# Partially-Perforated Self-Reinforced Polyurea Foams
## 3 Results and Discussion
The model is based on linear elasticity, which is sufficient to elucidate the stiffening effect. Given the lack of transitional interphase between the microsphere and the foam wall, as explicated from the inset of the micrograph presented in Figure 5a, the material entrapped in the wall of the microcell and that of the microspheres were taken to be the same. The material was assumed to have an elastic modulus of 10 MPa and a Poisson's ratio of 0.25. It is important to note that changes to the properties would linearly adjust the results given the underlying linear isotropy assumption in the model. The displacement resulting from the applied load was then calculated over one-quarter of the unit cell geometry. A schematic of a representative microcell with the reinforcing microspheres is shown in Figure 5b. Figure 6 shows an example of the analytical results demonstrating the difference in the mechanical response of a unit cell (i.e., microcell, see Figure 5b) without any reinforcement and with 144 uniformly distributed microspheres. In the simulation leading to the results in Figure 6, a modeled microcell without any reinforcing microspheres was first subjected to a load of 30 N distributed over an area defined by a zenith angle of 12\({}^{\circ}\) divided symmetrically. The microcell was assumed to have a diameter of 140 \(\upmu\)m, which is within the range reported by [PERSON] et al. [5]. Subsequently, the barren microcell was reinforced with microspheres with a diameter ratio of 0.1 (i.e., microsphere diameter/microcell diameter = 0.1), which was chosen to be in the same order of magnitude, observed by [PERSON] et al., while producing  
Figure 5: (**a**) High magnification SEM micrographs of the reinforcing microspheres on the cell walls of cured foam sample, and (**b**) a schematic representation of a microsphere-reinforced microcell. As can be seen from the inset, the cross-section confirms that the microspheres are dense. standalone polyurea microspheres [36]. All other loadings and boundary conditions were the same for both cases of the reinforced and barren microcells. The stiffening effect is substantiated based on three observations from Figure 6, namely, the deformation in the zenith and azimuth directions and the overall change in the deformed perimeter. The normalized zenith deformation in response to a 30 N applied load decreased by 35% upon introducing the reinforcing microsphere, where the normalized zenith radius changed from 0.71 to 0.81 for the barren and reinforced microcells, respectively. Similarly, the equator, i.e., normalized azimuth radius, exhibited a 63% reduction in the deformation for microsphere-reinforced microcell. Specifically, the equator for the barren case increased by 25%, while only expanding by 9% for the reinforced case. Therefore, the reduction in deformation under the same loading conditions signifies the increase in the stiffness, regardless of the direction. Remarkably, the addition of the reinforcing microspheres also changed the overall deformation behavior of the entire perimeter, such that the barren microcell experienced elastic instability resembled in the buckling of the wall. This is consistent with the plateau region in a typical foam response, the onset of which is associated with many deformation mechanisms, including elastic instability [2]. On the other hand, the reinforced microcell remained in the elastic region without showing any cell wall buckling, implying the ability of reinforced polyurea foam to manage the incoming low-velocity impact scenarios better than the smooth and barren microcell. [PERSON] et al. recently experimentally demonstrated this observation, where a polyurea foam with a relative density of 0.33 reported \(\sim\)71.7 kJ\(\cdot\)m\({}^{-3}\) of strain energy under the stress-strain curve. In comparison, a benchmark foam with barren microstructure only exhibited energy of 51.9 kJ\(\cdot\)m\({}^{-3}\), for the same strain history [5].