## 3 Numerical Results
### Microscale Analysis
#### 3.1.2 Homogenised Material Properties
The influence of PUF volume fraction on its homogenised elastic properties can be observed in Figure 9. Polynomial curves can be effectively fitted to each property as a function of volume fraction, indicating a predictable relationship. Also, FEM and NNRPIM predictions exhibit good agreement and follow similar trends. However, the most significant discrepancy is observed in Poisson's ratio, particularly at lower volume fractions. These findings pave the way for the development of macroscale models of sandwich beams or plates incorporating a PUF core. By employing the homogenised mechanical properties of PUF with varying volume fractions (due to the presence of cylindrical holes), such models can be used to analyse large-scale applications. Table 5 summarises the homogenised mechanical properties of PUF obtained from FEM-3n, NNRPIMv1, and NNRPIMv2 formulations for different volume fractions. For NNRPIM formulations, it is possible to observe that the homogenised Poisson's ratio for \(v_{f}=0.5\) is higher than \(\
u=0.5\). This observation aligns with findings reported by [PERSON] et al. (2021) for similar foams in sandwich plates [PERSON] et al. (2021). A Poisson's ratio greater than 0.5 violates the assumptions of plane strain, making macroscale applications utilising these homogenised properties impossible. However, alternative approaches such as 2D plane stress deformation and full 3D deformation theories remain viable options. Consequently, to maintain consistency in the numerical framework of the macroscale analysis, only the homogenised mechanical properties obtained from the FEM-3n formulation will be employed. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline  & \(\mathbf{v_{f}}\) & \(\mathbf{E_{x}}\) [MPa] & \(\mathbf{E_{y}}\) [MPa] & \(\mathbf{E_{z}}\) [MPa] & \(\mathbf{G_{xy}}\) [MPa] & \(\mathbf{v_{xy}}\) & \(\mathbf{v_{yx}}\) \\ \hline FEM-3n & 1.00 & 171.430 & 171.430 & 171.430 & 65.935 & 0.300 & 0.300 \\  & 0.95 & 149.211 & 149.212 & 162.913 & 56.481 & 0.300 & 0.300 \\  & 0.90 & 129.549 & 129.546 & 154.408 & 46.524 & 0.304 & 0.304 \\  & 0.85 & 111.759 & 111.757 & 145.651 & 36.680 & 0.311 & 0.311 \\  & 0.80 & 96.673 & 96.677 & 137.288 & 28.290 & 0.320 & 0.320 \\  & 0.75 & 82.702 & 82.702 & 128.669 & 21.019 & 0.333 & 0.333 \\  & 0.70 & 70.094 & 70.102 & 120.084 & 15.235 & 0.351 & 0.351 \\  & 0.65 & 58.514 & 58.513 & 111.435 & 10.757 & 0.374 & 0.374 \\  & 0.60 & 48.078 & 48.083 & 102.884 & 7.458 & 0.403 & 0.403 \\  & 0.55 & 38.870 & 38.865 & 94.580 & 5.110 & 0.437 & 0.437 \\  & 0.50 & 30.106 & 30.106 & 85.801 & 3.332 & 0.480 & 0.480 \\ \hline NNRPIMv1 & 1.00 & 171.430 & 171.430 & 171.430 & 65.935 & 0.300 & 0.300 \\  & 0.95 & 146.173 & 145.926 & 161.800 & 54.876 & 0.301 & 0.301 \\  & 0.90 & 125.357 & 125.447 & 152.704 & 44.252 & 0.306 & 0.306 \\  & 0.85 & 107.416 & 107.409 & 143.533 & 34.231 & 0.314 & 0.314 \\  & 0.80 & 92.007 & 92.139 & 134.751 & 25.747 & 0.324 & 0.324 \\  & 0.75 & 78.089 & 78.303 & 125.970 & 18.779 & 0.340 & 0.340 \\  & 0.70 & 65.625 & 65.713 & 117.131 & 13.302 & 0.360 & 0.360 \\  & 0.65 & 54.027 & 54.130 & 108.314 & 9.186 & 0.388 & 0.388 \\  & 0.60 & 43.397 & 43.367 & 99.035 & 6.163 & 0.422 & 0.422 \\  & 0.55 & 34.448 & 34.388 & 90.636 & 4.125 & 0.461 & 0.461 \\  & 0.50 & 25.769 & 25.720 & 81.410 & 2.587 & 0.511 & 0.511 \\ \hline NNRPIMv2 & 1.00 & 171.430 & 171.430 & 171.430 & 65.935 & 0.300 & 0.300 \\  & 0.95 & 146.234 & 146.098 & 161.801 & 54.882 & 0.301 & 0.301 \\  & 0.90 & 125.401 & 125.466 & 152.704 & 44.172 & 0.305 & 0.305 \\  & 0.85 & 107.391 & 107.438 & 143.537 & 34.173 & 0.314 & 0.314 \\  & 0.80 & 92.091 & 92.146 & 134.758 & 25.710 & 0.324 & 0.324 \\  & 0.75 & 78.180 & 78.321 & 125.977 & 18.750 & 0.340 & 0.340 \\  & 0.70 & 65.669 & 65.736 & 117.133 & 13.284 & 0.360 & 0.360 \\  & 0.65 & 54.072 & 54.162 & 108.317 & 9.169 & 0.387 & 0.387 \\  & 0.60 & 43.484 & 43.420 & 99.035 & 6.152 & 0.421 & 0.421 \\  & 0.55 & 34.466 & 34.421 & 90.639 & 4.116 & 0.460 & 0.460 \\  & 0.50 & 25.814 & 25.810 & 81.403 & 2.583 & 0.508 & 0.508 \\ \hline \hline \end{tabular}
\end{table}