## 4 Results and Discussion
### Detailed Mechanical Characterization of the 100 mm Diameter Membrane
#### 4.3.1 Measured Displacement Fields
Selecting diameters that correspond to grating-lines directions makes it easier to extract displacement information from the moire pattern. In fact, moire fringes imaged by the sensor actually are projections of spatial functions u, v and v onto the sensor image plane whose local reference system is centered in the entrance pupil of the camera and coincides just with the global XYZ reference system where displacement components are expressed. This makes coordinate transformations unnecessary. Figure 9 shows that the two moire setups obtained very similar results. The average difference between u-displacements determined with the present methodology and their counterpart obtained from IMPM was about 7.5% with a maximum value of about 12%. The most significant differences occurred near the constrained left side of the membrane (i.e., the part of the control path with x\(\leq\)0), the center of the membrane and the regions experiencing the maximum deformation. A similar behavior was observed for v-displacements: an average difference of about 6.3% and a maximum difference of about 10% near the constrained bottom side of the membrane (i.e., the part of the control path with \(y\leq 0\)), the membrane's pole and the most deformed regions of the tested specimen. As expected, the deformed shape taken by the inflated membrane was not perfectly symmetric with respect to the specimen's center. In fact, along the horizontal diameter of the specimen, the u-displacement ranged from \(-1.069\) mm at x\({}_{\rm m}=-3.101\) cm to 1.140 mm at x\({}_{\rm M}=3.171\) cm. Furthermore, on the vertical diameter of the specimen, the v-displacement ranged from \(-1.099\) mm at y\({}_{\rm m}=-3.004\) cm to 1.167 mm at y\({}_{\rm M}=3.086\) cm. This effect was better captured by the one-shot projection moire setup, which inherently accounts for the local curvature variations occurring for the deformed object. In fact, Figure 9 shows that the present moire model correctly located u = 0 and v = 0 positions on corresponding horizontal and vertical control paths right at points with x > 0 and y > 0, consistently with the conditions x\({}_{\rm M}>|\) x\({}_{\rm m}|\) and y\({}_{\rm M}>|\) y\({}_{\rm m}|\) observed in the experiments. The displacement curves plotted in Figure 9 appear somehow noisy, especially those relative to IMPM measurements. This may be due to the random variations in thickness occurring all over the membrane. In the processing of IMPM patterns, filter sizes were heuristically selected to try to predict the effect of thickness variations while the proposed one-shot projection moire model inherently attempted to correct diametrical strains based on the observed thickness variations along all analyzed diameters. Figure 10 compares the w-displacement distributions measured on horizontal and vertical diameters of the inflated specimen. Again, the results obtained by the two moire approaches were rather similar. However, the two-projectors, one-camera IMPM setup  
Figure 9: Comparison of in-plane displacements obtained by the proposed one-shot projection moire setup and the two-projectors, one-camera IMPM setup for the natural rubber membrane inflated at the maximum pressure of 2.13 kPa: (**a**) u-displacement; and (**b**) v-displacement. underestimated the magnitude of out-of-plane deformations: w-displacement peak values measured by the two methods at the pole of the inflated membrane differed by about 2.3% while the difference on measured w-displacements rose to almost 5% near the constrained edge of the membrane. The lack of information on parallax in the Y-direction caused by the fact that only vertical lines were projected onto the inflated membrane did not allow IMPM to correctly differentiate phase distribution with respect to the y-coordinate when the surface's slope and curvature components were determined. The out-of-plane displacement field was not symmetric about the pole of the inflated membrane.