2.1 Specimen for the fatigue test

For the fatigue tests, unnotched specimens of EN-GJS-400-18-LT were used, which were taken from a main frame of a wind turbine. The position of the main frame where the specimens have been taken had a wall thickness of approx. t = 85 mm. In the test volume, the specimens have a diameter of d = 15 mm and a length of l = 25 mm (Figure 1). The highly stressed volume^{22, 23} of the specimen is HBV_90% = 6.122 mm³. It should be noted that HBV_90% is not only in the test volume but also extends into the transition radius. The chemical composition is shown in Table 1. Static tests showed a tensile strength R_m = 363.2 MPa, a yield strength R_p0.2 = 225 MPa, and a breaking strain of A₅ = 26.6%. Metallurgical investigations²¹ showed a nodularity of 85.3 and a fraction of ferrite/pearlite/graphite of 82.2/4.6/13.3. Additional information is given in another publication.²⁰

2.2 Fatigue test under constant amplitude loading

The strain controlled tests were carried out at room temperature on a servo-hydraulic test rig with test frequencies between f = 0.1 and 20 Hz and a strain ratio of R_ε =  − 1 in orientation to ISO 12106.²⁴ The first 10 load cycles were always performed at a frequency f = 0.1 Hz. Then, in order to reduce the test duration, the frequency was increased as a function of the load levels. The tests were run until crack initiation. The crack initiation was detected by a stiffness decrease of the specimen. Based on the measured strain and the force on the specimen, a decrease of stiffness of 20% related to the initial state was used to detect the crack initiation. During the tests, the forces resp. nominal stresses were continuously recorded with a rate of 200 data points per cycle.

The number of cycles to crack initiation N_i and the elastic–plastic material behavior were determined according to ISO 12106.²⁴ The stress–strain curve²⁵ in the description according to (1) and the strain-life curve^26-29 according to (2) are shown in Figure 2.

(1)

(2)

The parameters of the strain-life curve, ε_f′ (fatigue ductility coefficient), σ_f′ (fatigue strength coefficient), b (fatigue strength exponent), and c (fatigue ductility exponent), shown in Table 2, are determined with a linear regression in double logarithmic ε_el-N space and double logarithmic ε_pl-N space. The parameters of the stress–strain curve, K (strength coefficient), and n (strain hardening) are determined by the compatibility conditions³⁰:

(3)

(4)

The fatigue tests under constant amplitude loading show cyclic hardening, that is, an increase in the stress amplitude at a constant strain amplitude (Figure 3). This cyclic hardening is dependent on the strain level. At low strain levels, the material shows only a low cyclic hardening. The maximum cyclic hardening is achieved at a strain amplitude of about ε_a = 0.25%. Higher strain amplitudes show a lower relative cyclic hardening, probably induced by a starting crack initiation, which leads to a decrease of the stress amplitude (Figure 4). The strain hardening h is defined according to (5), where N_i represents the number of cycles to crack initiation.

(5)

The increase of the test frequency after 10 cycles leads to an increase of the stress amplitude. This is most obvious in tests with low strain amplitudes, where the frequency difference between the first 10 cycles and the following cycles is highest.

2.3 Fatigue test under variable amplitude loading

The fatigue tests with variable strain amplitude were performed on the same servo-hydraulic test rig as those with constant strain amplitudes. In order to use realistic loads in the investigations, a load spectrum derived from a wind turbine²¹ was used that corresponds to a life of 20 years and has a strain ratio of R_ε =  − 1.

In order to obtain a suitable load-time history for the fatigue tests, the load spectrum is reduced from L_S = 2.8 · 10⁸ cycles to L_S = 5 · 10⁵ cycles by means of factorization and omission (Figure 5). This is necessary because the load-time history should be run through several times in Gaßner tests.³¹

The Gaßner-curves show a partially significant influence of the overload on the fatigue life (Figure 7). This is highest at low strain amplitudes since the residual stresses due to the overload are not reduced due to the subsequent load cycles. The local yield stress is never exceeded. At high strain amplitudes, the influence of the overload decreases since a partly relaxation of the residual stresses occurs. This leads to different slopes of the Gaßner-curves and a coincidence of the curves at about N = 5 · 10⁵ cycles. This is caused by a reduction of the residual stress at high strain levels. In the Gaßner-curves, following number of tests could be considered in the evaluation: without overload: 6, overload at the beginning: 12, and overload after one transit: 10.

2.4 Changes in the local mean respectively residual stresses

All effects are addressed when loading with significant plastic strains and a stress ratio of R ≠  − 1. The significance of the individual influences depends on the load-time-history.

3.1 Applied fatigue assessment approaches

When the strain-life approach is applied in industrial application, no transient material behavior is generally used in the fatigue assessment. Effects such as cyclic hardening or cyclic mean stress relaxation are not considered. A cyclically stabilized stress–strain behavior over the entire component lifetime is usually assumed. As it can be seen in Figure 3, the EN-GJS-400-18-LT, however, shows a load-dependent hardening. A stabilization of the stresses at a constant strain level does not take place. The hardening takes place continuously until failure.

To investigate the influence of the transient material behavior on the damage sum, two different damage calculations are carried out for the fatigue tests with variable amplitude loading (Figure 9). Input data for both approaches are the recorded strain-time course from the fatigue tests and the strain-life curve from the constant amplitude fatigue tests. In the first variant, the fatigue assessment is considering the transient material behavior, and the recorded stress amplitudes have been used. Subsequently, all transient effects are covered. This variant is referred to as the experimental variant in the following. In the second variant, the fatigue assessment is carried out without consideration of transient material effects, and the stress amplitudes were numerically derived based on the cyclically stabilized material behavior. This approach is common and often used when elastic–plastic stresses have to be evaluated, and it is referred to as the numerical variant in the following.

Since the Wöhler-curve and the Gaßner-curves were derived with the same unnotched specimen with a low surface roughness (R_Z < 6.3 μm), no support effects or surface influences need to be taken into account. The HCM³³ cycle counting algorithm was used to identify closed hysteresis. The mean stress was considered using the damage parameter P_B³⁴ (6) that is closely related to P_SWT.³⁵ P_B was used because it is easy to apply and it has the ability to control the mean stress sensitivity.

(6)

The parameter k (7) was used to consider the desired mean stress sensitivity.⁶

(7)

Different recommendations for mean stress sensitivities can be found in literature. According to the FKM guideline,⁴ a mean stress sensitivity of M = 0.21 is calculated for the EN-GJS-400-18-LT. However, a mean stress sensitivity of M = 0.5 is determined in fatigue test for EN-GJS-400-18-LT³⁶ and is used in the present fatigue analysis.

Next to P_SWT and P_B, there exist numerous other damage parameters, such as P_J.⁶ However, these are more complex in their application and not widely used in industrial application.

In order to visualize the influence of the mean stress sensitivity within the P_B parameter, stress amplitudes were adjusted to derive constant P_B-values at different mean stresses. Figure 10 shows examples of stress amplitudes and mean stresses for four different P_B-values. It can be observed that the slope of the curves at high mean tensile stresses is flatter at low P_B-values than at high ones. This means that the mean stress sensitivity depends on the load level.

The linear damage accumulation according Palmgren³⁷ and Miner³⁸ was used to calculate the damage sum. The two fatigue assessments differ only with regard to the stress behavior that occurs due to the measured strain-time history in the fatigue test.

3.2 Derived stresses for fatigue assessments

In the experimental variant, the transient material behavior is taken into account in the fatigue assessment. For this purpose, the actual strain from the extensometer and the actual force were recorded during the Gaßner tests. Based on the force, the nominal stress was calculated. Thus, no model of the stress–strain behavior was necessary in the damage calculation since the real measured time-dependent stress–strain behavior was used. This includes cyclic hardening, cyclic mean stress relaxation, and creep.

In the numerical variant, the corresponding stress is calculated on the basis of the actual strain with the cyclically stabilized material behavior and the Masing-Memory model.^{32, 33} The Masing-Memory model considers the shift of residual stresses caused by exceeding the yield stress, but does not consider cyclic hardening, cyclic mean stress relaxation, or creep.

The two variants are based on the identical strain-time history; only the underlying stress history is different due to the material behavior. A comparison of the stress curves is shown in Figure 11. In the scenario without overload, a deviation of the maximum stress can be seen at the beginning of the test. This is due to the not yet cyclically hardened stress–strain behavior of the transient material behavior. In the further course the deviations of the stresses are small but constant.

The compressive overload lead to measured residual stresses of σ_res,exp = 260 MPa and calculated residual stresses of σ_res,num = 320 MPa, independent of the time at which the overload occurs. These residual stresses were reduced by the following loads: in the scenario with overload at the beginning, the calculated stresses are significantly higher than the stresses derived based on the transient material behavior (σ_m,exp = 68 MPa vs. σ_m,num = 103 MPa). In the scenario with overload after one transit, the calculated mean stresses are also higher than the measured ones (σ_m,exp = 49 MPa vs. σ_m,num = 67 MPa).

In the second variant with numerically derived stresses, the resulting residual stresses due to the overload should be independent of the time they occur. At the beginning, it should lead to identical residual stresses as after one transit, because every time the same cyclically stabilized stress–strain behavior is used. In Figure 11, however, it can be observed that the calculated mean stresses differ (σ_m,num,start = 103 MPa vs. σ_{m,num,transit} = 67 MPa). These differences are due to control inaccuracy in the fatigue test (it should be noted that in both variants the measured strains from the fatigue test were used for the fatigue assessments). In the scenario with the overload after one transit, an overshoot of the controlled strains occurred directly after the overload, which led to a reduction of the tensile residual stresses. This is shown in Figure 12. So a direct comparison between the residual stresses for both scenarios, overload at the beginning and overload after one transit, cannot be performed, even for nominal identical strain amplitude levels. However, for the comparison of the impact of the transient material behavior, it is not necessary that the mean stresses are exactly the same because the fatigue assessments are evaluated with regard to their damage sum and the damage sum takes the mean stress into account.

In Figure 12, a jump in the stress after the overload cycles can be observed. This is due to the fact that the overload cycle was run with a very low test frequency (f = 0.01 Hz). At this low frequency, creep effects are already visible, which can be seen in a reduction of the maximum stress. The following load cycles are performed with a higher frequency (f = 8 − 20 Hz) and show a clear increase of the maximum stress. The variation of the test frequency was performed to find a compromise between test duration and control accuracy. The test frequency was varied in a range where no significant influence was expected. However, Figure 3 and Figure 12 show an influence. Though there is no significant influence on the main findings, since the same test frequencies were used for all scenarios, they are again comparable.

3.3 Results of the fatigue assessments

In the damage accumulation, higher mean stresses lead to higher damage fractions due to the mean stress sensitivity of the material. In the calculations, the influence of the mean stresses on the fatigue damage are assessed in the damage parameter according to Bergmann and are weighted with the parameter k.

Since the mean stresses in the experimental variant are lower than in the numerical variant, the damage sums in the numerical variant are higher than those determined with transient material behavior (Figure 13). With increasing deviation of the mean stress between both variants, the deviation in the damage sums also increases. This is caused by a high mean stress sensitivity. Figure 14 shows that with increasing mean stress sensitivity, the damage sums of the different overload scenarios also deviate from each other. This indicates that a mean stress sensitivity of M = 0.3 represents the different overload scenarios best, because there the smallest deviations between the damage sums of the overload scenarios are archived. The damage sum is not strongly depending on the mean stress or the occurrence of an overload; all fatigue tests fail at almost the same damage sum.

The calculated damage sums are in both cases clearly below the theoretical damage sum of D = 1, even if the transient material behavior with the measured stress amplitudes and mean stresses from the fatigue tests are taken into account (Figure 13). A dependence of the load level on the damage sum could not be determined (Figure 14).

The derived damage sums are depending on the overload scenario and the mean stress sensitivity. In the numerical variant, the average damage sums lie between D = 0.2 (without overload) and D = 0.6 (overload at the beginning with a mean stress sensitivity of M = 0.5). In the transient variant, the average damage sum lies between D = 0.06 (without overload) and D = 0.2 (overload after one transit with a mean stress sensitivity of M = 0.5). In literature, allowable damage sums for nodular cast iron vary. Most recommendations^39-41 are between D = 0.2 and D = 0.5; however, recommendations⁴ are also given with D = 1.