## 5 Results and Discussion
### Results for Non-Proportional (90\({}^{\circ}\) Out-of-Phase Tension and Torsion) Loading  
Figure 7a-c compares SIF obtained with the proposed method to the COD method (relative displacements of two opposite-to-crack points, Equation (14)) for modes I, II, and III, respectively, for the R-033 sample after 15,000 loading cycles. The LSM method was applied for this sample using one and seven terms of [PERSON] series. [PERSON] et al. [40] used 2, 3, and 4 terms, finding slight differences for mode I, and [PERSON] et al. [38] used up to 10 terms, finding that \(K_{I}\) stabilized after using as little as two terms. [PERSON] et al. [29] discussed how the number of terms can influence fracture mechanics parameters and FCG stability, but there is limited understanding of their physical meaning. Recently, [PERSON] et al. [41] used a similar approach, but used the stability of the energy release rate in a phase-field simulation as a benchmark. On the other hand, [PERSON] et al. [27] found that after nine terms, the differences in the results are negligible in photoelastic image analysis. So, to perform an analysis, one typically starts with one term in the Williams' series and gradually raises the number until SIF results stabilize. The attached algorithm includes the number of terms as a variable in the Supplementary Materials.  
Figure 6: R-031 (45\({}^{\circ}\) out-of-phase tension and torsion) results comparison between COD and LSM methods: (**a**) \(K_{I}\), (**b**) \(K_{II}\), (**c**) \(K_{III}\).  
On the other hand, SIFs do not show proportional behavior to either axial or torsional loads for most of the loading range. Such nonlinearity can indicate that crack blunting is present as the crack lips are displaced when the load is reduced. Additionally, one can see how sliding modes II and III stay about constant after the maximum axial load is reached. After the crack opens, hence disabling fatigue crack closure, the crack edges are free to slide, and the torque load keeps them open. However, \(K_{II}\) and \(K_{III}\) seem to reach zero at the smaller axial load. This may be attributed to rugosity and crack flank locking (attributed to the crack's curvature) interfering with in-plane sliding and being unable to overcome the friction. Finally, when comparing Figure 7a-c, one can see how, at maximum axial load, sliding modes are the largest but in opposite directions. As the axial load ramps down, \(K_{III}\) sharply falls, and when the axial load goes up, \(K_{II}\) rises. One must remember that under the applied fatigue loading, such continuous sliding would eventually produce crack smoothing [9], thus enhancing mode III.
Summarizing SIFs' evolution during a cycle obtained from displacements, one can say that for proportional loading in-phase tension-torsion, the SIFs' ratio remains quasi-constant other than what is attributed to accumulated plasticity, rugosity, crack flanks interlocking, and perhaps trapped debris. On the other hand, for the non-proportional load samples (90\({}^{\circ}\) out-of-phase and 45\({}^{\circ}\) out-of-phase tension and torsion), the SIFs change at different ratios. In these two cases, such extra difference is attributed to the constant change of applied loading.