Characterizing the Oblique Incidence Response and Noise Reduction Techniques for Luminescent Photoelastic Coatings

CHARACTERIZING THE OBLIQUE INCIDENCE RESPONSE AND NOISE REDUCTION TECHNIQUES FOR LUMINESCENT PHOTOELASTIC COATINGS

JOHN C. NICOLOSI JR.

A THESIS PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE

UNIVERSITY OF FLORIDA

2004

John C. Nicolosi Jr.

ACKNOWLEDGMENTS

I would like to thank my advisors, Dr. Peter Ifju and Dr. Paul Hubner, for all of their support and guidance. I would also like to thank Dr. Leishan Chen for all of the assistance he provided me, and Dr. Bhavani Sankar for his valuable advice. I would also like to thank my family members for all of the support they have given me over the years.

iii

TABLE OF CONTENTS

page

ACKNOWLEDGMENTS ...... iii

LIST OF FIGURES ...... vii

ABSTRACT...... ix

CHAPTER

1 INTRODUCTION ...... 1

1.1 Historical Background of Photoelasticity...... 1 1.2 Research History at the University of Florida ...... 3 1.3 Research Objectives...... 4

2 ELEMENTARY PHOTOELASTICITY...... 6

2.1 Behavior of Light...... 6 2.2 Polarized Light and Linear Polarizers ...... 7 2.3 Retardation Plates ...... 7 2.4 Polariscopes ...... 9 2.4.1 Plane Polariscope...... 9 2.4.2 Circular Polariscope ...... 11 2.4.3 Reflection Polariscope...... 12 2.5 Stress Optic Law...... 13

3 LUMINESCENT PHOTOELASTIC COATINGS ...... 17

3.1 Introduction...... 17 3.2 Excitation and Imaging Equipment ...... 18 3.3 Coating Formulation...... 19 3.3.1 Dual-Layer Coating Formulation ...... 19 3.3.2 Single-Layer Coating Formulation...... 21 3.4 Theory of Operation ...... 22 3.4.1 Dual-Layer Coating...... 23 3.4.2 Single-Layer Coating...... 29 3.5 Image Acquisition and Data Processing...... 31

iv 4 OBLIQUE INCIDENCE RESPONSE OF PHOTOELASTIC COATINGS ...... 33

4.1 Introduction...... 33 4.2 Principle of Operation...... 34 4.3 Experiments ...... 41 4.4 Results and Discussion ...... 43 4.4.1 Optical Fresnel Response ...... 43 4.4.2 Emission Strength...... 45

5 NOISE REDUCTION TECHNIQUES FOR THE LPC METHOD ...... 47

5.1 Introduction...... 47 5.2 Experimental Technique and Noise Determination...... 48 5.3 Testing Conditions...... 50 5.4 Preliminary Results and Discussion ...... 51 5.5 Results and Discussion ...... 57 5.6 Conclusions...... 60

6 FINAL RESEARCH CONCLUSIONS...... 61

6.1 Oblique Incidence Response of Photoelastic Coatings ...... 61 6.1.1 Optical Fresnel Response ...... 61 6.1.2 Emission Strength on Oblique Surfaces...... 61 6.2 Noise Reduction Techniques ...... 62

APPENDIX: NOMENCLATURE...... 63

LIST OF REFERENCES...... 68

BIOGRAPHICAL SKETCH ...... 70

LIST OF TABLES

Table page

2-1 Different circular polariscope arrangements ...... 12

5-1 Testing conditions ...... 50

5-2 Percent noise reduction ...... 59

5-3 Noise reduction for each intensity count level ...... 60

LIST OF FIGURES

Figure page

2-1 Plane polariscope...... 10

2-2 Circular polariscope ...... 11

2-3 Reflection polariscope...... 13

3-1 Strain measurement system...... 19

3-2 Excitation and emission for a luminescent coating...... 23

3-3 Plane polarized light entering the quarter wave plate ...... 24

3-4 Circularly polarized light entering the stressed coating...... 25

3-5 Elliptically polarized emission entering the analyzer ...... 26

3-6 Intensity response as a function of analyzer angle...... 29

4-1 Apparent strain observed on an unloaded cylindrical specimen ...... 34

4-2 Light vector obliquely striking the coating ...... 35

4-3 Fresnel coefficients as a function of surface inclination...... 37

4-4 Excitation striking the coating at an arbitrary polarization plane ...... 38

4-5 Plane of incidence ...... 39

4-6 Oblique excitation and emission of a coating ...... 39

4-7 Emission traveling through the analyzer...... 40

4-8 Experimental setup for measuring OFR...... 42

4-9 Experimental setup for measuring emission strength ...... 43

4-10 OFR with respect to surface inclination...... 44

4-11 Normalized intensity with respect to surface inclination...... 46

vii 5-1 Four-point bending apparatus...... 49

5-2 Plot of OSR for experiments with different imaging conditions ...... 50

5-3 OSR standard deviation as a function of the number of images ...... 52

5-4 OSR standard deviation as a function of the intensity count ...... 52

5-5 Intensity drift of the coating emission over a 90 minute time period ...... 54

5-6 Intensity drift measurement system...... 55

5-7 Intensity drift for the LEDs over a 90 minute time period...... 56

5-8 OSR standard deviation as a function of the number of images ...... 57

5-9 OSR standard deviation in microstrain versus the total number of images ...... 58

5-10 OSR standard deviation as a function of the intensity count ...... 59

viii

Abstract of Thesis Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Master of Science

CHARACTERIZING THE OBLIQUE INCIDENCE RESPONSE AND NOISE REDUCTION TECHNIQUES FOR LUMINESCENT PHOTOELASTIC COATINGS

John C. Nicolosi Jr.

August 2004

Chair: Dr. Peter G. Ifju Cochair: Dr. James P. Hubner Major Department: Mechanical and Aerospace Engineering

The method of Luminescent Photoelastic Coatings (LPC) was developed by researchers at the University of Florida in collaboration with the Visteon Corporation to measure the full-field, in-plane strain information on the surface of complex, three- dimensional components. This technology is a tool for determining failure modes and areas of high stress concentration in structural components under applied loads. The use of LPC can provide a better link between analytical models and physical testing because this method can be used to validate the boundary conditions for finite element analysis models. This technology is capable of detecting the overstressed regions on a component early in the prototype design cycle which can save industries time and reduce development and production costs.

Researchers at the University of Florida have conducted experiments and derived equations to characterize the oblique incidence response of LPC. The experimental data show that the intensity measurements made on the oblique surfaces of LPC are relatively

ix stronger than the oblique angle measurements made using the traditional photoelastic coatings. Measurements made on the oblique surfaces of luminescent coatings are subject to an apparent strain governed by the Fresnel equations. This effect, which has been termed the optical Fresnel response (OFR), is dependent on the surface inclination angle of the component. A technique has been developed that enables investigators to use the measured OFR to determine the index of refraction of the luminescent coating.

Knowing the coating index of refraction allows researchers to measure the surface inclination of any coated specimen. The surface inclination information is essential to correct the strain calculations for the OFR and will also be used in future attempts to decouple the in-plane principal strains.

Several imaging techniques have been employed to reduce the noise within the data acquisition system. The three noise reduction techniques that were investigated were as follows: increasing the number of analyzer positions, increasing the number of images taken at each analyzer position, and increasing the intensity count of each image.

Experiments were conducted to measure the amount of noise reduction for all of these techniques, and the optimum, time dependent, image acquisition conditions were determined.

x CHAPTER 1 INTRODUCTION

1.1 Historical Background of Photoelasticity

The method of photoelasticity is based on the physical behavior of transparent

noncrystalline materials. Many transparent noncrystalline materials that are optically

isotropic when free of stress become optically anisotropic and display characteristics of crystals when stressed [2]. These characteristics exist when loads on the material are maintained but vanish when the loads are removed. This behavior is known as temporary double refraction, or birefringence, and was first observed by Sir David Brewster in 1816.

Brewster found that clear stressed glass exhibited color patterns when examined with polarized light. The significance of Brewster’s discovery was not immediately recognized, and few practical engineering problems were investigated using photoelasticity prior to 1900 [4].

The theory of photoelasticity was well developed by Neumann, Maxwell,

Wertheim, and some other noted physicists [4]. These investigators formulated the concept that the optical retardation producing the color effect in a transparent material under stress is proportional to the difference between the principal stresses that exist in the material. The early work of Coker and Filon at the University of London enabled photoelasticity to develop into a viable technique for qualitative stress analysis [4].

In experimental stress analysis, polariscopes are used to perform full-field stress measurements. The circular polariscope is the most commonly used polariscope. In operation, a stressed photoelastic model is inserted between a pair of linear polarizers and

1 2

quarter wave plates. The model is illuminated through the first linear polarizer and

viewed through the second. The circular polariscope will be further discussed in section

2.4. The color contour, isochromatic fringe patterns observed on the surface of the

stressed model when viewed through a circular polariscope can be related to the shear

stress in the model. The use of photoelasticity with a circular polariscope requires that

the specimen of interest be both birefringent and transparent. Due to these restrictions,

this technique cannot be applied directly to metallic components, but only to transparent,

photoelastic models.

Photoelastic coatings were first developed in the 1950s as a nondestructive

technique of applying the principles of photoelasticity to three-dimensional components.

Traditional photoelastic coatings consist of a birefringent layer on top of a reflective layer. In the most commonly used method, sheets of reflective photoelastic coatings are bound to the surfaces of structural components via an adhesive. Although the coating thickness is usually relatively thin compared to the width of the component, thicker coatings can sometimes provide reinforcement to the specimen. Photoelastic coatings are assumed to be in a state of plane stress when load is applied to the specimen. It is also assumed that the surface strain from the specimen is transferred through the adhesive to the coating (assuming a perfect bond between the coating and the substrate). The original theory of photoelastic coatings neglected the influence of the coating reinforcement on the specimen, any possible mismatch in the Poisson’s ratio between the specimen and the coating, and the out-of-plane strain component. Methods to account for these influences

(which are typically small) were developed in the 1970s by Zandman et al. [14]. A more

3 thorough background of the theory and application of photoelastic coatings is given by

Zandman et al. [14], Redner [13], and Dally and Riley [2].

Although photoelastic coatings are capable of providing accurate full-field strain maps, there are several negative aspects of the technology. Photoelastic coatings are typically applied using prefabricated sheets and must be cut precisely to fit the geometry of the component. This process is very difficult and time consuming for specimens having complex geometries. The boundaries between two different sheets of coating can affect the quality of the data if the coatings are not applied with extreme care (especially if the boundary occurs on a region with a high strain gradient). The use of thicker photoelastic coatings can reinforce the substrate and may require multiple fringe counting and phase unwrapping to extract quantitative data. To avoid these issues thinner coatings can be used, but signal strength and sensitivity are sacrificed. The signal strength of photoelastic coatings is very low on surfaces orientated obliquely to the viewing plane.

1.2 Research History at the University of Florida

In the late 1990s researchers at the University of Florida, in collaboration with the

Visteon Corporation, developed a new full-field strain measurement technique called

Luminescent Brittle Coatings (LBC) [6, 7]. The operational theory behind the method of

LBC is based primarily on that of traditional brittle coatings and the photochemical process of luminescence. For this technique, a coating containing a luminescent dye is sprayed onto the surface of a component. The coating must be cured at specific temperature and humidity conditions. Random networks of microcracks form within the coating during the curing process. The specimen is excited with blue or ultra-violet light and the emission is captured by an optically filtered digital camera. An image is taken at each load level during the loading process. The applied strain field on the component

changes the microcrack morphology, which in turn, alters the amount of luminescent

emission from the coating. The change in relative intensity of the emission is then related

to the summation of the in-plane, principal strains on the surface of the specimen [6, 7].

If one were able to measure the difference between the in-plane principal strains,

that information could be used in accordance with the summation of the strains obtained using the LBC technique to decouple the in-plane principal strains. Also using the concept of luminescence, researchers at the University of Florida developed a new technology called Luminescent Photoelastic Coatings (LPC) that is capable of measuring the difference between the in-plane principal strains. This technology is based on the principals of photoelasticity, photoelastic coatings, and luminescence. By first testing a

component with the LBC method, and then testing it with the LPC method, full-field

maps of the individual in-plane principal strains can be created. The difficult process of

removing the LBC, and the humidity and temperature restrictions required for the LBC

technique, make this method of strain decoupling very troublesome. For this reason the

focus was placed on the new method of LPC. The theory behind Luminescent

Photoelastic Coatings will be discussed in chapter three of this thesis.

1.3 Research Objectives

The first objective of the research discussed in this thesis was to characterize the

oblique incidence response of LPC. The amount of emission captured by the camera

decreases as the surface inclination of the component (relative to the imaging system) becomes more oblique. This leads to less accurate strain measurements. Tests were performed by researchers at the University of Florida to measure the signal response of the LPC emission as the surface inclination of the component increases. This data was

compared to the oblique signal response of the traditional photoelastic coatings and is

presented in chapter four of this thesis.

An apparent strain was observed when making measurements on the oblique

surfaces of components. This apparent strain, which has been termed the optical Fresnel

response (OFR), must be properly accounted for to ensure accurate strain measurements.

Theoretical and experimental work has been conducted to help characterize the OFR.

The fact that this apparent strain exists has enabled researchers to develop a method of

determining the surface inclination of the components as well as the index of refraction of

the luminescent coatings. The ability to measure the surface inclination of a component is essential for accurate strain calculation and will be necessary for future attempts at full- field strain decoupling.

The second objective of this thesis was to characterize several imaging techniques that reduce noise within the data acquisition system. When using the LPC method, relatively long exposure times may be required to provide emission with adequate signal strength. It is important to reduce the imaging noise while still achieving a specified strain resolution. The goal of this study was to determine the optimum testing conditions for noise reduction while maintaining reasonable testing times. The three imaging techniques that were investigated are as follows: increasing the number of analyzer positions, increasing the number of images taken at each analyzer position, and increasing the intensity count of each image. The results of these noise reduction techniques are presented in chapter five of this thesis.

CHAPTER 2 ELEMENTARY PHOTOELASTICITY

2.1 Behavior of Light

The photoelastic effect, developed by Maxwell, can be described by the electromagnetic theory of light propagation [2]. In Maxwell’s theory, electromagnetic radiation is predicted as being a transverse wave motion composed of both electric and magnetic fields. Associated with the wave are oscillating electric and magnetic fields which can be described with electric and magnetic vectors. These vectors are in phase, perpendicular to each other, and are orientated normal to the direction of propagation.

All types of electromagnetic radiation propagate at the same velocity in free space.

Wavelength and frequency are the two parameters used to differentiate between various radiations. These quantities are related to the velocity of light, VL, by the relationship

VL = λf.

The electric field portion of the electromagnetic disturbance is associated with the

production of visible light, which is defined as the radiation that can be detected by the

human eye. The visible portion of the electromagnetic spectrum has a wavelength range

from approximately 400 to 700 nm. Within this range the eye interprets the wavelengths

as different colors. Light from a source that emits a continuous spectrum with nearly

equal energy at every wavelength is interpreted as white light. Light of a single

wavelength is defined as monochromatic light.

6 7

2.2 Polarized Light and Linear Polarizers

Light is considered polarized when the vector components of the light exhibit a preferential direction. There are three cases of polarization that are encountered in photoelasticity. They are referred to as elliptically polarized, linear or plane polarized, and circularly polarized. Elliptically polarized light occurs when the tip of the light vector describes an elliptical helix as the light propagates. Plane polarized light occurs when all components of the light vector lie in a single plane. This plane is known as the plane of polarization. Circularly polarized light occurs when the tip of the light vector describes a circular helix as the light propagates. Both plane and circularly polarized light are special cases of elliptically polarized light.

Plane polarized light is produced using a linear polarizer. Linear or plane polarizers are optical elements that block the components of the light vector vibrating normal to the axis of the polarizer (axis of polarization). When a light vector strikes a linear polarizer, the component of the light vector parallel to the axis of polarization is transmitted and the perpendicular component is blocked or absorbed. The production of elliptically and circularly polarized light requires the series combination of a linear polarizer and a retardation plate.

2.3 Retardation Plates

A retardation plate (also known as a wave plate) is an optical element that has the ability to resolve a light vector into two orthogonal components and to transmit the components at different velocities. Transparent materials exhibiting this property are known as doubly refracting or birefringent. A retardation plate has two principal axes.

As light enters a wave plate, it is split into two components, one of which travels along each of the principal axes of the plate. The light component that propagates along axis 1

has a velocity of v1 and the component traveling along axis 2 has a velocity v2 (where v1>v2). Axis 1 is often referred to as the fast axis and axis 2 the slow axis. Upon emerging from the wave plate, the component propagating along the slow axis is retarded in time relative to the component along the fast axis. This retardation produces a linear phase shift between the two components. The relative linear phase shift, δ, between the two light components traveling along axis 1 and axis 2 can be computed as

δ = h(n1- n2) (2.1) where h is the thickness of the wave plate and the n1 and n2 terms represent the respective indices of refraction along the fast and slow axes (when the material is stressed). The relative angular phase shift, ∆, between the two components as they emerge from the plate can be expressed as

2π 2πh ∆ = δ = (n - n ) (2.2) λλ21 where λ is the wavelength of the light. The relative angular phase shift produced by the retardation plate is dependant on the wavelength of the light, the thickness of the plate, and the difference in the indices of refraction of the plate. When a retardation plate is designed to have δ = λ/4 (∆ = π/2), it is called a quarter-wave plate. Wave plates designed to produce absolute retardations of δ = λ /2 and δ = λ are known as half-wave and full-wave plates respectively.

The type of polarization depends on the angular retardation and the orientation angle of the wave plate, β. The angle β is the angle between the fast axis of the wave plate and the axis of the polarizer. Light passing through a combination of a linear polarizer and a retardation plate always emerges polarized. If β = 0, and ∆ is not restricted in any sense, plane polarized light will emerge from the retardation plate. In

9 this case, the angular retardation has no effect on the polarization of the emerging light vector. Plane polarized light could also be obtained by setting β equal to any multiple of

π/2. Circularly polarized light can be created by orienting a quarter wave plate (∆ = π/2) at an angle β = π/4 with respect to the axis of the polarizer. The light vector described here has a constant magnitude and exhibits an emergence angle that linearly increases with time. When viewed from the positive direction of the light propagation, the tip of the vector traces out a circle as it rotates with a constant angular velocity in the counterclockwise direction. This type of light is known as left circularly polarized light.

Setting β = 3π/4 will produce right circularly polarized light, where the light vector rotates in a clockwise direction when viewed from the positive direction of the light propagation. If a quarter wave plate (∆ = π/2) is chosen and oriented at an angle β not equal to 0° or any multiple of π/4, the emerging light vector is elliptically polarized. As the light propagates, the tip of the light vector traces out the shape of an ellipse [2, 14].

2.4 Polariscopes

2.4.1 Plane Polariscope

The plane polariscope is the simplest optical system used in photoelasticity and consists of a light source and two linear polarizers as shown in Figure 2-1. The arrangement of the plane polariscope creates plane polarized light. The linear polarizer closest to the light source is simply called the polarizer. The second linear polarizer is referred to as the analyzer. The axes of the linear polarizers are always crossed in a plane polariscope to prevent the excitation light from transmitting through the analyzer, creating what is known as a dark field. The photoelastic model tested using a plane polariscope must be birefringent and transparent. The model is always viewed through the analyzer.

When a photoelastic model under an applied load is observed using a plane polariscope, the state of stress in the coating will produce two superimposed fringe patterns on the surface of the model. The first of the two groups is referred to as the isoclinic fringes and can be related to the principal stress directions. These fringe patterns represent lines along which the directions of the principal stresses have a constant angular inclination to an arbitrary reference axis. The second pattern observed are called isochromatic fringes and can be related to contour lines along which the difference in the principal stresses (the maximum in-plane shear stress) of the coating is constant in magnitude. The isoclinic fringes are black and the isochromatic fringes are colored. The color difference, combined with the fact that the isochromatic pattern remains stationary when the polarizer and analyzer are rotated together, allow the observer to distinguish between the two families of fringe patterns [2, 14].

Axis of Polarizer

Light Source Photoelastic Model

Polarizer

Analyzer

Axis of Analyzer

Figure 2-1 Plane polariscope

2.4.2 Circular Polariscope

The circular polariscope uses the same optical system as the plane polariscope, but

with the addition of two quarter wave retardation plates. A schematic of a circular

polariscope is shown in Figure 2-2. This setup is designed to create circularly polarized

light. Depending on the testing situation, the optical elements of a circular polariscope

can be arranged in the four different ways described in Table 2-1. The addition of the

two quarter wave plates eliminates the formation of the isoclinic fringe pattern while

leaving the isochromatic fringe pattern undisturbed. The color contours of the

isochromatic fringes can be related to the maximum in-plane shear stress of the material.

A more thorough explanation of the different types of polariscopes is given by Dally and

Riley [2].

Axis of Polarizer

Fast Axis Light Source Slow Photoelastic Axis 45° 45° Model Polarizer

Slow Quarter Wave Plate Axis Fast Axis 45° 45° Analyzer

Quarter Wave Plate

Axis of Analyzer

Figure 2-2 Circular polariscope

Table 2-1 Different circular polariscope arrangements Quarter Wave Plate Polarizer & Arrangement Field Orientation Analyzer 1 Crossed Crossed Dark 2 Crossed Parallel Light 3 Parallel Crossed Light 4 Parallel Parallel Dark

2.4.3 Reflection Polariscope

The reflection polariscope, shown in Figure 2-3, is nearly identical to the circular polariscope. The slight difference in the arrangement of optics is because the reflection polariscope was designed to measure fringe patterns produced by reflective photoelastic coatings, and not transparent photoelastic models. As discussed earlier, traditional photoelastic coatings contain a birefringent layer overtop of a reflective layer. The second quarter wave plate, analyzer, and camera of a reflection polariscope are arranged to capture the reflection off of the coating (not transmission through a transparent model).

The optical system used to measure strain for the Luminescent Photoelastic Coatings, is very similar to the reflection polariscope, as well as the grey-field polariscope [11], and will be discussed in chapter three.

Axis of Polarizer

Quarter Wave Plate Fast Light Source Axis Slow Axis 45° 45° Polarizer Reflective Photoelastic Coating Slow Fast Axis Axis 45° 45° Axis of Analyzer Camera Quarter Wave Plate

Analyzer

Figure 2-3 Reflection polariscope

2.5 Stress Optic Law

The stress-optic law is a theory developed by Maxwell in the mid-nineteenth

century that relates changes in the indices of refraction of a birefringent material to the

state of stress in the material [2]. Maxwell discovered that for a linearly elastic material,

the changes in the indices of refraction were linearly proportional to the principal

stresses. For the general three dimensional case, the stress-optic law equations are given

n-10 n = c1σ1 + c2(σ2+ σ3)

n-20 n = c1σ2 + c2(σ3+ σ1) (2.3)

n-30 n = c1σ3 + c2(σ1+ σ2)

where the c terms represent the stress optic coefficients and the σ terms represent the

principal stresses, along the respective axes. The n0 term represents the index of

14 refraction of the unstressed material. These equations indicate that the state of stress at a point can be determined by measuring the principal indices of refraction and establishing the direction of the three principal optical axes. Equation (2.3) describes the change in the indices of refraction of a material exhibiting temporary double refraction at a given stress state. Since the method of photoelasticity makes use of the relative changes in the index of refraction, Equation (2.3) is rewritten as

n-21 n = c(σ1- σ2)

n-32 n = c(σ2- σ3) (2.4)

n-13 n = c(σ31- σ ) where c = c2-c1 is referred to as the relative stress-optic coefficient [2]. Since transparent birefringent materials exhibit the same behavior as wave plates when loaded, Equation

(2.2) can be used to relate the relative angular phase shift to the difference between the principal indices of refraction. When polarized light strikes a photoelastic model, the relative retardations along each of the principal-stress directions can be obtained by separately substituting the three components of Equation (2.4) into Equation (2.2). This substitution yields

2πhc ∆ = (σ - σ ) 12 λ 1 2 2πhc ∆ = (σ - σ ) (2.5) 23 λ 2 3 2πhc ∆ = (σ - σ ) 31 λ 3 1

A state of plane stress (σ3 = 0) is typically assumed for photoelasticity applications causing Equation (2.5) to reduce to

2πhc ∆ = (σ - σ ) (2.6) λ 12

For this case, σ1 and σ2 are the in-plane principal stresses, and Equation (2.6) can be expressed in the following form:

Nf σ - σ = σ (2.7) 12 h where

∆ N = (2.8) 2π is the relative retardation in terms of cycles of retardation, otherwise known as the fringe order [2]. The material fringe value, fσ, is a property of the photoelastic material for a given wavelength and thickness and can be expressed as

λ fσ = (2.9) VL

For photoelastic models that exhibit perfectly linear elastic behavior, the difference in principal strains can also be measured. The stress-strain relation for a state of plane stress is given by

1 + ν ε - ε = (σ - σ ) (2.10) 12 E 1 2 where the ε terms represent the principal strains, ν is Poisson’s ratio, and E is the modulus of elasticity. Substituting Equation (2.10) into Equation (2.6) yields

2πhK ∆ = (ε - ε ) (2.11) λ 12 where the coating optical sensitivity, K, is defined as

cE K = (2.12) 1 + ν

The difference in principal strains can then be represented in the following form:

Nf ε - ε = ε (2.13) 12 h where the material fringe value in terms of strain, fε, is given by

1 + υ f = f (2.14) εE σ

The function of a polariscope in photoelasticity is to determine the fringe order, N, at each point in the model. Assuming the three material properties (E, ν, and fσ) of the photoelastic material are known, and by determining the fringe order experimentally, the difference in principal stress (or strain) can be calculated sufficiently [14].

CHAPTER 3 LUMINESCENT PHOTOELASTIC COATINGS

3.1 Introduction

The method of Luminescent Photoelastic Coatings (LPC) was developed by researchers at the University of Florida, in collaboration with the Visteon Corporation, to measure the full-field in-plane strain information on the surface of three-dimensional complex components [6, 8]. This technology is a great tool for determining failure modes and areas of high stress concentration in structural components under static loads.

The LPC method can be used to validate the boundary conditions used in finite element analysis models, hence, creating a better link between analytical modeling and physical testing. This technology is capable of detecting the overstressed regions on a component early in the prototype design cycle which can save industries time and reduce development and production costs.

The method of LPC is based on the principles of reflective photoelastic coatings

(RPC). The main difference between the two methods is that the emission of LPC is based on luminescence, not reflection. The optical system used for the LPC method is similar to the reflection polariscope, but with the addition of optical filters and the removal of the second quarter wave plate.

The LPC method has many advantages over the traditional reflective photoelastic coatings. Aerosol application makes the implementation of LPC much easier and quicker than that of RPC, especially for specimens having complex geometries. Luminescent coatings contain lower residual strain and provide less substrate reinforcement than

17 18 reflective coatings. LPC also provide higher spatial resolution, especially near the edges of the coating. Luminescent coatings are specifically designed to target the quarter fringe value, which allows for simpler data post-processing by eliminating the need for fringe counting and phase unwrapping [1, 8]. Using Equation (2.8), the quarter fringe value would correspond to N<1/4. The emission of luminescent coatings is diffuse which leads to higher relative intensity measurements on oblique surfaces (this topic will be further discussed in chapter four).

3.2 Excitation and Imaging Equipment

Blue LED lamps with a center wavelength of 465 nm are used to excite luminescent coatings. The lamps are coupled with a linear polarizer and a quarter wave plate to create blue, circularly polarized light. The quarter wave plates were specially designed to match the wavelength of the excitation source. The luminescent emission from the coating is captured using 16-bit charged-couple device (CCD) cameras. The

CCD cameras used for the LPC method imaging have excellent spatial resolution (512 x

512). The cameras are each filtered with a bandpass interference filter (550 nm center wavelength), an analyzer, and a standard or zoom lens. A schematic of the strain measurement system is shown in Figure 3-1. The emission filter is used to separate the emission from the excitation (this topic will be discussed in later sections). A rotation stage is used to rotate the analyzer to the desired angular orientation. The LED lamps, the CCD camera, and the rotation stage are automated and controlled using software developed with LabVIEW. Advanced image processing and analysis tools are used in the batch mode analysis to convert intensity images to full-field strain maps [8].

Specimen

Emission Coating Analyzer Filter CCD Camera

To PC

Quarter-Wave Plate Excitation Excitation Polarizer Source Filter

Figure 3-1 Strain measurement system

3.3 Coating Formulation

There are two different luminescent coating formulations, a dual-layer and a single- layer, and each will be discussed in this section. Regardless of the coating used, the component of interest must first be degreased and cleaned thoroughly (and sanded if necessary). A black primer coat is sometimes applied to the component (before the luminescent coating is applied) to prevent light from reflecting off the surface of the specimen. The thickness of the primer is applied in the range of 20-30 µm and the approximate cure time is 20 minutes.

3.3.1 Dual-Layer Coating Formulation

A dual-layer coating consists of a luminescent undercoat and a photoelastic overcoat. The luminescent undercoat is sprayed over the black primer to a target thickness of approximately 60 µm. The luminescent undercoat is formulated with a polymer binder and a dissolved luminescent dye which has the ability to retain the

20 polarization state of the excitation source [6, 8]. The luminescent dye, perylene diimide, absorbs light in the blue range (the excitation source has a center wavelength of 465 nm) of the visible light spectrum and emits light in the red range. The two emission peaks for perylene diimide are at approximately 545 nm and 580 nm in wavelength (the first peak being about 30% greater than the second peak). Via an optical filter, the excitation and emission can be separated. The separation of the excitation and emission, due to the process of luminescence and the use of an optical filter, is one of the main advantages of the LPC method.

After application, the undercoat must be cured with ultra-violet light at standard room temperature and humidity conditions. If the undercoat is not completely cured, the amount of strain transferred from the undercoat to the overcoat will be substantially reduced resulting is measurement error, especially in regions of high strain gradients [8].

An epoxy-based photoelastic overcoat is applied on top of the undercoat within a 200-

250 µm thickness range. Varying the overcoat thickness changes the quarter fringe value.

This enables the user to target the quarter fringe value which eliminates the need for fringe counting and phase unwrapping. The specimen reinforcement provided by the photoelastic coating is considered negligible. The photoelastic material of the overcoat is a methane monomer, and for simplicity, is referred to as BGM. The overcoat is applied in thin, multiple layers to better control thickness and to minimize the induced residual strain. A thixotropic agent is included in the overcoat to enable coating application to vertical surfaces without running [8]. After the application is completed, the epoxy overcoat must be cured with ultra-violet light for several hours.

To correct for the non-uniformity in the thickness of the overcoat layer, an absorption dye (dispersed orange 25) and a second luminescent dye are added to the undercoat. The second luminescent dye, osmium, emits at a higher wavelength than the perylene diimide (the osmium peak emission is at approximately 690 nm). The ratio of emission between the two luminescent dyes is used as a calibration factor to calculate the overcoat thickness. The addition of the absorption dye eliminates the thickness dependence of the luminescent undercoat (this concept will be discussed in the following section). The thickness calibration described above only applies to the photoelastic overcoat.

3.3.2 Single-Layer Coating Formulation

The single-layer LPC differs from the dual-layer coating because the luminescent dye and the absorption dye are both incorporated in the photoelastic layer, eliminating the luminescent undercoat. In the single-layer coating, a ruthenium based absorption dye is used to prevent the excess blue excitation from reaching the surface of the specimen. The absorption dye does not emit, but only acts as an attenuator to limit the penetration depth of the excitation light into the coating. By adjusting the concentration of the absorption dye, the penetration depth of the excitation light can be altered. As long as the thickness of the coating is larger than the penetration depth of the excitation, the coating is considered thickness independent [1]. Using the single-layer coating eliminates the issue of thickness correction for the overcoat and the possibility of shear lag between the two coating layers. The dual-layer coating is susceptible to bonding issues between the undercoat and overcoat if the undercoat is not fully cured. This problem is not encountered with the use of the single-layer coating. The application time for the single- layer coating is considerably less than for the dual-layer coating. However, a higher

22 signal response is measured using the dual-layer coating because the light is allowed to fully penetrate the photoelastic layer. The single-layer coating incorporates the absorption dye within the photoelastic material which inhibits the penetration of the excitation. The reduced amount of light penetrating the photoelastic material decreases the strength of the measured signal response.

3.4 Theory of Operation

When structural components containing an LPC are stressed, the state of stress on the surface of the specimen is transferred to the coating. Luminescent coatings are excited with blue, circular polarized excitation. A luminescent dye within the coating retains the polarized state of the excitation and emits a higher wavelength red-shifted emission (approximately 20% of the excitation light remains polarized due to the luminescent dye). Due to the birefringence effect, the emission emerges from the coating retarded and elliptically polarized. Because the excitation and emission are at different wavelengths, an optical filter can be used to block the excitation reflection off of the surface of the coating while capturing the emission signal. A schematic of the excitation and emission of a luminescent coating is shown in Figure 3-2. A detailed description of how the measured intensity is related to shear strain will be discussed in the remainder of this section.

Figure 3-2 Excitation and emission for a luminescent coating

3.4.1 Dual-Layer Coating

The theory of operation behind the dual-layer LPC is described by the electric field component of a light vector. The electric field component can be represented as

E=i Em sin ωt (3.1)

(refer to the appendix for the full description of the nomenclature). As the light vector propagates, it first penetrates the linear polarizer. The component of the light vector that is transmitted through the polarizer emerges linearly polarized and is described by the equation

E=P φt1Em sin ωt (3.2)

The linearly polarized light next passes through a quarter wave plate. When viewed from the positive direction of the light propagation, the fast axis of the quarter wave plate is oriented at an angle β = 45° relative to the axis of the polarizer, as shown in Figure 3-3.

Polarizer axis

Slow axis Fast axis

Quarter Wave Plate

Figure 3-3 Plane polarized light entering the quarter wave plate

The quarter wave plate retards the light component along the slow axis by ∆ = π/2 relative to the light component along the fast axis. The components along the fast and slow axes, respectfully, can be expressed as

φφt1 t2Em sin ωt E = φφEm sin ωt cos β = (3.3) f t1 t2 2

φφt1 t2Em cos ωt Es = φφt1 t2Em sin (ωt- π 2) sin β = (3.4) 2

The circular polarized light with excitation wavelength λex (blue) now penetrates the coating of the stressed component. The luminescent dye within the coating absorbs the blue excitation and emits light at a larger wavelength λem (red). The coating acts as a birefringent material and splits the light vector into two orthogonal components, one along each of the two principal stress directions. The first principal stress axis, σ1, is oriented at an angle θ with respect to the axis of the polarizer, as shown in Figure 3-4.

Polarizer axis

θ σ1 axis Stressed Component with Coating

σ2 axis

Figure 3-4 Circularly polarized light entering the stressed coating

The principal stress directions are chosen as arbitrary for this derivation. Upon entering the coating, the vector component of the light along the σ2 axis becomes retarded relative to the component along the σ1 axis by angular retardation, ∆in. When exiting the coating along the same path as the incoming light vector, the light vector is further retarded by

∆out. The total angular retardation through the coating is

* ∆ = ∆in + ∆out (3.5) where

2πhKγmax 2πhKγmax ∆in = and ∆out = (3.6) λex λem

The total angular retardation, ∆*, reduces to

2πhKγ ∆=* max (3.7) λ* where the effective wavelength, λ*, is given as

λλ λ* = ex ex (3.8) λex + λem

The equations that describe the vector components emerging from each of the principal stress axes are given as

φφφE cos (ωt - ∆*) φφφE sin ωt E= t1 t2 qm E= t1 t2 qm (3.9) σ1 2 σ2 2

The quantum efficiency, φq, results from the photochemical process of luminescence, not from absorption, and is independent of the intensity of the excitation source or the state of stress of the coating. This term also accounts for the intensity loss due to the nature of the diffuse emission. Due to the coating having a larger index of refraction than air, some of the emission is refracted away from the camera. The light next travels through the analyzer which blocks the light propagating perpendicular to the axis of the analyzer.

A schematic portraying the light vector penetrating the analyzer is shown in Figure 3-5.

The first principal stress axis, σ1, is oriented at an angle θ with respect to the axis of the polarizer. The analyzer orientation angle, α, represents the orientation of the analyzer axis with respect to the polarizer axis.

σ1 axis Polarizer axis α θ α’ Analyzer axis

Analyzer

σ2 axis

Figure 3-5 Elliptically polarized emission entering the analyzer

The portion of the light vector transmitted through the analyzer can be represented as

'' EA = Eσ1 cos α + Eσ2 sin α (3.10) where

α' = α - θ (3.11)

After the substitution of Equation (3.9) and performing trigonometric identities, Equation

(3.10) reduces to

' ' *' *' EA = E [(-sin ωt sin α ) + (cos ωt cos ∆ cos α ) - (sin ωt sin ∆ cos α )] (3.12) where

φφφE E' = t1 t2 qm (3.13) 2

The polarized intensity is proportional to the time integral of the square of the transmitted intensity and can be represented by the following expression:

1 TP I = E2 dt (3.14) P ∫ A TP 0

After performing the integration and using trigonometric identities to simplify the expression, the intensity can be expressed as

 *' IPR = I 1 + sin∆ sin 2α  (3.15) where

E' 2 I = (3.16) R 4

The total intensity IT is equal to the sum of the polarized intensity, IP, and the nonpolarized intensity, INP, and can be described by the equation

*' ITN = I P + IR + IRsin∆ sin 2α (3.17)

The average intensity Iavg can be represented as

Iavg = INP + IR (3.18)

Equation (3.17) can be rearranged and represented in the form

I = 1 + φ sin∆* sin 2α' (3.19) Iavg where the intensity coefficient is defined as

I φ = R (3.20) INP + IR

The optical strain response (OSR) is the measurement used to determine the value of the maximum shear stress, γmax, and is defined as

* IOSR = φ sin∆ (3.21)

The OSR is related to the shear strain via a calibration factor. The governing equation for the dual-layer coating is obtained by substituting Equations (3.11) and (3.21) into

Equation (3.19):

I = 1 + IOSR sin (2α - 2θ) (3.22) Iavg

During experimentation a sequence of images are taken at various analyzer angles. The measured intensity for each pixel of the image is normalized and plotted as a function of the analyzer angle. Using data at a minimum of four analyzer angles (in theory, only two could be used) a sine curve is fit through the data points using Equation (3.22). The amplitude and phase shift of the sine curve represent the OSR and principal strain directions, respectfully. In practice, the initial orientation of the analyzer is set arbitrarily, and the remainder of the analyzer sequence is referenced to the initial position. Regardless of the number of analyzer angles used in the sequence of images,

the angles are always equally spaced (although this is not theoretically required). An

example of the normalized intensity data fit with a sine curve is shown in Figure 3-6.

This curve is greatly exaggerated because, in actuality, the nonpolarized emission makes up approximately 80% of the measured intensity [6, 8].

Iavg

IOSR

Iavg

INP α

Figure 3-6 Intensity response as a function of analyzer angle

3.4.2 Single-Layer Coating

The governing equation for the single-layer coating is different from the dual-layer coating because the luminescent dye is dispersed throughout the photoelastic layer as opposed to underneath the photoelastic layer [1]. Both the relative luminescence and retardation become dependent on the coating thickness. The relative excitation intensity,

Iex, at a given coating depth, y, is modeled using Beer’s Law as shown in Equation (3.23)

-ay I(y)ex = I10 ex,o (3.23)

30 where a is the absorbitivity. Equation (3.24) models the attenuation of the intensity response at a specific depth, y:

I(ex y) -ay ' * = 10 1 + φsin(∆ ) sin(2α - 2θ) (3.24) Iavg where

2πKyγ ∆' * = (3.25) λ*

Integration of Equation (3.24) over the depth, h, yields the general governing equation for a single-layer coating:

** * -ah h a ln10 h γhh -10  sin * +  I 1 - 10-ah γγ h γ = + φ   sin(2α - 2θ) (3.26) I a ln10 * 2 avg a ln10 h  1+ γ  where the photoelastic depth, h*, is given as

λ* h* = (3.27) 2πK

There exists a penetration depth or threshold coating thickness in which the theoretical optical strain response is practically thickness independent [6]. As the thickness of the coating, h, approaches the penetration depth such that the term 10-ah approaches 0,

Equation (3.26) reduces to

γ  I η = 1 + φ sin (2α - 2θ) (3.28) I 2 avg γ 1 +  η where

η = a ln10 h* (3.29)

The nondimensional parameter η is a coating characteristic that relates the absorptivity per unit depth to the photoelastic depth and Iavg is the average intensity over an analyzer rotation of 180° [1].

3.5 Image Acquisition and Data Processing

All images acquired during a test are taken in a darkened testing environment to avoid contamination of the luminescent signal with unrelated background signals. A dark image (acquired before the LED is turned on) is taken to account for any undesired background excitation. The dark image also accounts for the thermally generated dark current noise of the CCD camera. Before load is applied to the specimen, an unloaded base image sequence is taken to account for any residual strain that might exist in the coating. The apparent strain caused by the Fresnel effect for oblique incidence is also captured by the unloaded base image sequence (the Fresnel effect will be discussed in chapter four). At any given load level (including the unloaded base image) images are taken for a sequence of at least four analyzer angles. Before analysis, the dark image is subtracted from the loaded images to eliminate the background excitation and the dark current noise of the CCD camera. Registration points are marked on the surface of the coating prior to running the experiment. During the post-processing, geometric transformations are used to properly align the registration points of each image to account for any shifting of the specimen during the loading process. For a series of analyzer images at a given load state, the raw intensity measurements at each pixel are numerically fitted to a sinusoidal function using a nonlinear Levenberg-Marquardt numerical routine and the respective governing equation. The coatings are designed to only provide

32 information for the first quarter of the sine curve (the quarter fringe value). This eliminates the need for fringe counting and phase unwrapping. At this point, a vector subtraction is performed to eliminate the residual strain information captured in the unloaded base images (the unloaded state is subtracted from the loaded states). The resulting raw intensity images are converted into two separate maps: one displaying the maximum shear strain and the other showing the principal strain directions [6, 8].

CHAPTER 4 OBLIQUE INCIDENCE RESPONSE OF PHOTOELASTIC COATINGS

4.1 Introduction

The objective of this research was to better characterize the oblique incidence response of LPC. Experiments have been performed that show that an apparent strain is observed on the oblique surfaces of unloaded specimens containing luminescent coatings.

One such experiment was performed on an unloaded cylindrical specimen. A side view of the cylinder specimen, with a color contour plot of the OSR, is shown in Figure 4-1.

Investigation into this phenomenon led researchers at the University of Florida to the conclusion that this apparent strain is governed by the Fresnel equations for oblique incidence, and therefore, has been termed the optical Fresnel response (OFR). The theory presented in this chapter will explain how the measured OFR can be used to determine the surface inclination of a specimen or the index of refraction of the luminescent coating. Knowing the surface inclination of the specimen is essential for correcting strain measurements due to the OFR and for future attempts at full-field strain decoupling.

The results from another experiment shows that the LPC method provides a relatively stronger, more diffuse emission signal at oblique viewing angles than the RPC method. The noise in the oblique angle strain calculations is lower for the LPC method because the signal strength on the inclined surfaces is higher. When using the RPC technique a large amount of the excitation light reflects off of the top layer of the coating.

This light never passes through the photoelastic layer and, therefore, does not supply any

33 34 strain information. Optical filters are used to eliminate this problem for LPC method because the excitation and emission have different wavelengths.

Figure 4-1 Apparent strain observed on an unloaded cylindrical specimen

4.2 Principle of Operation

When excitation is obliquely directed towards a specimen with a luminescent coating, one portion of the light vector is transmitted through the coating and the other portion is reflected from the surface. A diagram of a light vector obliquely striking the surface of the coating is displayed in Figure 4-2. The plane of incidence is defined as the plane containing the incident and reflected light vectors. The light vector is decomposed into components along the directions parallel and perpendicular to the plane of incidence.

The transmitted component of the excitation stimulates the luminescent dye, which emits red-shifted light at a higher wavelength. The luminescent dye acts as a diffuse, distributed light source which transmits the red emission out of the coating.

Surface Ei E || Normal

ςr Er E⊥ ςi

air: n = 1

ςt

Figure 4-2 Light vector obliquely striking the coating

The relationship between the angles of incidence and transmission for a light vector impinging on an interface between two media with different indices of refraction can be described by the law of refraction (also referred to as Snell’s law). The law follows from the boundary condition that a wave must be continuous across a boundary, which requires that the phase of the wave be constant on any given plane, resulting in

n1 sinς12 = n2 sinς (4.1) where n1 and n2 are the indices of refraction and ς1 and ς2 are the angles from the normal of the incident and transmitted waves, respectfully. For the case of the LPC method,

Snell’s law can be rewritten as

sinςi = ncsinςt (4.2)

The dielectric medium the light propagates through before striking the coating is air, so the n1 term reduces to unity. The n2 term represents the index of refraction of the coating,

and is replaced by the term nc. The subscripts 1 and 2 on the ς terms represent the angles of incidence and transmission, respectfully, and are replaced by i and t.

The equations derived by Augustin-Jean Fresnel describe the reflection and transmission of the electric field component of a light vector when passing through an interface of media with different indices of refraction [3]. The Fresnel equations give the reflection and transmission coefficients for waves parallel and perpendicular to the plane of incidence. The Fresnel coefficients (r for reflection and t for transmission) are functions of the refraction indices and the angles of incidence and transmission. The four

Fresnel terms are given as

2 sinςt cosςi t || = (4.3) sin(ςii + ςtt) cos(ς - ς )

2 sinςt cosςi t⊥ = (4.4) sin(ςi + ςt )

tan(ςi - ςt ) r || = (4.5) tan(ςi + ςt )

sin(ςi - ςt ) r⊥ = - (4.6) sin(ςi + ςt )

Equation (4.2) can be solved for the transmission angle, ςt, and substituted into the

Fresnel equations, putting the equations in terms of only the inclination angle, ςi, and the coating index of refraction, nc. After the substitution, the Fresnel coefficients can be viewed simply as efficiency terms that describe the amount of transmission and reflection as a function of the surface inclination angle and the coating index of refraction. The operational theory of LPC is only based on the transmitted portion of the light vector, so the Fresnel coefficients for reflections are not considered. A plot of the Fresnel

37 coefficients for an air-glass interface as a function of surface inclination angle is shown in

Figure 4-3. Notice that the transmission coefficients approach zero as the surface inclination approaches 90°.

1.0

0.5 r0 r90 0.0 t0 t90 -0.5

-1.0 0 153045607590 ζ

Figure 4-3 Fresnel coefficients as a function of surface inclination

For the theory of operation of LPC, only the electric field component of the light vector is considered. A top view of the light vector striking the coating at some arbitrary plane of polarization, ρ, is shown in Figure 4-4. The electric field component of the light vector is decomposed into parallel and perpendicular components. The parallel field vector is defined as the component of the light vector propagating in the plane parallel to the plane of incidence. The perpendicular field vector is defined as the component of the light vector propagating in the plane perpendicular to the plane of incidence. In Figure

4-4, the plane of incidence is parallel to the ρ|| axis and is normal to the plane of the page.

ρ||

E|| ρ E

E⊥

ρ ⊥

Figure 4-4 Excitation striking the coating at an arbitrary polarization plane

The plane of incidence is defined as the plane created by the incident light vector and the reflected light vector as shown in Figure 4-5. In the drawing shown in Figure 4-2 the plane of incidence is the plane of the page. For an arbitrary plane of polarization, the parallel and perpendicular field vectors can be expressed as

E = E cos(ρ) sin(ωt) || m (4.7) E=⊥ Em sin(ρ) sin(ωt) where Em and sin(ωt) represent the magnitude and wave motion of the original light vector, respectfully. When excitation passes through the coating, it stimulates the luminescent dye which emits red light out of the coating. Because the emission is diffuse, it is reasonable to assume that part of the emission travels back along the path of excitation, as shown in Figure 4-6. The electric field components of the emission can be expressed as

out ex em E|| = t|| t|| φqE|| (4.8) out ex em E⊥ = t⊥⊥tφqE⊥ where the Fresnel coefficients for transmission, tex and tem, account for both the excitation traveling into the coating, and the emission exiting the coating. The φq term is the

39 quantum efficiency of luminescence and accounts for the intensity loss due to the photochemical process of luminescence and to the nature of the diffuse emission.

Plane of Incidence E||

E ρ ρ

90°- ρ

E⊥

Figure 4-5 Plane of incidence

CCD ς LED i ςt

Figure 4-6 Oblique excitation and emission of a coating

When the light vector travels into the coating it is decomposed into two orthogonal components. As the emission leaves the coating and passes through the analyzer, the two components are recombined along the axis of the analyzer. Figure 4-7 represents the emission traveling through the analyzer. By projecting the field vector

40 components along an arbitrary analyzer axis α, the vector of light transmitted through the analyzer can be expressed as

out out EA = E|| cos(α - ρ||) + E⊥ sin(α - ρ||) (4.9)

Substituting Equations (4.7) and (4.8) into Equation (4.9) yields

ex em ex em EA = Emqφ tt|| || cos(α - ρ|| )cos(ρ) sin(ωt) + t⊥⊥t sin(α - ρ|| )sin(ρ) sin(ωt) (4.10)

ρ||

ou E||

α -ρ||

E α ou A E⊥

ρ ⊥

Figure 4-7 Emission traveling through the analyzer

The intensity is proportional to the magnitude of the electric field component of the light vector, so squaring Equation (4.10), and integrating ρ from 0 to π and t over time yields

 22 I = C cos (α - ρ||) + Tsin (α - ρ||) (4.11) where T is termed the luminescent transmittance ratio and C is a proportionality constant that incorporates the parallel Fresnel coefficients and the quantum efficiency. The term T can be represented as

22 ttex em  T = ⊥⊥  = cos4 (ςς - ) (4.12) ex em  i t tt|| ||  where, from Equation (4.2),

-1 1 ςt = sin  sinςi  (4.13) nc

The luminescent transmittance ratio is a product of the relative change of the perpendicular to parallel excitation intensity and the relative change of the perpendicular to parallel emission intensity due to light passing obliquely between the interface of two mediums having different indices of refraction [9]. Equation (4.11) can be rewritten in the following form

I1 - T = 1 + cos (2α - 2ρ|| ) (4.14) I1avg + T

The amplitude of Equation (4.14), referred to as the optical Fresnel response (OFR), is a function of both nc and ςi and is defined as

1 - T OFR = (4.15) 1 + T

The OFR and the parallel plane of incidence (ρ||) can be determined by exciting the coating with unpolarized illumination and acquiring the emission intensity using an analyzer optic through a range of analyzer angles. The phase shift, 2 ρ||, is relative to the initial analyzer position. For normal incidence (ςi = 0°), T is equal to unity and the OFR is zero. This result is expected, because for the case of normal incidence there is no relative change in the Fresnel transmission coefficients between the parallel and perpendicular components [9].

4.3 Experiments

The objective of the first experiment was to observe the optical Fresnel response on the oblique surfaces of both luminescent and reflective photoelastic coatings. Two cylindrical specimens were used for this test: one contained a .3mm thick single-layer

LPC and the other a .25mm thick prefabricated sheet of RPC. Cylinders were chosen for this experiment because the surface geometry of a cylinder is known and all of the possible inclination angles are present on the surface of a cylinder. A schematic of the system used to measure the OFR is shown in Figure 4-8. With the exception of the excitation optics, this system is identical to the system used for measuring strain.

Unpolarized light was used as the excitation source because polarized excitation would contain the residual strain information present in the coating. Unpolarized light represents all planes of polarization (0 ≤ ρ≤ π) without bias. Both specimens were tested without load and imaged at various analyzer positions.

Figure 4-8 Experimental setup for measuring OFR

The second experiment was conducted to measure the intensity signal strength on the oblique surfaces of both coatings. The two cylinders from the OFR experiment were used for this test. They were also imaged without the application of load. The

43 experimental measurement system was the same as the one used for the OFR experiment, but with the analyzer removed, as represented in Figure 4-9. The purpose of this experiment was to measure the strength of the intensity signal on the oblique surfaces of both coatings, so inclusion of the analyzer was not necessary.

Figure 4-9 Experimental setup for measuring emission strength

4.4 Results and Discussion

4.4.1 Optical Fresnel Response

The results of the OFR experiment are displayed in Figure 4-10. The data shown in the plot were taken along one row of each cylinder. The abscissa indicates the surface inclination angle, ςi, and the ordinate displays the optical Fresnel response. The inclination angle ςi = 0° corresponds to the point where the surface plane of the cylinder is parallel to the image plane of the CCD camera. The ± 90° inclination angles represent the edges of the cylinder with respect to the image plane.

Figure 4-10 OFR with respect to surface inclination

The RPC did not display an OFR until an inclination angle of approximately 70°, whereas the LPC first displayed a measurable response at an angle of 20°. The OFR of the luminescent coating continued to increase as the inclination angle became more oblique. Using Equations (4.12), (4.13) and (4.15) the theoretical OFR can be calculated as a function of the surface inclination. The fitted OFR curve (for n = 1.39) is shown as an overlay in Figure 4-10 and was calculated by minimizing the root-mean-square difference between the experimental and theoretical results for inclination angles in the range of ςi = ±20° to ±70° [7]. For angles less than ±20° the OFR of the LPC is very low and cannot be distinguished from the noise within the imaging system. At angles greater than 70°, the curves diverge due to errors in the OFR measurements due to the lack of emission strength.

The OFR information provided from the LPC is very useful because it can be used with Equations (4.12), (4.13) and (4.15) as a method of measuring the surface inclination or the coating index of refraction. Using the equations mentioned above, and the known surface inclination of the cylinder, the index of refraction of the coating was determined.

Now that the coating index of refraction has been determined, the surface inclination of any coated specimen can be calculated. Knowing the surface inclination of a specimen enables the strain calculations to be corrected for the OFR. The surface inclination will be required for future attempts at full-field strain decoupling [9].

4.4.2 Emission Strength

The objective of the second experiment was to observe the emission strength on the oblique surfaces of the cylinder specimen using both coatings. The data along one row of the cylinders for the LPC and RPC is shown in Figure 4-11. The abscissa indicates the surface inclination angle and the ordinate displays the measured intensity, normalized by the maximum (the maximum intensity occurred at ςi = 0°). The data clearly shows that measured intensity of the RPC sharply drops off as the surface inclination angle becomes more oblique. For the first three, 10° intervals of surface inclination, the relative decrease in measured intensity is approximately 50%. At an inclination angle of 30° the measured intensity of the RPC is only 11% of the maximum value. At an angle of 60° the measured intensity dropped to 4% of the maximum. At normal incidence the camera measures the reflection containing the strain information from the lower diffuse surface as well as the specular reflection from the upper surface. As the orientation angle becomes more oblique the reflected excitation from the upper surface of the coating

(specular reflection) is directed away from the camera resulting in the sharp decrease of the measured response. Specular reflection increases the measured intensity, but does not provide any useful information for strain measurement because it never passes through the photoelastic coating.

The LPC provided a more uniform response than the RPC. The signal strength at an angle of 30° is 88% of the maximum value. At 60°, the relative intensity is 49% of the

46 maximum. Because the excitation and emission of LPC are of different wavelengths, specular reflections can be filtered without reducing the emission signal. Based on the photonic shot noise of the CCD camera, a decrease in signal intensity at an inclination angle of 60° will decrease the signal-to-noise ratio by a factor of 5 for RPC, but only by a factor of 1.4 for LPC [9]. This is one of the more considerable advantages that LPC has over RPC when imaging at oblique angles. The time spent acquiring data is the only disadvantage the LPC method has compared to the RPC method because the exposure time for the CCD camera must be greater when using luminescent coatings.

Figure 4-11 Normalized intensity with respect to surface inclination

CHAPTER 5 NOISE REDUCTION TECHNIQUES FOR THE LPC METHOD

5.1 Introduction

There are several sources of noise involved with CCD measuring systems that cause errors in data analysis. Dark current noise is governed by the amount of heat within the CCD camera and can be minimized by either thermoelectric or cryogenic cooling [10]. The dark current noise can be measured and subtracted from the data.

Preamplifier noise, or read noise, is generated by the on-chip output amplifier of the CCD and can be greatly reduced by optimizing the operating conditions of the CCD camera’s electronics [10, 12]. Photonic shot noise is a fundamental property of the quantum nature of light and is unavoidable in imaging systems [12]. With the exception of the dark current noise, the user cannot account for these sources of noise produced by the CCD camera. However, the user can reduce the intensity measurement noise by employing certain imaging techniques. The content of this chapter will focus on the imaging techniques that can be used to reduce the noise in strain measurements using the LPC method. These methods are as follows: increasing the number of images taken at each analyzer position, increasing the number of analyzer positions, and increasing the intensity level of each image.

Statistical theory states that the arithmetic mean value can be taken as the best estimate of the true value of a set of experimental measurements [5]. This can be achieved by acquiring several images at each analyzer position and using the mean image for the data processing. In theory, as the number of images taken at each analyzer

47 48 position increases, the standard deviation of the measured intensity in the mean image should decrease, which would lead to more accurate strain calculations.

As previously discussed in chapter three, the OSR is calculated by fitting a sine curve through a plot of intensity data points for a sequence of analyzer angles. Increasing the number of analyzer positions during an experiment will enable a better fit of the sine curve to the data and will reduce noise. This will also lead to more accurate strain calculations.

The potential well of a CCD measures the electric charge produced by photons and converts that charge into an intensity value. As the potential well capacity of the CCD is approached, the signal-to-noise ratio of the intensity data increases proportionally to the square root of the intensity value. Increasing the exposure time of the CCD for the purpose of capturing higher intensity images will be the third noise reduction technique discussed in this chapter.

5.2 Experimental Technique and Noise Determination

The techniques discussed in the previous section were tested using a rectangular, aluminum bar loaded in a four-point bending apparatus. A single-layer coating was applied to the specimen and the standard OSR measurement system was used. A second

LED was added to the standard setup for the purpose of reducing the exposure time of the

CCD camera. The four-point bending apparatus produces a uniform state of strain on the bar specimen, so in theory, the surface strain on the component should be the same everywhere. The screw-driven, loading fixture, with the coated aluminum specimen, is shown in Figure 5-1.

Figure 5-1 Four-point bending apparatus

Although the theoretical surface strain on the specimen is constant, the calculated

OSR values on the coated region of the specimen will not be identical due to the noise within the imaging system and coating imperfections. The standard deviation of the OSR is related to the measurement error and is used as the parameter for determining noise.

Lower values of the standard deviation will indicate less measurement noise, hence, more accurate strain calculations. A visualization of noise level is shown in Figure

5-2. Each plot shows the OSR calculation for a row of data along the length of a specimen. It is clear that the standard deviation of the OSR is higher in the plot on the left than in the plot on the right. The data in the left plot obviously contains more noise.

0.125 0.125 0.120 0.120 0.115 0.115 0.110 0.110 R 0.105 R 0.105 OS 0.100 OS 0.100 0.095 0.095 0.090 0.090 0.085 0.085 0204060 0204060 relative length (pixels) relative length (pixels)

Figure 5-2 Plot of OSR for experiments with different imaging conditions

5.3 Testing Conditions

The first noise reduction technique investigated measured the effect of increasing the number of images at each analyzer position. The second technique measured the effect of increasing the number of analyzer positions. To simplify the description of these experiments, the following notation will be used: ‘A x B’ where ‘A’ represents the number of images taken at each analyzer position and ‘B’ represents the number of analyzer positions. For all of these tests, the intensity level of each image was approximately 55,000 counts (the potential well capacity of the CCD camera is 65,535 counts). The testing conditions for these experiments are given in Table 5-1.

Table 5-1 Testing conditions Number of Analyzer Positions Number of Images per 4 8 16 Analyzer Position 1 1 x 4 1 x 8 1 x 16 2 2 x 4 2 x 8 2 x 16 4 4 x 4 4 x 8 4 x 16 8 8 x 4 8 x 8 8 x 16 16 16 x 4 16 x 8 16 x 16

The third technique measured the effect of increasing the CCD intensity count level for each image. Changing the intensity level of the images was accomplished by adjusting the exposure time of the CCD camera. The targeted intensity count levels were 15,000,

25,000, 35,000, and 45,000 out of a possible value of 65,535 counts. This experiment was conducted using four, eight, and sixteen analyzer positions. The number of images taken at each analyzer position was held constant (at one image) for all three intensity count experiments.

5.4 Preliminary Results and Discussion

The first round of testing yielded unexpected trends in the data. The results of the first two noise reduction experiments are both shown Figure 5-3. The standard deviation

(by percentage of the measured OSR value) is represented on the y-axis of the plot. It was expected that the standard deviation would decrease as both the number of images and the number of positions increased, but not all of the data points followed this trend.

The results from the intensity count experiment are shown in Figure 5-4. In theory, the standard deviation should decrease as the intensity count increases, but the results of this test did not follow the expected trend either.

4.25

) 3.75 % (

n 3.25 o i

at 4 angles 2.75

evi 8 angles

D 2.25 d 16 angles r a

d 1.75 an t

S 1.25 0.75 0 5 10 15 20 Number of Images at each Analyzer Position

Figure 5-3 OSR standard deviation as a function of the number of images

4.5

) 4 % (

n 3.5 o i

at 4 angles 3

evi 8 angles

D 2.5 d 16 angles r a

d 2 an t 1.5 S 1 10000 20000 30000 40000 50000 Intensity Count

Figure 5-4 OSR standard deviation as a function of the intensity count

It was obvious from the preliminary results that there was some factor affecting several of the OSR calculations. It was assumed that the source of error was coming from the imaging system and not from a coding problem within the analysis software. To be certain, the analysis program was first examined thoroughly. It was determined that

53 the problem was definitely not due to the software, so investigation next moved to the imaging system. For the first two experiments, it was observed that the data points not fitting the expected trend were all from tests with long data acquisition times (on the order of 90 minutes or more). This raised the possibility that emission from the coating may have decreased during the course of the longer experiments. A simple test was conducted to measure the emission intensity of the luminescent coating as a function of time. It is essential that the excitation and luminescent emission remain constant during the entire sequence of analyzer positions to guarantee accurate strain calculations. For the emission intensity test, the analyzer was removed (to allow more emission to reach the CCD which reduces the testing time) and the specimen was tested without the application of load. Similar to the previous tests described above, the same two circular polarized, blue LEDs were used for this experiment.

The results of this emission test are shown in Figure 5-5. The ordinate represents the normalized intensity and the abscissa represents the time in minutes. It is clear that the emission intensity decreased as a function of time. After 90 minutes the emission dropped by approximately 2%.

y 0.995 it ns e t 0.99 In d e iz l 0.985 a m r o

N 0.98

0.975 0 20 40 60 80 100 Time (min)

Figure 5-5 Intensity drift of the coating emission over a 90 minute time period

This discovery raised several questions. It had to be determined whether this intensity drop was due to a decrease in the emission of the luminescent dye, or due to an intensity drift of the excitation source. The idea that the LED intensity was decreasing over time was first investigated. Both LEDs were tested separately. For this test, the

LED was directed towards the CCD camera. A piece of diffuse glass was placed in between the LED and the CCD to reduce the intensity. The analyzer was not required for the intensity test, so the optic was removed from the setup. Figure 5-6 represents a schematic of this optical measurement system.

Excitation Excitation Source Filter Diffuse Glass Emission Filter CCD Camera Polarizer Quarter-Wave Plate To PC

Figure 5-6 Intensity drift measurement system

Images were taken over the course of a 90 minute time period. The results for the

LED lamps are shown in Figure 5-7. The intensity of both LEDs decreased as a function of time. LED 1 dropped a little more than 1% during the 90 minute time period whereas

LED 2 dropped approximately 4.2%. The intensity drop experienced by LED 2 over the

90 minute time frame is quite significant and does not represent typical behavior of an

LED. A third LED was tested to check for a time dependent intensity drop (this LED was not used during any of the experiments discussed in this chapter). The intensity drop of LED 3 is also shown in Figure 5-7. LED 3 displayed the largest decrease of the three

LEDs, dropping by about 5.7% in 90 minutes.

0.99 y it ns

e 0.98 LED 1 Int

d 0.97 LED 2 e

liz LED 3 a 0.96 m r o N 0.95

0.94 0 20 40 60 80 100 Time (min)

Figure 5-7 Intensity drift for the LEDs over a 90 minute time period

In theory, LED drift should only affect the accuracy of the strain calculations and not the standard deviation. The OSR was calculated from a rectangular area located on the central region of the coating. For the first round of experiments, LED 1 and LED 2 were used to excite the coating. Due to the nature of the experimental setup, the right side of the rectangular region of interest was illuminated more by LED1 than LED 2.

The opposite holds true for the left side of the region. The fact that both LEDs experience different amounts of intensity drift created the discrepancy in the standard deviation values of the OSR calculations. If only one LED was used, the strain calculations would have been subject to error due to the amount of intensity drift, but the standard deviation of the calculations should have still fit the expected trend. To meet the objective of the noise reduction investigation, it was determined that only one LED should be used to complete the testing.

5.5 Results and Discussion

LED 1 was chosen for the re-testing of the noise reduction experiments because it experienced the smallest amount of intensity drift. The LED lamp was turned on approximately 20 minutes prior to testing to allow for the intensity drift to stabilize (the largest amount of drift occurs when the LED is first turned on). In the interest of time, the testing conditions involving sixteen images per analyzer position were dropped from the experiment. The intensity count level for all of the images taken during these tests was also reduced from 55,000 to 35,000 counts to reduce acquisition time. The results of these tests are shown in Figure 5-8. After discovering the problems caused by using two

LEDs simultaneously and the LED intensity drift, the results were as originally presumed. Although, the advantage of using sixteen analyzer positions over eight analyzer positions was less than expected.

7 4 angles 6 ) 8 angles %

( 16 angles

n 5 o i

at 4 evi

D 3 d r a

d 2 an t

S 1

0 0246810 Number of Images at Each Analyzer Position

Figure 5-8 OSR standard deviation as a function of the number of images

For these experiments, the required amount of loading using the four-point bending apparatus was calculated to produce a normal strain value of 1900 µε on the surface of the component. For an aluminum specimen (ν = .3), this normal strain value corresponds to a shear strain value of 2470 µε. The calculated OSR was converted to microstrain using the appropriate calibration factor. A plot of standard deviation (in terms of shear microstrain) versus the total number of images is shown in Figure 5-9.

The total number of images is defined as the product of the number of images per analyzer position and the number of analyzer positions. The total number of images is directly related to the total data acquisition time via the CCD exposure time. Figure 5-9 can be used to determine which testing configuration will provide the lowest noise level for a given data acquisition time. Table 5-2 represents the amount of noise reduction for each testing condition relative to the ‘1x4’ base condition.

4 analyzer positions 8 analyzer positions 16 analyzer positions 160

) 140 n n o i ai

r 120 at t s o evi 100 cr D i d r

m 80 a d

ear 60 an h t s ( S 40 20 0 8 16 24 32 40 48 56 64 72 80 88 96 104 112 120 128 Total Number of Images

Figure 5-9 OSR standard deviation in microstrain versus the total number of images

Table 5-2 Percent noise reduction Number of Analyzer Positions Number of Images per 4 8 16 Analyzer Position 1 - 27.84 30.03 2 8.29 34.01 36.51 4 14.27 39.85 51.88 8 56.62 70.66 73.51

In the interest of time, the intensity count experiment was retested only at the ‘1x8’ condition. The intensity levels used for this experiment were 5,000, 15,000, 25,000,

35,000, 45,000 and 55,000 counts. The CCD exposure time was adjusted after each test to ensure the emission intensity of the coating would reach each desired count level. The results from this experiment are shown in Figure 5-10. The standard deviation of the

OSR decreased as the intensity count increased, but after the intensity level of 35,000 the continued noise decrease became quite small. The noise reduction for each intensity count level, compared to the 5,000 count level, is displayed in Table 5-3.

9 8 ) %

( 7

ion 6 t a i

v 5 e

D 4 d r 3 da n

a 2 t S 1 0 0 10000 20000 30000 40000 50000 60000 Intensity Count

Figure 5-10 OSR standard deviation as a function of the intensity count

Table 5-3 Noise reduction for each intensity count level Intensity Count Level 5,000 15,000 25,000 35,000 45,000 55,000 Noise Reduction (%) - 21.68 35.04 59.35 62.44 64.04

5.6 Conclusions

The results in this chapter have shown that increasing the number of images, analyzer positions, and intensity count will reduce the noise in the measurement system.

The intensity count for all tests should be at a level no less than 50% of the potential well capacity of the CCD. For this CCD, that would correspond to an approximate intensity level of 35,000 counts. The tradeoff for the noise reduction is the amount of time spent acquiring data. The nature of some experiments may not require long exposure times, so in those cases it would be quite beneficial to increase all of the factors previously mentioned. The gain in noise reduction obtained from increasing the number of analyzer positions from eight to sixteen does not appear to be worth the additional acquisition time, but increasing the number of images per analyzer position from four to eight may be. The data shown in Figure 5-9 should be used as a guide to determine the optimum testing configurations for a given data acquisition time. Ultimately, the testing configuration used would depend on the time-dependent nature of the experiment and the discretion of the user.

CHAPTER 6 FINAL RESEARCH CONCLUSIONS

6.1 Oblique Incidence Response of Photoelastic Coatings

6.1.1 Optical Fresnel Response

The OFR observed on the oblique surfaces of specimens containing luminescent coatings is beneficial because it can be used as a method of measuring the surface inclination or the coating index of refraction. For a component with a known geometry, the method described in chapter four can be used to determine the index of refraction of the luminescent coating. Measuring the OFR, and knowing the coating index of refraction, enables researcher at the University of Florida to calculate the surface inclination of any structural component. The surface inclination information is crucial for correcting strain measurements for the OFR. In the future, the surface inclination of a component will be used in an attempt to decouple the principal strains.

6.1.2 Emission Strength on Oblique Surfaces

It has been shown through experimentation that the LPC method provides a relatively stronger, more diffuse emission signal at oblique viewing angles than the RPC method. The signal-to-noise ratio of the intensity measurements made on oblique surfaces is higher for luminescent coatings than for reflective coatings because the relative signal strength using the LPC method is higher. The higher signal-to-noise ratio enables more accurate strain calculations to be made using the LPC technique. The measured intensity using the LPC method also contains more strain information than the traditional RPC method because the reflection of the excitation is blocked using an

61 62 optical filter. The only trade off for using luminescent coatings is the longer CCD exposure times. For these reasons it has been concluded that the LPC method is better suited for oblique angle strain calculations than the RPC method.

6.2 Noise Reduction Techniques

The three noise reduction techniques investigated were successful in lowering the

OSR calculations, but all of these techniques require additional testing time. The intensity count for all testing images should be at least 50% of the potential well capacity of the CCD camera. Depending on the CCD exposure time and the nature of the experiment, different noise reduction approaches may be taken. The optimum testing configuration, with time under consideration, can be determined using the data in Figure

5-9. It appears that the noise reduction gained from increasing the number of analyzer positions from eight to sixteen is not worth the additional testing time (the acquisition time would be doubled). The testing configuration used during an experiment would depend on the required exposure time of the CCD camera and the discretion of the user.

Another area of research in regard to noise reduction is the effect of applying a coat of black primer under the luminescent coating. Experiments have shown that the measured intensity is affected by the surface features of the substrate. If the substrate is not primed with a black coating, the red emission may reflect off of the surface of the component and be captured by the CCD camera. The effects of this reflection have yet to be investigated.

APPENDIX NOMENCLATURE a absorptivity of the absorption dye

C coefficient that incorporates the parallel Fresnel coefficients and the quantum efficiency c relative stress optic coefficient

E modulus of elasticity

EA component of the light vector transmitted through the axis of the analyzer

Ef electric field component of the light vector along the fast axis of the quarter wave plate

Ei electric field component of the incident light vector

Em magnitude of the incident light vector

EP component of the light vector transmitted through the axis of the polarizer

Er component of the reflected light vector

Es component of the light vector along the slow axis of the quarter wave plate

E’ light vector coefficient defined in Equation (3.13)

E|| component of light vector propagating parallel to the plane of incidence entering the coating

E⊥ component of light vector propagating perpendicular to the plane of incidence entering the coating

out E|| component of light vector propagating parallel to the plane of incidence exiting the coating

out E⊥ component of light vector propagating perpendicular to the plane of incidence exiting the coating

Eσ1 component of the light vector along the first principal stress direction of the stressed coating

63 64

Eσ2 component of the light vector along the second first principal stress direction of the stressed coating f frequency of light fσ material fringe property in terms of stress fε material fringe property in terms of strain h coating thickness h* photoelastic depth

I intensity of the light vector

Iavg average intensity of the light

Iex(y) relative excitation intensity as a function of coating penetration depth

Iex,o initial relative excitation intensity

IOSR intensity of the optical strain response

IP intensity of the polarized light

INP intensity of the non-polarized light

IR intensity coefficient defined in Equation (3.16)

IT total intensity of the light

K coating optical sensitivity

LBC luminescent brittle coating

LPC luminescent photoelastic coating

N fringe order n index of refraction n0 index of refraction of the unstressed material n1 index of refraction of the stressed material along the first principal stress direction n2 index of refraction of the stressed material along the second principal stress direction

n3 index of refraction of the stressed material along the third principal stress direction nc index of refraction of the coating

OFR optical Fresnel response

OSR optical strain response

RPC reflective photoelastic coating r|| parallel reflection Fresnel coefficient r⊥ perpendicular reflection Fresnel coefficient

T luminescence transmittance ratio t time

TP period of the light vector t|| parallel transmission Fresnel coefficient t⊥ perpendicular transmission Fresnel coefficient

em t || parallel transmission Fresnel coefficient for emission

ex t || parallel transmission Fresnel coefficient for excitation

em t⊥ perpendicular transmission Fresnel coefficient for emission

ex t⊥ perpendicular transmission Fresnel coefficient for excitation v1 velocity of light along the fast axis of the quarter wave plate v2 velocity of light along the slow axis of the quarter wave plate

VL velocity of light y penetration depth of the light into the coating

α analyzer axis position relative to the axis of the polarizer

α’ angle between the axis of the analyzer and the principal stress direction

β angle between the fast axis of the quarter wave plate and the axis of the polarizer

γ shear strain

γmax maximum shear strain

∆ relative angular retardation

∆12 relative angular retardation between the light vectors traveling along the first and second principal stress axes of a photoelastic model

∆23 relative angular retardation between the light vectors traveling along the second and third principal stress axes of a photoelastic model

∆31 relative angular retardation between the light vectors traveling along the third and first principal stress axes of a photoelastic model

∆in angular retardation of the light entering the coating

∆out angular retardation of the light exiting the coating

∆* total angular retardation of the light entering and exiting the coating

∆’* coefficient defined in Equation (3.25)

δ relative linear retardation

ε1 first principal strain

ε2 second principal strain

η nondimensional parameter that relates the absorptivity per unit depth to the photoelastic depth

θ principal stress direction with respect to the axis of the polarizer

λ wavelength of light

λex excitation wavelength

λem emission wavelength

λ* effective wavelength

ν Poisson’s ratio

ρ plane of polarization position

ρ|| parallel polarization plane orientation

ρ⊥ perpendicular polarization plane orientation

ς surface inclination angle with respect to the incident light vector

ς1 inclination angle between the surface normal and the incident light vector (Snell’s law notation)

ς2 inclination angle between the surface normal and the transmitted incident light vector (Snell’s law notation)

ςi inclination angle between the surface normal and the incident light vector

ςr inclination angle between the surface normal and the reflected light vector

ςt inclination angle between the surface normal and the transmitted light vector

σ1 first principal stress or the first principal stress axis

σ2 second principal stress or the second principal stress axis

σ3 third principal stress

φ intensity efficiency term defined in Equation (3.20)

φt1 efficiency term for the light passing through the polarizer

φt2 efficiency term for the light passing through the quarter wave plate

φq quantum efficiency of luminescence

ω circular frequency of the light vector

LIST OF REFERENCES

1. Chen, L., Hubner, J. P., Liu, Y., Schanze, K., Ifju, P., Nicolosi, J. and El-Ratal, W., “Characterization of a New Luminescent Photoelastic Coating,” SEM X International Congress & Exposition on Experimental and Applied Mechanics, Paper #192, June 2004.

2. Dally, J. W. and Riley, W. F., “Experimental Stress Analysis,” 3rd ed., McGraw- Hill Inc. New York, NY, 1991.

3. Hecht, E., “Optics,” 2nd ed., Addison-Wesley Publishing Company, Reading, MA, 1989.

4. Hetenyi, M., “Handbook of Experimental Stress Analysis,” John Wiley & Sons Inc., New York, NY, 1950.

5. Holman, J. P., “Experimental Methods for Engineers,” 5th ed., McGraw-Hill Book Company, New York, NY, 1989.

6. Hubner, J. P., Ifju, P. G., Schanze, K. S., Jaing, S. Liu, Y., and El-Ratal, W., “Luminescent Strain Sensitive Coatings,” AIAA Journal, in press, AIAA Paper # 2003-1437, 2004.

7. Hubner, J. P., Ifju, P. G., Schanze, K. S., Jenkins, D. A., Carroll, B. F., Wang, Y., He, P., Brennan, A., and El-Ratal, W., “Full-field Strain Measurement using a Luminescent Coating,” Experimental Mechanics, vol. 43, no. 1, March 2003.

8. Hubner, J. P., Ifju, P. G., Schanze, K. S., Liu, Y., Chen, L., and El-Ratal, W., “Luminescent Photoelastic Coatings,” Proceedings of the 2003 SEM Annual Conference and Exposition, Paper #263, June 2003.

9. Hubner, J. P., Nicolosi, J., Chen, L., Ifju, P., and El-Ratal, W., “Oblique Incidence Response of Photoelastic Coatings,” SEM X International Congress & Exposition on Experimental and Applied Mechanics, Paper #193, June 2004.

10. LaBelle, R. D., and Garvey, S. D., “Introduction to High Performance CCD Cameras,” Photometrics Ltd., Tucson, AZ.

11. Lesniak, J. R., and Zickel, M. J., “Applications of Automated Grey-Field Polariscope,” Proceedings of Society for Experimental Mechanics Spring Conference, June 1998.

68 69

12. Photometrics, Inc., “The CCD Imager,” Author, Tucson, AZ, 1999.

13. Redner, A. S., “Photoelastic Coatings,” Experimental Mechanics, Vol. 20, No. 11, 1980, pp. 403-408.

14. Zandman, F., Redner, S., and Dally, J. W., “Photoelastic Coatings,” Iowa State University Press, Ames, IA, 1989.

BIOGRAPHICAL SKETCH

John C. Nicolosi Jr. was born on January 30 1980, in Dover, NJ. He lived in

Sellersville, PA, and Hackensack, NJ, before moving to Norwich, NY in the fall of 1989.

In June of 1998 he graduated with honors from Norwich High School. He began his undergraduate education in the fall of 1998 at Binghamton University. In May of 2002 he graduated with a Bachelor of Science in Mechanical Engineering. John continued his education by pursuing a Master of Science degree in the Mechanical and Aerospace

Engineering Department. He worked as a graduate research assistant in the

Experimental Stress Analysis Laboratory under the advisement of Dr. Peter Ifju and Dr. Paul Hubner. He conducted research to further develop the new full-field strain measuring technology named Luminescent Photoelastic Coatings.