3.3. Fatigue Tests
The fatigue tests are conducted under alternating tension-compression testing. This deduces a load stress ratio, lower load level to upper load level, of R
. Therefore, the mean stress is equal to zero. All fatigue tests are executed at a hydraulic testing machine with a testing frequency of 30 Hz until either specimen burst failure or run out at
10
load cycles. In line with the quasi-static tests, the fatigue tests at elevated temperature are performed using a heat chamber. The fatigue resistance in the finite life region is statistically evaluated by means of the ASTM E 739 standard [
41]. This methodology assumes a constant variance in the finite life region. Furthermore, the fatigue data in the run out region is statistically investigated based on the
methodology, see [
42]. This procedure approximates the probabilities of survival in the long-life region by a
function and is a proven methodology to statistically estimate the mean value and standard deviation of the fatigue strength and likewise minimum life, see [
43]. As proposed in [
44], the slope
of the run out region is assumed to be five times the slope
in the finite life region. This assumption is also verified in [
45,
46,
47]. All fatigue data has been normalized by the UTS at room temperature of EN AC-46200 -Pos #2, evaluated to 326 MPa, see
Table 3.
The tests are performed until a total number of
10
load cycles, because preliminary studies showed defect correlated specimen failure in the HCF region between
10
and
10
load cycles, see [
3,
48,
49]. All evaluated S/N-curves are displayed including the 10 and 90 % probability of survival scatter band. The statistical evaluation of scatter bands in the run out region is conducted by means of the
methodology [
42,
43]. The evaluated fatigue strength data for each sample position as well as the scatter band in the run out region is listed in
Table 4. In
Figure 5 the S/N-curve of EN AC-45500 with T6 heat treatment at Pos #1 at room and at elevated temperature is displayed. While the slope
is almost identical for RT and ET, the number of cycles
for the transition region rises with increased testing temperature from approximately
10
to about
10
load cycles. The investigated fatigue strength of EN AC-46200 with T5 heat treatment at Pos #1 states also a similar
at room temperature, see
Figure 6. On the other hand, the high-cycle fatigue strength at
10
load cycles
significantly decreases by about 21 % at 150
C. The investigation of the fatigue strength from EN AC-46200 with T6 heat treatment at Pos #2 assumes a slightly shallower S/N-curve at elevated temperature, represented by the slope in the finite life region
. Furthermore, the transition point
rises to
load cycles, with a decrease of about 7% in fatigue strength
, as seen in
Figure 7. Finally, EN AC-46200 with T6 heat treatment at Pos #3 shows a significant shallower slope in the finite life region
, while the evaluated fatigue resistance
decreases only by 2% at 150
C testing temperature, see
Figure 8. It must be pointed out that the depicted minor reduction of the fatigue strength in Pos #3 is within the scatter band of the S/N-curves.
To investigate the fracture-initiating defects, it is from utmost importance to analyze all tested specimens either by means of digital microscope as well as by SEM. The detected crack-initiating flaws are characterized by means of geometrical parameters such as the square root of the effective defect area. In line with the procedure proposed in [
20], the size of fatigue fracture-initiating defects is characterized by the square root of the projected area of the flaw, perpendicular to the load direction. This methodology is displayed in
Figure 9 using the example of a fracture-initiating heterogeneity at Pos #3. Furthermore, to characterize not only the fracture-initiating defects by means of fractography, selected specimens are investigated non-destructively with X-ray computed tomography (XCT). This methodology supports the holistic characterization of the defect population respectively its spatial extent and is further described in [
22,
48].
Figure 9 and
Figure 10 show a fracture surface with a crack origin at a micro pore. At room temperature, all tested EN AC-46200 specimens initiate from a such micro pores. On the other hand,
Figure 11 displays a different cause of failure. At ET, the stress intensity is enhanced in the defect-near area while the activation energy of slip-planes decreases due to the increased operating temperature. Therefore, specimens tested at a higher temperature activate a different failure mechanism.
Some specimens would reveal a crack initiation right at the surface along with crack propagation at large slip bands. Nevertheless, some investigated specimens revealed mixed defect mechanisms of micro pores and large slipping areas. At the latter ones, the crack initiates at an intrinsic inhomogeneity and possesses a stable crack growth as with increasing crack length, the stress intensity rises near the crack tip. If the stress intensity factor and the activation energy based on the thermal energy reach a certain threshold, the crack starts to slip over a slip-band area during one single load-cycle and therefore significantly increases the crack growth. As a result, the remaining fatigue strength is reduced compared to an arbitrary defect with a similar .
As the fractography results show, this failure mechanism occurs especially at EN AC-46200 T5 and EN AC-45500 T6, where a major part of the specimens at Pos #1 inherit a slip-band-induced failure. It must be noted that EN AC-45500 T6 Pos #1 even shows a slip-band-like failure mechanism also at room temperature. In
Figure 12, the different damage mechanism fractions of the corresponding positions and alloys are displayed.
As proposed by [
50] either Gumbel or GEV distributions are applicable for fatigue-initiating defects. The distribution parameters are evaluated by means of a maximum likelihood method, as presented in [
51].
The probability of occurrence of an arbitrary
, based on the cumulative density function of the GEV-fit for each sampling position is displayed in
Figure 13. The probability of occurrence of a
of 200
m is less than 10% for sampling positions #1 and #2, whereas at Pos #3 the probability of occurrence for the same equivalent diameter possesses a value of just above 97%.
The cumulative density function of the distribution is computed by means of Equation
2, using the equivalent circle diameter (
) from the most critical defects, whereas
is denoted as the shape
the scale and
the location parameter. The equivalent circle diameter
of one flaw can easily be derived by multiplying
with the factor
. The ratio of the equivalent circle diameter
to the maximum diameter
describes the shape
of the crack-initiating pores [
52], see Equation (
4). The shape factor
therefore ranges between zero and one, indicating the roundness of a crack-initiating pore. The lower
, the more complexly shaped is the defect. Thus, a circle-shaped flaw would possess a shape factor
1. The shape of representative defects with varying sphericity is presented in detail in [
53].
Both the evaluated distribution parameters as well as the mean shape of the defects are listed in
Table 5. The investigation of the different fracture-initiating defect sizes proposes a significant different micro pore size distribution at Pos #3, in line with the increased local sDAS. The mean shape
of the flaws in Pos #3 also reckon them to possess a more spherical shape.
3.4. Fatigue Assessment Model
To assess the fatigue strength of Al-Si alloys incorporating manufacturing process-based inhomogeneities, a defect size-based material model is set up. The main causes of failure, evaluated in both EN AC-46200 Pos #1 an EN AC-45500 Pos #1, are basically not based on micro pores. Therefore, the defect-based material model from Murakami [
19] is mainly set up for EN AC-46200 Pos #2 and Pos #3 as herein the main failure cause can be assigned to micro porosity. However, specimens with defect correlated crack initiation from Pos #1 of both alloys are also displayed in the adopted material model, see
Figure 14. A parameter set, containing the stress amplitude at
10
load cycles
of run-outs, the stress amplitude
and number of load cycles
N of tests in the finite life region, as well as the corresponding defect size
are required. To increase the applicable data a power-function like projection method for specimens failed in the finite life region is executed. This method is suitable to increase the data points in the run out region of
without significant falsification of the scatter band in the HCF region
, see [
21].
The original data from Murakami [
19] proposes a coefficient of
for the exponent of the defect size. The parameters
and
are material dependent constants. This approach provides reasonable defect-based material models for Al-Si alloys at room temperature. However, the exponent of the defect size can vary at elevated temperatures, as it represents the slope of the material model. Therefore, the coefficient
m is not further considered to be a constant, see Equation (
5).
This model is supplied with all parameter sets depending on their alloy, position, and testing temperature. Furthermore, the hardness of the corresponding positions at elevated temperature is estimated based on the experimentally evaluated yield strength, as discussed in
Section 3.2. Next, the parameters
and
as well as the slope
are estimated applying a non-linear solver, using the least square method. The evaluated parameters maintain a slope of
for specimens tested at room temperature, which agrees with the proposed constant of
in the original model, see Equation (
3). However, the data at elevated temperature leads to a change of the slope value. The least square method proposes a coefficient of
for specimens tested at an elevated temperature of 150
C. The defect-based material model for elevated temperatures with the 90 and 10% probability scatter bands is displayed in
Figure 14. The defect correlated fatigue strength is restricted by two major limits. On the one hand, the upper boundary is set by the fatigue strength of near defect-free material where the area of the crack-initiating inhomogeneities tends to zero. On the other hand, the lower boundary is determined by huge defects, such that the stress intensity factor along the internal crack meets the long crack threshold
, see Equation (
6).
Preliminary studies [
54,
55,
56] revealed that
rises in line with increasing testing temperature, until a critical temperature is met. This results from an increased plastic zone in front of the crack tip. The size of the monotonic plastic zone can be estimated by Irwin’s estimation, based on the YS of the material, see Equation (
7) [
57].
The relationship proposed by Irwin applies for a monotonic plastic zone with no crack closure occurring [
58]. To evaluate the cyclic plastic zone, incorporating crack closure effects,
K is superimposed by
, such that:
As the YS decreases with elevated testing temperature at an average of 10.7%, the size of the cyclic plastic zone
therefore rises by 25.4%, see Equation (
8). Hence, the plastic-induced crack closure effects significantly increase, in line with the extended plastic zone [
59,
60,
61], resulting in an elevated long crack threshold
at higher temperatures. As a result, defects with large spatial extent do not affect the fatigue strength at elevated temperatures as significantly as for room temperatures, see Equation (
6). The increased slope of the defect-based material model, represented by the coefficient
, thus is deduced by increased
at elevated temperatures.