## Abstract

In this paper, the property of panda polarization maintaining fiber Bragg gratings (PPM-FBGs) embedded in a composite laminate system (CLS) under a transversal force from 1 × 10^{5} Pa to 5 × 10^{5} Pa is explored. Both the wavelengths shift and the rotation angle of the principal axes of the PPM-FBGs are surveyed theoretically. We investigate the corresponding relation between the direction of external force and the rotation angle of principal axes of the PPM-FBGs and prove that the magnitude and direction of the strain distribution in the CLS can be monitored simultaneously, which can realize the detection of interlaminar damage of the CLS. We find that the strain, which corresponds to the shift of wavelengths of the PPM-FBGs, has a different angular dependence, and therefore the potential failure forms of the CLS can be differentiated.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

Optical fiber Bragg gratings (FBGs), which are compact, lightweight and of small sizes, have been frequently implemented in the field of structure health monitoring (SHM) and damage detection [1–3]. As one of the parts of SHM, monitoring the health of a composite laminate system (CLS), where an FBG is embedded inside the specimen, has been propelled to the forefront in recent years [4–9].

As shown in Fig. 1(a), a CLS is composed of substrates and adhesives [10] and the common failure forms of a CLS is delamination, e.g. interlaminar damage, which can reduce and damage the strength of a CLS seriously [11,12]. However, the process of delamination, which can be classified as delamination and intralaminar damage mechanisms, is quite complex and not easy to be observed. To predict the delamination precisely, many theories have been proposed, such as point stress criteria and fracture mechanics theory [11]. Based on these theories, the growth of the delamination has two processes: delamination initiation and delamination propagation, and three modes of failure forms exist in a CLS, for instance mode I, mode II and mode III [13]. The biggest difference between these three failure forms is the direction of the interlaminar stress. As for the failure forms of mode I and mode II, the direction of the interlaminar stress is normal and tangential to the interface, respectively. The failure forms of mode III can be regarded as the combination of mode I and mode II, and therefore it is significant to predict the failure forms of mode I and mode II. At present, an effective and simple method for predicting the failure forms of a CLS is to monitor the interlaminar stress by an FBG sensor directly. However, previous researches generally focused on the magnitude detection of the stress only [14–16] instead of evaluating the direction, which is critical to estimate the failure forms of a CLS [11,13]. Recently Pereira *et al* experimentally investigated a crack growth and damage event in a CLS by using an embedded FBG, and they demonstrated and validated that this technique is a design tool for SHM of a CLS [17–19]. However, this technology employed when an FBG is written in a traditional fiber relies on the change of the refractive index and the period of the FBG, and it is hard to measure the direction of the interlaminar stress sensitively. In order to improve the predictive capability effectively, in this paper, panda polarization maintaining fiber Bragg gratings (PPM-FBGs) embedded in a CLS are explored. The properties of PPM-FBGs embedded in the CLS, including different embedded directions of the PPM-FBGs and under different external forces of the CLS, are identified. In addition, the wavelengths shift of the PPM-FBGs and the rotated angle of principal axes, which correspond to the failure forms of the CLS, are extracted. And we proved that a PPM-FBG can be used to monitor the potential failure forms of a CLS, especially for the failure forms of mode I and mode II.

This paper is organized as follows. In section 2, the theory about measuring the magnitude and direction of the interlaminar stress of a CLS is explored scientifically. It can be seen in section 3 that the characteristics of a PPM-FBG embedded in a CLS are studied, and the failure forms of mode I and mode II are investigated according to the finite element method (FEM). Besides, we also testify that the PPM-FBGs can be utilized to monitor the failure forms of a CLS. The final part draws the conclusion, including suggestions for improving the accuracy of the PPM-FBGs to detect the interlaminar stress of a CLS.

## 2. Theory

As one of the traditional polarization maintaining fibers (PMFs), panda fibers, which are fabricated with stress applying parts (SAPs) [20], contain three parts: the cladding, the core and the SAPs. The main material of an SAP is composed of B_{2}O_{3}-doped silica [21]. Due to the differences of the thermal-expansion coefficients of these three parts, the internal thermal stress can be different during the cooling process in the manufacturing of a PMF [22]. Hence the birefringence is produced, and the birefringence *B* in the core region is defined as

*C*and

_{1}*C*are the elasto-optic coefficients of SiO

_{2}_{2},

*σ*and

_{0,max}*σ*stand for the stresses along the maximum and minimum principal strain directions, and

_{0,min}*n*and

_{0,F}*n*signify the refractive indices along the maximum and minimum principal strain directions. Based on the elasto-optic effect, the slow and fast axes of a Panda fiber are in the directions parallel and perpendicular to the principal strain. And the strain along the maximum principal strain and the minimum principal strain are

_{0,S}*ε*and

_{0},_{F}*ε*, respectively. Therefore, a PPM-FBG, which is formed in a Panda fiber, has two reflected peaks. The wavelengths of the two reflected peaks are

_{0,S}*λ*= 2

_{0,F}*n*and

_{F}Λ*λ*= 2

_{0,S}*n*, where

_{S}Λ*Λ*is the period of the PPM-FBG,

*n*and

_{F}*n*are the effective refractive indices of two orthogonal polarization modes corresponding to the fast-axis and slow-axis, respectively [20]. When external factors such as strain and temperature are exerted on the PPM-FBG, the two reflected peaks will shift [23]. In this paper, strain effect is investigated only and temperature effect can be rejected through superimposed gratings [24], a combination of an FBG and a fiber polarization-rocking filter [25] or high birefringence fiber gratings [26], but the temperature effect is not the focus of this paper.

_{S}As shown in Fig. 1(a), a PPM-FBG, with an angle *θ* between its slow axis and the *x*-axis, is embedded in a CLS. The refractive indices along the two principal strain directions are *n _{0,1}* and

*n*respectively, and the pre-existing strains along the

_{0,2}*x*,

*y*and

*z*axes are

*ε*,

_{0},_{x}*ε*and

_{0,y}*ε*, which can be expressed in the

_{0,z}*x*,

*y*reference frame using a standard stand 2 × 2 rotation matrix R

_{1}(

*θ*). Due to the elasto-optic effect, when an external force

*L*is applied to the CLS with an angle of

*α*to the

*x*-axis, the principal strains magnitude of the PPM-FBG will be changed to ${\epsilon}_{x}^{\text{'}}$, ${\epsilon}_{y}^{\text{'}}$ and ${\epsilon}_{z}^{\text{'}}$ [27–31], and the direction of the principal strains will be rotated by an angle of

*ϕ*relative to the

*x*axis simultaneously. Consequently, the strains of the PPM-FBG contain the residual strains (

*ε*and

_{0},_{x}*ε*) responsible for the initial birefringence and the strains (

_{0,y}*ε*and

_{L},_{x}*ε*) due to the loading

_{L},_{y}*L*. Based on the theory of solid mechanics, ${\epsilon}_{z}^{\text{'}}=-v\left({\epsilon}_{x}^{\text{'}}+{\epsilon}_{y}^{\text{'}}\right)$, where

*v*is the Poisson’s ratio of the fiber, the transverse strain can be considered only.

As a result, in the *x-y* reference frame (*x* and *y* in Fig. 1(b)), the total strains along the *x* axis and *y* axis will be changed to *ε _{x}* and

*ε*. And

_{y}*ε*and

_{x}*ε*can be expressed as [27]:

_{y}_{2}(

*ϕ*) is a stand 2 × 2 rotation matrix, and it is used to transform the strain caused by the force

*L*to the

*x-y*reference frame. At last, according to the Mohr’s circle theory [27,29,30], the principal strains components of the PPM-FBG can be determined as follows:

*γ*is the shear strain in the

_{xy}*x-y*plane. The direction of the new principal strains (

*x’*and

*y’*in Fig. 1(b)) will be rotated by an angle

*ϕ*relative to

*x*-axis or

*y*-axis, and

*ϕ*can be defined as:

*P*and

_{11}*P*are the strain-optic coefficients with the values of 0.121 and 0.270 respectively.

_{12}## 3. Results and analysis

Based on the analysis above, the FEM is employed in our simulations, and the PPM-FBG, which is embedded in a CLS with different *θ* and under different external forces *L*, are surveyed. The parameters used for the simulations are listed in Table 1 [32], where *n*, *r*, *E*, and *v* are the refractive index, radius, elastic modulus and Poisson’s ratio of the core, cladding, and SAP respectively, and *d _{SAP}* is the gap between the core and the center of SAP. To simplify the calculation process, the CLS is treated as one part with

*E*= 34.5GPa and

_{CLS}*v*= 0.48. Through finite element analysis, we can get

_{CLS}*n*= 1.44607,

_{0,1}*n*= 1.44572, and the birefringence

_{0,2}*B*is 3.49 × 10

^{−4}. In our simulation, the period of the PPM-FBG is 535.5 nm, and the two initial reflected peaks are

*λ*= 1548.74 nm,

_{0,F}*λ*= 1548.36 nm respectively.

_{0,S}First, a PPM-FBG is embedded in the CLS with *α* = 90° to detect the failure form of mode I of the CLS, and Fig. 2 shows the wavelengths shift and rotation angle *ϕ* of the PPM-FBG with different angles *θ* and magnitudes *L.* In Fig. 2(a), *θ =* 0° which means the fast axis aligns with the *x*-axis, and the red line and the green line represent the wavelengths shift of $\Delta {\lambda}_{x}^{\text{'}}$ and $\Delta {\lambda}_{y}^{\text{'}}$, respectively. It is obvious that with the increase of *L* from 1 × 10^{5} Pa to 5 × 10^{5} Pa, the wavelengths of $\Delta {\lambda}_{x}^{\text{'}}$ and $\Delta {\lambda}_{y}^{\text{'}}$ are blue-shifted linearly, and their difference, which is seen as another parameter to assess the quality of the CLS, increases simultaneously. The strain sensitivities of the PPM-FBG, corresponding to the slopes of the red line and the green line, are −3.91 nm/MPa and −9.27 nm/MPa, respectively. The blue curve in Fig. 2(a) shows the direction of the new principal strain, which means the polarization of the output beam from the PPM-FBG is changed from about −89.99981 to −89.99969° with the rise of external force *L.* The birefringence changes are negligible. However, the shift of wavelengths and polarization of output beam can be probed by using a polarization-dependent balanced detector. As a result, the magnitude and direction of the interlaminar stress can be calculated backwards respectively. Figure 2(b) shows the wavelengths shift and rotation angle *ϕ* of the PPM-FBG with *θ* = 45°, and we can find that the two center-reflected wavelengths of the PPM-FBG vary in good linearity with the applied force *L* as well, and the strain sensitivities are −1.42 nm/MPa and −7.32 nm/MPa, respectively. In Fig. 2(b), we can also find an inverse correlation between the direction of new principal strain and *L*, and the direction of new principal strain changes form −73.7696° to −87.0389°, which is more obvious than the case when *θ* = 0°. That can be interpreted in this way as for the first case, when an external force *L* is applied on the CLS, the directions of *ε _{L},_{x}* and

*ε*are opposite to

_{L},_{y}*ε*and

_{0},_{x}*ε*, and the new strain distribution results are obtained by subtracting the strain |

_{0,y}*ε*| and |

_{L},_{x}*ε*| from the initial strain

_{L},_{y}*ε*and

_{0},_{x}*ε*responsible for the initial birefringence. As a result, the birefringence obtained in comparison with the initial of it is smaller and the changing of

_{0,y}*ϕ*is not obvious. While when

*θ*= 45°, the directions of

*ε*and

_{L},_{x}*ε*are the same as

_{L},_{y}*ε*and

_{0},_{x}*ε*, and the new strain distribution is the total of |

_{0,y}*ε*|, |

_{L},_{x}*ε*|,

_{L},_{y}*ε*, and

_{0},_{x}*ε*. As a result, the changing of

_{0,y}*ϕ*is obvious. Figure 2(c) is the case when

*θ*= 90°, and the slope of $\Delta {\lambda}_{y}^{\text{'}}$ is −4.3 nm/MPa, which is similar to that of the first two cases, while $\Delta {\lambda}_{x}^{\text{'}}$ is not distinct. To clarify the detailed trend of $\Delta {\lambda}_{x}^{\text{'}}$, Fig. 2(c) is enlarged as shown in the inset picture in Fig. 2(c), and we can find that $\Delta {\lambda}_{x}^{\text{'}}$ is decreasing with a tiny slope of −1.0055 × 10

^{−2}nm/MPa. In Fig. 2(c), we can also find a positive correlation between

*ϕ*and

*L*. But the change is negligible, which means in this case the PPM-FBG is not sensitive to the change of

*L*. Figure 2(d) provides the condition when

*θ*= 135°. Compared with Fig. 2(b), the trend of $\Delta {\lambda}_{x}^{\text{'}}$and $\Delta {\lambda}_{y}^{\text{'}}$in Fig. 2(d) is consistent, while

*ϕ*is inversed.

From the analysis above, it can be found that a PPM-FBG embedded in the CLS with *θ* = 45° or *θ* = 135° is suitable for the monitor of the failure form of mode I of a CLS, and the decrease of $\Delta {\lambda}_{x}^{\text{'}}$ and $\Delta {\lambda}_{y}^{\text{'}}$ as a function of *L* can be modelled as a linear dependence given by the following expressions:

As for the rotation angle of the principal axes, they have a different *L* dependence and can therefore be modelled as an inhomogeneous dependence given by the following expressions:

With respect to the detection of failure form of mode II, we can set *α* = 0°, and therefore a shearing force will be applied on the surface of the CLS. This condition was also investigated as shown in Fig. 3. Figure 3 manifests the cases when *θ* = 0°, *θ* = 45°, *θ* = 90° and *θ* = 135° respectively, and the slopes of $\Delta {\lambda}_{x}^{\text{'}}$ and $\Delta {\lambda}_{y}^{\text{'}}$ are 4.28 nm/MPa and −4.28 nm/MPa, 9.95 nm/MPa and 2.96 nm/MPa, 4.28 nm/MPa and −4.28 nm/MPa, −2.95 nm/MPa and −9.98 nm/MPa, accordingly. As can be seen, the slope of $\Delta {\lambda}_{x}^{\text{'}}$ and $\Delta {\lambda}_{y}^{\text{'}}$ is the same when *θ* = 0° and *θ* = 90°, while the trend of *ϕ* is inversed (see Fig. 3). Comparing Fig. 3(b) with Fig. 3(d), we can also see that the change of *ϕ* is more obvious when *θ* = 0° and *θ* = 90°, and it means that *θ* = 0° or *θ* = 90° is more suitable for the monitor of failure form of mode II. The reason why the change of *ϕ* is not obvious when *θ* = 45° and *θ* = 135° is the same as the case of *θ* = 0° and *θ* = 90° when *α* = 90°. The trend of $\Delta {\lambda}_{x}^{\text{'}}$, $\Delta {\lambda}_{y}^{\text{'}}$ and the correlation between *ϕ* and *L,* when *θ* = 0° and *θ* = 90° can be modelled as the following functions, respectively:

To find the feature of the PPM-FBG embedded in a CLS obviously, the strain sensitivities of the PPM-FBG when *α* = 90° and *α* = 0° are listed in Table 2. It can be shown that the *θ* = 45° and *θ* = 135° are suitable for monitoring the failure form of mode I, while *θ* = 0° and *θ* = 90° are suitable for monitoring the failure form of mode II.

As mentioned before, the difference between $\Delta {\lambda}_{x}^{\text{'}}$ and $\Delta {\lambda}_{y}^{\text{'}}$ is also a crucial parameter to predict the health of the CLS, and Fig. 4 shows the difference between $\Delta {\lambda}_{x}^{\text{'}}$ and $\Delta {\lambda}_{y}^{\text{'}}$ when α = 90° and α = 0° respectively. In Fig. 4(a), the slopes are 4.21 nm/MPa, 3.78 nm/MPa, 5.92 nm/MPa, and 7.60 nm/MPa, respectively, denoting the ratio of the difference between $\Delta {\lambda}_{x}^{\text{'}}$ and $\Delta {\lambda}_{y}^{\text{'}}$ to the increase of external force *L*, which can also be used to describe the sensitivity of the PPM-FBG. The slopes in Fig. 4(b) are 8.45 nm/MPa, 9.03 nm/MPa, 8.46 nm/MPa and 4.87 nm/MPa, respectively.

At last, the result is presented when α changes from 0° to 350° with *L* = 5 × 10^{5} Pa. Figures 5(a), (b), (c) and (d) are corresponding to *θ* = 0°, *θ* = 45°, *θ* = 90° and *θ* = 135° respectively, and the solid line shows a sinusoidal fit. In Fig. 5, the trend of $\Delta {\lambda}_{x}^{\text{'}}$, $\Delta {\lambda}_{y}^{\text{'}}$and *ϕ* can be obtained, and therefore the interlaminar stress can be detected by embedding a PPM-FBGs in the CLS with different *θ* meanwhile the potential failure forms can be predicted.

Figure 6(a) plots the difference between $\Delta {\lambda}_{x}^{\text{'}}$ and $\Delta {\lambda}_{y}^{\text{'}}$ under different *α* when *θ* = 0°, *θ* = 45°, *θ* = 90° and *θ* = 135°. During the simulation, *L* has been fixed at 5 × 10^{5} Pa. Figure 6(b) shows the direction of the new principal strains *ϕ* with the changes of *α*, when *θ* = 0°, *θ* = 45°, *θ* = 90° and *θ* = 135° respectively. The trend of the difference between $\Delta {\lambda}_{x}^{\text{'}}$ and $\Delta {\lambda}_{y}^{\text{'}}$ and the new principal strain *ϕ* can also be used to improve the sensitivity of detecting the magnitude and direction of interlaminar stress. In Fig. 6(b), the changes of *ϕ* under different *θ* are not remarkable. To improve the sensitivity of detecting the failure forms of the CLS, we can select high birefringence fiber gratings, e.g. a bow-tie polarization maintaining fiber Bragg gratings.

## 4. Conclusion

The overall feasibility of PPM-FBGs embedded in a CLS is outlined, and a new method to monitor the potential failure forms of a CLS is proposed. The shift of wavelengths when the PPM-FBGs are embedded in a CLS under different angles and different external forces from 1 × 10^{5} Pa to 5 × 10^{5} Pa, is investigated. We investigate the corresponding relation between the direction of external force and the rotation angle of principal axes of the PPM-FBG, and have proved that the magnitude and direction of the strain distribution in the CLS can be monitored simultaneously. As a result, the potential failure forms of a CLS can be differentiated. This concept may be used in the field of condition monitoring-maintenance and model predictions of the residual life of a CLS. At last, to improve the sensitivity of the detection, we present that high birefringence fiber Bragg gratings can be selected to embed in a CLS.

## Funding

Natural Science Foundation of Anhui Province (1808085QF217, 1808085ME120).

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