## 3 Results and Discussion
### Mechanical Characterization of Cellular Structures and Insoles
#### 3.1.1 Insole Modeling and Apparent Compressive Stiffness via FEM for the Cellular Structures  
Figure 7a shows the surface obtained from the scanning process. The scanning process was performed on a person standing, of weight 70 kg and height 172 cm. The insole profile was obtained from the Gensole(r) software (see Figure 7b), where the insole profile was selected, and the top surface shape was generated from the surface obtained from the scanning process. Figure 7c shows the computational model of the insole; in the section view, the integration of the cellular structure in the internal part of the insole is displayed. Due to the thickness of the insole, the medial forefoot and medial midfoot areas were the only parts where the unit cells were fully integrated (there is no trim to fit the external geometry). In the thinnest part of the insole, the upper and lower surfaces were practically in contact; therefore, there were no unit cells in this area.  
The apparent Young's modulus (\(\overline{E}\)) of the cellular structures was calculated using the displacement results obtained from the FE models under the compressive loads described in Section 2.2; these were then fed to the equation also described in Section 2.2 and normalized with the modulus of the constituent material (\(\overline{E}/E\)). The material properties used were listed in Table 2. The resulting apparent Young's modulus and its relation to  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Printing Parameter** & **BCC** & **TPMS** \\ \hline Nozzle size (mm) & 0.4 & 0.2 \\ Nozzle temperature (°C) & 230 & 230 \\ Bed temperature (°C) & 50 & 50 \\ Layer height (mm) & 0.28 & 0.1 \\ Printing speed (mm/s) & 40 & 60 \\ Filling (\%) & 100 & 100 \\ Material & TPU & TPU \\ \hline \hline \end{tabular}
\end{table}
Table 3: Printing parameters for insole manufacturing.  
Figure 7: Inside 3D modeling. (**a**) Scanning process, (**b**) profile of the insole, and (**c**) final insole model.  
the corresponding relative density are summarized in Figure 8. Figure 9 includes the deformation of the structures; the structures with lower density have higher deformation.  
Figure 8 includes the stiffness-volume fraction results for both topologies studied. These types of curves enable the identification of the deformation mechanism by means of characterizing the scaling law that governs the property-volume fraction relationship [2]. For porous media, this relationship can be defined by the expression \(\overline{E}=CE\overline{p}^{n}\), where \(C\) is a constant that depends on the cellular material topology, and \(n\) is the scaling factor that takes a value \(n\
eq 1\) for bending-dominated structures [29]. Note from the interpolation included in Figure 8 that the BCC structures show a scaling factor of \(n\approx 2\), commonly obtained in other bending-dominated structures [1,9]. On the contrary, the primitive TPMS resulted in \(n=1\), commonly attributed to stretch-dominated topologies. Finally, for strut-based topologies, as the BCC, the [PERSON]'s criterion can be applied to corroborate its deformation mechanism. Taking [PERSON]'s criterion involves the calculation of the parameter \(M\) using the expression \(M=b-3j+6\), where \(b\) is the number struts in the  
Figure 8: Normalized apparent Young ’s modulus for BCC and TPMS topologies to different relative densities.  
Figure 9: Static simulation displacement results in cellular structure. (**a**) BCC 10.33%, (**b**) BCC 14.67%, (**c**) BCC 20.19%, (**d**) TPMS 10.33%, (**e**) TPMS 14.67%, and (**f**) TPMS 20.19%.  
unit cell and \(j\) is number the joints between struts. Now, if \(M<0\) the topology is bending dominated; here the BCC unit cell has eight struts and nine joints; therefore, \(M=-13\)[30]. Additionally, note from Figure 9a-c that struts bent upon the application of the compressive load. On the other hand, in Figure 9d-f, the constituent elements shorten. These results are in agreement to previously reported data for BCC and primitive porous structures [8; 31; 32; 33].