# Development of a Vibration Technique Based on Geometric Optimization for Fatigue Life Evaluation of Sandwich Composite Structures

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## Abstract

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## 1. Introduction

## 2. Materials and Methods

#### 2.1. Analytical Framework

#### 2.2. Establishing the Face Sheet Properties Values

_{x}, E

_{y}, E

_{z}, the shear modulus of elasticity, G

_{xy}, G

_{xz}, G

_{yz}, and the Poisson ratio, v

_{xy}, v

_{xz}, v

_{yz}. The influence these material properties have in predicting the natural frequencies was examined. A sensitivity analysis was performed in which each individual face sheet material property value was significantly increased and decreased (±60%) while the other values were held constant [39]. Core material property values available from the manufacturer were not varied during this process. The results showed that the face sheet properties E

_{x}, E

_{y}, G

_{xz}, G

_{yz}, and v

_{xy}, v

_{xz}, v

_{yz}had little or no influence in predicting the natural frequencies. Thus, the need to accurately obtain ten variables for FEA was reduced to three (ρ, E

_{z}, and G

_{xy}).

_{z}, and the transverse modulus of elasticity, G

_{xy}, were 1500 kg/m

^{3}, 8000 MPa, and 5500 MPa, respectively, yielding an initial objective function value of 0.02. This process was terminated after six iterations, when the objective function yielded a value of 6.27 × 10

^{−4}. The minimizing behavior of the objective function using this method is shown in Figure 3.

#### 2.3. Specimen Design

_{ult}), in which σ corresponds to the Von Mises stress and σ

_{ult}is the average ultimate stress of the several sandwich beams tested.

#### 2.4. Experimental Set-Up

_{ult})) of 0.77, 0.70, 0.62, and 0.55 was used for fatigue testing. The contact stress was minimized by using damping rubbers around the loading pins. A total of twenty specimens were tested.

## 3. Experimental Results

_{f}/t

_{c}= 0.08). In the original S-N curve generated, the effect of contact stress was not considered in the three-point bending stress ratio calculation. It is clearly evident that the fatigue life predicted with vibration testing yields a longer life than that predicted by three-point bending at the same stress level. One possible explanation for this behavior can be attributed to the jig-beam contact (see Figure 11) used to hold the three-point bending specimens. The vibration testing does not contain any contact points (see Figure 13). The Von Mises centerline stresses were calculated with a typical three-point bending test formulation that does not consider the effects of contact stress [43]. By considering contact stress [44,45], representative values of the Von Mises centerline stresses occurring in three-point bending can be estimated using Equation (2) [45].

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 8.**Stress profile pattern after several variations in the values of H1–H4 during the iterative process.

**Figure 10.**Comparison of the stress profiles obtained with Three-Point Bending and the final vibration test specimen (final Von Mises design profile).

**Figure 12.**Final sandwich specimens design used in fatigue vibration (total length 78 cm, end height 6 cm).

**Figure 13.**Experimental setup for composite specimen vibration using the double cantilever technique.

**Figure 16.**Comparison of sandwich composite fatigue life curves. Failures obtained during three-point bending test and vibration test.

**Figure 17.**Fatigue lifetime S-N curves for Vibrations and Three-Point Bending. The TPB stress ratio calculation takes into account the effect of contact stress.

Property | Face Sheet | Foam Core |
---|---|---|

Density (kg/m^{3}) | 1117 | 96 |

Longitudinal Modulus of Elasticity E_{Z} (MPa) | 40 × 10^{3} | 21.507 |

Transversal Modulus of Elasticity * E_{x}, E_{y} (MPa) | 40 × 10^{3} | 21.507 |

Longitudinal Shear Modulus of Elasticity G_{XY} (MPa) | 10 × 10^{3} | 8.994 |

Transversal Shear Modulus of Elasticity * G_{XZ}, G_{YZ} (MPa) | 10 × 10^{3} | 8.626 |

Poisson’s Ratio * ν_{xy}, ν_{xz}, ν_{yz} | 0.35 | 0.3 |

Stress Ratio | Experimental Result (Hz) | Predicted Frequency (Hz) | Average Number of Cycles to Failure (Vibrations) | Average Number of Cycles to Failure (Three-Point Bending) |
---|---|---|---|---|

0.77 | 32.5 | 31 | 5954 | 4507 |

0.70 | 31.95 | 31 | 18,078 | 14,726 |

0.62 | 32.25 | 31 | 85,461 | 66,741 |

0.55 | 32.5 | 31 | 186,876 | 166,742 |

**Table 3.**Experimental results from Three-Point Bending and Vibration tests, considering the contact stress.

Corrected Contact Stress Ratio | Average Cycles to Failure Using Three-Point Bending | Predicted Cycles to Failure Using Vibrations | Error Difference (Using TPB as Base) |
---|---|---|---|

0.78 | 4507 | 4742 | 5.22% |

0.71 | 14,726 | 15,332 | 4.12% |

0.63 | 66,741 | 58,586 | 12.22% |

0.56 | 166,742 | 189,365 | 13.56% |

Frequency, Hz | Time per Data Point | Time Required for Obtaining 10 Data Points | Time Required for Curve Four different Stress Ratios (10 Data Points/Stress Ratio) |
---|---|---|---|

5 | 5.5 h | 2.29 days | 9.16 days |

32 | 52 min | 8.67 h | 1.45 days |

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**MDPI and ACS Style**

Menegozzo, M.; Just-Agosto, F.A.; Serrano Acevedo, D.; Shafiq, B.; Cecchini, A.; Valencia Bravo, J.M.; Vaidya, U.K. Development of a Vibration Technique Based on Geometric Optimization for Fatigue Life Evaluation of Sandwich Composite Structures. *Appl. Sci.* **2022**, *12*, 16.
https://doi.org/10.3390/app12010016

**AMA Style**

Menegozzo M, Just-Agosto FA, Serrano Acevedo D, Shafiq B, Cecchini A, Valencia Bravo JM, Vaidya UK. Development of a Vibration Technique Based on Geometric Optimization for Fatigue Life Evaluation of Sandwich Composite Structures. *Applied Sciences*. 2022; 12(1):16.
https://doi.org/10.3390/app12010016

**Chicago/Turabian Style**

Menegozzo, Marco, Frederick A. Just-Agosto, David Serrano Acevedo, Basir Shafiq, Andrés Cecchini, Joaquín M. Valencia Bravo, and Uday K. Vaidya. 2022. "Development of a Vibration Technique Based on Geometric Optimization for Fatigue Life Evaluation of Sandwich Composite Structures" *Applied Sciences* 12, no. 1: 16.
https://doi.org/10.3390/app12010016