# Fiber Bragg Gratings Based Cyclic Strain Measuring of Weld Toes of Cruciform Joints
## 2 Theoretical Context
### FBG Strain Sensor  
FBG is a periodic modulation of the index of refraction, which is inscribed into the core of a single-mode optical fiber [37]. Essentially, it works as a wavelength filter [38]. When a broadband light source is injected into the fiber, the FBG reflects a narrowband portion of the incident light with a specific wavelength, while the rest portion of the broadband light passes through, as shown in Figure 4.  
Generally, the center wavelength of the reflection spectrum is called the Bragg wavelength (\(\lambda_{\text{B}}\)), which is correlated with the grating period (the distance between the consecutive grooves).  
\[\lambda_{B}=2n_{eff}\Lambda \tag{9}\]  
where \(n_{eff}\) is the effective refractive index of the optical fiber, and \(\Lambda\) is the periodicity of the grating.  
When the FBG is subjected to the external environment (e.g., strain and temperature), the grating period increases or decreases, and then the Bragg wavelength increases or decreases, respectively. Thus, the peak shift of the Bragg wavelength introduced by the strain variation (\(\Delta\varepsilon\)) and the temperature variation (\(\Delta T\)) can be expressed as:  
\[\frac{\Delta\lambda_{B}}{\lambda_{B}}=(1-P_{e})\cdot\Delta\varepsilon+\left( \alpha_{f}+\xi\right)\Delta T \tag{10}\]  
where \(\Delta\lambda_{\text{B}}\) is the shift of the Bragg wavelength, \(P_{e}\) is the photo-elastic coefficient of the optical fiber, \(\alpha_{f}\) is the thermal expansion coefficient, \(\xi\) is the thermal-optic coefficient.  
Typically, the fractional wavelength changes of \(\lambda_{\text{B}}\) are on the order of 10 pm/\({}^{\circ}\)C and 1.2 pm/\(\mu\varepsilon\) for FBG with a center wavelength of 1550 nm [37]. So, the Bragg wavelength shift can be considered linear to the strain changes under constant temperature condition.  
It should be noted that the optical fiber is a cylindrical filament composed of a cladding layer with an outer diameter of 125 \(\upmu\)m and a core with a diameter of 9 \(\upmu\)m. In order to enhance the mechanical properties of the optical fiber, a protective layer with a certain thickness may be coated outside the optical fiber. Therefore, the fiber (including grating) sectional dimension is about 0.25 mm, which is much smaller than the base size of the strain gauge. Therefore, it is possible to arrange multiple FBG sensors along the longitudinal direction of the weld toe. In addition, the optical fiber with secondary coating protection has good flexural deformation resistance and can be bonded along the weld toe transition arc even across the weld toe characteristic point. Since the wavelength signal collected by the FBG sensor is the maximum value of the Bragg wavelength in the grating, for the strain testing for weld joints, it can be considered that the obtained strain data is just the exact value at the most severe position ( which refers to weld toe local position) on the certain weld section. So, the FBG is a kind of ideal strain sensors which can be used for local strain measurement at the weld toe.  
Figure 4: Schematic diagram of working principle of Fiber Bragg Grating (FBG).