# A Strain-Transfer Model of Surface-Bonded Sapphire-Derived Fiber Bragg Grating Sensors
## 1 Introduction  
Fiber-optic strain, twist, and torsion sensors have played an important role in structural-integrity monitoring because their compact size, light weight, and immunity to electromagnetic interference represent significant advantages [1; 2; 3; 4; 5; 6; 7]. Between these, strain sensors are generally embedded into or bonded to the surface of the host material. The strain on the host material transfers to fiber sensors through adhesive and polymeric coating, and is then measured by the sensitive structure in the fiber core. Unfortunately, strain loss inevitably occurs in the transfer process because of shear deformation of the intermedium between fiber and host material. Thus, strain-transfer theory was developed to describe the actual strain-transfer process [8; 9; 10; 11; 12; 13; 14; 15; 16; 17; 18; 19; 20; 21; 22; 23]. In the case of embedded strain sensors, an analytical model based on force equilibrium and compatibility was developed to present the real strain on the host material on the basis of a number of realistic assumptions, such as materials behaving in a linear elastic manner, and the existence of a perfect bond [8]. The multilayer-interface strain-transfer mechanism is usually analyzed using simplified geometrical and physical functions [9]. The transfer model was improved by considering the influence of the sensor on strain distribution in the host material [10]. Strain-transfer analysis was conducted with the viscoelastic properties of the host material considered to describe the interfacial shear-stress relationship [16]. To investigate the interfacial-failuremechanism, the strain-transfer model was also extended to diagnose local interfacial debonding [18]. In the case of surface-bonded sensors, various stacking parameters of carbon-fiber-reinforced polymer laminates were analyzed by establishing a finite-element (FE) model [20]. The sensor was tested under fatigue load to derive the transfer formula in response to dynamic strain [23]. A strain-transfer model was also presented to study the influence of coating [24]. However, the above models were all based on the common single-mode fiber, and are probably not suitable for strain sensors containing sapphire-derived fibers (SDFs). The SDF is a specific fiber with an alumina dopant in the fiber core [25], and the sapphire-derived fiber Bragg grating (SDFBG) showed potential for strain measurement in temperatures as high as 950 \({}^{\circ}\)C [26; 27; 28; 29]. The Young's modulus of alumina (about 500 GPa) is much higher than that of silica (about 72 GPa); therefore, deformation resistance of a core with a high alumina content is much higher than that of pure silica cladding, leading to the shear deformation of the cladding and strain loss. Although many investigations were conducted on the strain-transferring mechanism between optic fiber and host material, they were all based on the assumption that the fiber possessed an approximately homogeneous elastic property, which is probably reasonable because both the core and cladding of the fiber are mainly made of silica. The pure silica approximation, however, could not be used in our case because of the alumina dopant in the core of the SDF. Therefore, the application of traditional strain-transfer models is quite limited in SDFBG-based strain measurement. In this paper, an improved four-layer strain-transfer model for surface-bonded SDFBG sensors was theoretically developed and experimentally demonstrated. The four layers were a core, cladding, an adhesive, and a host material, and the cladding was analyzed as an individual layer of the medium in order to reflect strain loss. When the core's radius increased to that of the cladding, however, the cladding layer disappeared, and the four-layer model regressed to a three-layer model. The FE model and the static-tensile-strain experiment were conducted using a universal testing machine to verify both the four- and three-layer models. Good agreement was achieved between theoretical predictions and experiment results.