# A Strain-Transfer Model of Surface-Bonded Sapphire-Derived Fiber Bragg Grating Sensors
## 2 Theoretical Approach  
The four-layer sensing model for the surface-bonded SDFBG test system had a core, cladding, an adhesive, and a host material, and the grating was located in the center of the core, as shown in Figure 1. The homocentric cylindrical optic-fiber cladding and core were mounted on the host material, with the adhesive filled in between the two. When the core's radius increased to that of the cladding, the cladding layer disappeared, and the four-layer model regressed to a three-layer model that only had a core, an adhesive, and a host material, as shown in Figure 2. Like traditional models [8; 11; 20; 22], the three-layer model treated the core and cladding as essentially the same homogeneous material. The stress state in the four-layer model is presented in Figure 3. Some assumptions were made before the analytical model could be built:  
1. The SDF core, cladding, adhesive layer, and monitored substrate were to be made of linear, elastic, homogeneous materials. 2. The adhesive layer was to be closely bonded to the SDF and substrate materials, with no relative slip. Shear stress and displacement at the interface between the two different media were to be at the same amplitude. 3. Both the adhesive layer and the cladding layer were to only be subject to shear strain. This is because the Young's modulus of the adhesive layer and the cladding layer would be much smaller than that of the heavily doped alumina core and the substrate material. In this model, subscripts \(co\), \(cl\), \(a\), and \(m\) represent core, cladding, adhesive, and host material, respectively. \(\tau\), \(\sigma\), \(\varepsilon\), and \(\gamma\) are shear stress, normal stress, shear strain, and normal strain, respectively; \(G\) and \(E\) represent shear modulus and Young's modulus; and \(r\), \(L\), \(h\), \(\theta\), and \(w\) are radius, half-bond length of the adhesive, height, angle, and width, respectively. The Young's modulus of the SDF core was determined by its alumina content, and it could be expressed as percentage \(m\) of its volume in the Al\({}_{2}\)O\({}_{3}\)-SiO\({}_{2}\) binary system. Thus, by using the addition model [30], the Young's modulus of the core could be calculated as  
\[E_{co}=mE_{Al_{2}O_{3}}+(1-m)E_{SiO_{2}}, \tag{1}\]  
where \(E_{Al_{2}O_{3}}\) and \(E_{SiO_{2}}\) are the [PERSON]'s moduli of alumina and silica, respectively. As shown in Figure 3, difference in the displacement along the optic-fiber axis (x- co-ordinate axis) between the core-cladding and the cladding-adhesive interfaces could be described as  
\[u(r_{cl},\theta,x)-u(r_{co},\theta,x)=\int_{r_{co}}^{r_{cl}}\gamma_{cl}(r, \theta,x)\mathrm{d}r=\frac{1}{G_{cl}}\int_{r_{co}}^{r_{cl}}\tau_{cl}(r,\theta,x)\mathrm{d}r \tag{2}\]  
Figure 1: Four-layer model of surface-bonded sapphire-derived fiber Bragg grating (SDFBG) test system. Figure 3: Stress state in four-layer model. Figure 2: Three-layer model regressed from four-layer model. According to Assumption 2, shear stresses at the interface between core and cladding are equal in amplitude, namely, \(\tau_{cl}(r_{co},\theta,x)=\tau_{co}(r_{co},\theta,x)\). Shear stresses in the cladding were assumed to be inverse to the radius, and could be given by  
\[\tau_{cl}(r,\theta,x)=\tau_{cl}(r_{co},\theta,x)\frac{r_{co}}{r}=\tau_{co}(r_{ co},\theta,x)\frac{r_{co}}{r}. \tag{3}\]  
According to [PERSON] and [PERSON] (2014), displacement continuity at the interface between cladding and core could be derived from Equation (2) using the stress relationship in Equation (3), and given as  
\[u_{cl}(r_{cl},\theta,x)=u_{a}(r_{cl},\theta,x)=\frac{r_{cl}}{G_{cl}}.\ln\left( \frac{r_{cl}}{r_{co}}\right)\tau(r_{cl},\theta,x)+u_{co}(r_{co},\theta,x), \tag{4}\]  
where \(u_{co}(r_{cl},\theta,x)\) can be replaced with \(u_{co}(r_{cl},x)\) because the displacement of the core was regarded as independent from the angle.