## 2 Theoretical Model of the Strain Transfer Ratio  
The measurement structure of an FBG sensor is shown in Figure 1. The structure consists of an optical fiber, adhesive layer and CFRP laminates. The optical fiber is composed of fiber core, cladding, and coating. The fiber core and cladding are both made of silica and have the same mechanical properties [26]. Thus, they can be regarded as silica core. The silica core is protected by polyimide coating. The optical fiber and CFRP laminates are bonded together using an epoxy resin adhesive layer. The length of the adhesive layer is \(2L\), and the width of the adhesive layer is \(D\). \(h_{\text{a}}\) represents the thickness of adhesive above the optical fiber. CFRP laminates can be regarded as an anisotropic multilayered structure [31; 32]. Each layer of carbon-fiber fabrics is orthotropic and its elasticity modulus along the fiber direction is significantly higher than the elasticity modulus along other directions. The configuration of CFRP laminates is shown in Figure 2. The fabrics are placed with certain stacking sequences  
Figure 1: Measurement structure of a Fiber Bragg Grating (FBG) sensor bonded to the surface of carbon fiber reinforced polymer (CFRP) laminates. their fiber directions as \([\theta_{1}/\theta_{2}/\cdots/\theta_{n}]_{s}\), where the symbols in the square bracket indicate the stacking sequence and directions (angles in degree) of the carbon-fiber fabrics in CFRP laminates and the subscript s means symmetric arrangement. Each fabric layer has three principal axes represented by 1, 2 and 3. The principal axes 1 and 2 are both in the fabric layer, and the principal axis 1 is along the fiber direction. The principal axis 2 is perpendicular to the principal axis 1. The principal axis 3 is perpendicular to the fabric layer. There is no relative slip between two fabric layers for intact CFRP laminates. The measurement structure is abstracted into four parts, which are the silica core, coating, adhesive layer, and CFRP laminates. Two hypotheses are applied to this study as treated in some published references [24,33]:  
**Hypothesis 1**. _Silica core, coating, and adhesive layer are linear elastic isotropic materials._  
**Hypothesis 2**. _The bonds between all the interfaces are perfect and there is no relative slip._  
The Hypothesis 1 describes the mechanical properties of the optical fiber and adhesive layer, and the Hypothesis 2 indicates the integrity of the measurement structure. If the hypotheses do not hold, it manifests that the adhesive layer is not fully cured or there are cracks in the measurement structure. Theoretically, the mathematical model of the measurement structure can be discontinuous in this case, and there will be no specific functional relation between the measured strain and the original strain, and this uncertainty should be avoided in the measurement. A coordinate system is set up for the measurement structure as shown in Figure 3. The \(x\) axis is set along the central axis of the optical fiber and the \(y\) axis is along the vertical symmetry axis of the measurement structure. Only the first quadrant is shown in the figure because this structure is symmetric. \(L\) represents the half length of the adhesive layer; \(y_{s}\) and \(y_{c}\) are the radiuses of the silica core and coating; \(y_{a}\) is the distance between the \(x\) axis and the lower surface of the adhesive layer; \(k\) is a positive integer, and \(y_{k}\) is the distance between the \(x\) axis and the lower surface of the \(k\)-th fabric layer. The CFRP laminates are subjected to an original normal stress \(\sigma_{m}\). Correspondingly, \(\sigma_{s}\), \(\sigma_{c}\), and \(\sigma_{a}\) represent the normal stress of the silica core, the coating, and the adhesive layer, respectively; \(\tau_{s}(x,y_{s})\) is the shear stress at the interface between the silica core and the coating; \(\tau_{c}(x,y)\), \(\tau_{a}(x,y)\), and \(\tau_{\text{CF}}(x,y)\) are the shear stress in the coating, the adhesive layer, and the CFRP laminates, respectively.