## 3 Numerical Results
### Microscale Analysis
#### 3.1.2 Homogenised Material Properties  
One of the main goals of the present research work is to study the influence of the volume fraction (\(v_{f}\)) of the PUF on the final homogenised mechanical properties. Therefore, the increasing volume fractions presented in Table 4 were considered for the PUF core. Then, applying Equation (55), for each volume fraction considered, a corresponding radius for the circular hole was calculated; see Table 4.  
Concerning the nodal discretisation to adopt for this analysis, the results of the previous convergence test, Figure 5, allowed us to understand that a nodal density with approximately 4000 nodes for \(v_{f}=0.75\) is sufficiently dense, permitting us to obtain solutions very close to the theoretical converged value. Hence, the RVEs of each volume fraction were discretised with nodal meshes respecting a proportional nodal density; see Figure 8.  
All the constructed models follow the same geometry of Figure 3, with \(L=D=10\) mm and unitary thickness, and the same bulk PUF mechanical properties: \(E=171.43\) MPa, \begin{table}
\begin{tabular}{l c c c c c c c c c c c} \hline \(v_{f}\) & 0.500 & 0.550 & 0.600 & 0.650 & 0.700 & 0.750 & 0.800 & 0.850 & 0.900 & 0.950 & 1.000 \\ R [mm] & 3.989 & 3.785 & 3.568 & 3.338 & 3.090 & 2.821 & 2.523 & 2.185 & 1.784 & 1.262 & 0.000 \\ \hline \end{tabular}
\end{table} 
Table 4: Relation between the volume fraction and the circular hole radius. \begin{table}
\begin{tabular}{l c c c c c c} \hline \hline  & \(E_{x}\) [MPa] & \(E_{y}\) [MPa] & \(E_{z}\) [MPa] & \(G_{xy}\) [MPa] & \(v_{xy}\) & \(v_{yx}\) \\ \hline NNRPIMv1\({}^{*}\) & 78.545 & 78.525 & 126.072 & 18.995 & 0.340 & 0.340 \\ NNRPIMv2\({}^{*}\) & 78.587 & 78.560 & 126.073 & 19.022 & 0.339 & 0.339 \\ \hline  & \(C_{11}\) & \(C_{12}\) & \(C_{21}\) & \(C_{22}\) & \(C_{33}\) & \(C_{44}\) \\ \hline NNRPIMv1\({}^{*}\) & 101.004 & 42.389 & 42.393 & 100.978 & 151.881 & 18.995 \\ NNRPIMv2\({}^{*}\) & 100.976 & 42.293 & 42.285 & 100.941 & 151.857 & 19.022 \\ \hline \end{tabular}
\end{table} 
Table 3: Elastic homogenised mechanical properties and constitutive constants of the RVE with \(v_{f}=0.75\) obtained with NNRPIMv1\({}^{*}\) and NNRPIMv2\({}^{*}\) formulations considering an higher-order integration scheme. \(C_{ij}\) in MPa. Figure 7: Von [PERSON] equivalent stress distribution obtained with NNRPIM considering an higher-order integration scheme: NNRPIMv1 with \(2\times 2\) integration scheme per sub-area (**a**) \(\tilde{\mathbf{f}}_{100}\), (**b**) \(\tilde{\mathbf{f}}_{010}\), and (**c**) \(\tilde{\mathbf{f}}_{01}\). NNRPIMv2 with \(2\times 2\) integration scheme per sub-area (**d**) \(\tilde{\mathbf{f}}_{100}\), (**e**) \(\tilde{\mathbf{f}}_{010}\), and (**f**) \(\tilde{\mathbf{f}}_{001}\). \(G=57.810\) MPa, and \(\
u=0.30\). Then, using the same homogenisation procedure and considering plane strain deformation theory, the homogenised mechanical properties for each volume fraction \(v_{f}\) were calculated. The results are shown in Figure 9.  
Figure 8: Discretisation meshes used with the NNRPIM and FEM analyses. (**a**) \(v_{f}=0.50\), \(2671\) nodes, and \(5178\) triangular elements. (**b**) \(v_{f}=0.55\), \(2937\) nodes, and \(5696\) triangular elements. (**c**) \(v_{f}=0.60\), \(3205\) nodes, and \(6214\) triangular elements. (**d**) \(v_{f}=0.65\), \(3472\) nodes, and \(6732\) triangular elements. (**e**) \(v_{f}=0.70\), \(3739\) nodes, and \(7249\) triangular elements. (**f**) \(v_{f}=0.75\), \(4005\) nodes, and \(7767\) triangular elements. (**g**) \(v_{f}=0.80\), \(4273\) nodes, and \(8285\) triangular elements. (**h**) \(v_{f}=0.85\), \(4539\) nodes, and \(8803\) triangular elements. (**i**) \(v_{f}=0.90\), \(4807\) nodes, and \(9321\) triangular elements. (**j**) \(v_{f}=0.95\), \(5073\) nodes, and \(9839\) triangular elements. Figure 9: Influence of the volume fraction \(v_{f}\) on the homogenized elastic mechanical properties: (**a**) \(E_{x}\), (**b**) \(E_{y}\), (**c**) \(G_{xy}\), and (**d**) \(v_{xy}\).