Introduction

GCr15 bearing steel (similar to AISI 52100, JIS-SUSJ2 and 100Cr6) is one of the most widespread applied materials. With quenching and tempering at a low temperature, GCr15 bearing steel would possess good wear and corrosion resistance. Corrosion resistance is one of the most important factors for choosing bearing materials [1] due to the typical failure mode of corrosion [2,3,4]. It is well known that corrosion can affect the life of mechanical parts dramatically [5]. Pitting corrosion is one of the most dangerous situations as it may lead to severe stress concentration and facilitates crack initiation [6,7,8]. Study on the stress distribution around corrosion pits is beneficial for the early identification of dangerous areas that are more likely to be crack initiation regions. In addition, fatigue propagation may start from such an initial region under cyclic loads. Thus, it is important for fatigue life prediction to focus on the formation, mechanism and effects of pits on stress concentration factors.

Recently, experimental investigations and numerical simulations have been conducted to explore the stress distributions around corrosion pits [9,10,11,12]. Huang et al. performed a series of investigations to study the effect of pitting holes on the fatigue life of aluminum alloy 7075-T6. Firstly, they investigated the quantitative correlation between geometric parameters and stress concentration factors of corrosion pits [13]. Then, the equivalent crack size model for pre-corrosion fatigue life prediction was proposed [14]. Furthermore, the interaction effect of stress concentration between two neighboring pits was investigated considering relative position parameters and pit depth [15]. The critical corrosion regions bearing high susceptibility to crack initiation were determined for intersecting and adjacent pits. On the other hand, the interaction of porosity and fatigue crack evolutions was characterized with both X-ray computed tomography and finite element analysis (FEA) [16]. The formation and propagation of fatigue cracks were dominated by the highest local stress–strain concentration induced by the presence of porosity adjacent to the free surface and the crack tip, respectively.

Fracture due to very high cycle fatigue is a special behavior of materials, which has drawn much attention recently [17,18,19,20]. During the fracture process, the initial cracks usually start from an internal defect rather than a surface defect [21,22,23]. The surfaces of very high cycle fatigue specimens should be very smooth, as to ensure the fatigue initiation is located at an internal defect rather than a surface defect. Specimens in a corrosion environment may generate corrosion pits which would influence their surface roughness. Thus, in a corrosive environment, the interaction between the corrosion pits and the defects plays an important role for crack initiation [24,25,26,27]. However, to the best of our knowledge, study on the interaction between the corrosion pits and defects is ignored and should be paid more attention.

In this work, finite element simulations were carried out to investigate the interaction between surface corrosion pits and internal defects in hourglass-shaped specimens. The structure characteristics of corrosion pits and defects were obtained through observing the outer surfaces of the pre-corroded specimens and the fracture surface of specimens after very high cycle fatigue testing. Then, the stress distributions around the corrosion pit and the internal defect were simulated and plotted using the finite element method. The stress concentration factor induced by a single defect was estimated and compared. The interaction between a corrosion pit and an internal void was investigated. The quantitative relationship between location parameters and stress concentration factors was proposed. The obtained quantitative information would be useful for understanding crack initiation and fatigue life prediction of engineering components.

Experimental Results and Simulation Details

Preparation of Specimens

The material of samples was high-chromium bearing steel of grade GCr15. The chemical composition (wt.%) of the material was C 1.01, Si 0.18, Mn 0.37, S 0.0033, P 0.020, Cr 1.64, Ni 0.016 and Fe balance. The hourglass-shaped specimen for fatigue testing was machined from the annealed steel bar with a certain amount of finishing margin. The specimens were heated at 1133 K for 30 min. Then, they were quenched in oil and tempered at 453 K for 120 min and finally air-cooled. Those specimens were ground with 400–1000-grit abrasive paper and polished with 1-μm diamond paste. Finally, they were cleaned with distilled water and absolute ethyl alcohol. The microstructure of the specimen was tempered martensite. The tensile strength was 2200 MPa. Figure 1 shows the diagram of an hourglass-shaped specimen.

Fig. 1
figure 1

Shape and dimensions of hourglass-shaped specimen (units: mm)

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Before the fatigue test, a pre-corrosion process was carried out to form corrosion damage in the specimens, i.e., the specimens were immerged into a neutral 3.5 wt.% NaCl solution for 120 min. After that, the specimens were cleaned with 1000-grit abrasive paper to remove the oxide layer. Then, the pre-corroded specimens were polished with a 1-μm diamond paste and cleaned with distilled water and absolute ethyl alcohol. The very high cycle fatigue tests were carried out with the pre-corroded specimens.

Morphology of the Pits in the Specimens

Surface corrosion morphologies of the pre-corrosion specimens were observed with a stereo microscope (Smartzoom5, Zeiss, Germany). Figure 2a illustrates the top view of typical pits on the outer surface of a pre-corroded GCr15 specimen. The corrosion pit appears to be a circle on the surface. To acquire the depth of corrosion pits and dimensions of defects accurately, fatigue experiments under fully reversed tension (stress ratio R = − 1) were performed on the pre-corroded specimens. The tests were conducted with a Shimazu USF-2000 ultrasonic fatigue testing machine in an open environment at room temperature. Fracture morphologies of the failed pre-corrosion specimens were also carefully observed using a scanning electron microscopy (Supra55, Zeiss, Germany). The typical morphology of the pit on the fracture surface in profile view is shown in Fig. 2b. The corrosion pit appears to be a hemi-ellipse on the cross section. Thus, they can be idealized as hemi-ellipsoid pits, which are consistent with previous works [10, 28, 29]. On the other hand, surface morphologies of internal defects were observed and the typical morphology is shown in Fig. 2c. Considering their typical shape, the internal defects can be idealized as spherical voids in this investigation.

Fig. 2
figure 2

Typical morphologies of (a) a corrosion pit on the outer surface, (b) a corrosion pit and (c) an internal inclusion on the fracture surface of a pre-corroded GCr15 specimen

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Finite Element Models

To further understand the effect of corrosion pits and internal defects on the specimen, stress concentration analysis was performed using the finite element method. Three hourglass models, which contain surface corrosion pits and internal voids, were established. Model I contained a single corrosion pit on the outer surface of the specimen; Model II contained a single internal void in the center part of the specimen while both a corrosion pit and an internal void were constructed in Model III. Shapes and dimensions of established hourglass shape models are the same as those of the hourglass-shaped specimens shown in Fig. 1. The pit and the void were all localized at the narrow section of the specimen. Considering the symmetry of the spatial distribution, only a half-hourglass geometry was modeled along the loading direction for finite element simulations. Details of the models for a surface corrosion pit and an internal void are illustrated in Fig. 3. As shown in Fig. 3a, the element size near the concentration zones was as fine as necessary to calculate the real stress concentration. The boundary conditions were set as follows: The end of surface A1 was fully fixed, and a uniaxial tensile face loading was acted on the other end of surface A2.

Fig. 3
figure 3

(a) Finite element meshing with refined mesh around the corrosion pit and the internal void; diagrams of (b) an internal void and c a corrosion pit, a and h denote the radius and depth of the corrosion pit, respectively

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Considering that the stress concentration usually exists at the narrow section, and stress concentration factor is independent of the applied loading; a mechanical loading with specific value was applied on the end of surface A2. Under this load, the nominal stress at the narrow section of the specimen is to be 1 MPa. Then, the stress concentration factors can be directly evaluated by the stress distribution results.

In this work, according to the experimental results mentioned in “Morphology of the Pits in the Specimens” section, corrosion pits were idealized as hemi-ellipsoid pits in which two of the three semi-axes are equal (a = 7.5 μm). The internal defects were idealized as spherical voids (a = 7.5 μm). For all of the models, a mesh convergence was performed to confirm the accuracy of the finite element simulation. For the case with an internal void (a = 7.5 μm) and a corrosion pit (a = 7.5 μm, h = 0.75 μm), the stress concentration factor was convergent when the element size of the defect surfaces is about 0.5 μm. Thus, the defects of all models were meshed with 0.3 μm to ensure the simulation accuracy in all finite element simulations.

Results and Discussion

To investigate the effects of the corrosion pit and the internal void on the stress concentration, a series of finite element simulations were carried out. The stress concentration factors, which can be calculated by the ratio of the local maximum stress to the nominal stress, were obtained for all of the models.

Stress Concentration around a Single Defect

Model with a Single Corrosion Pit

To analyze the effect of corrosion pit depth on the stress concentration quantitatively, the dimensionless parameter depth-to-length ratio Rd was defined as the ratio of pit depth-to-the length of the semi-axis parallel to tensile loading. According to the results of the previous work [13], the dimensionless parameter Rd of a corrosion pit ranges from 0.26 to 4.8. Considering the complexity of corrosion pits, the finite element models were carried out with a much wider range of Rd (0.05–20).

The simulation results are illustrated in Fig. 4a. The stress concentration factor increases with the pit depth increasing. The increasing rate of the stress concentration factor is as high as 1.2 in the range of Rd = 0.05–0.5. When Rd = 0.5, the stress concentration factor becomes one and a half of that with Rd = 0.05. This indicated that during this range, a very small increase in depth of the pit would give rise to a dramatic increase in the stress concentration in the sample, thus, reducing the fatigue life of materials to a great extent. However, when Rd reaches the range of 2.5–20, the increasing rate of the stress concentration factor becomes slow. And finally, the stress concentration factor tends to be a constant. This phenomenon may be due to the geometry structural characteristic of the corrosion pit. As the depth of the pits increase, the curvature of the pit bottom decreases accordingly. However, the deeper the pit is, the slower the decreasing rate of the curvature becomes. Then, the growth of the stress concentration factor around the corrosion pits slow down. Similar results were also reported for a cuboid model containing a corrosion pit [13]. An empirical equation has been proposed to correlate the stress concentration factors around a single corrosion pit to the dimensionless parameter Rd [13]. In this work, a modified equation was proposed to estimate the relationship of the stress concentration factors and Rd for the hourglass shape model:

Fig. 4
figure 4

The stress concentration factors as a function of (a) the depth-to-length ratio Rd of corrosion pits and (b) the distance-to-radius ratio D/a of internal voids

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$$K_{{t,{\text{cor}}}} = \frac{{R_{d} }}{{0.5023 + 0.4875R_{d} }} + 1$$
(1)

Based on Eq 1, the stress concentration factor induced by a single corrosion pit can be estimated when the dimensionless parameter Rd is obtained through experimental observation. This equation is suitable for estimating the stress concentration factors induced by surface pits. According to Eq 1, the stress concentration factor induced by a single corrosion pit approaches to 1 when the dimensionless parameter Rd approaches to 0. It is in good agreement with values obtained using the finite element method. However, the value of the stress concentration factor is 0 using the empirical equation mentioned in Ref. [13]. Thus, the modified equation is more suitable for estimating the stress concentration factors induced by shallow corrosion pits.

Model with a Single Internal Void

For the models with a single internal void, the location effect of the void on the stress concentration was investigated. A single internal void with dimension of a = 7.5 μm was designed as the testing target, which was located on the narrow section of the hourglass-shaped specimen. For the location-dependent property discussion, the distance-to-radius ratio D/a was introduced to express the location of the void, where D denoted the distance from the center of the internal void to the outer edge of the model along the radius direction. It reflects the relative size of the distance (i.e., the thickness between the void and the free surface) and the void size. It would have great effect on the stress distribution according to the previous works about two adjacent corrosion pits [15, 30] and cracks [31]. Considering the symmetry and dimensions of the hourglass-shaped specimen, it can fully represent all the conditions. In addition, in this work, distance D ranges from 8.25 to 1500 μm, i.e., the distance-to-radius ratio D/a ranges from 0.1 to 200.

Figure 4b shows the variation of stress concentration factors with the distance-to-radius ratio D/a. The stress concentration factor decreases with the distance-to-radius ratio increasing. Particularly, in the range of D/a = 1.1 to 2, the decreasing rate of the stress concentration factors is rather high. This indicates that in this range the stress concentration factors suffer from dramatically decreasing with a very minor increment of D/a. However, the decreasing rate of the stress concentration factor becomes very slow when D/a > 2 and the stress concentration factor are approaching to a constant. In addition, if the void is close to the outer surface of the specimen, a very large stress concentration factor could be introduced.

Drawing on the experience of the exponential equation proposed by previous works [12, 32] for the interacting effect between two adjacent corrosion pits, in this work, a modified exponential relation was proposed to estimate the relationship between the stress concentration factor and the distance-to-radius D/a of an internal void:

$$K_{{t,{\text{inc}}}} = \frac{1.9649}{{D\left( {\frac{D}{a} - 1} \right)^{0.7365} }} + 1.9649$$
(2)

According to Eq 2, the stress concentration factor can be calculated when the proximity D is obtained. When the distance-to-radius D/a is rather high, the stress concentration factor is approaching to that induced by the void located on the center of narrow section. When D/a approaches to 1, i.e., the minimum thickness of base material between the void and the free surface approaches to 0, and the stress concentration factor becomes infinity. Considering the rationality under critical condition, Eq 2 can be used to evaluate the effect of location parameters on the stress concentration factors effectively.

Interaction Between a Corrosion Pit and an Internal Void

For the models with both a corrosion pit and an internal void, the cooperative effect of the two defects should not be ignored. Besides the individual geometry parameters of the defects, parameters for analysis also include the distance (D2) between the center of the corrosion pit and the center of the internal void; the difference (D2 − h) between the distance D2 and the pit depth h. The value of D2 − h denotes the distance from the center of the internal void to the bottom of the corrosion pit (i.e., the free surface of the specimen). In this work, the value of (D2 − h) was set from 8.25 to 1500 μm, i.e., the distance-to-radius ratio (D2 − h)/a ranges from 0.1 to 200.

Figure 5a shows the stress concentration factor varies with the distance-to-radius ratio when Rd = 1. It can be found that the stress concentration factor increases dramatically as the ratio (D2 − h)/a approaches to 1. This indicates that the proximity degree of the corrosion pit and the void plays an important role in crack initiation and fatigue damage. As the internal void becomes closer to the corrosion pit, the localized stress fields around the internal void will cross with that around the corrosion pit. It will generate influences on the stress distribution. The closer the corrosion pit is to the void, the more serious the interacting effect is. Thus, the stress concentration factor is rather high when the internal void is very close to the corrosion pit.

Fig. 5
figure 5

(a) Variation of the stress concentration factors with the distance-to-radius ratio (D2 − h)/a for internal voids; (b) the contrast of stress concentration factors induced by the corrosion pit depths and internal voids

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When (D2 − h)/a > 2, the stress concentration factor has a very low value. The value is approximately the corresponding value of the stress concentration factors induced by a single defect (shown in Fig. 5b). Actually, in these circumstances, the stress concentration factor around the defect is depending on either the single corrosion pit or the single internal void. If the stress concentration is located in the corrosion pit, the single corrosion pit plays the major role in determination of the stress concentration factor and vice versa. This indicated that in this range, the interaction of the corrosion pit and the internal void has little contribution to the stress concentration factor. A similar phenomenon was also observed for the interacting effect of two adjacent corrosion pits on the stress concentration factor [12, 15].

Considering that the stress concentration factors are determined by a certain single defect when (D2 − h)/a > 2, it is necessary to contrast the stress concentration factors induced by the corrosion pit with that induced by the internal void. As shown in Fig. 5b, the stress concentration factor induced by the corrosion pit increases with increasing pit depth, while the stress concentration factor induced by the internal void is a constant except for the internal void near the surface. When Rd = 1, the stress concentration factor induced by these two types of defects are close to each other as shown in Fig. 6a. For a single spherical pit, the stress concentration factor induced by the corrosion pit is slightly larger than that induced by the internal void. The stress concentration is located on the corrosion pit when Rd ≥ 1, while it is located on the internal void when Rd < 1 (shown in Fig. 6b).

Fig. 6
figure 6

The comparison of stress distributions with different depth-to-length ratios when (D2 − h)/a = 4

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Furthermore, quantitative analysis of the stress concentration factors which were dependent on both the distance-to-radius ratio (D2 − h)/a and depth-to-length ratio Rd was carried out. A series of models with different relative position parameters and pit depths were established. The stress distributions of these models under uniaxial tensile loading were obtained. Figure 7a shows the stress concentration factors with different corrosion pit depths (Rd = 0.05–10) as a function of the location parameter of internal voids [(D2 − h)/a]. One can see that the stress concentration factors are relatively high for all of the models when the distance-to-radius (D2 − h)/a approaches 1. It indicates that approaching of the corrosion pit to the internal void is most likely to be the fatigue initiation position and is detrimental to the very high cycle fatigue performance of the specimens.

Fig. 7
figure 7

(a) The stress concentration factors with different Rd as a function of the location parameter (D2 − h)/a of internal voids; corresponding estimated results for (b) Rd ≥ 1and (c) Rd < 1

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As mentioned above, the determinants of the stress concentration factors could be classified into two types according to the stress concentration location when (D2 − h)/a > 2. For Rd ≥ 1, an exponential equation could be proposed to estimate the relationship between the stress concentration factor and the structure parameters as shown in Eq 3. The simulating results and corresponding estimated results are illustrated in Fig. 7b.

$$K_{t} = \frac{1.8662}{{(D_{2} - h)\left( {\frac{{D_{2} - h}}{a} - 1} \right)^{0.8739} }} + K_{{t,{\text{cor}}}}$$
(3)

As shown in Eq 3, the stress concentration factors were determined by two parts which were dominated by parameters of (D2 − h)/a and Rd, separately. There is no interaction between these two parts. For each value of Rd, an obvious declining tendency of the stress concentration factors can be observed with the increase in the distance-to-radius ratio (D2 − h)/a. For the void remote from the corrosion pit (i.e., the distance-to-radius ratio is relatively large), the stress concentration factor is close to a constant (i.e., Kt ≈ Kt,cor).

For Rd < 1, the stress concentration factor is dominated by the internal void. The following modified exponential equation was proposed to correlate the stress concentration factor and the structure parameters. The corresponding predicted results are shown in Fig. 7c.

$$K_{t} = 1.9649\left( {\frac{1}{{(D_{2} - h)\left( {\frac{{D_{2} - h}}{a} - 1} \right)^{\alpha } }} + 1} \right)$$
(4)
$$\alpha = 0.6690 - 0.3089R_{d} + 0.5127\sqrt {R_{d} }$$
(5)

where α is the parameter varies with depth-to-length ratio Rd. Values of α are listed in Table 1.

Table 1 The estimated results of α with various Rd, the corresponding results of α obtained with Eq 5 and the relative error
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For this situation, the stress concentration factor is dominated by the interaction of (D2 − h)/a and Rd as shown in Eqs 4 and 5. For a particular value of Rd, the downward tendency of stress concentration factor could also be observed with the increase in (D2 − h)/a. The reduction rate of the stress concentration factor increases with increasing Rd. When the internal void is far away from the corrosion pit, the (D2 − h)/a is relatively high and the stress concentration factor is close to a constant. It is in good agreement with the tendency mentioned above. The correlation between the parameter α and the depth-to-length ratio Rd is estimated using Eq 5. The maximum relative error is 0.54% as shown in Table 1.

Conclusions

In summary, in the present work, the interacting effects of internal voids and corrosion pits on the stress concentration factor in hourglass-shaped specimens were simulated. Roles of the structure parameters of the specimens including the depth-to-length ratio Rd and the distance-to-radius ratio (D2 − h)/a were considered. The main results are summarized as follows.

  1. (1)

    Stress concentration factor increased with the depth-to-length ratios Rd for models with a single corrosion pit. The rate of the stress concentration factor increase becomes slower as the Rd increasing and the stress concentration factor finally tends to a constant.

  2. (2)

    For models with a single internal void, the stress concentration factor decreased with increasing distance-to-radius ratio. When the internal void is present near the surface the stress concentration factor increased dramatically. The stress concentration factor approaches to a constant when the internal void is far from the surface.

  3. (3)

    The effects of the distance-to-radius ratio (D2 − h)/a and depth-to-length ratio Rd on the stress concentration factors were investigated, and corresponding equations were proposed to evaluate these quantitative relations. For Rd ≥ 1, the stress concentration factor was determined by two parts. These two parts were dominated by (D2 − h)/a and Rd, separately. For Rd < 1, the stress concentration factor was dominated by the interaction of (D2 − h)/a and Rd.

  4. (4)

    The stress concentration factor dramatically increases with the decrease in the distance-to-radius ratio (D2 − h)/a when (D2 − h)/a approaches 1. Thus, the corrosion pit adjacent to the internal void is most likely to be the initiation position and may significantly affect the very high cycle fatigue performance of the specimens.