1. Introduction
The presence of small defects is common in many real engineering components (for instance connecting rods, truck wheels, suspensions arms) made of casting metals such as high-strength steels [
1,
2,
3,
4,
5], cast irons [
6,
7,
8,
9,
10,
11] and aluminium alloys [
12,
13]. Such defects, usually arising from the manufacturing process, are present in the form of non-metallic inclusions, slag inclusions, shrinkage porosities, gas pores, oxides and dross defects [
14,
15]. Moreover, these defects (also named solidification defects) can be encountered with different sizes (ranging from several µm to few mm) and shapes (spherical, elongated, or complex 3D geometry) and can be located both at the surface and in the bulk of the metallic components. Note that in the last years some optimisation procedures [
16,
17] have been developed in order to partially avoid the formation of solidification defects during the casting process, even if such defects cannot be completely removed.
Several studies conducted over the last forty years have demonstrated that solidification defects have a deleterious effect on mechanical properties under both monotonic and cyclic loading conditions [
15]. As far as cyclic loadings are concerned, the solidification defects represent the preferential fatigue crack initiation sites and, consequently, metals containing such defects experience not only shorter fatigue life but also a larger scatter in fatigue strength with respect to their counterpart free from defects [
13]. Since in real situations structural components can be subjected to cyclic loading, it is crucial to take into account the defect effects in the fatigue strength assessment in order to ensure a correct fatigue design and an appropriate in-service durability.
In such a context, three typical approaches for fatigue assessment of defective metals are available in the literature [
18], and more precisely:
- -
approach (a): the defects are treated as notches and the fatigue limit is calculated on the basis of the elastic stress concentration factor [
19,
20]. Note that, since the actual value of such a factor is difficult to be determined in presence of defects with an irregular shape, Mitchell [
19] proposed to consider the defect equivalent to a hemispherical pit;
- -
approach (b): the defects are considered as pre-existent cracks and the fatigue limit is determined by using the stress intensity factor concept [
18,
21]. The main drawback of this approach is related to the computation of the threshold stress intensity factor range, especially in presence of defects with a complex shape;
- -
approach (c): the defects are regarded as small cracks and the fatigue limit depends on the defect size [
15]. Such an approach is also known with the name of
-parameter model, proposed in the past by Murakami [
22] and Yanase et al. [
23].
According to the most relevant literature (see for instance the recent researches reported in Refs [
24,
25]), the
-parameter model is the most employed approach in practical applications since only two quantities are required for the fatigue limit determination. Such quantities are the Vickers hardness (representative material parameter) and the defect size (representative geometrical parameter for defects), whereas no information regarding the shape of defects is required. Moreover, the model proposed by Murakami [
22] and Yanase et al. [
23] has proved to be in agreement with the experimental evidences that the size of the biggest defect is the controlling parameter of material fatigue strength [
14].
Due to its unique features, the
-parameter model has been implemented in several multiaxial fatigue criteria [
1,
14,
24,
25,
26,
27,
28,
29,
30], providing satisfactory estimations in terms of fatigue strength of defective materials. For instance, Nadot and co-workers [
14,
26] analysed the accuracy of both the
-parameter model and the Crossland multiaxial fatigue criterion in estimating the fatigue strength of steel specimens containing small defects with different shapes; subsequently, the Authors proposed a modified version of the Crossland criterion for defective materials. The
-parameter model has been also implemented in the Dang Van fatigue criterion in order to simulate experimental tests performed on steel specimens with superficial and artificial spherical defects [
27]. Moreover, Endo and Ishimoto [
28] proposed a multiaxial fatigue criterion for the strength estimation of defective steels (with cylindrical defects) subjected to in-phase and out-of-phase cyclic loading. In particular, the fatigue limit under normal loading has been estimated by using the
-parameter model by Murakami [
22]. Recently, the research group of Araújo [
1,
25] has employed the
-parameter model in order to calibrate the material constants of some well-known fatigue criteria available in the literature and based on the so-called critical plane approach (i.e., Fatemi-Socie, Smith-Watson-Topper, Findley and Modified Wöhler Curve Method criteria).
By following a similar approach of that proposed in Ref. [
1], Vantadori and co-workers [
29,
30] have theoretically investigated the fatigue behaviour of defective metals (both a high-strength steel with non-metallic inclusions and a ductile cast iron with micro-shrinkage porosity) by employing a procedure based on the joined application of the
-parameter model [
22,
23] and the multiaxial fatigue criterion by Carpinteri et al. [
31,
32,
33,
34,
35,
36]. More precisely, the fatigue limits under both normal and shear loading (i.e., criterion parameters) have been calculated by using the
-parameter model, whereas the fatigue strength has been assessed by means of the Carpinteri et al. multiaxial fatigue criterion, based on the critical plane approach. It is important to point out that, since the defects were more than one in the analysed specimens, a defect content analysis has been performed through a statistical method deriving from the Extreme Value Theory (EVT) [
37,
38]. Moreover, such a statistical method has been also employed to determine an optimised return period for fatigue limit calculation.
It deserves to point out that the above defect content analysis can be easily performed by using machine learning techniques. As a matter of fact, the machine learning techniques have been recently employed to successfully predict the fatigue strength of both conventional and additive manufactured metals (see for instances Refs [
39,
40]).
By taking as starting point the procedure developed by Vantadori et al. [
29,
30], the goal of the present paper is to discuss the accuracy and reliability of such a procedure in simulating fatigue tests available in the literature and performed on three different Ductile Cast Irons (DCIs) containing solidification defects, in terms of micro-shrinkage porosity [
6,
41]. In particular, the fatigue strength of such DCIs subjected to both uniaxial (rotating bending or torsion) and biaxial (combined tension and torsion) cyclic loading is evaluated and compared with the experimental results.
The paper is structured as follows.
Section 2 is dedicated to the description of the fundamentals of the present procedure for defective materials. The experimental campaign available in the literature is described in
Section 3, whereas the defect content analysis together with the computation of the optimised return period are presented in
Section 4. The discussion of the obtained results is reported in
Section 5 and the main conclusions are summarised in
Section 6.
3. Experimental Fatigue Data Examined
The procedure for fatigue strength estimation of defective metals presented in
Section 2 is hereafter applied to a set of data available in the literature and related to infinite life fatigue tests [
6,
41]. In particular, both uniaxial (rotating bending or torsion) and biaxial (combined tension and torsion) fatigue tests are analysed.
The investigated materials were three DCIs, and more precisely:
- -
a ferritic DCI with 14% graphite nodules in a white ferrite matrix, identified as EN-GJS-400-18, according to the European designation;
- -
a ferritic/pearlitic DCI with a typical bulls-eye structure of 14% graphite nodules in a matrix of 46% ferrite and 40% dark pearlite, identified as EN-GJS-600-3 DCI;
- -
a pearlitic DCI with 13% graphite nodules in a matrix of 62% pearlite and 25% ferrite, identified as EN-GJS-700-2 DCI.
The chemical compositions of such DCIs are reported in
Table 1, whereas the mechanical properties are listed in
Table 2. Note that, the present values of the Vickers hardness, HV, were measured by considering a load of 98.1 N.
The small cylindrical smooth specimens, made by turning and milling from DCI large sections, were characterised by a diameter of the gauge section equal to 10 mm. The geometrical sizes of the specimens are reported in the original work by Endo and Yanase [
6].
A specimen surface finishing with an emery paper was performed and, subsequently, a surface layer of about 30 µm was removed by electro-polishing. Moreover, in order to avoid that the surface defects would grow further during the electro-polishing, specimen surface was refinished with alumina paste (up to a depth of about 10 µm) and, then, 1–2 µm of the surface layer were removed by a second electro-polishing.
Regarding uniaxial fatigue tests, a rotating bending testing machine of uniform moment type (operating speed equal to 57 Hz) was used, whereas torsional and biaxial tests were performed by means of an MTS servo-hydraulic testing machine with an operating speed ranging from 30 Hz to 45 Hz.
All the fatigue tests, performed under the condition of constant amplitude loading with a sinusoidal waveform, were characterised by a loading ratio, , equal to −1. The ratio of the applied shear stress amplitude to the applied normal stress amplitude, , was equal to: (rotating bending), (combined tension and torsion) and (torsion). The phase shift, , between tensile and torsional loading was either 0° or 90° for EN-GJS-400-18 and EN-GJS-700-2 DCIs, whereas only proportional biaxial loading was considered for EN-GJS-600-3 DCI. Note that, the run-out condition was assumed when a specimen survived more than cycles.
From the uniaxial fatigue data, the fully reversed normal fatigue limits were computed [
42,
43], and more precisely:
for EN-GJS-400-18 DCI,
for EN-GJS-600-3 DCI and
for EN-GJS-700-2 DCI. Regarding the fatigue limits under fully reversed shear loading, no data are available in the original research works [
6,
41,
42,
43].
Finally, as far as the rotating bending tests are concerned, one specimen of each analysed DCI was employed for the analysis of the defects. In particular, the fracture surface, normal to the specimen longitudinal axis, was examined by using a Scanning Electron Microscope (SEM), characterised by an inspection area,
, equal to 0.5 mm
2. It was observed that the largest micro-shrinkage porosity was usually the preferential site for fatigue crack nucleation. Therefore, the largest defect inside
was detected and the square root of such a defect area,
, was measured by considering the area of the circumscribed circle. Then, the above operation was repeated 50 times on each fracture surface of the three DCIs being examined. At the end of the above procedure, the values of
(with
) for each DCI were determined. Note that, as reported by the Authors in Refs [
6,
41], it was not possible to measure a micro-shrinkage porosity with a value of
smaller than 25 µm, due to the limits of the SEM.
Regarding torsional fatigue tests, measurements of the area of the largest micro-shrinkage porosity are not available in Refs [
6,
41].
4. Defect Content Analysis and Return Period Optimisation
By using the experimental data in terms of
[
6,
41], a defect content analysis (based on a statistical method deriving from the EVT [
37,
38]) together with an optimisation procedure of the return period are hereafter performed for each of the examined DCI. Note that, due to the lack of experimental data, the obtained results for normal uniaxial cyclic loading are extended to shear one. In other words, the value of
is assumed to be equal to
value, that is,
.
Firstly, in order to identify the defect statistical distribution, the experimental values of
are arranged in ascending order for each DCI analysed:
Then, for each DCI, a reduced variable,
, is calculated as follows:
Such a reduced variable is plotted against
values, obtaining the probability graph of the defect distribution for the three DCIs, as shown in
Figure 1. Independently of the analysed material, it can be noted that the probability graphs have a linear trend and linear regressions of the above defect distributions can be obtained (see the red dashed lines in
Figure 1). More precisely, such regressions are given by the following equations:
with:
where
is the return period related to normal uniaxial cyclic loading.
According to Equation (9),
value is a function of
value, which in turn depends on the volumes considered in the defect content analysis. More precisely, the probability of finding larger defects increases within the volume. The
value is, hence, defined as the ratio of two volumes, that is:
where
is the prediction volume related to the specimen useful cross-section and
is the standard inspection volume related to normal uniaxial cyclic loading. The volume
is calculated by assuming that the inspection area
is characterised by a given thickness,
, and more precisely:
with:
According to the experimental data reported in Refs [
6,
41], the value of
is equal to
µm for EN-GJS-400-18 DCI,
µm for EN-GJS-600-3 DCI and
µm for EN-GJS-700-2 DCI; consequently, for
equal to
mm
2 (as reported in
Section 3),
turns to be equal to
mm
3 for EN-GJS-400-18 DCI,
mm
3 for EN-GJS-600-3 DCI and
mm
3 for EN-GJS-700-2 DCI.
By following the optimisation procedure proposed in Refs [
29,
30], different values of the prediction volume,
, are hereafter employed in order to define an optimised return period,
, for the computation of
and, hence, of the fatigue limits
and
for the three DCIs analysed. The considered values of
together with the corresponding
values (Equation (11)) are listed in
Table 3,
Table 4 and
Table 5 for EN-GJS-400-18 DCI, EN-GJS-600-3 DCI and EN-GJS-700-2 DCI, respectively.
According to
Table 3,
Table 4 and
Table 5, the first four values of
are computed as a function of the volume of the specimen gauge section (length
mm and diameter
mm) [
6,
41], and more precisely:
- -
mm3 is the volume of the specimen gauge section for which the return period is ;
- -
mm3 is five times the volume of the specimen gauge section (i.e., ) for which the return period is ;
- -
mm3 is ten times the volume of the specimen gauge section (i.e., ) for which the return period is ;
- -
mm3 is fifty times the volume of the specimen gauge section (i.e., ) for which the return period is .
Moreover, also the typical volume of a crankshaft (that is,
mm
3) is considered and the corresponding return period is named
(see
Table 3,
Table 4 and
Table 5). Finally,
represents a fictitious prediction volume obtained by considering the experimental fatigue limit
of each DCI analysed (see the data reported in
Section 3). More precisely, by employing
instead of
in Equation (1), the corresponding
value is derived for each DCI and, then, the fictitious prediction volume
is obtained from Equation (9) together with Equation (10). Finally, the corresponding return period (see Equation (11)) is indicated as
in
Table 3,
Table 4 and
Table 5.
Once the
values are obtained, the
values are derived from Equation (9) for the three DCIs examined. Then, by assuming that
(that is,
), the fatigue limits
and
are computed by means of Equations (1) and (2), and the values are listed in
Table 3,
Table 4 and
Table 5. Finally, for each pair (
,
), the Carpinteri et al. multiaxial fatigue criterion (
Section 2.2) is applied and the mean value of the error index,
, is deduced through Equation (6) (see
Table 3,
Table 4 and
Table 5).
The
values vs. the
values are plotted in
Figure 2 for each DCI; it can be observed that the above points are well interpolated by logarithmic curves, obtaining the following relationships:
For each DCI, an optimised value of the return period,
, is obtained from the above equations by equating the mean value of the error index to zero (i.e.,
), as reported in
Table 3,
Table 4 and
Table 5. Then, by exploiting Equation (9),
is equal to:
µm for EN-GJS-400-18 DCI,
µm for EN-GJS-600-3 DCI and
µm for EN-GJS-700-2 DCI; the fatigue limits
and
at
loading cycles are, hence, computed for the three DCI being examined (see
Table 3,
Table 4 and
Table 5).
5. Discussion of the Results
By taking as starting point the fatigue limits obtained at the end of the optimisation procedure of
Section 4, the fatigue strength of the three DCIs containing solidification defects is evaluated by means of the Carpinteri et al. multiaxial fatigue criterion. In particular, the fatigue endurance condition of Equation (5) is represented by the
against
plot, reported in
Figure 3,
Figure 4 and
Figure 5. For each DCI, the ellipse with semi-axis equal to
and
(solid line in
Figure 3,
Figure 4 and
Figure 5) represents Equation (5); in the above Figures, the dashed and dot-dashed lines are the error bands equal to
and
, respectively. Moreover, the points
correspond to the analysed fatigue data, being the full and empty symbols, respectively, the experimental failures and run-outs. Note that, if the above points lie out of the ellipse, the present procedure allows to predict fatigue failures; on the other hand, run-out conditions are estimated when such points are inside the elliptical domain.
From
Figure 3,
Figure 4 and
Figure 5, it can be observed that the present procedure is in general able to correctly predict both the experimental failures and run-outs, independently of the loading condition (that is, uniaxial and biaxial loading). In particular, for EN-GJS-400-18 DCI (see
Figure 3), the following remarks can be made:
- -
for uniaxial fatigue tests (i.e., and ), the above procedure is able to correctly estimate the fatigue failures (being all the full symbols outside the ellipse), whereas the experimental run-out conditions are not capture, with the exception of one data relating to torsion. However, the empty symbols lie on or very close to the failure curve (within the error band equal to ), thus representing a condition of incipient failure;
- -
for proportional biaxial fatigue tests (i.e., with ), all the experimental failures and one of the two run-outs are perfectly predicted by means of the present procedure;
- -
for non-proportional biaxial fatigue tests (i.e., with ), all the empty symbols are inside the elliptical domain, in agreement with the experimental observations. On the other hand, two of the three experimental failures are not capture by means of the present procedure, falling the results into the error band equal to .
From
Figure 4 related to EN-GJS-600-3 DCI, it can be observed that:
- -
for rotating bending fatigue tests (i.e., ), the fatigue failure conditions are perfectly estimated, whereas incipient failure conditions (i.e., the corresponding points lie very close to the failure curve) are obtained, even if run-outs were observed;
- -
for torsion fatigue tests (i.e., ), the present procedure allows to correctly estimate the experimental run-out and one of the two experimental fatigue failure, while for the other it is possible to predict an incipient failure condition;
- -
for proportional biaxial fatigue tests (i.e., with ), experimental run-outs are correctly represent by the empty symbols falling inside the elliptical domain; regarding fatigue failures, only one is perfectly predicted by means of the present procedure, whereas incipient failure conditions are achieved for the other two.
As far as EN-GJS-700-2 DCI is concerned (
Figure 5), it can be pointed out that:
- -
for rotating bending fatigue tests (i.e., ), the full symbols falling outside the elliptical domain correctly represent experimental failures, whereas the empty symbols should be located inside the ellipse, since run-outs were experimentally observed. However, such points lie very close to the failure curve (inside the error band equal to ), and, consequently, the procedure allows to estimate conditions of incipient failure;
- -
for torsion fatigue tests (i.e., ), the experimental failure and run-out conditions are perfectly predicted by means of the present procedure;
- -
for proportional biaxial fatigue tests (i.e., with ), all the full symbols are outside the elliptical domain, in agreement with the experimental outcomes. An incipient failure instead of a run-out is predicted, since the empty symbol is outside the ellipse, but very close to it;
- -
for non-proportional biaxial fatigue tests (i.e., with ), the present procedure allows to correctly capture the run-outs but not the fatigue failures, even if the full symbols lie inside the error band equal to .
Finally, the effectiveness of the present procedure in estimating fatigue strength of defective DCIs can be evaluated from
Figure 6,
Figure 7 and
Figure 8. As a matter of fact, by considering only the experimental data related to fatigue failures, the results in terms of equivalent uniaxial stress amplitude,
, are plotted together with the fatigue limit
of each DCI (see
Table 3,
Table 4 and
Table 5), represented by a black solid line in the above Figures. Moreover, the experimental fatigue limits under fully reversed normal stress,
, (reported in
Section 3) are also depicted with red dashed lines in
Figure 6,
Figure 7 and
Figure 8 for the materials being analysed.
It can be observed that the use of
in the fatigue endurance condition of the present procedure (see Equation (5)) provides more conservative results with respect to those obtained when the experimental fatigue limit
is employed, and this holds true for all the three DCIs. More precisely, for EN-GJS-400-18 DCI (
Figure 6),
and
of the results are conservative when
and
are, respectively, used in the fatigue endurance condition. Regarding EN-GJS-600-3 DCI (
Figure 7),
and
of conservative estimations are achieved if
and
are, respectively, employed. Finally, for EN-GJS-700-2 DCI (
Figure 8), the present fatigue endurance condition in terms of
provides
of conservative results, whereas only
is obtained by employing
.