## 2 Interaction Effect and Component Design  
It is a well known fact that during axial crushing of foam-filled tubes the crushing force is higher than the sum of the individual crushing forces of the empty tube and the foam crushed separately. This is mainly due to two mechanical phenomena, namely: (i) the friction between both materials generates additional resistance to crushing, and (ii) the foam works as a constraint for the tube walls and reduces their buckling length, thus increasing the number of folds generated during axial crushing, and therefore the energy absorbed by plastic deformation of the tube. This constraint can also change the global folding mode developed in the tube quite radically, as shown by [PERSON] et al. (2018). The interaction effect in filled tubes is illustrated in Figure 1 using conceptual force-displacement curves.  
In order to shed some more light on the origin on this interaction, validated numerical models of two components consisting of an aluminium extrusion in alloy AA6063-T5 with circular and square cross-sections filled with two different cellular materials were run and analysed. The filling cellular materials were an agglomerate of natural cork (Amorim Corecork NL25) and a polyethylene terephthalate (PET) polymeric foam (Armacell ArmaFORM PET/W AC 135). The cross-sectional dimensions of the aluminium walls and the density of the PET foam were variable in order to investigate their influence on the interaction effects. A schematic representation of the studied components is shown in Figure 2, and the bounds of the variable properties are given in Table 1. Notice that the diameter of the circular extrusion and the side of the square extrusion were bounded in a way such that both tubes had the same perimeter.  
\begin{table}
\begin{tabular}{c c c c} \hline
**Cross-Section** & **Dimension** & **Lower Bound** & **Upper Bound** \\ \hline 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 \\ \hline \end{tabular}
\end{table}
Table 1: Lower and upper bounds for the design parameters of the circular and square tubes.  
Figure 1: Conceptual illustration of the interaction effect in the force-displacement curves of axially crushed tubes with foam filler.  
The use of validated numerical models helps in understanding the source and magnitude of the interaction effect in the current problem, because the energies dissipated by each material in a filled component can be immediately quantified and compared. Moreover, the energy dissipated by friction is also readily available from the simulations.  
Alternatively, the assessment of the interaction between the aluminium tubes and the inner cores can also be done applying the model proposed by [PERSON] et al. [22], where the average crushing force of a component is given by  
\[F_{\text{avg}}=F_{\text{avg}}^{0}+A_{\text{r}}\sigma_{\text{r}}+C_{\text{avg} }\sigma_{\text{r}}^{\alpha}\sigma_{\text{y}}^{(1-\alpha)}b_{\text{m}}^{\beta} t^{(2-\beta)}, \tag{1}\]  
where \(F_{\text{avg}}\) is the average crushing force of the filled component, \(F_{\text{avg}}^{0}\) is the average force of the aluminium tube, \(\sigma_{\text{y}}\) is the yield stress of the aluminium alloy, \(b_{\text{m}}=b-t\) where \(b\) represents the outer diameter (circular tube) or side (square tube) of the aluminium extrusion, \(A_{\text{r}}\) is the area of the reinforcing material, and \(\sigma_{\text{r}}\) refers to its plateau stress. The parameters \(C_{\text{avg}},\alpha,\) and \(\beta\) account for and quantify the additional crushing force stemming from the interaction effect, and are to be fitted to the results of an experimental campaign or a set of numerical simulations.