## 2 Theoretical Model of the Strain Transfer Ratio
Figure 2: Configuration of CFRP laminates. Each fabric layer in CFRP laminates is orthotropic. According to [PERSON]'s law, the relationship between stress and strain can be defined as  
\[\left\{\varepsilon\right\}_{m}=\left[C\right]_{m}\left\{\sigma\right\}_{m} \tag{5}\]  
where \(\left\{\varepsilon\right\}_{\text{m}}=\left[\varepsilon_{11}\ \varepsilon_{22}\ \varepsilon_{33}\ \gamma_{12}\ \gamma_{23}\ \gamma_{31}\right]\) and \(\left\{\sigma\right\}_{\text{m}}=\left[\sigma_{11}\ \sigma_{22}\ \sigma_{33}\ \tau_{12}\ \tau_{23}\ \tau_{31}\right]\) are the strain and stress matrixes, respectively. The elastic matrix \(\left[C\right]_{\text{m}}\) is described as follows:  
\[\left[C\right]_{m}=\left[ \begin{array}{cccccc}\frac{1}{E_{1}}&-\frac{\mu_{21 }}{E_{2}}&-\frac{\mu_{31}}{E_{3}}&0&0&0\\ -\frac{\mu_{12}}{E_{1}}&\frac{1}{E_{2}}&-\frac{\mu_{32}}{E_{3}}&0&0&0\\ -\frac{\mu_{12}}{E_{1}}&-\frac{\mu_{23}}{E_{2}}&\frac{1}{E_{3}}&0&0&0\\ 0&0&0&\frac{1}{C_{12}}&0&0\\ 0&0&0&0&\frac{1}{C_{23}}&0\\ 0&0&0&0&\frac{1}{C_{31}}\end{array} \right] \tag{6}\]  
where \(E_{1}\), \(E_{2}\) and \(E_{3}\) are the elasticity modulus of fabric layer along principal axes 1, 2 and 3. \(G_{12}\) is the shear modulus in the plane formed by principal axes 1 and 2. \(G_{23}\) is the shear modulus in the plane formed by principal axes 2 and 3, and \(G_{31}\) is the shear modulus in the plane formed by principal axes 3 and 1. \(\mu_{12}\), \(\mu_{21}\), \(\mu_{31}\), \(\mu_{13}\), \(\mu_{23}\), and \(\mu_{32}\) represent the corresponding [PERSON]'s coefficients. If the fiber direction of a fabric layer is \(\theta\) in the coordinate system, the transformation matrix \(\left[T\right]\) can be expressed as  
\[\left[T\right]=\left[ \begin{array}{cccccc}\cos^{2}\theta&\sin^{2}\theta&0&- 2\sin\theta\cos\theta&0&0\\ \sin^{2}\theta&\cos^{2}\theta&0&2\sin\theta\cos\theta&0&0\\ 0&0&1&0&0&0\\ \sin\theta\cos\theta&-\sin\theta\cos\theta&0&\cos^{2}\theta-\sin^{2}\theta&0 &0\\ 0&0&0&0&\cos\theta&\sin\theta\\ 0&0&0&0&-\sin\theta&\cos\theta\end{array} \right] \tag{7}\]  
According to (6) and (7), the elastic matrix can be described in the coordinate system as  
\[\left[C\right]=\left[T\right]^{T}\left[C\right]_{m}\left[T\right] \tag{8}\]  
Figure 3: Coordinate system of the measurement structure. The external forces are parallel to the \(x\) axis, and the measurement structure is symmetrical about the \(x\)-\(y\) plane. Hence, \(\tau_{23}\) is equal to 0. Equation (8) is introduced to [PERSON]'s law, and the relationship between shear strain and shear stress in \(x\)-\(y\) plane can be derived as follows:  
\[\gamma_{\rm CF}=(\frac{\sin^{2}\theta}{G_{23}}+\frac{\cos^{2}\theta}{G_{31}}) \tau_{\rm CF} \tag{9}\]  
where \(\gamma_{\rm CF}\) and \(\tau_{\rm CF}\) represent the shear strain and shear stress of the fabric layer. Besides this, mechanics equations can be established in the coordinate system for each part of the measurement structure according to their stress state. For the silica core, the equation can be derived as  
\[\tau_{\rm s}(x,y_{\rm s})=-\frac{y_{s}}{2}\frac{d\sigma_{s}}{dx} \tag{10}\]  
In the coating, the equation is obtained as  
\[\tau_{c}(x,y)=-\frac{y_{s}^{2}}{2y}\frac{d\sigma_{s}}{dx}-(\frac{y}{2}-\frac{y _{s}^{2}}{2y})\frac{d\sigma_{c}}{dx} \tag{11}\]  
In the adhesive layer, the equation can be described as  
\[\tau_{a}(x,y)=-\frac{2\pi y_{s}^{2}}{D}\frac{d\sigma_{s}}{dx}-\frac{\pi(y_{c} ^{2}-y_{s}^{2})}{D}\frac{d\sigma_{c}}{dx}-(h_{a}+y_{c}+y-\frac{\pi y_{c}^{2}} {D})\frac{d\sigma_{a}}{dx} \tag{12}\]  
In the CFRP laminates, the equation can be derived as  
\[\tau_{\rm CF}(x,y)=\tau_{a}(x,y_{a})-(y-y_{a})\frac{d\sigma_{m}}{dx} \tag{13}\]  
The CFRP laminates are very thin, and there is no shear stress on the lower surface of the laminates. Hence, the boundary condition can be applied as follows:  
\[\tau(x,y_{n})=0 \tag{14}\]  
where \(y_{n}\) is the thickness of the CFRP laminates.