Measurement error analysis of surface-bonded distributed fiber-optic strain sensor subjected to linear gradient strain: Theory and experimental validation

Strain transfer analysis is an important means of correcting the measurement error of embedded or surface-bonded distributed fiber-optic sensors, but the effect of host strain patterns has not been well elucidated. Here, a theoretical model for strain transfer analysis of surface-bonded multi-layered fiber-optic sensor subjected to a linear gradient strain was established. Closed-form solutions were obtained for both singleand bi-linear strain distributions, and, in particular, a simple method was described for determining the strain transfer coefficient at the turning point of a bi-linear type strain. The presented model was validated through laboratory testing with high-spatial resolution strain profiles acquired by optical frequency-domain reflectometry. Furthermore, parametric analyses were performed to investigate the influences of mechanical and geometrical properties of protective and adhesive layers on the strain transfer performance, shading light on the design, installation, and measurement error correction of fiber-optic sensors after accounting for the effect of host strain distribution.


Introduction
improved the model in ref. [16] based on the assumption that the strain gradient at the 23 midpoint of each layer of FO sensor was approximately equal; the derived result was 24 closer to the actual situation [17]. On this basis, strain transfer mechanisms in FBG 25 sensors under nonaxial uniform strains were studied [22]. By introducing Goodman's 26 hypothesis, Wang et al. further considered the influence of host viscoelasticity and 27 ambient temperature on the strain transfer coefficient, which enriches the research on 28 the strain transfer mechanism of embedded FO sensors [23]. 29 Different from that of embedded FO sensors, analyzing the strain transfer for 30 surface-bonded FO sensors should take into extra consideration the impacts of 31 geometric and physical properties of the adhesive layer [24]. Wan et al. introduced an 32 axisymmetric model of surface-bonded FBG sensor to investigate the influence of 33 adhesive layer width and bottom thickness on the strain transfer coefficient, and the 34 reliability of the model was validated through experiments and finite element analysis 35 [25]. Considering the possible gap between FO cable and adhesive layer, Her et al. 36 proposed an elaborate analytical model for strain transfer analysis of surface-bonded 37 FO sensors [26,27]. Xin et al. derived a strain transfer model in the polar coordinate 38 system and discussed the strain transfer phenomenon observed in crack detection [28]. 39 Billon et al. developed a strain transfer function for concrete crack monitoring and the 40 function was validated by the high-performance distributed FO sensing technology-41 optical frequency-domain reflectometry (OFDR) [29]. By also employing OFDR, 42 Zhang et al. systematically investigated the effects of mechanical parameters and 43 bonding method of FO cable on the strain transfer efficiency from both theoretical and experimental sights [30]. More recently, Falcetelli et al. developed a strain transfer 45 model of multi-layered FO cable and obtained the distribution of strain transfer 46 coefficient for a nonzero boundary condition; the theoretical analyses were more consistent with actual observations [31]. 48 From the above literature review, it can be found that current strain transfer   surface-bonded FO sensor with an n-layered structure subjected to a nonuniform strain 70 in the host material was established (Fig. 1). The proposed model is based on the 71 following assumptions:

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(1) Both the core and cladding of the sensor are silica, which can be regarded 73 collectively as a single layer named fiber core.

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(2) The fiber core, adhesive layer, and protective layers are all linear elastic 75 materials; bonding conditions among different layers are good with no relative slippage.

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(3) Only the shear stress transfer process among various layers within the bonded 77 sensor length is considered.

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The analytical model is established in the polar coordinate system where x 79 represents the position along the axis of the sensor, r the radial position, and  the 80 angle between the boundary point of the adhesive layer and the horizontal direction (see 81 Fig. 1(a)). Referring to Fig. 1(b), the mechanical equilibrium of a fiber core element 82 can be expressed as: where rc is the outer radius of the fiber core layer, c  denotes the normal stress on the between the fiber core and the first protective layer.

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Eq. (1) can be readily reduced to the following: According to assumption (3), the force equilibrium of the first protective layer 90 leads to: where ( , ) x r  represents the shear stress at the interface between the first and second 93 protective layers. By combining Eqs. (2) and (3), ( , ) x r  one gets: Because the fiber core and each protective layer are assumed to behave linearly 96 elastically during the strain transfer process (assumption (2)), the shear strain ( , ) x r  97 at the interface between the first and second protective layers, according to the Hooke's 98 law, can be expressed as: where 1 G represents the shear modulus of the first protective layer, Ec is the Young's 101 modulus of the fiber core, and c  denotes the normal strain of the fiber core. Since the 102 radial displacement is far less than the axial displacement u, Eq. (5) can be alternatively 103 written as: Then, the axial displacement on the boundary of the first protective layer can be 106 obtained, by integrating Eq. (6) from rc to r1, as follows:  where u1 and uc represent the axial displacement at the outer boundary of the fiber core 110 and the first protective layer, respectively.

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The same derivation is made for the other protective layers and the adhesive layer, 112 and the following equation can be obtained: 113 2 a n 2 1 a n n-1 2 1 1 where h u represents the axial displacement on the interface between the adhesive 115 layer and the host; n r and n G represent the radius and shear modulus of the nth 116 protective layer, respectively; a G is the shear modulus of the adhesive layer; and a r 117 is the equivalent radius of the adhesive layer, which can be calculated according to the 118 geometric characteristics of the model ( Fig. 1(a)):

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  π n a n n 2 cos where t represents the thickness of the adhesive layer from the sensor bottom to the host 121 surface (see Fig. 1(a)). Here, a shear lag coefficient k is introduced, and then Eq. (9) 122 can be simplified as: where the coefficient k has the following form: 125 2 a n 2 1 a n n n-1 2 1 1 π 2 Since the first derivative of axial displacement with respect to x is the axial strain,

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Eq. (11) can be converted to: represents the strain distribution in the host material. The solution of Eq.

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(13) is obtained by solving the second order linear nonhomogeneous differential 131 equation with constant coefficients: 133 where 1 C and 2 C represent the integration constants that can be determined 134 according to appropriate boundary conditions.

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Finally, the strain transfer coefficient of the surface-bonded FO sensor can be 136 defined as the ratio of the fiber core strain to the host strain, which is given by: When a cantilever beam with a uniform cross section is subjected to a point load at the 142 free end, the strain distribution of the beam will be a single linear gradient. Consider 143 such a strain distribution as shown in Fig. 2, the corresponding strain transfer coefficient, 144 with the boundary conditions ( ) 0 z L   , can be derived as: (denoted as 2 low L ), the strain transfer performance in its middle portions will be good.

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Therefore, in practical applications, the bonded sensor length should be longer than 2 159 low L to avoid poor data quality. By contrast, when the shear lag coefficient k is small 160 (corresponding to a poor strain transfer performance), the strain transfer profile will be 161 directly affected by the strain distribution in the host material. Notably, the curves will 162 incline to the side with a lower host strain, exacerbated by steeper gradients (see Fig.   163 3). These results collectively indicate that when the shear lag coefficient k is small at a   In a numerical example we assumed that the host strain distribution was as follows: Besides, we assumed that the strain transfer coefficient at the turning point was 0.95 to 184 look at the strain transfer coefficient distribution (Fig. 5). It can be seen that the strain 185 transfer profiles for single linear and bilinear gradient strains were of the same pattern.

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However, in actual applications, strain transfer coefficients at turning points remain 187 unknown. Here, a simple method was proposed to solve this problem, which can be 188 described as following. First, the host strain distribution along the entire sensor length 189 is assumed to have a gradient equivalent to the longer sensor section (e.g., the section 190 0-L2 in Fig. 4). Next, a hypothetical strain transfer distribution is obtained according to tube installed with a 0.9 mm diameter tight-buffered FO strain sensing cable (Fig. 6); 210 the test setup is shown in Fig. 7. Table 1 summarizes the materials and parameters of 211 the cable's components.

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The FO cable was surface-adhered along the axial direction of the tube with epoxy 213 resin. After the glue was cured, the inclinometer tube was symmetrically placed on two supports, and five dial gauges were installed at different positions above the pipe to  Table 1. These two strain profiles were compared 234 (Fig. 9) The test setup is shown in Fig. 10. Two FO cables AB (orange) and ab (red) were  Fig. 13. The parameters of the FO cable and 282 adhesive layer used in the theoretical analysis were the same as those listed in Table 1 283 (except for the jacket). It can be seen from Fig. 13 that the two coefficient curves 284 coincided with each other, hence validating the proposed theoretical model.                [17,18,[32][33][34][35]).

Materials Parameter Symbol Value Unit
Fiber core Silica