## 2 The Sensing Theory of FBG Dynamometry Bolt  
### Optical Fiber Sensing Principle  
In this paper, the sensing transmission mechanism of the FBG sensor bonded on the surface of the matrix structure via adhesives is studied. According to the shear lag theory proposed in the literature [13] to analyze the internal stress transfer of composites, it was considered that the strain was transmitted from the substrate to the fiber grating in the form of shear strain through the adhesive layer. On the one hand, the FBG sensor packaging materials are mostly polymers. The magnitude order difference between the elastic modulus of polymers and the elastic modulus of the optical fiber and the substrate results in a non-synergistic deformation between the response strain and the actual strain. Onthe other hand, most polymers are viscoelastic, which can show the mechanical properties of elastic solids and viscous liquids under the response of an external force, that is, their mechanical properties change over time. The study of the strain transfer rules of the FBG sensors over time can provide practical and reliable theoretical analyses for parameter calibration, accuracy analysis, error estimation, and correction of sensors, so that the sensors can maintain good performance for long-term service in engineering practice. A strong ultraviolet laser was used to change the structure of the germanium-doped fiber core, which can form fringes with periodic changes in the refractive index and are uniformly distributed along the axial direction inside the fiber; a diffraction grating is then formed. The FBG sensors are composed of the fiber core, cladding layer, coating layer, reinforced fiber layer, and optical fiber protective sleeve from the inside to the outside. The fiber without a reinforced fiber layer and fiber protective sleeve is called bare fiber optics. The structure is shown in Figure 1. The main component of the fiber core is silica, which is doped with a very small amount of germanium. The main component of the cladding layer is pure silica. The coating layer is generally a polymer material, such as epoxy resin or silicone rubber. As shown in Figure 2, when a wide spectrum incident light passes through the fiber grating, the light waves meeting the Bragg condition of the fiber grating are reflected, and the rest of the light waves continue to be transmitted through the optical fiber. The reflected light is demodulated by the optical element and the peak of the reflected light wavelength is called the center wavelength \(\lambda_{B}\). When the external physical quantity changes, the grating period \(\Lambda\) and effective refractive index \(\Delta n_{\text{eff}}\) of the grating part are changed simultaneously, and then the reflected wavelength changes accordingly. The change of the reflected wavelength and the external physical quantity follow a fixed law, so that the value of the external physical quantity can be reflected by the change of the central wavelength. When the optical fiber is only subjected to stress, the change of the central wavelength of the reflected light has the following relationship with the axial strain of the grating [15]:  
\[\frac{\Delta\lambda_{\text{B}}}{\lambda_{\text{B}}}=(1-P_{\text{e}})\varepsilon, \tag{1}\]  
Figure 1: Schematic diagram of the FBG sensor’s structure. Figure 2: Principle of fiber Bragg grating sensing. where \(\Delta\lambda_{\rm B}\) is the drift value of the central wavelength of the reflected light, \(\varepsilon\) is the axial strain of the grating, and \(P_{\rm e}\) is the elasto-optical coefficient. For silica fiber, \(P_{\rm e}\) is equal to 0.22. When the optical fiber is only affected by temperature, the change of the central wavelength of the reflected light has the following relationship with the temperature:  
\[\frac{\Delta\lambda_{\rm B}}{\lambda_{\rm B}}=(\alpha+\zeta)\Delta T, \tag{2}\]  
where \(\alpha\) is the thermal expansion coefficient, \(\zeta\) is the thermo-optic coefficient, and \(\Delta T\) is the temperature variation.