# Band Gap Analysis for Materials with Cookie-Shaped Auxetic Microstructures, Using Finite Elements
## 4 Results
### Simulation with Different Alterations of the Inner Ellipse Size  
A parametric investigation with changes in the size of the inner ellipses, as well as the thickness of the outer four cross rods that surround the geometry, has been performed. In particular, the dimensions (mm) of the inner ellipse are the following: 170 \(\times\) 30, 170 \(\times\) 60, 170 \(\times\) 80, 170 \(\times\) 90. The thickness of the surrounding rods takes the values 3, 6 and 12 mm. The results are shown in Figures 5-15 and compared in Tables 2-6. Figure 4: Response along the IBZ of the RVE, for (**a**) with and (**b**) without circular inclusion. Figure 3: Representative element microstructure (**a**) with and (**b**) without circular inclusion. Figure 5: Response for Inner ellipse 170 \(\times\) 30, rod thickness 3 mm. (**a**) microstructure, (**b**) response for first eigenfrequencies. Two band gaps are identified. Between 5 th and 6 th eigenfrequency (1160–1800 Hz) and between 7 th and 8 th eigenfrequency (1800–2460 Hz). Figure 6: Response for inner ellipse 170 \(\times\) 30, rod 6 mm with one band gap, between 5 th and 6 th eigenfrequency (1520–2400 Hz). (**a**) microstructure, (**b**) response for first eigenfrequencies. Figure 7: Response for inner ellipse 170 \(\times\) 30 and rod thickness 12 mm. (**a**) microstructure, (**b**) response for first eigenfrequencies. Two band gaps between 3 rd and 4 th eigenfrequency (1720–2320 Hz) and a marginal between 5 th and 6 th eigenfrequency (2680–2700 Hz). Figure 8: Response for inner ellipse 170 \(\times\) 60, rod 3 mm. (**a**) microstructure, (**b**) response for first eigenfrequencies. Two band gaps are shown, between 5 th and 6 th eigenfrequency (1120–1775 Hz) and between 7 th and 8 th eigenfrequency (1805–1835 Hz). Figure 10: Response for inner ellipse 170 \(\times\) 60 and rod thickness 12 mm. (**a**) microstructure, (**b**) response for first eigenfrequencies. One band gap exists between 3 rd and 4 th eigenfrequency (1420–1680 Hz). Figure 9: Response for inner ellipse 170 \(\times\) 60 and rod thickness 6 mm (**a**) microstructure, (**b**) response for first eigenfrequencies. One band gap between 5 th and 6 th eigenfrequency (1440–1780 Hz). Figure 11: Response for inner ellipse 170 \(\times\) 80, rod 3 mm. (**a**) microstructure, (**b**) responce for first eigenfrequencies. One band gap exists between 5 th and 6 th eigenfrequency (970–1740 Hz). Figure 12: Response for inner ellipse 170 \(\times\) 80, rod 12 mm. (**a**) microstructure, (**b**) responce for first eigenfrequencies. One band gap exists, between 3 rd and 4 th eigenfrequency (1050–1530 Hz). Figure 3:0 Response for inner ellipse 170 \(\times\) 90, rod 3 mm. (**a**) microstructure, (**b**) responce for first eigenfrequencies. One band gap exist, between 5 th and 6 th eigenfrequency (850–1570 Hz). Figure 3:1 Response for inner ellipse 170 \(\times\) 90, rod 6 mm. (**a**) microstructure, (**b**) response for first eigenfrequencies. Two band gaps exist, between 3 rd and 4 th eigenfrequency (840–1160 Hz) and between 5 th and 6 th eigenfrequency (1220–1570 Hz). Figure 3:2 Response for inner ellipse 170 \(\times\) 90, rod 12 mm. (**a**) microstructure, (**b**) response for first eigenfrequencies. One band gap is shown, between 3 rd and 4 th eigenfrequency (950–1530 Hz). \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Inner Ellipse** & **Rod Thickness** & **Rod Thickness** & **Rod Thickness** \\
**170 \(\times\) 80 (mm)** & **3 mm** & **6 mm** & **12 mm** \\ \hline
1 st bandgap & 5 th–6 th (1120–1775 Hz) & 5 th–6 th (1440–1780 Hz) & 3 rd–4 th (1420–1680 Hz) \\
2 nd bandgap & 7 th–8 th (1805–1835 Hz) & - & - \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Parametric investigation. Inner ellipse 170 \(\times\) 60.