## 2 The One-Shot Projection Moire: Model and Experimental Setup
### One-Shot Projection Moire Measurements and IMPM Variant
The 78.1 um pixel size was judged a precise calibration value because the standard deviation of calibrated pixel sizes computed over 10 different positions of the image was only 0.1 um. In order to have an independent measure of in-plane displacements, a variant of the one-projector IMPM setup of [34] (also recalled in Section 2.1) was developed. For that purpose, a reference pattern of dots with pitch 2 mm (practically the same in-plane sensitivity achieved by the projected cross-grating patterns) was also printed on the tested 100 mm diameter rubber membrane. The optical setup of Figure 4 was modified so as to reproduce the projection from infinity condition realized by the one-projector moire setup of [34], where a grating of vertical lines was projected onto the specimen by a collimated light beam. Since the specimen analyzed in this study was 2.5 times larger than the one analyzed in [34] (i.e., 100 mm vs. 40 mm), the single projection optical setup could not be realized with the available equipment. Hence, projection from infinity was realized here by using symmetric double illumination and gratings with only vertical lines: the setup hence sensed the same parallax vector as in [34]. The layout of the two-projectors IMPM optical setup (including characteristic distances and illumination angles) and the pitch of the two gratings were obviously the same as those selected for the one-shot projection moire. Figure 5 shows typical images recorded by this setup. The moire data were processed as described in [34], and the corresponding displacement fields were taken as the target for evaluating the performance of the one-shot projection moire setup utilized in this study. The images in Figure 5 recorded by the two-projectors IMPM setup for the inflated membrane refer to the following: the printed dot grating (Figure 5a); the projected gratings by left and right projectors (Figure 5c,d); and the central region of the projected line patterns (Figure 5b). In Figure 5b, the high-frequency lines correspond to projected vertical lines by the two projectors, while the low-frequency fringes are produced by the interfering wave fronts originating from the two projectors. Figure 4: Optical setup of the one-shot projection moiré technique used in the mechanical characterization process of the natural rubber membrane: (**a**) 3D assembly view; (**b**) Schematic. Since the one-shot projection moire developed in this study utilized cross-gratings, it was projected onto the natural rubber membrane in a dot pattern similar to that printed on the specimen in the case of the two-projectors IMPM setup described above. The dot structure of the projected pattern onto the tested specimen originated from the combination of vertical and horizontal lines included in the two gratings. However, the present setup can inherently retrieve curvature information because it utilizes cross-gratings [31; 35; 36], while the two-projectors IMPM setup is insensitive to curvature changes in the Y-direction unless the deformed shape is fully axi-symmetric. Such a requirement may not be satisfied in the case of a variable thickness membrane like the one analyzed in this study. Moire fringes are obtained by mutually subtracting the recorded images of reference and modulated gratings. However, moire patterns must be transformed into full-field continuous displacement maps by computing the phase difference \(\Delta\phi\) between unloaded and loaded configurations [31]. Displacements are hence determined for each selected pressure level as:  
\[u,v=(\Delta\phi_{u,v}/2\pi)\cdot\Delta s_{u,v} \tag{11}\]  
\[w=(\Delta\phi_{w}/2\pi)\cdot\Delta s_{w} \tag{12}\]  
where subscripts \(u\), \(v\) and \(w\) refer to the different displacement components and \(\Delta\phi/2\pi\) is the fractional fringe order for the corresponding unwrapped phase distribution \(\Delta\phi\).