3.1. Homogenisation Results
The engineering constants obtained from the explicit simulations are reported in
Table 6. As shown, the E
1 simulations led to the estimation of Poisson’s ratios of the cores as well. Moreover, a slight anisotropy was observed for the T45 core. While the triangular honeycombs are known to have isotropic in-plane properties [
30], the difference between the computed E
1 and E
2 is smaller than 5%, and thus, it was considered acceptable. The analytical results are also reported in
Table 7. As shown, the results demonstrate quite a good agreement between the two approaches. The 30% discrepancy obtained for the R20 core is more likely due to the very low value of its modulus, which makes a small difference of 9 MPa significant.
Figure 9a,b showcases the comparison between the experimentally measured and the computationally calculated stiffnesses for the T45 and the R20 specimens, respectively. The numerical values are also reported in
Table 7. As shown, the 2D simulations consistently show some degree of underpredictions, especially for the low span length. This was expected because the 2D analyses did not simulate the presence of the lateral walls, which have a non-negligible impact on the specimens’ stiffness. This is also supported by the fact that the full 3D simulations show a very good agreement with the experimental results.
Considering the results reported above and the fact that the considered prosthetic component presents a relatively long span (see
Section 2.5), it was decided that the obtained engineering constants for the design of the component should be used.
3.2. Design and Optimisation of the 3D Printed Components
Laminated carbon fibre composites generally display better mechanical properties than 3D-printed glass fibre ones. Therefore, the newly designed 3D-printed component must be significantly thicker than the reference one. A thicker 3D-printed composite would also behave as a composite sandwich component, which is generally a very efficient lightweight structure [
25].
A preliminary analytical optimisation was thus performed to identify the best geometrical parameters for the new component. The complex shape of the laminated component prevents an analytical optimisation over its whole shape. For this reason, the analytical optimisation was performed on an equivalent cantilever beam which was representative of a portion of the component highlighted in
Figure 9. The highlighted portion indeed behaves as a cantilever beam, whereas the stiffer vertical portion behaves similar to a constraint, and the contact with the floor introduces the load.
For a cantilever sandwich beam, the skins and the core determine the bending and shear rigidities, respectively. Therefore, the bending and shear rigidities
Kb and
Ks can be obtained via Equations (4) and (5), respectively:
where
Ef and
Gc are the elastic moduli of the fibres and the shear modulus of the core, respectively;
b,
t, and
c are the width of the panel, the thickness of the skins, and the thickness of the core, respectively. Moreover, the beam flexibility, namely the ratio of the tip displacement
δ at a given load
P over that load, and weight
W are calculated as:
where
l is the length of the beam,
g is the acceleration of gravity, and
ρs and
ρc are the densities of the skins and core, respectively. Note that the cores’ densities were calculated using the volume occupied by the Onyx in a single cell times the Onyx density over the total volume of the cell. Moreover, the shear moduli of the cores were obtained via the homogenisation procedure described in
Section 3. Regarding the skins, the properties were assumed to be equal to those of the glass fibres (see
Table 1).
The design variables of the optimisation procedure are
t/l and
c/l, namely the thicknesses of the skins and the core when normalised with the beam of the length. The optimisation constraint imposes that the new design has the same stiffness as the reference prosthesis, which is obtained via the FE simulations described in
Section 2.5. The objective of the optimisation was to minimise the weight. The parameters adopted are reported in
Table 8. Two optimisations were performed, considering a T45 and an R20 core. Therefore, two different designs of the same component will be obtained and further evaluated.
The results of the optimisations are reported in
Figure 10a,b for the T45 and R20 cores, respectively. In the graphs, the iso
–weight curves appear linear, while curves relative to the stiffness requirements are hyperboles. In both figures, the points labelled A show the optimal design variables to minimise the weight. The relative results are reported in
Table 9. As shown, the skin’s thickness is not a multiple of 0.8 mm, which is the width of the glass fibre filaments specified by the supplier [
27]. Both points A are, therefore, unfeasible solutions. A more realistic design had to be selected on the same stiffness curve. Points B were thus considered, the dimensions of which are also reported in
Table 9; the weight increase with respect to the optimal points was found to be less than 2% in both cases.
The resulting designs of the 3D-printed prosthetic component, considering a T45 or an R20 core, are reported in
Figure 11a,b, respectively. Note that these were obtained by leaving the shape of the lower surface unaltered, using the updated core and skin thicknesses.
Two-dimensional FE simulations were then performed on the newly designed prostheses to compare them with the reference laminated one. The boundary conditions, mesh size, and types are the same as those used for the reference prosthesis simulations described in
Section 2.5. The material parameters adopted for the core are the ones obtained from the homogenisation procedure and reported in
Table 6; the glass fibre properties are reported in
Table 10.
The load–displacement curves obtained from this comparison are reported in
Figure 12. As shown, both 3D-printed prosthesis components show a stiffness that is comparable to that of the laminated reference.
3.3. 3D Printed Prosthesis Components and Further Design Work
The FE analyses performed in this work show that 3D-printed components can meet the stiffness requirements of prosthetic feet. This is an encouraging result because stiffness is one of the major requirements for this kind of structure [
26]. Moreover, further improvements can still be applied to the proposed designs. First of all, glass fibres were considered in this work as a continuous reinforcement instead of carbon fibres to reduce the cost of the prosthetic device. However, the use of the more performing carbon fibre can further increase the mechanical properties of a prosthetic component, thus allowing smaller thicknesses and a slenderer shape.
Another possible improvement to a 3D-printed prosthesis would be the manufacturing of an integrated prosthesis. As shown in
Figure 8, the reference prosthesis is composed of two main components: a common solution for laminated prostheses [
32]. However, the increased manufacturing flexibility allowed by AM can be exploited to design a one-component prosthesis. Such prosthesis would thus integrate the functionalities of adequate contact to the ground and the connection to the upper pylon. This would further reduce the cost of the final products, thanks to reduced assembly costs.
Finally, while the potential of an additively manufactured foot is demonstrated here, further work is required for it to reach the market. In particular, it must be proven that the prosthetic component can also resist fatigue loads in both standard and extreme environmental conditions (considering a wide range of temperatures and humidity). This requires further work on the different 3D-printed materials because their fatigue characterisation and modelling are still an active research field [
14,
33,
34]. To reach commercialization, further analysis is needed in the framework of the biomechanical behaviour of the prosthesis, similar to the analysis of the roll-over shape and the assessment of the energy storage and releasing capability.