## 3 Material Tests and Constitutive Modelling
### Aluminium Alloy 6063-T5
Because some cracks were reported in the walls of the aluminium extrusions in the experimental campaign [29], we have included a damage model based on the Cockcroft-Latham (C-L) criterion [34], which was calibrated from the tensile tests data. Moreover, since the thickness of the profiles and the strength of the filling materials was varied in the current investigation, accounting for failure seemed reasonable. According to the C-L criterion, the damage variable \(D\) is computed at each integration point as  
\[D=\int\frac{1}{W_{\text{c}}}\langle\sigma_{1}\rangle\,\text{d}p, \tag{6}\]  
where \(W_{\text{c}}\) is an experimental parameter, \(\sigma_{1}\) is the first -major- principal stress, and \(\langle\bullet\rangle\) represents the Macaulay brackets defined as \(\langle x\rangle=\max\left\{0,x\right\}\). Thus, damage is only accumulated when the first principal stress is positive. The parameter \(W_{\text{c}}\) was obtained from the inverse model of the tensile tests with a refined mesh of solid elements. Despite its simplicity, this damage model successfully predicted damage in a variety of different loading scenarios and aluminium manufacturing routes such as extruded profiles [35], die-cast alloys [36], or additive manufactured aluminium parts [37]. During axial crushing, the walls of the extrusion are subjected to a combination of membrane and bending loads. Membrane loading of ductile materials implies that strains localise at a certain point before failure, while localisation does not take place under bending deformation [38]. Therefore, it is reasonable to modulate the macroscopic ductility according to the ratio between bending and membrane loads in each element. To that end, we incorporated the damage regularisation strategy for shell elements proposed by [PERSON] et al. [35]. This implies that elements under membrane loading--susceptible to strain localisation--get their macroscopic ductility \(W_{\text{c}}\) reduced, while the fracture parameter determined from inverse modelling of tensile tests with a fine solid element mesh is assigned to those elements under pure bending. Furthermore, the adopted damage regularisation model alleviates the mesh dependency of the failure parameter by scaling it according to the element size. This means that smaller elements will have a higher \(W_{\text{c}}\) than larger elements. The regularisation scheme evaluates the bending-to-membrane loading ratio in each element by means of a local indicator \(\Omega\), defined as  
\[\Omega=\frac{1}{2}\frac{\left|\varepsilon_{\text{p,33}}^{\text{T}}-\varepsilon_ {\text{p,33}}^{\text{B}}\right|}{\max\left\{\left|\varepsilon_{\text{p,33}}^{ \text{T}}\right|,\left|\varepsilon_{\text{p,33}}^{\text{B}}\right|\right\}}, \tag{7}\]  
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**). where \(\varepsilon_{\text{P},33}^{\text{T}}\) and \(\varepsilon_{\text{P},33}^{\text{B}}\) are the through-thickness plastic strains at the top and bottom integration points of the element, respectively. Notice that the \(x_{3}\)-axis is aligned with the thickness direction of the shell element in the adopted co-rotational system. Equation (7) bounds the indicator \(\Omega\) between 1 for pure bending and 0 for pure membrane loads, thus the local \(W_{\text{c}}\) parameter for each element can be obtained from the weighted sum of the parameters under pure bending \(W_{\text{c}}^{\text{b}}\) and pure membrane \(W_{\text{c}}^{\text{m}}\),  
\[W_{\text{c}}=\Omega W_{\text{c}}^{\text{b}}+(1-\Omega)\,W_{\text{c}}^{\text{m}}. \tag{8}\]  
In turn, the element ductility under membrane loads depends on the element length-to-thickness ratio \(l_{\text{e}}/t_{\text{e}}\),  
\[W_{\text{c}}^{\text{m}}=W_{\text{c}}^{\text{l}}+\left(W_{\text{c}}^{\text{s}}- W_{\text{c}}^{\text{l}}\right)\exp\left[-c\left(\frac{l_{\text{e}}}{t_{\text{e}}}-1 \right)\right]. \tag{9}\]  
It was shown in [PERSON] et al.