## 2 Numerical Analysis  
### Determination of the Relieved Displacements  
The hole-drilling problem requires the determination of calibration coefficients after having imposed the analytical form of the solution. The determination of these coefficients is usually performed by running different load cases and collecting the displacement results. In order to solve this problem with high resolution and accuracy within a reasonable computation time, a plane axisymmetric harmonic model is strongly recommended. As shown in Figure 1a, the radial displacement due to the relief of a circular volume and an equibiaxial stress distribution can be seen as the difference between the hole without the stress at the boundary and the hole (before drilling) with the stress at the boundary. The difference of these two loadings is equivalent to an inverse stress at the hole bound ary and with no far-field stress. A similar approach can be applied to the shear loading (see Figure 1b), in which the tangential traction needs to be applied to the hole boundary along with the radial one and both stresses show an angular variability according to either \(\cos(2\theta)\) or \(\sin(2\theta)\), still with no far-field boundary conditions. The radial and the tangential displacements follow the same angular dependency. This \(\theta\) variability for the two loadings is in principle only valid in a half-space geometry, which is often a reasonable assumption. On the contrary, specific analyses are required on more complex geometries, requiring dedicated 3D modelling, for example with the cylindrical components in [11]. In this paper, the half-space was assumed to be representative of the analysed geometry under investigation. Following the concept described in Figure 1, it is evident that the load reduces to a traction at the cylindrical hole without far-field stresses. For this reason, the geometry reduces to a cylinder in a half-space with the far boundaries not loaded. Moreover, as the angular dependency is either constant or related to sinusoidal functions in terms of \(2\theta\), the FE analyses can be performed with axisymmetric harmonic elements, e.g., PLANE25 in Ansys. According to this modelling, only a section with a plane through the axis of the hole is required, without the need to introduce any cumbersome and heavy 3D analysis. The FE model and its features are presented in Figure 2. The half-space is modelled as a large cylindrical geometry with respect to the hole. Otherwise, the outer boundary can play a role, as discussed in [41]. As there are small features and large global model dimensions in the same model, a strong mesh size variation is required from the hole region to the periphery. As shown in the figure, the hole radius was set equal to 1 mm. The region near the hole was imposed with a mesh of 0.01 mm and the mesh size was increased with a factor of \(\times 2\) at each transition, which was repeated five times from the hole region to the periphery. Despite only the radius \(R=1\) mm was analysed, a rescale of the problem can be applied, as discussed below, allowing different sizes to be easily considered with almost equal degrees of accuracy. Figure 1: Relieved displacement distributions obtained as the difference between the holed and the non-holed configurations for (**a**) equibiaxial stress and (**b**) shear stress. According to the ASTM E837 standard [4; 5], the residual stresses can be well approximated by piecewise-constant distributions. Under this assumption, and with the principle of superposition being valid (as the system is linear), the effect at the surface in terms of displacements can be obtained as the sum of the single stress contributions \(\sigma_{i}\) along the depth. Each of these needs to be multiplied by its weight \(\overline{U}_{si}\) for a general component of displacement; this weight is the displacement generated by a unitary \(\sigma_{i}\) and applied individually (see Figure 3a). In principle, this can be considered satisfactorily, especially when small steps are imposed.