**3. Results**

### *3.1. Experimental Characterization of the Residual Stresses for Di*ff*erent Input Parameters*

The residual stresses have been obtained experimentally by means of the hole drilling method (ASTM E837-13 standard) [43], in which the material is drilled and the resulting deformation as the material is removed is measured by a precision strain gage up to 1 mm thickness (HBM K-RY61-1.5/120R-3 prewired). The specimens subject to LSP treatment were obtained from a 5 mm thickness rolled Mg AZ31B plate. Four different treatment strategies were implemented experimentally, covering a treated squared area of 900 mm<sup>2</sup> (Figure 9 shows a schematic representation):

(1) Setting a laser spot diameter of 1.5 mm, with a distance between successive pulses of 0.66 mm, developing in an Equivalent Overlapping Density, EOD [44], of 225 pp/cm2, and an Equivalent Local Overlapping Factor, ELOF [44], of 4. The peening direction (PD) is set coincident with the transverse direction, TD.

(2) Setting a laser spot diameter of 1.5 mm, with a distance between successive pulses of 0.66 mm, developing in an Equivalent Overlapping Density, EOD, of 225 pp/cm2, and an Equivalent Local Overlapping Factor, ELOF, of 4. The peening direction (PD) is set coincident with the rolling direction, RD.

(3) Setting a laser spot diameter of 1.5 mm, with a distance between successive pulses of 0.50 mm, developing in an Equivalent Overlapping Density, EOD, of 400 pp/cm2, and an Equivalent Local Overlapping Factor, ELOF, of 7. The peening direction (PD) is set coincident with the transverse direction, TD.

(4) Setting a laser spot diameter of 1.5 mm, with a distance between successive pulses of 0.50 mm, developing in an Equivalent Overlapping Density, EOD of 400 pp/cm2, and an Equivalent Local Overlapping Factor, ELOF, of 7. The peening direction (PD) is set coincident with the rolling direction, RD.

**Figure 9.** (**a**) Schematic representation of treatments (1) and (3) (PD = TD); (**b**) Schematic representation of treatments (2) and (4) (PD = RD).

The EOD corresponds to the number of applied pulses per unit of area, while the ELOF represents the average number of pulses at a given location. All the studied configurations can be considered as low-density treatments according to their respective EOD's and ELOF's. This is consistent with the proposed material model, in which the cyclic plasticity modelling is neglected. Higher density treatment-modelling would require further research in the proper calibration of stress–strain asymmetry.

The experimental results from the material's surface up to 1 mm for configurations (1), (2), (3) and (4) are represented in Figures 10 and 11. In these figures, the in-depth minimum, *Smin*, and maximum in plane, *Smax* are represented with their corresponding uncertainties, calculated with the aid of Hdrill software. Several conclusions can be extracted from the experimental results:

(1) For all the experimental measurements developed, the minimum stress, *Smin*, is almost coincident with the stress along the PD, while the maximum in-plane stress, *Smax*, is similar to the stress along a perpendicular direction to the ND and PD This result is independent of the relative orientation between the PD and the RD.

(2)While significant differences are observed between treatment (1) and the others, grea<sup>t</sup> similarities are presented between configurations (2), (3) and (4). This suggests that the residual stresses tend to reach a saturation value for relatively low-density treatments, which is as expected considering the relatively low yield stress of the present alloy in comparison with most of metallic materials.

(3) Setting the peening direction (PD) coincident with the rolling direction (RD) leads to greater peak compressive residual stresses for low-density treatments (Figure 10).

**Figure 10.** (**a**) Experimental residual stresses for EOD = 225 pp/cm2, PD = TD. (**b**) Experimental residual stresses for EOD = 225 pp/cm2, PD = RD.

**Figure 11.** (**a**) Experimental residual stresses for EOD = 400 pp/cm2, PD = TD; (**b**) Experimental residual stresses for EOD = 400 pp/cm2, PD = RD.

### *3.2. Realistic Modelling Results for Extended Surface High-Coverage LSP Treatments*

The advances in computational resources in the last decade have motivated an increase in the number of publications regarding numerical predictions of extended LSP treatments in steel, aluminum and titanium alloys, with di fferent loading strategies. However, there is still a lack of results concerning magnesium alloys which may be caused by the significant anisotropy presented. In this section, the corresponding numerical predictions of four particular treatments are obtained by means of two di fferent models: (1) Considering the anisotropic model presented in Section 2.2, in which the stress–strain behaviour along ND, RD and TD directions is fully characterized. (2) Considering a purely isotropic model, in which both the plastic loading and unloading are modelled by the compressive stress–strain curve in ND direction. The purpose of comparing these two models is to show the impact in numerical predictions of the explicit consideration of anisotropy, since conventional LSP models consider a unique stress–strain curve. Results confirm that representative di fferences are predicted between the anisotropic model and the isotropic one. Better agreemen<sup>t</sup> with experimental results is obtained by means of the anisotropic model.

In realistic high-coverage extended treatments, such as the ones presented in this study, the peening direction motivates anisotropy in the residual stress predictions even in isotropic materials. Concretely, higher compressive residual stresses are usually predicted numerically and obtained experimentally along the PD [45]. This e ffect is also presented in the considered anisotropic alloy, in which results show that the minimum principal stress direction is coincident with the PD. The relative orientation between the PD and RD motivates variations in *Smax* and *Smin* which cannot be predicted by purely isotropic hardening models.

The numerical predictions have been extracted and averaged from a representative squared area of 2 mm<sup>2</sup> up to 1 mm depth, which represents a similar volume than the one removed by means of the hole drilling method. Figures 12–15 show a comparison between analytical predictions and experimental results for strategies (1), (2), (3) and (4) respectively. Table 4 shows a comparison between experimental results and the analytical predictions obtained by means of the anisotropic model. (*dmax*)*i* is the depth at which the maximum peak compressive residual stress is presented. *min*(*Smin*)*i* represents the peak compressive residual stress. *i* = *isot* and *i* = *anisot* corresponds to the analytical predictions of isotropic and anisotropic models respectively, and *i* = *exp* to the experimental results.

**Figure 12.** Experimental results vs. numerical predictions for strategy (1). (**a**) Analytical isotropic model predictions. (**b**) Anisotropic model predictions.

**Figure 13.** Experimental results vs. numerical predictions for strategy (2). (**a**) Analytical isotropic model predictions. (**b**) Anisotropic model predictions.

**Figure 14.** Experimental results vs. numerical predictions for strategy (3). (**a**) Analytical isotropic model predictions. (**b**) Anisotropic model predictions.

**Figure 15.** Experimental results vs. numerical predictions for strategy (4). (**a**) Analytical isotropic model predictions; (**b**) Anisotropic model predictions.


**Table 4.** Experimental results vs. analytical predictions for treatments (1), (2), (3) and (4).

Regarding the principal stress directions, the PD sets the minimum stress curve, *Smin*. The maximum in plane stress, *Smax*, is perpendicular to the ND and the PD. This is predicted correctly by both models. However, the effect of the relative orientation between the PD and the RD in the stress curves, which is significant between the low-density treatments (strategies (1) and (2)), can only be simulated by the anisotropic model.

Concerning the comparison between experimental results and the analytical predictions, the anisotropic model provides a reasonably accurate calculation the effect of the relative orientation between the PD and RD for strategies (1) and (2). A comparison between Figures 12 and 13 shows that setting the PD coincident to the RD leads to greater peak compressive residual stresses. In addition, the depth at which this peak value is achieved also increases.

The anisotropic model provides, in general, more accurate predictions than the isotropic one. In particular, the isotropic model predicts high compressive residual stresses at material's surface, which is not consistent with experimental results. In treatments where the EOD is set equal to 400 pp/cm<sup>2</sup> (strategies (3) and (4)), experimental evidence shows that the residual stresses tend to saturate. Reasonably good agreemen<sup>t</sup> is obtained by means of the anisotropic model for strategy (3) while an overestimation of the compressive residual stresses is predicted for configuration (4). This is not surprising since the model is conceived for low-density treatments, which are the most suitable ones for the present alloy.
