**4. Conclusions**

The critical in-plane strains and effective strain at fracture of incremental tube expansion are greater than those of conventional tube expansion by rigid tapered conical punches. However, the conclusion that the gains in formability are due to the fact that strain paths go beyond the fracture forming limit (FFL) line of the tube material is erroneous because it does not account for the non-proportional, cyclic nature of the strain paths and because it ignores the FFL being a material property that is independent of any type of applied loading.

Finite element modelling of incremental tube expansion considering material strain hardening and non-proportional, cyclic paths resulting from the real tool trajectory combined with the utilization of appropriate integral forms of stress triaxiality allows understanding that the gains in formability result from a shift of the loading paths towards the left side in the effective strain vs. stress triaxiality space. Moreover, the results also show that this shift of material flow is of paramount importance to ensure compatibility between the critical strains at fracture and the threshold admissible values of the material FFL.

The necessity of using effective strain vs. stress triaxiality evolutions based on average stress triaxiality to ensure compatibility with the FFL in incremental tube expansion is understandable because the individual locations of the plastically deformed tube surface oscillate cyclically from shearing to biaxial stretching, as the single point hemispherical tool approaches, contacts and moves away from these locations during its trajectory.

The fact that average stress triaxiality is not required to handle the formability of conventional tube expansion is compatible with the match between stress triaxiality and the integral forms of stress triaxiality (e.g., average stress triaxiality) when material is subject to near-proportional loading paths. This last conclusion is no less important than the previous ones because it justifies the successful utilization of McClintock's fracture criterion to analyze the onset of fracture by tension in conventional tube forming processes over the past few decades.

**Author Contributions:** Conceptualization, G.C., C.V. and P.A.F.M.; methodology, M.B.S. and G.C.; software, C.S.; validation, C.S. and G.C.; formal analysis, G.C. and M.B.S.; investigation, C.S., M.B.S. and G.C.; resources, C.V., M.B.S. and P.A.F.M.; writing—original draft preparation, C.S. and G.C.; supervision, G C. and C.V.; project administration, G.C. and C.V.; funding election and management, G.C. All authors have read and agreed to the published version of the manuscript.

**Funding:** The authors would like to express their gratitude for the funding received through the Grant number US-1263138 US/JUNTA/FEDER\_UE within "Proyectos I+D+i FEDER Andalucía 2014-2020" and the Fundação para a Ciência e da Tecnologia of Portugal, through IDMEC under LAETA, project UIDB/50022/2020.

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** Not applicable.

**Acknowledgments:** The authors would like to acknowledge the collaborations of Jaime Blanco Vital in the preliminary development of the numerical modeling and Valentino Cristino in the experimentation.

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