PSI - Issue 32

Andrey Yu. Fedorov et al. / Procedia Structural Integrity 32 (2021) 194–201 A.Yu. Fedorov et al. / Structural Integrity Procedia 00 (2021) 000–000

196

3

Quarz cladding

Bragg grating

Broadband spectrum

Transmitted spectrum

L

fiber core

Reflected spectrum

Fig. 1. Principle of operation of the Bragg grating.

A change in the Bragg grating length leads to a change in the wavelength of the reflected signal ∆ λ . Under isother mal conditions, the relationship between the change of reflected wavelength and the fiber strain in the zone of the Bragg grating is defined by the following relations (Luyckx et al. (2010))

1 2 n

2 ( p

∆ λ 1 λ ∗ = ε 3 − ∆ λ 2 λ ∗ = ε 3 −

11 ε 1 + p 12 ( ε 2 + ε 3 )) ,

(1)

1 2 n

2 ( p

11 ε 2 + p 12 ( ε 1 + ε 3 )) ,

where ε 3 is the strain along the fiber; ε 1 , ε 2 are the principal strains in the plane perpendicular to the optical fiber; ∆ λ 1 = λ 1 − λ ∗ , ∆ λ 2 = λ 2 − λ ∗ are the di ff erences in the resonance wavelengths of the reflected spectrum at the current ( λ 1 , λ 2 ) and initial ( λ ∗ ) instants of time, and p 11 , p 12 are the Pockels coe ffi cients. In the uniaxial stress state of the optical fiber, which does not interact with the environment, the strain in the optical fiber is ε 1 = ε 2 = − νε 3 , where ν is Poisson’s ratio for the optical fiber. In this case, ∆ λ 1 = ∆ λ 2 = ∆ λ and ∆ λ λ ∗ =   1 − n 2 1 2 ( p 12 − ν ( p 11 + p 12 ))   ε 3 , (2) or ∆ λ/λ ∗ = k · ε 3 . For the optical fibers used k = 0 . 78. Thus, when the fiber is in a complex stress state there are two resonant peaks in the reflected spectrum (1). Then, the problem of unambiguous determination of the strain in the fiber core in terms of a change in the wavelength of the reflected spectrum is unsolvable. In the case when an optical fiber is in the uniaxial stress state it is possible to uniquely determine the strain along the fiber through the change in the wavelength of the reflected spectrum (2). Therefore, the development of fiber-optic sensors is associated, as a rule, with a search for the variant of implementation of the uniaxial stress state in the Bragg grating zone. One of the variants of such implementation is the installation of the fiber-optic sensors on a substrate. Numerical analysis of strain changes caused by the installation of the substrate-bonded fiber optic sensors on the surface of a structure was performed for a square plate with a rectangular substrate of a fiber optic sensor located on its surface (Fig. 2). The plate is subjected to distributed tensile forces P x and P y applied to its side edges. The dimensions of the plate are determined by the length of its sides a and thickness t . The dimensions of the substrate of the fiber optic sensor are determined by the length l x , width l y and thickness t 1 . In the study, the dimensions of the substrate were fixed. The dimension a was also fixed, but in such a way as to capture the entire zone of influence of the substrate placed on the surface of the plate. From the computational model (Fig. 2), it is evident that with the substrate dimensions fixed, the ratios of the mechanical characteristics of the structure and substrate materials, as well as the plate thickness t are the parameters, which will determine the strain changes near the substrate. In this case, 3. Mathematical model and its finite element implementation