## 2 Theoretical Model of the Strain Transfer Ratio
By substituting (12) and (14) into (13), an equation can be derived as follows:  
\[\tau_{\rm CF}(x,y)=-(1-\frac{y-y_{a}}{y_{n}-y_{a}})[\frac{2\pi y_{s}^{2}}{D} \frac{d\sigma_{s}}{dx}+\frac{\pi(y_{c}^{2}-y_{s}^{2})}{D}\frac{d\sigma_{c}}{ dx}+(h_{a}+y_{c}+y_{a}-\frac{\pi y_{c}^{2}}{D})\frac{d\sigma_{a}}{dx}] \tag{15}\]  
Based on the Hypothesis 1, [PERSON]'s law \(\sigma=E\varepsilon\) is suitable for the silica core, the coating, and the adhesive layer. The adhesive layer thickness is much smaller than the gauge length, and bonds between all the two interfaces are intact according to the Hypothesis 2. Hence, the strain gradients of the four parts are expected to be of the same order [24], which can be described as  
\[\frac{d\varepsilon_{\rm s}}{dx}\cong\frac{d\varepsilon_{\rm c}}{dx}\cong \frac{d\varepsilon_{\rm a}}{dx} \tag{16}\]  
where \(\varepsilon_{\rm s}\), \(\varepsilon_{\rm c}\), and \(\varepsilon_{\rm a}\) are the normal strain of the silica core, the coating, and the adhesive layer, respectively. The relationship between displacement \(u\) and shearing strain \(\gamma\) can be derived based on shear-lag theory [34; 35; 36] as follows:  
\[\left\{ \begin{array}{l}u(x,y_{c})-u(x,y_{s})=\int_{y_{a}}^{y_{c}}\gamma_{c} (x,y)dy\\ u(x,y_{a})-u(x,y_{c})=\int_{y_{c}}^{y_{a}}\gamma_{a}(x,y)dy\\ u(x,y_{k})-u(x,y_{k-1})=\int_{y_{k-1}}^{y_{k}}\gamma_{\rm CF}(x,y)dy\end{array} \right. \tag{17}\]where \(k\) is a positive integer and not greater than \(n\); \(y_{0}\) is equal to \(y_{a}\); \(\gamma_{\rm c}\) and \(\gamma_{\rm a}\) are the shear strain of the coating and the adhesive layer; \(\gamma_{\rm CF}\) is the shear strain of the fabric layer. Then, Equation (9) is substituted into (17). [PERSON]'s law \(\tau_{c}=G_{\rm c}\gamma_{c}\) and \(\tau_{a}=G_{a}\gamma_{a}\) are applied to the coating and adhesive layer, where \(G_{c}\) and \(G_{a}\) represent the shear modulus of coating and adhesive layer. By taking the derivative with respect to \(x\), it can be obtained as  
\[\left\{\begin{array}{ll}\frac{du(x,y_{c})}{dx}-\frac{du(x,y_{s})}{dx}= \varepsilon(x,y_{c})-\varepsilon(x,y_{s})=\frac{1}{\rm G_{c}}\int_{y_{s}}  
The average strain transfer ratio is defined as the average value of the strain transfer distribution ratio along the whole gauge length [38] as calculated in (24). The average strain transfer ratio can evaluate the difference between the strain measured by FBG sensors and the original strain in the host material for the whole gauge length. The value of the average strain transfer ratio varies from 0 to 1. When the average strain transfer ratio is close to 1, most of the strain in the host material is transferred to FBG sensors, and a relatively accurate strain measurement result will be obtained. If the average strain transfer ratio is close to 0, most of the strain is lost for the strain transfer, and the strain measurement accuracy is relatively low. Therefore, the average strain transfer ratio indicates the strain transfer efficiency. In addition, as the average strain transfer ratio is the ratio between the strain measured by FBG sensors and the strain in the host material, the ratio can be used to compensate the strain transfer loss during the strain measurement.