# Strain Transfer Analysis of a Clamped Fiber Bragg Grating Sensor
## 5 Experimental Verification  
Based on the strain transfer rate model, a clamped FBG sensor is designed for experimental tensile tests. A bare FBG sensor, as shown in Figure 7, is bonded into a capillary tube along the axis, with a tube diameter of 1 mm, a length of 25 mm, and a gauge length of 12.5 mm, respectively. The matrix material is shown in Figure 8. The other experimental parameters are adopted by the default parameters from Table 1. Epoxy adhesive 353 ND is used, curing at 70 \({}^{\circ}\)C for half an hour. As can be seen from Figure 8, a strain gauge, a bare FBG sensor and a clamped FBG sensor are bonded on the matrix with a section size of 10 mm \(\times\) 40 mm.  
Figure 6: Interlayer thickness vs. strain transfer rate of a clamped FBG sensor.  
Figure 7: Bare FBG sensor.  
Loading steps by universal testing machine are as follows:  
1. Specimen is clamped by supports on the testing machine. The pre-loading applied to the specimen is 30 kN, and the step loading is 5 kN every ten seconds until the total loading of 60 kN.
2. In ten seconds pause time, the resistances of the strain sensor and FBG demodulators are acquired as the immediate data.  
Since the regression correlation coefficient R\({}^{2}\) is more than 0.99, the calibration curve of the clamped FBG sensor measurement is linear. The linear calibration equation is expressed as \(y=0.00086x+1528.15\), as shown in Figure 9. The calibration equation of the bare FBG sensor measurement is \(y=0.00123x+1528.02\) as shown in Figure 10.  
Figure 8: Matrix material.  
Figure 9: Calibration curve from tests on a clamped FBG sensor.  
The average strain transfer rate of the clamped FBG sensor is equal to the encapsulated FBG sensor strain divided by the matrix strain, which is measured by the demodulator. \(\alpha\), \(\epsilon_{\epsilon}\), \(\epsilon_{m}\), \(\Delta\lambda_{e}\), \(K_{\epsilon}\) are the test value of the average strain transfer rate of a clamped FBG sensor, the encapsulated FBG sensor strain, the matrix strain, the clamped FBG sensor wavelength shift, and the strain sensitivity coefficient of a bare FBG sensor, respectively. The experimental test value of the average strain transfer rate of a clamped FBG sensor can be obtained as follows:  
\[\alpha=\frac{\epsilon_{\epsilon}}{\epsilon_{m}}=\frac{\Delta\lambda_{e}}{K_{ \epsilon}\cdot\epsilon_{m}}=\frac{0.00086}{0.00123}=69.9\%\]  
The theoretical value of the average strain transfer rate can be calculated from the following formula, by using the parameters in Table 1.  
\[\overline{\alpha}=\frac{1}{a}\bigg{\{}1-\frac{\sinh(\sqrt{a}\cdot k\cdot\text{ L}_{1})}{\sqrt{a}\cdot k\cdot\text{L}_{1}\cdot\cosh(\sqrt{a}\cdot k\cdot\text{L}_{1})} \bigg{\}}=64.6\%\]  
The average strain transfer rate of the clamped FBG sensor from the experimental test is 69.9% and the theoretical result is 64.6%. The error is less than 5%, indicating a good agreement, and the theoretical model is demonstrated to be suitable for the strain transfer analysis of the clamped FBG sensor. However, the possible reasons why the theoretical result is a little smaller than the test result are discussed as follows:  
1. Parameter error: The strain transfer rate of the theoretical model involves the elastic modulus of the optical fiber, shear modulus of the interlayer, Poisson ratio, etc. This may generate errors because of the difference between the used and actual values in testing conditions.
2. Model error: Some assumptions of the theoretical model lead to model errors. For example, the defect of the \(n\)-segment optical fiber appears in the writing process, which could alter the mechanical properties.
3. Test error: This is one important error source including sensor zero drift, asynchronous acquisition and the artificial reading error.