## 3 Material Tests and Constitutive Modelling
### Aluminium Alloy 6063-T5  
The circular extrusions were made of AA6063 in temper T5. We extracted tensile specimens with the dimensions provided in Figure 2(a) using electrical discharge machining (EDM) in a pool  
Figure 2: Dimensions of the studied components, given in millimetres. of de-ionized water to prevent the material from altering its microstructure due to heating. Three specimens were tested at a constant cross-head velocity of 1 mm/min, as described in [29]. Strains were measured with digital image correlation (DIC) using a Prosilica GC2450 camera to take pictures of the gauge's surface every second. This surface was previously painted with a speckle pattern to enable the DIC software eCorr [30] to compute the strain fields. The mechanical behaviour of this alloy was modelled with an hypoelastic-plastic constitutive law, adopting an elastic modulus of 70 GPa and a Poisson's ratio of 0.33 for the elastic part. The yield function \(f\) can be written as  
\[f=\sigma_{\text{eq}}-(\sigma_{0}+R)\leq 0, \tag{2}\]  
where \(\sigma_{\text{eq}}\) is the equivalent stress, \(\sigma_{0}\) is the initial yield stress, and \(R\) represents the isotropic hardening. We adopted [PERSON] yield criterion with an associated flow rule, so the equivalent stress in Equation (2) is given in terms of the deviatoric stress tensor \(\sigma_{ij}^{\prime}\) as \(\sigma_{\text{eq}}=\sqrt{\frac{3}{2}\sigma_{ij}^{\prime}\sigma_{ij}^{\prime}}\), and the direction of the plastic flow is determined by adopting \(f\) as the flow potential. Therefore, the components of the plastic strain rate tensor \(\varepsilon_{ij}^{\text{P}}\) are computed as  
\[\varepsilon_{ij}^{\text{P}}=\lambda\frac{\partial f}{\partial\sigma_{ij}}, \tag{3}\]  
where \(\lambda\) is the plastic multiplier computed from the consistency condition \(\dot{\lambda}\dot{f}=0\). A Vocc work-hardening rule with three terms was fitted to the experimental data. Thus, the strain-hardening \(R\) is represented by  
\[R=\sum_{i=1}^{3}Q_{i}\left[1-\exp\left(-\frac{\theta_{i}}{Q_{i}}p\right) \right], \tag{4}\]  
where \(Q_{i}\) represent the saturation stresses, \(\theta_{i}\) are the initial hardening moduli, and \(p\) is the equivalent plastic strain, defined through its rate \(\dot{p}\) by  
\[\dot{p}=\frac{\sigma_{ij}\varepsilon_{ij}^{\text{P}}}{\sigma_{\text{eq}}}= \dot{\lambda}=\sqrt{\frac{2}{3}\varepsilon_{ij}^{\text{P}}\varepsilon_{ij}^{ \text{P}}}, \tag{5}\]  
since the equivalent plastic strain rate is work-conjugate to the equivalent stress. Notice that the last part of Equation (5) is valid for Mises plasticity only. While a set of parameters for the work-hardening rule was already provided in [31], we refined the calibration using inverse modelling of the tensile tests. The tensile specimens were modelled with 10 solid elements through the thickness in Abaqus/Standard 2019 [32], and we used LS-OPT version 5.2 [33] to find the best-fitting set of strain-hardening parameters. Table 2 provides the values obtained from this calibration. The representative experimental engineering stress-strain curve is compared to the results of the calibrated inverse model in Figure 3b, where an adequate fit is observed. A virtual extensometer with a length of 15 mm was used for both the DIC experimental data and the numerical simulation. \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \(\sigma_{0}\) & \(Q_{1}\) & \(\theta_{1}\) & \(Q_{2}\) & \(\theta_{2}\) & \(Q_{3}\) & \(\theta_{3}\) & \(W_{\text{c}}^{\text{b}}\) & \(W_{\text{c}}^{\text{s}}\) & \(W_{\text{c}}^{\text{I}}\) & \(c\) \\ \hline
205.1 & 3.42 & 12060 & 57.99 & 1215 & 2.78 & 92.27 & 146.43 & 74.44 & 32.81 & 0.93 \\ \hline \end{tabular}
\end{table} 
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.