# A Strain-Transfer Model of Surface-Bonded Sapphire-Derived Fiber Bragg Grating Sensors
## 3 Finite Element Analysis  
The FE model is usually generated to validate analytical models. Unlike in previous studies [10; 22; 231] that mainly focused on the strain-transfer medium except for the fiber, the FE simulation in this section was used to study the effects of an alumina dopant in the fiber core. In order to obtain the appropriate adhesive size for the FE model, however, the half-bond length and the bottom thickness of the adhesive layer had to first be analyzed using the three-layer model (\(r_{\infty}=r_{d}\) in Equation (14) with no alumina dopant (\(m=0\))). The relevant physical characteristics of the materials and the size of each layer are listed in Tables 1 and 2, respectively. Results are shown in Figure 4. A half-bond length of \(L=5\)mm and a bottom thickness of \(h_{a}=0.1\)mm were selected, as a relatively high ASTR (over 90%) had been obtained in these conditions. The FE model was then established using COMSOL Multiphysics 5.3 (COMSOL Co., Ltd., Stockholm, Sweden) commercial software. Mesh generation is shown in Figure 5. \begin{table}
\begin{tabular}{c c c c c} \hline  & **Sapphire** & **Silica** & **Adhesive** & **Host material** \\ \hline \(E\)(GPa) & 503 & 72 & 10 & 195 \\ \(\mu\) & 0.25 & 0.17 & 0.34 & 0.24 \\ \hline \end{tabular}
\end{table} 
Table 1: Physical characteristics of materials used in strain-transfer model. \begin{table}
\begin{tabular}{c c c c c} \hline  & **\(r_{co}\)** & \(r_{cl}\) & \(h_{m}\) & \(w_{m}\) \\ \hline Size (mm) & 0.0625 & 0.0625 & 1.5 & 4.4 \\ \hline \end{tabular}
\end{table} 
Table 2: Size of each layer in strain-transfer model. The influence of the fiber core's radius and alumina content on the ASTR was studied. Figure 6 shows the simulated results calculated from both the analytical four-layer model and the FE model. When the core's radius increased to the cladding's radius (62.5 \(\upmu\)m), the ASTRs of the four-layer model for different alumina contents were approximately equal to the corresponding rates from the three-layer model (see black rectangle in Figure 6). Simulated results indicated that ASTRs generally experienced a decreasing trend as the core's radius increased from 5 to 62.5 \(\upmu\)m. For example, when the alumina content was 50 mol %, ASTRs from the four-layer and FE models decreased from 97.2% to 77.6% and 92.1% to 76.4%, respectively. This could mainly be attributed to two factors. First, cladding thickness decreased with the core's increasing radius, which was beneficial to the efficient transfer of strain from the adhesive to the core. Second, increasing radius reduced the normal stress on the cross section of the core because normal stress is approximately inversely proportional to the core's radius, as shown in Equation (11). Although both factors mitigated each other, the latter was proven to be stronger, which led the ASTR to decrease with the core's increasing radius. In addition, increasing alumina content also led to a decrease in ASTRs. When the core's radius was fixed at a certain value, such as 30 \(\upmu\)m, ASTRs from the four-layer and FE models decreased from 95.3% to 84.1% and 91.5% to 85.9%, respectively. Significant strain loss resulted from large shear deformation that occurred in cladding. Figure 4: Influence of half-bond length and adhesive-bottom thickness on average strain-transfer rate (ASTR) calculated from three-layer model. Figure 5: Finite-element (FE) mesh of strain transfer. Errors of the four- and three-layer models from the FE model are shown in Figure 7. For the cases where there was no- and low-alumina content (\(m=0\) mol % and \(m=4\) mol %), the three-layer model achieved smaller error percentages than those of the four-layer model. This was reasonable because the three-layer model assumed homogeneous material within the fiber region, which was consistent with the fact that the fiber was still mainly composed of silica. Therefore, the three-layer model was more suitable than the four-layer model for predicting transfer efficiency when there was no or low alumina dopant in the core.