# A Tunable Metamaterial Joint for Mechanical Shock Applications Inspired by Carbon Nanotubes
## 5 Evaluation and Characterization of the Compressive Behavior of the Joint
### Dynamic Impact Analysis  
In this section, the impact behavior of the joint is investigated for various compressive shock loadings uniformly applied on the upper face. The numerical tests were associated with relatively small impact frequencies \(f_{\text{imp}}\) of up to 50 Hz, which resembled low-velocity impact cases. The same constraints were hypothesized as in the static simulations. The considered impact load could be represented by a half-sine compressive stress pulse [38], using the following time-dependent function:  
\[\sigma_{z}(t)=\left\{ \begin{array}{cc}\sigma_{z\text{max}}\sin(2\pi f_{\text{ imp}}),&0\leq t\leq 0.5f_{\text{imp}}^{-1}\\ 0,&t>0.5f_{\text{imp}}^{-1}\end{array} \right. \tag{12}\]  
where the maximum level of compressive stress is taken to be equal to \(\sigma_{z\text{max}}=0.5\,\text{MPa}\) in order to avoid crack initiations and fractures in the polymeric wireframe. The specific stress considering the area \(A_{\text{u}}=\pi\text{r}_{\text{u}}^{2}\) of the upper face of the joint corresponded to a maximum impact force of about 100 kg. Figure 10: Relation between twisting and the vertical axial deflection of the middle flywheel due to the joint axial compression. Figure 9: Joint just before fracture: (**a**) first two points of failure in the deformed mesh and (**b**) contours of rotational displacement on the flywheel. The implicit dynamic numerical simulations of the joint were conducted by assuming that a non-structural mass per unit length \(\mu_{\rm add}\) was uniformly added around the perimeter of the flywheel, which was equal to \(2\pi R_{\rm m}\). A variety of perimetrically distributed mass cases were tested, i.e., \(\mu_{\rm add}=0,\,1,\,2,\,3,\,4\,\)Kg/m, to find the optimal design scenario, in terms of the flywheel mass, that offered enhanced impact protection. The relevant conclusions could be indirectly reached by observing the permanent plastic deformations in the load-carrying wireframes. The dynamic simulations were carried out for a time interval equal to 20 times the impact load duration (\(10f_{\rm imp}^{-1}\)) to reach a steady state for the joint. The time-variation of the axial vertical deflection of the upper face of the joint, with respect to the added mass magnitude around the perimeter of the flywheel, is depicted in Figure 11a-c for three different impact frequencies \(f_{\rm imp}\) of 30, 40, and 50 Hz, respectively. Similarly, the time-dependency of the flywheel torsion angle with respect to the added mass magnitude around its perimeter is depicted in Figure 12a-c considering impact loads governed by a 30, 40, and 50 Hz frequency, respectively. Generally, the numerical data corresponding to the steady-state deformation of the joint at \(t\approx 10f_{\rm imp}^{-1}\) revealed that the increase in the flywheel perimetric mass led to a distinct decrease in both the vertical upper face deflection as well as the flywheel twisting. These types of deformations corresponded to the permanently deformed configuration of the joint that remained after the finalization of the impact-induced vibration. This \"spring-back\" phenomenon was observed due to the special microstructure of the triple-wall tubular wireframes. The flywheel was forced to rotate by the two wireframes and thus absorb the impact's kinetic energy. Then, most of the temporarily saved energy was conversely transferred from the flywheel to the upper face of the joint, and vice versa. However, this vibrational response and resulting energy exchange effectively decayed with time because of the special damping mechanisms provided by the architecture and the inherent damping of the metamaterial. Figure 13 illustrates the time variation of the total strain energy in the tubular wireframes considering the simulation of the joint under the impact load at 50 Hz. To better demonstrate the effect of the mass magnitude around the flywheel, the mass case \(\mu_{\rm add}=0\) was set to contrast with the mass case \(\mu_{\rm add}=4\,\)Kg/m.