**5. Conclusions**

This paper provides a complete guideline to model anisotropic behaviour of metallic materials subject to LSP treatments. Experimental evidence, confirmed by the corresponding numerical predictions, leads to the identification of the material's stress–strain anisotropy as a key factor to appropriately compute residual stress predictions in low-density treatments. The main conclusions are as follows:

(1) The relevance of including anisotropic behaviour in material modelling in residual stress predictions has been demonstrated in the particular case of Mg AZ31B alloy. Conventional isotropic models are not able to predict accurately the in-depth residual stress distribution. In addition, incorrect principal stress directions are obtained by means of purely isotropic models.

(2) The experimental results show that the anisotropy in residual stresses after treatment are motivated mainly by the treatment strategy. The minimum stress, *Smin*, is similar to the stress along the PD, while the maximum in plane residual stress, *Smax*, is similar than the one presented along a perpendicular direction to the ND and the PD. Both the implemented isotropic and anisotropic models predict properly this e ffect. Nevertheless, better agreemen<sup>t</sup> between numerical predictions and experimental results is achieved by the proposed anisotropic model.

(3) *Smin* and *Smax* curves are influenced by the relative orientation between the PD and the RD. Setting the PD coincident to the RD leads to greater compressive residual stresses for low-density treatments (EOD = 225 pp/cm2). This e ffect cannot be computed properly by isotropic models. Therefore, the anisotropic hardening model is required for this purpose.

Regarding further developments, several issues may be considered:

(1) Modelling the material loss during target irradiation. The application of successive pulses leads to the evaporation of already deformed surfaces, which may modify residual stresses especially at material's surface.

(2) Analyzing the e ffect of the anisotropic modelling in residual stress predictions with a large variety of di fferent input parameters, including additional relative orientations between the PD and material's anisotropic directions.

**Author Contributions:** Conceptualization, Á.G.-B. and J.L.O.; Data curation, F.C.; Investigation, J.A.P. and M.D.; Methodology, I.A.; Project administration, J.L.O.; Software, I.A. and F.C.; Supervision, J.L.O.; Validation, J.A.P., Á.G.-B. and M.D.; Writing—original draft, I.A. and F.C.; Writing—review and editing, I.A. and F.C. All authors have read and agreed to the published version of the manuscript.

**Funding:** Work partly supported by MINECO (Spain; Projects MAT2012-37782 and MAT2015-63974-C4-2-R).

**Conflicts of Interest:** The authors declare no conflict of interest.
