**4. Discussion**

This paper provides a complete methodology to obtain residual stress predictions in anisotropic materials subject to low-density LSP treatments. In summary: Firstly, the material's stress–strain curves in ND, RD and TD directions were properly modelled both at low and high strain rates. Then, the spatial-temporal pressure pulse distribution was calibrated with the aid of experimentally determined deformation profiles. Finally, the average in-depth residual stresses were obtained for four different treatment strategies, showing the effect of the treatment density and the relative orientation between the PD and RD in results. A reasonably good agreemen<sup>t</sup> is presented between the experimental results and the analytical predictions obtained by means of the proposed anisotropic model.

Experimental results sugges<sup>t</sup> that the PD sets the minimum principal stress direction, which implies that the anisotropy in residual stress distribution is mainly motivated by the treatment strategy itself. In fact, the minimum stress, *Smin*, is obtained along the PD for the four treatment strategies experimented. This is properly simulated by both the isotropic and the anisotropic models. Nevertheless, the relative orientation between the PD and the RD motivates differences in the minimum stress curve, *Smin*, and the maximum stress, *Smax*, in low-density treatments. This effect cannot be predicted by the isotropic model, since a unique stress–strain curve is defined.

Regarding the comparison between the numerical predictions and the experimental results, better agreemen<sup>t</sup> is obtained by means of the implemented anisotropic model. This is observed specially at material's surface, where the isotropic model predicts high compressive residual stresses that are not presented in experimental results. The residual stress distribution from 200 μm to 700 μm is predicted

with grea<sup>t</sup> accuracy for configurations (1), (2) and (3). The greatest di fferences between the analytical predictions of the anisotropic model and the experimental results are presented near to the material's surface, where numerical results tend to predict higher tensile residual stresses than the ones observed experimentally. This may be caused by the evaporation of the material's surface during the application of successive pulses, leading to removal of already deformed material. This e ffect is not considered in LSP simulations [25–27,32,34,45].

The presented procedure is particularized for Mg AZ31B alloy. However, it is applicable to any material of interest subject to relatively low-density LSP treatments. In addition, the e ffect of the explicit consideration of mechanical anisotropy may di ffer depending on the selected material, since it is motivated by variations in the stress–strain curves along di fferent loading paths.
