Numerical Investigation of Stress-Strain State Effects on Strain Measurements with Fiber Bragg Grating Sensors

This study investigates the behaviour of resonant wavelengths of Fiber Bragg Gratings (FBG) inscribed within optically isotropic fibers under transverse loading, both in free and embedded conditions. A numerical-analytical approach is employed, utilizing the finite element method to calculate strain tensor components in the optical fiber core, followed by an analytical determination of resonant wavelengths and reflected FBG spectrum shape. The research demonstrates the influence of the ratio of host material and optical fiber elastic moduli on the birefringence level in FBG area under transversal loading. Based on analytical model of FBG spectrum simulation the discrepancy between analytically calculated and experimentally recorded resonant wavelength shifts in FBG embedded within isotropic material under varying transverse load levels is demonstrated.


Introduction
Currently, fiber-optic sensors (FOSs) based on fiber Bragg gratings (FBG) are in high demand for monitoring the mechanical state of structures [1][2][3].These sensors, due to their compact size, the ability to be embedded into the structure, resistance to electromagnetic interference and high sensitivity, have found widespread applications in such areas as civil engineering [4,5], aerospace [6,7] and geotechnical [8] industries.
When utilizing FBG sensors, especially those embedded within the structure of the material, several challenges arise.These include the redistribution of the stress-strain state in the vicinity of the embedded optical fiber [9,10] and the issue of strain transfer from the material to the optical fiber [11,12].
One of the most critical challenges in using Bragg grating based FOSs is ensuring the reliability of strain value evaluations calculated from the recorded physical quantities.The established correlations between the physical parameters identified by the interrogation system and the strain tensor components in the Bragg grating area can only be uniquely solved for the longitudinal strain along the optical fiber, provided that the optical fiber is under a uniaxial stress condition.However, optical fibers embedded in materials typically experience complex stress state [13], thus calibration is necessary for the measured strain values.Despite this, controlling the calibration of transverse strain remains challenging, as discussed in detail in [14].Work [15] highlights that embedded FBGs under large radial strain levels exhibit significant deviations in measured strain.It is suggested that after embedding in the material, calibration of the sensors must be performed in cases when a strain tensor component along the fiber axis is not predominant.
The reflected FBG spectrum is influenced by both longitudinal and transverse strain, as well as the distribution of strain along the sensor length [16,17].It is known that significant transverse strain can alter the spectrum's shape, causing it to split.This characteristic not only presents a potential source of measurement error but can also be utilized for simultaneous measurement of longitudinal and transverse strain [18,19].
In this paper, the behavior of resonant wavelengths of FBG inscribed in an initially optically isotropic optical fiber is studied, where birefringence is induced by a load acting perpendicular to the longitudinal axis of the optical fiber.A numerical-analytical approach is employed, wherein the strain tensor components in the optical fiber core are calculated by solving the elasticity theory problem using the finite element method.Subsequently, the resonant wavelengths and the shape of the reflected FBG spectrum are determined analytically.The study focuses on cases involving an optical fiber subjected to a transverse load and the case of loading a material volume with an embedded optical fiber.

Methodology and Analysis
A fiber Bragg grating represents a periodic variation of the refractive index within a specific region of the optical fiber core.Such grating filters the light that is transmitted through the fiber waveguide in such a way that major part of the optical signal's spectrum passes through the grating and only a narrow range of wavelengths are reflected back to the interrogation unit (figure 1).The resonant (Bragg) wavelength λ B of the reflected signal can be expressed as the product of the effective refractive index n of the fiber-optic core in the grating region and the geometric period of index modulation Λ: In general, the Bragg wavelength is a function of mechanical stresses and temperature, as both terms in equation (1) depend on these parameters.Thus, the shift or incremental change of the resonant wavelength relative to the initial state can be expressed as follows: In this equation, the first term corresponds to a change in the resonant wavelength with an increase in mechanical stresses Δσ and a constant temperature, while the second term corresponds to a change in temperature ΔT under a constant level of mechanical stresses.
In this paper, the influence of temperature on the shift of the resonant wavelength is not considered, so the second term in equation ( 2) and the index T in the following relations can be omitted.By substituting equation (1) into equation (2), and taking into account that both the effective refractive index n and the Bragg grating period Λ depend on mechanical stresses, and transforming the equation from stress to strain, taking into account that the relative change in the grating period is the axial strain component 3  , the following relation can be derived: In accordance with the theory of photoelasticity, the application of mechanical strain leads to a change in the refractive index of the material: The component notation of the equation, which represents the change in the refractive index of the material due to mechanical strain is as follows: where ij p are the components of the photoelasticity tensor.For an isotropic material, the photoelasticity tensor contains two independent components 11 p and 12 p .By substituting equation (4) into equation (3) the dependence of the shift of the resonant wavelength on the components of the strain tensor can be expressed by the following relations: where  are the principal strain in the plane perpendicular to the optical fiber.
Thus, the difference between two resonant wavelengths of the reflected spectrum 21   is independent of the longitudinal component of the strain tensor and depends solely on the principal strain components in the plane perpendicular to the direction of the optical fiber.Consequently, an increase in transversal strains in the optical fiber core in the region of the FBG will lead to an increase in the difference between the resonant wavelengths, resulting in a split of the reflected spectrum.
The most common is application of the relations for calculating longitudinal strain based on wavelength shift for the case of a uniaxial stress state, where For the studied silica glass optical fibers 0.78  k .The applicability of equation (7) for determining strain for complex stress state cases is questionable as it may cause measurement errors.
To investigate the pattern of changes in resonant wavelengths 12 , in the presence of significant transverse strain in the core of the optical fiber, the problems of loading applied perpendicular to the fiber's longitudinal axis and the problem of loading a volume of material with an embedded optical fiber with a transverse load are considered.To address this problem, a numerical-analytical approach is employed, in which the strain tensor components are calculated by solving linear elasticity theory problem using the finite element method, implemented in Ansys.The shift of resonant wavelengths is determined from relations (5).

Transversal load applied to optical fiber
The finite element method was used to solve the problem of optical fiber compression in the direction perpendicular to its longitudinal axis.Cases of plane stress and plane strain states, as well as cases of optical fiber in the presence and absence of a protective polyimide coating with a thickness of 12.


. Loading is carried out by transmitting a force from the horizontal plate to the optical fiber using contact interaction.On the bottom plate, the displacement boundary condition is set to zero.The loading scheme is shown in figure 2.   calculated by finite element analysis for the cases of an optical fiber with (figure 3a) and without (figure 3b) a polyimide coating.Based on the strain tensor components obtained in the center of the optical fiber

 
on the applied load for the plane stress and plane strain states is shown in figure 4a, the cases of an optical fiber with and without a protective coating are shown in figure 4b.From the obtained results, it can be concluded that the nature of changes in the resonant wavelengths for plane stress and plane strain cases is significantly different, primarily because, for the plane strain case 3 0   .This result qualitatively corresponds to the results presented in work [21], where an analytical solution to the stress-strain state problem is provided.The effect of a protective polyimide coating with a low modulus of elasticity is found to be insignificant.

Transversal load applied to embedded optical fiber
The case of an optical fiber embedded in a volume of material holds practical significance, as an optical fiber with fiber-optic sensors can be effectively integrated into the structure of various isotropic and anisotropic materials during the manufacturing stage.This creates opportunities for evaluation of the mechanical state of structures both during operation and at the manufacturing stage.However, significant transverse strains may occur at the location of FOSs due to the specifics of the manufacturing process and the material formation.A high level of residual strains can result in a significant change in the initial spectrum of the FBG sensor, causing it to split and exhibit two resonant wavelengths.This, in turn, can lead to errors in the measurement of longitudinal strain.
For numerical calculation of the stress-strain state using the finite element method the linear elastic material behavior was assumed with perfect contact of all adjacent parts of the model.Due to symmetry, only a quarter of the model with the corresponding boundary conditions was considered.A mesh refinement in the vicinity of the optical fiber was used (figure 5).
In numerical simulations, the elastic modulus for the host material in which the optical fiber is embedded varied in the range E h = 10÷110 GPa with a constant Poisson's ratio . The dimensions of the parallelepiped are chosen in such a way to exclude the influence of boundaries on the results.The cross-section of the model is square shaped with a side equal to 40 optical fiber diameters.As boundary conditions on the side surface, the displacement 0 UU  is set in accordance with the diagram in figure 5.
To compare the considered cases, the birefringence value B [21] was used: nn  represents the refractive index change with polarization in the direction of axis 1 and 2 after applying a load, 0 n is the effective refractive index in the FBG region in the initial, unloaded state.Figure 6 illustrates the dependence of the birefringence value for an optical fiber embedded in an isotropic material under the considered boundary conditions on the ratio of the elastic modulus of the host material to the elastic modulus of the optical fiber.From the obtained results, it follows that embedding an optical fiber into a stiffer material contributes to the splitting of the spectrum at a lower strain level.
The two resonant wavelengths obtained from relation (5) do not always imply their experimental determination, since in a real experiment the resonant wavelength is estimated by the main peak in the reflected spectrum.However, the reflected FBG spectrum has a certain width at half maximum (FWHM) and amplitude, depending on parameters such as the FBG length and the amplitude of refractive index modulation in the FBG region.Experimentally, the splitting of the spectrum does not occur immediately but follows the initial broadening of the spectrum.To demonstrate this, an analytical approach to modeling the reflected spectrum of a uniform FBG based on the coupled-mode theory [21,22] was employed.The reflected spectrum of the FBG with birefringence induced by loading can be described as the sum of the reflected spectra for polarization axes 1 and 2:   are the resonant wavelengths of the FBG under load.
In order to analyze the change in the total spectrum of FBG embedded into material under transverse external load, the results of numerical modeling, relations (5) for calculating the wavelength shift, and equations ( 9)- (11) for calculating the spectrum of the reflected signal were used according to the following algorithm: 1.The finite element analysis is used to calculate strain components ε 1 , ε 2 , ε 3 in the optical fiber core under specified boundary conditions.2. Δλ 1 and Δλ 2 are calculated using relations (5).Having Δλ 1 and Δλ 2 , new resonant wavelengths λ 1 and λ 2 can be obtained for the grating under load.It is assumed that the strain distribution along the entire length of the FBG is uniform.3.For new values of resonant wavelengths, the reflected signal spectra are calculated based on equations ( 9)- (11).
Figure 7 shows the spectrum evolution of the FBG embedded in an isotropic material as the load acting perpendicular to the longitudinal axis of the optical fiber increases.In the graph, the spectrum maxima are marked with red dots, and the values of λ 1 and λ 2 calculated by equations ( 5) are indicated with black marks.It should be noted that for this problem, a force boundary condition distributed over the side surface was used.The mechanical properties of the material in which the optical fiber is embedded are E = 30 GPa, 0.2  .Based on the obtained results, it can be concluded that the spectrum of the embedded FBG under transverse external load undergoes a complex change, involving a simultaneous shift of the spectrum due to the presence of longitudinal strain, a gradual broadening of the spectrum, and subsequent splitting with the emergence of two separate peaks.In this case, at the initial stage of loading, the values of resonant wavelengths λ 1 and λ 2 calculated analytically from equation ( 5) do not coincide with the maximum amplitude values of the spectrum.

Conclusions
The problems of transverse loading of a free and embedded optical fiber in an isotropic material are considered.Using a numerical-analytical approach, results on changes in the resonant wavelengths of the FBG are obtained.It is shown that an increase in the elastic modulus of the material in which the optical fiber is embedded contributes to the spectrum splitting at a lower strain level.By employing the analytical model of the reflected FBG spectrum, the pattern of changes in the spectrum of embedded FBG in an isotropic material with an increase in the transverse load level and the possible discrepancy between the analytically calculated and experimentally recorded values of the resonant wavelength shift are demonstrated.

Figure 1 .
Figure 1.The scheme of FBG operation.

3 
, and  represents the Poisson's ratio of the optical fiber.In this case 12         , and instead of two equations with three unknowns, a single equation with one unknown is defined:

Figure 2 .
Figure 2. Loading scheme of optical fiber compression in the plane perpendicular to its longitudinal axis.

Figure 3
Figure 3 illustrates the distribution of the strain component 2  calculated by finite element analysis

Figure 4 .
Figure 4. Dependence of the resonant wavelengths shift 12 ,  from the applied load for the plane stress and plane strain states (a), for the cases of an optical fiber with and without a protective coating (b).

Figure 5 .
Figure 5. Finite element model of the host material with embedded optical fiber and applied displacement boundary conditions.

Figure 6 .
Figure 6.Dependence of the birefringence value for an optical fiber embedded in an isotropic material on the ratio of the elastic modulus of the host material to the elastic modulus of the optical fiber.

Figure 7 .
Figure 7. Evolution of the FBG spectrum embedded in an isotropic material with an increase in the load acting perpendicular to the longitudinal axis of the optical fiber.