# A Tunable Metamaterial Joint for Mechanical Shock Applications Inspired by Carbon Nanotubes
## 2 Design and Microstructure of the Joint  
The microstructure of the proposed metamaterial-based joint is illustrated in Figure 1. The design of the mechanical joint is adapted to the specific frequency range and size of the impact loadings that are investigated here. The joint is fully composed of a polymeric material, i.e., a UV-curable resin, which is frequently used in stereolithography-based 3D printing techniques. Figure 1: Design of the proposed metamaterial-based joint. The basic load-carrying component of the joint is a triple-wall tubular wireframe, shown in Figure 2, that is inspired by the nanostructure of the well-known CNTs. The triple-wall wireframe is--in all places--characterized by a wire diameter \(d=1\) mm, while it consists of three concentric cylindrical hexagonal lattices. The architecture of the central tubular wireframe is analogous to the atomistic discrete geometry of the single-wall CNT (SWCNT) of the armchair type \((\,n\,,\,n\,)=(10\,,\,10\,)\). For the purpose of the present research, the mean radius of the central tubular wireframe is chosen to be equal to \(R_{\rm c}=20\) mm, while its nodal positions may be defined in a global Cartesian coordinate system \((x\,,\,y\,,\,z)\) using the following equation [31]:  
\[(x_{\rm c}\,,\,y_{\rm c}\,,\,z_{\rm c})=\left(R_{\rm c}\cos\!\left(\frac{X}{R_{ \rm c}}\right),\,r_{\rm c}\sin\!\left(\frac{X}{R_{\rm c}}\right),\,Y\right) \tag{1}\]  
where \((\,X\,,\,Y\,)\) are the corresponding in-plane nodal coordinates for a two-dimensional (2d) graphene-like honeycomb lattice lying in the \(x-y\) plane, defined by the following hexagonal side length [31]:  
\[l_{\rm hex}=\frac{2\pi}{3n}R_{\rm c} \tag{2}\]  
The required key attribute of the developed joint is its ability to transform an amount of the axial compressive strains into torsional ones, providing structural protection in the impact loading direction (vertical \(z\)-axis). To provide such a property, two inclined, parallel, wire reinforcements are added in each hexagonal microstructure, as Figure 2 illustrates. For the \((\,10\,,\,10\,)\) SWCNT-based microstructure, these wire additions lead to the creation of twenty right-handed helixes (spirals) of a helix angle of about \(\theta_{\rm c}=30^{\rm o}\)[31]. Consequently, the developed angular structural inhomogeneity along the longitudinal axis of the tubular wireframe results in the appearance of twisting phenomena under axial tension or compression. The total length of the triple-wall wireframe, which may be measured parallel to the axis of the helixes, is chosen in such a way so that one complete helix turn is obtained, i.e.,:  
\[L=2\pi R_{\rm c}\tan\theta_{\rm c} \tag{3}\]  
Figure 2: The architecture of the triple-wall wireframe. The inner and outer wireframes are chosen to have a similar tubular hexagonal structure as the central one, and they are designed to have a radius of \(R_{\rm i}=0.75R_{\rm c}\) and \(R_{\rm o}=1.25R_{\rm c}\), respectively, while their nodal positions are defined, respectively, by:  
\[(x_{\rm i},\,y_{\rm i},\,z_{\rm i})=(0.75x_{\rm c}\,,\,0.75y_{\rm c}\,,\,z_{\rm c}) \tag{4}\]  
\[(x_{\rm o},\,y_{\rm o},\,z_{\rm o})=(1.25x_{\rm c}\,,\,1.25y_{\rm c}\,,\,z_{\rm c}) \tag{5}\]  
To assure that the triple-wall wireframe presents transverse structural coherence, a suitable interconnection should be generated between the three concentric tubular lattices. Therefore, as Figure 2 illustrates, the nodes of the central wireframe are radially connected with the nodes of the inner and outer wireframes lying in the same horizontal planes via the introduction of straight wire links that have a length of \(l_{\rm link}=0.25r_{\rm c}\).