# Effect of Loading Frequency on the Fatigue Response of Adhesive Joints up to the VHCF Range

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## Abstract

**:**

^{9}cycles follow the same trend: at 25 Hz and 50 Hz, the run-out data were found at 10 MPa, increasing to 15 MPa at 20 kHz. The P–S–N curves showed that frequency effects have a minor impact on the experimental variability and that standard deviation values lie in the range of 0.3038–0.7691 between 5 Hz and 20 kHz. Finally, the trend of fatigue strengths at 2·10

^{6}cycles with the applied loading frequency for selected probability levels was estimated.

## 1. Introduction

^{®}-1277—aluminium 7075 T6) were subjected to symmetrical tension–compression loading (R = −1) tests. The non-singular specimen was accurately designed through analytical relationships based on the inverse Bogy formulation [40], numerically verified with finite element (FE) procedures, and thus tested under fatigue conditions.

^{6}cycles was found to follow a linear trend in a bi-logarithmic chart. Data scattering is present among the test conditions, but this effect had a minor impact on experimental outcomes.

## 2. Materials and Methods

#### 2.1. Material and Testing Devices

^{®}-1277 epoxy resin. This adhesive is adopted for bonding structural components in transportation and general industry. Its characteristics make it a good choice for repair purposes in combination with spot welding or riveting. Glass spheres were pre-mixed by the producer in the resin tube, and these are responsible for maintaining the 0.3 mm thickness.

^{®}-1277 epoxy resin were tested in compliance with [53,54], respectively, whereases values for Ti-6Al-4V were already known from a past study [50]. The viscosity value was assumed to be valid based on the datasheet [55] and no additional tests were performed regarding the variation in the quantity with respect to the temperature. Extracted values for SikaPower

^{®}-1277 refer to a hardening period of 24 h at 23 °C, as suggested by the adhesive supplier.

^{®}-1277 epoxy resin was assessed in [43], where dedicated tensile tests were performed at 2.5 mm/min. The load–displacement curves were comparable, and processed stress data were in line with those extracted from the bulk adhesive in Table 1. Moreover, specimens failed fully cohesively without the macroscopic presence of adhesive detachments, further validating the test results.

^{®}8801 (Norwood, MA, USA) hydraulic testing machine with loading frequencies below 100 Hz. The low-frequency fatigue tests were load-controlled and performed at 5, 25, and 50 Hz.

#### 2.2. Specimen Mechanical Design

#### 2.2.1. Removal of the Stress Singularity

#### 2.2.2. VHCF Specimen Design: FE Model Development and Analyses

- The numerical model of the titanium horn is first analyzed through a modal analysis under free–free conditions to collect the closest natural frequency and mode shape to the UFTM operative range. An operative frequency of ${f}_{n}=\mathrm{20,195}$ Hz is thus selected and adopted for design purposes.
- The 1D elastic wave equations [58] reported in Equation (4) permit the definition of the specimen total length $\left({L}_{tot}\right)$, where ${f}_{n}$ is the frequency of interest and $E,\rho $ are the mechanical properties of the aluminium. This calculation results in ${L}_{tot}=125.5$ mm:$${L}_{tot}=\frac{1}{2{f}_{n}}\sqrt{E/\rho}$$
- A frequency response analysis (FRA) is run at ${f}_{n}=\mathrm{20,195}$ Hz, where the current numerical model is composed using the ${L}_{tot}$ of the titanium horn and the aluminum bar, with a displacement excitation at the horn base in the range of 2.2–18 $\mathsf{\mu}\mathrm{m}$, which corresponds to the capability of the UFTM. Longitudinal stress distributions are collected along the aluminium specimen axis and are depicted in Figure 5. This assessment was useful and necessary to explore the stress magnitudes applied to the whole specimen.
- A 0.3 mm thick adhesive layer is inserted into the mechanical system, thus dividing the aluminium bar in two separated components, now adhesively joined. The adhesive is placed at a suitable distance in order to impose the desired stress magnitude on the adhesive layer. In this work, the stress range from 5.5–46 MPa was selected. In this way, the ${L}_{tot}$ is divided into ${L}_{1}$ = 114.15 mm and ${L}_{2}$ = 11.35 mm, as shown in Figure 5.
- A final numerical re-assessment is performed through an FRA.

## 3. Experimental Results

^{®}8801 servo-hydraulic testing machine. A standard procedure suggested by the adhesive manufacturer was followed to obtain the highest adhesive strength. In particular, abrasive sandpapers were used to prepare the substrate surface to increase the surficial roughness, followed by a cleaning phase with acetone for removing the aluminum powder and impurities. The thin layer of adhesive was thus applied between the substrates and spread to reach a uniform bonding area. Suitable alignment devices were designed to guarantee the substrate co-axiality. The joint curing required one day under a constant load at room temperature and, once the adhesive was fully dried, the surplus was removed from the substrate edges with a cutter. In this way, it was possible to maintain the 0.3 mm adhesive layer without resin concentration effects on the experimental results.

^{®}grips.

^{9}cycles in ultrasonic VHCF tests and to 2 × 10

^{6}cycles in servo-hydraulic low-frequency tests. This difference is mainly attributable to the testing time required by the hydraulic machine working below 100 Hz, which is unable to test specimens in VHCF within a reasonable timeframe.

^{®}-1277 adhesive.

^{®}R2023a script and an optimization toolbox that exploits the Nelder–Mead simplex algorithm in order to identify the entire $\mathsf{\theta}$ vector of the unknowns.

^{6}cycles were collected for three probability levels, namely, 10%, 50%, and 90%. Data were interpolated through a power law scheme, i.e., $\sigma =A\xb7{f}^{b}$, where $\sigma $ is the experimental fatigue strength, $f$ is the applied test frequency, and $A,b$ are the model parameters to be determined. The calculation was performed with the least squares method. The R

^{2}factor was higher than 0.995 for all investigated cases. As demonstrated by the interpolation, in a bi-log chart, the data were quite well approximated. This information is particularly useful, and the results demonstrated a power law increase in the frequency effect. Moreover, this experimental evidence can be considered when predicting strength values not covered by the experimentation.

## 4. Discussion

^{6}were selected for three probability levels (P = 10, 50, 90%), and fatigue strengths were well-approximated using a power law model.

## 5. Conclusions

- At a fixed applied load, specimens tested at an ultrasonic frequency are prone to fail in the very high cycle fatigue range (N > 10
^{7}), whereas specimens subjected to common frequencies fail in the high cycle fatigue range, thus confirming the presence of a frequency effect. - Specimens tested at an ultrasonic frequency express run-out data below 15 MPa compared to 10 MPa in the case of 25 and 50 Hz frequencies. Specimens at 5 Hz do not present run-outs.
- There is an increase in fatigue performance among specimens tested in the low-frequency range, and this is visible through the S–N curve at the 50% probability level.
- Data scattering was investigated through standard deviations, and the experimental results demonstrated a minor impact on fatigue properties (i.e., 0.3038 at 5 Hz, 0.5763 at 25 Hz, 0.4687 at 50 Hz, and 0.7691 at 20 kHz);
- Interpolating data at N = 2 × 10
^{6}for three probability levels (P = 10, 50, 90%) showed the possibility of strength prediction.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Nomenclature

$E$ | Modulus of elasticity |

${\mathrm{f}}_{\mathrm{Y}}$ | Probability density function of the fatigue life |

${\mathrm{F}}_{\mathrm{Y}}$ | Cumulative distribution function of the fatigue life |

${\mathrm{F}}_{\mathrm{Y}|\mathrm{x}}$ | Conditional distribution of fatigue life |

$\mathrm{L}\left[\xb7\right]$ | Maximum likelihood function |

${\mathrm{n}}_{\mathrm{f}}$ | Number of failures |

${\mathrm{n}}_{\mathrm{r}}$ | Number of run-outs |

$R$ | Tension–compression loading ratio |

$r$ | Radial coordinate |

BCs | Boundary conditions |

FE | Finite element |

FRA | Frequency response analysis |

ML | Maximum likelihood |

PID | Proportional–integrative–derivative |

SD | Standard deviation |

UFTM | Ultrasonic fatigue testing machine |

V | Viscosity |

VHCF | Very high cycle fatigue |

$\alpha ,\beta $ | Dundurs parameters |

$\theta $ | Local material angle |

$\mu $ | Shear modulus |

${\mu}_{Y}$ | Mean stress distribution |

$\nu $ | Poisson’s ratio |

$\rho $ | Density |

$\mathsf{\sigma}$ | Stress vector |

${\sigma}_{uts}$ | Ultimate tensile strength |

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**Figure 1.**The UFTM equipment [43].

**Figure 4.**VHCF specimen—Global model [40].

**Figure 5.**Stress distributions resulting from the application of 2.2–18 μm of displacement excitation [43].

**Figure 6.**Adopted local model for FE analyses [19].

**Figure 12.**Strength distributions for 10%, 50%, and 90% probability levels at N = 2 × 10

^{6}cycles.

Quantity | Symbol | Units | Materials | ||
---|---|---|---|---|---|

Ti-6Al-4V | Al. 7075 T6 | SikaPower®-1277 | |||

Modulus of elasticity | E | MPa | 115,000 | 71,955 ± 224 | 2280 ± 533 |

Poisson’s ratio | $\nu $ | / | 0.34 | 0.3 | 0.33 |

Ultimate tensile strength | ${\sigma}_{uts}$ | MPa | 950 | 610 ± 15 | 35.7 ± 1.4 |

Density | $\rho $ | g/cc | 4.395 | 2.8 | 1.1 |

Loss factor | LF | / | 2.96 × 10−4 | 3.5 × 10−3 | 2 × 10−2 |

Viscosity | V | Pa·s | / | / | 430 (at 20 °C) |

$\mathbf{Adhesive}\mathbf{Angle}{\mathsf{\theta}}_{1}[\xb0]$ | $\mathbf{Substrate}\mathbf{Angle}{\mathsf{\theta}}_{2}[\xb0]$ | |
---|---|---|

Plane Stress | Plane Strain | |

10 | 179.6 | 179.6 |

20 | 179.3 | 179.2 |

30 | 178.8 | 178.7 |

40 | 177.9 | 177.5 |

50 | 175.5 | 171.8 |

60 | 121.2 | 70.28 |

70 | 61.35 | 51.09 |

80 | 51.68 | 45.7 |

90 | 48.17 | 43.5 |

100 | 46.97 | 42.64 |

110 | 47.78 | 42.65 |

120 | 60.60 | 44.83 |

Failures at 5 Hz | Failures at 25 Hz | Failures at 50 Hz | Failures at 20 kHz | |
---|---|---|---|---|

Standard deviation (SD) | 0.3038 | 0.5763 | 0.4687 | 0.7691 |

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## Share and Cite

**MDPI and ACS Style**

Pederbelli, D.; Goglio, L.; Paolino, D.; Rossetto, M.; Tridello, A.
Effect of Loading Frequency on the Fatigue Response of Adhesive Joints up to the VHCF Range. *Appl. Sci.* **2023**, *13*, 12967.
https://doi.org/10.3390/app132312967

**AMA Style**

Pederbelli D, Goglio L, Paolino D, Rossetto M, Tridello A.
Effect of Loading Frequency on the Fatigue Response of Adhesive Joints up to the VHCF Range. *Applied Sciences*. 2023; 13(23):12967.
https://doi.org/10.3390/app132312967

**Chicago/Turabian Style**

Pederbelli, Davide, Luca Goglio, Davide Paolino, Massimo Rossetto, and Andrea Tridello.
2023. "Effect of Loading Frequency on the Fatigue Response of Adhesive Joints up to the VHCF Range" *Applied Sciences* 13, no. 23: 12967.
https://doi.org/10.3390/app132312967