# An Innovative Adaptive-Dimension Accordion Honeycomb for Circumferential Deformation
## Mechanical Property Analysis
### Unit Cell Stiffness \(k^{11}\)  
The theoretical analysis describing the deformation properties outlined in Section 3.1 is based on [PERSON]'s second theorem, with the analytical model assuming that the cell wall undergoes both axial and flexural deformations. When an angular deformation \(\Delta_{\theta}\) is applied on the unit cell (Figure 1), there is a proportional displacement of the beam \(CD\) and  
Figure 3: Deformation mode of the honeycomb. a fixed constraint of beam \(AB\). The circumferential displacements at points \(C\) and \(D\) can be expressed as follows:  
\[\delta_{c}=\Delta\rho r \tag{18}\]  
\[\delta_{d}=\Delta_{\theta}(r+l) \tag{19}\]  
Considering the dimensional difference between beams AE, CE, BF, and DF (Figure 1), it is sufficient to analyze one side of the unit cell. As depicted in Figure 4, point C experiences the force \(P_{c}\) and the moment \(M_{sc}\) due to circumferential displacement. A local coordinate system for beam CE is established at point C, where the angle between \(e_{r}\) and \(\eta_{1}\) can be expressed as \(\varphi\). Here,  
\[\varphi=\frac{\theta}{2}+\frac{\phi}{2} \tag{20}\]  
The components of the force \(P_{c}\) in the local coordinate system are  
\[ \begin{pmatrix}P_{\eta 1}\\ P_{\eta 2}\end{pmatrix}  = \begin{bmatrix}\cos\varphi&-\sin\varphi\\ \sin\varphi&\cos\varphi\end{bmatrix}\begin{pmatrix}0\\ P_{c}\end{pmatrix}  \tag{21}\] \[= \begin{pmatrix}-P_{c}\sin\varphi\\ P_{c}\cos\varphi\end{pmatrix} \]  
From the equilibrium of the moments and the forces on beam \(CE\), the force and moment at an arbitrary section can be expressed as  
\[F_{\eta}(x)=-P_{c}\sin\varphi \tag{22}\]  
\[M_{\eta}(x)=P_{c}x\cos\varphi-M_{sc} \tag{23}\]  
As shown in Figure 5, a local coordinate system for beam \(AE\) is established at point \(E\), and the angle between \(\eta_{1}\) and \(\psi_{1}\) can be expressed by \(\phi-\pi\). As a result, the components of the force \(P_{c}\) in the local coordinate system are  
\[ \begin{pmatrix}P_{\phi 1}\\ P_{\phi 2}\end{pmatrix}  = \begin{bmatrix}-\cos\phi&-\sin\phi\\ \sin\phi&-\cos\phi\end{bmatrix}\begin{pmatrix}P_{\eta 1}\\ P_{\eta 2}\end{pmatrix}  \tag{24}\] \[= \begin{pmatrix}P_{c}\sin(\varphi-\phi)\\ -P_{c}\cos(\varphi-\phi)\end{pmatrix} \]  
Figure 4: The local coordinate systems of beam \(CE\) and corresponding forces and moments. Similarly, the force and moment at an arbitrary section on beam \(AE\) can be expressed as  
\[F_{\varphi}(x)=P_{c}\text{sin}(\varphi-\phi) \tag{25}\]  
\[M_{\varphi}(x)=-P_{c}x\text{cos}(\varphi-\phi)+P_{c}w_{1}\text{cos}\ \varphi-M_{sc} \tag{26}\]  
The deflection angle can be obtained by integrating the moment equation:  
\[ \begin{array}{ll}\gamma_{c}(\odot)&=\int_{0}^{w_{1}}\frac{M_{\eta}+M_{\phi}} {EI}dx\\ &=\frac{1}{EI}\left(-\frac{1}{2}P_{c}w_{1}^{2}\text{cos}(\varphi-\phi)+\frac{3} {2}P_{c}w_{1}^{2}\text{cos}\ \varphi-2M_{sc}w_{1}\right)\end{array}  \tag{27}\]  
In Equation (27), \(E\) is [PERSON]'s modulus of the material, \(w_{1}\) is the calculated by Equation (1), and \(I\) is the second moment of area of the beam, which can be expressed as  
\[I=\frac{1}{12}Ht^{3} \tag{28}\]  
In geometric linear theory, the deflection angle at the turning point of the unit cell is zero.