## 3 Material Tests and Constitutive Modelling
### Cork Agglomerate  
The mechanical behaviour of cork agglomerate Croccork NL25 was modelled with the hyperelastic compressible constitutive law proposed by [PERSON] [39] and [PERSON] [40], where the strain energy density function \(U\) is in the form  
\[U=\sum_{i=1}^{N}\frac{2\mu_{i}}{\alpha_{i}^{2}}\Bigg{[}\lambda_{1}^{\alpha_{i }}+\lambda_{2}^{\alpha_{i}}+\lambda_{3}^{\alpha_{i}}-3+\frac{1}{\beta_{i}} \left(J^{-\alpha_{i}\beta_{i}}-1\right)\Bigg{]}, \tag{10}\]  
where \(N\) determines the number of terms, \(\mu_{i}\), \(\alpha_{i}\) and \(\beta_{i}\) are material parameters to be calibrated, \(\lambda_{i}\) are the principal stretches, and \(J\) is the volume ratio computed as the determinant of the deformation gradient. The volume ratio is used in the second part of the strain energy density function equation because cork is a collapsible material and hyperelastic incompressibility cannot be assumed. This model was successfully applied to impact in a similar cork agglomerate by [PERSON] et al. [41], the difference being they coupled a Mullins dissipation model for cyclic loading. Since the current investigation deals with proportional loading only, such dissipative extension is not necessary. The \(\beta_{i}\) parameters are related to the material's Poisson's ratio, and therefore govern the compressibility of each term. It can be shown that this relation reads  
\[\beta_{i}=\frac{\
u_{i}}{1-2\
u_{i}}. \tag{11}\]  
A [PERSON]'s ratio of 0.1 was experimentally determined from uniaxial compression tests, therefore we set \(\beta_{i}\) to 0.125 for all terms. The parameters \(\mu_{i}\) and \(\alpha_{i}\) were identified from uniaxial compression tests on cylindrical samples of cork agglomerate with a nominal diameter of 50.5 mm and a nominal thickness of 12 mm. The tests were conducted using an HBM-C2 50 kN load cell with a gain of 2 mV/V and two Vishay HS50 linear variable differential transformers (LVDT). The actuator moved at a constant velocity of 1.48 mm/min. Thin sheets of polytetrafluoreten were placed between the specimens and the machine to minimise friction. The test set-up is displayed in Figure 4a. Engineering stress-strain curves were computed from the tests and used for the identification of the parameters \(\mu_{i}\) and \(\alpha_{i}\). To that end, the first Piola-Kirchhoff stress -or engineering stress- was derived from the strain potential in Equation (10). In principal axes, the first Piola-Kirchhoff stress in direction \(j\), \(P_{j}\), is obtained as  
\[P_{j}=\frac{\partial U}{\partial\lambda_{j}}=\frac{2}{\lambda_{j}}\sum_{i=1}^{N }\frac{\mu_{i}}{\alpha_{i}}\Big{(}\lambda_{j}^{\alpha_{i}}-J^{-\alpha_{i}\beta _{i}}\Big{)}. \tag{12}\]  
The volume ratio under uniaxial compression is \(J=\lambda_{3}^{1-2\
u}\), where \(\lambda_{3}\) is the stretch in the compression direction. By inserting this expression in Equation (12), applying Equation (11) for a constant Poisson's ratio \(\
u\) and simplifying the expression, the first Piola-Kirchhoff stress in the compression direction \(P_{3}\) reads  
\[P_{3}=\frac{2}{\lambda_{3}}\sum_{i=1}^{N}\frac{\mu_{i}}{\alpha_{i}}\Big{(} \lambda_{3}^{\alpha_{i}}-\lambda_{3}^{-\alpha_{i}\
u}\Big{)}. \tag{13}\]  
If we transform the stretch into engineering strain \(e\) by doing \(\lambda_{3}=1+e_{3}\), we can fit Equation (13) to the experimental data from the uniaxial compression tests and thus identify \(\mu_{i}\) and \(\alpha_{i}\). This fit was done for \(N=1,\ 2,\ 3\), and the comparison with the representative experimental engineering stress-strain curve is shown in Figure 4b, where compressive stress and strain are plotted with positive sign for clarity. As it can be seen, the curves are quite similar in the three cases. However, a Drucker stability check was run for each case showing that only the parameters fitted with \(N=2\) produced a Drucker-stable material for all strain levels and states. Therefore, we opted for including two terms in the strain energy density function.