## 3 Material Tests and Constitutive Modelling
### Pet Foam  
The foam ArmaFORM PET/W AC 135 is a PET-based foam with anisotropic plastic behaviour. The constitutive model and parameters for this material have been borrowed from [31], where the isotropic yield surface proposed by [PERSON] and [PERSON] [42] with non-associated plastic flow was calibrated for both the extrusion and transverse directions. Since the foam is primarily loaded in uniaxial compression during axial crushing, we have considered only the calibration in the extrusion direction. [PERSON]'s yield surface reads  
\[f=\sqrt{3J_{2}+\left(\frac{\alpha I_{1}}{3}\right)^{2}}-B=0, \tag{14}\]  
where \(J_{2}\) is the second invariant of the stress deviator, \(\alpha\) is a dimensionless parameter governing the shape of the yield surface, \(I_{1}\) is first invariant of the stress tensor, and \(B\) is a parameter controlling the size of the yield locus. Therefore, the yield surface is deviatoric- and pressure-dependent through von Mises' equivalent stress and the hydrostatic stress, respectively. 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. Work-hardening was modelled through a direct input of the material stress-strain data under uniaxial compression provided in [31]. Plastic flow is governed by the flow potential \(g\), defined as  
\[g=\sqrt{3J_{2}+\left(\frac{\beta I_{1}}{3}\right)^{2}}=0. \tag{15}\]  
Notice that the flow potential is similar to the yield function in Equation (14) except for the parameter \(\beta\) instead of \(\alpha\) and the absence of the scalar \(B\). Therefore, \(\beta=\alpha\) would give associated plastic flow. If we introduce a parameter \(\
u^{\mathrm{P}}\) as  
\[\beta=\frac{3}{\sqrt{2}}\sqrt{\frac{1-2\
u^{\mathrm{P}}}{1+\
u^{\mathrm{P}}}}. \tag{16}\]  
The parameter \(\
u^{\mathrm{P}}\) was obtained from [PERSON] et al. [31], where low strain-rate, uniaxial compression tests were run until densification of the foam. The value of \(\
u^{\mathrm{P}}\) was computed by measuring the permanent lateral deformation of the post-mortem specimens and dividing it by their permanent axial deformation. All parameters required for the constitutive model of the PET foam were borrowed from [31], and are summarised in Table 4.  
A representative experimental true stress-strain curve is compared with the numerical model in Figure 6 for the extrusion direction, with positive sign. The minor softening exhibited by the material at low strain levels was neglected in exchange for a smoother computation of the plastic flow. The density of an extruded PET foam can be set during its fabrication. Higher densities imply lower void ratios and thus higher crushing strength, in exchange for a heavier material. Although [PERSON] et al. [31] only reported material tests on the 135 kg/m\({}^{3}\) version of this foam, the supplier offers this material in five other densities between 80 kg/m\({}^{3}\) and 200 kg/m\({}^{3}\). The relationship between density and strength for this material can be approximated by  
\[\sigma_{0}^{\mathrm{y}}\ [\mathrm{MPa}]=0.0221899\rho\ [\mathrm{kg/m^{3}}]-0.7219274, \tag{17}\]  
where \(\sigma_{0}^{\mathrm{y}}\) is the initial yield stress and \(\rho\) is the density expressed in kg/m\({}^{3}\)[31]. The work-hardening was assumed to be multiplied by the same ratio as the initial yield stress in the current investigation. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Property or Parameter** & **Transverse Direction** & **Extrusion Direction** \\ \hline Density, \(\rho\) [t/m\({}^{3}\)] & 0.135 & 0.135 \\ [PERSON]’s modulus, \(E\) [MPa] & 20.41 & 59.01 \\ Elastic Poisson ratio, \(\
u\) & 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 \(\
u^{\mathrm{P}}\) & 0.09 & 0.07 \\ \(\beta\) & 1.84 & 1.72 \\ \hline \hline \end{tabular}
\end{table}