## 4 Materials and Methods  
Figure 3 depicts a blueprint of one of the three S235 steel thin tubes (2.6 mm thickness) with a 42.4 mm outside diameter with a 14 mm long and 4 mm wide slit. The three tubes were loaded with sinusoidal in-phase tension and torsion (R-030), 90\({}^{\circ}\) out-of-phase tension and torsion (R-031), and 45\({}^{\circ}\) out-of-phase tension and torsion (R-033). In any case, the stress inversion ratio was \(-1\), as shown in Table 1. Specimens were photographed to be processed using 3D-VIC(r) from Correlated Solutions (Columbia, SC, USA) with the DIC technique. The load was applied at 10 Hz frequency but lowered to 0.1 Hz for the DIC recording using two Allied Vision Tech. (Stadtroda, Germany) GT 2450 cameras with 2448 \(\times\) 2050 resolution, giving a 3.45 \(\upmu\)m \(\times\) 3.45 \(\upmu\)m pixel size. Finally, all tests used through-thickness cracks, with SIFs computed from recorded displacement fields. Extensive details concerning the testing experimental details (instruments, subset, correlation criteria, crack tip position, estimation, and more), as well as an explanation of the calculation methods for \(K\), \(\Delta K\) crack growth rate, and crack lengths, are described elsewhere [7; 39]. Finally, a MATLAB(r) 2021 (Natick, MA, USA) routine was written to test the method whose pseudocode is available in the web version. Figure 3: Thin tubes, dimensions in mm. From [7]. The implemented algorithm takes data input as a MATLAB, \({}^{*}\).mat, extension file. In that case, the data is an array (m \(\times\) n \(\times\) q) of coordinate matrices X, Y, Z, and displacement fields (both in mm), as well as displacements matrices u, v, w, and coordinate matrices x, y, z (both in pixels) as schematized in Figure 4a, whereas the steps are illustrated in Figure 4b. [PERSON] series has been used in curved samples with cracks smaller than the curvature. On the other hand, the LSM is a tool for solving over-determined systems. It finds the solution that minimizes the sum of the squared errors between the measured and the predicted responses. [PERSON] series for displacements were deducted for a planar surface, but the samples analyzed here were cylindrical. Therefore, if one assumes the distance between two points located in the same circumference of a cylinder as a straight line, there would be a mistake because of the specimen's radius. Such inaccuracy is reduced by maintaining those two points close [10], which are CTL and displacement data. Hence, the AOI was determined by picking two \(1\times 1\) mm squares that were situated roughly 1 mm below the fracture tip because the compressive applied load produces negative stress forward of the crack tip, potentially producing negative \(K_{l}\) values. These AOIs were selected in order to exclude accumulated plasticity, which could result in nonlinear behavior and unrealistic SIF values both before and around the crack tip. Furthermore, in an effort to reduce the accumulated plasticity ahead of the crack tip, displacement data was collected behind the crack tip, mimicking a COD clip gauge. That short distance minimizes the effects of accumulated plasticity, the uncertainty derived from the low correlation criteria given around the crack lips (which may provide unrealistic strains), and keeping the distance between AOI and CTL as short as possible to minimize