# An Innovative Adaptive-Dimension Accordion Honeycomb for Circumferential Deformation
## 5 Results and Discussion
### Deformation Behavior of the \(\{\mathbf{m},\mathbf{n}\}\) Honeycomb  
Figure 9 shows the comparison between the theoretical predictions and the numerical simulation for the circumferential stiffness vs. \(m\) and \(n\) when the \(\{m,n\}\) honeycomb is subjected to PD with a strain of \(0.02\). The number of unit cells in the radial direction, \(m\), varies from \(1\) to \(10\), while the number of unit cells in the circumferential direction, \(n\), ranges from \(1\) to \(6\). It is apparent that the variation in stiffness conforms to Equation (42), and the stiffness \(k^{mn}\) calculated by SPM shows a very close agreement with FEM simulation for \(n=1,\ 2,\ 3\), with the maximum error being within \(2\%\).  
As shown in Figure 9, the theoretical descent gradient of \(k^{mn}\) is lower than that of numerical simulation with an increase in \(n\), resulting in the theoretical stiffness being larger than the simulated stiffness for \(n=1,2\) and smaller than the simulated stiffness for \(n>2\). The relative errors are \(-1.94\%\) and \(21.93\%\) for the \(\{1,1\}\) and \(\{1,6\}\) honeycombs, respectively. However, as \(m\) increases, the theoretical increase in the gradient of \(k^{mn}\) is opposite to \(n\). For \(\{1,1\}\) and \(\{10,0\}\) honeycombs, the relative errors are \(-1.1098\%\) and \(-1.1100\%\), respectively. For \(\{1,6\}\) and \(\{10,6\}\) honeycombs, the relative errors are \(22.96\%\) and \(10.30\%\), respectively.  
The main reason for this phenomenon is that in the assumption of parallel theory, the unit cells in the radial direction have no constraints on each other, but they are actually constrained by adjacent radial cells in FEM. Consequently, the effect on circumferential stiffness is the same, as the increase in degrees of freedom in theory is larger than that in FEM. Additionally, it is assumed that the unit cells in the circumferential direction are subjected to the same boundary conditions, while no constraints are applied to the intermediate layers in FEM. The increase in the degrees of freedom, in theory, is lower than that in FEM, resulting in a slow decrease in circumferential stiffness.  
Figure 8: FEM and theoretical predictions of circumferential stiffness \(k^{11}\) vs. \(\phi\) and \(\theta\).  
Figure 10 shows the results of the FEM simulation for the Poisson ratio \(v\) vs. \(m\) and \(n\). [PERSON]'s ratio increases with increasing \(n\) and is almost unaffected by \(m\). The largest value of [PERSON]'s ratio is \(8.37\times 10^{-5}\) for the \(\{10,6\}\) honeycomb, which is a very small value. As a result, the zero Poisson ratio assumption can be validated in a substantial way by the FEM simulation results. Figure 11 shows the Mises stress contour of the \(\{4,3\}\) honeycomb; the stress on the radial honeycomb wall is almost \(0\), the stress level on the internal walls decreases with an increase in radius, the maximum stress occurs on the smallest cell structure, and the stress value is \(3.754\) MPa.  
Figure 9: FEM and theoretical predictions of circumferential stiffness \(k^{mn}\) vs. \(m\) and \(n\): (a) stiffness; (b) relative error.