**2. Procedure for the RS evaluation by the HMs**

In general the application of the HMs (HDM and RCM) consist on:


**Figure 1.** *Ring-Core Method: general notations and groove depth increments.*

*Use of Hybrid Methods (Hole-Drilling and Ring-Core) for the Analysis of the RS on Welded… DOI: http://dx.doi.org/10.5772/intechopen.102051*

**Figure 2.**

*Typical Experimental setup used by the HDM for the experimental RS analysis.*

4.computing the three strain components *pi, qi, ti* by the simple formulas:

$$p\_i = \frac{\varepsilon\_{ci} + \varepsilon\_{ai}}{2};\tag{1}$$

$$q\_i = \frac{\varepsilon\_{ci} - \varepsilon\_{di}}{2} \tag{2}$$

$$t\_i = \frac{\varepsilon\_{ci} + \varepsilon\_{ai} - 2\varepsilon\_{bi}}{2} \tag{3}$$

computing the corresponding three stress components *Pi, Qi,Ti* (*i = 1,2 … n*) by the iterative relationships (Eqs. 4, 5 and 6):

$$P\_i = \frac{1}{a\_{ii}} \left[ p\_i c\_E c\_\nu - \sum\_{j=1}^{i-1} a\_{ij} P\_j \right] \tag{4}$$

$$\mathbf{Q}\_{i} = \frac{\mathbf{1}}{b\_{ii}} \left[ q\_{i} \mathbf{c}\_{E} \mathbf{c}\_{\nu} - \sum\_{j=1}^{i-1} b\_{ij} \mathbf{Q}\_{j} \right] \tag{5}$$

$$T\_i = \frac{1}{b\_{ii}} \left[ t\_i c\_E c\_\nu - \sum\_{j=1}^{i-1} b\_{ij} T\_j \right] \tag{6}$$

where *aij* and *bij* (*i = 1,2 … n*, *j* = 1,2 … *i*) are the well known influence coefficients obtained by proper numerical simulations [6, 7, 14], whereas *cE* and *c<sup>ν</sup>* represent the elastic corrective coefficients for the actual material characteristics (*E,v*) of the examined component, that are equal to *E/E0* and (*1 + ν0*)/(*1 + ν*) respectively, being *Eo* and *vo* the material characteristics used in the numerical simulations.

Computing the principal residual stresses *σ*1*i*,2*<sup>i</sup>*(*i = 1,2 … n*) and the relative orientation *θ<sup>i</sup>* (*i = 1,2 … n*) as (Eqs. 7 and 8):

$$
\sigma\_{1i,2i} = P\_i \pm \sqrt{{\mathcal{Q}\_i}^2 + {T\_i^2}};\tag{7}
$$

$$\theta\_i = \frac{1}{2} \arctan\left(\frac{-T\_i}{Q\_i}\right) \tag{8}$$

In the following **Figure 1** the experimental setup of the RCM is depicted, along with the relative general notations.

**Figure 3.** *Modern equipment used for the RS analysis by the HDM (a) and RCM (b).*

It is note that the experimental setup of the HDM differs from that of that of the RCM (compare **Figures 1** and **2**) simply for the different rosette used, a special rosette with separate grids that allows to drill a centered hole for a HDM, instead a stacked rosette that allows to minimize the dimension of the groove as well as to avoid the electrical disconnection of the rosette during the successive execution of the depth increments, performed by an annular cutter.

Recently the application of the HMs is performed by using automatic systems that use high speed drilling methods (obtained by proper miniaturized air turbines) and accurate centering systems that involve optical microscopes and step-by-step electric motors. As an example **Figure 3a** shows a diffuse system used for the practical application of the HDM, whereas **Figure 3b** shows a similar system used for the RCM.

It is important to note that the use of such modern systems allows the user to realize the hole or the groove by limiting significantly the further RS introduced by machining (thanks to the high speed machining), as well as center accurately the rosette thanks to the use of the optical microscope and step-by-step electric motors used to move accurately the cutter in the plane parallel to the component surface (see **Figure 3**).
