## 2 Numerical Analysis
### Determination of the Relieved Displacements
This issue can be apparently circumvented by deducing the IFs from the cumulative displacement; however, this is inappropriate, as in an FE model any distribution of stresses can be seen as a superimposition of loads at the nodes, making it again a superimposition of the Dirac delta function solutions. Figure 3: (**a**) Piecewise-constant distribution of the residual stress and displacement obtained as a summation of unitary weights. (**b**) Integral definition of a generic influence function. (**c**) Extraction of the influence function values by introducing a Dirac delta function. An example of IFs along the radial path at the hole region is provided in Figure 5, in which the IFs are shown at single depth values of the Dirac delta function. From this graph, it is evident that large displacements can only be found close to the hole rim. For comparison with the strain gage method, the overall size of the usual grid is shown in the figure overlapped with the IF distributions. It is evident that the grid is relatively far from the hole rim where the IFs are lower. More precisely, the dependency of the displacements along the radial direction is not actually the IF itself, which on the contrary is the variability of these displacements depending on the depth position of the Dirac delta function. The other loading type (component \(Q\)) is shown in Figure 6, where for each point there is a radial displacement and a tangential displacement. The former follows \(\cos(2\theta)\), while the latter follows \(\sin(2\theta)\); thus, the maximum tangential displacement can be found at the angle \(\vartheta=45^{\circ}\). The sign of the influence functions shown in Figures 6 and 7 are in agreement with a positive \(P\) residual stress component and a positive pure shear \(Q\) residual stress component. The difference between the two components is due to the fact that the displacement is not a constant function of the displacement. Figure 4: Dirac delta loadings at different points along the depth for (**a**) \(P\) and (**b**) \(Q\) components and numerically defined influence functions at the surface nodal points. Figure 5: Example of the displacements induced by the load component \(P\) along the radial coordinate for different depths of the Dirac delta function, and position and size comparison with a rosette strain gage grid. stress, while the scales of these distributions mainly depend on the Young's modulus and the depth of the hole. Each IF actually depends on several terms: the material elastic properties (Young's modulus and Poisson's ratio, denoted as \(E\) and \(\
u\), respectively), the radial position at which the displacements are evaluated, the hole depth \(H\), and finally the depth position \(z\), which is the location at which the Dirac delta function can be applied to explicitly found the IF. The hole radius \(R=D/2\) is another parameter that the IFs depend on; however, this latter geometrical variable is discussed later, and the hole depth is initially assumed to have \(R=1\) mm. Among these parameters, the only variability is the \(z\) coordinate, along which the integral needs to be performed after multiplying by the IF for the corresponding stress component according to the following. \[U_{r0} =\int_{0}^{H}P(z)f_{Ur0}(z)\mathrm{d}z \tag{2}\] \[U_{r2} =\int_{0}^{H}Q(z)f_{Ur2}(z)\mathrm{d}z\] \[U_{\theta 2} =\int_{0}^{H}Q(z)f_{U\theta 2}(z)\mathrm{d}z\]  
When the stress directions are not the principal ones, a third stress component needs to be considered. This is the load \(T\), which is again a pure shear, although now at an angular offset of \(45^{\circ}\) with respect to the load \(Q\). As this is again a shear stress, the same IFs need to be multiplied. \[U_{r2}^{\prime} =\int_{0}^{H}T(z)f_{Ur2}(z)\mathrm{d}z \tag{3}\] \[U_{\theta 2}^{\prime} =\int_{0}^{H}T(z)f_{U\theta 2}(z)\mathrm{d}z\]  
These numerical integrations can be easily obtained according to the strategy reported in Figure 7a.