## 3 Fabrication of the PP Based OFBG Strain Sensor (Processing Techniques)  
### Interface Strain Transfer of PP and OFBG  
The idea of the paralleled semi-extension rule based algorithm is as follows: firstly, the algorithm decomposes the maximum terms space of the clause set into several partial maximum terms spaces, which convert the SAT problem of the clause set into the SAT problem of the partial maximum terms spaces. If there is a certain partial maximum terms space that is satisfied, then the clause set is satisfied. If all the partial maximum terms spaces are unsatisfied, then the clause set is unsatisfied. In other words, the clause set is satisfied. In the following, the concept of the partial maximum terms space is discussed.  
Figure 1: Variation of apparent viscosity of PP. matrix (0%-15%) with temperature.  
Figure 2: Variation of apparent viscosity of PP. matrix (5%) with shear rate.  
The internal force of PP and OFBG can be calculated as follows:  
\[\sigma_{p}=\frac{\varepsilon_{sp}}{\frac{1+C}{E_{p}}+\frac{A_{p}}{A_{f}E_{f}}} \tag{1}\]  
where: \(\sigma_{p}\), \(E_{P}\) and \(A_{p}\) represent the internal stress, modulus and sectional area of PP matrix, respectively; \(\varepsilon_{sp}\) is the free shrinkage strain of PP matrix without any restraint. \(E_{f}\) and \(A_{f}\) refer to the modulus and sectional area of optical fiber, respectively, \(C\) is creep coefficient.  
Considering the size of sensor designed in this study, \(\sigma_{p}\) should be very small, which is about 0.0832 Pa. The creep of the sensor should not be significant and the strain in the sensor can remain the same level as that in the concrete or asphalt pavement. Therefore, it can test the real strain changes.  
According to the model given by [PERSON] and [PERSON] [16], the strain transfer performance of the interface between the fiber and PP as shown in Figure 3 can also be expressed by the following equation:  
\[\lambda^{2}=\frac{2G_{\mathrm{pp}}}{E_{e}r_{\mathrm{c}}^{2}\ln\frac{r_{\mathrm{ pp}}}{r_{\mathrm{c}}}} \tag{2}\]  
where \(\lambda\)--strain transfer eigenvalue, \(E_{c}\)--modulus of OFBG, \(r_{c}\)--radius of OFBG, \(G_{\mathrm{pp}}\)--shear modulus of the matrix, \(r_{\mathrm{pp}}\)--radius of the matrix.  
So, the error rate of the strain transfer of the interface between OFBG and matrix (\(\eta\)) and correction factor (k) can be described as:  
\[\eta=\frac{\mathrm{ch}(\lambda_{t})\!-\!1}{\lambda_{t}\;\mathrm{sh}(\lambda_{ t})} \tag{3}\]  
\[k=\frac{1}{1\!-\!\eta} \tag{4}\]  
The shear modulus of PP ranges from 0.32 to 0.75 GPa, and the diameter of sensor is 5\(\sim\)20 mm. \(E_{\mathrm{c}}\) is about 70 GPa and \(r_{\mathrm{c}}\) is about 6.25 \(\times\) 10\({}^{-2}\) mm, the l\({}_{\mathrm{f}}\) is 10 mm. The relationship of \(\eta\) and \(G_{\mathrm{pp}}\) and \(r_{\mathrm{pp}}\) is shown in Figures 4 and 5, respectively.  
With the \(G_{\mathrm{pp}}\) (0.4 Gpa) and \(r_{\mathrm{c}}\) (14 mm), the error rates of the strain transfer of the interface between OFBG and matrix (\(\eta\)) and correction factor (k) are 0.126898 and 1.145342, respectively, and the strain sensing coefficient is 1.04 pm/\(\upmu\).  
Figure 3: Bare FBG directly embedded in structure.