Damage Assessment in Composites using Fiber Bragg Grating Sensors

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PRABHUGOUD MOHANRAJ. Damage Assessment in Composites using

Fiber Bragg Grating Sensors. (Under the direction of Assistant Professor Kara J. Peters).

This dissertation develops a methodology to assess damage in composites using

fiber Bragg grating (FBG) strain sensors. First, a strain-transfer model using the finite

element (FE) method is developed to simulate the response of an embedded FBG to the

applied loading. This FE model is also able to calculate birefringence in the FBG due to

applied transverse loads. The model is validated considering the two-dimensional problem

of diametrical compression of polarization-maintaining fibers. A modified T-matrix model

is then formulated to simulate the response of an embedded FBG due to an applied axial

strain field. The response of FBGs surface mounted on PMMA and two-dimensional woven

composites subjected to multiple low velocity impacts is experimentally investigated. The

complex spectral response is related to the residual strains after impact in the PMMA

specimens and the surface strain to failure in the two-dimensional woven composites. The

feasibility of using FBGs to measure internal strain in woven composites during damage

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by

Mohanraj Prabhugoud

A dissertation submitted to the Graduate Faculty of North Carolina State University

in partial satisfaction of the requirements for the Degree of

Doctor of Philosophy

Mechanical Engineering

Raleigh

2005

Approved By:

Dr. Eric C. Klang Dr. Tasnim Hassan

Dr. Kara J. Peters Dr. Mohammed A. Zikry

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Biography

MOHANRAJ PRABHUGOUD was born in Bangalore, India. He was brought up in

Ban-galore, where he attended Cambridge School for his secondary education. He majored in

Science at S. Nijalingappa College. He opted for a career in Engineering and attended M.

S. Ramaiah Institute of Technology, Bangalore, from September 1995 to September 1999.

He graduated with a Bachelor of Technology degee in Mechanical Engineering.

In January 2001 he moved to the United States for his graduate studies at North

Carolina State University, Raleigh, NC. He obtained a Master of Science in Mechanical

Engineering in 2002. In Spring 2003, he started working towards his Ph.D. in Mechanical

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Acknowledgements

I would like to thank Dr. Kara Peters for her support, encouragement and invaluable

guidance throughout the course of this work. She was one of the first people to formally

introduce me to the wonderful world of fiber optics and has since motivated me to pursue

a research career in this field.

I am extremely thankful for the financial support provided by National Science

Foundation, which enabled me to pursue this doctoral program. I would like to thank

Advanced Composites for donating the composite twill and Corning Inc., for donating the

gratings used for the experiments.

I would like to specially thank Dr. Mohammed Zikry for his valuable inputs for

the plexiglass experiments. I would like to thank James D Pearson for being available at all

times (with untiring enthusiasm) and helping me out with the experiments especially using

the drop tower.

I would like to thank my committee members, Dr. Eric Klang, and Dr. Tasnim

Hassan for serving on my committee. I would like to extend my thanks to Sharon Kiesel,

Mike Sitar, Luke Davis and Sarah Wilson for their help with the experiments. I would also

like to thank Rufus (Skip) L. Richardson and Mike Breedlove for their help in machining

components required for conducting the experiments.

Graduate study at NCSU has been a wonderful experience, during which I made

a number of dear friends. I would like to thank all my friends for their support and help.

I would like to thank my family members for their moral support and encouragement. My

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List of Figures

2.1 Classification of optical fibers. . . 4

2.2 FBG written into a PM fiber, subjected to multi-axis loading. . . 9

2.3 Typical optical fiber cross-sections: (a) circular core, step-index (no

birefrin-gence), (b) elliptical core, step-index (geometrical birefrinbirefrin-gence), (c) bow-tie

fiber with pre-stressed regions (geometrical and residual stress birefringence). 10

3.1 Schematic of the procedure for calculation of FBG spectral response for a

sensor embedded in a host material system. . . 16

3.2 Discretization of optical fiber into FBG sensor elements. Also shown are

definition of local polarization axesp andq, global polarization axes, X and

Y, and local principle strain axes, 1−2−3. z is the direction of propagation

along the optical fiber. . . 17

3.3 Definition of triangular element with nodal coordinates. . . 23

3.4 Global index ellipse showing variation ofneff for a given mode with rotation

of global axes X −Y. X0 −Y0 corresponds to global propagation axes of

optical fiber. . . 24

3.5 Cross-section of circular core, step-index fiber with applied transverse load.

Core size is exaggerated to show dimensions. . . 27

3.6 Variation of effective index of refraction with normalized frequency for

cir-cular cross-section, step-index optical fiber. Circles represent exact solution. Triangles represent results of coarse mesh simulations and squares represent

results of fine mesh (270 core elements, 452 cladding elements) simulations. 28

3.7 Variation of Bragg wavelength with applied diametrical load for FBG in

circular core, step-index optical fiber. Squares represent result of center

strain approximation. FE result is plotted as a solid line. . . 29

3.8 Geometry of PM fiber types considered: (a) elliptical core fiber; (b) D-fiber;

(c) elliptical core SAP fiber; (d) Bow-Tie fiber; (e) Panda fiber. The slow

and fast axes are also indicated. All dimensions shown inµm. . . 30

3.9 Variation of Bragg wavelength with applied load for FBG in elliptical core

fiber of figure 3.8 (a). Results from both the finite element and center strain

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3.10 Variation of Bragg wavelength with applied load for FBG in D-fiber of fig-ure 3.8 (b). Results from both the finite element and center strain solutions

are plotted. (a) γ = 0◦; (b)γ = 90◦. . . 34

SAP fiber of figure 3.8 (c) with outer diameter d = 125 µm. Results from

both the finite element and center strain solutions are plotted. (a) γ = 0◦;

(b)γ = 90◦. . . 35

SAP fiber of figure 3.8 (c) with outer diameter d = 80 µm. Results from

both the finite element and center strain solutions are plotted. (a) γ = 0◦;

(b)γ = 90◦; (c) γ= 36◦; and (d)γ = 72◦. . . 36

3.13 Variation of Bragg wavelength with applied load for FBG in bow-tie fiber of figure 3.8 (d). Results from both the finite element and center strain solutions

are plotted. (a) γ = 0◦; (b)γ = 90◦; (c)γ = 36◦; and (d)γ = 72◦. . . 37

3.14 Variation of Bragg wavelength with applied load for FBG in panda fiber of figure 3.8 (e). Results from both the finite element and center strain solutions

are plotted. (a) γ = 0◦; (b)γ = 90◦; (c)γ = 36◦; and (d)γ = 72◦. . . 39

3.15 Variation of the orientation of global index ellipse with applied load for FBG in panda fiber of figure 3.8 (e). Results from both the finite element and

center strain solutions are plotted. (a) γ = 0◦; (b)γ = 90◦; (c)γ = 36◦; and

(d)γ = 72◦. . . 40

3.16 Variation of Bragg wavelength with applied load for multiple orientations of

applied loading angle,γ, relative to geometrical axes of Panda fiber. Results

from both the finite element and center strain solutions are plotted. . . 41

3.17 Variation of Bragg wavelength with applied temperature (a) elliptical core

fiber (d = 125 µm); (b) bow-tie fiber; (c) panda fiber. Results for each axis

calculated using the center strain approximation (CSA) are also plotted. . . 43

4.1 Response of a uniform grating subjected to an applied (a) linear strain field.

(b) Reflected spectra due to the applied linear strain field along the grating length: dashed line is Runge-Kutta simulation of the coupled mode

equa-tions; solid line is T-matrix simulation using Λ(z); dotted line is T-matrix

simulation using ˜Λ(z). . . 52

4.2 Response of a uniform grating subjected to an applied (a) quadratic strain

field. (b) Reflected spectra due to the applied quadratic strain field along the grating length: dashed line is Runge-Kutta simulation of the coupled

mode equations; solid line is T-matrix simulation using Λ(z); dotted line is

T-matrix simulation using ˜Λ(z). . . 53

4.3 Response of a uniform grating subjected to an applied (a) linear strain field

with high strain gradient. (b) Reflected spectra due to the applied linear strain field along the grating length: dashed line is Runge-Kutta simulation

of the coupled mode equations; solid line is T-matrix simulation using Λ(z);

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4.4 Response of a uniform grating subjected to an applied (a) exponentially varying strain field with high strain gradient. (b) Reflected spectra due to the applied exponentially varying strain field along the grating length: dashed line is Runge-Kutta simulation of the coupled mode equations; solid line is

T-matrix simulation using Λ(z); dotted line is T-matrix simulation using ˜Λ(z). 55

4.5 Response of a chirped grating: (a) ²0(z) = 1000×10−6 and ²(z) = (1×

10−10/mm)z; (b) ²0(z) = 1000×10−6 and ²(z) = (1×10−16/mm2)z2; and

(c) ²0(z) = (1×10−10/mm)z and ²(z) = (1×10−16/mm2)z2. Runge-Kutta

simulations are plotted as dashed lines and T-matrix simulations using ˜Λ(z)

are plotted as dotted lines. . . 57

4.6 Reflection spectra for a grating surface mounted on an aluminum diaphragm.

(a) Experimentally measured spectra (from Chang and Vohra [84]), (b) Spec-tra simulated including sSpec-train gradient in the effective period formulation, and (c) Spectra simulated without strain gradient in the effective period

for-mulation. . . 60

4.7 Cross-section of PM fiber with applied transverse load. Core size is

exagger-ated to show dimensions. . . 63

4.8 Azimuthal variation of global effective refractive index forP = 10 N/mm and

γ = 54◦ (squares). Ellipse oriented at θ with major and minor axes nmax_eff

and nmin

eff is plotted as a solid line. . . 65

4.9 (a) Variation of Bragg wavelength with applied diametrical load atγ = 36◦

for PM fiber. Squares represent result of center strain approximation, while FE results are plotted as solid line. (b) Variation of the orientation of global

index ellipse with the applied load. The variation ofθis plotted over a greater

range ofP to demonstrate asymptotic behavior. . . 66

4.10 (a) Variation of Bragg wavelength with applied diametrical load atγ = 72◦

for PM fiber. Squares represent result of center strain approximation, while FE results are plotted as solid line. (b) Variation of the orientation of global

index ellipse with the applied load. The variation ofθis plotted over a greater

range ofP to demonstrate asymptotic behavior. . . 66

4.11 (a) Linear variation of applied diametrical load along the length of the FBG. FBG spectra obtained before (dashed line) and after (solid line) application

of transverse load at (b) γ = 36◦ and (c)γ = 72◦. . . 67

5.1 (a) PMMA specimen with surface mounted FBG. (b) Schematic

representa-tion of PMMA specimen. Also indicated are the locarepresenta-tion of the optical fibers

and FBGs for each specimen and the impact point. All dimensions in mm. 74

5.2 Instrumentation for FBG sensor. . . 75

5.3 Drop tower for impact loading of the specimens . . . 75

5.4 Variation of strain with strike # for FBG surface mounted on specimen A58. 76

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5.10 Variation of strain with strike # for FBG surface mounted on specimen A79. 79 5.11 Measured transmitted spectra of a FBG surface mounted on specimen A81

at two different load levels. (a) 300 lbs, maximum load; (b) 300 lbs, zero

load; (c) 1000 lbs, maximum load; (d) 1000 lbs, zero load . . . 80

5.12 Variation of strain with load for an FBG surface mounted on specimen A80. 81

5.13 Variation of strain with load for an FBG surface mounted on specimen A81. 82

5.14 Variation of the maximum residual strain with the radius for 90◦ and 45◦

orientations. Solid and dashed lines are the quadratic fit for 90◦ and 45◦

orientations respectively. . . 83

5.15 Curing cycle for the fabrication of 2D woven composites. . . 86

5.16 Photograph of two-dimensional woven composite specimen (a) before and (b)

after impact. . . 88

5.17 Cross-section of two-dimensional woven composite specimen under a optical

microscope (a) before and (b) after impact. . . 89

5.18 Instrumentation for continuous FBG sensing. Also shown is the 5 Hz

inter-rogator from Micron Optics. . . 90

5.19 Schematic representation of specimen C5 along with optical fiber and impact

location. All dimensions in mm. . . 90

5.20 Global measurements for specimen C5. (a) Variation of peak acceleration

with strike #. (b) Variation of dissipated energy with strike #. . . 91

5.21 Variation of strain with strike # for an FBG surface mounted on specimen C5. Strains computed from both the reflection and transmission spectra are

plotted. . . 91

5.22 Schematic representation of composite specimens. Also shown are the lo-cation of FBGs for C6,C17,C18 and C15 along with impact lolo-cation. All

dimensions in mm. . . 92

5.23 Global measurements for C6. (a) Variation of peak acceleration with strike

#. (b) Variation of dissipated energy with strike #. . . 92

5.24 Variation of strain with strike # for an FBG surface mounted on C6. Strains

computed from both the reflection and transmission spectra are plotted. . . 93

5.26 Variation of strain with strike # for an FBG surface mounted on C17. Strains

computed from the reflection spectra of (a) End 1 and (b) End 2 are plotted. 94

5.27 Measured reflection spectra from End 1 for a FBG surface mounted on

spec-imen C17 after strikes #: (a) 11, (b) 14, (c) 17, and (d) 20. . . 95

5.28 Measured reflection spectra from End 2 for a FBG surface mounted on

spec-imen C17 after strikes #: (a) 11, (b) 14, (c) 16, and (d) 18. . . 96

5.29 Global measurements for specimen C18. (a) Variation of peak acceleration

with strike #. (b) Variation of dissipated energy with strike #. . . 97

5.30 Variation of strain with strike # for an FBG surface mounted on specimen C18. Strains computed from both the reflection and transmission spectra are

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5.31 Measured spectral response of FBG surface mounted on specimen specimen

C18 after strikes #: (a) 6, (b) 9, (c) 10, (d) 14, (e) 15, (f) 16, and (g) 20. . 105

5.33 Cross-section of two dimensional woven composite specimen embedded with

optical fiber under optical microscope. . . 106

5.34 Schematic representation of specimen C49 along with optical fibers and

im-pact location. All dimensions in mm. . . 107

5.35 Schematic representation of specimen C50 along with optical fibers and

im-pact location. All dimensions in mm. . . 107

5.36 Measured spectra for FBG-A embedded in specimen C49 (a) after embedding,

after strikes #: (b) 1, (c) 6, and (d) 8. . . 108

5.37 Measured spectra for FBG-B embedded in specimen C49 (a) after embedding,

5.38 Measured spectra for FBG-C embedded in specimen C50 (a) after embedding,

5.39 Measured spectra for FBG-D embedded in specimen C50 (a) after embedding,

5.40 Cross-section of two dimensional woven composite specimens under optical microscope: (a) specimen C29, (b) specimen C30, (c) specimen C32, and (d)

specimen C34. . . 112

A.1 Measured spectral response of FBG surface mounted on A58 for impact

strikes 1-5. . . 124

strikes 1-7. . . 125

strikes 8-15. . . 126

strikes 16-19. . . 127

strikes 1-6. . . 128

strikes 7-14. . . 129

strikes 15-23. . . 130

strikes 23-27. . . 131

A.9 Measured spectral response of FBG surface mounted on A69 at the end of

impact loading. . . 132

strikes 1-4. . . 133

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strikes 8-16. . . 135

strikes 17-25. . . 136

strikes 26-34. . . 137

strikes 35-36. . . 138

strikes 1-4. . . 139

strikes 1-7. . . 140

strikes 8-16. . . 141

strikes 17-20. . . 142

A.20 PMMA specimens surface mounted with FBGs. . . 143

A.21 Measured spectral response of FBG surface mounted on A80 for a static

loading from 0-700 lbs. . . 144

A.22 Measured spectral response of FBG surface mounted on A80 for a static

loading from 800-1300 lbs. . . 145

A.23 Measured spectral response of FBG surface mounted on A81 for a static loading from 0-400L lbs. Notation L and U stand for loading and unloading

respectively. . . 146

A.24 Measured spectral response of FBG surface mounted on A81 for a static load-ing from 400U-800U lbs. Notation L and U stand for loadload-ing and unloadload-ing

A.25 Measured spectral response of FBG surface mounted on A81 for a static load-ing from 900L-1100U lbs. Notation L and U stand for loadload-ing and unloadload-ing

B.1 Dimensions of the mold used to fabricate the composite specimens. All

di-mensions in inches. . . 150

B.2 Progression of damage in a typical two-dimensional woven composite specimen.151 B.3 Measured spectral response of FBG surface mounted on C17 for impact

strikes 1-7. . . 152

B.4 Measured spectral response of FBG surface mounted on C17 for impact

strikes 8-13. . . 153

strikes 14-17 . . . 154

strikes 18-20. . . 155

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strikes 8-16. . . 157

strikes 17-22. . . 158

B.10 Measured spectral response of FBG embedded in C15 for impact strikes 1-7. 159 B.11 Measured spectral response of FBG embedded in C15 for impact strikes 8-14. 160 B.12 Measured spectral response of FBG-A embedded in C49 for impact strikes 1-4.161 B.13 Measured spectral response of FBG-A embedded in C49 for impact strikes

5-13. . . 162

B.14 Measured spectral response of FBG-B embedded in C49 for impact strikes 1-4.163 B.15 Measured spectral response of FBG-B embedded in C49 for impact strikes

5-13. . . 164

B.16 Measured spectral response of FBG-B embedded in C49 for impact strikes

14-18. . . 165

B.17 Measured spectral response of FBG-C embedded in C50 for impact strikes 1-4.166 B.18 Measured spectral response of FBG-C embedded in C50 for impact strikes

5-12. . . 167

B.19 Measured spectral response of FBG-C embedded in C50 for impact strikes

13-16. . . 168

B.20 Measured spectral response of FBG-C embedded in C50 for impact strikes

17-24. . . 169

B.21 Measured spectral response of FBG-D embedded in C50 for impact strikes 1-4.170 B.22 Measured spectral response of FBG-D embedded in C50 for impact strikes

5-10. . . 171

B.23 Measured spectral response of FBG-D embedded in C50 for impact strikes

11-18. . . 172

B.24 Measured spectral response of FBG-D embedded in C50 for impact strikes

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List of Tables

3.1 Parameters of scalar wave equation for fundamental LP modes. . . 20

3.2 Parameters of circular core, step-index fiber. . . 26

3.3 Material Properties of Modeled Optical Fibers. . . 31

3.4 Number of Elements Meshed for PM Fiber Types. . . 32

3.5 Comparison of Sensitivity to Applied Load for PM Fiber Types. . . 42

3.6 Comparison of Sensitivity to Applied Temperature for PM Fiber Types. . . 42

4.1 Parameters of PM fiber. . . 64

5.1 Configurations of PMMA Specimens . . . 70

5.2 Configurations of 2D woven composite specimens . . . 84

5.3 Configurations of 2D woven composite embedded with polyimide coated fibers without FBGs. . . 100

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Chapter 1 Introduction

The goal of a damage assessment system is to measure failure initiation and

pro-gression in a structure. From this one estimates the remaining life time of the damaged

structure. This subsystem therefore forms an integral part of long-term structural health

monitoring system. The concept of damage assessment is particularly important for fiber

reinforced composites subjected to low velocity impacts due to the different failure

mech-anisms involved. The interaction of these various mechmech-anisms during the progression of

damage poses a challenge to the assessment system.

The first step towards building an effective damage assessment system is the choice

of the sensor(s). Due to its ability to measure strain distributions as well as with

multiplex-ing capabilities fiber Bragg gratmultiplex-ing (FBG) sensor is an effective tool for a damage assessment

system. In addition, FBGs can be embedded into many host structures to measure internal

damage. After selecting the sensor, the next step is to model the strain transfer from the

host material to the sensor and the sensor response due to the physical change in the host

structure incorporating this transfer. One of the challenges posed by a damage assessment

system is to be able to distinguish between sensor failures and host material failures from

the complex sensor response.

This dissertation eliminates the need for a reduced strain transfer model by using

a finite element (FE) formulation to model the response of an embedded FBG sensor.

A linear strain-optic effect is applied to model the local opto-mechanical response of the

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FBG sensors are surfaced mounted and embedded in two-dimensional woven composites.

The developed FE model for the sensor response and the analysis of experimental data is

important towards (i) better understanding the sensor response to be able to distinguish

between sensor failure and host material failure and (ii) integrating into existing FE models

for structural behavior.

This thesis is organized as follows: Chapter 2 presents a literature review of existing

models for strain transfer and motivations for developing the FE formulation. Chapter 3

presents the FE formulation for the complex strain transfer using a linear strain-optic

effect. The model is applied to a two-dimensional problem of diametrical compression of

FBGs written in PM fibers for validation. Chapter 4 presents the spectral response of

the FBG sensor using a modified T-matrix model including the effects of strain gradients.

Chapter 5 presents experimental results and analysis of the FBG sensor response when

surface mounted and embedded in two-dimensional woven composites. We also present

the results of FBG’s surface mounted on PMMA specimens. Finally Chapter 6 presents

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Chapter 2 Background

2.1 FIBER BRAGG GRATINGS

Figure 2.1 shows a classification of optical fibers based on the diameter of the core

into single mode or multi mode fibers. For small core diameters (4-10µm) the fiber supports

only the first mode. The diameter of the claddding is typically 125µm. Single mode fibers

are further classified into circular core and polarization maintaining fibers. In polarization

maintaining fibers the first linearly polarized mode, LP01, is split into two modes each

having a specific propagation constant. Thus two polarization axes are established. PM

fibers are further classified as low birefringent and high birefringent fibers based on whether

the birefringence is due to geometry or a stress applying part (SAP) [18].

The first fiber optic sensor was introduced by Butter and Hocker in 1978 [1]. They

used an interferometric technique to measure the strain on the surface of a cantilever beam.

They related the change in phase of a lightwave propagating through the optical fiber, ∆φ,

to the axial strain applied to the fiber by

∆ ={β−1

2βn

2_[(1₋_ν₎_p

12−νp11]}²L (2.1)

whereβ =nk0 is the propagation constant,nis the effective refractive index,k0 = 2π/λis

the wavenumber, λ is the wavelength of light, L is the sensing length of the optical fiber,

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Optical fibers

Multi-mode Single-mode

Circular core PM fiber

Low birefringent High birefringent Elliptical core

D-fiber

Panda fiber

Figure 2.1: Classification of optical fibers.

strain-optic effect one arrives at the Butter and Hocker formulation. The coefficients p11

andp12are the strain optic coefficients for the optical fiber. Later Hocker, showed that the

sensitivity to temperature is higher than the pressure sensitivity [2].

In 1988, Bertholds and Dandliker developed a technique to experimentally

deter-mine the strain optic coefficients for an isotropic, mechanically homogeneous optical fiber

[3]. They applied axial strain to the fiber and used (2.1) to compute the difference in strain

optic coefficients [5]. They applied torsional strain to the fiber and related the change in

polarization of light to the difference in strain optic coefficients [4]. Using these two

inde-pendent measurements they computed the individual strain optic coefficients. The values of

strain-optic coefficients computed arep11= 0.113 andp12= 0.252. These values have been

universally accepted and widely used, including in this thesis. The main disadvantages

to applying with the interferometric techniques to measure strain are the high vibration

sensitivity and the need to count fringes which becomes laborious at high strains.

The formation of permanent gratings in an optical fiber was first demonstrated

by Hill et al. in 1978 at the Canadian Communications Research Centre (CRC) [8]. This

achievement was an outgrowth of research on the nonlinear properties of germania-doped

silica fibers. Currently, these gratings are used extensively as filters for telecommunication

purposes and as strain sensors. A Bragg grating is a periodic modulation of the core index

of refraction along a segment of optical fiber. It is formed usually by means of exposure to

an interference pattern of intense ultraviolet light (' 245 nm). The optical back-reflected

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occurs at the Bragg wavelength λB, which can be related to the effective core refractive

index, n, and the period of the index modulation, Λ, by the relation [6]:

λB= 2nΛ (2.2)

This Bragg wavelength will shift with changes to either n or Λ, thus measuring the

wave-length of this narrow-band spectrum will determine the strain or temperature to which the

optical fiber is subjected. The application of these gratings to measure axial strain was

demonstrated by Measures in 1992 [6]. Measures formulated the change in Bragg

wave-length, ∆λB, to the applied axial strain,², by the relation

∆λB

λB

= (1−pe)² (2.3)

wherepe= 0.22 is the effective photoelastic constant. This linear relation has been widely

used to measure axial strain. Since then, Bragg gratings have been used for many

applica-tions including aircraft health monitoring [19], and spacecraft [12].

Bragg gratings can be classified based on their length as short period (≤500 nm)

and long period gratings. From the modeling point of view, in a short period grating the

coupling is between a forward propagating core mode and a backward propagating core

mode. In a long period grating the coupling is between a forward propagating core mode

and a forward propagating cladding mode [20, 21]. For a review of the formation, modeling

and applications of fiber Bragg gratings see Kashyap [9] and Othonoset al. [10]. For more

specific reviews, Hillet al. from CRC present an excellent review on the formation of Bragg

gratings in [7]; Kerseyet al. from the Naval Research Laboratory review the intrumentation

for Bragg gratings sensors in [17]; and Erdogan reviews the modeling of Bragg grating sensor

response in [13].

To model the Bragg grating sensor response, Erdogan first assumes light to be an

electromagnetic wave propagating through a weakly guiding waveguide (optical fiber) [13].

One notes that this approximation is valid for all optical fibers with a difference in core and

cladding refractive index of less than 2%. Using the above approximation, one solves the

wave equation to obtain the guided electromagnetic fields in the optical fiber. An important

point to note is that the exact solution to the wave equation exists in terms of modified

Bessel functions due to the geometrical symmetry of the fiber. The electromagnetic fields

are defined in terms of modified Bessel function of first kind, and in the cladding in terms

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as a perturbation to the effective refractive index,nof the guided mode(s) as [14]:

δneff =δneff(z){1 +νcos[2π

Λz+φ(z)]} (2.4)

whereδneff is the “dc”index change spatially averaged over a grating period, ν is the fringe

visibility of the index change, andφ(z) describes the grating chirp. This perturbation causes

the forward propagating core mode to couple to the backward propagating core mode. This

coupling is maximum at the Bragg wavelength where in energy from the forward propagating

core mode is transfered to the backward propagating core mode. The coupling between the

modes is well described by the coupled-mode theory [15, 16]. The spectral response of the

Bragg grating sensor can then be derived using coupled-mode theory [13]. This modeling

approach allows one to consider the response of the Bragg grating to more general strain

distributions than the constant axial strain assumed in (2.3).

2.2 APPLICATION OF BRAGG GRATING STRAIN

SEN-SORS TO COMPOSITES

McKenzie et al. used optical fiber sensors to study the feasibility of optical fiber

sensors for health monitoring of bonded repair systems [25]. They surface mounted FBGs

to an aluminum-boron epoxy patch to monitor fatigue crack growth and used the finite

element method to determine the optimal placement of sensors. This work demonstrates

that FBGs can be used to detect crack growth. The problems encountered were debonding

of the fibers and a distortion in the response of FBGs due to strong strain gradients.

Murukeshan et al. used fiber Bragg gratings for cure monitoring of composites

[26]. They considered different composite material systems embedded with FBGs. The

results demonstrated the use of FBGs to measure residual strain during curing and also the

linear response of the sensor to bending. An average strain was computed from the spectral

response of the FBG, however an attempt to analyze sensor response was not made.

Takedaet al. embedded Bragg gratings in Carbon Fiber Reinforced Plastic (CFRP)

laminates to detect transverse cracks [27]. They performed tensile tests on laminates

em-bedded with FBG’s to induce transverse cracks. They performed both theoretical and

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fiber coating on strain transfer from the laminate to the embedded optical fiber by applying

the widely used shear lag theory for the strain transfer to numerically simulate the spectral

response of the embedded grating [31]. The authors also demonstrated the use of small

diameter fiber Bragg grating sensors to detect transverse crack and delamination in CFRP

laminates [33, 34].

Kuang et al. embedded Bragg gratings in advanced composite materials and fiber

metal laminates (FML) [28]. They measured the residual stress during fabrication and

attributed the split in the Bragg reflection peak to transverse loading. Afterwards they

subjected the FML’s to tensile loading and observed an increase in intensity of one Bragg

peak with respect to one another. They called this “wavelength hopping”phenomenon.

This wavelength hopping phenomenon could be due to stress relief during loading. One

observes high strain gradients induced during loading from the spectral response. However,

a detailed analysis is lacking in [28]. They also studied the spectral response of FBG’s,

embedded in FML’s, subjected to low velocity impacts [29, 32]. They observed spectrum

distortion during impact loading and also a drop in intensity. They measured the S-shape

trend in the shift in Bragg peak with strike number. A similar observation was made by

Guemes and Men´endez [30]. Kuang et al. also demonstrated the use of FBG’s, embedded

in FML’s, to monitor damage induced during cyclic loading [35]. Fiber Bragg gratings have

also been used to measure internal strain in textile composites [36, 37, 38].

2.3 MODELING OF EMBEDDED BRAGG GRATING STRAIN

SENSORS

As mentioned above, optical fiber Bragg grating (FBG) sensors have been widely

embedded in composite material systems for the measurement of curing stresses,

interlam-ina stresses, delaminterlam-ination, crack growth, and other phenomena [22]. Their unique ability to

be embedded within fiber-reinforced composites with a minimum perturbation to the

sur-rounding host material makes them attractive for the above applications. One particular

field of current interest is long-term health monitoring of composite structures, for

exam-ple aircraft or FRP reinforced concrete structures. Therefore, a clear understanding of the

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A variety of models have been developed for the stress transfer in isotropic

ma-terials embedded with optical fibers [39, 40, 41]. The goal of such models is to calculate

the strain in the optical fiber, and hence the sensor response, due to loading applied to the

host material system. Koll´ar and Van Steenkiste developed a model of the strain transfer

between a laminated composite and an optical fiber embedded between two laminae [42].

The optical fiber is treated as an elliptical inclusion in the composite laminate cross-section

and the average fiber strain due to the surrounding strain in the laminate is calculated

analytically. Later, Prabhugoud and Peters modeled a unidirectional composite by

apply-ing a combination of the finite element method and optimal shear-lag theory [43]. While

applying varying levels of detail to the host structure, each of these models considers the

optical fiber as an isotropic or orthotropic, homogeneous fiber with constant strain across

the cross-section of the fiber. Although this assumption produces excellent results for the

mechanical response of embedded fibers [44], it is not evident that the same is true for the

optical response of the FBG due to the optical non-homogeneities of the optical fiber.

Once the strain field applied to the optical fiber is known, the shift in Bragg

wavelength of the FBG, ∆λB, is calculated as [6],

∆λB = ∆(2Λneff) = 2Λ∆neff + 2neff∆Λ (2.5)

where Λ is the period of the grating and ∆neffis the change in effective refractive index of the

fiber cross-section due to the applied strain. For example, in the case of an unconstrained

fiber subjected to an axial load,

∆λB=λB²z

h

1−n

2 eff

2 {p11−ν(p11+p12)}

i

(2.6)

where ²z is the axial strain in the fiber, p11 and p12 are photoelastic constants for silica,

and ν is the Poisson’s ratio of the fiber [1].

Although the term ∆Λ in (2.5) can be directly related to the resulting axial strain

in the FBG, the term ∆neff is more difficult to calculate. This change is due both to

geometrical changes in the cross-section and the strain-optic effect. In addition, induced

birefringence in the fiber separates the single Bragg reflection peak into two peaks [45].

This birefringence is highly dependent on the index of refraction and material distribution

throughout the fiber cross-section. Examples of typical optical fiber cross-sections are shown

in figure 2.3. The circular and elliptical step-index cross-sections shown in figure 2.3(a)

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P

₃

P

₁

P

₂

Figure 2.2: FBG written into a PM fiber, subjected to multi-axis loading.

ratio νcore = νclad. Optically, they are non-homogeneous with two index of refraction

regions: ncore in the core and nclad in the cladding. Whereas the circular fiber propagates

light at one propagation constant, β1, per mode, the elliptical fiber propagates modes

along two orthogonal axes with two different propagation constants, β1 and β2. The third

optical fiber, referred to as the polarization maintaining (PM) fiber, shown in figure 2.3(c),

includes regions of a separate material (and is therefore both mechanically and optically

non-homogeneous) called the stress applying part (SAP). The purpose of these regions is

to provide stress to the fiber core as the fiber is drawn, inducing residual birefringence due

to the thermal expansion mismatch of the silica and SAP. Several other configurations for

the PM fiber exist other than the bow-tie form shown in figure 2.3(c).

To calculate ∆neff due to an applied strain field, Kimet al. considered the fibers

shown in figure 2.3 to be optically and mechanically homogeneous with isotropic or

transver-sly isotropic material properties [46]. Based on the model of an inclusion in a composite

laminate, the remote strains were analytically linked to the principle strains in the fiber

core and ∆β in the direction of propagation calculated. As in later models, the

assump-tion is made that most of the energy of the fundamental mode propagating in the fiber is

contained in the core, therefore the principle strains at the center of the fiber are sufficient

to estimate ∆neff. In addition, none of the above models account for the birefringence due

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(a) (b) (c) n , E

clad clad

n , E_core _core

n , E

clad clad

n , E_core _core

n , E

clad clad

n

Ecore_core n E

SAP

Figure 2.3: Typical optical fiber cross-sections: (a) circular core, step-index (no birefrin-gence), (b) elliptical core, step-index (geometrical birefrinbirefrin-gence), (c) bow-tie fiber with pre-stressed regions (geometrical and residual stress birefringence).

the strain-optic effect. Sirkis related the change in Bragg wavelengths, ∆λB,1 and ∆λB,2,

to the principle strains at the center of the core,²1,²2, and ²3, as [47]

∆λB,1 λB,1

=²1−n

2

o

2 (p11²2+p12²3+p12²1) (2.7)

∆λB,2 λB,2

=²1−n

2

o

2 (p12²2+p11²3+p12²1)

where ²2 and ²3 are in the plane of the fiber cross-section, ²1 is in the axial direction and

no is the effective refractive index of the fundamental mode. Wagreichet al. demonstrated

the linear dependence of ∆λB,1 and ∆λB,2 on the applied load for a circular core fiber

(figure 2.3(a)) under diametrical compression [45]. Lawrence et al. [48] modeled the

me-chanical non-homogeneities in a PM fiber with elliptical SAP using a finite element analysis

to calculate the strains at the center of the fiber due to applied transverse loading. Their

main goal was to calculate the calibration matrix for the transverse sensitivity of a FBG

sensor in a PM fiber. Later Bosiaet al. [49] also modeled the mechanical non-homogenieties

in a bowtie type PM fiber using finite element analysis to calculate the principle strains at

the center of the core and hence the shift in Bragg wavelength due to applied transverse

loading using (2.7). Both the experimental and numerical studies of [48] and [49]

demon-strated that for a PM fiber the shift in Bragg wavelength is nonlinear with transverse load

for certain loading angles. Gafsi and El-Sherif expanded the center strain formulation to

include variations of refractive indices along the axis of the fiber by introducing (2.7) into

the coupled mode equations describing the spectral response of the FBG [50]. However,

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and theoretical sensitivity to applied strain.

2.4 MOTIVATION

As described in the above section there exists handful of models to simulate the

response of a FBG subjected to arbitrary loading. One should note that all the models use

the center strain approximation (CSA). None of the models described in the above section

can therefore account for the variation of the refractive index in the cross-section of the

fiber due to the applied strain.

The goal of this thesis is to present a methodology to calculate the response of a

FBG subjected to arbitrary loading as shown in figure 2.2. We formulate a finite element

(FE) model to calculate the birefringence effect due to the applied transverse load. The

linear strain-optic law is assumed to calculate the change in refractive index due to the

ap-plied transverse load. One can integrate this model into existing FE models for structural

behavior. We will consider a diametrical compression of PM fibers to validate the model.

As described in the above section PM fibers are both mechanically and optically

nonho-mogeneous. We will also formulate a modified T-matrix model to simulate the response of

FBG subjected to longitudinal strain. This model considers the effect of strain gradients

in the response and is computationally efficient. Afterwards, the response of FBGs surface

mounted on PMMA and two-dimensional woven composites subjected to low velocity

im-pacts are analyzed. Finally, FBGs are embedded in two-dimensional woven composites to

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Chapter 3 Finite Element Model for

Embedded Fiber Bragg Grating

Sensor

The goal of this chapter is to derive a finite element (FE) formulation to predict

the optical response of an embedded FBG sensor as a function of the loading applied to

the host structure. The formulation incorporates both the mechanical and optical

non-homogeneities of the optical fiber. Firstly, the FE fomulation calculates the change in index

of refraction distribution throughout the cross-section of the fiber due to the resulting

mechanical stresses. From the updated index of refraction distribution, the propagation

constants of the fundamental modes, as well as the propagation axes, are obtained. Previous

work by Huang has approached a similar problem for planar waveguides analytically [51].

As can be observed from [51], analytical solutions are only obtainable for a few loading

conditions. In the current formulation, the propagation constants are then introduced into

a discretized version of the coupled mode equations to determine the spectral response of

the FBG.

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systems for which extensive FE modeling has already been performed. Examples include

fiber reinforced composites and concrete structures. The current model also allows one to

accurately calculate the sensitivity of the FBG to transverse strains and is applicable to

various fiber types including bowtie and panda PM fibers. The formulation also includes

the effect of rotating polarization axes due to significant strain amplitudes especially when

the fiber is embedded near a stress concentration or failure location.

3.1 FINITE ELEMENT FORMULATION

The propagation of a given guided mode through an optical fiber can be

char-acterized through the mode distribution in the cross-section of the optical fiber and the

propagation constant, β, for a given frequency. The mode propagation constant is related

to the effective index of refraction, neff, for the particular mode through

β = 2π

λ neff (3.1)

whereλis the propagating wavelength [52]. Exact solutions for the propagation

character-istics of optical fibers, obtained by solving wave equations, are limited to relatively simple

geometries (e.g., circular or elliptical cross-sections) with an axisymmetric index of

refrac-tion distriburefrac-tion in the core. Thus, to calculate the propagarefrac-tion characteristics of an optical

fiber with an arbitrary cross-sectional shape or arbitrary variation of refractive index in the

core, cladding, and SAP, one needs to adopt a numerical method such as the finite element

method [53]. Current finite element methods for optical fiber waveguides can be classified

into vector methods and scalar methods.

Vector finite element methods are applicable to all values of refractive index

differ-ence between the core and the cladding. The main disadvantages of these methods are the

large computational effort required and the appearance of spurious modes in the solution.

The spurious modes can be eliminated using a penalty approach [53]. Different variations

of the vector formulation are based on the components of the electric field,E~, or magnetic

field, H~, considered. For example, in the formulation of Yeh et al., the axial components

of E~ and H~ field are considered [54]. All other components are then expressed in terms of

these axial components using Maxwell’s equations. A minimizing functional is obtained by

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interface. In the work of Koshiba, the minimizing functional is obtained from the complete ~

E orH~ field satisfying the vector wave equation [53]. Different approaches have also been

proposed to address the open boundary problem for example, applying the FE formulation

to the core and appropriate boundary condition to the core-cladding interface. This reduces

the number of elements required in the cladding to obtain an accurate solution [55, 56].

Scalar finite element methods, on the other hand, are only applicable to weakly

guiding fibers, i.e. for which the variation of the refractive index is negligible over a distance

of one wavelength [52]. However, such an assumption is reasonable for most fibers into which

FBGs are written, within their elastic strain limit. The advantages of a scalar method are

that no spurious solutions appear (since only linearly polarized modes are captured) and

only one component of the E~ or H~ field is considered, reducing the size of the required

system of equations to solve for the propagation constant. For this reason, in this chapter

we derive a sensor element based on a scalar formulation without imposing the assumption

of axisymmetry.

3.1.1 Overview

In the current analysis, the prediction of the FBG spectral response is performed

through the following steps (see figure 3.1):

• The surrounding host composite material and optical fiber sensor are meshed using

a commercial FE package (e.g., ANSYS for the current work). The chosen sensor

mesh is shown in figure 3.2, where the fiber is divided into segments of length ∆z in

the axial direction and each cross-section is meshed using 2D plane stress triangular

elements. An example element is also shown in figure 3.2. For the purpose of later

calculations, this “3D propagation”element is characterized by its stiffness properties,

indices of refraction, and length, ∆z. Axis 1 is along the propagation direction and

coincides with the global propagation axis,z. Axes pand q are the local optical axes

of propagation.

• Using the thermo-mechanical FE model, the nodal displacements are obtained due to

the external applied loads. From the nodal displacements, strain components in each

element are also calculated.

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due to the applied strain field is calculated in local optical axes using a linear

strain-optic law (see section 3.1.2). The updated indices of refractions are then transformed

from the local axes to the global structural axes.

• For each sensor segment, the propagation constant for the optical fiber about the

global structural axis is calculated using the optical FE formulation including the

updated index of refraction distribution and the nodal displacements.

• The propagation constant/effective index of refraction at two other angles with respect

to the global structural axis is calculated. From these values the maximum and

mininum propagation constants are calculated for the cross-section as well as the

global optical axes corresponding to these extrema.

• The FBG spectral response is calculated from the local axial strain, effective indices

of refraction, and curvature of each segment using the modified T-matrix method (see

section 3.1.4).

The details of some of these calculations are given below.

3.1.2 Calculation of indices of refraction for an element

Each element is assumed to be optically isotropic with an index of refraction in

the unstressed state ofno_e. The displacement field vector for light propagating through the

element in the 1-direction is given as [46],

{D}=Ap{sp}sin

·

ωt− 2πn

p e

λ x1

¸

+Aq{sq}sin

·

ωt−2πn

q e

λ x1

¸

(3.2)

wheresp andsqare orthogonal unit vectors in the 2-3 plane in the direction of the principle

optical axes, npe and nqe are the element index of refractions about these axes, ω is the

angular frequency of the wave, and Ap and Aq are the amplitudes of the displacement

vector components. We can write the wave equation for this displacement field vector as

[46],

{s} ×({s} ×[B]{D}) + 1 (ne)2{D}

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Structural Model (geometry, material properties, loading)

Nodal Displacements

Element Indices of Refraction and Local Optical Axes

ANSYS

Transfer Matrix Formulation Finite Element Formulation

Segment Global Optical Axes and Principle Propagation Constants

Strain - Optic Law

Segment Propagation Constants at Orientations α, β, and γ

FBG Spectral Response Global Index Ellipse

Figure 3.1: Schematic of the procedure for calculation of FBG spectral response for a sensor embedded in a host material system.

where ne = npe or nqe, {s} is the unit vector in the propagation direction, and [B] is the

material dielectric impermeability tensor,

[B] =

   

B1 B6 B5

B6 B2 B4

B5 B4 B3

  

 (3.4)

Writing{D} in terms of its components {D}= (0, D2, D3), {s}= (1,0,0), and evaluating

(3.3) yields the matrix equation,



 B2−1/n2e B4

B4 B3−1/n2e

 

  

D2 D3

 

= 0 (3.5)

The non-trivial solutions to (3.5) npe and nqe are,

1 (np,qe )2

= (B2+B3)±

p

(B2−B3)2+ 4B₄2

(32)

direc

tion of pr

opaga

tion

∆ z

q

1

p

2 3

X

Z

Y

Figure 3.2: Discretization of optical fiber into FBG sensor elements. Also shown are

def-inition of local polarization axes p and q, global polarization axes, X and Y, and local

principle strain axes, 1−2−3. z is the direction of propagation along the optical fiber.

These solutions correspond to the indices of refraction about the principle optical axes p

and q in the 2−3 plane shown in figure 3.2. These axes will be determined later. For an

optically isotropic material, B1 = B2 = B3 = 1/(noe)2, B4 = B5 = B6 = 0. Therefore,

npe =nqe=noe.

Once strain is applied to the element, the dielectric impermeability tensor change

is defined by the linear strain-optic equation,

∆Bi =

6

X

j=1

pij²j (3.7)

where [p] is the strain-optic tensor and the compact notation is used for the strain

compo-nents (²1 =²11, ²2 =²22, ²3 =²33, ²4 =γ23, ²5 =γ31, ²6 =γ12) [46]. Expanding the solution

of (3.6), writing Bi =Bio+ ∆Bi, and applying the isotropic properties to Bio, we find,

1 (np,qe )2

= 1

(no e)

+(∆B2+ ∆B3)

2 ±

1 2

q

(33)

For an optically isotopic material, the strain-optic tensor, [p], reduces to,

[p] =

             p11 p12 p12 0 0 0 p12 p11 p12 0 0 0 p12 p12 p11 0 0 0 0 0 0

(p11−p22)/2 0 0 0 0 0 0

(p11−p22)/2

             (3.9)

Substituting (3.9) into (3.7) into (3.8) yieldsnpe andnqe for an element in the stressed state.

1 (np,qe )2

= 1

(no e)2

+p12²1+ (p11+₂ p12)(²2+²3)

±(p11−p12)

2

q

(²2−²3)2+²24 (3.10)

Kim et al. considered the same formulation for the optical fiber as a single

ho-mogeneous element and derived a linearized form of (3.10) to calculate the sensitivity of

the FBG to transverse strain [46]. To model a polarization maintaining fiber (such as

fig-ure 2.3(c)) they considered the fiber to be initially optically orthotropic. This approach

produces identical results to the center strain approximation of (2.7).

For FBG sensor problems including thermal loading, (3.6) can be expanded to

include a linear thermo-optic effect [46],

∆Bi=Wi∆T+pij(²j−αj∆T) (3.11)

where{α}are the coefficients of thermal expansion of the sensor in the local optical

coordi-nates. For an isotropic sensor (α1=α2 =α3 =α and α4=α5 =α6 = 0). The coefficients

Wi, defined as,

Wi=

Ã

∂Bi ∂T

!

σ=const.

(3.12)

are measured during iso-stress conditions. For an optically isotropic material, the non-zero

coefficients are thus evaluated as,

W1 =W2=W3 = ∂

∂T

Ã

1 (no

e)2

!

=− 2

(no e)3

Ã

∂no_e ∂T

!

(3.13)

A typical value for the thermooptic coefficient (∂no

e)/(∂T) for silica is given by Kimet al.

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(3.10)) would thus be modified to

1 (np,q_eff)2 =

1 (no

e)2

+p12²1+(p11+₂ p12)(²2+²3)−₍_n2_o

e)3

Ã

∂no_e ∂T

!

± (p11−p12)

2

q

²2₄+ (²2−²3)2 (3.14)

The angle of orientation of the element principle optical axes are identical to the

principle strain directions [57]. Although the index of refraction is not a true tensor quantity,

it can be represented by an ellipse in the 2-3 plane with the major and minor axes of length

npe and nqe in the principle strain directions [57]. Therefore to calculatene about the global

axesX and Y, we find

nX_e = n

p enqe

p

(npecosψe)2+ (nqesinψe)2

nY_e = n

p enqe

p

(npesinψe)2+ (nqecosψe)2

(3.15)

whereψe is the angle required to rotate thep axis to theX axis.

3.1.3 Calculation of propagation constants for a sensor segment

Once the index of refraction for each element is known about both the X and Y

axes, the propagation constantsβmax andβmin and the orientation of the principle optical

axes for the complete cross-section must be calculated. The propagation characteristics for

linearly polarized (LP) modes propagating through a waveguide of arbitrary cross-section

and arbitrary variation of refractive index are determined by solving the scalar wave

equa-tion over the cross-secequa-tion of the fiber, here defined as the region Ω,

∂ ∂x

h

pzx_∂x∂ (pxΦ(x, y))

i

+ ∂

∂y

h

pzy_∂y∂ (pyΦ(x, y))

i

+

(qk2₀−β2)Φ(x, y) = 0 (3.16)

where the field Φ(x, y) and the coefficients px, pzx,py, pzy, and q are defined in table 3.1

for the fundamental LP_x01 andLP_y01 modes.

The scalar wave equation defined in (3.16) is derived as follows:

Maxwell’s equations for source free, time harmonic fields are [52]

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Table 3.1: Parameters of scalar wave equation for fundamental LP modes.

Mode Φ px pzx py pzy q

LP_x01 Ex n2x 1/n2z 1 1 n2x

LP_y01 Ey 1 1 n2y 1/n2z n2y

∇ ×_H~ =−jω²0[²]E~ (3.18)

∇ ·_D~ ₌_²₀_{∇ ·}_([_²_r_]_E~_{) = 0} _(3.19)

∇ ·_H~ = 0 (3.20)

where E~(x, y, z) = E~(x, y)e−jβz and H~(x, y, z) = H~(x, y)e−jβz are the electric field and

magnetic fields respectively, ²0 and µ0 are the free space permittivity and permeability

constants, and [²r] is the material permittivity tensor given by

[²r] =

   

n2_x 0

0 0

n2_y 0

0

n2_z

  

 (3.21)

Taking the curl of (3.17) and substituting (3.18) we obtain

∂2Ey

∂x∂y −

∂2Ex

∂y2 −

∂2Ex

∂z2 +

∂2Ez

∂x∂z =k

2

0n2xEx (3.22)

∂2Ex

∂x∂y−

∂2Ey

∂x2 −

∂2Ey

∂z2 +

∂2Ez

∂y∂z =k

2

0n2yEy (3.23)

∂2Ex

∂x∂z −

∂2Ez

∂x2 −

∂2Ez

∂y2 +

∂2Ey

∂y∂z =k

2

0n2zEz (3.24)

wherek0= 2π/λ. Noting that for theLP_x01 mode, Ey = 0 and using (3.19) yields

Ez =− j

βn2_z ∂ ∂x(n

2

xEx) (3.25)

Substituting (3.25) into (3.22) we obtain

∂ ∂x

h ₁

n2_z ∂ ∂x(n

2

xEx)

i

+∂

2_E

x

∂y2 +k

2

0n2xEx−β2Ex= 0 (3.26)

Similarly for the LP_y01 mode,Ex = 0 and

∂ ∂y

h ₁

n2_z ∂ ∂y(n

2

yEy)

i

+ ∂

2_E

y

∂x2 +k

2

(36)

Thus, the scalar wave equation allows us to solve for the fields Ex and Ey independently.

Although, the scalar wave equation derived above is similar to the one derived by Koshiba

[53], here we include the gradient of the refractive index in the formulation.

The functional for (3.16) is given by,

F = Z Z Ω δΦ n _∂ ∂x h

pzx(x, y)_∂x∂ (px(x, y)Φ(x, y))

i

+

∂ ∂y

h

pzy(x, y)_∂y∂ (py(x, y)Φ(x, y))

i

+ (q(x, y)k2₀−β2)Φ(x, y)

o

dxdy (3.28)

Taking the first variation of (3.28) and reducing, one obtains

δF =

Z Z

Ω

h

pzx ∂ ∂x(pxΦ)

∂

∂x(δΦ) +pzy ∂ ∂y(pyΦ)

∂ ∂y(δΦ)

+(β2−qk₀2)(δΦ)Φ

i dxdy − Z Γ n δΦ h

pzx_∂x∂ (pxΦ) +pzy_∂y∂ (pyΦ)

io

dΓ = 0 (3.29)

where Γ is the boundary of the region Ω. Since we are only concerned with propagated

modes, i.e. modes that are fully contained in the optical fiber, we apply the boundary

condition Φ = 0 on Γ. Discretizing the region into triangular elements as shown in figure 3.2

and noting that the coefficients px,pzx,py,pzy, and q are constant for each element, (3.29)

reduces to X e Z Z Ωe h

pe_zxpe_x∂Φ ∂x

∂

∂x(δΦ) +p e zypey

∂Φ

∂y ∂ ∂y(δΦ)

+(β2−qek2₀)(δΦ)Φ

i

dxdy= 0 (3.30)

where the superscript e refers to the values for a given element. We expand Φ in each

element,

Φe= [_N₁ _N₂ _N₃]

    Φ₁ Φ₂ Φ₃   

={N}T{Φ} (3.31)

whereN1,N2, andN3 are element shape functions and Φ1, Φ2, and Φ3 are nodal values of

Φ. Substituting (3.31) into (3.30), we obtain the global matrix equation,

Damage Assessment in Composites using Fiber Bragg Grating Sensors

Contents

List of Figures

List of Tables

Chapter 1

Introduction

Chapter 2

Background

2.1

FIBER BRAGG GRATINGS

2.2

APPLICATION OF BRAGG GRATING STRAIN

SEN-SORS TO COMPOSITES

2.3

MODELING OF EMBEDDED BRAGG GRATING STRAIN

SENSORS

P

3

P

1

P

2

2.4

MOTIVATION

Chapter 3

Finite Element Model for

Embedded Fiber Bragg Grating

Sensor

3.1

FINITE ELEMENT FORMULATION

direc

tion of pr

opaga

tion

₃

₁

₂