# A Probabilistic Fatigue Strength Assessment in AlSi-Cast Material by a Layer-Based Approach

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## Abstract

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## 1. Introduction

#### 1.1. Statistical Analysis of Defects

- if $\xi \to 0$, a Gumbel or Type I distribution is defined; see Equation (2);
- if $\xi >0$, a Fréchet, or Type II distribution is defined;
- if $\xi <0$, a Weibull, or Type III distribution is defined.

#### 1.2. Consideration of Defects in Fatigue Design

#### 1.3. Measurement of Defects

#### 1.4. Outline of the Work

## 2. Materials and Methods

## 3. Experimental Results

#### 3.1. Computed Tomography

^{®}. The cumulative distribution function (CDF) of the GEV is given in Equation (1). A Kolmogorov–Smirnov (KS) test was conducted at a significance level of 5% to determine the goodness of fit for the statistical assessment of the distribution [76]. The empirical distribution of the samples was therefore compared with a hypothesized distribution. Thus, a value of ${p}_{KS}=1.00$ indicates a matching compliance between fit and data. In addition, the parameters for the distributions in this paper were statistically evaluated using the maximum likelihood estimation [28]. A summary of the statistical assessment within the above mentioned layers for Positions HP and LP is given in Figure 9. The trend of the pore size depending on the distance from the surface is plotted for the projected $\sqrt{area}$ for a probability of occurrence ${P}_{Occ}=50\%$ as a solid line with data points for the middle of each layer. Additionally, the scatter band for ${P}_{Occ}=10\%$ and ${P}_{Occ}=90\%$ of the corresponding distribution function is plotted as a dotted line and a dashed line, respectively, in Figure 9. The pore size for ${P}_{Occ}=50\%$ ranges from 46 to 555 µm for the HP specimens and from 102 to 424 µm for the LP specimens. Particularly, in the HP layers, an increased scatter band of the values for ${P}_{Occ}=10\%$ and ${P}_{Occ}=90\%$ can be observed. The reason for that is the occurrence of few very large defects in these sections, leading to a flatter curve of the GEV fit. In contrast, layers inheriting smaller defects result in a steep course of the cumulative distribution function and a lower scatter band of the data.

#### 3.2. Fatigue Strength

#### 3.3. Fractography

## 4. Layer-Based Fatigue Assessment Methodology

## 5. Conclusions

- The clustering of porosity obtained by computed tomography revealed a significant impact on the statistically evaluated parameters of the defect distribution given a high porosity, such as in sponge-like imperfect sections. The effect of clustering may increase the projected size of the largest defect distribution up to 100 µm in such cases; however, if there is a low pore density, clustering has a negligible effect on defect distribution, resulting into a deviation of only a few microns.
- Although the SDAS is quite constant across the whole investigated cross section, significant changes regarding pore size and distribution may occur due to local casting process conditions.
- In order to reduce effort regarding micro-computed tomography for large scan volumes, the applicable distribution function can be also obtained by scanning a smaller, albeit representable, volume and proposing the final distribution parameters scaled by the return value ratio. As illustrated, a deviation of only 11 µm for a defect size of 728 µm and 12 µm for a defect size of 134 µm is observed.
- The previously developed methodology using a probabilistic Kitagawa–Takahasi diagram was extended using defect distributions of computed tomography enabling a non-destructive determination of the fatigue strength.
- The layer-based assessment methodology leads to a sound estimation of the thick-walled specimen fatigue strength at strongly varying casting conditions with deviations in a range of −2.6% up to 12.9%. The introduced methodology enables the probabilistic calculation of the most damaged layer, both dependent on cyclic loading and local casting conditions.

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

$\sqrt{area}$ | Defect size of Murakami’s approach |

$\alpha $ | Return period of the highly stressed volume |

$\kappa $ | Weibull factor |

${\sigma}_{LLF}$ | long-life fatigue strength |

${\sigma}_{LLF,{V}_{0}}$ | long-life fatigue strength of the reference volume ${V}_{0}$ |

${\sigma}_{LLF,{V}_{1}}$ | long-life fatigue strength of the $\alpha $-times enlarged volume ${V}_{0}$ |

$\Delta $ | Deviation of model to experiment |

${\sigma}_{0}$ | Fatigue range of near-defect-free material |

${\sigma}_{b}$ | Bending stress |

$\delta $ | Scale parameter of the GEV distribution |

${\delta}_{\alpha}$ | Scale parameter of the GEV distribution for the $\alpha $-times enlarged volume ${V}_{\alpha}$ |

$\mu $ | Location parameter of the GEV distribution |

${\mu}_{\alpha}$ | Location parameter of the GEV distribution for the $\alpha $-times enlarged volume ${V}_{\alpha}$ |

$\xi $ | Shape parameter of the GEV distribution |

${\xi}_{\alpha}$ | Shape parameter of the GEV distribution for the $\alpha $-times enlarged volume ${V}_{\alpha}$ |

${\nu}_{i}$ | Weighting factor for crack closure effect i |

${l}_{i}$ | Crack elongation, where the crack closure effect ${\nu}_{i}$ is completely developed |

$\Delta {K}_{th,lc}$ | Long crack threshold range |

$\Delta {K}_{th,\Delta a}$ | Crack threshold range in respect to the crack extension |

${a}_{SL}$ | Surface layer thickness |

h | Specimen height |

${M}_{b}$ | Bending moment |

$\Delta {K}_{th,eff}$ | Effective crack threshold range |

$\Delta a$ | Crack extension |

a | Crack length |

${a}_{m}$ | Crack length of the reference volume ${V}_{0}$ for a probability of occurrence of 50% |

${a}_{m,\alpha}$ | Crack length of the reference volume ${V}_{\alpha}$ for a probability of occurrence of 50% |

P | Probability |

${P}_{Occ}$ | Probability of occurrence |

${P}_{S}$ | Probability of survival |

${P}^{\alpha}$ | Defect distribution of $\alpha $-times enlarged volume ${V}_{\alpha}$ |

${V}_{90,0}$,${V}_{90,1}$ | 90% highly stressed volumes |

${V}_{\alpha}$ | $\alpha $-times enlarged highly stressed volume |

${p}_{ks}$ | p-value of the Kolmogorov–Smirnov test |

Y | Geometry factor |

${k}_{1}$ | Inverse slope of the S/N curve in finite life region |

${T}_{S}$ | Fatigue scatter band of the S/N-curve |

${N}_{T}$ | Transition knee point of the S/N-curve |

$\Psi $ | Sphericity |

${A}_{pore}$ | Surface area of pore |

R | Load ratio |

R-curve | Cyclic crack resistance curve |

HSV | Highly stressed volume |

SDAS | Secondary dendrite arm spacing |

GEV | Generalized extreme value distribution |

CDF | Cumulative distribution function |

FEM | Finite element method |

HCF | High cycle fatigue |

HIP | Hot isostatic pressing |

CT | Micro-computed tomography |

CP | Local casting process conditions |

CL | Clustering of defects |

GEV | Generalize extreme value distribution |

LP | Low porosity layer |

HP | High porosity layer |

SL | Surface layer |

LEVD | Largest extreme value distribution |

KTD | Kitagawa–Takahasi diagram |

HT | Heat treatment |

E | Young’s modulus |

TS | Tensile strength |

YS | Yield strength |

A | Elongation at fracture |

BM | Base material strength of near-defect-free material |

FDD | Focus detector distance |

FOD | Focus object distance |

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**Figure 1.**Setup of the Kitagawa diagram incorporating size effect (applying modification of El-Haddad).

**Figure 2.**Comparison of SDAS values at Positions A–D for high-porosity (HP) and low-porosity (LP) regions in each specimen.

**Figure 4.**Thin-walled specimen for fatigue testing with a plate thickness of 1.5 mm. (

**a**) Illustration of the thin-walled specimen geometry with main dimensions in millimetres. (

**b**) Numerical analysis of stress concentration factor depicted as the maximum principal stress.

**Figure 5.**Comparison of AlSi-cast specimen cross sections in machined and vibratory finished conditions.

**Figure 7.**Representation of an exemplary CT scan. (

**a**) CT scan of a thick-walled specimen. (

**b**) Projection of clustered pore.

**Figure 8.**Sphericity over the projected $\sqrt{area}$ perpendicular to the load direction of the investigated positions evaluated by $\mathsf{\mu}$CT analysis. (

**a**) CT results of thick-walled specimen Pos. A and C. (

**b**) CT results of thick-walled specimen Pos. B.

**Figure 9.**Distribution of the porosity size in a cross section of the thick-walled specimen geometry for extracting Positions A to C.

**Figure 10.**CDF of $\sqrt{area}$ perpendicular to the load direction for all layers of the thick-walled specimen.

**Figure 11.**Comparison of normalized S/N-curves for ${P}_{Occ}=50\%$ experimentally determined within the framework of this study.

**Figure 13.**SEM fracture surface analysis in the HP and LP specimens. (

**a**) Fracture initiating defect in the HP specimen. (

**b**) Fracture initiating defect at specimen LP.

**Figure 15.**Relation between measured and calculated cumulative defect distributions. (

**a**) Cumulative defect distributions of LP samples. (

**b**) Cumulative defect distributions of HP samples.

**Figure 17.**Calculated course of the fatigue strength over the cross section of the thick-walled cast specimen.

Alloy | Si [%] | Cu [%] | Fe [%] | Mn [%] | Mg [%] | Ti [%] | Al [-] |
---|---|---|---|---|---|---|---|

Standard [64] | 7.5–8.5 | 2.0–3.5 | max. 0.8 | 0.15–0.65 | 0.05–0.55 | max. 0.25 | balance |

Experiment | 7.96 | 3.13 | 0.43 | 0.32 | 0.25 | 0.13 | balance |

HT | E [GPa] | TS [MPa] | YS0.1 [%] | A [%] |
---|---|---|---|---|

T6 | 73.6 | 272 | 258 | 0.19 |

HIP + T6 | 74.7 | 319 | 244 | 1.78 |

Parameter | Value |
---|---|

Voltage [kV] | 111 |

Intensity of current [µA] | 142 |

Number of images [-] | 785 |

FDD [mm] | 1000 |

FOD [mm] | 75 |

Exposure time [ms] | 1000 |

Voxel size [µm] | 15 |

Pos. | Clustered | ${\sqrt{\mathit{area}}}_{{\mathit{P}}_{\mathit{Occ}}=50\%}$ [µm] | ${\sqrt{\mathit{area}}}_{{\mathit{P}}_{\mathit{Occ}}=10\%}$ [µm] | ${\sqrt{\mathit{area}}}_{{\mathit{P}}_{\mathit{Occ}}=90\%}$ [µm] |
---|---|---|---|---|

LP | no | 131 | 191 | 108 |

LP | yes | 134 | 190 | 120 |

HP | no | 626 | 858 | 459 |

HP | yes | 728 | 1117 | 523 |

Position | HT | Loading Type | ${\mathit{k}}_{1}$ [-] | ${\mathit{\sigma}}_{\mathit{LLF},50\%,\mathit{norm}}$ [-] | ${\mathit{N}}_{\mathit{T}}$ [-] | ${\mathit{T}}_{\mathit{S}}$ [-] |
---|---|---|---|---|---|---|

LP | T6 | tension | 4.6 | 0.49 | 920,000 | 1:1.3 |

HP | T6 | tension | 3.7 | 0.35 | 530,000 | 1:1.19 |

SL | T6 | bending | 4.8 | 0.47 | 1,300,000 | 1:1.24 |

BM | HIP + T6 | tension | 1.01 | 1.00 | 330,000 | 1:1.06 |

Pos. | Investigation | ${\sqrt{\mathit{area}}}_{{\mathit{P}}_{\mathit{Occ}}=50\%}$ [µm] | ${\sqrt{\mathit{area}}}_{{\mathit{P}}_{\mathit{Occ}}=10\%}$ [µm] | ${\sqrt{\mathit{area}}}_{{\mathit{P}}_{\mathit{Occ}}=90\%}$ [µm] |
---|---|---|---|---|

LP | Frac. | 173 | 275 | 101 |

LP | CT | 134 | 190 | 120 |

HP | Frac. | 740 | 977 | 512 |

HP | CT | 728 | 1117 | 523 |

**Table 7.**Comparison of statistically evaluated $\sqrt{area}$-parameters for 50% probability of occurrence considering the effect of HSV.

Pos. | $\sqrt{\mathit{area}}\left({\mathit{V}}_{0}\right)$ [µm] | $\sqrt{\mathit{area}}\left({\mathit{V}}_{\mathit{\alpha},\mathit{exp}.}\right)$ [µm] | $\sqrt{\mathit{area}}\left({\mathit{V}}_{\mathit{\alpha},\mathit{calc}.}\right)$ [µm] | $\mathit{\Delta}$ [µm] | $\mathit{\Delta}$ [%] |
---|---|---|---|---|---|

LP | 104 | 134 | 122 | −12 | −9.8 |

HP | 522 | 728 | 716 | −11 | −1.7 |

Pos. | Model [-] | Experiment [-] | $\mathsf{\Delta}$ [%] |
---|---|---|---|

SL | 0.42 | 0.47 | 11.9 |

HP | 0.3 | 0.34 | 12.9 |

LP | 0.5 | 0.49 | −2.6 |

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## Share and Cite

**MDPI and ACS Style**

Oberreiter, M.; Fladischer, S.; Stoschka, M.; Leitner, M.
A Probabilistic Fatigue Strength Assessment in AlSi-Cast Material by a Layer-Based Approach. *Metals* **2022**, *12*, 784.
https://doi.org/10.3390/met12050784

**AMA Style**

Oberreiter M, Fladischer S, Stoschka M, Leitner M.
A Probabilistic Fatigue Strength Assessment in AlSi-Cast Material by a Layer-Based Approach. *Metals*. 2022; 12(5):784.
https://doi.org/10.3390/met12050784

**Chicago/Turabian Style**

Oberreiter, Matthias, Stefan Fladischer, Michael Stoschka, and Martin Leitner.
2022. "A Probabilistic Fatigue Strength Assessment in AlSi-Cast Material by a Layer-Based Approach" *Metals* 12, no. 5: 784.
https://doi.org/10.3390/met12050784