# An Innovative Adaptive-Dimension Accordion Honeycomb for Circumferential Deformation
## 5 Results and Discussion
### Geometric Nonlinear Stiffness of \(\{\mathbf{5,2}\}\) Honeycomb  
Since it is not necessary for the SPM to fulfill small deformation, FEM simulations are employed to validate the accuracy of the circumferential stiffness in this section while considering geometric nonlinear deformation. Specifically, considering a \(\{5,2\}\) honeycomb, the FEM simulations of the honeycomb and five unit cells are conducted, and the parameters are listed in Table 1. The stiffnesses of the [5,2] honeycomb and each unit cell are obtained by numerical simulations. The theoretical predictions of the [5,2] honeycomb are calculated by substituting the stiffness of five unit cells into Equation (41). Figure 11: The Mises stress contour of the \(\{4,3\}\) honeycomb with \(\varepsilon=0.02\). Figure 10: FEM simulation for the Poisson ratio \(v\) vs. \(m\) and \(n\). Table 2 presents the different stiffnesses of five unit cells vs. the circumferential strains ranging from \(-0.2\) to \(0.2\). There is a positive correlation between the stiffness and strain of unit cells when geometric nonlinearity is taken into account. In comparison to linear theoretical predictions, nonlinear stiffness increases by \(16.68\%\) in PD (\(\varepsilon=0.2\)) and decreases by \(8.31\%\) in ND (\(\varepsilon=-0.2\)). In addition, the geometric nonlinear effects on different unit cells are numerically approximated. The main reason for this is that FEM is continuously iterated during the solution process, resulting in the angles \(\theta\) and \(\phi\) increasing in PD and decreasing in ND. According to Section 5.1, it is reasonable that the variation in nonlinear stiffness is consistent with the influence of \(\theta\) on the stiffness because \(\theta\) has a larger effect on stiffness than \(\phi\). Figure 12 present the comparison between the theoretical predictions and the numerical simulation for the stiffness of the \(\{5,2\}\) honeycomb vs. boundary conditions in which the strain ranges from \(-0.2\) to \(0.2\) (equivalent to an angle, \(\Delta_{\theta}\), range of \(-3.0^{\circ}\) to \(3.0^{\circ}\)). For the \(\{5,2\}\) honeycomb, the variation in stiffness considering geometric nonlinearity is similar to that of unit cells, increasing with increasing \(\Delta_{\theta}\) in PD and decreasing with increasing \(\Delta_{\theta}\) in ND. Compared to the linear theoretical predictions, the stiffness values obtained by nonlinear theoretical predictions and numerical simulation are in good agreement. Using the simulation results as a reference, the error of linear theoretical predictions ranges from \(0.49\%\) to \(9.20\%\) during a strain range of \(-0.2\)\(\sim\)\(0\) (ND) and from \(-1.75\%\) to \(-13.35\%\) during a strain range of \(0-0.2\) (PD). Meanwhile, the error of nonlinear theoretical predictions is in ranges of \(0.44\%\) to \(0.12\%\) (ND) and \(0.55\%\)\(\sim\)\(1.09\%\) (PD). The errors between linear theoretical predictions and nonlinear finite element analysis can be attributed to two factors. On the one hand, the errors arise from the difference between the theoretical model and FEM as described in Sections 5.1 and 5.2. On the other hand, the errors are caused by geometric nonlinear analysis, which varies with the simulated input conditions. For large deformations, such as those shown in Figure 12, the errors due to geometric nonlinearity are clearly the main source. However, the errors between nonlinear theoretical predictions and numerical simulation appear as almost horizontal lines, indicating that the effect of geometric nonlinearity is eliminated. Therefore, it can be explained that SPM is not affected by geometric nonlinearity.