# A Tunable Metamaterial Joint for Mechanical Shock Applications Inspired by Carbon Nanotubes
## 5 Evaluation and Characterization of the Compressive Behavior of the Joint
### Static Elastoplastic Analysis  
In this section, the full elastoplastic compressive behavior of the joint is under investigation. Here, a static compressive load was considered to act at the upper circular lamina of the joint. The static load was applied incrementally in order to treat the inherent material non-linearity, while a Newton-Raphson iterative algorithm, controlled by the residual force and incremental displacement norms, was introduced between subsequent increments to obtain equilibrium and convergent solutions. The variation of the axial compressive stress \(\sigma_{z}\) with respect to the axial strain \(\varepsilon_{z}\) of the joint is presented in Figure 7. Linear elastic behavior was achieved for strains up to 5%. Then, mild strain-hardening regions were observed until the fracture. The compressive stress-strain behavior of the joint was also estimated by developing and using an experimental simplified setup, consisting of a medium-size tensile machine and a digital camera, for capturing both the axial translational and twisting deformations of five 3D-printed prototypes during their quasi-static compression. It should be noticed  
Figure 6: Critical buckling: (**a**) deformed meshed geometry of the joint and (**b**) contours of the strain energy density of the triple-wall wireframe (in the range of 0–100 arbitrary units). Figure 7: Stress–strain curve of the joint under axial compression. that the five prototypes of the joint had additionally been cured for 4.5 h after being manufactured by the LCD 3D printer. Figure 8a-c present illustrations regarding the curing process of a joint prototype, the undeformed configuration of a printed joint, and the compressed state of a similar part within the testing machine. Good agreement between the theoretical and average measured data may be observed in Figure 7. Some differences were perhaps due to the small structural imperfections in the final printed parts caused by the technological limitations of the utilized printing machine and technique. Furthermore, as Figure 9a illustrates, the failure initiation of the joint occurred in a hexagon wire side, located at both the upper and bottom inner-wall wireframes. Note that the specific pre-failure deformed state did not correspond to an instability mode shape but to a statically deformed configuration just before fracture. Moreover, Figure 9 presents the deformed shape of the joint in the true scale and not in a magnified one. Thus, the intensive strains of the flywheel observed in Figure 6 are not apparent in Figure 9. The contours of rotational displacement on the flywheel at its maximum twist state are depicted in Figure 9b. The maximum compressive stress was found to be equal to \(\sigma_{\mathrm{zmax}}=0.576\,\mathrm{MPa}<\sigma_{\mathrm{zcr}}\) and appeared under a strain of \(\varepsilon_{\mathrm{zmax}}=15.45\%\), implying a ductile overall mechanical behavior, which offered sufficient energy-absorbing properties. It was very interesting to observe the twisting phenomena occurring at the flywheel due to the overall compression of the main load-carrying parts of the joint, i.e., the upper and lower triple-wall wireframes. As it may be seen from Figure 10, the design of the joint resulted in achieving almost the same tangential deflection around the flywheel perimeter \(\delta s_{\mathrm{m}}\) as the vertical axial deflection at the flywheel center \(\delta z_{\mathrm{m}}\). Again, good agreement between the present numerical and average experimental estimations could be observed. Note that, according to the FEM results, as the compression load increased, the specific linear relation between tangential and vertical deflection in the flywheel was violated in favor of the first one. This implies that the possible use of a hyperelastic material such as an elastomer for the construction of the wireframes would lead to even a more intensive flywheel rotation.