## 6 Conclusions  
A method to establish \(K_{III}\) using full-field displacement data, obtained with the Digital Image Correlation technique, was proposed and implemented. However, the method could be used with displacement fields obtained in any manner. SIF calculation is more robust when using more than two data points. The procedure was applied to three thin tubes subjected to axial and torsional loads: one in-phase and two out-of-phase. The comparison of results with the COD method shows an agreement for one case of proportional and two cases of non-proportional loads. The pseudocode reflecting the proposed method is available on the web version of the article. The proposed method captures the accumulated plasticity, providing less noisy results. Finally, the SIF evolution during proportional and non-proportional loading was described, showing a pronounced nonlinearity attributed to the constant change of applied loading for the latter case.  
The proposed method data fits displacements to [PERSON] solution for mode III, which is an infinite plate. When approximating the distance from the crack tip to the measured points, the curvature error induced by the camera measurements introduced an error using a straight line instead of the perimeter. However, in this manuscript, the distance between the crack tip and the measured area was kept as close as 1 mm, so the error was minimized, always ensuring that only elastic strains were used to compute SIF. An approach to solve this could use the proposed approximations for \(K_{I}\) based on curvature as described in [8; 40], which could provide more accurate results. However, there are no solutions for sliding modes that allow further improvement. Finally, optimizing the area of interest (AOI) selection could mitigate the impact of accumulated plasticity and enhance the accuracy of SIF calculations.  
Although not tested in this paper, the method could be implemented with displacement fields from any source, such as other experimental techniques, or from numerical simulations. The latter case has advantages as the crack tip location is known in advance, thus reducing the uncertainty in SIF calculation. Furthermore, numerical simulations deliver through-the-thickness displacements, so the maximum SIF could be calculated instead of only surface displacements as presented and tested here.  
The following supporting information can be downloaded at: [[https://www.mdpi.com/article/10.3390/app15063404/s1](https://www.mdpi.com/article/10.3390/app15063404/s1)]([https://www.mdpi.com/article/10.3390/app15063404/s1](https://www.mdpi.com/article/10.3390/app15063404/s1)).  
Conceptualization, J.G.D.-R.; Methodology, J.G.D.-R.; Software, A.R.-T.; Validation, C.H.V.-N.; Formal analysis, J.G.D.-R.; Investigation, C.H.V.-N.; Data curation, C.H.V.-N.; Writing--original draft, J.G.D.-R.; Writing--review & editing, A.R.-T.; Visualization, A.R.-T.; Funding acquisition, J.G.D.-R. All authors have read and agreed to the published version of the manuscript.  
This research received no external funding.  
Data Availability Statement: The original contributions presented in this study are included in the article. Further inquiries can be directed to the corresponding author.  
The authors express their gratitude to [PERSON] at TU-Darmstadt for facilitating the DIC data and the personnel at the Photomechanics laboratory at PUC-Rio.