# Energy Absorption of Aluminium Extrusions Filled with Cellular Materials Under Axial Crushing: Study of the Interaction Effect

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## Abstract

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## 1. Introduction

## 2. Interaction Effect and Component Design

## 3. Material Tests and Constitutive Modelling

#### 3.1. Aluminium Alloy 6063-T5

#### 3.2. Cork Agglomerate

#### 3.3. Pet Foam

## 4. Component Modelling and Interaction Analysis

#### 4.1. Component Design and Simulation

#### 4.2. Analytical Study Using Hanssen’S Interaction Formula

#### 4.3. Energy Assessment

#### 4.4. Failure and Folding Modes

## 5. Discussion

## 6. Conclusions

- The axial quasi-static crushing of filled circular and square thin-walled extrusions with different cross-section dimensions and thicknesses was successfully simulated using a validated numerical model. The results from the simulations were used to adjust an empirical formula to accurately predict the average crushing force of the components within the studied design space, accounting for the interaction effect.
- The values of the parameters from the adjusted Hanssen formula showed that a strong interaction effect exists in the axial crushing of foam- and cork-filled specimens. Higher energy dissipation gains were obtained with the square foam-filled cross-sections, whose constrained collapse yielded an average increase of 55% in the energy dissipation of the full component.
- The values for $\alpha $ and $\beta $ for circular tubes indicated a high contribution from the strength of the core material and the cross-section perimeter to the increase in energy dissipation per unit volume. Foams with higher densities exerted a stronger constraint on the tube fold length thus dissipating more energy through bending.
- Square components showed the highest increase in energy dissipation from the interaction between both materials. The side length had no clear effect on the increase in energy dissipation by the full component, although the interaction effect was increased with thinner tube walls as shown by the low values of $\beta $. As with circular tubes, higher density foams favoured a stronger interaction effect. Cork cores experienced buckling in square components with low thickness and long sides, which led to smaller gains in the energy dissipation by the component.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Conceptual illustration of the interaction effect in the force-displacement curves of axially crushed tubes with foam filler.

**Figure 3.**Dimensions of the aluminium tensile specimens machined from the tubes, in millimetres, adapted from [29] (

**a**), and engineering stress-strain curves of the representative experiment and the inverse model (

**b**).

**Figure 4.**Experimental set-up used in the compression tests of cork (

**a**) and different fits of Equation (13) to a representative experimental test for different number of terms N (

**b**).

**Figure 5.**Scanning electron microscopy images of the cork agglomerate and the PET with density 135 kg/m${}^{3}$ foam at different magnification levels. The arrows in subfigures (

**c**,

**d**) indicate the extrusion direction of the foam.

**Figure 6.**True stress-strain curves from a uniaxial compression test and simulation for Armaform PET/W AC 135 foam in the extrusion direction. Adapted from [31].

**Figure 7.**Force-displacement curves for an empty circular tube with $D=130$ mm and $t=1.9$ mm. Experimental and numerical results.

**Figure 8.**Sampling data points, fitted Hanssen interaction formula, and Hanssen interaction term for the cork-filled square tube tube.

**Figure 9.**Sampling data points, fitted Hanssen interaction formula, and Hanssen interaction term for the foam-filled circular tube with a foam density of 80 kg/m${}^{3}$ and 200 kg/m${}^{3}$.

**Figure 10.**Increase of energy dissipation of the full component versus thickness and diameter, and contribution from the aluminium extrusion and friction for the foam-filled circular tube. Sampling data and interpolated surfaces.

**Figure 11.**Increase of energy dissipation of the full component versus thickness and diameter, and contribution from the aluminium extrusion and friction for the cork-filled circular tube. Sampling data and interpolated surfaces.

**Figure 12.**Increase of energy dissipation of the full component versus thickness and foam density, and contribution from the aluminium extrusion and friction for the foam-filled square tube. Sampling data and interpolated surfaces.

**Figure 13.**Increase of energy dissipation of the full component versus thickness and side length, and contribution from the aluminium extrusion and friction for the cork-filled square tube. Sampling data and interpolated surfaces.

**Figure 14.**Deformation comparison between empty and foam-filled circular tubes ($D=130$ mm, $t=1.9$ mm, foam density = 135 kg/m${}^{3}$).

**Figure 15.**Deformation comparison between empty and foam-filled square tubes ($C=102$ mm, $t=2$ mm, foam density = 135 kg/m${}^{3}$).

**Figure 16.**Deformation comparison between foam-filled and cork filled square tubes ($C=115$ mm, $t=1.5$ mm, foam density = 200 kg/m${}^{3}$).

Cross-Section | Dimension | Lower Bound | Upper Bound |
---|---|---|---|

Circular | Diameter (D) | 110 mm | 150 mm |

Wall thickness (t) | 1.0 mm | 3.0 mm | |

Square | Side (C) | 86.5 mm | 118 mm |

Wall thickness (t) | 1.0 mm | 3.0 mm |

**Table 2.**Initial yield stress, work-hardening and failure parameters of the calibrated constitutive model for the AA6063-T5 aluminium extrusion. All quantities are given in megapascals except c, which is dimensionless.

${\mathit{\sigma}}_{0}$ | ${\mathit{Q}}_{1}$ | ${\mathit{\theta}}_{1}$ | ${\mathit{Q}}_{2}$ | ${\mathit{\theta}}_{2}$ | ${\mathit{Q}}_{3}$ | ${\mathit{\theta}}_{3}$ | ${\mathit{W}}_{\mathbf{c}}^{\mathbf{b}}$ | ${\mathit{W}}_{\mathbf{c}}^{\mathbf{s}}$ | ${\mathit{W}}_{\mathbf{c}}^{\mathbf{l}}$ | c |
---|---|---|---|---|---|---|---|---|---|---|

205.1 | 3.42 | 12060 | 57.99 | 1215 | 2.78 | 92.27 | 146.43 | 74.44 | 32.81 | 0.93 |

N | ${\mathit{\mu}}_{1}$ [MPa] | ${\mathit{\alpha}}_{1}$ | ${\mathit{\beta}}_{1}$ | ${\mathit{\mu}}_{2}$ [MPa] | ${\mathit{\alpha}}_{2}$ | ${\mathit{\beta}}_{2}$ |
---|---|---|---|---|---|---|

2 | 3.067 | 9.276 | 0.125 | 0.001 | −6.092 | 0.125 |

**Table 4.**Material properties and model parameters of ArmaFORM PET/W AC 135 PET-based foam in the extrusion and transverse directions. An asterisk indicates that the value was obtained assuming the same $\alpha $ coefficient obtained for the transverse direction of the foam. Only the extrusion direction is considered in the present work. Adapted from [31].

Property or Parameter | Transverse Direction | Extrusion Direction |
---|---|---|

Density, $\rho $ [$\mathrm{t}/{\mathrm{m}}^{3}$] | 0.135 | 0.135 |

Young’s modulus, E [MPa] | 20.41 | 59.01 |

Elastic Poisson ratio, $\nu $ | 0.1 | 0.1 |

${\sigma}_{0.3}^{\mathrm{C}}$ [kPa] | 982.98 | 2529.97 |

${\sigma}_{0.3}^{\mathrm{H}}$ [kPa] | 1383.30 | * 3465.71 |

$\alpha $ | 0.73 | * 0.73 |

Plastic Poisson’s ratio ${\nu}^{\mathrm{p}}$ | 0.09 | 0.07 |

$\beta $ | 1.84 | 1.72 |

**Table 5.**Parameters and fitness of Equation (1) for the reinforced tubes.

Cross-Section | Core | ${\mathit{C}}_{\mathbf{avg}}$ | $\mathit{\alpha}$ | $\mathit{\beta}$ | RMSE (kN) | ${\mathbf{R}}^{2}$ |
---|---|---|---|---|---|---|

Circular | Cork | 0.444 | 0.968 | 2.000 | 2.169 | 0.997 |

Foam | 1.155 | 0.999 | 1.869 | 5.163 | 0.993 | |

Square | Cork | 5.020 | 0.233 | 0.833 | 4.190 | 0.975 |

Foam | 2.188 | 0.272 | 1.098 | 4.919 | 0.991 |

**Table 6.**Average energy dissipation per unit volume for the aluminium extrusions and filling materials. All values in J/cm${}^{3}$.

Tube | Filling | |||||
---|---|---|---|---|---|---|

Cross-Section | Core | Original | Simulation | Original | Simulation | |

Circular | Cork | 50.134 | 55.971 (+11.62%) | 1.633 | 1.844 (+12.92%) | |

Foam | 50.747 | 54.986 (+8.35%) | 1.749 | 2.331 (+33.28%) | ||

Square | Cork | 39.347 | 50.957 (+29.51%) | 1.491 | 1.703 (+14.22%) | |

Foam | 38.774 | 60.230 (+55.33%) | 1.793 | 2.788 (+55.49%) |

**Table 7.**Percentage of failed integration points and average element bending-to-membrane loading ratio. Average value and standard deviation.

Cross-Section | Core | Integration Point Failure (%) | Bending-to-Membrane Loading Ratio, $\mathbf{\Omega}$ |
---|---|---|---|

Circular | Cork | 5.270 ± 1.840 | 0.794 ± 0.035 |

Foam | 4.214 ± 2.020 | 0.885 ± 0.023 | |

Square | Cork | 8.215 ± 2.338 | 0.777 ± 0.027 |

Foam | 5.161 ± 1.967 | 0.812 ± 0.011 |

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**MDPI and ACS Style**

Paz, J.; Costas, M.; Delgado, J.; Romera, L.; Díaz, J. Energy Absorption of Aluminium Extrusions Filled with Cellular Materials Under Axial Crushing: Study of the Interaction Effect. *Appl. Sci.* **2020**, *10*, 8510.
https://doi.org/10.3390/app10238510

**AMA Style**

Paz J, Costas M, Delgado J, Romera L, Díaz J. Energy Absorption of Aluminium Extrusions Filled with Cellular Materials Under Axial Crushing: Study of the Interaction Effect. *Applied Sciences*. 2020; 10(23):8510.
https://doi.org/10.3390/app10238510

**Chicago/Turabian Style**

Paz, Javier, Miguel Costas, Jordi Delgado, Luis Romera, and Jacobo Díaz. 2020. "Energy Absorption of Aluminium Extrusions Filled with Cellular Materials Under Axial Crushing: Study of the Interaction Effect" *Applied Sciences* 10, no. 23: 8510.
https://doi.org/10.3390/app10238510