Figure 3.13 compares the fracture surface topologies of four specimens with internal defects. Besides the failure in one crack plane (cf. Figure 3.13a), several parallel planes of stable crack growth (cf. Figure 3.13b) were observed. The shape of each respective crack growth could be described by a section of an ellipse.

As shown in Figure 3.13c, ragged fracture surfaces were observed in some cases. In these cases, an approximately elliptical area differing from the defect and residual fracture could be identified as well. This failure pattern occurred with branched defects mostly located in the clamping area of the tensile specimen. Generally, the area of stable crack growth could be represented in a good approximation by an ellipse (cf. Figures 3.12a and c and Figure 3.13a). Only in special cases (cf. Figure 3.13d) was stable crack growth limited to one location of the defect. In addition, a comparison of Figures 3.13a and d illustrates the dependence between the expansions of the residual fracture surface and the applied stress level. As expected, the residual fracture area increased with increasing loads.

**Figure 3.13:** Fracture topologies of: a) U20O20 ∆σmic = 241 MPa; b) U20O17 ∆σmic = 248 MPa; c) U22O6 ∆σmic = 459 MPa; d) U22M7 ∆σmic = 384 MPa

In Figure 3.14, the necking behaviors of both materials are compared. Both specimens had internal defects of similar size and were subjected to similar loading conditions. In the case of material 1.6220, a distinctive necking could be identified, as illustrated by the dashed reference lines. The specific residual fracture surfaces were honeycombed and showed pronounced shear lips. On the other hand, for material 1.6760, no distinctive necking could be observed, and the residual fracture surface was less indicative of ductile failure.

**Figure 3.14:** a) Necking for 1.6220 specimen U20O2 at ∆σ = 280 MPa; b) No pronounced necking for 1.6760 specimen U22M6 at ∆σ = 280 MPa

### **<IP_ADDRESS> Fatigue Strength**

Emphasis was put on the statements that could be derived directly from statistical analysis. A deeper understanding of the failure mechanisms and an estimation of the proportions of crack initiation and growth were based on numerical simulations in Section 4.1. For a basic understanding of the decisive correlations, only the regression curves were determined and shown in the following diagrams. These curves were determined according to the evaluation procedure of EN 1993-1-9:2010 [156] with ∆σ as the independent variable. The regression curves were highlighted in the ∆σ range and were extended to the range from 1 · 104 to 2 · 107 cycles (grey dotted lines) for a better comparability. If not specified, the evaluation was performed with a variable gradient k and is indicated within the diagrams. ∆σ50% corresponds to the mean value of the stress range at 2 · 106 cycles to failure. In addition, the standard deviation SD is given as an indicator of the scatter. The blanket exclusion of individual tests that exceed a limit number of cycles was initially dispensed within these fundamental analyses. Characteristic resistance curves were determined according to EN 1993-1-9:2010 [156].

Figure 3.15a shows the results of all experiments related to the nominal stress range ∆σnom. The data are presented separately for material and specimen type. The regression curve resulted from a common evaluation of all data points. The data showed a large scatter quantified by SD = 0.51. On that basis, no clear dependence on the investigated material could be identified. Type M specimens tended to show lower fatigue resistance. However, numerous outliers disturbed this correlation.

Figure 3.15b shows an evaluation of the experiments separated by material and as a function of the defect size Afail,mic. For simplification, the specimens are presented in clusters depending on the reduction of the cross section. The limit values are defined in Table 3.5. The icon size in Figure 3.15b illustrates the defect size associated with the clusters. For each cluster, a regression curve was determined independently from the material and is plotted in Figure 3.15b, including the descriptive parameters.

**Figure 3.15:** Test results of nominal stress SN data related to the material and a) the specimen type; b) the defect size (small icon = Afail,mic/Asec < 5 %; medium icon = Afail,mic/Asec 5 % - 10 %; large icon = Afail,mic/Asec > 10 %)

The loading is indicated by the nominal stress ranges. The hypothesis that larger defects result in lower fatigue resistance could be confirmed. Nevertheless, the data still scattered considerably (SD = 0.37−0.43), and the gradients k of the SN curves deviated significantly from each other (k = 2.3−4.1). A further subdivision into clusters with narrower boundaries did not lead to more distinctive results. Thus, the defect size is not the only influencing factor.

The influence of the reduction in the load-carrying cross section and the resulting increasing stresses were eliminated by the evaluation of the net stress range according to Eq. (3.1). A consideration based on ∆σmic was useful with regard to real components, since the casting defects that occurred in the test specimens attenuated large parts of the cross sections despite small defect dimensions. This procedure was also used in [10] with a similar justification. The test data were evaluated in this manner, separated by material and presented in Figure 3.16. For both materials, two separate populations could been identified. These populations correlated with the defect location, specifically with internal and surface defects. These differences were not limited to the stress level, but also applied to the gradients k of the SN curves.

**Figure 3.16:** Test results evaluated by ∆σmic for a) 1.6220 and b) 1.6760

This indicates either that different mechanisms dominate the damage process or that notch severities strongly deviate. The gradient k of the surface defects corresponds approximately to the Paris exponent m determined in Section 3.1.2. Given the explanations of [66], it becomes apparent that crack growth is the predominant failure case. A verification is carried out in Section 4.1. The failure of internal defects seems to be coupled with a crack initiation phase, causing the deviation in the gradients k. The scattering was reduced by both the consideration of ∆σmic and the evaluation according to the defect location.

It should be pointed out that the "internal" and "surface" defect locations do not refer to the cast components according to Figure 3.4a, but to the machined specimens in Figure 3.4b. Thus, both defect positions have a comparable origin, and the differences in their lifetimes must result from mechanical effects. Nevertheless, the surface defects were often caused by larger defects, which had ventilated vertically due to the horizontal casting position. In some cases, these showed up through gas channels with a narrow cross section. In Figure 3.16a, specimen U20M9 is marked as an outlier. According to the defect location, it was assigned to the internal defect group in the failure plane. However, the pore pierced the surface of the specimen at a plane approximately 20 mm away in the longitudinal axis of the specimen (Annex C). In this specific case, the shape of the fracture surface was similar to those of surface defects. Despite the effects of assigning it to the internal defects being minor (k = 5.6; ∆σ50% = 252 MPa and SD = 0.37), this outlier was not considered in the evaluation. In Figure 3.17a the results of the tests on both materials were evaluated collectively. This showed a material independence while the two populations (internal defects and surface defects) remained almost unaffected.

**Figure 3.17:** a) Combined evaluation of the test results for both materials separated by defect location; b) Statistical evaluation of the test results according to EN 1993-1-9:2010 [156]

The combined evaluation led to a larger scatter compared to the evaluation according to material 1.6220. Compared to the evaluation of material 1.6760, the SD was reduced due to the larger number of samples. Figure 3.17b shows the results of the complete statistical evaluation according to EN 1993-1-9:2010 [156]. Following this concept, endurance strength was defined at 5 · 106 cycles to failure. In addition to the previously determined regression curves with free gradients, the evaluation was carried out with a fixed gradient k. For the evaluations with fixed gradients, the 95 % survival probability is indicated in the diagram as the characteristic resistance. The gradients k were chosen with the following justifications: k ≈ 3 for surface defects, as crack growth is the failure mechanism that causes failure according to Section 3.1.2, which applies k = m. In addition, this corresponds to the gradient of most notch details in EN 1993- 1-9:2010 [156] and facilitates its application in a design concept. Internal defects were evaluated using k = 5.0, which is in line with EN 1993-1-9:2010 [156] and represents the behavior more realistic than k = 3.0 would do.

The significant influence of surface defects is illustrated in Figure 3.18 using the two representative specimens U20M4 and U22M5. In both cases, failure was caused by small surface defects despite larger internal cavities. The CT records are shown in the projection, the section of the maximum cross-sectional weakening and the section in the failure plane. The figure is supplemented by microscopic images of the fracture surfaces. The stress ranges of the test results including the data points marked by "failure" are given as ∆σmic. Whereas the nominal stress range in the plane with the maximum cross-sectional weakening determined by CT ist marked as "theoretic" and calculated by Eq. (3.2). In this way, a theoretically tolerable number of cycles was determined from the regression curve of the internal defects and is marked in the diagram.

$$
\Delta\sigma\_{\text{Amax,CT}} = \frac{\Delta\text{F}}{A\_{\text{sec}} - A\_{\text{max,CT}}}.\tag{3.2}
$$

**Figure 3.18:** Dominating effect of surface defects on service life despite larger internal defects

The same procedure was used for further specimens with comparable geometric properties. The dotted lines connected real failure states with states that were theoretically determined for the internal defects. The increasing factor of the number of load cycles was related to the actual failure state and is shown in the diagram. Despite the partly considerably larger cross-sectional weakening of the internal defects, the theoretically determined lifetimes of these internal defects were up to 29 times higher than the cycle numbers that actually occurred. This ratio depended on the load level. The fact that lower stresses occurred in some cases with internal defects resulted from the described differences between Amic and ACT. Endurance limit was not considered, when calculating the theoretically tolerable cycles to failure for the internal defects.

In Figure 3.19a, the test results are presented normalized to the tensile strength Rm and seperated according to material and defect location. Rm corresponds to the mean value of the experimental investigations presented in Table 3.1. The gradients k of the regression curves correspond to the evaluations of the individual materials according to Figure 3.16. For a better overview, the comparrison is limited to P50%, limited to the mean values for 2 · 106 cycles to failure. It became apparent that the higher-strength material 1.6760 could only activate a significantly lower proportion of its nominal tensile strength under fatigue loading. The exception was the range of less than 1 · 105 cycles to failure, as the yield strength of 1.6220 was exceeded in this case and specimen of 1.6760 led to higher lifetimes. Additionally, the characteristic resistance according to FKM-Rili:2012 [162] of the flawless base material is added to the chart. In the case, Rm corresponds to the minimum requirements according to the decisive technical delivery conditions. All tests of material 1.6220 led to higher resistance. For material 1.6760, the mean value of the surface defects was below this resistance. "Coupling" the fatigue resistance with the tensile strength is not considered effective if defects are present.

**Figure 3.19:** a) Test results related to tensile strength; b) Test results compared with literature data

Figure 3.19b classifies the test results in the context of the experimental investigations from literature presented in Section 2.3. The comparison was based on ∆σmic, corresponds to the representations of published values for faulty components and follows the methodology of the present discussion. If the specimens from the literature showed geometrical notches, the comparison was based on the maximum stress range instead of nominal or net stress ranges to consider stress gradients. This resulted in higher lifetimes due to the supporting effect (see Section 2.2.3). The literature data covered the entire range of results obtained in these experimental investigations. Some of the literature data showed shorter lifetimes. The results of the specimens with internal defects were of the same order of magnitude as the flawless base material in the literature data. The gradients of the regression curves of the internal defects were approximately the same. The difference between the internal and surface defects corresponds to the order of magnitude and ratios of [15], indicated in the diagram as (Vishnevsky 1966). The results of the present investigations are classified as representative to slightly above average. The material independence of fatigue strength determined here is underlined by the comparison, and the necessity of a separate evaluation of internal and surface defects becomes clear.

#### **<IP_ADDRESS> Effects of Defect Size**

According to Eq. (2.5), ∆K directly depends on the defect size in the order of √ a. As an infinite plane is considered in Eq. (2.5), effects caused by reduced cross sections are not decisive. In crack propagations according to Eq. (2.10), a change in crack size a, and thus in ∆K, has a disproportionate effect on the tolerable number of cycles due to the exponent m. These effects can be quantified according to Eq. (2.7) on three-dimensional problems via the root of the crack surface √ area. Using simplified approaches, the effects of these correlations on the experimental results were evaluated. On this basis, the scatter could be explained. In the first step, the investigations were limited to surface defects. Finally, an analogous evaluation of the internal defects was carried out.

Figure 3.20a shows that the detected size and shape of the surface defects varied significantly regardless of the material. If multiple surface defects or both internal and surface defects were present, only the largest surface defect (Afail,mic,surf) was considered in the evaluation. Minimum, maximum and mean values are marked in the diagram. The situation is illustrated by two representative fracture surfaces of a large and a small defect. Fracture mechanics-based SN curves were determined for an elliptical surface crack with a/c = 1 in a cylinder according to Annex B.1.3. The selected defect sizes were chosen in a way that the minimum and maximum defect sizes Afail,mic,surf were mapped separately by material. The lifetime was determined at three load levels with ten iterations each. The integration was carried out according to approach 3 from Figure 2.4. The applied loads were considered by ∆σmic. This eliminated the effects of a reduced cross section due to a larger initial defect, and the effects of the defect magnitude on the stress intensity factors were considered individually. The crack propagation parameters were set according to Table 3.2 no. (1) and no. (3).

**Figure 3.20:** a) Sizes of surface defects and representative fracture surfaces for small and large surface defects; b) Comparison of fracture mechanics-based SN curves with test results for surface defects depending on defect size

Figure 3.20b shows the results and compares them with the test data. The specimens are presented as two clusters of smaller and larger defects, respectively, and are considered separately by material. The results of the crack propagation analysis as well as the test data showed a dependence of crack size on service life. The gradients and magnitudes of the calculations showed good agreement with the test data. This indicates that crack growth is the dominant damage mechanism for surface defects under HCF. The deviations between the experiments and calculations are attributed to the simplified model of a semicircular crack. In the real specimens, the major part of the defect was placed in the center of the specimens and not eccentrically on the surface (see Figure 3.20b). For material 1.6220, deviations in the calculated service life showed a factor of two between the small and large defects. These differences increased for material 1.6760 to a factor of five due to the much smaller defect sizes. Again, it is pointed out that the effects of a reduced stress cross section were eliminated and would amplify these differences. The √ area law according to [66] cannot be applied, as its validity is limited to surface defects where √ area < 1000 µm.

For the investigation of the internal defects, a circular concentric crack in a cylinder was considered according to Annex B.1.4. The explanations are limited to material 1.6220, as the investigations on 1.6760 led to similar results due to the comparable defect sizes and crack propagation parameters. Figure 3.21a shows the sizes of the internal defects. Fracture mechanics-based SN curves were calculated for three defect sizes, as given in Figure 3.21b. The experimental data are also presented in three size clusters, and the regression curve is added. The effects of the different defect sizes, which can be seen in the presentation of the test data, were confirmed by this calculation. Nevertheless, the

results of the crack propagation and test data differed significantly in both the gradient and order of magnitude. Thus, in the case of internal defects, a dominant share of the total service life must be attributed to crack initiation in addition to the considered crack growth. It is assumed that the extent of the crack initiation, in the HCF range, depends on the load level.

**Figure 3.21:** a) Sizes of internal defects; b) Comparison of fracture mechanics-based SN curves with the test results

### **<IP_ADDRESS> Effects of Defect Detection Methods**

Differences in the defect sizes between CT and microscopy images increase by up to 12 %, as shown in Figure 3.9. The influence of these deviations on stress range and fatigue strength was evaluated. This investigation was limited to failure cases I and II due to the necessity for available CT data of the defect in the failure plane. Only internal defects were considered. As shown in Figure 3.22a, the stress ranges ∆σmic resulting from the fracture surface analysis were on average 5 % higher than

$$
\Delta\sigma\_{\text{Afail,CT}} = \frac{\Delta\text{F}}{\text{A}\_{\text{sec}} - \text{A}\_{\text{fail,CT}}}.\tag{3.3}
$$

An analysis of the load cycle shown in Figure 3.22b indicated that the variance of 5 % on ∆σ causes a deviation in lifetime of approximately 23 %. This consisted only of the failure cases I and II specimens and was not sufficiently statistically verified. More important was the fact that statements can only be made on a phenomenological level and not a mechanical basis for the examined geometric conditions. Thus, all further evaluations were based on the defect area measured by microscopy.

**Figure 3.22:** a) Deviations between the stress ranges determined by CT data and microscope images; b) Evaluation of the service life according to CT data and microscope images

In later recommendations for the definition of a grading, the projected area of the defect is conservatively considered. When linking defect sizes to NDT displays, this gains special importance but is not within the scope of this work.

## **<IP_ADDRESS> Residual Fracture**

The purpose of these investigations was to describe residual fracture behavior. In particular, for applications at low temperatures (e.g., bridges) this has a significant influence on the overall safety level and robustness of a structure. A failure behavior that is indicated by large deformations is preferred to a sudden failure. In Figure 3.23,

$$\text{Net section max stress at failure} = \sigma\_{\text{res}} = \frac{\mathbf{F}\_{\text{o}}}{\mathbf{A}\_{\text{res}}} \tag{3.4}$$

is related to the material's yield and tensile strength mean values according to investigations pressented in Table 3.1. The results are presented separately by steel grade and are restricted to specimens with internal defects. This restriction resulted from the explanations of the fracture surfaces, since it was not possible to make a clear distinction between residual fractures and stable crack growth in surface defects.

Figure 3.23a shows that σres is 10 % to 90 % higher than the material's mean yield ReH strength for all specimens of material 1.6220. Therefore, the failure mechanism was ductile and strongly dominated by inelastic effects. In many specimens, σres even exceeded the mean tensile strength Rm. In those specimens where this was not the case, the growing crack reached the surface and additional bending occurred that was not captured by this simple approach. Due to the present stress state, σres could exceed the material uniaxial tensile strength and [33] confirmed that the maximum applicable load of tensile specimens with notches was higher compared to standard tensile tests of the same cross sections. In tensile specimen sharper notches cause higher fracture loads at lower elongations. The influence on the fracture strain is assigned to stress triaxiality (Tσ = σH/σeq) and was quantified in [84]. Newer research as [7] identified a discontinuity in this relation at Tσ ≈ 0.4 and further influencing factors that are neglected in this context. A numerical calculation on a circular internal crack indicated Tσ ≈ 0.95 at the crack tip. According to [7, 84], this value is associated with low elongation at the fracture and thus higher loads.

For the specimens of material 1.6760, σres was between 60 % and 80 % of the yield strength, except for three outliers, and did not exceed the tensile strength in any case. It is assumed that the failure mechanism is dominated by crack tip stresses. This finding coincides with the results of the fracture surface analyses. In the specimens of material 1.6220, necking occurred and pronounced shear lips were observed (cf. Figure 3.14). These effects were considerably less pronounced or not recognizable for specimens of material 1.6760. For a more detailed analysis, a circular crack of area Acrack was assumed and placed centrally in a cylinder with the same diameter as the considered specimen. The appearing stress intensity factors were determined according to Annex B.1.4. The analysis of the failure state was carried out by the FAD as shown in Figure 3.24a. The fracture toughness KIc was selected according to Table 3.3, with KIc = 7747 N/mm−3/2 for 1.6220 and KIc = 4206 N/mm−3/2 for 1.6760. All effects resulting from an eccentric defect position, asymmetrical crack growth and deviations from the circular crack shape were not taken into account in these simplified considerations.

**Figure 3.23:** Stresses in fracture states determined in net sections and related to yield and tensile strength for material a) 1.6220 and b) 1.6760

Figure 3.24a illustrates the assumed differences between the fracture states of both materials. The conditions of the 1.6220 specimens were clearly attributed to the plastic limit state, as they exceeded the right border of the limit curve. States outside the limit curve can occur due to local multiaxial stress states. The determined failure states of the 1.6760 specimen were within the no-failure area of the FAD. Thus, plastic failure was ruled out. One explanation for this is the influencing factors that reduce fracture toughness KIc. As described in [23], a three-dimensional stress state causes a reduction in KIc. This occurred inside thick components or at the crack tips of the internal cracks considered within this context. The qualitative dependency is shown in Figure 3.24b. Although the thickness effect is eliminated by the specimen shape for the determination of KIc, the resulting stress state is fundamentally different from that present at the crack tip under consideration. In addition, a significant influence on the fracture resistance of KIc is attributed to the temperature and loading speed in [23]. Since the tests were carried out at about 20 °C, temperature effects are negligible at this point. From the tests in [94], differences between static and dynamic KIc values at a factor in the range of factor 2.0 (S355) to 2.3 (high-strength steel) were derived. According to [23], the dynamic minimum value occurs for steel at about dKI/dt > 105 Nmm−3/2/s. Based on the previous simplifications of a circular crack lying centrically in a shaft, dKI was calculated in the range of 800 to 1985 Nmm−3/2/cycle. At a testing frequency of f = 70 Hz, dKI/dt was in the range of 0.6 · 105 to 1.4 · 105 Nmm−3/2/s. The crack resistance present in the specimen must approach its minimum value. The effects of the test frequency on the experimentally determined number of load cycles are considered small, since crack growth in this range increases strongly and over proportionally and very few cycles occur. If stress state and loading speed are taken into account with a reduction factor of 2.8 on KIc, this results in the modified states shown in Figure 3.24b. A clear distinction can be made between the ductile and brittle failure states of the two materials. This corresponds to the results for monotonic loading from [103].

**Figure 3.24:** a) FAD for tensile specimens in fracture state; b) Influencing factors of KIc see [23]

## **3.2.3 Summary**

These experimental investigations were carried out on tensile specimens from the materials G20Mn5 (1.6220) and G22NiMoCr5-6 (1.6760) with solidification shrinkage cavities. The investigations aimed at a description of casting defect characteristics and the quantification of fatigue resistance dependent on the defect properties. Finally, conclusions were drawn on the relevant damage processes as well as on the residual fracture behavior. The core statements are summarized by:

*Geometric Characteristics of the Casting Defects:*

## *Fracture Surfaces:*

## *Influence of Casting Defects on Fatigue Resistance:*

• The location of the defect (internal or surface defect) had a decisive influence on the fatigue resistance. On average, the resistance ∆σ50% at the reference value 2 · 106 load cycles of the internal defects ∆σ50%,internal = 268 MPa (with resp. to the net section) was 1.9 times higher than that of the surface defects ∆σ50%,surface = 139 MPa. Distinct differences in the gradients of the regression lines were present. For internal defects k ≈ 6 whereas for surface defects it resulted in k ≈ 3 and let to the assumption of different dominating damage mechanisms.

## **3.3 Component Tests with Artificial Defects**

The three main objectives of the component tests were to:

With these intentions, components of different wall thicknesses and representative geometric notches were tested under cyclic tension. The influence of the interaction of geometrically caused stress concentrations and casting defects was examined by adding representative artificial defects to the roots of the notches. Within this chapter, only the experimental investigations and directly derived results are discussed. An extended numerical evaluation is presented in Section 4.2. This presentation follows the same structure as the description of the tensile fatigue specimen in Section 3.2, meaning that the methodology and results are discussed separately.

## **3.3.1 Methodology**

## **<IP_ADDRESS> Specimens**

Figure 3.25 gives an overview of the investigated specimens. Three wall thicknesses were chosen, and the specimens were cast by three different foundries. These investigations were limited to material 1.6220. The chemical composition and mechanical properties according to inspection certificates 3.1 are summarized in Annex A.

**Figure 3.25:** Geometry of the component test specimens

The specimens were identified using the following concept:

$$\begin{array}{l} \text{G-(size)-(foundry)-(n)}\\ \text{G} & \text{identifier for component tests} \\ \text{(size)} & \text{K}=\text{small}; \text{M}=\text{medium}; \text{G}=\text{large} \\ \text{(foundry)} & \text{A, B, C according to Annnex A} \\ \text{(n)} & \text{consecutive number} \end{array}$$

## *Geometric Notch:*

The notch parameters were chosen based on Finite Element Analysis (FEA) in a way that the resulting SCF was approximately 2.0. This definition was based on different published examples of executed cast components and further engineering requirements for an adequate fatigue design. In [112], SCFs are presented in a range from 1.2 to 4.7, and 2.7 is not exceeded in the majority of cases. These examples resulted from nodes typical in offshore structures in the 1950s. Using state-of-the-art Computer-Aided Design (CAD) and FEM-supported design, SCFs lower than 2.0 shall be aimed at in fatigue-loaded components nowadays. [35] came to a similar conclusion that the SCFs of optimized cast steel nodes are almost exclusively lower than 2.0 and supported this statement with published examples.

The geometric proportion of the notches as well as their relation to the wall thickness were kept constant for all specimen sizes, as shown in Figure 3.25. Doing so, identical stress distributions, gradients and thus supporting effects were achieved independent of component size. This condition is illustrated in Figure 3.26a. It shows identical axial stresses in the center of the specimens that were related to ∆σnom along an evaluation path connecting both notches. Additionally, the stress distribution on the component surface is shown. Compared to the center, the stresses as well as the magnification were lower due to the multiaxial stress state.

**Figure 3.26:** a) Comparison of the normal stress distributions depending on the component type; b) Manufacturing sequence of the geometric notch

Since the geometric notch is representative of cast components in terms of its effect but not its actual dimensions, it could not be cast directly. This is due to heating of the molding material in the corner areas and considerable deviations from the nominal geometry in the notch area. In order to avoid this but at the same time achieve the typical surface condition of a cast component in the notch area, the production sequence was adjusted. The raw casting contour (see Figure 3.26b) coarsely reproduced the geometric shape. After normalization, mechanical processing was carried out in which the final notch shape was milled. The subsequent hardening and tempering created the required conditions in the notch area. Subsequently, quality class 1 according to EN 12680-1:2003 [144] was verified in the notch area. The design tolerance class DCTG 12 according to ISO 8062-3:2008 [157] was defined. Finally, the artificial defects were symmetrically induced as semi-elliptical surface defects in both notches of each specimen.

## *Artificial Defects:*

The defects were induced by die-sinking erosion perpendicular to the principal stress. The size and shape of the artificial defect was derived to represent the largest and most unfavorable crack that can be overlooked by NDT. Caused by thickness of electrode used in the die-sinking erosion process, the defects had a lateral extension of about 1.1 mm.

According to FKM-Rili-BM:2009 [161], near-surface defects must be reproduced conservatively by surface defects due to an interaction with the free surface and thus increased crack tip stresses. This justified a translation of the UT requirements for the rim zone to artificial surface defects (definition of core and rim zone acc. to [144]). According to EN 12680-1:2003 [144] (UT), the maximum permissible diameter of an equivalent Flat-Bottom Hole (FBH) is limited to 3 mm both in the core and in the rim zone for quality grade 1. FKM-Rili-BM:2009 [161] provides an approximation method for converting the equivalent FBH to a worst-case crack. The dimensions of this semi-elliptical surface crack are in the ratio a/c = 0.4, where the crack depth a is determined according to Eq. (3.5). This resulted in a crack size of a = 2.7 mm and c = 6.7 mm.

$$\mathbf{a} = \frac{2\mathbf{FBH}}{\sqrt{5}}\tag{3.5}$$

According to EN 1369:2013 [148] (MT), the dimension of the smallest MT-detectable defect depends on the surface condition. In this case, the comparative surface pattern according to BNIF or SCRATA is weighed. According to EN 1370:2012 [149], the sand molding process is assigned to SCRATA class A2/A3 for medium and small castings. Thus, the dimension of the smallest recognizable indication is 2 mm. If the component is ground, a surface condition of 1S2 (BNIF) and thus a detection limit of 0.3 mm could be assumed according to EN 1370:2012 [149]. In summary, the crack defined by UT became decisive.

## **<IP_ADDRESS> Measuring Concept**

In addition to the recording of common parameters, such as nominal stress range respectively load range and cycles to failure, all specimens were equipped with additional measuring techniques. Both conventional strain gauges and Electromagnetic Acoustic Transducers (EMATs) were used to get additional information on the actual stress state and the crack initiation and propagation behavior.

## *Strain Gauges:*

Two uniaxial strain gauges and a rosette strain gauge were arranged as shown in Figure 3.27a on both sides of each specimen. This enabled the stress state shown and aimed for in Figure 3.26a to be checked and enabled any deviations to be quantified. Besides, additional bending moments resulting from geometric deviations or eccentric clamping in the testing machine could be mapped. Due to the high stress gradients in the notch area, the evaluation was carried out as an average value over the respective measurement width. Recording over the entire test duration made it possible to map not only the initial stress situation but also the crack propagation, load redistributions and the description of which initial defect dominates. The definition of the relevant geometric parameters and coordinate specifications are listed in the introductory remarks of Annex D. The types and exact positions of the strain gauges as well as the development of the strain ranges over the test period are documented in Annex D for each specimen separately.

**Figure 3.27:** a) Arrangement of strain gauges and EMATs; b) Identification of representative regions on the fracture surface

## *Electromagnetic Acoustic Transducers (EMATs):*

The aim was to track the ratio of the crack initiation phase to the total service life and the crack propagation velocities. For this purpose, a measurement method developed by Fraunhofer IZFP based on the propagation time measurement of electromagnetically induced surface waves[68, 108] was applied. Fraunhofer IZFP carried out this conception and evaluation in the context of ErStaGu [116]. The principal idea is based on the time of flight of surface waves. These are excited by the transmitter by an electromagnetic field in the component and have a direction. The time of flight and the intensity of the waves until they reach the receiver are recorded. If a crack propagates in the observed area, the propagation path is extended, and a change in the propagation time and amplitude can be measured. The measuring rate was chosen in accordance with the test frequency so that approximately 5 to 10 measurements were made per load cycle. In this specific case, the transmitter and receiver were arranged around the notch area as shown in Figure 3.27a. In select cases, two units were used, and thus the behavior of both notch areas resp. both artificial defects was observed. As these data brought only limited benefit to the overall objective, the results are not considered within this context.

## **<IP_ADDRESS> Fatigue Testing**

The force-controlled fatigue tests were carried out with a stress ratio of R = 0.1 under cyclic tension. Depending on the specimen size, calibrated servo-hydraulic testing machines at the KIT Research Center for Steel, Timber and Masonry of different load ranges were used. These are summarized in Table 3.6, including the corresponding test frequencies. The considered nominal stresses ∆σnom relate the applied force range to the individually measured cross sections in the notch areas; the defect area was deduced. In specimen GKC3, beach marks were induced according to the procedure described in Section <IP_ADDRESS>.

**Table 3.6:** Details of the component tests

## **<IP_ADDRESS> Fracture Surface Analysis**

A fracture surface analysis was carried out for all specimens and is documented in Annex D. Figure 3.27b shows the key areas that were identified. Special focus was put on the quantification of the residual fracture as well as the non-uniformity of the crack growth.

## **3.3.2 Results**

### **<IP_ADDRESS> Fatigue Strength**

Figure 3.28 shows the nominal stress range ∆σnom separately for each component size. As major differences were observed between the specimens of foundry A and those of foundries B and C, these were evaluated separately. The mean value ∆σ50% was calculated with no restrictions to the gradient k according to EN 1993-1-9:2010 [156] at 2 · 106 cycles and is presented along with k in the diagrams.

**Figure 3.28:** Tests results for a) small, b) medium and c) large specimens

Both populations showed considerable differences in service life independent of component size. This was particularly pronounced at low load levels and resulted in large deviations in the gradients of the regression curves. The comparison of a population over the different component sizes showed an almost identical load level. ∆σ50% of foundries B and C spread between 87 MPa and 91 MPa, while for foundry A, this dispersion was greater. The reason for this scatter in the population itself is not considered a primarily physical phenomenon but is rather due to the small sample size and missing data points due to outliers in the low load level. The gradients of the regression curves behaved analogously. In the specimens from foundries B and C, this was in the range k = 2.8 to 3.1 and thus corresponds very well with the Paris exponent m = 3.0 determined in Section 3.1.2. The connection established in Section 3.2.2 with the effect of surface defects as crack-like defects is underlined.

Residual stresses resulting from a missing or ineffective stress relief heat treatment in foundry A were assumed to be the reason for the differences between the populations. Their determination and evaluation according to ASTM E837:2013 [134] is documented in Annex F. The measurements carried out on large components showed major differences. Significant compressive residual stresses in the relevant areas were identified for the specimens from foundry A. In addition, differences in the manufacturing processes

and thus different metallurgical compositions in non-regulated elements could increase the effects of residual stresses.

Due to the limited number of specimens, component GKA1 was tested after reaching 1 · 107 load cycles without any detectable crack (∆σnom = 85 MPa) at an increased load level (∆σnom = 220 MPa) until failure. In the case of the large specimens, GGB2 had to be excluded from the evaluation based on ∆σnom, since the origin of failure was a grinding groove on an edge, which as shown in Section 4.2.2 showed additional stresses due to large deviations from straightness.

Figure 3.29a shows all component tests collectively. Based onEN 1993-1-9:2010[156] and the correlation with the Paris exponent, the statistical evaluation of the regression curves for the specimens from foundries B and C was carried out with a predefined gradient k = 3.0. Despite the same defect size, all curves are almost identical for different component cross sections. Even the collective evaluation shown in Figure 3.29b did not increase the scatter so significantly that the physical processes taking place must be defined. Figure 3.29b also shows a comparison with the results of the tensile specimens with real casting defects from Section 3.2. The influence of the notch effect became apparent in this context. Compared to the tensile specimens with surface defects, the results of the component tests showed an offset to lower strength with a comparable overall behavior. The effects of the SCF on fatigue behavior were determined after the evaluation of the strain gauges and the numerical determination of the actual stress states in Section 4.2.

**Figure 3.29:** a) Common presentation of all component tests and evaluations with k = 3.0 and a predefined Nendu = 5 · 106 cycles according [156]; b) Comparison of the test results with the tensile specimens from Section 3.2

## **<IP_ADDRESS> Beach Marks**

Figure 3.30a shows the fracture surface of the beach mark test GKC3 with ∆σnom = 150 MPa. Up to the first marking, the specimen was loaded with 100,000 cycles. Between all other lines, 50,000 cycles were applied. The dominance of the crack in the lower part of the picture is clearly visible at the fracture surface. This resulted from geometrical deviations or the clamping in the testing machine. It was examined more closely, as shown in the microscope image in Figure 3.30b. All markers were clearly visible and were measured for further evaluation. Due to the lateral extension of the artificial defect, there were two competing stress hotspots per defect at each of the two interfaces in the specimen. In the present example, the crack started from the left side of the upper edge and from the right side of the lower edge and only united later to form a common crack front. This could be recognized by the diagonally running fragment on the fracture surface and explained by the crack front, which was not perfectly elliptical. In addition, it could be seen from the course of the residual fracture surface that an additional moment led to higher stresses on the left side of the specimen.

**Figure 3.30:** Evaluation of the beach mark test GKC3: a) Full scale; b) Detailed view

The figure illustrates that the predominant part of the service life is attributable to short crack states. Crack growth up to a doubling of the initial crack width from c = 6.7 mm to c = 13.4 mm was considered. This corresponds to half of the minimum distance that must be maintained between two adjacent defects according to Section 5.2.6 and FKM-Rili-BM:2009 [161]. As marked in Figure 3.30b, this condition lies between the fourth and fifth marking lines. In the specific case, 250,000/348,279 ≈ 70 % to 300,000/348,279 ≈ 85 % of the total lifetime was passed until this state was reached.

A closer look at the shape of the crack front reveals a uniform expansion along the crack contour at the beginning. The interaction of the stress distribution caused by the notch with its maximum at the component surface and the additional moment led to comparable crack tip stresses along the crack contour. In the case of a purely semi-elliptical surface crack on a flat plate under tensile stress, a stress hotspot is present in the apex area. With increasing crack size in these real components, this behavior changed and the increments at the surface exceeded those in the apex area by a factor of approximately 1.5. The reason for this was the increase in additional moments resulting from a displacement of the surface center of gravity and the interaction with the component's edge.

#### **<IP_ADDRESS> Evaluation of the Strain Measurement**

Figure 3.31a shows a typical strain history over the test duration. The labels of the strain gauges are defined in Figure D.1. Some of the curves shown in detail in Annex D contain interferences. A few measurements had to be aborted early due to technical problems. However, the basic statements could be derived in most cases and were not influenced by this evaluation. Figure 3.31a clearly and quantifiably shows the biaxial bending stress that is assumed when considering the residual fracture surfaces. In the present example, the "south" defect experienced higher stresses from the beginning. In addition, a small difference between "east" and "west" can be seen. A recalculation of these moments to substitute loadings and local stresses was done with the help of a numerical model given in Section 4.2. A straightforward approach to the calculation was not possible due to the superimposed notch effect. In addition to the strain ranges, Figure 3.31b considers changes with respect to the initial value.

In both diagrams, the qualitative crack propagation could be derived from the sequence of the failing strain gauges. The strains of the more-stressed crack increased continuously until failure. On the side of the less-stressed crack, the strains decreased continuously until just before failure. As soon as the strains of the side facing the crack started to grow disproportionately, a disproportionate decrease occurred on the side of the

**Figure 3.31:** a) Strain range history of component tests; b) Definition of stain ranges from the initial values

less-stressed crack. The strain gauges located in the center, labeled "east" and "west", did not experience this decrease and remained mostly unaffected by crack growth. In [121], the strain states were assigned to crack sizes by means of numerical crack propagation calculations. The share of large-area crack growth in the total service life can be evaluated based on the strain data. In the evaluation of the beach marks, the decisive part of the service life could be attributed to the area until twice the initial crack size was reached.

For simplicity's sake, ∆N was defined as the cycles between the first strain gauge tear-off and component failure. This was approximately the range between mark five and failure in the beach mark test shown in Figure 3.30. Figure 3.32a relates ∆N to the total fatigue life N determined in the test. This is a measure of the proportion of long-range crack growth that is clearly visible from the outside. The results are shown separately according to component size and foundry, analogous to the previous illustrations. The marked outlier is specimen GGB2. Here, the crack originated at the edge of the component and can therefore not be compared with the other components. In all other components, ∆N accounted for less than 12 % of the total service life. The proportions decrease as the load is applied (i.e., the number of cycles increases). However, this only occurs to a very small extent. A much more pronounced trend can be seen between the different component sizes. The proportion is considerably higher for the small samples than for the large ones. The effects are examined in more detail in Section 4.2.2 in connection with the crack tip stresses that depend on the component size. The size dependence is disturbed for the "large" components from

**Figure 3.32:** a) Proportion of cycles between the first strain gauge tear-off and failure, expressed as a proportion of the total life; b) FO related to the residual fracture surface

foundry B. As shown in Section 4.2.2, additional moments from geometric deviations acting perpendicular to the other cases were present here. The lower level of the samples from foundry A underlines the assumption of a delayed crack initiation (e.g., due to residual stresses).

## **<IP_ADDRESS> Residual Fracture**

Similar to the evaluation of the tensile specimens in Figure 3.32b, the upper loads are related to the residual cross section and are compared to the nominal yield strength or tensile strength in Figure 3.32b. As observed and explainedin Section <IP_ADDRESS> formaterial 1.6220, the stresses in the residual section also exceed both the yield strength and the tensile strength. The fracture surfaces show a honeycomb-like structure and reinforce the ductile residual fracture behavior. The present magnitude is considerably larger than that determined for the tensile specimen. A differently pronounced multiaxial stress state at higher stress triaxialities is assumed as the underlying cause.

## **3.3.3 Summary**

These experimental investigations were performed on specimens with wall thicknesses between 35 mm and 100 mm made of the material G20Mn5 and conditions representative for real components. The investigations aimed at an integral consideration of the technological thickness effect as well as statements on the interaction between geometrically induced stress increases (notches) and casting defects. In addition, these investigations served to validate the design concept from Chapter 5. The specimens were designed with two symmetrically located notches (SCF ≈ 2.0) and artificial semielliptical surface defects were introduced into each notch root. The dimensions of the artificial defects were defined independently of the component dimensions in the size of the UT acceptance limit.

Due to deviations from straightness, additional bending moments acted in the test cross section. As a result, one of the two artificial defects was subjected to higher stresses, resulting in asymmetric crack growth. The stress conditions were mapped by strain measurements and the strain changes were evaluated over the test duration. This confirmed that only a small proportion of the total service life is due to large-scale crack growth. The regression curves of the component test showed a gradient k ≈ 3.0 comparable with the SN curves of the surface defects from Section 3.2. The fatigue resistance related to the nominal stress range at the reference value 2 · 106 cycles was ∆σ50% = 86MPa. The reduction compared to the results from Section 3.2 was attributed to the local stress increase caused by the geometric notches. The influence of component thickness on fatigue resistance was not significant. Residual fracture behavior was classified as ductile and was consistent with the tensile specimens of G20Mn5 material from Section 3.2. In one series of specimens, beneficial compressive residual stresses were present, so these results were excluded from further considerations.

# **4 NumericalAnalyses**

This chapter covers an extended evaluation of the experiments from Chapter 3 based on numerical simulations. The aim of these investigations was to generate a comprehensive understanding of the damage process caused by real casting defects. For this purpose, the tensile specimens from Section 3.2 were subjected to high-resolution crack propagation calculations. The second objective was to quantify the actual stress states present in the component tests from Section 3.3 (e.g., due to geometric deviations) and thus enable a more profound evaluation of these tests. The calculation approaches adopted within this chapter provide information essential to the simplified design concept developed in Chapter 5, but can also be used for a simulation-based component design. In all simulations, the commercial FEM software Ansys Workbench 2019 R2 [3] was used.

## **4.1 Tensile Specimens**

In Section 3.2.2, a significant difference in the fatigue behavior between internal and surface defects was observed. At internal defects, a pronounced crack initiation phase was found, whereas surface defects seemed to behave as initial cracks. This hypothesis was based on the gradient of the SN curves and was underlined by simple analytical approaches. Within this chapter, these effects are confirmed and quantified. In doing so, different modeling depths and analysis strategies are compared. Any effects of eccentric defect locations for both internal and surface defects were directly taken into account. The focus was placed on material 1.6220 and selectively transferred to material 1.6760.

Following the assumptions of Section 2.1, these simulations purely focused on the phase of stable crack growth. The remaining part compared to the experimentally determined service life was assigned to a crack incubation phase. The research partners in [116] carried out further attempts to map the crack initiation using a combination of several online NDT methods [69] or numerical by means of damage models.

Figure 4.1a illustrates the basic procedure of these simulations. An FEM model was generated, evaluated and validated for each specimen considered. The casting defects were introduced into the FEM model at different modeling depths, and the crack tip stresses present in each crack growth increment were calculated.

Separated from this stress analysis, crack growth was evaluated. For this purpose, crack geometries and number of cycles were determined, and failure criteria were checked in newly developed Matlab [60] routines. After passing through this incremental calculation procedure, the results of the individual specimens were considered and evaluated in the overall context. The essential influences, modeling approaches, calculation strategies and results of these three blocks are described in detail in the following subsections.

## **4.1.1 Modeling**

The Finite Element simulation was only used to determine the crack tip stresses at every individual crack growth increment. Its presentation focuses on the fundamental parameters, such as the boundary conditions, material models and especially the modeling of the casting defects.

## **<IP_ADDRESS> Overall Model and Boundary Conditions**

The calculation was performed on the model of the tensile specimens shown in Figure 4.1b. Modeling the entire specimen was necessary to represent any additional stresses caused by eccentricities of the defects. The boundary conditions and stiffness ratios were taken into account according to the test fixture schematically shown in the same figure. The loading was force controlled and corresponded to the axial load range ∆Fx,FEM applied in the experimental investigation.

**Figure 4.1:** a) Basic procedure of numerical crack growth calculations; b) Specimen geometry, boundary conditions and installation of the specimen in the testing machine

## **<IP_ADDRESS> Modeling the Casting Defects**

Since the numerical models were based on the CT records from Section 3.2.1, only specimens corresponding to failure cases I and II defined in Figure 3.6, could be considered. The three modeling variants shown in Figure 4.2 were evaluated and compared. The highest modeling depth was achieved by considering the defect volume and a planar initial crack surrounding the defect in the failure plane. Analogous to [103], a constant expansion increment of 0.1 mm starting from the surface of the defect volume was applied. In this way, potential crack-like fragments in the dissolved microstructure were considered conservatively. In a second model, an identical initial crack front was assumed, but the defect volume was neglected. By comparing these approaches, the influence of the 3D stress field caused by the casting defects was investigated. The application of this model was limited to a small number of selected specimens with internal defects at different load levels. The most far-reaching simplification was achieved through the approach of an elliptical substitute crack with an identical surface area in the failure plane. This aimed at a simplified and more cost-efficient consideration and should correspond to the elliptical failure state observed in the fracture surface analysis in Section <IP_ADDRESS>.

Figure 4.2 compares the stress intensity factors of all three modeling approaches in the specimen U20O12 exemplar. Regions with no crack growth due to ∆KI < ∆Kth are marked in dark blue. The comparison of the results of models 1 and 2 illustrates shadowing effects caused by the defect volume, since high crack tip stresses were reduced to fewer regions of the crack front. Second, the stress trajectories are deflected by the defect in such a way that they acted as relief notches and reduced the crack tip stresses at identical locations. The maximum stress intensity factors of the elliptical crack front were significantly lower compared to the first two models. The reason for this was the lack of curvature in the undercut areas. These local effects thus dominated the shielding effects from the volume defects.

**Figure 4.2:** Modeling approaches for crack growth at casting defects

## *Model 1* − *Volumetric defect:*

Figure 4.3 shows the applied steps for the generation of an FEM model for the determination of the crack tip stresses based on CT data. These records were transformed within the pore analysis into surface models as described in Section <IP_ADDRESS>. The resulting STL file data was not suitable for a direct FEM meshing, as some triangles were inverted and the surface showed multiple indentations that would cause stress singularities. In particular, a defined planar initial crack as a prerequisite for these simulations was not included in this data.

**Figure 4.3:** Necessary steps from CT records to the FE model

In a first step, the failure plane of each individual specimen determined within the experimental investigations was transferred to CT records. To eliminate measurement inaccuracies, the resulting defect contour in the section plane was compared with the fracture surface analysis. By iteratively adopting the axial location of the failure plane in the CT data, the best match between the observed failure and the simulation model was ensured. Additionally, pores aside the cavity that caused failure or defects that did not significantly affect the local crack tip stresses were removed.

In a second step, the remaining parts were smoothened and re-meshed in such a way that the characteristic effects of the pore on the stress field remained but the local singularities vanished. These data were transferred to volumetric bodies and exported as IGS files. Additionally, the defect contour in the failure plane was extracted as a list of nodes. The defined contour line was expanded by a uniform increment of 0.1 mm using the procedure described in Figure 4.6. The resulting contour line formed the initial crack front.

In the last step, both the volume and the crack front were induced into the FEM model. The pore volume was subtracted from the geometry of the tensile specimen shown in Figure 4.1b at its precise location. Afterwards, the geometry of the initial crack front was imprinted on both sides of the failure plane. Specimens with several failure planes could not be precisely reproduced with this approach, so a dominant plane had to be selected. Finally, the upper and lower parts of the specimen were merged together in all regions outside the crack front.

## *Model 2* − *Crack contour:*

The crack contour of this model corresponded to that of model 1; only the defect volume was deactivated for these calculations.

## *Model 3* − *Substitute ellipse:*

Figure 4.4a shows an example of a substitute ellipse. This was determined in such a way that it corresponded to the area weakened by the induced crack from model 1. The remaining geometry parameters were set in such a way that:

**Figure 4.4:** Substitute ellipse: a) Definition of geometric properties; b) Incremental crack growth

## **<IP_ADDRESS> Material Model, Meshing and Determination of KI**

The analysis was intended to cover stable crack growth only. To apply Linear Elastic Fracture Mechanics (LEFM), an isotropic linear elastic material model where E = 210,000 MPa and ν = 0.3 was chosen analogous to [103]. The meshing was carried out using volume elements with a quadratic approach and an element size of 0.1 to 0.2 mm in the failure plane. The stress intensity factors were determined by Ansys [3] in a post-processing routine by the "interaction integral method" [4] according to [106].

## **<IP_ADDRESS> Validation**

The general applicability of the procedure was validated through comparative calculations of simpler geometries such as those presented in Annex B.1.4. For the extension to more complex conditions, the different crack modeling approaches, which were implemented in Ansys were applied and compared. The procedure was confirmed by obtaining results that were close enough according to engineering judgment.

During the actual simulations, the convergence behavior of six evaluated integration paths was monitored for all specimens in all crack growth increments. Only solutions that converged completely along the entire crack path were transferred to the propagation calculation. The comparison of the results with previous and subsequent crack configurations as well as generally valid criteria (e.g., for element distortion) formed the final quality assurance step of the simulations.

## **4.1.2 Crack Growth**

The description of the crack growth process in this section is divided into the material data, the crack growth approach and the applied failure criterion.

## **<IP_ADDRESS> Crack Growth Parameters**

Figure 4.5 shows the crack growth parameters from Section 3.1.2 supplemented by the underlying experimental data and separated by material for a stress ratio of R = 0.1. Since any effect of the component thickness has no statistically significant influence on the crack growth parameters, the parameters listed in Table 3.2 were used in these simulations. In addition to this classical approach, a two-stage growth law was derived in accordance with the basic concept presented in BS 7910:2013 [138]. In particular, the crack propagation rate, which is considerably lower in reality when ∆KI is close to ∆Kth, was to be represented. The parameters used in the crack propagation calculation are summarized in Table 4.1. The values for "stage B" corresponded to the parameters of the classical Paris law. The cusp in the experimental data is discussed in the evaluation and derivation of the crack propagation parameters in [116]. Here, this effect is attributed to roughness-related crack closing effects.

**Table 4.1:** Summary of two-stage crack growth parameters

for da/dN in mm/cycle and ∆K in Nmm−3/2

**Figure 4.5:** Test results of (one-stage) crack growth parameters determined according to ASTM E647:2015 [132] for R = 0.1 from [116] and a two-stage approach for material a) 1.6220 and b) 1.6760

## **<IP_ADDRESS> Growth Procedures**

A preceding analysis showed that ∆KII and ∆KIII attributed to the remaining crack opening modes [28] were negligibly low, and thus a planar consideration of the crack propagation based on ∆KI was considered sufficiently accurate. In the area of stable crack growth, this coincided with the findings of the fracture surface analysis in Section <IP_ADDRESS>. The proposed procedures are presented separately according to the modeling depths. The approaches for model 1 and model 2 correspond to each other.

## *Model 1 and 2* − *Crack Contour:*

Figure 4.6 summarizes all the relevant steps of the applied crack growth procedure. The starting point of each increment i was a disordered set of m points describing the crack front. In the first increment (i = 0), this was derived from the CT data as described in Section 4.1.1. In all further steps (i > 0), it resulted from the nodal coordinates of the FEM simulation. For each propagation increment (i > 0), the following calculation process was applied:

Sorting: A sorting algorithm developed within this work organized the node set in such a way that a closed contour line was formed in ascending order. For this purpose, the neighboring node with the minimum distance in each case was determined beginning from a random starting point. When selecting the output data, it was important to ensure that the mesh size of the FEM model sufficiently resolved the geometries of all existing outgrowths in the crack front. In special cases, manual adjustment was necessary.

Crack Growth: The updated contour line was derived in three stages.

Criterion and Intensity: The local crack growth increment da(n) was determined as a function of the local ∆K(n) according to the Paris law Eq. (2.10). For this purpose, an individual damax(i) was applied for the respective crack growth increment i. First, the number of cycles of the currently considered increment i was determined by

$$\text{dN}(i) = \frac{\text{da}\_{\text{max}}(i)}{\text{C} \cdot \Delta \text{K}\_{\text{max}}(i)^{\text{m}}} \tag{4.1}$$

and stored for the calculation of the total number of cycles. Then, a local crack growth increment was determined for each node n along the contour line by

$$\mathbf{d}\mathbf{a}(n) = \mathbf{d}\mathbf{N}(i) \cdot \mathbf{C} \cdot \Delta \mathbf{K}(n)^{\mathbf{m}}.\tag{4.2}$$

For ∆K(n) < ∆Kth, the local increment was set to da(n) = 0 by default. If the two-stage crack growth law was applied, the Paris parameters C and m were selected according to the assignment of ∆K(n) to stage A or stage B, according to Table 4.1. To create the initial crack front ati = 0, a constant value of da = 0.1 mm was applied at all nodes. In the course of the calculation or with an increasing approximation of a circular shape, damax(i) tended to increase. If local peaks in the stress intensity factors occurred along the contour line, large increments damax(i) led to a breakthrough of the crack front and energetically impossible states.

Direction: Each local crack growth increment da(n) was applied in the direction of the bisector between the connection vectors of three adjacent nodes da( ~ n).

Orientation: The sign of each expansion vector da( ~ n) was ultimately determined using calculus of variations to maximize the area covered by the crack.

The procedure developed within these investigations allowed the consideration of arbitrarily shaped crack contours, undercuts, convex zones and concave zones in the defect contour.

FEM: The updated crack contour acted as the starting point for the numerical determination of the stress intensity factor for the next increment i + 1 using the FEM model according to Section 4.1.1.

After the failure criterion was reached, the total number of cycles Ntot was calculated by

$$\mathbf{N}\_{\text{tot}} = \sum\_{i=0}^{\dot{\mathbf{i}}\_{\text{tot}}} \text{dN}(i). \tag{4.3}$$

In the actual calculations, typically 25 to 45 crack growth increments (itot) were applied. As the changes between ∆K(n)(i) and ∆K(n)(i+1) were negligible and the new crack contour was unknown before applying the propagation procedure, iteration approach 1 according to Figure 2.4 was applied in these simulations.

**Figure 4.6:** Steps in each crack growth increment for arbitrary contours

In the case of surface defects, an analogous procedure was performed. Additionally, the intersection between the crack contour and the specimen contour was specifically determined and shifted so that the intersection was perpendicular to the surface. This was necessary to eliminate numerical problems that occur at other angles, as discussed in [14, 77], and that are not yet solved in general.

## *Model 3* − *Substitute Ellipse:*

The fundamental idea of deriving the local crack growth increments da(n) from the local ∆K(n) using the Paris law was the same as in the previous procedure. However, in this approach, the crack contour was not described by multiple points but by an ellipse equation. Its characterization was done using the orientation α, the two main axes h1, h2 and the coordinates S of the centroid.

The local crack expansion da1−4 was determined as shown in Figure 4.4b by evaluating ∆K1−4 at the four vertices orthogonal to the crack contour. From the coordinates of these newly determined vertices, updated ellipse shape parameters were calculated. Thus, eccentric growth of non-centrically arranged defects could be represented in a simplified way. The calculation of the stress intensity factors in the individual increments was carried out using the FEM model described in Section 4.1.1.

Due to the predefined crack contour having an elliptical shape, a successive outgrowth of the undercut areas was omitted. In this way, the stress intensity factors were more uniform along the crack contour, and larger crack growth increments could be selected. In the actual calculations, 5 to 10 increments had to be used. For the determination of the number of cycles dN(i) associated with each increment, the application of iteration approach 3 according to Figure 2.4 was required. Surface defects were not considered in this approach.

## **<IP_ADDRESS> Failure State**

The objective of the simulation was to map the experimental investigations. For this purpose the calculation was terminated when the area enclosed by the crack contour corresponded to that determined in the fracture surface analysis (Asim = Acrack according to Section <IP_ADDRESS>). Additionally it was checked if dN/da → 0 was valid. Ko < KIc with KIc = KId was checked for dynamic effects according to Section <IP_ADDRESS>.

## **4.1.3 Results**

## **<IP_ADDRESS> Comparison of the Calculated Crack Propagation with Fracture Surface Analysis**

Figure 4.7 shows the fracture surfaces of three representative specimens. On these, the crack contours of the individual calculation increments according to model 1 using the single-stage Paris law are marked. Figure 4.7a shows U20O12, representing a specimen with small, branched, centrally arranged casting defects. Figure 4.7b illustrates a large, eccentrically arranged defect, and U20M10 in Figure 4.7c appears for specimens with large surface defects.

**Figure 4.7:** Comparison of the fracture surface analysis with the calculated crack contours for specimens a) U20O12, b) U20M7 and c) U20M10

In all cases, a good agreement was found between the calculated crack expansion and the area that was assigned to stable crack growth in the fracture surface analysis. The observations from Section 3.2.2 of an elliptical or approximately circular crack surface in the fracture state could be confirmed with these results. The largest stress intensity factors occurred in the indentations of the crack contour, provided that no shadowing effects were present. As a result, these grew out with a comparatively small increase in cracked area per growth increment, and a smooth crack contour was quickly obtained. Deep undercuts in the crack front led to local compressive stress states. This was caused by strain limitation and appeared both in model 1 and in model 2. As a result, negative K-factors could be observed locally. In these cases, subsequent crack configurations caused a closure of these undercuts by joining the edges. In these computationally determined areas, fragments of another fracture process could be spotted on the real fracture surface. The shifted center of gravity at eccentrically arranged defects led to additional stresses. They consequently grew toward the closest surface. Furthermore, the interaction (cf. Section <IP_ADDRESS>) between crack tip stresses and the specimen surface intensified this behavior. In the case of surface defects, the propagation along the surface was dominant, and bending effects rapidly increased. The calculations on surface defects were limited to two examples. In all other cases, either failure case III according to Figure 3.6 was present or the defects were so small that the LEFM lost its validity.

## **<IP_ADDRESS> Comparison of the Calculation Approaches**

Figure 4.8 shows the correlation between the crack propagation behavior and modeling depth exemplar for specimen U20O19. Considering model 1 (Figure 4.8a), the crack front grew eccentrically despite a centric arrangement. The area marked on the crack surface was in a plane with a longitudinal offset of approximately 1 mm. The calculation showed that this section of the crack front was subject to lower stress intensity factors in the considered failure plane. The reasons were shadowing effects that caused crack growth in another parallel plane. The unification of both cracks in the fracture state could be attributed to burrs on the fracture surfaces. In addition, deviations between the initial defects based on the CT data and the resolved microstructure observed on the fracture surface could be detected.

In the case of the contour model (Figure 4.8b), an almost concentric crack growth was observed. Undercuts quickly grew into an elliptical crack shape, which then expanded concentrically. For most specimens, the agreement between model 2 and the fracture surface was considerably poorer than the results of model 1. The third model based on the approach of a substitute ellipse is shown in Figure 4.8c. Its behavior was similar to that of model 2. In cases of eccentrically arranged defects, a clear directivity to the component surface was detected. This was caused by interaction effects between the crack tip and the specimen surface as well as additional bending moments that resulted from a shift of the centroid.

Figure 4.9a shows the evolution of crack growth for specimen U20O19 discussed in Figure 4.8. Herein, √ Acrack is plotted as a function of the calculated number of cycles. This operation eliminates the almost quadratic increase of the crack surface resulting from the elliptical shape. The resulting representation corresponds to the more familiar da/dN evaluation. The points in the graphs indicate the individual increments and are connected to each other by straight lines. The calculation according to model 1 led to significantly higher lifetimes compared to the two planar considerations. The behavior of the elliptical substitute defect and model 2 corresponded very well both qualitatively, as shown in Figure 4.8, and in absolute values Figure 4.9b. Due to the simple geometric description, the non-existent local peaks in the stress intensity factors and the considerably lower number of iteration steps, the calculation effort of the substitute ellipse was a fraction of that of the highly resolved crack contour. Independent of the calculation model, it became obvious that the major part of the calculated lifetime could be assigned to crack configurations close to the initial state. Besides the growth paths of the three calculation models, the limit state Acrack derived from the fracture surface analysis is marked in the plot. The second horizontal line corresponds to the size of a crack surface in which the nominal stress in the remaining cross section reaches the yield strength σy at Fo. Generally, the differences in the calculated service life were less than 5 %. So, besides the findings on the residual fracture in Section <IP_ADDRESS>, σy acts as a useful and not overly conservative criterion (e.g., in the evaluation in the FAD). The negligible differences result from the increase in ∆K and its potentiation in the Paris equation.

**Figure 4.8:** Comparison of the crack propagation approach for specimen U20O19: a) Volume; b) Contour; c) Ellipse

## **<IP_ADDRESS> Comparison of the Calculated Lifetimes with those Determined in the Experiments**

Figure 4.9b compares the service life calculated based on crack growth with the load cycles determined in the experiments. The different modeling variants of the considered specimens of material 1.6220 are grouped for better clarity. Specimens marked with "(S)" had surface defects. For comparison, the regression curve of the considered internal defects is shown in addition for the test results and the results of model 1. All other calculation approaches are supplemented and led to lower calculated service lives. On the other hand, the calculation of specimens with surface defects, regardless of stress level, corresponded very well with the service life observed in the experimental investigations. This underlines the hypothesis from Section 3.2.2 that as long as surface defects meet the minimum dimensions for fracture mechanics consideration, the damage behavior can be described by crack growth. For smaller defects, the approaches established in [66] apply. Since permissible defects of this size cannot be detected by common NDT on untreated cast surfaces, these approaches lose their practical relevance for application and are not examined further. In the case of low load amplitudes, stress intensity factors between ∆Kth and the transition point specified in Table 4.1 occurred locally so that higher lifetimes resulted from the two-stage crack growth law.

**Figure 4.9:** a) Evaluation of crack size development over the service life; b) Comparison of simulated and experimentally determined SN data

Figure 4.10a illustrates the dependence of the proportion Nsim/Ntest of numerically determined stable crack growth Nsim on the cycles to failure detected in the experimental investigations Ntest as a function of the experimentally determined service life. Additionally, the behavior is compared with that of material 1.6760. In the case of high loads, the stable crack growth determined by model 1 could represent almost the total service life determined in the experiments. At the lowest considered load levels, this dropped to about 20 %. Thus, a design model purely based on crack growth is increasingly conservative for lower loads and the same initial defect size. Despite the sharp-edged crack contour, the behavior indicated a pronounced crack initiation phase corresponding to the observations in [42]. Specimen U20M7 was outside the trend of

the other tests, which resulted from a large, eccentrically located casting defect. In this case, the crack propagated exclusively in the direction of the nearest surface, causing additional moments and pronounced interaction effects. For statements on the effect of the material, three specimens with internal defects of different sizes and stress levels were examined. The calculations were limited to model 1 using the one-stage Paris law. The proportions of the total service life were between 20 % and 30 % and thus were considerably lower than those of material 1.6220. The tendency that high stresses lead to a large proportion of crack growth during the total service life was still present, but the expression was significantly lower. Otherwise, no further differences between the two materials under consideration could be determined in these calculations.

### **<IP_ADDRESS> Quantification of the Calculation Models**

Figure 4.10b provides a more detailed analysis of the effects of the calculation model. For this purpose, the calculated cycles were placed in relation to model 1 with a single-stage crack growth law. The results are illustrated as a function of the number of cycles determined in the experiments and indirectly relate to the applied load level. The approach of a planar crack, regardless of whether it followed the exact crack contour or was represented by a substitute ellipse, provided between 50 % and 80 % of the service life calculated with model 1. A dependence on the load level could not be found. In addition, the influence of the two-stage crack growth model did not depend on the load level in the considered examples.

**Figure 4.10:** a) Share of the calculated number of cycles on the experimentally determined service life; b) Comparison of the calculation models

Since the maximum ∆K in the first crack growth increment was already above the transition point between stage A and stage B, the two-stage law had only an indirect influence on the service life by a slightly modified crack shape. The resulting gain in the calculated service life was in the range of 5 % to 20 %. In this case, the characteristics of the local crack contour and the shadowing effects of the 3D volume defect were the main factors. In later crack growth states, ∆KI exceeded the transition point along the entire crack contour. Therefore, the two-stage law was no longer decisive.

## **4.1.4 Summary**

The numerical crack propagation calculations aimed at a deeper understanding of the dominant damage mechanisms and formed the basis of a numerical component design. For this purpose, algorithms for crack propagation calculations of the branched casting defects were developed. The computational models were derived from the CT records and different modeling depths were considered.

The calculations confirmed the different effects of internal and surface defects. In the case of the internal defects, a pronounced crack initiation phase was met, especially at lower load amplitudes. The portions ascribed to crack growth seem to be small and overly conservative at first glance for a design model. A consideration in a double logarithmic diagram and the transfer to tolerable stress amplitudes reduces the effects to an order of magnitude that lies within the scatter known for tests on welded specimens.

Any initial crack geometry grew over an ellipse into an approximately circular state. The volumetric characteristic of the defects or the resulting 3D stress field influenced the crack growth in form, direction and simulated load cycles. A planar consideration was always conservative and the simplified consideration of a substitute ellipse was sufficient. With the methodology shown and the approach of an elliptical plane initial defect, a component design in the HCF range could be carried out numerically. Compared to the method derived in Chapter 5, load transfers and actual stress distributions could be considered with this method. For material 1.6760, the share of crack growth within the total service life was lower, so a design approach based on fracture mechanics is more conservative. However, this statement does not include the residual fracture behavior discussed in Section <IP_ADDRESS> or the brittle fracture behavior investigated in [103]. Furthermore, the simulation results suggested that the disintegrated microstructure surrounding the casting defects did not act as a sharp crack. If this was the case, the crack growth would have occupied a much larger proportion of the total service life.

## **4.2 Component Tests**

In the component tests discussed in Section 3.3, non-uniform crack growth was detected. Nevertheless, the previous evaluations were based exclusively on the nominal stress ranges. Additional stresses resulted from geometric deviations such as non-straightness, the position of the holes in the components, but also from a not entirely centric clamping in the testing machines. The aim of these investigations was to derive the actual stress situation for the initial crack geometry at the beginning of the fatigue tests. Their representation was done using substitute moments, which led to a condition comparable with the measurements. This allowed the derivation of SN curves based on the maximum stress range. Furthermore, the effects of geometric deviations could be quantified. Finally, the main findings on the crack propagation behavior from [121] based on these models and presented in this section are shown.

## **4.2.1 Modeling**

The calculations were performed on the model cutout shown in Figure 4.11 for the "small"-type specimen. This contained the geometric notch, significant influences from the transition areas and the artificially introduced surface defects. The definition of the boundary conditions, designations, the selection of the coordinate system and the transfer to other specimen types are shown in Figure D.1. After a preliminary study, the dimensions and boundary conditions were selected in a way that the resulting stress state corresponded to that of the entire component. The meshing was carried out by volume elements with a quadratic approach (Solid 186 & 187 according to [3]), whereby the element sizes were set in such a way that a converged mesh was obtained in the notch area.

For the stress and strain analyses, two models with different objectives were set up. In model 1, the surface defects were introduced as semi-elliptical cracks to derive the surface strains with consideration of the defect size. In addition, model 1 was used to determine the stress intensity factors. In model 2, the initial defects were not considered so that a regular mesh in the investigation cross section could be achieved. The aim of this model was to generate a homogeneous equivalent stress distribution. From this, the maximum stress ranges were determined. This approach corresponds to a typical model used in the design phase.

The loading was force controlled. All calculations were made on the nominal geometry, and the axial load was scaled by a ratio of the actual and nominal cross-sectional areas. Exemplary investigations on the effect of the actual geometry in [121] justified this procedure and showed that the local geometrical deviations of the notch and connection areas had only a negligible effect on the stresses in the investigated cross section. Young's modulus was varied in Section 4.2.2 for the comparison of the measured and simulated strain states within the range discussed in Section 3.1.

## **4.2.2 Calculation of Substitute Loads and Actual Stresses**

The calculation of the substitute loads was used to quantify the notch stresses present in the investigated cross section. This enabled the determination of SN curves based on ∆σmax, corresponding to the approach in a design process and served to validate the design concept in Chapter 5. In addition, the substitute loads could quantify the effects of geometric deviations. On the one hand, the mechanical effects of the investigated specimens could be classified according to ISO 8062-3:2008 [157]. The relevance of the consideration of geometric deviations in a design concept was underlined. Finally, the substitute loads were used as a starting point for crack propagation analyses in [121].

## **<IP_ADDRESS> Calculation Procedure**

The procedure is summarized in Figure 4.12. The starting point was the strain measurements from Section 3.3. The sequential application of both FEM models from Figure 4.11 was necessary for the reconstruction of the stress states. In the first step, a test section was selected in the form of a cycle range in the measured strains. In this selection, the initial settlement must have subsided; the values of all measuring channels had to be as constant as possible in this cutout. To compensate for fluctuations and peaks, the strain ranges in this interval were averaged and form the reference value for the calibration of substitute stresses. An initial value of the substitute stress Mx and the final value of the load ∆Fx,FEM were determined directly from these values using the actual dimensions by

$$
\Delta \mathbf{F}\_{\mathbf{x}, \text{FEM}} = \Delta \mathbf{F}\_{\text{test}} \frac{\mathbf{A}\_{\text{nom}}}{\mathbf{A}\_{\text{act}}}.\tag{4.4}
$$

**Figure 4.12:** Procedure for determining the substitute loads and actual stress distributions

The determination of My had to be done iteratively. A direct calculation was not possible due to the superposition with the notch stress being coupled with the disproportionate effects of small positional deviations of the strain gauges in the high-gradient areas. Therefore, My was adjusted, and the calculated distribution of the surface strains was compared with the measurements. In order to minimize the errors of the strain gauges in the high-gradient areas, the comparison was carried out with mean values from the FEM. The calculated strains were averaged over a length corresponding to the strain gauge width. The respective positions of the strain gauges were taken into account and documented in Annex D. The substitute loads Mx and My that led to the smallest deviation between the calculated and measured values were used for further calculations.

In a third step, the substitute loads were applied to model 2. From this, the stress field in the undisturbed investigation cross section was determined and compared to a state without substitute loads. The parameters ∆σmax, ∆σmax,N-only, ∆σmax,defect and SCF were determined according to the following definitions.

The results of steps 2 and 3, including the substitute loads and parameters, are listed for all test specimens in Annex D. When recalculating the strains in step 2, it was found that a choosing of Young's modulus E = 200,000 MPa resulted in the smallest deviation between the calculated and measured strains. The deviations in the strains with E = 210,000 MPa were about 5 % higher. Since the loading was force controlled both in the test and in the simulation, there was no significant effect of Young's modulus on the calculated stresses or on ∆K.

## **<IP_ADDRESS> Results**

Figure 4.13 compares representative stress states for the components "small," "medium" and "large − Foundry B." The colored plane represents the actual stress state, whereas the red mesh represents a state without any additional bending moment. In the case of the "small" and "medium" components, the effects of My dominated, whereas Mx played only a minor role. One artificial defect was loaded more strongly compared to the stress distribution without an additional My loading. As can be seen from the fracture surfaces in Section <IP_ADDRESS>, the crack subjected to higher stress propagated in an almost straight line toward the opposite side. In the case of the "large – Foundry B" component, deviations from straightness led to a dominance of Mx. In this case, the defects were located in the neutral fiber related to Mx and were only influenced during propagation, resulting in asymmetrical crack propagation. The situation for the "large − Foundry A" specimen corresponded to those of the "small" and "medium" specimens.

**Figure 4.13:** Typical stress distributions for a) small, b) medium and c) large components

An individual evaluation of the SCFs and SCFmax,def for all specimens is shown in Figure 4.14a. The differences between SCF and SCFmax,def were mostly negligible with the exception of GGB1, GGB2 and GGB3. In these cases, SCF was higher by a factor of up to 1.3. In GGB2, this led to the starting point of the crack growth at the component edge where a grinding groove was located. SCFmax,def was relevant for both crack tip stress at the artificially introduced defects and crack growth and was thus considered exclusively. This was in the range of 2.1 to 2.3 and thus slightly above the design objective of 2.0 due to the additional stresses.

Figure 4.15a shows∆σmax,defect as a function of the number of cycles to failure determined in the experiments. Additionally, the corresponding values for ∆σnom were added to this plot. The regression curve of both parameters according to EN 1993-1-9:2010 [156]

was limited to the samples from Foundry B and C for identical reasons, as discussed in Section 3.3.2. The difference between ∆σ50% corresponded approximately to the mean value of the ∆σmax,defect. Gradients and standard deviations were not significantly influenced by this consideration.

The comparison of ∆σmax,defect with the test results of the tensile specimens with surface defects from Section 3.2 in Figure 4.15b shows that ∆σmax,defect led to higher stress ranges. The ∆σ50% value of the notched specimen was about 50 % greater than the ∆σnet of the tensile specimens. This was due to the supporting effect of the geometric notches and illustrates the potential of considering the notch effect in the design and not performing the check purely based on maximum values.

Figure 4.14b compares the effect of the additional bending moment Mx and My with the effects of deviations from straightness according to ISO 8062-3:2008 [157]. For this purpose, the axial stresses σx,Mx, σx,My resulting from the substitute moments were related to the nominal stress caused by the pure tensile force σx,N by

$$\frac{\sigma\_{x,M}}{\sigma\_{x,N}} = \frac{M/W}{N/A}.\tag{4.5}$$

This indicates the extent to which the normal stress was increased by the additional bending moment on a square reference section. In this simplified approach, the notch effect was not taken into account. The different behavior of the component types is also visible in the graph. In the case of the "small," "medium" and "large − Foundry A" components, the shares of My dominated and led to a related additional stress σx,M σx,N of

**Figure 4.14:** a) Resulting SCFs; b) Determination of additional stress caused by the substitute moments and their comparison with the straightness tolerances according to ISO 8062-3:2008 [157]

5 % to 10 % and in exceptional cases, up to 23 %. The effects of Mx were in the order of 1 % to 5 % and reached 10 % in exceptional cases. In the case of the "large − Foundry B" component, the fractions from Mx dominated and considerably exceeded all other comparison values with a related additional stress σx,M σx,N of 30 % to 60 %. The effects of My were in the same order of magnitude as the other specimens.

Figure 4.14b also contains the limit values for the straightness tolerance group GCTG 6 according to ISO 8062-3:2008 [157] for the respective component dimensions. GCTG 6 is the most stringent requirement that can be economically applied to steel castings produced by hand molding [157]. The length of the component determines which deviation from straightness e is permissible. ("small," l < 300 mm: e = 1.4 mm; "medium," l = 1350 mm: e = 2 mm; "large" l = 1500 mm: e = 2 mm) The underlying background and conversions are described in detail in Section 5.2.7. As an additional comparative value, GCTG 5, the highest group for steel casting at mechanically supplemented sand forming is given. Most of the components were well below the GCTG 6 tolerance limits and were in the range of GCTG 5; only GGB(1-3) exceeded the values significantly. In summary, the large components were representative components. However, the geometric imperfection or effect of eccentric clamping in the testing machine was below the admissible limits according to ISO 8062-3:2008 [157]. Therefore, the test results cannot be used directly for component design, and the approaches in Section 5.2.7 are essential to a generally applicable design method.

**Figure 4.15:** a) Comparison of nominal stress range and maximum stress range in artificial defects; b) Comparison with tensile specimens with surface defects

## **<IP_ADDRESS> Influence of Component Size on Crack Tip Stress**

The small scatter between the lifetimes of different component sizes (see Figure 3.29) and the wall thickness independence of the Paris parameters shown in Section 3.1.2 are leading to the assumption that the influence of wall thickness plays a minor role at surface defects and that the absolute defect size dominates the effects on service life.

As the artificial defects have been defined according to UT detection limits, they had identical dimensions for all component sizes. Secondly, the components were designed such that the stress concentrations and gradients were consistent. This resulted in the situation shown in Figure 4.16a. The defects extend into the interior of the component with different percentages of component thickness. Therefore, the crack vertex of the small component was located in areas of lower stress compared to the larger components. The calculated stress increase at this point in a defect-free component would correspond to SCFdefect,tip,small = 1.3. For the large components, the substitute defects were almost completely located in the highly stressed area and resulted in SCFdefect,tip,large = 1.8.

The consideration carried out in Figure 4.16a for the nominal dimensions without considering the additional bending moment showed that up to 40 % higher stresses occurred at the crack front of "large" components. These differences should be reflected in ∆Kdefect,tip due to the linear relationship between σ and K according to Eq. (2.5).

Figure 4.16b shows that these effects were not fully transferred to the actual maximum stress intensity factors. For this purpose, the maximum stress intensity factors occurring under the respective substitute loads were numerically evaluated. Despite the superposition of notch effects and bending stresses, the location of the highest

**Figure 4.16:** a) Comparison of the nominal stress range and the maximum stress range in artificial defects; b) Effect of crack tip stress in the initial state

stress intensity factors in all specimens was at the vertex inside the components and not on their surfaces. The stress intensity factors determined in this way are shown as a function of ∆σmax,defect and thus of a virtual stress on the component surface. The regression curve of all data points as well as a scattering of ± 20 % complete the representation and serve to classify the SCFlarge and SCFsmall, which differ by 40 %.

It can be seen that the "large" components occupied the top line of the scatter band and the small ones occupied the bottom line. Nevertheless, the difference between the regression curves for "large" and "small," with a deviation of < 25 % in the parameter range under consideration, was smaller than the stress analysis would suggest. An analytical estimation according to Annex B.1.1 showed that the effect of component width in the considered parameter range on ∆Kdefect,tip was less than 1 % and could not be the reason for the contrary effect.

If the major part of the service life were assigned to crack growth in the region after the initial failure, the crack tip stresses could be qualitatively transferred to the total service life. According to Eq. (5.9) and m = 3, the service life of "small" components would be more than twice as long as that of "large" components.

The observation of the beach mark test in Figure 3.30 and the results from [121] confirm the dominance of crack growth near the initial defect in the total service life. Nevertheless, a changein the crack shape was visible, which resultedin a disproportionate increase of crack width c caused by the component width (interaction with the surface of the finite component width see influence of c/w in Appendix **??** & stress gradient). Thus, the small differences between the component sizes in the total service life could explain and indicate an independence of the dimensions of the surface defects from the wall thickness in practical application.

## **4.2.3 Crack Propagation**

Within the framework of these investigations, crack propagation calculations were carried out on the component tests in [121]. The calculations were based on the models and boundary conditions presented in Section 4.2.1. Modeling approaches as well as calculation procedures correspond to the data given herein. The simulations were limited to stable crack growth; crack initiation and plastic effects were not considered. The aim of these investigations was to quantify crack propagation behavior and evaluate essential influencing factors.

This presentation is limited to the findings relevant to this work, for a comprehensive study of the background the reader is referred to [121]. It was shown that the crack propagation of the large component tests could be recorded both qualitatively and quantitatively by calculation. The growth behavior was divided into different areas. At the beginning, an elliptical crack growth was considered to start from the artificially introduced initial defects. When the crack flanks reached the lateral surfaces of the component, a crack front propagating as a straight line was assumed. Since only a small part of the total lifetime was attributed to the transition period between these two states, this transition period was neglected.

A closer examination of the two competing initial defects in Figure 3.27b showed that the load on and consequently the growth rate of one of the two crack was reduced by the additional moment and came to a standstill in the course of growth. The highly loaded crack grew much faster from the beginning, so the center of gravity moved toward the less-loaded crack. The crack tip stresses that occurred in relation to the more heavily loaded crack resulted in increasingly smaller local crack growth increments with subsequent calculations. Finally, a compressive stress state acted on that crack and stopped its propagation. The behavior calculated by [121] corresponds well with the observations on the crack surfaces. A comparison of the calculated and experimentally determined surface strains (cf. Figure 3.31a) showed that the simulations qualitatively represented the behavior very well. Effects such as the cusp in the strain curve on the side of the lower-stressed defects were present. However, the substitute moments acting constantly during the entire crack growth led to increasing deviations from the measured data. This resulted in greater calculated stress intensity factors and consequently lower calculated lifetimes.

The sensitivity analyses in [121] showed that the selected crack propagation parameters dominated. On the other hand, the deviations between the actual and nominal geometry had no decisive influence on crack tip stress or crack growth. The formulation of the stress state at the beginning of the test by substitute moments for a crack propagation calculation was less suitable due to the facts described above.

For an actual component design, it can be derived from [121] that the mathematical evaluation of surface defects in components with stress gradients can be conservatively represented in the HCF range by a crack growth calculation. In most cases, a focus on crack growth in the immediate vicinity of the initial defect is sufficient. Its numerical representation is comparatively simple using tools such as "semi-elliptical cracks" in Ansys [3].

## **4.2.4 Summary**

The aim of these investigations was to quantify the notch effect and the additional bending moments resulting from deviations from straightness from Section 3.3. The calculations were performed on representative model sections. The stress states that occurred in the test sections of the individual specimens were determined iteratively based on the strain measurements and the actual dimensions. For a more accurate assessment of damage mechanism and crack propagation, the artificially introduced defects were subjected to numerical fracture mechanics evaluation.

The stress states derived from the strain measurements were converted into substitute moments. The evaluation of the stress states confirmed the targeted SCF ≈ 2.0. The evaluation of the fatigue resistance using local maximum stress ranges under consideration of the substitute moments, showed ∆σ50% = 196 MPa. This fatigue resistance was a factor of 1.5 higher than the value for tensile specimens with surface defects and approximately constant stress distribution from Section 3.2. This illustrated the effect of the stress gradient on the crack propagation and consequently on the fatigue resistance.

A comparison of the effects of the additional bending moments with the tolerance groups for permissible deviations from straightness according to ISO 8062-3:2008 [157] showed that the specimens were considerably below the GCTG 6 quality requirements, which are the highest for hand-molding processes. The substitute moment My resulted in one of the two artificial defects experiencing higher stresses. Compared to a condition without My, this usually resulted in a local stress increase of 5 % to 10 % and in one exceptional case up to 23 %. Only one series (GKB1-3) showed significant deviations. In contrast to the other specimens, in these cases Mx dominated and led to a stress increase on the undisturbed sides of the components(no notch and no artificial defects). Their magnitude exceeded the threshold values according to GCTG 6 and in one case led to a failure that did not originate from one of the artificial defects.

The substitute defect sizes selected independently of the wall thickness resulted in their vertices being stressed to different extents depending on the component type. Thus, the stresses in the region of the defect vertex of the "large" specimens were 40 % higher than in the case of the "small" specimens. As the calculations showed, this increase was not completely transferred to ∆K. The deviations of the crack tip stresses between the two component types amounted to values < 25 %. From this approach, the wall thickness dependent residual cross-section between the opposing defects and interaction effects between defects and free surfaces, an explanation could be found for the insignificant influence of wall thickness on load cycles observed in the tests for wall thickness independent crack propagation parameters.

# **5 DesignConcept**

Within this chapter, the previous findings on defects' effects on fatigue behavior are transferred into a design concept. The objective is a concept that is easy to apply for the special requirements of the construction industry as defined in Section 1.2. The following summary of the dominating features provide the starting point for a generalized concept:

The derivation of a design concept based on the quality grades according to EN 1559- 2:2014 [152] is not target oriented for the following reasons:

• As discussed in Section 2.1, different acceptance limits for casting defect sizes of the same quality grade are allowed depending on the NDT method. Since the defect size is the decisive influencing factor apart from its location, a generalization is not possible.

**Figure 5.1:** Comparison of the maximum permissible single defect sizes on the surface and inside the component with ∆σ = 60 MP and ∆K = ∆Kth

## **5.1 Basic Idea**

The design concept is outlined by the following principles:

Figure 5.2 provides an overview of the core issues and quality requirements the designer needs to give to the foundry. The RCs act as a link and communication medium between the designer thinking in fatigue classes (FAT) and the foundry providing evidence with respect to quality requirements and defect properties.

**Figure 5.2:** Resistance categories linking engineering stage with execution stage

In an application of this concept, the following additional points must be considered:

## **5.2 Development of the Concept**

The design concept is developed according to the steps illustrated in Figure 5.3. Each of these topics is discussed in an individual section. Useful FAT classes are selected from EN 1993-1-9:2010 [156] based on engineering requirements. Minimum material properties are discussed in the context of the present investigations and the literature. Following the concept of [103], simplified reference cases are specified and form the basis of all investigations. These cases are used to generate the link between fatigue resistance and tolerable defect sizes. In further steps, influencing factors such as additional ultimate limit state (ULS) loading, stress gradients, the effect of multiple defects and geometric deviations are investigated. Finally, the design concept and derived defect sizes are specified.

**Figure 5.3:** Procedure for the development of the design concept

## **5.2.1 Definition of the Loading Classes**

EN 1993-1-9:2010 [156] provides 14 FAT classes downwards with a spacing of √ 5 2 . The derivation of quality requirements for each individual class is not feasible, but Table 5.1 summarizes a well-founded selection.

**Table 5.1:** Selection of FAT classes

## **5.2.2 Input Parameters**

The material parameters to be used for the development are summarized in Table 5.2. These correspond to the minimum requirements for material 1.6220 according to EN 10340:2008 [142]. Parameters that are not directly regulated are discussed and chosen conservatively or are limited via substitute parameters. The Paris coefficients, material strength and the results of the fatigue tests in Chapter 3 exceeded these minimum requirements by far and underline the conservatism. The ductile failure mode observed in Section <IP_ADDRESS> is not taken into account, as the minimum required KV value could cause brittle fracture as identified in [103]. Material 1.6760 according to SEW 520:2017 [164] needs to fulfill higher values and thus is covered by the following considerations. Only in the case of brittle fracture tests under ULS loads, special considerations are necessary for this material. This is caused by higher permissible stresses under static loading.

A generally applicable selection of the crack propagation parameters or a coupling to existing minimum requirements according to the technical delivery conditions is not possible. The aim of the following considerations is to select an approach based on historical data that is as universal as possible and not overly conservative. A mandatory determination of the Paris parameters in each individual case is not considered an objective and contradicts this general approach.

The results from ErStaGu [116] (no. 1 in Table 3.2), a simplified approach according to BS 7910:2013 [138] (no. 16 in Table 3.2) and the approach for a more specified use according to BS 7910:2013 [138] (no. 12 in Table 3.2) are compared. Both approaches in BS 7910:2013 [138] are not explicitly derived for cast steel, but rather aim at a much wider range of applications.

**Table 5.2:** Summary of material parameters (minimum requirements, characteristic or design values)

For a detailed comparison, the fracture mechanics-based SN curves resulting from the considered parameters are extracted from Figure 3.2b and shown in Figure 5.4a. In addition, selected reference values are presented and discussed. Figure 5.4b shows the maximum permissible defect sizes in order to achieve 2 · 106 cycles to failure at ∆σ = 71MPa. The defect sizes are based on a crack propagation calculation. In all cases, semi-elliptical surface cracks are considered according to the situation in Figure 3.2b. For a classification into quality grades according to EN 1369:2013 [148] (MT), references are supplemented in the diagram.

The characteristic values for the fracture mechanics parameter (m, C(R), ∆Kth(R)) determined in the experimental investigations of Section 3.1.2 lead to significantly longer service lives and considerably larger permissible initial defects compared to the values from BS 7910:2013 [138]. In thicknesses t > 30 mm [87, 155], these correspond to AM 5−6 according to EN 1369:2013 [148]. For a generally applicable design concept, the number of samples for the da/dN tests carried out in ErStaGu [116] is too small for a profound statistical evaluation, and a generally applicable conservative approach cannot be derived.

The simplified approach according to BS 7910:2013 [138] leads to extremely low resistances and is strong overly conservative compared to all other literature data. According to BS 7910:2013 [138], the objective of this definition was to provide "preliminary screening assessments" for a general initial assessment with low requirements on the material properties. The permissible initial defect sizes derived from this cannot be implemented economically and contradict the experiences summarized in Section 2.3 and Chapter 3.

Table 3.2 shows identical values according to HSE:1995 [166] and BS 7910:2013 [138] at different stress ratios. The values according to HSE:1995 [166] for R = 0.1 correspond to the values according to BS 7910:2013 [138] for R < 0.5. Compared with the results from

ErStaGu [116], this corresponds to the behavior at R = 0.5. Furthermore, the values according to HSE:1995 [166] for R = 0.5 are assigned to R > 0.5 in BS 7910:2013 [138]. The results from the experimental investigations of [44] for R = 0.1 tend to behave as determined in ErStaGu [116]. The approach according to BS 7910:2013 [138] for R < 0.5 seems to be calibrated conservatively by results for R = 0.5. The derivation of the design value using a 2SD shifting is less than the difference between the test results for R = 0.1 and the mean value according to BS 7910:2013 [138] for R = 0.5.

For further calculations, the following approach is used as indicated in Table 5.2, using the mean value according to BS 7910:2013 [138] for R < 0.5 for an assessment at R = 0.1. According to Figure 3.2b, good agreement is achieved with the assumption in [35]. An additional validation via ∆Kth of these characteristic values is done in Section 5.2.3.

**Figure 5.4:** a) Comparison of Paris parameters using fracture mechanics-based SN curves; b) Permissible defect sizes for surface defects at ∆σ = 71 MPa and 2 · 106 cycles to failure from crack growth calculations

## **5.2.3 Derivation of Basic Resistance Values and Defect Sizes**

Figure 5.5 illustrates the applied procedure to link fatigue resistance curves to maximum tolerable defect sizes. Analogous to [103], the basic investigation cases shown in Figure 5.6a and Figure 5.9a form the foundation of the following procedures. Due to the different effects of internal and surface defects, both are studied separately using analytical crack growth calculations. Loads causing nominal stress ranges according to FAT 100, 71 and 45 at 2 · 106 cycles are applied to the basic investigation cases with variable wall thicknesses of 16 mm, 35 mm, 50 mm, 75 mm and 100 mm. The initial crack geometry is selected to represent the most unfavorable state with respect to crack tip stresses by a/c = 0.4 [28]. The maximum initial defect sizes Acg result from crack

growth calculations and are defined by a resulting lifetime of 2 · 106 cycles to failure at ∆σ = ∆σc(FAT). Based on investigations on endurance behavior, the maximum initial defect sizes are updated to AKth. Finally, the resulting defect sizes are compared to quality requirements according to EN 12680-1:2003 [144] and EN 1369:2013 [148].

**Figure 5.5:** Procedure to link fatigue resistance to defect sizes for basic cases

These investigations purely focus on single defects and are based on the material parameters for R = 0.1 according to Table 5.2. Any effects of stress concentration caused by additional bending moments or geometric shapes are not considered within these investigations. A generalization follows in the subsequent sections.

## **<IP_ADDRESS> Surface Defects**

## *Step 1: Definition of basic cases*

Figure 5.6a illustrates the basic situation for the investigations of surface defects. Stress intensity factors are calculated according to Annex B.1. The plate geometry defined by the ratio 2w t = 7.5 is chosen according to [103] and [54]. The crack growth calculation follows the concept described in Section 2.2.4 using the stress intensity factors at the surface ∆KI,c and the internal apex ∆KI,a. The step size is set to an increment of damax = 0.01 mm at the point of maximum crack tip stress. The remaining increment is calculated according to Eq. (2.10). Both the exceedance of the limit curve in a FAD and the wall breakthrough of the crack define failure.

*Step 2: Determination of the maximum tolerable initial crack size Acg using crack growth* Figure 5.6b summarizes the resulting defect sizes that cause failure after 2 · 106 cycles under an applied nominal stress rage of 100, 71 and 45 MPa. The calculations are marked as dots and are interpolated linearly. As a reference, the quality requirements according to EN 12680-1:2003 [144] (MT) for surface defects and according to EN 1369:2013 [148] (UT) for subsurface defects in the rim zone are illustrated. In both cases, the NDT displays are transformedinto defect geometries using the equal surface concept according to FKM-Rili-BM:2009 [161].

**Figure 5.6:** Surface defects: a) basic investigation case; b) maximum defect surface sizes determined by crack growth simulations

In contrast to the requirements of the NDT standards, the calculations show a dependency on wall thickness. The defect sizes that result for a nominal stress range of 100 MPa are in the range of the highest MT requirements for an untreated casting surface. For lower FAT classes, the resulting crack sizes distinctively exceed the NDT requirements, which is in line with the findings of [35].

#### *Step 3: Behavior in the near threshold regime*

In this fracture mechanics approach, endurance is not defined a priori by a fixed number of cycles but by the loading that causes ∆K ≤ ∆Kth. To set the cycles to failure Nendu at the endurance limit and check its conformity with EN 1993-1-9:2010 [156], the behavior in the threshold region is investigated more thoroughly. Nendu is calculated according to the procedure discussed in Section <IP_ADDRESS> to determine fracture mechanics-based SN curves. This was done for all wall thicknesses, FAT-classes and the associated maximum defect sizes Acg(FAT). In all cases, Nendu varies from 4 · 106 to 3.5 · 107 cycles and thus is higher than defined in EN 1993-1-9:2010 [156] in most cases. The resulting ∆σ where ∆K = ∆Kth are shown in Figure 5.7a. Additionally, the stress ranges that are associated with the considered FAT classes at 5 · 106 and 1 · 107 cycles to failure are shown. These are determined by extrapolation according to Eq. (5.1) and m = 2.88.

$$
\Delta \sigma\_{\rm a} = \Delta \sigma\_{\rm c} \cdot \left(\frac{2 \cdot 10^6}{\rm N\_{\rm a}}\right)^{\frac{1}{\rm m}} \tag{5.1}
$$

**Figure 5.7:** a) Nominal stress ranges at ∆Kmax = ∆Kth and Acg; b) SN curves with ∆σ(∆Kth) at different reference points compared to the results of crack growth simulations (t = 50 mm)

The comparison of the wall thickness and defect dependent values with ∆σ at Nendu = 5 · 106 deviates from the FAT class and shows that cracks still propagate at stress ranges that would be considered endurant according to EN 1993-1-9:2010 [156]. This means that the maximal tolerable initial crack sizes derived from the crack growth simulation and the definition of Nendu = 5·106 according to EN 1993-1-9:2010 [156] underestimates the resistance for low stress ranges. It is pointed out that ∆Kth and the considered Paris parameters have a different background. This is assumed to be the reason for these differences but adds robustness as the worst of two approaches is considered. Choosing Nendu = 1 · 107 is in line with the concept of [96] and most of the studies discussed in Section 2.3.1. With this adaption, theoretically unsafe design scenarios are limited to wall thicknesses of less than ≈ 30 mm. This condition is addressed in Section 5.4.3.

## *Step 4: Adoption of SN curves based on* ∆*Kth*

To eliminate these uncertainties, a ∆Kth-based approach to derive the maximum initial defect sizes is applied. At first, the endurance limit is defined according to the results of step 3 at 1 · 107 . Then, ∆σ(∆K = ∆Kth) is calculated and provides an anchor point for the following SN curves. The gradient of the so-defined SN curves is set to m = 3 in accordance with EN 1993-1-9:2010 [156]. Exemplary results for defect sizes Acg(FAT 100), Acg(FAT 71) and Acg(FAT 45) for t = 50 mm are presented in Figure 5.7b. These curves are compared to the fracture mechanics-based curves from step 2. To illustrate the effects of the definition of NKth, the state for NKth = 5 · 106 cycles is also shown. These ∆Kth-based curves provide increasingly conservative results with an increasing initial defect size (i.e., with decreasing FAT class). An approach based on NKth = 5 · 106 cycles provides overly conservative results. As a result of these investigations, 1 · 107 is defined as the number of cycles from which the endurance limit

is reached. Further service life curves are derived with the gradient m = 3 based on ∆σ(∆Kth).

*Step 5:Update of the maximum initial defect sizes based on* ∆σ(∆*Kth*) To derive maximum defect sizes associated with the FAT classes that are targeted in Section 5.2.1, the stress ranges ∆σendu are evaluated by

$$
\Delta \sigma\_{\rm endu} = \Delta \sigma\_{\rm c} \cdot \left(\frac{2 \cdot 10^6}{1 \cdot 10^7}\right)^{\frac{1}{5}} \tag{5.2}
$$

and form the load cases for a static calculation of the stress intensity factors. Iteratively, the defect sizes AKth are determined for all wall thicknesses in such a way that ∆Kmax = ∆Kth for the individual ∆σendu. The results are shown in Figure 5.8.

**Figure 5.8:** a) Comparison of Acg and AKth; b) Classification of AKth into NDT requirements

The comparison in Figure 5.8a between AKth and Acg shows that initial defect sizes defined by ∆Kth are smaller than those defined by crack growth for wall thicknesses larger than 30 mm. The design approach of [103] generally requires the highest quality grades for wall thicknesses t < 30 mm and quotes [87, 155] for approval. The present approach tries to avoid this limitation, so a combined approach is used for t < 30 mm, as discussed in Section 5.4.3.

#### *Step 6: Comparison with existing NDT requirements*

In Figure 5.8b, a comparison with the quality requirements according to the existing NDT standards is made analogous to Figure 5.7a.  The newly defined maximum initial defect sizes AKth are of equal dimensions to common manufacturing qualities. The permissible defect sizes for RC 5 with FAT 45 are classified as large for designs subject to fatigue loading and are not extended to lower FAT classes and larger permissible initial defects.

### **<IP_ADDRESS> Internal Defects**

Internal defects are considered according to the geometrical situation shown in Figure 5.9a. The procedure and essential statements are identical to those for the surface defects. The representation is reduced to the comparison to existing quality requirements analogous to Figure 5.8b. Intermediate steps discussed in detail in the context of surface defects are given in Annex E without further comments.

**Figure 5.9:**Internal defects: a) Basic investigation case; b) Classification of AKth into the requirements of NDT standards

## **5.2.4 Interaction of Static and Cyclic Loading**

The approach developed herein is based on the assumption of a crack-like initial defect. This defect grows in the course of the calculated service life until a critical state is reached. In common structural applications, static forces act in addition to the fatigue loading. These can occur during the entire service life as mean stress (see Section 5.2.8) more or less frequently as well as in exceptional design situations only. The most unfavorable condition is shown in the lower part of Figure 5.10 and is reached when the design load case for brittle fracture occurs at the end of the calculated service life. In this concept, the maximum allowable defect size required for the brittle fracture analysis (e.g., according to [103]) must be considered as the final state of crack growth from fatigue loading. This approach was also implemented in [35]. In addition, Figure 5.10 shows an example of a component that is exclusively subjected to fatigue loads. In this case, the state in the FAD at σo or the wall breakthrough defines the failure state.

**Figure 5.10:** Influence of the load situations on the critical crack size relevant for failure

According to EN 1993-1-10:2010 [154], the actions for brittle fracture design are to be determined as an exceptional load case. Thus, σEd,brit < 0.75 σy is generally applicable. Both EN 1993-1-10:2010 [154] and [103] consider the brittle fracture resistances in clusters of σEd,brit = 0.50 to 0.75 σy, 0.25 to 0.50 σy and 0 to 0.25 σy. The condition σo = σEd corresponds to pure fatigue loading and values of σo > σEd is not considered. For each FAT class and material to be tested, the scenarios shown in Table 5.3 cover the entire parameter range.

For each scenario, crack growth was calculated according to the procedure on the reference components defined in Section 5.2.3. The calculations were carried out on components with a representative wall thickness of 50mm, and themaximum permissible initial defect sizes AKth for RC 1, RC 2 and RC 5 deviated from ∆Kth, as discussed in Section 5.2.3. The results of these calculations were load cycles Nbrit at which the crack reached the critical state according to the FAD under brittle fracture stress σEd,brit or wall breakthrough. To determine the comparison value Nfat, the calculation was carried out until the critical state was reached under the effect of σo. KIc was selected according to Table 5.2 and considers a design temperature of TEd = -30 °C.

Figure 5.11a shows a representative FAD for both fatigue loading and σEd,brit. In the present example, the calculated failure state at σo did not occur due to plastic or brittle fracture but due to a wall breakthrough as shown in Figure 5.11b. Herein, both Nbrit and Nfat are marked, and the critical crack dimension for a failure at σEd,brit can be determined by the dimensions a and c.

**Table 5.3:** Summary of the cases under investigation

**Figure 5.11:** Failure criterion for crack growth simulation: a) FAD for fatigue and ULS load, b) Crack dimensions depending on the load cycles

## **<IP_ADDRESS> Surface Defects**

These investigations were carried out for both materials 1.6220 and 1.6760 with σy, according to the technical delivery conditions and similar crack growth parameters used in Section 5.2.2.

Figure 5.12 shows the ratio Nbrit/Nfat depending on the utilization clusters σEd,brit/σy and on the ratio σo/σEd,brit for surface defects and both materials separately. In the cases of σo/σEd,brit = 1.0, Nbrit/Nfat = 100 % follows by definition. Values of 100 % for other σo/σEd,brit are possible and occur when the fracture state is not defined by the FAD but is caused by a wall breakthrough. At a low utilization of σEd,brit/σy = 0.25, both materials behave similarly, and their service lives are not influenced by σEd,brit as the geometric boundaries of the crack are dominant in the failure state. For material 1.6220, this is also valid for σEd,brit/σy = 0.5. Only for the stress cluster σEd,brit/σy = 0.75 larger deviations in the service life are noted and more pronounced with an increasing initial defect size. For defects with an initial size that corresponds to RC 1, failure due to σEd,brit occurs at 93 % of the service life for pure fatigue loading. For an initial defect according to RC 5, this value decreases to 72 %.