## 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  
The last step of the present study was to determine the hyperelastic constants and the stiffness distribution of the 100 mm diameter natural rubber membrane subjected to inflation. For this purpose, the inverse problem stated by Equation (1) was reformulated as:  
\[\left\{ \begin{array}{l}Min\left[\Omega(\overset{\rightarrow}{a_{10}}, \overset{\rightarrow}{a_{01}},\overset{\rightarrow}{t})=\sqrt{\frac{1}{N_{ CNT}}\sum\limits_{k=1}^{N_{CNT}}\left(\frac{u_{Au}k-\overline{u_{Au}k}}{u_{Au}k} \right)^{2}}\right]\\ a_{10}\leq a_{10}\leq a_{10}\lx@note{footnote}{$\ast$}\\ a_{01}\leq a_{01}\leq a_{01}\lx@note{footnote}{$\ast$}\\ a_{01}\leq a_{01}\leq a_{01}\lx@note{footnote}{$\ast$}\\ t_{\text{min}}\leq t_{s}\leq t_{\text{max}}\end{array} \right. \tag{35}\]  
If the membrane has homogeneous material properties and constant stiffness, the inverse problem (35) obviously includes only two variables, the MR constants \(\lx@note{footnote}{$\ast$}\\ a_{10}\) and \(\lx@note{footnote}{$\ast$}\\ a_{01}\). However, if mechanical properties and stiffness (strictly related to thickness) are not uniform, local values of MR parameters, and membrane thickness should be stored in the vectors \(\overset{\rightarrow}{a_{10}},\overset{\rightarrow}{a_{01}},\overset{ \rightarrow}{t}\) and selected as design variables. In order to cover this general scenario, the membrane can be divided in the 108 sectors shown in the schematic of Figure 11, and the terms \(\{a_{10}a_{01},t\}\)s describing local properties of the generic sth sector may be included as design variables. Membrane subdivisions were strictly related to the locations where thickness was measured along the six diameters corresponding to the 12 loading directions of the preliminary in-plane equi-biaxial tests (i.e., 0, \(\pm 1/8\), \(\pm 1/4\), \(\pm 3/8\), \(\pm 1/2\), \(\pm 5/8\), \(\pm 3/4\), \(\pm 7/8\) and \(\pm 1\) of the radius, see Section 4.2). These locations correspond to the green dots sketched in Figure 11 (the loading directions of in-plane tests are schematized in the figure with etched lines): (i) the center of the membrane (i.e., the vertex common to the 12 inner triangular sectors of the membrane schematization, numbered from 1 to 12); (ii) the centers of 84 quadrilateral sectors located from \(\pm\) 1/8 to \(\pm\) 7/8 of the membrane radius (sectors numbered from 13 to 96); and (iii) 12 points on the external boundary of the membrane (sectors numbered from 97 to 108). The radial dimension of the 12 inner sectors and the 12 outer sectors is half of the radial dimension of the remaining 84 sectors. In summary, sectors are centered or symmetric about thickness measurement points. Figure 11: Sectors defined in the solution of the inverse mechanical characterization problem for the 100 mm diameter natural rubber membrane. Three variants of the inverse problem (35) were considered. The first variant corresponds to the most general formulation including a total of 324 (i.e., 108 \(\times\) 3) design variables. The second variant is a simpler problem formulation with only 110 design variables: 2 unknown MR parameters and 108 unknown thickness values. The last variant of the inverse problem solved here assumes the measured thickness distribution and hence includes only two unknown MR parameters as design variables. Additional constraints (e.g., positive definiteness of tangential material stiffness) ensured stability of nonlinear material. The lower bounds of hyperelastic constants were 100 kPa for a\({}_{10}\) and 1 kPa for a\({}_{10}\), while upper bounds were 1 MPa and 200 kPa, respectively. Thickness could vary between 0.9t\({}_{\text{to}}\) and 1.1t\({}_{\text{to}}\) (t\({}_{\text{to}}\) = 1 mm is the nominal thickness of the membrane), a slightly larger range than the one observed experimentally. The measured total displacements of the inflated membrane \(\overline{u_{tot}}\) = \(\sqrt{u^{2}+v^{2}+w^{2}}\) were compared with those computed by an FE model simulating the inflation test.