# A Model of Damage for Brittle and Ductile Adhesives in Glued Butt Joints

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## Abstract

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## 1. Introduction

## 2. Analytical Method for Damage Prediction

#### 2.1. Classical Solution for Perfect Contact between the Adherents

#### 2.2. Generalized Equilibrium Solution for Imperfect Contact between the Adherents

- The stress field (1), which is continuous across the adhesive and equilibrated by the applied loads;

#### 2.3. Micro-Cracking Damaging Adhesive Model

#### 2.4. Damage Evolution

#### 2.5. Stress–Strain Response

## 3. Numerical Implementation

## 4. Results and Discussion

#### 4.1. Simulation of Pure Tensile Tests

- ${\rho}_{0}=1.14$ for the QS case;
- ${\rho}_{0}=1.07$ for the HR case.

#### 4.2. Simulation of Pure Torsion Tests

- $a=1.58\times {10}^{-3}$ MPa${}^{-1},$ $b=1.05\times {10}^{-6}$ MPa${}^{-4},$ ${\tau}_{0}=50.25$ MPa for the QS case;
- $a=1.50\times {10}^{-3}$ MPa${}^{-1},$ $b=1.78\times {10}^{-7}$ MPa${}^{-4},$ ${\tau}_{0}=64.97$ MPa for the HR case.

- ${\rho}_{0}=0.55,$$\eta =2296.70$ Ns/m, $\omega =-679.34$ N/m for the QS case;
- ${\rho}_{0}=0.51,$$\eta =0.90$ Ns/m, $\omega =-1135.64$ N/m for the HR case.

#### 4.3. Simulation of Combined Tensile–Torsion Tests

## 5. Summary

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Appendix A

## References

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**Figure 1.**(

**a**) Sketch of the tube-to-tube joint with the loading configuration. (

**b**) Longitudinal and traversal sections with dimensions.

**Figure 2.**Stress–strain curves in pure loading conditions: (

**a**) Stress–strain curves of the adhesive layer under a pure tensile load obtained with the proposed model for quasi-static and high-rate loading (solid lines) compared with experimental curves by [9] (dashed lines). (

**b**) Stress–strain curves of the adhesive layer under a pure torsion load obtained with the proposed model for quasi-static and high-rate loading (solid lines) compared with experimental curves by [9] (dashed lines).

**Figure 3.**Stress–strain curves in combined tensile–torsion loading conditions: (

**a**) Tensile stress–strain curves of the adhesive layer obtained with the proposed model for quasi-static and high-rate loading. (

**b**) Torsion stress–strain curves of the adhesive layer obtained with the proposed model for quasi-static and high-rate loading.

**Table 1.**Geometrical parameters of the joint specimen [9].

Quantity | Symbol | Value | Unit |
---|---|---|---|

Outer diameter | D | 26.0 | mm |

Inner diameter | d | 20.0 | mm |

Adhesive thickness | $\epsilon $ | 0.3 | mm |

**Table 2.**Mechanical properties of the joint materials [9].

Quantity | Symbol | Value | Unit |
---|---|---|---|

Adhesive Young’s modulus | ${E}_{0}$ | 4.53 | GPa |

Adhesive Poisson’s ratio | ${\nu}_{0}$ | 0.36 | – |

Adherents’ Young’s modulus | E | 200.00 | GPa |

Adherents’ Poisson’s ratio | $\nu $ | 0.30 | – |

**Table 3.**Experimentally estimated tensile and torsional (shear) strengths of butt-joint specimens studied in [9].

Loading Rate | Loading Angle [deg.] | Tensile Strength [MPa] | Shear Strength [MPa] | Failure Strain in Tension [%] | Failure Strain in Shear [%] |
---|---|---|---|---|---|

QS | 0 | 61.8 | – | 4.0 | – |

HR | 0 | 90.0 | – | 6.1 | – |

QS | 90 | – | 53.2 | – | 37.0 |

HR | 90 | – | 70.0 | – | 32.0 |

QS | $18.0$ | 61.0 | 19.9 | 2.15 | 2.85 |

HR | $15.5$ | 92.1 | 25.5 | 3.55 | 3.69 |

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**MDPI and ACS Style**

Raffa, M.L.; Rizzoni, R.; Lebon, F. A Model of Damage for Brittle and Ductile Adhesives in Glued Butt Joints. *Technologies* **2021**, *9*, 19.
https://doi.org/10.3390/technologies9010019

**AMA Style**

Raffa ML, Rizzoni R, Lebon F. A Model of Damage for Brittle and Ductile Adhesives in Glued Butt Joints. *Technologies*. 2021; 9(1):19.
https://doi.org/10.3390/technologies9010019

**Chicago/Turabian Style**

Raffa, Maria Letizia, Raffaella Rizzoni, and Frédéric Lebon. 2021. "A Model of Damage for Brittle and Ductile Adhesives in Glued Butt Joints" *Technologies* 9, no. 1: 19.
https://doi.org/10.3390/technologies9010019