When analyzing and generalizing experimental data on the estimation of the size of short cracks at the endurance limit of metals and alloys, the comparison of the crack size in smooth specimens with that in specimens with stress raisers is crucial.

The data presented in publications are limited and contradictory. For example, in [1] it is assumed that the size of fatigue cracks, which corresponds to the endurance limit of smooth specimens and specimens with stress raisers, is the same; in [2,3,4], the dependence of the threshold stress intensity factor for small cracks on the crack size is assumed; in [5,6,7], it is shown that the crack size at the endurance limit of specimens with stress raisers decreases with increasing theoretical stress concentration factor. However, the question of what this crack size as compared to that at the endurance limit of a smooth specimen remains open.

In [4, 8,9,10,11, et al.], relationships for the threshold stress intensity factor ΔK th and the endurance limit σ R , which take into account the size of small defects (cracks), are proposed on the assumption that the value of endurance limit is determined according to the non-propagation conditions of fatigue cracks, which appeared near small defects, by taking into account the dependence of the threshold stress intensity factor for small cracks on the crack size and generalizing a large body of experimental data obtained in the study of the dependence of endurance limits on initiated defects of different size and shape.

The value of the square root of the projection of defect or crack area onto a plane perpendicular to maximum normal stresses, \( \sqrt{area} \), was adopted as a defect size characteristic.

Figure 1 shows the dependence of ΔK th on \( \sqrt{area} \), obtained in [11] as a result of a generalization of the investigation of numerous materials under circular bending and tension–compression. The values of ΔK th were calculated from the following relation:

$$ \Delta {K}_{th}=0.65{\Delta \upsigma}_R\sqrt{\uppi \sqrt{area}}. $$
(1)
Table 1. Investigated Materials and Defect Types that are Assessed Using the Parameter \( \sqrt{area} \)
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Fig. 1.
figure 1

Dependence of ΔK th on \( \sqrt{area} \) for specimens with different defects and cracks. [The investigated materials (1–14) and defect types are listed in Table 1.]

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In this study, the following defects were analyzed: very small drill-holes 40–500 μm in diameter and over 40 μm deep [12,13,14,15,16,17,18], very shallow notches 5–300 μm deep [19,20,21,22,23,24,25,26,27,28,29], very shallow semicircular cracks 30–260 μm deep [30], defect 72 μm in size generated by Vickers indentation [13].

The data presented in Fig. 1 indicate that the dependence of ΔK th on \( \sqrt{area} \) in the range of values \( \sqrt{area} \) ≤ 1000 μm for the investigated materials is described in logarithmic coordinates by the linear dependence:

$$ \Delta {K}_{th}\propto {\left(\sqrt{area}\right)}^{1/3}. $$
(2)

Moreover, for the specimens with very short flat cracks of length 2a, the dependence

$$ \Delta {K}_{th}\propto {(2a)}^{1/3} $$
(3)

exists.

The simultaneous solution of Eqs. (2) and (3) yields:

$$ \sqrt{area}\propto (2a). $$
(4)

In [4], the important conclusion was drawn that the specimens with small fatigue cracks and defects in the form of drill holes and notches at the same value of the parameter \( \sqrt{area} \) have the same endurance limit. That is at the same endurance limit:

$$ {\left(\sqrt{area}\right)}_c={\left(\sqrt{area}\right)}_{dh}={\left(\sqrt{area}\right)}_n, $$
(5)

where \( {\left(\sqrt{area}\right)}_c \), \( {\left(\sqrt{area}\right)}_{dh} \), and \( {\left(\sqrt{area}\right)}_n \) are values for fatigue cracks and small defects in the form of drill holes, and notches, respectively.

In Fig. 2, a plot of the endurance limit σ R vs \( \sqrt{area} \) was constructed for specimens with fatigue cracks and defects in the form of drill holes and notches, which supports the above conclusion.

Fig. 2.
figure 2

Comparison of the endurance limit σ R with the value of \( \sqrt{area} \) for specimens with small fatigue cracks and defects: (1) crack, (2) drill hole, (3) notch.

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This conclusion is also substantiated by the data listed in Table 2 [4].

Table 2. Comparison of the Endurance Limits of Specimens with Small Cracks and Small Defects at the Same Values of \( \sqrt{area} \) (Carbon Steel)
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The above conclusion and the proportional relationship between the parameter \( \sqrt{area} \) and the crack size 2a, which is described by Eq. (4), allow one to calculate the size of non-propagating cracks at the endurance limit using experimental data for a wide range of metals and alloys, obtained in studies by Murakami [4, 8,9,10,11,12,13,14,15,16], on the threshold stress intensity factor ΔK th , the endurance limit σ R and defects of different size and shape, which are evaluated with the parameter \( \sqrt{area} \), on the one hand, and on the geometry of fatigue cracks at the endurance limit under relevant test conditions (semicircular, semielliptical cracks, etc.), on the other hand. Let us consider some examples.

In the study [8], the effect of small defects (drill holes 40–200 μm in diameter and depth on the endurance limit of S10C and S45C carbon steels under the circular bending of specimens 10 and 6 mm in diameter, respectively, was investigated. It was noted that at the endurance limit, which is defined as maximum nominal stresses at a number of cycles to fracture of 107, non-propagating cracks were observed in all specimens (Fig. 3). The size of non-propagating surface crack is denoted by (2a) exp . During a circular bending fatigue test of cylindrical specimens, cracks of semielliptic geometry with the ratio of the ellipse semiaxes b/a = 0.75 generally nucleate and propagate [32].

Fig. 3.
figure 3

Size of non-propagating cracks originating from drill holes at the endurance limit: (a) S10C steel, (b) S45C steel [8].

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The data on the size of non-propagating cracks, (2a) exp , presented in Fig. 3 allow one to assume, in view of relations (4) and (5), that \( {\left(\sqrt{area}\right)}_{dh} \) for a defect as a drill hole and \( {\left(\sqrt{area}\right)}_c \) for a surface crack of the size (2a) calc are equal:

$$ {\left(\sqrt{area}\right)}_{dh}={\left(\sqrt{area}\right)}_c. $$

For a semielliptical crack with the ratio of the ellipse semiaxes b/a = 0.75, we shall have

$$ {\displaystyle \begin{array}{c}{\left(\sqrt{area}\right)}_{dh}={\left(\sqrt{1/2\uppi {(a)}_{calc}b}\right)}_c=1.0851{a}_{calc},\\ {}{(2a)}_{calc}=1.843{\left(\sqrt{area}\right)}_{dh}=1.843\sqrt{hd-{d}^2/4\sqrt{3}}.\end{array}} $$

Comparison of the calculated crack size values (2a) calc with the experimental ones (2a) exp (Table 3) for all cases presented in Fig. 3 shows them to differ only slightly. The maximum discrepancy between these values is not over 15%.

Table 3. Comparison of Calculated Crack Size Values with Experimental Ones
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It should be noted that the error in the determination of the crack size (2a) calc is primarily due, in our view, to uncertainty in finding the value of b/a, especially at small crack size (2a ≤ 0.2 mm). This can be confirmed, e.g., by plots of the ratio of the geometrical crack dimensions, b/a, vs surface crack length [32] (Fig. 4). Carbon steel containing 0.43% C, investigated in the study [32], is close to the materials considered above in mechanical characteristics and grain size.

Fig. 4.
figure 4

Dependence of the ratio of the geometric crack dimensions, b/a, on the crack length on the surface for fine-grained (1) and coarse-grained (2) 0.43 C steel.

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As is seen from Fig. 4, a considerable scatter of the ratio b/a is observed in the crack size region 2a ≤ 0.2 mm. The average value of b/a in the crack size range 2a = 0–0.2 mm is 0.95. Taking into account this fact, the error in the determination of crack size (Table 3) can be reduced.

Figure 5 shows, with the use of data presented in [8, 9, 14], the dependence of the endurance limit of S10C and S45C steels on the value of \( \sqrt{area} \) and the surface crack size 2a proportional to it, calculated by the procedure described above. The values of \( \sqrt{area} \) and 2a correspond to detects in the form of drill holes with the ratios h/d = 0.5, 1.0, 2.0 and d = 40–500 μm.

Fig. 5.
figure 5

Dependence of the endurance limit of S10C (a) and S45C (b) steels on the value of \( \sqrt{area} \) and the surface crack size 2a proportional to it: (∆) h/d = 0.5, (○) h/d = 1.0, (□) h/d = 2.0 \( \left(\sqrt{area}=\sqrt{hd-{d}^2/4\sqrt{3}},\kern0.5em 2a=1.843\sqrt{area}.\right) \)

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The data presented in Fig. 5 suggest the following. The endurance limit of specimens with defects decreases with increasing defect of the size \( \sqrt{area} \) and surface crack size 2a proportional to it. The endurance limit of S10C steel with a defect of the size \( \sqrt{area} \) = 37 μm (2a = 68 μm), \( \sqrt{area} \) = 46 μm (2a = 85 μ), and \( \sqrt{area} \) = 60 μm (2a = 110 μm) is equal to that of smooth specimen (σ R = 181 MPa). The endurance limit of S45C steel with a defect of the size \( \sqrt{area} \) = 37 μm (2a = 68 μm) and \( \sqrt{area} \) = 46 μm (2a = 85 μm) differs from that of smooth specimen (σ R = 245 MPa) by 4.9 MPa.

The critical values \( {\left(\sqrt{area}\right)}_{cr} \) and (2a) cr at the points of intersection of the horizontal straight line corresponding to the endurance limits of smooth specimen and the plot of σ R vs \( \sqrt{area} \) (2a) for S10C steel are 65 and 119 μm and for S45C steel 32 and 59 μm, respectively. These results indicate that the endurance limits of the above steels are independent of the defect of the size \( {\left(\sqrt{area}\right)}_{cr} \) ≤ 65 μm and ≤ 32 μm, as well as (2a) cr ≤ 119 μm and ≤ 59 μm, respectively. The critical crack size (2a) cr is in satisfactory agreement with the critical size of non-propagating cracks, which are observed at the endurance limit of smooth specimen (100 μm for S10C steel and 50 μm for S45C steel). Thus, the crack size at the endurance limit of specimens with defects firstly decreases with nominal stresses and, secondly, always remains larger than the size of non-propagating crack, which is observed at the endurance limit of a smooth specimen.

Figure 6 shows, with the use of the results presented in [15, 16, 18], several similar calculated and experimental data for different classes of materials. It can be seen that the presence of small defects (cracks) more strongly affects the reduction in the endurance limit of high-strength materials. Whereas the critical size of a non-propagating crack (2a) cr = 110 μm for the low-strength 2017-T4 aluminum alloy and 70/30 bronze does not affect the reduction in endurance limit, this crack size greatly reduces it for high-strength S45C steel in the hardened state. The critical crack size for this steel is 20 μm. This conclusion is in good agreement with the known fact that high-strength steels are very sensitive to notches and defects.

Fig. 6.
figure 6

Dependence of the endurance limit of different materials on the value of \( \sqrt{area} \) and the surface crack size 2a proportional to it: (1) 2017-T4 aluminum alloy, (2) 70130 bronze, (3) S45C steel (hardening), (4) S45C steel (hardening + tempering) \( \left(\sqrt{area}=\sqrt{hd-{d}^2/4\sqrt{3}}\kern0.5em \mathrm{and}\kern0.5em 2a=1.843\sqrt{area}.\right) \)

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In Fig. 7, the results presented in Fig. 5 and 6 are shown in the coordinates endurance limit–logarithm of the crack size corresponding to the endurance limit. As can be seen, the set of experimental data for all investigated materials corresponds to a single straight line. This indicates that the crack size at the endurance limit of specimens with defects firstly always remains larger than the size of non-propagating crack, which corresponds to the endurance limit of smooth specimen, and, secondly, decreases with increasing nominal stresses and tends in the limit to the crack size at the endurance limit of smooth specimen.

Fig. 7.
figure 7

Dependence of the endurance limit of different materials on the value of 2a for S10C steel (a), S45C steel (b), 70/30 bronze (c), 2017-T4 aluminum alloy (d), S45C steel in the hardened state (e), and in hardened and tempered state (f): (▲) h/d = 0.5, (●) h/d = 1.0, and (■) h/d = 2.0.

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Based on the above data, let us consider the effect of stress concentration and gradient on the crack size corresponding to the endurance limit of different materials.

Table 4 lists values of theoretical stress intensity factors (K T ) and relative stress gradients (\( \overline{\upeta} \)) for stress raisers in the form of drill holes with different ratio of h/d. The numerical values of K T and \( \overline{\upeta} \) were calculated by the finite element method using the ANSYS software product. It also lists surface crack sizes (2a) calc , which correspond to the endurance limits of different materials in the presence of such stress raisers.

Table 4. Theoretical Stress Concentration Factors, Relative Stress Gradients, and Crack Sizes at the Endurance Limit for Stress Raisers with the Different Ratio h/d
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Figure 8 illustrates the dependence of crack size at the endurance limit of S10C steel on relative stress gradient and theoretical stress concentration factor at the different ratio h/d. It can be seen that the crack size corresponding to the endurance limit of S10C steel specimens with stress raisers decreases with relative stress gradient and theoretical stress concentration factor and tends in the limit (at K T = 1 and \( \overline{\upeta} \) = 0) to the crack size at the endurance limit of a smooth specimen, (2a)0.

Fig. 8.
figure 8

Dependence of the crack size at the endurance limit of S10C steel on the relative stress gradient (a) and theoretical stress concentration factor (b): (1, 4) h/d = 0.5, (2, 5) h/d = 1.0, and (3, 6) h/d = 2.

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Results similar to those presented in Fig. 8 have also been obtained for all other materials investigated in studies [4, 8,9,10].

Conclusions

  1. 1.

    At the endurance limit, which is defined as maximum nominal stresses at a number of cycles to fracture of 107, non-propagating cracks were observed on all specimens.

  2. 2.

    The value of the square root of the projection of defect area onto a plane perpendicular to maximum normal stresses, \( \sqrt{area} \), has been adopted as a defect size characteristic.

  3. 3.

    A proportional relationship between the parameter \( \sqrt{area} \) and the crack size 2a has been established: \( \sqrt{area} \) ∝2a.

  4. 4.

    It has been found that the specimens with small fatigue cracks and defects in the form of drill holes and notches with the same value of the parameter \( \sqrt{area} \) have the same endurance limit.

  5. 5.

    A procedure for the calculation of the size of the size of non-propagating cracks at the endurance limit of specimens with stress raisers in the form of drill holes has been proposed. Using experimental data for a wide range of metals and alloys, obtained in studies by Murakami, on the endurance limit σ R and the size of defects of different size and shape, which are evaluated with the parameter \( \sqrt{area} \), on the one hand and on the geometry of fatigue cracks at the endurance limit under relevant test conditions (semicircula, semielliptical cracks, etc) on the other hand, the size of non-propagating cracks at the endurance limit of these materials has been calculated.

  6. 6.

    The effect of the stress concentration and gradient level on the crack size at the endurance limit of different materials has been considered.

  7. 7.

    It has been shown that the crack size at the endurance limit of specimens with defects firstly decreases with nominal stresses and, secondly, is always larger than the size of non-propagating crack at the endurance limit of smooth specimen.

  8. 8.

    It has been found that the crack size corresponding to the endurance limit of specimens with stress raisers in the form of a drill hole decreases with the relative stress gradient and theoretical stress concentration factor and tends in the limit (at K T = 1 and \( \overline{\upeta} \) = 0) to the crack size at the endurance limit of a smooth specimen.