## 2 Numerical Analysis
### Determination of the Relieved Displacements
The FE database was performed for a unique value of the Young's modulus \(E\); this does not limit the generality of the problem, as all the displacements are linearly related to \(E\). On the contrary, the variation of the Poisson's ratio modifies the numerical results of the displacements in a way that cannot be exactly predicted. For this reason, five Poisson's ratios were considered, and the full FE simulation set was repeated for these values. A piecewise linear interpolation was again performed for intermediate values of the Poisson's ratio (see Figure 8b). In order to avoid extrapolations, this procedure is not recommended for values of the Poisson's ratio outside the range of the simulated values, i.e., from \(\
u=0.25\) to \(\
u=0.45\), which is expected to include most materials of structural interest. According to the procedure presented so far, the hole radius is \(R=1\) mm, and this would be a strong limitation if the radius were to be imposed. However, the scaling approach can be applied for this variable, which is an exact procedure. If the radius is different than unitary as modelled in the FE analysis, a simple scalar (dimensionless) parameter can be calculated for scaling the entire model: \(\lambda=D/(2\ \text{mm})=R/(1\ \text{mm})\), where \(D\) and \(R\) are the actual hole diameter and radius, respectively. When the stress load distribution corresponding to a Dirac delta loading is applied, assuming that the problem is geometrically scaled by \(\lambda\) and considering the same applied traction, the problem simply scales with respect to the original configuration in which the diameter has a unitary radius. Accordingly, the corresponding points share the same solution in terms of strains; thus, the displacements are scaled by the same factor \(\lambda\). The corresponding points are those obtained by scaling the coordinates, again by the \(\lambda\) factor. Reconsidering the IFs in this situation, the whole integral along the depth is higher or lower than the reference geometry by a factor of \(\lambda\), as the \(z\) integration is scaled accordingly. This is due to the Dirac delta load being a limit distribution and the described scaling  
Figure 8: (**a**) Definition of the simulated configurations in terms of hole depth and Dirac delta function applications points along with possible intermediate interpolations; (**b**) different databases with the same resolution and loading points obtained with different values of the Poisson’s ratio. rule being valid for any instance of this limit. In conclusion, the displacements and the Dirac delta functions share the same scaling factor; therefore, the Dirac delta function with unitary integral generates the same displacements at the corresponding points, which in turn are just the influence function values. In the end, the IFs do not scale with \(\lambda\); while the corresponding point coordinates scale instead. According to this rule, it is possible to retrieve the IFs at the surface points after scaling their radial coordinates from the original FE model. Finally, the integration of Equations (2) and (3) can be solved using the same method shown in Figure 7.  
An example of the application of IF integration and determination of the displacements around the hole is shown in Figure 9, in which the parameters of the load, geometry, and material properties are similar to the experimental configuration reported later.