## 3 Materials and Methods
### Theoretical Framework and Implementation of Digital Image Correlation  
Digital image correlation is a well-established technique to precisely measure displacements through image processing. The method is based on the presence of random features in the images, which can be tracked through a dedicated algorithm. These features can be naturally present on the specimen (e.g., surface roughness or texture) or can be added in the form of random speckle patterns to improve robustness. Due to the printing process, the 3D-printed specimen used in this paper presents a repetitive texture which is not suitable for DIC analysis; thus, black speckles were sprayed on the top white surface before the measurements. When such a random pattern is present, it is possible to perform DIC measurements by comparing the deformed image to the undeformed one. Many different DIC algorithms are available in the literature, most of them based on a two-step procedure involving rough displacement estimation at the pixel level through correlation followed by fine subpixel measurements through interpolation schemes [47]. In this paper, the Digital Image Correlation and Tracking algorithm was used [48]. To accomplish the first step, a measurement grid can be arbitrarily defined on the image. An example grid is shown in Figure 16a with red dots. Then, a square subset is defined around each measurement point, e.g., the red box in Figure 16a. The algorithm then iteratively moves the subset in the deformed image, as exemplified in Figure 16b with the green box. The correlation between the moved subset and the original subset in the reference image is then computed using the following equation [48]:  
\[C(u,v)=\frac{\sum_{i,j}(r(i,j)-r))(d(i-u,j-v)-\bar{d})}{\sqrt{\sum_{i,j}(r(i,j)- \bar{r}))^{2}\sum_{j,j}(d(i-u,j-v)-\bar{d})^{2}}} \tag{9}\]  
Figure 15: Stress distributions along the thickness of the plate: (**a**) at the centre of the plate, showing the perturbation effect due to the bottom contact and (**b**) at a slightly lateral position, where the solution is more similar to the beam theory distribution and is expected to be more reliable.  
where \(u\) and \(v\) refer to the iterative tentative displacements along the horizontal and vertical directions, respectively. The indexes \(i\) and \(j\) refer to the image pixels, while \(r\) and \(d\) represent reference and deformed image subsets, respectively. Additionally, the terms \(r\) and \(\bar{d}\) represent the mean value of both reference and deformed subsets, and are defined for normalization purposes. The correlation result \(C\) is a matrix that contains information about the correlation value, i.e., a scalar between 0 and 1, for each tentative displacement \(u\),\(v\). It is worth noting that because the images are discrete matrices, the values of \(u\) and \(v\) at this step must be integers to allow for the selection of the pixel intensity values. The resulting correlation map can be used to detect the maximum correlation at the pixel level, which represents the rough displacement between the reference and deformed images. To improve the method's sensitivity, it is possible to refine the measurements for subpixel accuracy. In particular, a sub-array of \(C\) is defined by selecting eight points around the rough maximum, for a total of nine measurement points. These points are then used to define a two-variable quadratic polynomial fitting function, which allows for analytical detection of its maximum. The peak of this polynomial function is assumed to be the actual maximum of the correlation map, and the fitting procedure allows for an accuracy between 1/10 and 1/100 of a pixel.