# Detection of Thermal Strain in Steel Rails with BOTDA
## Laboratory Test
### Measurement Principle  
Distributed optical fiber sensing based on Brillouin scattering takes advantage of light interaction with the acoustic phonons propagating in the fiber core [9]. The frequency shift of Brillouin-scattered light is proportional to the acoustic phonons velocity, and can be given by:  
\[\
u_{B}=\frac{2n_{eff}V_{a}}{\lambda}, \tag{1}\]  
where \(n_{eff}\) is the effective refractive index of the optical fiber, \(\lambda\) is the free-space wavelength of the pump light, and acoustic wave velocity \(V_{a}\) depends on the local density and tension of the glass that contribute to the temperature and strain dependence of the Brillouin frequency shift.  
The temperature response of the loose jacketed fiber results from two effects, including the Brillouin frequency shift with the temperature of the bare fiber and the thermal strain generated by the difference between thermal-expansion coefficients in bare fiber and its jacket material [6]. The temperature variation-induced frequency shift of \(\
u_{B}\) can then be written as the following [10]:  
\[d\
u_{B}/dT=C_{T}+C_{e}d\epsilon/dT, \tag{2}\]  
Here, \(C_{T}\) is the Brillouin frequency shift with the temperature of the bare single-mode fiber (including fiber coating) that is calibrated to be at a constant value, with a value of 1.01 MHz/\({}^{\circ}\)C [7], strain coefficient \(C_{e}\) was measured to be 0.048 MHz/\(\mu\)e [11], and \(d\epsilon\) represents the axial strain along the fiber caused by the thermal-expansion mismatch of the fiber and the jacket material. The jacket effects are commonly modeled by Lame solutions [12], in which the jacketed fiber is considered to have a multilayer structure as indicated in Figure 3. In our case, a two-layer model, including the bare single-mode fiber and an external jacket layer, was utilized. Following the Lame formulas for the strain components in a cylindrical fiber subjected to a differential temperature, and assuming no external force to the jacketed fiber, the induced axial thermal force can be given by [13]:  
\[P=\pi E_{2}(r_{2}^{2}-r_{1}^{2})\overline{\pi}dT, \tag{3}\]  
where \(E_{2}\) is the Young's modulus of the jacket material, \(\overline{\pi}=\frac{1}{\Delta T}\int_{T_{0}}^{T}\alpha_{2}(\tau)d\tau\) is the effective thermal expansion coefficient of the jacket material at the given temperature, \(\alpha_{2}(T)\) is the instantaneous thermal-expansion coefficient, \(T_{0}\) is the zero-strain temperature of the jacket material, and \(\Delta T=T-T_{0}\). Then, we can get the axial strain variation with temperature and, hence, the Brillouin frequency variation with temperature on the single-mode fiber:  
\[d\epsilon/dT=\frac{E_{2}}{E_{1}}\frac{(r_{2}^{2}-r_{1}^{2})}{r_{1}^{2}} \overline{\pi}, \tag{4}\]  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Fiber** & **Fiber A** & **Fiber B** & **Fiber C** \\ \hline Jacket diameter (\(\mu\)m) & \(850\pm 50\) & 900 & \(155\pm 5\) \\ Jacket material & Polyvinyl chloride & Flame-retardant polyvinyl chloride & Carbon/PYROCOAT \\ \hline \hline \end{tabular}
\end{table}
Table 1: Diameters of the jacketed fiber.  
Here, \(E_{1}\) is the Young's modulus of the bare fiber. We can see that the Brillouin frequency shift of the jackted fiber is determined by the Young's modulus, thermal-expansion coefficient, and the radius of each layer.  
Considering the fiber glued on the rail specimen, there is external force generated by the thermal expansion or contraction of the rail specimen to the optical fiber. Strain by the rails, the effects of the jacket material, and the optical fiber itself may contribute to the measured Brillouin frequency shift under different temperatures. As a result, each single contribution has to be determined by considering the Young's modulus of the materials, and the dimension of both the rail and the jackted fiber for thermal-force measurement.