# A Tunable Metamaterial Joint for Mechanical Shock Applications Inspired by Carbon Nanotubes
## 4 Results
### Mesh, Material Properties, and Boundary Conditions  
The triple-wall wireframe was modeled by using exclusively 3D, two-noded, linear, straight, thick beam elements, which included shearing deformations and had six degrees of freedom (DoFs) at each node, i.e., three translations and three rotations. Everywhere in the model, two beam elements were utilized for the division of each straight wireline, while a circular cross-section of a diameter \(d=1\) mm was assumed for each element. Specifically, 7080, 4800, and 3040 beam elements were utilized for the discretization of the sides of all hexagons, the inclined reinforcements, and the inter-wall radial links, respectively. The lower face, the middle flywheel, and the upper face were modeled by using appropriate 3D, four-noded, linear, quadrilateral, thick shell elements with six DoFs per node. As a result, the corresponding circular areas in the planes \(z=0\), \(z=L\), and \(z=2L\) were discretized by using 527, 767, and 527 shell elements of an equivalent thickness of 2 mm, 4 mm, and 2 mm, respectively. A representative mesh of the adopted finite element model is illustrated in Figure 5.  
To express the elastoplastic mechanical behavior of all finite elements, a Von Mises potential material model was assumed [34] according to the average experimental stress-strain curve shown in Figure 4. The utilized Von Mises stress potential function includes a nonlinear hardening behavior approximated by using a series of straight-line segments. Following the present experimental evidence, the almost isotropic elastic response of the 3D-printed material may be computationally described by an elastic modulus and Poisson's ratio of \(E=842\) MPa and \(\
u=0.35\), respectively. The damping properties of the investigated material were of high importance for the accurate numerical prediction of the joint performance under mechanical shock conditions. To simulate the energy dissipation characteristics of the 3D-printed polymeric material, a Rayleigh damping model was introduced [35]. According to the Rayleigh model, the damping matrix of the system is expressed as a linear combination of the mass and stiffness matrix, i.e.,  
\[\mathbf{C}=a\mathbf{M}+b\mathbf{K} \tag{9}\]  
Figure 5: Discretized joint with beam and shell 3D finite elements. For the requirements of the present work, a mass and stiffness constant of \(a=3.673\) and \(\beta=2.446\times 10^{-3}\) were adopted, respectively. The specific values were calculated by using experimental evidence regarding the loss factor \(\zeta\), also known as \(\tan\delta\), of the 3D-printed material at a frequency level \(f_{1}=1\,\mathrm{Hz}\). The loss factor of the Anycubic UV resin at such a frequency was estimated at \(\zeta_{1}=0.3\)[32; 36; 37]. Assuming that at a higher frequency of \(f_{2}=50\,\mathrm{Hz}\) the material typically exhibits a 30% increase in its loss factor [36; 37], i.e., \(\zeta_{2}=1.3\zeta_{1}\), one may estimate the alpha and beta constants required in the Rayleigh damping model via the following equations:  
\[\alpha=4\pi f_{1}f_{2}\frac{\zeta_{1}f_{2}-\zeta_{2}f_{1}}{f_{2}^{2}-f_{1}{}^ {2}} \tag{10}\]  
\[\beta=\frac{1}{\pi}\frac{\zeta_{2}f_{2}-\zeta_{1}f_{1}}{f_{2}^{2}-f_{1}{}^{2}} \tag{11}\]  
It should be mentioned that, according to the approach developed here, the damping Rayleigh model was not holistically applied for the entire structure, a fact that would pre-require a complete modal analysis of the joint to determine its natural frequencies. Here, the experimental loss factor of the pure raw material was implemented for two different frequencies, i.e., 1 and 50 Hz, which were representative of the investigated frequency range. Then, based on these two data points, the energy dissipation of the system was indirectly approximated by assigning the internal friction of the solidified resin (as a separate material property), which was reasonably approximated via a Rayleigh-like damping ratio curve within the tested frequency domain.