# Validation Study on the Statistical Size Effect in Cast Aluminium

^{*}

## Abstract

**:**

## 1. Introduction

- The influence of disconnected highly-stressed volumes as statistical size effect based on accumulated highly-stressed volumes.
- The impact of the highly-stressed volume on the defect distribution and its associated parameters is verified for samples with not-yet investigated casting process conditions. This enhances the existing database and strengthens the a priori established model framework of probabilistic fatigue strength design.
- The effect of the element size during numerical evaluation of the highly-stressed volume is studied and supports recommendations for engineering applicability.
- The work validates the prior developed statistical size effect approach which depends not only on the return period of the highly-stressed volume but takes also the defect distribution of the fractograpic analysis and the material resistance as probabilistic values into account.

## 2. Investigated Alloy

## 3. Experimental Results

#### 3.1. Fatigue Strength

#### 3.2. Fractography

## 4. Verification of Size-Effect Related Fatigue Strength

## 5. Conclusions

- Based on a numerical parameter study, a deviation factor of about 0.03 is recommendable for numerical evaluation of the highly-stressed volume (HSV) in engineering applications.
- If several independent HSVs with the same microstructural properties are attached as one component and loaded simultaneously, the failure of each HSV leads to failure of the whole component. Hence, the aggregated sum of disconnected HSVs has to be considered as size effect in fatigue strength design. But in the case of varying microstructures between the individual highly-stressed volumes, the local microstructure has to be considered as well.
- The conducted validation of the aforesaid defect based probabilistic fatigue assessment model, originally published in Reference [3], is based on samples with a return period of about two. The results confirm that the model assesses the fatigue strength in terms of statistical size effect best by applying the local Weibull factor $\kappa $ depending on the return period $\alpha $ and defect population ${\mu}_{0}$. Thus, the verified probabilistic approach is recommendable for engineering design of complex parts, whereat the HSV has to be linked to the local microstructural properties for proper fatigue strength design.

## Author Contributions

## Funding

## 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 ${V}_{0}$ |

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

${\sigma}_{LLF,50}$ | Estimated long life fatigue strength with 50% probability of survival |

${\sigma}_{\ast ,Ps50}$ | Experimental long life fatigue strength at position * with 50% probability of survival |

$\Delta $ | Deviation of model to experiment |

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

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

${\delta}_{0}$ | Scale parameter of the GEV distribution for the reference volume ${V}_{0}$ |

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

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

${\mu}_{0}$ | Location parameter of the GEV distribution for the reference volume ${V}_{0}$ |

${\mu}_{\alpha}$ | 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 build-up |

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

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

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

$\Delta {K}_{eff}$ | Effective stress intensity factor range |

${K}_{max}$ | Maximum stress intensity factor |

${K}_{op}$ | Opening stress intensity factor |

$\Delta a$ | Crack extension |

a | Crack length |

${a}_{0,eff}$ | Intrinsic crack length |

${a}_{0,lc}$ | Crack length at the transition to long crack behaviour |

${a}_{m}$ | 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% |

h | Segment height of a circle |

L | Chord length of the segment |

n | Number of elements on circumference |

P | Probability |

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

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

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

${V}_{0}$,${V}_{1}$ | Highly stressed volume of specimen A and B |

${V}_{90,0}$,${V}_{90,1}$ | 90% highly stressed volume of specimen A (${V}_{0}$) and B (${V}_{1}$) |

${V}_{\infty}$ | Threshold volume |

${V}_{\alpha}$ | $\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 |

${k}_{2}$ | Inverse slope of the S/N-curve in long life region |

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

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

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 |

KTD | Kitagawa Takahashi diagram |

ECD | Equivalent circle diameter |

FE | Finite element |

HCF | High cycle fatigue |

HIP | Hot isostatic pressing |

## Appendix A. Fatigue Failure Hypothesis

**Figure A1.**Schematic representation of the specimens manufactured from a cube possessing a homogeneous defect distribution and sketch of expected fatigue strength results.

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**Figure 1.**Schematic set up of the Kitagawa Takahashi diagram with its modifications and exemplary defect distributions of a volume ${V}_{0}$ and an enhanced volume ${V}_{\alpha}$.

**Figure 5.**Finite Element (FE) analysis of specimens A and B with 90% HSV determined with C3D20R elements.

**Figure 13.**Weibull factor $\kappa $ depending on the return period $\alpha $ and the defect population ${\mu}_{0}$ and evaluated point of the current test series of specimen B.

**Table 1.**Nominal chemical composition of the investigated cast alloy in weight percent [63].

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

EN AC-46200 | 7.5–8.5 | 2.0–3.5 | 0.8 | 0.15–0.65 | 0.05–0.55 | 0.25 | balance |

Specimen | HT | Volume | ${\mathit{k}}_{1}$ [-] | ${\mathit{\sigma}}_{\mathbf{LLF},\mathbf{50}\%}$ [-] | ${\mathit{N}}_{\mathit{T}}$ [-] | ${\mathit{T}}_{\mathit{S}}$ [-] | |
---|---|---|---|---|---|---|---|

A | T6 | ${V}_{0}$ | 7.84 | 1.00 | 1,100,000 | 1:1.08 | |

B | T6 | ${V}_{1}$ | 10.73 | 0.96 | 1,900,000 | 1:1.23 |

**Table 3.**Statistically evaluated distribution parameters of the generalized extreme value distribution (GEV).

Position | Volume | $\mathit{\mu}$ [$\mathsf{\mu}$m] | $\mathit{\delta}$ [$\mathsf{\mu}$m] | $\mathit{\xi}$ [-] | $\sqrt{\mathit{area}}\left({\mathit{P}}_{\mathbf{Occ}=0.5}\right)$ [$\mathsf{\mu}$m] | ${\mathit{p}}_{\mathbf{ks}}$ [-] | |
---|---|---|---|---|---|---|---|

A | ${V}_{0}$ | 95.1 | 20.1 | 0.43 | 103 | 0.93 | |

B | ${V}_{1}$ | 118.4 | 28.1 | 0.36 | 129 | 0.69 | |

B (model) | ${V}_{1}$ | 111.3 | 27.1 | 0.43 | 122 | 0.58 |

**Table 4.**Parameters resulting from crack propagation tests in position A and B for a probability of occurrence of ${P}_{Occ}=50$%.

$\mathbf{\Delta}{\mathit{K}}_{\mathit{th},\mathit{lc}}\phantom{\rule{3.33333pt}{0ex}}\left[\mathbf{MPa}\sqrt{\mathbf{m}}\right]$ | $\mathbf{\Delta}{\mathit{K}}_{\mathit{th},\mathit{eff}}\phantom{\rule{3.33333pt}{0ex}}\left[\mathbf{MPa}\sqrt{\mathbf{m}}\right]$ | ${\mathit{\nu}}_{1}$ [-] | ${\mathit{\nu}}_{2}$ [-] | ${\mathit{l}}_{1}$ [mm] | ${\mathit{l}}_{2}$ [mm] | |
---|---|---|---|---|---|---|

3.95 | 1.06 | 0.4 | 0.6 | 0.03 | 0.75 |

**Table 5.**Comparison of the normalized fatigue strength resulting from different Weibull parameters $\kappa $ using a return period of $\alpha =1.98$ (specimen B in this study).

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**MDPI and ACS Style**

Oberreiter, M.; Pomberger, S.; Leitner, M.; Stoschka, M.
Validation Study on the Statistical Size Effect in Cast Aluminium. *Metals* **2020**, *10*, 710.
https://doi.org/10.3390/met10060710

**AMA Style**

Oberreiter M, Pomberger S, Leitner M, Stoschka M.
Validation Study on the Statistical Size Effect in Cast Aluminium. *Metals*. 2020; 10(6):710.
https://doi.org/10.3390/met10060710

**Chicago/Turabian Style**

Oberreiter, Matthias, Sebastian Pomberger, Martin Leitner, and Michael Stoschka.
2020. "Validation Study on the Statistical Size Effect in Cast Aluminium" *Metals* 10, no. 6: 710.
https://doi.org/10.3390/met10060710