# Microstructured Phononic Crystal Isolates from Ultrasonic Mechanical Vibrations
## 5 Discussion  
Dispersion and transmission analyses exploited in the design phase and described in Section 2 have been repeated on the actual geometries of the three microstructures of Figure 3 to compute the expected attenuation properties. For the supporting glass the elastic modulus \(E=72\) GPa, the Poisson's coefficient \(\
u=0.23\) and the mass density \(\rho=2500\) Kg/m\({}^{3}\) have been employed, while for the resin, typical parameters (\(E=2.3\) GPa, \(\
u=0.42\) and \(\rho=1200\) Kg/m\({}^{3}\)) available in the literature [45] were assumed. In Figure 6, transmission diagrams computed for the three fabricated metastructures are reported. Bandgaps estimated through dispersion analyses under the hypothesis of infinite periodic structure are also reported as shaded areas. The small discrepancies can be ascribed to the finite dimension of the fabricated metastructures, to the presence of the supporting glass and of the top plate. Note that the apparent interruption of the bandgap shown for the unit cell dimension of 100 \(\upmu\)m and 200 \(\upmu\)m is only due to flat modes (as described in, e.g., [46; 47]) that do not compromise the correct functioning of the metastructure and that will be neglected in the following. Figure 5: Signal transmitted by the micro-phononic structures at varying frequency for the 100 \(\upmu\)m, 150 \(\upmu\)m and 200 \(\upmu\)m unit cell dimension. Comparing numerical predictions of Figure 6 with measurements reported in Figure 5, one can observe a qualitative agreement for the three fabricated structures. The quantitative agreement is instead obtained only for the bigger structure (\(a=200\) um). To explain this behavior several aspects should be considered. Firstly, it is worth mentioning that the 2 PP technique fabricates layered structures in the printing direction, i.e., \(z\)-axis, with the number of overlapping layers determined by the voxel dimension and by the chosen spacing in the printing direction, as explained in [44]. For the fabrication process employed in this work, the \(z\)-axis voxel dimension was about 6 um, while the \(z\)-axis slicing was 1 um. To evaluate the influence of this layered structure, we supposed that the bulk mechanical properties of the resin was obtained through at least three superimposed layers and that, as an example, a 25% and 50% reduction in the Young's modulus was obtained in the presence of two superimposed layers and just one layer, respectively. Horizontal beams inside the unit cell must hence be considered layered, as reported in Figure 7a. Each beam was divided into three regions where, (i) one superimposed layer (orange), (ii) two superimposed layers (green) and (iii) at least three superimposed layers (grey) were present. By considering a Young's Modulus equal to 2.3 GPa in the central area, to 1.725 GPa in the _two-layers_ area (1 um) and to 1.15 GPa in the _one-layer_ area (1 um) according to our assumption and running again dispersion analyses in COMSOL Multiphysics for the three unit cells, the original bandgaps (reported as grey bars in Figure 7b) shifted towards smaller frequencies (orange bars in Figure 7b). The influence of the stratigraphy was different for the three cells and higher for the smaller ones, with the voxel dimension being the same and the beam thickness different (Table 1). Due to the uncertainties on the mechanical properties of each layer and to the number of superimposed layers needed to obtain fully developed material properties, it is not possible to make a quantitative estimation. However, on the basis of the test case shown here, we can state that layers play an important role in terms of reduction of the opening frequency at the fixed bulk Young's modulus, especially for smaller structures. Figure 6: Transmission diagrams numerically computed for the three fabricated metastructures. Bandgaps estimated through dispersion analyses are reported as shaded grey areas.