## 3. Results and Discussion
### Strain Sensing Information from Numerical Results
#### 3.2.3 Torsion  
Numerical results from this loading case are evaluated and discussed based on the strain contours of FOS #2, as it is closer to the surface and has higher strain values than FOS #1. Figure 11b shows that FOS #2 under torsion is in complex three-dimensional strain states, where both in-plane and out-of-plane strains contribute to its maximum principal strains. For example, as shown in Figure 11c, the \(E_{23}\) strain component dominates in mid-line crossing areas at both ends, while in the center of the longitudinal span, positive and negative \(E_{23}\) strain components cancel out. In peak areas, the maximum principal strains result from both \(E_{13}\) and \(E_{11}\) strain components, which are illustrated in Figure 11d,e, respectively. Furthermore, three major patterns can be derived from the comparison between the maximum principal strain values in peak areas and those in vertical portions in Figure 11b: (1) at the loading end, the peak-area-to-vertical-portion ratio is around 1.5; (2) in the center of the longitudinal span, the values in peak areas are more than four times those in vertical portions; (3) the fixed end sees a less than threefold difference between them. However, due to the complex strain states, it is difficult to relate the strains along the FOS cores to strain components in host materials. The resulting strain contours of the FOS cores from torsion agree with expectations and show specific patterns that are different from those in tension and cantilever bending. While the results from this loading case can also imply the capabilities of dual-sinusoidal FOSs to sense different strain components along their paths and to provide complementary strain measuring areas, to more accurately interpret the strains measured along sinusoidal FOSs under torsion, further correlations need to be established between numerical and experimental results. To go somewhat toward the validation of the concept suggested in the current article, an analytical approach has been suggested in previous work (3). Glass-fiber-reinforced composite material manufactured by pultrusion was considered and specimens (800 mm \(\times\) 30 mm \(\times\) 20 mm) were cut for bending tests. Fiber-optic sensors (diameter 125 microns 80 mm microns acrylate coating) were embedded as double-sinusoids (Figure 12). The specimens were heated at 40 \({}^{\circ}\)C to ensure appropriate bonding and then mechanically tested under 3-point bending and displacement controlled. FOS signals were recorded continuously with a LUNA OBR-4600-type optical back-scattering reflectometer (LUNA company, Roanoke, VA, USA). Figure 10: Cantilever bending case: (**a**) Iso view of the numerical model under cantilever bending; (**b**) the maximum principal strain contours of FOS cores. Figure 11: _Cont._  
It is worthwhile to note that the analytical result listed hereafter is not a full validation of numerical modeling but is a verification that the assumptions upstream of our new concept are making sense. Sinusoidal placement has the advantage of covering a much larger surface of a structure with a single optical-fiber sensor than a linearly placed FOS by recording strain in the three dimensions. This tendency is much more reinforced when using double-sinusoid placement. The result pictured in Figure 13 emphasizes strain measurements under bending obtained with the double-sinusoid placement technique. It is clearly shown that both FOSs are able to measure bending same-strain signals (Both FOSs shows are recording the same value around 1200 \(\upmu\)m/m), which is an encouraging result. However, we should avoid excessive satisfaction since additional experiments and deep analysis have to be performed so that the described concept of using double-sinusoid placed sensors be more convincing, either for scientists or users.