel 718 at 20, and 90 Hz testing frequencies are equal to 548 and 545 MPa as well as 545 MPa for VHCF strength. Basquin’s endurance limit was determined to be equal to 471, 504, and 516 MPa for 20, 90, and 1000 Hz, respectively. Such a result can be due to the high slope value of the S-N curve within the Basquin’s region as well as the transition to the endurance limit which is observed to be rapid in the case of the nickel-based superalloy, Inconel 718 (Ref 18).

3.2 Fracture Surface Analysis

For all the specimens tested at 20, 90 and 1000 Hz frequencies, fatigue cracks in aluminum alloy, AA2024-T3, initiated at or near the surface region. Figure 4 depicts the SEM overview of the fracture surface of the AA2024 alloy specimen tested at 20 and 1000 Hz frequencies with a maximum stress level of 300 MPa. Figure 4(A) and (B) represents the crack initiation site near the surface of the specimen, which is typical fracture behavior for most of the tested specimens with 20 and 90 Hz of AA2024. The surface exhibits a planar topography, and the observed fractures are transgranular in nature, exhibiting multiple layers of macroscopic striations formed along the crack propagation path. The escalation of plasticity levels is the underlying mechanism driving the initiation and propagation of the fractures. In Fig. 4(C), the dimple structure can be observed which is caused by significant plastic deformation in the surrounding region. Such fracture behavior well correlates with the investigations on aluminum alloys performed in previous studies (Ref 11, 16, 35, 36). However, single and multiple surface crack initiations were identified and depicted in Fig. 4(D) and (E) for the 1000 Hz testing frequency. Such a behavior started to appear for AA2024-T3 showing that with an increase in testing frequencies multiple surface crack initiations can be observed (Fig. 4D). High-frequency loading can be a contributing factor in the development of multiple crack initiations in AA2024-T3. This loading condition may accumulate small plastic strains that lead to high-cycle fatigue and the formation of multiple microcracks, which may result in a failure. This type of fatigue failure is known as ‘multiple-site damage’ and is commonly observed in aircraft structures subjected to high-frequency loading (Ref 37).

Fig. 4
figure 4

Fracture surface of AA2024 alloy tested at 20 Hz (A, B, C) and 1000 Hz (D, E, F) testing frequencies and at 300 MPa maximum stress level

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Three different types of fracture behavior were identified on the surfaces of the Ti-6Al-4V alloy (Fig. 5). Figure 5(A, B and C) shows the initiation of a surface and near the surface crack, which are depicted in Fig. 5(D), (E) and (F), leading to the failure of the specimen. These two types of crack initiations were observed in most of the titanium alloy specimens tested for all of the testing frequencies: 20, 90, and 1000 Hz. The surface crack initiation occurred between 104 and 107 cycles to failure. On the other hand, near the surface crack initiation was observed between 107 and 109 cycles to failure. Third type of crack initiation from an interior site was also identified for the Ti-6Al-4V alloy in HCF and VHCF testing between 107 and 109 cycles to failure in the case of 1000 Hz, which is shown in Fig. 5(G), (H), and (I). It can be identified that the crack is located further away from the surface than near the surface crack initiation, depicted in Fig. 5(E), (F). The characteristic feature of the internal crack initiation is a small bright spot at the crack origin, which is also called a 'fish-eye' pattern (Fig. 5H, I). The fish-eye formation mechanism can be attributed to several factors and one of them is the non-metallic inclusions as stress concentrators (Ref 30, 38). However, if the material does not contain a sufficient amount of inclusions or scratches with high-stress concentrations the crack initiates from a region of larger grain (Ref 39). Within the region of a crack origin, a dark area can be observed via an optical microscope and it is called an optically dark area (ODA) (Ref 38, 40). According to previous studies, the ODA regions have rough morphology and the size of it can be increased with an increase in the fatigue strength of the material (Ref 38). In the present study, the internal crack initiation behavior dominates between 107 and 109 cycles to failure, which demonstrates negative effect on the VHCF strength of the alloy (Fig. 3C). A similar conclusion was made in (Ref 41, 42) performing VHCF testing at 120 and 20 kHz of Ti-6Al-4V alloy. The studies observed a degradation in the fatigue strength in VHCF regime due to internal crack initiation under mean stress conditions. A combination of several factors may be the reason for such fatigue fracture behavior. One of them is that the cleavage strength of each grain is not the same across the volume and the interior crack initiations occur on the assumption that there are no active defects in the surface rim area (Ref 5, 43). In addition, although the specimens were tested in an ambient air environment, interior sites were surrounded by a vacuum-like environment which might decrease the short crack growth rate significantly, as reported in (Ref 44). However, if the surface crack or defect site is present, the amount of time it is subjected to an ambient air environment varies for all the testing frequencies 20, 90, and 1000 Hz, which can also affect the fatigue performance of the material.

Fig. 5
figure 5

Fracture surface with three different crack initiations or defects observed in Ti-6Al-4V alloy: surface defect (A, B, C), near the surface defect (D, E, F), and an interior crack initiation, so-called ‘fish eye’ pattern (G, H, I) tested at 1000 Hz testing frequency and 860 MPa maximum stress level

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During the fatigue testing of Ti-6Al-4V alloy at the maximum stress level of 860 MPa for Weibull analysis some of the specimens tested at 90 Hz frequency showed plastic deformation in fracture surface of the material Fig. 6(A), (B). These specimens showed very low fatigue strength and fatigue fracture occurred before 1 × 105 load cycles, where other specimens survived up to 4 × 106 load cycles for 90 Hz testing frequency. Although the experiments were conducted at ambient temperature such a fatigue behavior can be caused by the relatively low creep resistance of the Ti-6Al-4V alloy, especially if the testing load is close to the yield strength of the material, which is the case in the current study. Such fracture behavior well correlates with previous studies at low or ambient temperatures for the Ti-6Al-4V alloy (Ref 14, 32).

Fig. 6
figure 6

Ti-6Al-4V specimen (A) and its fracture surface (B) tested at 90 Hz testing frequency and maximum stress equal to 860 MPa with fatigue fracture at 3.9 × 104 cycles

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Crack initiation sites for all specimens of Inconel 718 start at the surface or very close to the surface regardless of the testing frequency (Fig. 7). The material did not show any interior crack initiation, which would affect the fatigue behavior during the experiments. Figure 7(A), (B) and (C) represents multiple crack initiation sites on the surface of the material, which are common for the samples of Inconel 718 (Ref 6, 18,19,20). It can be seen that the growing path of the fatigue crack is regular, showing an elliptical or semi-elliptical propagation. Fatigue striations and dimple region can be identified on Fig. 7(D) and (E), respectively. Multiple surface crack initiations were identified for 20, 90 and 1000 Hz testing frequencies in case of Inconel 718. The behavior does not show any frequency influence or dependency. Such fracture behavior correlates well with previously published results (Ref 19, 20). The ratio of the fatigue and residual fracture was also investigated for the samples of AA2024, Ti-6Al-4V, and Inconel 718. The result shows that there is no noticeable effect of frequency on the ratio of the fracture.

Fig. 7
figure 7

Fracture surface of Inconel 718 tested at 20 Hz testing frequency and at 650 MPa of maximum stress level

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3.3 Weibull Analysis

It is challenging to quantify the influence of the testing frequencies on the material using an S-N curve. Therefore, the Weibull distribution was utilized using Eq 2. Table 2 represents the shape and scale parameters of AA2024-T3, Ti-6Al-4V, and Inconel 718 with respect to different testing frequencies. The maximum testing stress level was determined to be 300, 860, and 650 MPa to stay within the HCF regime in case of all the testing frequencies for AA2024-T3, Ti-6Al-4V, and Inconel 718, respectively. Table 2, as well as Weibull distribution for AA2024-T3 alloy, Ti-6Al-4V alloy, and Inconel 718, depicted in Fig. 8(A), (B) and (C) do not show any relationship between shape parameters and testing frequencies. However, the increase in the scale parameter can be observed for some of the alloys with respect to different testing frequency.

Table 2 Weibull parameters with respect to testing frequencies
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Fig. 8
figure 8

Weibull analysis for AA2024-T3 (A), Ti-6Al-4V (B) and Inconel 718 (C) for 20 Hz, 90 Hz and 1000 Hz of testing frequencies. Three types of identified crack initiations of the titanium alloy (B): internal (◊), near the surface (○), and surface crack sites (rest)

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Although, no significant increase in the scale parameter was detected for AA2024-T3 and Inconel 718 at 20 and 90 Hz, with a higher frequency of 1000 Hz the increase is seen for all of the testing materials. The change in the scale parameter in the case of AA2024-T3 alloy from 20 or 90 to 1000 Hz can be identified; however, it is not so pronounced and can be within the range of error. This confirms that the aluminum, AA2024-T3, alloy is not so sensitive to the testing frequency.

In the case of the titanium alloy Ti-6Al-4V alloy, an increase in the scale parameter was observed between 20 and 90 Hz testing frequency (Table 2). A drastic change in the scale parameter was also observed for the Ti-6Al-4V alloy tested at 1000 Hz testing frequency. This fact indicates very high frequency dependency, which correlates well with the fact that the titanium alloy is sensitive to the testing environment and oxidation (Ref 13, 14). With a higher testing frequency, the time when the crack is exposed to the surrounding environment is less, which might affect the crack propagation rate and increase the fatigue performance of the material. In addition to the environmental effect, the strain-hardening effect of the sample can occur in the LCF and HCF regimes when the maximum stress level approaches the yield stress. The increase in testing frequency can cause such a shift in the fatigue strength of the material. Figure 8(B) shows a relatively high scatter for the 1000 Hz testing frequency compared to 20 and 90 Hz, which might be caused by different fatigue fractures including near the surface and internal crack initiations, affecting the crack growth rate. Internal crack initiation (Fig. 5G, H and I) for the specimen with the failure between 108 and 109 load cycles reduced the fatigue crack propagation rate and increased the fatigue performance. Strain-hardening of the material and environmental effects such as vacuum-like conditions surrounding the internal crack can be the reasons for such fatigue behavior. Similarly, the specimen with the failure between 107 and 108 load cycles had a near the surface crack initiation (Fig. 5D, E and F), which resulted in a high number of cycles of failure. Where surface crack initiations (Fig. 5A, B and C) were identified for the samples with fatigue fracture between 104 and 106 load cycles Fig. 8(B).

Inconel 718 did not exhibit any notable changes in frequency sensitivity at 20 Hz and 90 Hz. Nevertheless, Fig. 8(C) displays an improvement in fatigue performance within the Weibull distribution chart for 1000 Hz. The scale parameter showed a slight increase in the fatigue strength between 20 and 90 as well as 20 Hz and 1000 Hz (Table 2). Figure 7(A), (B) and (C) represents fracture behavior of the specimen with multiple surface crack initiations which were observed for Inconel 718 at a 20 Hz testing frequency. Multiple or single crack initiations were identified on the fracture surface of the nickel-based superalloy with no noticeable dependency on the testing frequency.

3.4 Threshold Stress Intensity Factor Range Determination

In order to determine the threshold stress intensity factor range and the fatigue crack initiation behavior Eq 3 was fitted to an experimentally derived S-N curve with 20 Hz testing frequency in order to quantify the values of the fatigue strength constants α and c. The value of the threshold SIF range for 20 Hz of testing frequency was taken from the literature for all the materials (Ref 32, 35, 44,45,46,47,48,49,50). Once the fatigue strength constants were derived for 20 Hz they were used to determine the threshold stress intensity factor range for the other testing frequencies, 90 and 1000 Hz.

The fatigue limit for the alloys is determined based on the S-N curve depicted in Fig. 3(A), (B) and (C) with respect to the testing frequencies. Aluminum alloy AA2024-T3 showed no endurance limit up to 109 load cycles (Fig. 3A), however, the fatigue strength at different testing frequencies was determined to be equal to 230, 225 MPa at 107 load cycles to failure for 20 and 90 Hz as well as 155 MPa at 109 load cycles to failure for 1000 Hz of testing frequency. The determination of the threshold stress intensity factor range is based on the S-N curves up to 107 cycles to failure for all testing frequencies in case of AA2024. The yield strength, shear modulus, grain size, and Poisson’s ratio are displayed in Table 3 (Ref 51, 52). The threshold stress intensity factor for 20 Hz of testing frequency was equal to 2.9 MPa m1/2 (Ref 45,46,47). The fatigue strength constants α and c were determined for 20 Hz to be equal to 0.5 and 3, respectively, and applied for the analysis of the rest of the testing frequencies. The determined values of the threshold SIF range were 3.5 MPa m1/2 and 4.5 MPa m1/2 for 90 and 1000 Hz of testing frequencies in the case of AA2024 (Table 4). S-N curves fitted by Eq 3 for the aluminum alloy at 20, 90, and 1000 Hz of testing frequencies showed a relatively good agreement (Fig. 9A). AA2024 demonstrated no significant increase in identified threshold SIF range values, which can also be within the error range due to the lack of tested specimens at each load level.

Table 3 Material properties for threshold stress intensity factor range determination for AA2024-T3, Ti-6Al-4V, and Inconel 718 alloys
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Table 4 Threshold stress intensity factor ranges tested at 20 Hz (taken from literature), 90 Hz and 1000 Hz for AA2024-T3, Ti-6Al-4V, and Inconel 718 alloys
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Fig. 9
figure 9

Experimentally determined S-N curves together with predicted S-N curve of AA2024-T3 (A), Ti-6Al-4V (B), and Inconel 718 (C) with respect to 20 Hz, 90 Hz, and 1000 Hz testing frequencies

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The fatigue strengths of 820, 845 MPa at 107 load cycles for 20 and 90 Hz as well as 700 MPa at 109 load cycles for 1000 Hz of testing frequency were used as input parameters for the threshold SIF range determination for titanium Ti-6Al-4V alloy. Yield strength, shear modulus, grain size, and Poisson’s ratio were equal to 898 MPa, 44 GPa and 5 µm and 0.36, respectively (Table 3) (Ref 53,54,55,56). The threshold stress intensity factor range was assumed to be equal to 3.8 MPa m1/2 (Table 4) for a testing frequency of 20 Hz (Ref 35, 44, 57). The fatigue strength constants α and c were determined for 20 Hz and equal to 0.43 and 40, respectively. The threshold stress intensity factor ranges for Ti-6Al-4V for 90 and 1000 Hz of testing frequencies were equal to 3.2 and 12 MPa m1/2, respectively (Table 4). In the case of 1000 Hz of the testing frequency, the fitted curve is valid only up to 107 load cycles due to different fracture behavior, which affects the fatigue performance in the VHCF regime (Fig. 9B). Before 107 load cycles cracks initiate from the surface for most of the samples, however, after 107 load cycles internal crack initiations start to appear (Fig. 3C). Therefore, the S-N curve up to 107 cycles to failure was utilized for the threshold SIF range values determination. The resulting threshold SIF range values for 20 and 90 Hz show no frequency dependency, but the increase exists if the values are compared for 20 and 1000 Hz of testing frequency. The obtained result may be attributed to the frequency sensitivity of the material, which incorporates several factors, including its strain hardening behavior when subjected to a testing frequency of 1000 Hz and a stress level near the yield strength. Moreover, the fatigue crack growth rate could be affected by the exposure time of the material or the crack to the surrounding environment, which is shorter when tested at higher frequencies than 20 Hz (Ref 14). This reduced exposure time could potentially influence the fatigue crack growth rate.

The values of fatigue strength of nickel-based superalloy, Inconel 718 were determined to be equal to 548, 545 MPa at 107 load cycles to failure for 20 and 90 Hz as well as 545 MPa for 1000 Hz of testing frequency. Yield strength and shear modulus of Inconel 718 is equal to 529 MPa (Table 3) and 80 GPa respectively, the grain size was equal to 9.5 µm and Poisson’s ratio to 0.31 (Ref 58, 59). The values of the threshold SIF range were assumed to be equal to 6.5 MPa m1/2 for 20 Hz of testing frequencies (Ref 48,49,50). Based on Equation 3 the values of fatigue strength constants α and c were determined to be equal to 1.3 and 0.001, respectively. The nickel-based superalloy, Inconel 718, showed a good agreement between the experimental and determined values depicted in Fig. 9C. The values of the threshold stress intensity factor range were equal to 8 and 11 MPa m1/2 for 90 and 1000 Hz, respectively (Table 4). With the decrease of the testing frequency from 1000 to 20 Hz the fatigue strength of the material slightly decreases, which can also be observed by the Weibull distribution (Fig. 8C) as well as on the S-N curve within the Basquin’s region shown in Fig. 3(E), (F).

4 Conclusion

HCF and VHCF tests were carried out at 20, 90 and 1000 Hz testing frequencies to investigate the effect of testing frequencies on the fatigue behavior of AA2024-T3, Ti-6Al-4V and Inconel 718. Experiments with 20 and 90 Hz were performed within the high cycle regime (up to 107 load cycles to failure) and 1000 Hz was utilized in the VHCF regime (up to 109 load cycles to failure) for each material. Basquin’s equation and Weibull distribution were used to describe and quantify the fatigue behavior of the material as a function of different frequencies. Threshold stress intensity factor range as well as fatigue crack initiation were analyzed to shed light on the frequency effect on fatigue behavior. The main conclusions are as follows:

Aluminum alloy AA2024 showed no noticeable frequency dependency in terms of fatigue performance. Based on Weibull distribution an increase was observed in the fatigue performance between 20, 90, and 1000 Hz testing frequency, however, for 20 and 90 Hz no noticeable effect was identified. Fracture surface analysis showed single crack initiations at the surface of the specimens for 20 and 90 Hz. With an increase in the testing frequency, multiple surface crack initiations were observed.

Titanium alloy Ti-6Al-4V demonstrated an increase in fatigue performance with an increase in testing frequency. Weibull distribution analysis quantified the difference. During the fracture surface analysis of the alloy, three different crack initiation types were identified: surface, near the surface, and internal (‘fish eye’ pattern) crack sites which affected the fatigue behavior of the material. Only surface and near the surface sites appeared during testing at 20 and 90. For 1000 Hz all three types of crack initiations were observed. The internal sites were identified in the HCF and VHCF regimes and can be the reason for the change in the slope of the S-N curve and the increase in fatigue performance.

Nickel-based superalloy Inconel 718 showed relatively low frequency sensitivity. The increase in fatigue strength was not so pronounced during the Weibull distribution analysis. Fracture surface evaluation demonstrated no frequency dependence and single or multiple surface crack initiation sites were observed for all of the testing frequencies.

The determination of the range of the threshold SIF was based on the utilization of Eq 3, which was fitted to experimental data. The results showed that there was an increase in the value of ΔKth as the testing frequency was increased for AA2024, Ti-6Al-4V, and Inconel 718. In the case of AA2024, the increase in the value of ΔKth exists; however, it is not significant and can be within the error range. The threshold SIF range determined with respect to the testing frequency for Inconel 718 shows a noticeable change. However, it does not affect drastically the S-N curve as it does in the case of Ti-6Al-4V. With an increase in testing frequency (1000 Hz), the fatigue performance of the alloy increases demonstrating a good agreement between the ΔKth changes of Ti-6Al-4V. The observed increase in ΔKth may have been caused by several factors, such as environmental effects that occur when a crack is exposed to the surrounding environment for different durations, which could be a result of the shift in testing frequency from 20 to 90 and 1000 Hz. Other potential contributing factors include strain-hardening mechanisms resulting from variations in strain rate relative to testing frequency, as well as low creep resistance of the alloy at stress levels approaching the yield strength. The resulting threshold stress intensity factor range values of AA2024, Ti-6Al-4V, and Inconel 718 alloys can be used to predict the fatigue behavior for 20, 90, and 1000 Hz of testing frequencies within the HCF regime.