## 4 Results and Discussion
### Detailed Mechanical Characterization of the 100 mm Diameter Membrane
#### 4.3.2 Solution of the Inverse Problem: FE Analysis and Metaheuristic Optimization
Measured displacements \(\overline{u_{tot}}^{k}\) and computed displacements \(u_{tot}^{k}\) were compared at N\({}_{\text{CNT}}\) control points, and the former quantities were the target of the optimization process because moire data are gathered without knowing material/structural parameters a priori. Conversely, hyperelastic constants were given in input to the FE solver to compute displacements; moire data and numerical results match only if true MR constants are given in input to FE model. As mentioned in Section 4.1, the natural rubber membrane specimen characterized in this study has a diameter of 100 mm, 2.5 times larger than the 40 mm diameter analyzed in [34]. To account for the significantly lower stiffness of the present specimen and to deform the inflated membrane in the nonlinear regime without affecting its structural integrity, inflation pressure was reduced to 2.07 kPa from the 15.25 kPa of [34]. Interestingly, the maximum in-plane deformations measured in the present case (i.e., \(\text{u}_{\text{max}}\) = 1.140 mm and \(\text{v}_{\text{max}}\) = 1.167 mm) practically coincided with the \(\text{u}_{\text{max}}\) = \(\text{v}_{\text{max}}\) = 1.1 mm value (axis-symmetric deformation) reported in [34] for the 40 mm diameter membrane; maximum deformations occurred near (based on thickness variations as highlighted in Section 4.2) or precisely (in the case of axi-symmetric deformation) at \(\pm\) 0.6R independent of diameter size. However, the maximum out-of-plane deformation of the 100 mm diameter membrane increased to 13.932 mm from the 8.850 mm value measured in [34], in spite of the fact that the inflation pressure was reduced by 7.5 times. The total displacement of the inflated membrane was driven by the out-of-plane component w, which became more and more predominant as bending stiffness decreased, such as occurred passing from the 40 mm to the 100 diameter membrane. However, including also the u- and v-displacements in the definition of the error functional \(\Omega\) allowed us to test the robustness of the metaheuristic algorithm used for solving the inverse problem (35). The ANSYS(r) 2022 R2 Student general purpose FE code developed by ANSYS Inc. (Canonsburg, PA, USA) was utilized in this study; the Student version was selected in order to prove implementation easiness of proposed framework. Figure 12a shows the FE model of natural rubber membrane used in the inverse problem (35). The mesh included 3072 4-node SHELL181 hyperelastic elements with 3137 nodes; convergence analysis was performed to obtain mesh-independent solutions. The pressure load acting on the membrane and the kinematic constraints on the membrane's circular edge also are indicated in Figure 12b. The nonlinear FE analysis option was selected by switching on the NLGEOM command to account for the large deformations caused by inflation. ANSYS(r) adaptively increased the load up to the 2.13 kPa applied pressure with the automatic time-stepping option based on the convergence history of the FE solution. The optimization problem (35) was solved with the HS-JAYA metaheuristic algorithm described in Section 3. The error functional was computed as \(\Omega\) = \(\Omega_{1.07\text{kPa}}\) + \(\Omega_{1.33\text{kPa}}\) + \(\Omega_{1.60\text{kPa}}\) + \(\Omega_{1.87\text{kPa}}\) + \(\Omega_{2.13\text{kPa}}\), summing over the values of \(\Omega\) for each pressure level. Experimental data and numerical results were compared along the six control paths indicated in Figure 12b. These control paths coincided with the loading directions selected in the planar equi-biaxial tests. A total of N\({}_{\text{CNT}}\) = 769 control points were considered, sampling each control path in 128 points, i.e., an eight-times-higher sampling frequency than the one selected for thickness measurements. Since the chosen element size for the ANSYS(r) model was 10 times larger than the pixel size of the recorded images, it was easy to match nodes of the FE model with the image pixels of moire patterns.