## 2 Sensor Design
### Strain Gauge Factor  
In conformity to the guideline for strain sensors based on fiber Bragg gratings [3], the patches were analyzed concerning their strain gauge factor and the temperature dependence of the strain gauge factor. Previous investigations have shown that there are quite large disagreements between specifications of the manufacturers of commercially available fiber Bragg grating patches and the observed characteristics [2]. The strain gauge factor and the temperature dependence of the strain gauge factor of the designed patches were therefore characterized. The patches with embedded acrylate-coated FBGs were applied together with resistance strain gauges on stainless steel and aluminum tensile test coupons. The resistance strain gauges were chosen for referencing the strain gauge factor of the patches, while the coupons were only statically loaded with relative small loads (up to 2000 um/m). At first, the tensile test coupons were loaded up to a maximum load of 15 kN in three cycles to pre-stress the coupons. After that, the coupons were loaded in steps of 1.5 kN up to the maximum load, and down again also in three cycles. The maximum load of 15 kN correlates to a strain of 740 um/m of the stainless steel coupons and a strain of 1800 um/m of the aluminum coupons. During the whole test, the temperature was kept constant at 23 \({}^{\circ}\)C. All the investigated patches showed linear strain behavior over the total loading cycle, as shown in Figure 8. The strain gauge factor was determined to be 0.7799. To analyze the temperature dependence of the strain gauge factor, the tensile test coupons were loaded in the same way as for the tests to determine the strain gauge factor at room temperature. These tests were carried out at temperatures of \(-40\)\({}^{\circ}\)C, \(-20\)\({}^{\circ}\)C, \(70\)\({}^{\circ}\)C and \(80\)\({}^{\circ}\)C. Although the patches were manufactured at a curing temperature of \(60\)\({}^{\circ}\)C, it was necessary to determine how and under what temperature and load the patches start to degrade. Before starting with the test, the patches were post-cured at a temperature of \(82\)\({}^{\circ}\)C. The loading cycles started when the temperature chamber reached thermal equilibrium. In Figure 9, it can be observed how the strain gauge factor varies with temperature. Between \(-40\)\({}^{\circ}\)C and \(+70\)\({}^{\circ}\)C, the strain gauge factor is stable. At \(+80\)\({}^{\circ}\)C, the strain gauge factor amounts to \(0.6896\pm 0.0451\). The strain gauge factor dropped nearly \(12\%\) compared to the strain gauge factor at room temperature. Additionally, it can be observed that at higher temperatures, the single data points spread farther from the ideal linear fit curve than at lower temperatures. This is probably due to an initiation of material changes of the patch carrier near the post-curing temperature. Figure 8: Determination of the strain gauge factor from the change in Bragg wavelength compared to resistance strain gauges. Figure 9: Change in Bragg wavelength of a patch compared to the strain determined with resistance strain gauges at different temperatures. In case of all other temperatures, the strain gauge factor is in a reasonable range, as also determined for room temperature. This can be concluded from Table 1.  
It can be concluded from these experiments that the developed patches with embedded acrylate-coated FBGs show a linear and constant strain behavior over a temperature range of \(-40\)\({}^{\circ}\)C to \(+70\)\({}^{\circ}\)C, if a post-curing has been carried out over 80 \({}^{\circ}\)C. The strain gauge factor is slightly temperature dependent, but the corresponding gauge factor error over the whole temperature range is \(1.4\times 10^{-4}\) 1/K. It is assumed that this effect is not significant in this temperature range, and for practical applications, the average strain gauge factor for the designed patches is calculated to be \(0.7874\pm 0.0081\).