Strain transfer efficiency model and finite element simulation experiment for end-bonded substrate-type FBG sensors

Abstract

FBG sensors are widely used in damage monitoring of wind turbine blades. Due to the adhesive layer, strain transfer loss occurs when blade strain is transmitted to the fiber Bragg grating. This paper proposes a strain transfer efficiency model for the end-bonded method of substrate-type FBG sensors, and conducts numerical analysis on the elastic modulus, Poisson’s ratio, and thickness of the bonding adhesive layer, as well as the gauge length of the FBG sensors. Finally, finite element simulation is used to verify the accuracy of the derivation. The results of numerical analysis and finite element simulation show that the proposed model can describe the law of strain transfer in the end-bonded structure of substrate-type FBG sensors and reveal the influence mechanism of each key parameter on the transfer efficiency.

Introduction

It is reported that the total installed capacity of wind energy will reach 414 GW by 2030¹. As the core components for converting wind energy into electrical energy, wind turbine blades directly affect power generation efficiency and operational stability. Therefore, effective condition monitoring of wind turbine blades is essential for ensuring the safe and reliable operation of wind turbines.

Current monitoring methods for wind turbine blades mainly include noise analysis^2–4, vibration analysis^5,6, acoustic emission techniques^7,8, and image recognition methods^9–11. Due to their advantages of miniaturization, light weight, and immunity to electromagnetic interference^12,13, fiber Bragg grating (FBG) sensors have been increasingly applied in structural damage monitoring of wind turbine blades.

In recent years, several studies have explored the application of FBG sensors in this field. Wang et al.¹⁴ attached multiple FBG sensors to wind turbine blades to identify impact locations. Yan et al.¹⁵ employed FBG sensors for crack detection, while Wu et al.¹⁶ proposed a Z-shaped FBG sensor capable of simultaneous strain and temperature measurement. However, when FBG sensors are surface-bonded to blades, the presence of an intermediate adhesive layer affects measurement accuracy. Liu et al.¹⁷ demonstrated that the length and thickness of the adhesive layer significantly influence strain transfer efficiency. Zhu et al.¹⁸ experimentally investigated strain response under various loading conditions and found that both the size and material of the adhesive layer affect strain transfer, although no analytical model was established.

In addition, studies on strain transfer mechanisms of bonded optical fiber sensors have been conducted. Chen et al.¹⁹ derived strain transfer models for substrate-free and small-diameter sensors and analyzed the effects of adhesive properties and structural parameters. Falcetelli et al.²⁰ investigated the influence of bonding length and loading conditions on strain transfer efficiency in surface-bonded optical fiber sensors. Nevertheless, existing models are not directly applicable to substrate-type packaged FBG sensors. Beyond linear elastic models, the viscoelastic behavior of adhesives has also been considered in strain transfer analysis^21–23. Furthermore, environmental factors such as hygrothermal aging have been shown to affect adhesive joint performance and FBG strain transfer accuracy^24,25. In the context of composite structures, FBG sensors have been increasingly applied for damage detection in composite wind turbine blades^26,27.

Nevertheless, most existing strain transfer models—including the multilayer model proposed by Wang et al.²⁸. for surface-bonded FBG sensors—assume a continuous bonding configuration, i.e., the adhesive layer is present along the entire length of the sensor. In practical calibration and measurement scenarios, however, substrate-type FBG sensors are often end-bonded: the sensor is attached to the host structure only at its two ends, leaving the central grating region free from direct adhesive contact. This end-bonded configuration fundamentally changes the shear stress distribution and boundary conditions compared to continuous bonding. As a result, existing continuous-bonding models are not directly applicable to end-bonded substrate-type FBG sensors; applying them would lead to inaccurate predictions of strain transfer efficiency. This end-bonded mounting method is particularly relevant for calibration of substrate-type FBG sensors, where the sensor is typically mounted onto a reference substrate using only two end adhesive joints to enable repeatable and traceable strain transfer.

To address this limitation, this paper proposes a strain transfer efficiency model for end-bonded substrate-type FBG sensors, which better reflects their structural characteristics. First, the strain transfer model is derived analytically. Then, numerical analysis is performed to evaluate the effects of adhesive layer material properties and thickness, as well as sensor gauge length, on strain transfer efficiency. Finite element simulations are subsequently conducted to validate the proposed model. Finally, based on the combined results of numerical analysis and simulation, recommended parameter ranges for the adhesive layer and sensor gauge length are provided.

To clearly highlight the distinctions between the proposed end-bonded model and the existing continuous-bonding models in the literature, Table 1 summarizes their key differences in terms of bonding configuration, shear stress distribution, strain transfer path, boundary conditions, and applicability.

Table 1.

Comparison of key assumptions, bonding configurations, and applicability between existing continuous-bonding models and the proposed end-bonded model.

Feature	Existing continuous-bonding models	Proposed end-bonded model
Bonding configuration	Adhesive layer present continuously along the entire sensor length	Adhesive joints only at the two ends of the substrate; central grating region free of adhesive
Shear stress distribution	Distributed continuously along the bonding interface	Concentrated at the two end-bonded joints; central region transfers strain via substrate tension
Strain transfer path	Host structure → adhesive layer → FBG sensor (directly or via thin substrate)	Matrix → end adhesive joints → substrate → fixed adhesive layer → FBG sensor
Uniform strain region in FBG	May be affected by end effects over a larger proportion of gauge length	A clearly defined central uniform strain region exists due to end-concentrated shear transfer
Boundary conditions for strain	ε(x = ± L) = 0 at both free ends of the adhesive layer	Same form but with distinct shear-lag parameters; additionally, substrate strain is non-uniform
Primary applicability	Surface-bonded FBG sensors with full adhesive coverage	Substrate-type FBG sensors in calibration or end-mounted scenarios
Key assumptions	Linear elasticity, perfect bonding, small strains, uniform axial stress across layer thickness	Same core assumptions, but with two-stage shear-lag analysis (matrix→substrate→FBG)

Strain transfer efficiency model for end-bonded substrate-type FBG sensors

Figure 1a is a schematic diagram of the model’s axial cross-section, and Fig. 1b is a schematic diagram of the model’s radial cross-section. One of the matrices serves as a fixed platform, while the other matrix can move freely along the x-axis to stretch or compress the substrate-type FBG sensor on the matrices. The FBG sensor is only in the axial strain field, and the generated theoretical strain value can be calculated by Eq. (1):

Fig. 1.

Schematic diagram of the end-bonded cross-section of a substrate-type FBG sensor.

1

where L₀ denotes the initial position of the matrix for stretching or compression, L_c denotes the position of the matrix after movement, and 2L_g denotes the gauge length of the substrate-type FBG sensor.

Owing to its special structure, this method concentrates all axial stress on the central axis of the substrate; meanwhile, the annular structure on the substrate facilitates the compression or stretching of the sensor when subjected to force. For the convenience of analysis, the subscripts g, n, c, j, and h represent the FBG sensor, the fixed adhesive layer between the FBG sensor and the substrate, the substrate, the bonding adhesive layer between the substrate and the matrix, and the matrix. The cross-section of the groove for placing the optical fiber is semicircular, with its radius denoted as r_n; the radius of the FBG is r_g; The distance of the fixed adhesive layer between the bottom of the substrate-type FBG sensor and the bottom of the groove is d; the width of the substrate is b (Fig. 2).

Fig. 2.

Strain transfer diagram of each layer of the substrate-type FBG sensor.

For the convenience of analysis, the following assumptions are made^17,19,20: (1) All bonding interfaces are continuous without relative slip; (2) All adhesive materials are linearly elastic and isotropic; (3) Since the thickness of each layer is relatively small, the axial stress distribution across the layer cross-section can be considered uniform; (4) The strain distribution of the base body remains unchanged after bonding; (5) The strain variation gradient of each layer is consistent.

The above assumptions follow common practices in strain transfer analysis. Assumption 1 holds under small strains and well-bonded conditions; excessive loads or aging may cause local debonding. Assumption 2 is adequate for epoxy and metal within elastic limits but fails under viscoelastic or large-strain conditions. Assumption 3 is justified by the small thickness-to-length ratio (< 0.01 in this study), though end effects exist within about one thickness distance. Assumption 4 requires that the sensor/adhesive stiffness be negligible (< 5% of host), which is satisfied here; for flexible substrates, stiffening effects should be considered. Assumption 5 follows from deformation compatibility; large modulus mismatch (e.g., FBG vs. adhesive) may enhance shear lag. Overall, the model is applicable to small-strain, linear-elastic conditions.

When analyzing the bonded interface between the FBG sensor and the matrix, the differential elements of the substrate and the bonding adhesive layer are first taken as the research objects. Based on the principle of force equilibrium, the force balance equation for the bonding adhesive layer is obtained as follows:

2

Where σ_j is the axial stress of the bonding adhesive layer, b is the width of the sensor, h_j is the thickness of the bonding adhesive layer, τ_cj is the shear stress between the substrate and the bonding adhesive layer, and τ_jm is the shear stress between the bonding adhesive layer and the matrix.

The force equilibrium equation of the substrate micro-element segment is as follows:

3

where σ_c is the axial stress of the substrate, h_c is the thickness of the substrate, and r_n is the radius of the fixed adhesive layer.

By combining Eqs. (2) and (3), we obtain:

4

Since the bonding adhesive layer and the substrate deform synchronously, they have the same strain gradient, that is:

5

where ε_c is the strain of the substrate, and ε_j is the strain of the bonding adhesive layer.

By substituting Eq. (5) and the relational expression between axial stress and axial strain into Eq. (4), the following can be obtained:

6

where E_c is the elastic modulus of the substrate, and E_j is the elastic modulus of the bonding adhesive layer.

Since the substrate is made of stainless steel, its elastic modulus is much larger than that of the bonding adhesive layer, thus, the following holds:

7

Thus, Eq. (6) can be simplified to:

8

Since the length of the substrate is much larger than its thickness and width, only its axial deformation is considered. From the relationship between shear stress and shear strain, the following can be obtained:

9

where G_j is the shear modulus of the bonding adhesive layer, γ_jm is the shear strain of the bonding adhesive layer, and du is the axial deformation of the adhesive.

By substituting Eq. (9) into Eq. (8) and integrating both sides, the following can be obtained:

10

Simplifying Eq. (10) yields:

11

where u_h is the axial displacement of the matrix, and u_c is the axial displacement of the substrate.

Taking the derivative of both sides of Eq. (10) gives:

12

where ε_h is the strain of the matrix, j² is a coefficient determined by the following formula:

13

The general solution of this differential equation is:

14

where C₁ and C₂ are integral constants determined by the boundary conditions.

Since the two axial end faces of the substrate are in a free state and not constrained by external forces, the boundary conditions are as follows: . Based on these boundary conditions, the following can be obtained:

15

where L₁ is the distance from the bonding adhesive layer to the center of the optical fiber.

Therefore, the strain expression of the substrate and its derivative form are as follows:

16

17

where L is half of the gauge length.

Using Eq. (11), the displacement difference between the matrix and the substrate at position L can be obtained as follows:

18

Thus, the strain in the middle part of the substrate is:

19

20

Then, the FBG sensor and the fixed adhesive layer are taken as the research object for force analysis, and the force equilibrium equation of the fixing adhesive layer is as follows:

21

The force equilibrium equation of the FBG sensor is as follows:

22

By combining Eqs. (21) and (22), the following can be obtained:

23

Since the FBG sensor and the fixed adhesive layer deform synchronously, they have the same strain gradient, thus, the following holds:

24

By substituting Eq. (24) and the relational expression between axial stress and axial strain into Eq. (23), the following is obtained:

25

where E_g and E_n are the elastic moduli of the FBG sensor and the fixed adhesive layer material, respectively. The two usually differ by more than a factor of tens, thus, the following holds:

26

Thus, Eq. (25) can be simplified to:

27

Since the aspect ratio of the FBG sensor is very large, only the axial strain is considered, thus, the following holds:

28

where G_n is the shear modulus of the fixed adhesive layer, γ_nc is the shear strain of the adhesive. y_gn is the coordinate of the contact point between the lowest point of the FBG sensor in the groove and the fixed adhesive layer, and y_nc is the coordinate of the bonding point between the lowest point of the fixed adhesive layer and the matrix.

By substituting Eq. (28) into Eq. (27) and integrating both sides, the following can be obtained:

29

Simplifying Eq. (29) gives:

30

Let (where ). Physical meaning of K: The parameter K characterizes the shear lag effect in strain transfer from the substrate to the FBG sensor. It depends on the shear stiffness of the fixed adhesive layer (G_n), the geometry of the adhesive contact (r_n and d), and the axial stiffness of the FBG sensor (). A larger K indicates stronger shear interaction and more efficient strain transfer over a shorter bonding length, while a smaller K implies weaker coupling and greater strain loss.

Taking the derivative of Eq. (30) gives:

31

The general solution of this differential equation is as follows:

32

where C₃ and C₄ are constants determined by the boundary conditions; since both ends of the FBG sensor are free interfaces with zero strain , the following can be obtained:

33

Thus, the strain expression of the FBG sensor and its derivative form can be written as:

34

35

Using Eq. (30), the displacement difference between the substrate and the FBG sensor at position L_g can be obtained as follows:

36

where Δl_cg is the displacement difference between the substrate and the FBG sensor at position L_g.

Then, the strain of the FBG sensor can be expressed as:

37

Substituting Eq. (20) into Eq. (37) gives:

38

It can be seen from Eq. (38) that the strain of the FBG sensor is related to the material properties and dimensions of the sensor. Once various parameters are determined, the true strain on the FBG can be calculated. Based on the above analysis, substituting Eq. (1) into Eq. (38) gives the strain on the FBG sensor as:

39

where β is the strain transfer efficiency, expressed as:

40

Numerical analysis of the influence of various parameters on the strain transfer rate

In practical applications, once the FBG sensor is selected, its radius, elastic modulus, and Poisson’s ratio remain constant^23,29. Therefore, combined with Eq. (40), the following analysis focuses on the influences of the gauge length, thickness, elastic modulus, and Poisson’s ratio of the bonding adhesive layer on the strain transfer efficiency. The initial values of each layer are presented in Table 2.

Table 2.

List of initial values of parameters.

Material parameter	Symbol	Value
Radius of FBG sensor/mm	r g	0.0625
Elastic modulus of FBG sensor/GPa	E g	72
Poisson’s ratio of FBG sensor	γ g	0.3
Thickness of bonding adhesive layer/mm	h j	0.18
Elastic modulus of bonding adhesive layer/GPa	E j	3.6
Poisson’s ratio of bonding adhesive layer	γ j	0.35
Gauge length of FBG sensor/mm	2Lg	17

Substituting the initial parameters in Table 1 into Eq. (40), the strain transfer efficiency is 93.74%.

Figure 3 shows the variation of strain transfer efficiency with the elastic modulus of the bonding adhesive layer under Poisson’s ratios of 0.30, 0.35, and 0.40. As the elastic modulus increases, the strain transfer efficiency rises rapidly at first and then gradually approaches a stable value. In the low elastic modulus range, a small increase in modulus significantly enhances strain transfer. However, when the elastic modulus exceeds approximately 2 GPa, the growth rate of efficiency slows down, indicating that the influence of elastic modulus weakens beyond this point. By comparing the curves corresponding to different Poisson’s ratios, it is observed that their overall trends are highly consistent, with only minor numerical differences. This suggests that, within the scope of this study, the effect of Poisson’s ratio on strain transfer efficiency is relatively insignificant, whereas the elastic modulus of the bonding layer is the dominant influencing factor.

Fig. 3.

Effect of the bonding adhesive layer’s elastic modulus on strain transfer efficiency under different Poisson’s ratios of the adhesive layer.

Figure 4 clearly shows the effect of adhesive layer thickness on strain transfer efficiency. As the thickness increases from 0.10 mm to 0.34 mm, the strain transfer efficiency decreases approximately linearly. This indicates that adhesive layer thickness has a significant impact on the strain transfer performance of the FBG sensor, and excessive thickness hinders effective strain transfer. In practical applications, to ensure accurate strain monitoring of wind turbine blades using FBG sensors, the adhesive layer thickness should be properly controlled. While satisfying structural bonding strength requirements, strain transfer efficiency must also be considered.

Fig. 4.

Effect of adhesive layer thickness on strain transfer efficiency.

Figure 5 illustrates the effect of FBG sensor gauge length on strain transfer efficiency. As the gauge length increases from 16 mm to 26 mm, the strain transfer efficiency increases from 93.35% to 95.91%. At smaller gauge lengths, the efficiency increases more rapidly, whereas the rate of increase gradually decreases as the gauge length becomes larger. This indicates that increasing the gauge length improves strain transfer efficiency. In practical applications, the strain transfer performance of substrate-type FBG sensors can be optimized by appropriately increasing the gauge length.

Fig. 5.

Effect of FBG sensor gauge length on strain transfer efficiency.

Based on the numerical results presented in Figs. 3, 4 and 5, the relative importance of the four parameters on strain transfer efficiency can be ranked as follows. Within the typical ranges examined, the elastic modulus of the bonding adhesive layer exhibits the most pronounced effect: increasing it from 0.5 GPa to 2 GPa raises the efficiency from approximately 85% to 93%, after which the gain diminishes. The adhesive layer thickness shows a nearly linear inverse relationship: a 0.24 mm increase (from 0.10 mm to 0.34 mm) reduces efficiency by about 4%. The FBG sensor gauge length has a moderate positive effect: increasing from 16 mm to 26 mm improves efficiency by approximately 2.6%. Poisson’s ratio of the adhesive layer, within the range of 0.30 to 0.40, has a negligible influence (< 0.2% variation). Therefore, for design optimization of end-bonded substrate-type FBG sensors, priority should be given to selecting an adhesive with sufficiently high elastic modulus (≥ 2 GPa) and controlling the adhesive layer thickness within 0.15–0.25 mm. Gauge length can be adjusted as a secondary factor, while Poisson’s ratio can be treated as of minor concern within the studied range.

Finite element simulation of the strain transfer model for surface-bonded FBG sensors in the non-grating region

To verify Eq. (40) and the numerical results, the model shown in Fig. 1 was established and analyzed using finite element simulations in Ansys Workbench. The resulting model is presented in Fig. 6.

Fig. 6.

Finite element simulation model diagram.

The finite element simulation result of the strain distribution contour is shown in Fig. 7. After assigning the initial parameters to each component of the model, the mesh was generated using the automatic method with an element size of 0.4 mm. All components were meshed using 20‑node hexahedral elements (SOLID186). A mesh convergence study was performed by refining the element size from 0.6 mm to 0.2 mm in steps while monitoring the maximum strain in the FBG sensor. The results showed that the strain variation between the 0.4 mm mesh and the finer 0.2 mm mesh was less than 2%, confirming that the 0.4 mm mesh provides sufficient accuracy for the strain transfer analysis. To simulate perfect bonding without relative slip, bonded contacts (Bonded type in Ansys Workbench) were defined between each pair of adjacent components. One end of the model was fixed, and a prescribed axial displacement was applied to the other end.

Fig. 7.

Finite element model simulation of strain distribution contour plot.

The strain contour plot shows non-uniform strain distribution at the interface between the FBG sensor and the fixed adhesive layer. For clearer observation, the strain contour of the FBG sensor alone is presented in Fig. 8.

Fig. 8.

Strain distribution along the axial direction of the FBG sensor.

It can be observed from Fig. 8 that the strain along the FBG sensor after stretching is not uniformly distributed. At both ends, the strain exhibits sharp variations due to bonding boundary effects and other factors, which do not accurately reflect the strain of the main structure. In contrast, the central region shows relatively stable and uniform strain, effectively minimizing measurement deviations caused by complex stress disturbances at the ends. Therefore, using the average strain of the middle region as the representative value of the FBG sensor provides stable and reliable measurements that better reflect the actual strain state of the structure. A longer gauge length increases the absolute length of the uniform strain region (the effective sensing zone), thereby reducing the relative contribution of the edge transition zones and improving the accuracy of strain averaging. Conversely, if the gauge length is too short, the entire FBG may lie within the boundary‑affected regions, leading to significant underestimation of the true substrate strain.

As shown in Fig. 3, when the elastic modulus exceeds approximately 2 GPa, the rate of efficiency increase slows; therefore, the finite element simulation considers an elastic modulus range from 2.2 to 4.3 GPa. Figure 9 presents the simulation results of strain transfer efficiency as a function of the adhesive layer’s elastic modulus under Poisson’s ratios of 0.30, 0.35 and 0.40. The results indicate that strain transfer efficiency increases with the elastic modulus regardless of Poisson’s ratio. Comparing the curves, efficiency is slightly higher at a Poisson’s ratio of 0.30 and slightly lower at 0.40, suggesting that a smaller Poisson’s ratio marginally improves strain transfer. When the elastic modulus exceeds 2 GPa, the strain transfer efficiency stabilizes at a relatively high level, consistent with the numerical analysis results.

Fig. 9.

Effect of the elastic modulus on strain transfer efficiency under varying Poisson’s ratios of the bonding adhesive layer.

Figure 10 shows the effect of bonding adhesive layer thickness on the strain transfer efficiency of FBG sensors, along with the relative error between simulation and theoretical results. The analysis indicates that when the adhesive layer thickness is approximately 0.15 to 0.25 mm, the relative error remains within ± 2%. This range represents the optimal thickness for the bonding layer, providing both reasonable strain transfer efficiency and high agreement between finite element simulations and theoretical predictions. As the adhesive layer thickens, the assumptions of uniform axial stress across the layer cross-section and negligible shear deformation within the adhesive become less accurate. The theoretical model, which simplifies the adhesive as an ideal shear-lag medium, underestimates the actual strain gradient and shear lag effect that become more pronounced in thicker bonds. In contrast, the finite element simulation captures the full three-dimensional stress distribution, including edge effects and non-uniform shear strain through the adhesive thickness. Consequently, the divergence between the two methods grows when the thickness exceeds approximately 0.30 mm. These results offer guidance for the optimal design of bonding adhesive layer thickness in FBG sensor applications for wind turbine blade damage monitoring, enhancing the accuracy and reliability of strain measurements.

Fig. 10.

Comparison of simulation and numerical analysis results for the effect of bonding adhesive layer thickness on strain transfer efficiency.

Figure 11 illustrates the effect of FBG sensor gauge length on strain transfer efficiency, along with the relative error between finite element simulation and numerical analysis results. Compared with the influence of the bonding adhesive layer, the effect of gauge length on strain transfer efficiency is smaller, with the relative error remaining within ± 0.5% across the analyzed range. When the gauge length is approximately 16 to 20 mm, the relative error is around ± 0.2%. This range represents the optimal selection for FBG sensors, ensuring reasonable strain transfer efficiency and high consistency between simulation and theoretical results. The recommended gauge length range of 16–20 mm is well aligned with commercially available FBG products, where 8 mm, 12 mm, 18 mm, and 22 mm are standard lengths^30–33. Custom lengths are generally available upon request. Thus, the recommended range is both theoretically validated and practically feasible.

Fig. 11.

Comparison of simulation and numerical analysis results for the effect of FBG sensor gauge length on strain transfer efficiency.

Basis for optimal parameter ranges: The recommended adhesive thickness of 0.15–0.25 mm is selected because (i) within this range, the relative error between theoretical and FEM results is ≤ ± 2% (Fig. 10), which is acceptable for typical blade strain measurements; (ii) thicknesses below 0.15 mm are difficult to achieve reliably in practice due to surface roughness and clamping tolerances, while thicknesses above 0.25 mm lead to rapidly increasing error ( > ± 5% at 0.34 mm). The recommended gauge length of 16–20 mm is selected because (i) within this range, the relative error is ≤ ± 0.2% (Fig. 11), indicating excellent model accuracy; (ii) shorter lengths (< 16 mm) suffer from edge effect contamination, while longer lengths (> 20 mm) offer diminishing accuracy gains and may exceed practical packaging constraints. These ranges thus balance accuracy, manufacturability, and geometric compatibility.

These findings provide a reference for optimizing gauge length to achieve more accurate strain monitoring in FBG sensor applications for wind turbine blade damage detection. Based on the numerical analysis and finite element simulation results, the recommended ranges for the bonding adhesive layer’s material parameters and thickness, as well as the FBG sensor gauge length, are summarized in Table 3.

Table 3.

Table of the optimal parameter ranges.

Material parameters	Symbol	Reference range
Thickness of bonding adhesive layer/mm	h j	0.15 ~ 0.25
Elastic modulus of bonding adhesive layer/GPa	E j	2 ~ 4
Poisson’s ratio of bonding adhesive layer	γ j	0.3 ~ 0.4
Gauge length of FBG sensor/mm	2Lg	16 ~ 20

The parametric analysis above treats each factor individually. In practice, the influences of adhesive layer thickness, elastic modulus, and FBG gauge length on strain transfer efficiency are not completely independent but may interact with each other. As can be inferred from the trends in Figs. 3, 4 and 5 and a thicker adhesive layer reduces transfer efficiency, while a higher elastic modulus or a longer gauge length improves it. It is reasonable to hypothesize that when two parameters vary simultaneously, their combined effect might be approximately additive, given that the strain transfer efficiency expression in Eq. (40) has a weakly coupled structure. However, a rigorous demonstration of this superposition would require additional systematic simulations or experiments, which are beyond the scope of the present work. For instance, increasing the adhesive layer thickness from 0.15 mm to 0.25 mm causes a certain loss in efficiency, but this loss can be partially compensated by selecting an adhesive with a higher elastic modulus (e.g., from 2 GPa to 4 GPa) or by moderately increasing the gauge length (e.g., from 16 mm to 20 mm). Conversely, if the adhesive layer is too thin (below 0.10 mm), even a high elastic modulus may lead to excessive shear stress concentration at the end bonds, which is not captured by the isolated parameter study. Therefore, the optimal parameter ranges recommended in Table 3 serve as a practical starting point, and future work should investigate potential parameter interactions in more detail.

Conclusion

This paper presents an in-depth investigation of the strain transfer behavior of substrate-type FBG sensors for wind turbine blade damage monitoring, yielding the following key findings. A strain transfer efficiency model for end-bonded substrate-type FBG sensors was established, and numerical analysis was conducted on critical parameters including the elastic modulus, Poisson’s ratio, and thickness of the bonding adhesive layer, as well as the FBG sensor gauge length. Finite element simulations were performed to verify the results, revealing the influence of each parameter on strain transfer efficiency. Specifically, increasing the elastic modulus of the adhesive layer improves strain transfer efficiency, Poisson’s ratio has a minor effect, and increasing the adhesive layer thickness exacerbates strain transfer loss. For the FBG sensor gauge length, an optimal range exists that minimizes the relative error between simulation and numerical analysis results.

Practical contribution: The study provides a theoretical basis and practical design guidelines for optimizing end-bonded substrate-type FBG sensors. The recommended parameter ranges (Table 3) can assist engineers in selecting appropriate adhesive materials, controlling adhesive layer thickness, and choosing suitable sensor gauge length to reduce strain transfer losses and improve measurement accuracy in structural health monitoring of wind turbine blades. To implement these recommendations in practice: control adhesive thickness to 0.15–0.25 mm using shims or precision dispensing; select a commercially available FBG gauge length of 16–20 mm; ensure clean, abraded bonding surfaces and proper alignment; and follow recommended curing protocols. Under these conditions, the end-bonded configuration yields a predictable strain transfer efficiency with ± 3–5% uncertainty, suitable for sensor calibration and damage monitoring. In the broader context of bonded composite systems and structural health monitoring, the understanding of strain transfer mechanisms is essential for reliable damage detection and durability assessment^34,35.

Limitations: The proposed model is developed under idealized assumptions: linear elasticity, perfect bonding, small static strains, and constant temperature. It does not account for several complexities inherent to real wind turbine blade applications, including:

1. The anisotropic nature of composite blade materials;

2. Environmental factors such as humidity, thermal cycling, and adhesive degradation over time;

3. Cyclic, fatigue, or dynamic loading conditions (the current model only considers static loading);

4. Temperature effects on material properties and strain transfer;

5. Practical bonding imperfections, including curing-induced residual stresses, partial debonding, voids, and misalignment between the sensor and the substrate.

Literature on bonded composite interfaces has shown that environmental aging and fatigue loading significantly degrade joint performance and alter interfacial stress distributions^36–38. Consequently, the predicted strain transfer efficiency should be interpreted as a reference for design optimization under controlled or quasi-static laboratory conditions, rather than an absolute value for all field scenarios.

Expected influence of environmental aging, fatigue, and viscoelasticity: For wind turbine blade applications, these factors can notably affect strain transfer accuracy. Hygrothermal aging reduces adhesive stiffness and strength, potentially lowering strain transfer efficiency by an estimated 5–15% under severe conditions. Viscoelastic creep under sustained static load may cause a drift in FBG strain readings of approximately 4–7%, depending on bonding length. Fatigue/cyclic loading leads to progressive stiffness degradation and reduced transfer reliability compared to static conditions, with the damage initiation life typically occupying a substantial portion of the total fatigue life. These estimates are illustrative ranges derived from adhesive joint and bonded FBG sensor studies; a precise quantification for the end-bonded configuration requires dedicated future work.

Consequently, the predicted strain transfer efficiency should be interpreted as a reference for design optimization under controlled or quasi-static laboratory conditions, rather than an absolute value for all field scenarios.

To support practical applicability, we provide the following quantitative error estimates when some of the idealized assumptions are partially violated :

Perfect bonding: For a 10% end-debonding, strain transfer efficiency drops by 8–12%; under well-bonded conditions, error ≤ ± 2%.

Linear elastic adhesive: Within typical blade strains (< 2000 µε), error < 1% relative to viscoelastic model; above 5000 µε or > 50 °C, error may reach 5–10%.

Uniform axial stress across thickness: For adhesive thickness h_j = 0.15–0.25 mm, theoretical vs. FEM error ≤ ± 2% (Fig. 10); for h_j > 0.30 mm, error ≤ ± 5–8%.

Negligible sensor stiffening: For steel substrates, error < 0.5%; for composites, error 2–5%.

Under recommended conditions, the overall uncertainty in predicted strain transfer efficiency from these idealized assumptions is estimated as ± 3–5%, supporting the model‘s utility for design optimization and calibration of end-bonded substrate-type FBG sensors.

Future research: Future work should extend the model to account for dynamic and fatigue loading, incorporate the effects of temperature and humidity, and investigate the influence of anisotropic substrates. Experimental validation under realistic service conditions, including long-term aging and cyclic loading, is also needed. Furthermore, the impact of bonding quality—such as curing protocols, residual stresses, and alignment errors—should be systematically studied to bridge the gap between idealized models and practical sensor installations.

Author contributions

X.L. and J.C. conceived the theoretical framework, performed the simulations and experiments, and drafted the original manuscript. B.Z. and J.C. managed the research project and supervised the overall research progress. K.C. provided theoretical guidance and critical suggestions for the research design. L.P.and B.T. revised and polished the manuscript, and provided valuable comments for improvement. B.C. assisted in the simulation experiments and data processing.

Funding

National Key Research and Development Program of China (NO. 2023YFF0611203), Science and Technology Project of Tibet Autonomous Region of China (NO. XZ202401YD0002).

Data availability

The derived formula for strain transfer efficiency and initial values of the relevant material parameters are listed in Table 1, with their value ranges directly obtainable from the corresponding curves in the manuscript. This research relies solely on pre-existing theoretical models and parameter datasets, and no supplementary raw experimental data are needed.

Declarations

Competing interests

The authors declare no competing interests.