## 2 Design and Development of 3D Curvature Sensor of FBG
### Sensor Numerical Simulation Study  
In order to verify the correctness of curvature--strain relationship, finite element simulation is carried out by Ansys workbench. By adding a fixed constraint at one end of the flexible substrate and deflecting the other end by a certain angle along the direction of the \(X\)-axis, the curvature radius of the curvature part of the three-dimensional FBG curvature sensor model is made to be 100 m, 200 m, 300 m, 400 m and 500 m. The relationship between the strain characteristics of the FBG string at the bending side of the pressure sensor and the curvature at the bending side is analyzed. After building the model and dividing the grid, 1,425,803 units and 306,242 nodes were generated. The outer diameter of the sensor is 80 mm, the inner diameter is 60 mm, the thickness is 10 mm, the elastic modulus is E = 6.11 MPa, and the Poisson ratio \(\mu\) = 0.49. The model of FBG string is NZS-FBG-S, and the main material of bare fiber is glass silica. The elastic modulus of bare fiber is E = 72 GPa, and the Poisson ratio \(\mu\) = 0.17. Figure 5a-e respectively show the strain characteristics of sensor fiber grating strings under different curvature states.  
1. Since the simulation study in this paper is the strain characteristics of the FBG string built in the sensor under the condition of small curvature, the string mode of FBG is studied, respectively, on the theory of neutral axis and axis-bearing, and the strain characteristics of FBG string under different curvature modes are simulated. The results show that the strain of the FBG string on the axis-bearing string should be one order of magnitude larger than that of the FBG disguised as a relative neutral axis, which fully indicates that the sensor beam designed with a circular section satisfies the neutral axis theory.
2. It can be seen from the figure that the bending direction of the FBG string is consistent with the bending direction of the flexible substrate of the sensor, which fully indicates that the FBG string inside the sensor can maintain the same deformation as the sensor substrate, and proves the rationality of the selection of the sensor substrate and filling material.
3. The maximum compressive strain of the last beam of FBG under different curvature modes is 0.00041, 0.000205, 0.000137, 0.0001 and 0.0000807, almost consistent with the theoretical calculation results, which proves the correctness of the sensor theory of the developed fiber Bragg grating curvature sensor. Table 1 below shows the theoretical and simulation results of the relationship between curvature and strain. The variation curve of FBG strain with curvature is shown in Figure 6.  
Figure 5: _Cont._Figure 5: Strain characteristics of sensor fiber gratings in different curvature states. (**a**) The radius of curvature is 100 m. (**b**) The radius of curvature is 200 m. (**c**) The radius of curvature is 300 m. (**d**) The radius of curvature is 400 m. (**e**) The radius of curvature is 500 m.  
\begin{table}
\begin{tabular}{c c c c c c} \hline
**Concrete Parameters** & \multicolumn{5}{c}{**Numerical Value**} \\ \hline Radius of Curvature & 100 m & 200 m & 300 m & 400 m & 500 m \\ \hline Curvature & 0.01 m\({}^{-1}\) & 0.005 m\({}^{-1}\) & 0.00333 m\({}^{-1}\) & 0.0025 m\({}^{-1}\) & 0.002 m\({}^{-1}\) \\ \hline Theoretical strain & 0.0004 & 0.0002 & 0.000133 & 0.0001 & 0.00008 \\ \hline Simulated strain & 0.00041 & 0.000205 & 0.000137 & 0.0001 & 0.0000807 \\ \hline \end{tabular}
\end{table}
Table 1: The relationship between theoretical and simulation strain and curvature.  
Figure 6: Curve of sensor curvature—strain relationship.