# Experimental Evidence of Specimen-Size Effects on EN-AW6082 Aluminum Alloy in VHCF Regime

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## Abstract

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## Featured Application

**The work provides a robust methodology for the determination of fatigue life expectation in the VHCF regime of the EN-AW6082 aluminium alloy, taking into account the influence of specimen size. In this way, correct fatigue life expectation for EN-AW6082 aluminium alloy full-scale components can be obtained.**

## Abstract

^{9}cycles, by means of the ultrasonic fatigue testing machine developed by Italsigma

^{®}(Italy). Three specimens groups were considered, with a diameter in the middle cross-section ranging from 3 mm up to 12 mm. The stress field in the specimens was determined numerically and by strain gauge measurements in correspondence of the cross-section surface. The dispersion of experimental results has been accounted for, and data are reported in P-S-N diagrams. The decrease in fatigue resistance with increasing specimen size is evident. Theoretical explanation for the observed specimen-size effect is provided, based on Fractal Geometry concepts, allowing to obtain scale independent P-S*-N curves. The fatigue life expectation in the VHCF regime of the EN-AW6082 aluminium alloy full-scale components is rather overestimated if it is assessed only from standard small specimens of 3 mm in diameter. Experimental tests carried out on larger specimens, and a proper extrapolation, are required to assure safe structural design.

## 1. Introduction

^{10}cycles in a very short time in comparison with traditional testing techniques, such as servo-hydraulic or rotating bending machines [8]. In addition, another important issue regarding the VHCF field is the specimen-size effect on the fatigue resistance in the ultralong life regime. In fact, VHCF tests are commonly performed on specimens with a standard diameter of 3 mm, so that the fatigue resistance of full-scale components has to be extrapolated through theoretical models, which nevertheless have not yet been fully validated experimentally by using ultrasonic fatigue testing machines. To this aim, Furuya [9] performed VHCF tests on high-strength steel specimens with a diameter in the middle cross-section ranging from 3 mm up to 8 mm. From these experimental campaigns, he found that the larger the risk volume, the lower the fatigue limit, where the risk volume is usually defined as the region where the stress is higher than the 90% of the peak value [10,11]. Moreover, Murakami et al. [12,13] found that the internal fatigue cracks origin is the largest defect within the risk-volume. In addition, the probability to find defects with larger sizes increases with the risk-volume, so that a decrease in the fatigue resistance in the VHCF region can be expected. A few years ago, Tridello et al. [14,15,16] designed an innovative specimen shape with a Gaussian profile, which can be used to enlarge the risk volume of the specimen. Therefore, they were able to test a high-strength steel, AISI H13 steel, and to analyze results from a wide range of different risk volumes from 194 mm

^{3}up to 2300 mm

^{3}[17,18]. More recently, in order to increase the range of tested risk volume, the same authors carried out VHCF experiments on AISI H13 steel hourglass and gaussian samples with risk volumes of 55 mm

^{3}and 5000 mm

^{3}, respectively [19]. Tridello et al. [20,21] also investigated the VHCF response of SLM AlSi10Mg and SLM Ti6Al4V gaussian specimens with a large loaded volume, pointing out that specimen-size effects on the VHCF region are observed in non-ferrous metallic materials, like aluminum alloy, as well. Xue et al. [22] performed fatigue tests on Al-Si-Cu cast alloy specimens of different sizes beyond ${10}^{9}$ cycles with an ultrasonic fatigue testing machine operating at 20 kHz and $R=-1$. Firstly, they observed that pores within the material act as a preferential site for the fatigue crack initiation. Secondly, they observed experimentally that, for a certain number of cycles, the larger the risk volume of the specimen, the lower the fatigue life. In the present investigation, fully reversed ultrasonic fatigue tests up to ${10}^{9}$ cycles were performed on hourglass specimens made of EN-AW6082-T6 aluminum alloy, which is characterized by good tensile strength, weldability, and an excellent corrosion resistance. Furthermore, in order to investigate the specimen-size effects on this material, three different dimensions of specimens were considered, with diameter ranging between 3 mm and 12 mm. Firstly, the optimal geometry of the specimens was designed based on numerical finite element simulations. Secondly, strain gauge calibrations were carried out before the tests to validate the assumed stress distribution in the designed hourglass specimens. Furthermore, the experimental results were analyzed, and P-S-N curves were obtained for the three different specimens. Finally, by exploiting lacunar fractality concepts [23], the specimen-size effect on fatigue resistance was theoretically assessed, and scale invariant generalized P-S*-N curves provided.

## 2. Materials and Methods

#### 2.1. Materials

#### 2.2. Ultrasonic Fatigue Testing Machine

^{®}(UFTM MU90) and it is shown in Figure 1. The UFTM MU90 is equipped with an ultrasonic generator (Branson

^{®}DCX Series S 4 kW), which provides an electric signal at the piezoelectric transducer at 20 kHz (Branson

^{®}CR-20). Therefore, thanks to the converse piezoelectric effect, the latter converts the electric signal in a sinusoidal mechanical vibration with the same frequency of 20 kHz. In addition, it is possible to vary the amplitude of the mechanical vibration of the piezoelectric converter in the 5.3–21.7 μm range, by changing the output voltage setting of the ultrasonic generator. The piezoelectric converter is rigidly connected to a booster (Branson

^{®}2000X Series Gold) through a screw connection, which provides fixed support to the whole mechanical system and amplification of the mechanical vibration equal to 1.5. Finally, a catenoidal horn (Branson

^{®}126-192), characterized by an amplification factor of 2.0, is assembled in line with the booster with the aim of magnifying the displacement provided to the specimen, which is connected with an M6 screw. Therefore, the mechanical vibration is transmitted from the piezoelectric transducer to the specimen using the two mechanical amplifiers, so that an amplitude variable from 16 μm up to 50 μm is guaranteed at the free end of the specimen, where the maximum allowed value is evaluated in order to prevent unwanted failures of the horn. Furthermore, the mechanical vibration amplitude at the bottom of the specimen is monitored through an eddy current sensor (Micro-Epsilon

^{®}eddyNCDT 3300/3301) to keep the strain amplitude in the middle section of the specimen constant by adjusting the power setting of the ultrasonic generator. Notice that, since the piezoelectric transducer works in the 19.5–20.5 kHz range, all the components of the resonant system, as well as the specimen, must be designed in order to have an axial fundamental frequency included within the same interval. It is also worth noting that, when the fatigue crack propagates inside of the specimen up to a critical amount, the fundamental frequency falls below the lower limit of 19.5 kHz. Consequently, the test is automatically interrupted, since the piezoelectric transducer is no longer able to excite the mechanical components. Furthermore, an air-cooling system is used to control the temperature increment in the sample which could occur due to internal heat arising from high-speed deformations in ultrasonic fatigue tests [27]. At the same time, the temperature is monitored during the test with a pyrometer (Optris

^{®}CT LT22CF), which is characterized by an accuracy of ±1 °C, a resolution of 0.1 °C, and a measurable temperature range comprised between −50 °C and 975 °C. Besides, the ultrasonic fatigue testing machine (MU90) can operate discontinuously, in a pulse-pause mode, in order to avoid excessive heating of the specimen. Finally, the UFTM is equipped with a load cell able to provide a maximum constant preload of 1.5 kN, so that it is possible to carry out tests with non-zero mean stresses ($R>-1$).

#### 2.3. Specimens Design

^{®}Workbench. The axis-symmetry of the problem can be exploited to set up a 2D reduced model, discretized with 8-node axisymmetric quadrilateral elements, so that the computational cost of the analysis has been drastically reduced compared to a fully 3D finite element simulation. At first, a modal analysis was carried out to check that the resonance frequency of the specimens was as close as possible to the working frequency of the machine, i.e., 20 kHz. Subsequently, an harmonic analysis was performed to determine the stress concentration factor, ${k}_{t}$, which is evaluated according to [14]. In addition, the risk-volume has been evaluated, ${V}_{90}$, which is usually defined as the region where the stress amplitude is higher than the 90% of the peak value [9,11,28]. The above-mentioned dynamic mechanical properties of three different specimens, as well as the risk-volume, are summarised in Table 3, whereas Figure 2 shows a photograph of the specimens.

## 3. Results

#### 3.1. Experimental Results and Fracture Surface Analysis

#### 3.2. P-S-N Curves

#### 3.3. Theoretical Interpretation of the Specimen-Size Effect on Fatigue Resistance

^{*}-N curve, which is represented by the following equation:

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

^{®}for the collaboration and its kind availability. SEM fractography was carried out at the J-Tech center of the Politecnico di Torino, whose collaboration is deeply acknowledged.

## Conflicts of Interest

## Abbreviations

VHCF | Very High Cycle Fatigue |

SLM | Selective Laser Melting |

AISI | American Iron and Steel Institute |

P-S-N | Probabilistic Stress-Life |

UFTM | Ultrasonic Fatigue Testing Machine |

SEM | Scanning Electron Microscope |

CDF | Cumulative Distribution Function |

MLM | Maximum Likelihood Method |

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**Figure 3.**Specimen of 3 mm in diameter: (

**a**) specimen geometry; (

**b**) results of the numerical analysis.

**Figure 4.**Specimen of 6 mm in diameter: (

**a**) specimen geometry; (

**b**) results of the numerical analysis.

**Figure 5.**Specimen of 12 mm in diameter: (

**a**) specimen geometry; (

**b**) results of the numerical analysis.

**Figure 9.**SEM images details of the propagation zone and striation marks of the specimen of 12 mm in diameter.

Element | Si | Mg | Mn | Cu | Fe | Cr | Zi | Ti |
---|---|---|---|---|---|---|---|---|

Min% | 0.70 | 0.60 | 0.40 | |||||

Max% | 1.30 | 1.20 | 1.00 | 0.10 | 0.50 | 0.25 | 0.20 | 0.10 |

Diameter of Bar | $\mathit{\rho}$ (kg/m${}^{3}$) | ${\mathit{E}}_{\mathit{d}}$ (GPa) | ${\mathit{\sigma}}_{\mathit{y}}$ (MPa) | ${\mathit{\sigma}}_{\mathit{u}}$ (MPa) | ${\mathit{\u03f5}}_{\mathit{u}}$ (%) | Brinnel Hardness (HB) |
---|---|---|---|---|---|---|

20 mm | 2713 | 72.3 | – | – | – | 91$\phantom{\rule{0.166667em}{0ex}}\pm \phantom{\rule{0.166667em}{0ex}}1.4$ |

30 mm | 2700 | 70.1 | 357$\phantom{\rule{0.166667em}{0ex}}\pm \phantom{\rule{0.166667em}{0ex}}1.3$ | 375$\phantom{\rule{0.166667em}{0ex}}\pm \phantom{\rule{0.166667em}{0ex}}0.2$ | 11$\phantom{\rule{0.166667em}{0ex}}\pm \phantom{\rule{0.166667em}{0ex}}1.0$ | 91$\phantom{\rule{0.166667em}{0ex}}\pm \phantom{\rule{0.166667em}{0ex}}1.7$ |

**Table 3.**Dynamic mechanical properties and risk-volume of the investigated three different specimens.

Specimen | ${\mathit{k}}_{\mathit{\sigma}}$(MPa/m) | ${\mathit{f}}_{\mathbf{FEM}}$ (Hz) | ${\mathit{k}}_{\mathit{t}}$ (–) | ${\mathit{V}}_{90}$ (mm${}^{3}$) |
---|---|---|---|---|

3 mm | 7.474 | 20,213 | 1.021 | 42 |

6 mm | 4.233 | 20,047 | 1.025 | 301 |

12 mm | 3.350 | 19,947 | 1.046 | 1731 |

Diameter | $\overline{\mathit{\alpha}}$ | $\overline{\mathit{\beta}}$ |
---|---|---|

3 mm | 12.2091 | 1.0423 |

6 mm | 3.7579 | 1.1596 |

12 mm | 3.9214 | 1.1508 |

GoF Statistics Tests | ${\mathit{\chi}}^{2}$ | A-D | C-vM | K-S |
---|---|---|---|---|

Actual values | 7.792 | 0.701 | 0.143 | 0.242 |

Critical values | 7.815 | 0.728 | 0.214 | 0.316 |

GoF Statistics Tests | ${\mathit{\chi}}^{2}$ | A-D | C-vM | K-S |
---|---|---|---|---|

Actual values | 7.000 | 0.475 | 0.077 | 0.168 |

Critical values | 7.815 | 0.731 | 0.215 | 0.331 |

GoF Statistics Tests | ${\mathit{\chi}}^{2}$ | A-D | C-vM | K-S |
---|---|---|---|---|

Actual values | 0.600 | 0.156 | 0.019 | 0.107 |

Critical values | 7.815 | 0.731 | 0.215 | 0.338 |

$\Delta {\sigma}_{0,50\%}^{\ast}$(N mm${}^{-1.85})$ | ${\mathit{d}}_{\mathit{\sigma}}$ | n | ${\mathit{R}}^{2}$ | ${\overline{\mathit{\alpha}}}^{\ast}$ | ${\overline{\mathit{\beta}}}^{\ast}$ |
---|---|---|---|---|---|

1221 | 0.15 | 18.5 | 0.966 | 3.8522 | 1.1428 |

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**MDPI and ACS Style**

Invernizzi, S.; Montagnoli, F.; Carpinteri, A.
Experimental Evidence of Specimen-Size Effects on EN-AW6082 Aluminum Alloy in VHCF Regime. *Appl. Sci.* **2021**, *11*, 4272.
https://doi.org/10.3390/app11094272

**AMA Style**

Invernizzi S, Montagnoli F, Carpinteri A.
Experimental Evidence of Specimen-Size Effects on EN-AW6082 Aluminum Alloy in VHCF Regime. *Applied Sciences*. 2021; 11(9):4272.
https://doi.org/10.3390/app11094272

**Chicago/Turabian Style**

Invernizzi, Stefano, Francesco Montagnoli, and Alberto Carpinteri.
2021. "Experimental Evidence of Specimen-Size Effects on EN-AW6082 Aluminum Alloy in VHCF Regime" *Applied Sciences* 11, no. 9: 4272.
https://doi.org/10.3390/app11094272