## 3 Numerical Results
### Macroscale Analysis
#### 3.2.1 Sandwich Cantilever Beam
(**e**) \(v_{f}=0.9\). (**f**) \(v_{f}=1.0\). Figure 12: Variation in the normal stress \(\sigma_{xx}\) along the PUF core, for distinct PUF cores. (**a**) \(v_{f}=0.5\). (**b**) \(v_{f}=0.6\). (**c**) \(v_{f}=0.7\). (**d**) \(v_{f}=0.8\). (**e**) \(v_{f}=0.9\). (**f**) \(v_{f}=1.0\). Figure 13: Variation in the normal stress \(\sigma_{xx}\) along the aluminium bottom face sheet for distinct PUF cores. (**a**) \(v_{f}=0.5\). (**b**) \(v_{f}=0.6\). (**c**) \(v_{f}=0.7\). (**d**) \(v_{f}=0.8\). (**e**) \(v_{f}=0.9\). (**f**) \(v_{f}=1.0\). Figure 14: Variation in the shear stress \(\tau_{xy}\) along the aluminium top face sheet for distinct PUF cores. (a) \(v_{f}=0.5\). (b) \(v_{f}=0.6\). (c) \(v_{f}=0.7\). (d) \(v_{f}=0.8\). (e) \(v_{f}=0.9\). (f) \(v_{f}=1.0\). Figure 15: Variation in the shear stress \(\tau_{xy}\) along the PUF core, for distinct PUF cores. (**a**) \(v_{f}=0.5\). (**b**) \(v_{f}=0.6\). (**c**) \(v_{f}=0.7\). (**d**) \(v_{f}=0.8\). (**e**) \(v_{f}=0.9\). (**f**) \(v_{f}=1.0\). With Figures 11 and 13 (normal stress \(\sigma_{xx}\) at the top and bottom aluminium layer, respectively), it is possible to observe that FEM-4n and NNRPIMv1 produce very close solutions. Regarding NNRPIMv2, the results are not so different from the FEM-4n solutions, as Table 8 initially suggested. With Figures 11 and 13, it is possible to understand that because the integration points of FEM-4n are closer to the top (and bottom) surface of the top (and bottom) aluminium layers, they allow for higher magnitudes for \(\sigma_{xx}\) than the NNRPIM solutions. Regarding the PUF core, Figure 12, it is possible to visualise that for lower PUF densities, NNRPIMv1 is closer to FEM-4n, but for higher PUF densities, NNRPIMv1 starts to approximate the NNRPIMv2 solution. It is also visible that the NNRPIMv2 solution is disturbed in the interface between the PUF core and the top (and bottom) aluminium layers. This stress perturbation is due the nature of the influence domains. The integration points belonging to the PUF core near the interface start to possess nodes inside their influence domains that belong to the aluminium layer. As the PUF core rigidity starts to increase (by increasing the PUF core density), the effect starts to dissipate because the rigidity of the core material starts to approximate the rigidity of the face sheet material. Nevertheless, the magnitude of the normal stress \(\sigma_{xx}\) level within the PUF core is very low, with the highest magnitudes observed in the top and bottom aluminium layers. Regarding the shear stress \(\tau_{xy}\) distribution along the beam thickness, the results show that the highest values can be found at the top and bottom aluminium layers (Figures 14 and 16). Notice that, as for \(\sigma_{xx}\), also shear stress \(\tau_{xy}\) on the integration points closer to the top (and bottom) surface of the top (and bottom) aluminium layers present lower magnitudes than FEM-4n. This can be explained with the position of the integration point. The integration points of FEM-4n are closer to the beam's top (and bottom) surfaces, allowing us to predict higher stress magnitudes. Concerning the PUF core, shown in Figure 15, as expected, the distribution is approximately parabolic and all the formulations allow us to obtain very similar results. The same disturbance effect observed for the normal stress \(\sigma_{xx}\) near the aluminium/PUF interface is visible although highly attenuated.