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Evaluation of Fiber Optic Bragg Grating Sensors in Monitoring the Integrity of Structures Repaired with Bonded Patches

by

Jerry J.S. Chwang

A thesis submitted in conformity with the requirements for the degree of Master of Applied Science as detertnined by the

University of Toronto Institute for Aerospace Studies

O Copyright by Jeny I.S. Chwang January 1999

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Acknowledgments

1 would like to thank first and foremost, Dr. Raymond Measures for providing the opportunity to pursue a Master's degree and for his many ideas, and support which allowed me to complete this thesis. 1 would also like to thank Photonics Research Ontario (formerly the Ontario Laser and Lightwave Research Center) for providing financial support, without which this endeavour could not have been completed.

1 also would like to thank my fellow students in the Fiber Optic Smart Structures Lab at the University of Toronto Institute for Aerospace Studies for answering al1 my questions and providing comic relief dunng the difficult times. Lastly, 1 wish to thank my parents and Julia for always k i n g there when 1 needed them.

Evaluation of Fiber Optic Bragg Grating Sensors in Monitoring the

Integrity of Structures Repaired with Bonded Patches

by Jerry J.S. Chwang

Su bmi t ted to the Aerospace Engineering on 25 January 1999, in partial MUment of the

requirements for the degree of Master of Appled Science

Abstract

Bragg grating sensors are evaluated as tools to determine crack growth in structures repaired with bonded patches, Linear elastic Facture rnechanics theory, as weU as the 1-dimensional theory of bonded joints are incorporated into a computer model which simulates crack growth in t hin plates loaded in plane stress. The model suggests an elegant method of tracking crack length regardles of load magnitude based on the location of the peak strain measured dong an ctvis perpendic-dar to the crack. This strain can be theoreticdy measured using short Bragg grating sensors positioned perpendiculariy 2 mm away fiom a 4 cm long crack in a 7 c m wide, thin aluminum plate subject to loads of 2000 to 5000 N.

Research Head: Raymond M. Measures Tit le: Director, Fiber Optics Smart Structures Laboratory

Contents

1 Introduction 9

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Patches 9

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 .4 dhesive Bond Inspection 10

. . . . . . . . . . . . . . . . . . . . . . 1.3 Fiber Optic Sensors for Smart Structures 11

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Fiber Optic Sensors 13

. . . . . . . . . . . . . . 1.4.1 Fiber optic sensing for composite smart structures 13

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.2 Debond Detection 13

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.3 Crack Detection 14

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.4 Interferometers 14

. . . . . . . . . . . . . . . . . . . . . . 1.4.5 Distributed Bragg Grating Sensors 15

2 Theoretical Analyses 17

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Interferometric Sensors 17

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Bragg Grating Semors 18

. . . . . . . . . . . . . 2.2.1 The Bragg Condition of Constructive Interference 18

. . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.2 The Reflection Spectrurn 19

. . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.3 Grating Response to Strain 20

. . . . . . . . . . . . . . . . . . . . . . 2.3 The Nature of Stresses in Adhesive Joints 23

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.1 Classical Theones 23

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.2 Recent Theories 27

. . . . . . . . . . . . . . . . 2.3.3 Elast-Plastic -4nalysis of a Single Lap Joint 36

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.4 Service Life 41

. . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Linear Elastic Fracture Mechanics 41

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.1 Brittle Fkacture 42

. . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.2 Fatigue Design Philosophies 42

. . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.3 Stresses Near Crack Tips 46

3 Computer Mode1 49

. . . . . . . . . . . . . . . . . . . . . . . . 3.1 Case 1: Unpatched Plate/Static Load 50

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.1 Assumptions 50

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.2 Case 1 Input 50

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.3 Case 1 Results 52

3.1.4 Case 1 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

. . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Case 2: Patched Plate/Static Load 60

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1 Assumptions 60

. . . . . . . . . . . . . . . . . . . . . . . . . . . 3-22 Case2Input andResults 61

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Case 3: Dynamic Load 61

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.1 Assumptions 61

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.2 Case 3 Input 61

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.3 Case 3 Results 64

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.4 Case 3 Analysis 64

4 Experimental Results 67

4.1 Unpatched Single-Edge Cracked Specimen . . . . . . . . . . . . . . . . . . . . . . 67

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.1 Bragg Grating Sensors 67

4.1.2 Static Load Strain Profiles at the Crack Tip . . . . . . . . . . . . . . . . . 71

4.1.3 Dynamic Load and Crack Growth . . . . . . . . . . . . . . . . . . . . . . 71

5 Conclusions 73

A Computer Program 77

A . 1 Graphical User Interface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

A.2 Computer Code . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

B Raw Data Output 93

. . . . . . . . . . . . . . . . . . . . . B . 1 Crack length versus Norrnalised peak strain 94

B.2 Crack length versus Peak strain location dong y- avis . . . . . . . . . . . . . . . 97

List of Figures

1-1 Fatigue strength of riveted versus adhesively bonded joints . . . . . . . . . . . . . 10

- 2 Typical defects in adhesively bonded joints . . . . . . . . . . . . . . . . . . . . . 11

1-3 Fiber Optic Michelson Interferorneter Configuration on Patch . . . . . . . . . . . 15

1-4 Fiber Optic Fabry-Perot Interferorneter Configuration on Patch . . . . . . . . . . 16

2-1 Cross-section of a Bragg grating having uniforrn amplitude . index modulation

and period . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2-2 Theoretical plot of reflectivity as a Function of wavelength for a uniforrn Bragg

grating (L = Icm. AB = 1550nm. Rm, = 80% . . . . . . . . . . . . . . . . . . . 21

2-3 Some cornmon engineering adhesive joints . . . . . . . . . . . . . . . . . . . . . . 24

2-4 Exaggerated deformations in loaded single-lap joint: (a) with rigid adherends:

(b) with elastic adherends . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

. . . . . . 2-5 Bending moments induced in the outer adherends of a double-lap joint 26

2-6 Iilustrating a way of representing the Goland and Reissner bending moment

factor geometrjcally . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

2-7 Stress distribution in a single lap joint . . . . . . . . . . . . . . . . . . . . . . . . 28

2-8 Tensile stress in adherend and shear stress in adhesive (each per unit load) plotted

against x for a 1 in x 1 in single-lap joint . . . . . . . . . . . . . . . . . . . . . . 29

3-9 Deformation in single-lap joint due to lateral straining . . . . . . . . . . . . . . . 30

2-10 Transverse direct stress per unit load (uiZ/P) in the adherend plotted against z

for a 1 in x 1 in single-lap joint at different positions dong the length of the joint:

- approximate analytical solution (exact at x = 0. 1 in ); 0 finite-difference

solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

2-11 Diagrammatic lap joints to show adhesive layers with (a) square edge; (b) spew

fiflet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

2-12 (a) Comparison between calculated and experimental displacements of the sili-

cone rubber mode1 . (The black crosses are the finite-element predictions of the

intersections of the grid Lines of the model.) (b) Principal stress pattern for

. . . . . . . . . . . . . . . . . . . . . . silicone rubber mode1 showing end effects 32

2- 13 Finite-element prediction of the principal stress pattern at the end of an adhesive

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . layerwith0.57nmspe-x 33

2-14 Typical crack on loading double-lap joint made with spew fillets (joint has been

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . cut and polished) 34

. . . . . . . . . . . . . . . . . . . . 2-15 Diagram of failure surfaces of single-lap joint 34

2-16 Variation of adhesive principal ( ( T ~ ~ ~ ) , peel (O,,), and shear ( T ~ ) stresses across

the adhesive thickness (y) at various distances (x) from the overlap ends . . . . . 35

. . . . . . . . . . . . . . . . . . 2- 17 Single-lap joint geometry and finite-element mesh 36

. . . . . . . . . . . . . . . . . . . . . . . 2-18 Adhesive shear stresses in bonded joints 37

. . . . . . . . . . . . . . . . . . . . . . . 2-19 Adhesive stresses in flawed bonded joints 38

2-20 Adhesive stresses in flawed bonded joints . . . . . . . . . . . . . . . . . . . . . . . 38

2-11 Influence of lap length on bond stress distribution . . . . . . . . . . . . . . . . . 39

. . . . . . . . . . . . . . . . . . . . . . . . . . 2-22 Design of double-lap bonded joints 40

2-23 A schernatic representation of the effect of water on an internally stressed adhe-

sive joint to a high energy substrate . . . . . . . . . . . . . . . . . . . . . . . . . 41

2-24 Interaction of critical parameters influencing the brittle behaviour of machine

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . components 43

2-25 Fatigue design philosophies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

2-26 Coordinates measured from the Ieading edge of a crack and the stress components

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . in the crack tip stress field 46

2-27 The basic modes of crack surface displacements . . . . . . . . . . . . . . . . . . . 47

3-1 Specirnen dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

3-2 Theoretical strain perpendicular to crack and 2 mm from crack tip . . . . . . . . 51

3-3 Theoretical strain perpendicular to crack and 4 mm from crack tip . . . . . . . . 52

. . . . . . . . . . . . . . . . . . . . . . 3-4 Theoretical 2-d strain field near crack tip 53

. . . . . . . . . . . . . . . . . . . . . . . 3-5 Isochromatic Ennge pattern: a/W = 0.4 55

3-6 Crack lengt h versus normalized peak st tain (peak strain/ far- field strain) for 2000

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . N static load 56

3-7 Crack length versus normalized peak strain (peak strain/far-field strain) for 5000

IV static load . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

3-8 Crack length versus peak strain location along y- axis for 2000 N static load . . 58 3-9 Crack length versus peak strain location along y- axis for 5000 N static load . . 59

. . . . . . . . . . . . . . . . . . . . . . . . 3-10 Crack behaviour due to cyclic loading 62

. . . . . . . . . . . . . . . . . . . . . . . . . . . 3-11 7075-T6 Crack propagation rates 63

. . . . . . . . . . . . . . . . . . . . . . . 3-12 Crack growth due to 2000 N cyclic load 64

. . . . . . . . . . . . . . . . . . . . . . . 3-13 Crack growth due to 3000 N cyclic load 65

3-14 Crack growth due to 4000 N cyclic load . . . . . . . . . . . . . . . . . . . . . . . 65

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-1 Characteristics of grating #1 68

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-2 Characteristics of grating #2 69

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-3 Bragg grating sensor Iocations 70

List of Tables

1.1 The capabiiities of NDT methods for defects in adhesively bonded joints . . . . . 12

Chapter 1

Introduction

As stated in [l?], aircraft suffer £rom cracks due to fatigue loading, environmental deterioration,

and bird strikes. For bird strikes, the damage is usualiy visuaiiy apparent and critical areas

~ ; f an aircraft are designed to withstand a statistically determined impact. allowing the plane

to land safely. Fatigue and environmental damage may not be so easily determined however.

since cracks can forrn in interior surfaces. In these cases, a non-destructive inspection (NDI)

technique is needed to determine the extent of damage.

Tliese cracks must be taken into account in the design process. Different design philosophies

have been used over the years to determine the service life of a component. These philosophies

include Safe Life (SL), Retirement for Cause (RFC): and Fail Safe (FS). For these criteria,

crack lengths must be knom. These can be determined by visual inspection but cracks can

f ~ r m in interior surfaces. In these cases, a non-destructive inspection (N'DI) technique is needed

to determine the extent of damage.

1.1 Patches

Patches of either aluminum, or composite materials can be used to repair cracked structures.

Adliesive bonding is proving to be a superior method for fastening these patches because it

provides a smoother load transfer across the crack, results in longer joint service life; especially

L.ldhesive joint is the prevailing term used to describe any two materiais bonded together using adhesives. A patch can be viewed in this manner and this term will be used throughout this report.

Figure 1-1: Fatigue strength of riveted versus adhesively bonded joints

for cyclic loading as seen in Fig.l-1, [4], and it offers the benefits of sealing the crack £rom the

environrnent2. The integrity of the bond is a h quite insensitive, at least initially, to relatively

large rnanufacturing flaws (see Section 2.3.3) However, this fact makes it diflicult to detect any

defects that do occur since there is no discernible los of joint strength. However, the joint will

weaken at a faster rate and defects m u t be detected and tracked if adhesives are to be used on

a wide scale in the aerospace industry where failure could have severe consequences.

1.2 Adhesive Bond Inspection

St ringent tests can be performed during the rnanufacturing process of adhesive joints. However,

these joints cannot be tested reliably with curent techniques. This leads to large factors of

safety and a non-optimal joint design. The different defects present in bonds cari be categorized

as leading to either cohesive or adhesive failures. Cohesive failure occurs when a defect exists

wit hin the adhesive and adhesive failure is the term used for the destruction of the bond between

adhesive and adherend [4]. The common types of defects are shown in Fig.l-2, [25]. Voids,

porosity, cracking and poor cure reduce the mhesive strength of the bond whereas disbonds

'Other factors such as economics and preparation of the adherends must be analysed on a case by case basis. These factors are summarhed in !QI, p. 12-13.

Poor cure Void -

Figure 1-2: Typical defects in adhesively bonded joints

affect the adhesive strength. There exist several methods for detecting cohesive defects as

detailed in [25]. Current methods do not detect adhesive defects reliably. Table 1.1, [25]

reviews the abilities of current NDT methods. Any NDT test requires special inspections and

complicated procedures which increase aircraft down-tinie. It is this lack of any in-service

monitoring that prevents any widespread usage of adhesive bonding in structures. A sensor

that could detect the onset of debonding would be beneficial by reducing or even eliminating

inspection time. The sensor could be used in a 5mar t Patch' that could continually monitor its

own integrity and perhaps even provide real-time information concerning its own stress state.

Fiber Optic Sensors for Smart Structures

The benefits of optic fibers (OF) embedded in materials for use as sensors has been detailed

in [33]. As stated in [II], different types of optic fiber sensors. such as Michelson and Fabry-

Perot interferometers, as well as Bragg grating sensors, are being studied to determine their

suitability for smart patches In order to continue the development of a smart patch sensor, a

better understanding of the stress state in adhesive bonded joints is needed. Specifically, an

appropriate location for placing these sensors and the magnitude of the stresses and strains

they are likely to encounter d l aid in the choice of an appropriate type of sensor and the

f Disbonds Voids, Poor Poor porosity cohe- adhe-

sion sion Ultrasonic techniques Normd incidence cornputeriseci wave 1 1 I Spectroscopy 1 1 2 R Ultrasonic oblique incidence technique 1 1 2 R Interfacial waves 1 1 2 2 R Bondtesters 1 1 3 1

O t her techniques Lon- frequency vibration 1 1 Holography 1 I/R

U I J

Thermography 1 3 I/R YvIFtI 3 3 2 I/R

1 Dielectric measuring 3 3 2 2 R -- -- - -- - - - - - - - - -

Key: 1 - Dcrrccablc (aiihough ihcm may be iimits on the size of defect capable of k ing detcctcd) 2 - Technique currently k i n g developed and showing some promise 3 - Application limited 1 - Industriai technique R - Rescmh tool UR - Spcialist usc in industry and cumntly k i n g m m h e d

Table 1.1: The capabilities of MIT methods for defects in adhesively bonded joints

signal demodulation techniques to be empioyed. A means of promoting deb~nding needs to be

determined and the ability of the sensors to detect this failue will be evaluated.

Fiber Optic Sensors

Optic fibers have been studied before for integration into engineering materials as an integrated

sensing system [23]. They have been used to measure temperature, strain, pressure, displace-

nient and other measurands. Sensors that detect strain Mil be the focus of this thesis. There

are several types of strain sensors, including point, distributed or truly distributed sensors. A

point sensor rneasures strain at only one point. A distributeci sensor is a set of point sensors

niultiplexed together in a single opticai fiber. This ailows a rough determination of the strain

field along the fiber to be determined. However, spatial resolution is in the centimeter range.

This is alIowable if the strain does not vary considerably between sensors and the structure

is relatively large compared to each sensor. A truly distributed sensor offers the capability of

measuring strain continuously along one sensor. The spatial resolution is considerably improved

and is in the nanometer range.

1.4.1 Fiber optic sensing for composite smart structures.

They offer the advantages of being linearly responsive, extremely sensitive, insensitive to electri-

cal noise. conipatible within concrete or composite ply Iayers: and in certain configurations are

lead insensitive. Different configurations of fiber optic sensors will be examined to see which

combinations best measure patch integrity. The primary focus of the experimentation is to

detect the location of a debond in a patch and any subsequent growth of this debond. This tviil

riecessi t at e several specimens, each mounted wit h a different sensor arrangement, and a means

to promote crack growth in each.

1.4.2 Debond Detection

Previous work by h w e r y has examined the axial strain experienced at the surface of the

reinforcement in a double iap joint due to an axial load. [22]

A long Bragg grating in an optic fiber was used to measure the strain distribution in the

axial direction. Debonds resulted in a large drop in measured strain at the location of the

debond. Results were encouraging enough to further pursue the concept of the smart patch.

The main disadvantage of Lowery's system was the assurnption that the debond is uniforrn and

across the entire width of the plate, which would not be true in an actual patch. This can be

overcome by measuring the strain in the transverse direction, across the widt h of the plate. By

sensing the transverse strain near the edge of the reinforcement, cracks; which usually initiate

at the edge of a patch, can be detected. This transverse strain is directly related to the axial

loading due to Poisson's effect. Any drop in measured strain will indicate a debond.

1.4.3 Crack Detection

Early work on crack detection utilized intensiometric sensors. Optic fibres were bonded to or

imbedded within structures. If a crack of sufficient size passes through the fiber, the fiber Ml1

break. This leads to several measurable pheno~nenon. The power of the light detected will drop

dra~tically~ light wiil be scattered at the location of the crack and light ni11 be reflected back to

the source from the crack. The first approach is limited by attenuation due to contamination of

the fiber ends or coupling losses, both of which are not due to a break in the fiber. The second

approach was used in the detection of cracks inside a composite leading edge of an airplane

wing in [21]. The method is limited to areas that can be visually inspected. The third method

in\yolves opt ical t ime domain interferomet ry (OTDR)

Several fiber optic interferometric approaches could be used to obtain the necessary sensi-

tivity to strain. Long Bragg gratings could be used by placing t hem in the transverse direction,

but are currently rather expensive. Other devices will first be used to illustrate the feasibility

of this approach.

1.4.4 Interferometers

Several configurations studied include the Michelson and Fabry-Perot interferometer. -4 Michel-

son interferometer requires a reference fiber arm which can be bonded to the adherend. The

sensing arm is bonded to the reinforcement. Both arms could be embedded in a shallow groove

in the duminum to ensure a better strain transfer. The two sensing arms should be placed as

close together to ensure other factors besides strain do not affect the length of the fibers. This

To strain 3 dB Coupler Single Mode h r c d Fiùer

meter ,\ 7 *tic ~ i k \ P d

Figure 1-3: Fiber Optic Michelson Interferorneter Configuration on Patch

also requires that the sensing arrns and coupler be taped down so that vibration is not a fac-

tor. Mirrored ends reflect light from a laser and together these reflections form an interference

pattern that can be seen on a photodetector. Displacement wiii form a number of fringes that

can be related to the change in displacement of the reinforcement. The instrurnented patch is

shown in Fig.1-3.

-4 Fabry-Perot interferorneter does not require a reference arrn and instead uses a mirrored

end and a short Bragg grating to act as a semi-reflective second mirror. Each sensing arm is

localized and is lead insensitive. Therefore, several sensing a r m could be used at different axial

locations to measure crack progression by using one system and connectors to interrogate each

urrn. as shown in Fig.1-4.

1.4.5 Distributed Bragg Grating Sensors

Recent work by Duck[l5], has led to the development of Bragg grating sensors which can

measure distributed strain. This offers the benefits of measuring a tw-dimensional strain field.

i3y monitoring the strain field in front of a crack, its growth can be tracked over time.

To sirain red Fiber

Figure 1-4: Fiber Optic Fabry-Perot Interferorneter Configuration on Patch

Chapter 2

Theoret ical Analyses

Since the number of sensors to be applied to a structure should be minirnized, an optimum

location for a Bragg grating sensor must be determined. The theories of adhesive bonded joints

and linear elastic kacture mechanics (LEFM) can be used to facilitate this. These theories

predict crack behaviour in reinforced structures, thus providing a theoretical optimum location

for the sensor and serving as a cornparison to actual experimental results. Firstly, the behaviour

of Bragg grating sensors wiU be discussed.

2.1 Interferometric Sensors

Using the principles of interference of light waves, interferometric sensors can be used to measure

strain over the length of a fiber bonded to a structure. When two light bearns interfere, a h n g e

pattern is produceci. The maxima of this pattern occur when the phase diiferences between

the beams is a multiple of 27r. Any change in the phase of either of the bearns can be detected

by a shift in the fringe pattern. Optoelectronic techniques can measure this shift to about

10-" of the fkinge spacing . This shift could be as Little as 1/100'~ of a wavelength. which for

visible light is 5 nm. Thus, strain in a structure can be measured by attaching an appropriate

configuration of fiber optic sensors. The strain produces a change in the physical length of the

fiber and which modulates the phase of light passing through the fiber.

2.2 Bragg Grating Sensors

The Bragg intra-grating sensor offers a t d y distributed strain measuring capability. This type

of sensor can measure the continuous distributed strain as a function of distance dong a Bragg

grating with a spatial resolution of better than 1 mm [Ml.

Bragg gratings are periodic structures in the core of a single moded optic fiber which have a

different index of refraction from the rest of the core. The structures can have differing indices

of refraction, reflect differing amplitudes and be spaced at regular or increasing intervals. When

light is sent into the optic fiber, each grating structure will scatter a portion of the light. For one

wavelength of light, what is known as the Bragg condition will be satisfied. At this wavelength,

the successive scattered light from each structure will add constructively to form a coherent,

narrow band reflection. [31]

2.2.1 The Bragg Condition of Constructive Interference

This condit ion is a form of the conservation-of-momentum condition or phase matching condi- +

tion. For an incident wave with wavevector kl reacting with a Bragg grating Mth wavevector

2. a wavevector & is reflected. As seen in Fig.2-1 [8],

The rnagnit udes of these wavevectors are:

Since the incident and reflected light must have the same wavelength, Xi = X2 = AB, which

is the wavelength of light that satisfis the Bragg condition. nef! is the effective modal index

of refraction of the fiber at this Bragg wavelength. A is the period of the grating. Rearranging

Eqn.2.1 we have:

/ - k* k,- Fi ber Cladding I

Figure 2-1: Cross-section of a Bragg grating having uniform amplitude, index modulation and period.

Thus, the Bragg condition is:

AB = 2nef A

This condition states that light with a center wavelength of AB, travelling through fiber with

an effective index of refraction of ne,! will reflect off a grating with a period of A. As a grating

is strained, the Bragg wavelength shifts as detailed in Section 2.2.3

2.2.2 The Reflection Spectrum

For a uniform grating in a fiber with refractive index no, the refractive index will vary dong

the grating as [8]:

n,f f = no + An - cos - (21;1) where An is the amplitude of the index change in the grating. Using coupled mode theory,

the reflected wavelength for a grating with uniform period and index modulation is:

kr2 sinh2 (SL) R (L, A) = ap2 sinh2 (SL) + r;2 cosh2 (SL)

O R is the grating reflectivity as it varies due to input wavelength and position in the grating

O L is the Iength of the grating

O K is the coupling constant given by r;: = nAn/A

r 40 is given by A 0 = /3 - ( r / A )

r f is the fiber propagation constant in the core of the fiber given by ,d = y r s is given by S = b2 - M2

Light at the Bragg wavelength X E will produce Ap = O and the reflectivity Eqn.2.5 is

reduced to:

R (L, A) = tanh2 (SL) (2.6)

A plot of this function using L = lcm, AB = 1550nm and peak reflectivity R,, = 80% is

shown in Fig.2-2 [8].

2.2.3 Grating Response to Strain

Changes in the Bragg wavelength are affected by changes to the period of the index modulation

in the Bragg grating. Strain, teniperature and wavelength will al1 affect this period. By

expanding Eqn.2.3 in terms of these variables, we have:

By assuming temperature and input wavelength remain constant, we can concentrate on

changes to the Bragg wavelength due to strain. Strain wiil cause the period of the grating A

Figure 2-2: Theoretical plot of reflectivity as a function of wavelength for a uniform Bragg grating (L = lcm, AB = 1550nm, R,, = 80%

to change, thus changing the Bragg wavelength. As weil, due to the strain-optic effect, the

effective refractive index of the grating n,f j wiIl also change. This leaves us with:

Given that strain E = y, the equation can be written as:

Defining %-AL = Anelj and A

The strain optic effect appean as a change in the optical indicatrix (101

8 Sj is the strain vector

pi3 is the strain-optic tensor

For longitudinal strain in the x direction, the strain vector is

For an isotropic, homogenous medium, the strain-optic tensor only has two numerical values

n-hich we can cal1 pl1 and pi?. This leaves us with a change in the optical indicatrix of

Substituting back into Eqn.2.10 with a A / a L = AIL gives us the change in the Bragg

The shift in the Bragg wavelength in relationship to the applied st.rain t o the grating is thus

where the effective strain optic constant P e f f is

For the single-mode optic fiber used in the experiments, pl1 = 0.113, pl;? = 0.352, v = 0.16

and n,ff = n o = 1.4818. This gives a p,f i = 0.213. For 1550 n m light, this gives a wavelength

sensitivity to strain of 1.22 p n / p . . ~ .

2.3 The Nature of Stresses in Adhesive Joints

There exist several factors that affect the strength of adhesive joints. These include the mag-

nitude of the loads, the joint's structural geometry and the mechanical properties of both the

adhesive and the adherends. The structure of the joint will determine the type of stresses that

will dominate the behaviour of the joint under load. The magnitude of these loads will deter-

mine what approximations cari be used and thus the material properties over the range of loads

must be known. Several complexities arise when we examine adhesive joints. The joint does not

consist of simple: separate? elastic materials ~ i h a clear, mathematical geometry. The surface

of the adherend is rough. It has an indeterminable elastic modulus and thickness. There is

usualIy an oxide layer on the surface or primer may be applied there. The adhesive may have

different properties depending on whether it is a thin film or in bulk form. Simple geometries

such as the single and double lap joints shown in Fig.2-3 [4] will be analysed in an attempt to

underst and the mechanisms involved.

2.3.1 Classical Theories

There are two classical linear elastic analyses of adhesive joints. One is by Volkersen [33j and

the other was put forward by Goland and Reissner (161.

Volkersen's Analysis

A single-lap joint shown in Fig.2-4 (41 as used in Volkersen's analysis. Two important factors

were not accounted for in this analysis: the directions of the loading P are not collinear. This

will lead to a bending moment on the joint. Because the adherends are not rigid, they will bend

Figure 2-3: Some common engineering adhesive joints

Figure 2-4: Exaggerated deforrnations in loaded single-lap joint: (a) with rigid adherends: (b) with elastic adherends

due to this force and this will lead to a non-linear problem, since the joint dispiacements are no

longer proportional to the applied load. With a syrnrnetrical double-lap joint, there is no net

bending moment. However, since the loads are applied by the adhesive to the adherends away

from their neutral axes, the joint will experience interna1 bending. This is shown in Fig.2-5 [4].

The double lap joint can thus be treated as two single lap joints.

Goland and Reissner's Analysis

Goland and Reissner accounted for bending by using a bending moment factor, k. This factor

relates the bending moment a t the end of the overIap, filo, to the in-plane loading using:

where P is the applied load and t is the adherend thickness.

For small loads, the overlap region will not rotate and the load line, shown in Fig.2-6(a) [4J,

will pass close to the edge of the adherends at the ends of the overlap. For this case, MO - Pt /2

Figure 2-5: Bending moments induced in the outer adherends of a double-lap joint

Figure 2-6: Illustrating a way of representing geomet rically

the Goland and Reissner bending moment factor

and k 2: 1. For larger loads, the overlap rotates and brings the load line closer to the centre-line

of the adherends, as shown in Fig.2-6(b). This will reduce the value of the bending moment

factor. These two loading approximations are known as Goland and Reissner's fkst and second

approximations. For metal-on-metal patches, the second approximation is more applicable.

The range of validity as set by Goland and Reissner for their second approximation is:

where E, G and t are as before and 1 and 3 represent the adherends and adhesive respectively.

Benson 171 has shown that there is another fundamental Iimitation on this theory. k: is

derived assurning that the overlap region in the joint can rotate, but the interna1 stresses in the

joint are derived assuming no rotation. Therefore, the calculated adhesive shear and the tensile

stresses are strictly true only for k = 1.0, i.e. small loads. Parametric studies by Kutscha and

Hofer (201 showed that the widt h and load on the joint were not separable from the functions

for the shear and normal stress. This is due to the change in the value of k as the load is

increascd.

The results from these two analyses have been used in later work: most notably in the

PABST ' programme (See Section 2.3.3)

2.3.2 Recent Theories

These early theories by Volkersen, Goland and Reissner were limited because the peel and shear

stresses across the thickness of the adhesive were assumed constant, the shear at the overlap ends

were assumed to be maxima, and shear deformation of the adherends was neglected. However,

the shear stress at the overlap ends must be zero because they are free surfaces. This ïeads

to results similar to Fig.2-7 [25]. Work by Benson [7], and Peppiatt [29] have taken this fact

into account, but bending in the joint was neglected. Renton and Vinson [30] and Allman [5]

have produced analyses where bending, shear and normal stresses are present in the adherends.

In addition, Allman assumeci a linear variation of the peel stress across the thickness of the

' Primary XdhesiveIy Bonded Structure Tdiu>!r>gy

27

i 1

Mean Stress 1

i I

1-b Distance almg overiap

4

Figure 2-7: Stress distribution in a single lap joint

Figure 2-8: Tensile stress in adherend and shear stress in adhesive (each per unit load) pIotted against s for a 1 in x 1 in single-lap joint

adhesive. Section 2.3.2 discusses other, non-linear variations.

Transverse Stresses

Another factor that hasn't been mentioned yet, is the existence of stresses across the width of

the zdhesive joint. This was shown by Adams and Peppiatt (21 who accounted for the shear

stresses in the adhesive layer and direct stresses in the adherends acting at 90° to the direction

of the applied load. These direct stresses are the result of Poisson's Effkct.

There are no exact solutions to the equations obtained hom this analysis, but approximate

solutions can be found. Finite differencing can also be used to obtain results. The approximate

and finite difference solutions for the tensile stress in the joint as a function of x are shown in

Fig.2-8 (21. The coordinate system is as shown in Fig.2-9 [4]. The transverse direct stresses

are shown in Fig.2-10 [2].

Figure 2-9: Deformation in single-lap joint due to lateral straining

Figure 2-10: Transverse direct stress per unit load (u l , /P) in the adherend plotted against z for a 1 i n x 1 in single-lap joint at different positions along the length of the joint: - approximate analytical solution (exact at x = 0 , l in ); 0 finite-difference solution

Figure 2-11: Diagrammatic lap joints to show adhesive layers mith (a) square edge; (b) spew fillet

End Effects

The closed form analyses predicted that the maximum stresses and strains are near the end of

the overlap. This was assuming a square edge at the end of the adhesive. (See Fig.2-il [3])

This is not the case in real life because the adhesive layer almost never has a square edge.

Instead, the adhesive is squeezeci out of the joint when it is compressed and cured2.

No attempt has been made to determine a closed-form analytical solution for the effects

of the spew. Finite element techniques (FET) have been used though. Adams and Peppiatt

[3] treated the problem as one of plane strain and a physical mode1 using rubber to mode1 the

adhesive was constructed to veri& the results of the finite element mode1 (FEM). Fig.2-12(a)

and (b) [3] show the experimental and FEM results. The absolute magnitude of these principal

'This fiilet area is known as a spew of adhesive.

Figure 2-12: (a) Comparison between calculated and experimental displacements of the silicone rubber model. (The black crosses are the finite-element predictions of the intersections of the grid lines of the model.) (b) Principal stress pattern for silicone rubber model showing end effects

Figure 2-13: Finite-element prediction of the principal stress pattern at the end of an adhesive layer with 0.5 mm spew

stresses were at lest 3.6 tirnes the shear stress in the rubber. The stresses in the fillet are mostly

tensile ranging up to 3.5 times the shear stress in the rubber. Fig.2-12(b) also illustrates that

the spew and end of the adhesive layer is rnostly in tension at an angle of 45O to the loading.

-4 stress concentration in the spew occurs at the corner of the unloaded adherend.

Further experiments with alurninum bonded to aluminurn using low-ductility adhesives

showed that cracks formed at approximately right-angles to the directions of the maximum

principal stresses shown in Fig.2-13 [3]. Adams presented the idea that M u r e in lap joints

is initiated by high tensile stresses in the spew. In well bonded structures, cohesive failure

occurs. If a bond is weak, adhesive failure will occur where the spew is pulIed away from the

loaded adherend by interna1 tensile stresses. Fig.2-14 [4] shows a typical cohesive failure in a

double-Iap joint. When the lap joint fails completely, an initial crack in the spew propagates

along the adhesive-adherend interface until it meets a similar crack propagating from the other

end of the joint. The result is s h o m in Fig.2-15 (41.

Centrai adharand

Figure 2-14: Typical crack on Ioading double-lap joint made with spew fiIlets (joint has been cut and polished)

Figure 2-15: Diagram of failure surfaces of single-hp joint

Ylmrn YIrnrn Ylmm

Figure 2-16: Variation of adhesive principal (opnn), peel (O,), and shear (T-) stresses across the adhesive thickness (y) a t various distances (x) from the overlap ends

Through the Thickness Adhesive Stresses

Crocombe and Adams [ ~ 2 ] have studied the stress distribution across the adhesive thickness.

Most other analysis have assurned the stresses here to be constant. Fig.2-16 [131 shows the

results as they vary across the adhesive thickness at different distances from the end of the bond

line. The variation in stress across the adhesive thickness may be due to the discontinuity caused

by the corner of the unloaded adherend. The load transfer does not occur perpendicularly to the

adhesive thickness, but instead works its way through the adhesive at an angle until it reaches

the corner of the unloaded adherend. The positions of the peak stresses probably occur on this

load line. The maximum principal stresses occur a t the unloaded adherend corner and act at

45" to the longitudind axis of the joint. The largest peel stress occurs a t the overlap edge as

well, just within the adhesive. The maximum shear stress is located at the adherend-adhesive

interface, and a small distance kom the overlap. These points: A, B, and C respectiveiy

are shown in Fig.2-17 [13] and do not vary considerably with different geometries. Thus an

Figure 2-17: Single-lap joint geometry and finite-element mesh

analysis of a bonded joint must t.ake into account the spew not only because the stresses in the

joint are reduced considerably, but the location of maximum stress has shifted from the loaded

adherend-adhesive interface to the corner of the unloaded adherend.

2.3.3 Elasto-Plastic Analysis of a Single Lap Joint

Al1 the eiastic analyses presented so far have one drawback in that they fail to predict faihre.

This is because modern adhesives such as those based on rubber-modified epoxies have large

plastic strain to failure and this causes a redistribution of the load transfer. Testing on adhesives

has shown them to be both viscoelastic and plastic, within a possible time dependence on the

plastic properties [17]. These adhesives are also strong enough to cause yielding in the adherends

as well. If an adherend yields at the loaded end, the differential straining shown in Fig.2-4(b)

will be increased. The shear stress peaks near the ends of the bond-line will thus be increased.

This increase in stress is offset by an opposite effect where the yielding in the adherend allows

it to rotate, resulting in a load that is more collinear. This decreases the Goland and Reissner

Joint factor k and the stresses in the joint.

Two main approaches to plasticity have been taken so far. One is continuum mechanics,

ADHESIVE SHEAR STRESS

MN/m2 (AT ROOM

TEMPERATURE)

1

Figure 2-18: Adhesive shear stresses in bonded joints

while the second uses FET. Continuum mechanics is a continuation of the shear-lag analysis

used by Volkersen [33] and the two theories of Goland and Reissner (161. This analysis was

used extensively in the PABST study [32j. Raults for shear stress in several joints, each with

different amounts of disbond are shown in Fig.2-18 through 2-20 [6]. Anot her notable

result of the study was the relative insensitivity of the joint strength to the presence of large

rnanufacturing flaws. The ductile FM-73 was able to redistribute the load to compensate for

voids, porosity and variable bondline thickness.

Fig.2-21 il71 shows the effect of lengthening the overlap region in double-lap joints where

the adhesive is in plastic deformation. With increasing overlap, an elastic region in the middle of

the overlap is seen to increase. This elastic region, it is argued by Hart-Smith, is not structurally

inefficient, but is necessary in order to overcorne creep a t the ends due to fatigue loading. By

designing for the worst operating condition of high temperature and high humidity, the criteria

for the overlap region are listed in Fig.2-22 [17].

Cyanmid expoxide adhesive

ADH€SIVE WEAR STRESS

M N / d (AT ROOM

1 EMPERAfURE)

: I t S L M / i i , 1 ,- l 7

(1 wo Uni.)+ 1 + A

1 9 - i I~ O.l, mm (O.m Y. J o.U mm (0.025 W.b 12.7 mm (OS M.) ! û O N O M W

. -

Figure 2-19: Adhesive stresses in flawed bonded joints

(KSI)

i ! ADHESIVE i SHEAR STRESS

I MN/m2 (AT ROOM

1 TEMPERATURE)

1 I !

1

Figure 2-20: Adhesive stresses in fiawed bonded joints

' t + F i MAXIMUM (CONSTANT) STRENGTU ZONE

L E f F f C I OF UC LEMGTU ON A D H E S I M - W N D C D JOINT STRINGTH

JOINT CROSS SECTION . --_1- P

C. L O N G O V E R U P

Figure 2-21: Influence of lap length on bond stress distribution

a PUSTIC ZONES LONG ENOUGH FOR ULTIMATE LOAD

E U m c TROUGH WIDE ENOUGH TO PREVENT CREEP AT MIDDLE

a CHECK FOR AOEQUATE STRENGTH

--

Figure 2-22: Design of double-lap bonded joints

Figure 2-23: -4 schematic representatiori of the effect of water on an internally stressed adbesivc joint to a high energy substrate

2.3.4 Service Life

I t ha.s been found that moisture penetration into the adhesive is a major cause of joint failure in

well-bonded joints. Bowditch [9] reviewed the effects of water on adhesive joints. He concluded

that three possibilities for joint strength as a function of water content existed. These are

shown graphically in Fig.2-23 [9]. The lower curve represents water at tacking the adhesive-

adherend interface only. Assuming catastrophic M u r e does not occur, the joint strength falls

gradually as the water content increases. There is, however, a limit wherc higher water content

cannot gain access to the interfacial area and the joint strength stubilizes. T h upper cur1.c

stiows behaviour of joints which seem to beconie stronger with moisture irigress. This can be

txplained by relief of interna1 stresses as the adhesive phsticizes. F'urther moistiire content wiI1.

however, weaken the joint. In practice, the middle c u v e seems to be the most common type

of behaviour. Here, plasticization in the adhesive is balanced by interfacial weakening at the

bondline.

2.4 Linear Elastic Fracture Mechanics

LEFM is a basis for predicting crack growth in materials with relatively low fracture resistance.!24]

These niaterials include higli strength materials used in the aerospace iridustry such as 7075-T6

aluminum alloy. Differing design philosophies use these predictions t o determine s d e operat ing

lives for cornponents. These design philosophies are dependent on non-destructive inspection

techniques to venfy crack behaviour.. Results from the theoretical predictions can then be

compared to measurements made by optic fiber sensors.

LEFM provides a quantitative means for determining:

a The likelihood that a structure will fail by brittie fracture

a Conditions of crack initiation, propagation and arrest

o Crack growth in structures subject to dynarnic loads

2.4.1 Brittle Fkacture

For brit tle fracture, the main parameters in the design of a component are:

a KI, the fracture toughness of the material

rn Q the configuration factor depending on the geometry of the cracked cornponent

a a, the maximum allowable design stress

a a, the allowable crack size dependent on fatigue crack growth data and accuracy of

ND1 techniques [24]. Improving this measurement of crack length will heip improve

component design.

Fig.2-24 [24]: shows the interaction of these parameters. From these parameters, differing

design philosophies are used to determine safe operating lives for components. These design

philosophies are dependent on non-destructive inspection techniques to verify crack behaviour.

Results from the theoretical predictions can then be cornpared to measurements made by optic

fiber sensors.

2.4.2 Fatigue Design Philosophies

-4 flowchart showing how different design philosophies relate to each other is s h o w in Fig.2-25.

FRAC1 URE TOUGHNESS

INFLUENCE0 BY: 0 TEMPERATURE

STRAIN RATE ENVlRONnENTAL INTERACTIONS METALLURGICAL EFFECTS

-GRAIN SIZE -INCLUSION -LATTICE

STRUCTURE

CONFIGURATION FACTOR I

FUNCTION OF CRACKEO COMPONENT GLOMETRY m n TECHNIQUES ARE ConnoNLY

HAXlHiJM ALLOWABLE DESIGN STRESS 1

-

STRESS CONCENT RATION LOAO INTERACTION STRESS TRlAXlALlTY SERVICE REQUIREMENT

I MAXIMUM ALLOWAeLE CRITICAL CRACK SITE

INFLUENCED BY FATIGUE CRACK GROWTt DATA AND ACCURACY OF NO1 TECHNIQUES AVAILABLE

Figure 2-24: Interaction of critical parameters influencing the brittlc behaviour of niachine coniponents

- MODlFlED FATIGUE SAFE t l FE TRADITIONAL FATICUE i

DESIGN PHILOSOPHY ' '

C R I 1 ERlON DESIGN PHlLOSOPHY , ! ,

DAMACE TOLERANCE

I LE FM EPFM

ACCURATE MEASUREMENT OF CRACKS

r

ACCURATE PREDICTIOU OF FATIGUE CRACK CROWTH

I

- - - - - - - - - - -

Figure 2-25: Fatigue design philosophies

Safe Life (SL)

,A cornponent is taken out of service if a crack of a given size is detected. An optic fiber sensor

could be used to detect cracks in components which wodd otherwise be inaccessible. The life

of a component can be predicted by comparing its critical design features to specirnens with

experirnentally determined S-N curves where S is the net-sectional stress amplitude applied

to the specimen and N is the number of loading cycles to failure. A rnean-life expectancy is

determined experimentaily based on the load spectrum of the component. There is uncertainty

with this method since it is based on experimental results so condition monitoring philosophies

were developed.

Retirement for Cause (WC)

This philosophy assumes that

components contain pre-eusting cracks

a the fatigue crack growth rate can be predicted with long crack propagation (LCP) or

short crack propagation (SCP) relationships

a that cracks can be monitored accurately using XDI techniques.[24]

A periodic inspection of components is conducted and if detected cracks are below a critical

size. the cornponent is put back into service. This is a damage tolerance approach where the

designer has to show that the crack will not propagate to M u r e before the next inspection.

The inspection periods are thus very dependent on the accuracy of the XDI techniques used. A

truly distributed stress measurement in front of a crack could give very accurate location and

behaviour of a growing crack.

Fail-safe (FS)

If the condition of a cracked component cannot be accurately predicted and/or measared,

the fail-safe criterion must be used by the designer. Structurai redundancy, load redistribution

and/or minimized damage to the system are some of the ways of ensuring this. In the caçe of load

redistribution, the load transferred by the failed member is redistributed to ot her rnembers. The

Figure 2-26: Coordinates measured from the leading edge of a crack and the stress components in the crack tip stress field

cffects of this increased load on the other members must be accounted for. In-service monitoring

n-ould help to e ~ a l u a t e load transfer from the cracked component to ot.her components.

2.4.3 Stresses Near Crack Tips

Using LEFM and the stress intensity approach developed by Irwin [19]. the elastic stress field

i!mr a crack tip depends on a stress tensor rr,, where

and 7,. 8 are cylindrical polar coordinates of an element ahead of the crack tip as seen in Fig.2-

26 i28! and h.0.t. are higher order terms in r . The K terms are stress intensity factors wliich

correspond to one of the three basic modes of crack growth îs shown in Fig.2-27.

Figure 2-27: The basic modes of crack surface displacements

Modes of Crack Tip Formation

m II*, - opening mode where crack surfaces move apart

0 K I I - sliding mode where surfaces shear in-plane along a direction perpendicular to Ieading

edge of crack

rn KrrI - tearing mode where surfaces shear anti-plane along a direction parallel to the

leading edge of crack

Cracks usually are mixed mode or become mixed niode as they groxv. Hoivever. 1,lode 1 is

usually the most severe so the other modes are ignored in the analysis. T h e h.0.t. are ignored

as iveil. since nre are interested in the stress distribution near the crack tip.

Westergaard Stress Function: Elastic Stress Field Equat ions

Irwin deidopcd the stress intensity factor from procedures used by IVestergaard to solve cert aiil

classes of plane-strain/stress crack problems. The Westergaard stress function f34! is a special

case of the cornplex potential formulations of Muskhelishvili [27] for problems with x - a i s syrn-

met ry (loading and geometry). For al1 geoinetries with this symmet ry and subject to a biasial

tensile stress 00. the stresses near the crack tip can be written as:

with oz, = Y (oz, + aw)fOr plane strain and a,, = O for plane stress.

Kr is the opening mode of the stress intensity factor and can be defined as:

which is valid for r << a where

0 a is the crack length

O cro is the g r o s stress.

0 Q is the correction factor which accounts for different geometries

For an infinite plate with a center crack of length 2a, Q is equai to 1. The Q factor takes into

account the fact that a unifom compression, -go, exists and can be added to the value of a,,

[19j. This result can be obtained frorn anaiysis. For finite sized cornponents with different crack

locations, there are generally few closed form solutions. Q is usually determined empirically.

Some of these results can be found in handbooks such as [26].

Knowing the stresses a t the crack tip, we can easily determine the strains. For plane stress:

wtiere u is Poisson's ratio for the material.

Chapter 3

Computer Mode1

Using the preceding analyses of LEFM and bonded joints, a cornputer program was written to

calculate the theoretical behaviour of a cracked plate, both unpatched and patched, subjected

to c-arious static and dynamic loads. A copy of the program can be found in Appendk A. The

program can determine theoretical 1-d and 2-d strain distributions near the crack tip dependent

on user-defined variables. These variables are chosen using a graphical interface as s h o m in

Xppendix A. The controls determine:

t.he geonietry of plate

0 whether the plate is patched or unpatched

the plate material

the reinforcement material

0 the type of adhesive

0 whether the plate is subject to static or dynamic loading

0 the magnitude of the loading

0 the Iocation of strain measurement

0 the coordinate system for display of strain

Only several cases w r e examineci for the purposes of this thesis.

3.1 Case 1: Unpatched Plate/Static Load

For an unpatched structure, only the LEFM analysis is needed.

3.1.1 Assumptions

A very small plastic region exists ahead of the crack tip. The rest of the material is loaded

elast ically

0 The plate is thin and in plane stress

0 The crack tip is infinitely thin

The validity of the results decreases away from the crack tip. For r / a < 0.16 calculated

stresses are accurate to approxirnately 10% [24]

0 Q correction factors are accurate to approicimately 0.5% for a / W < 0.6 where F.V is the

width of the plate[24]

Because of these last two limitations, the crack length a must be large enough that the

distance to the optic fiber sensor r is relatively srnail and yet, a has to remain short compared

to the width of the plate W . The optic fiber cannot be placeci too close to the crack tip without

the bond line affecting crack growth. A larger sample would be desirable, but CIi is limited by

the size of the loading apparatus.

3.1.2 Case 1 Input

Dimensions s h o m in Fig.3-1

0 Single edge crack - 40.0 mm

1000 N static tensile load

Figure 3-1: Specimen dimensions

Figure 3-2: Theoretical strain perpendicular to crack and 2 mm fiom cr

51

ack tip

Figure 3-3: Theoretical strain perpendicular to crack and 4 mm fiom crack tip

3.1.3 Case 1 Results

A plot of the strain in the y-direction for x = 2 mm is shown in Fig.3-2. The simulation was

limited to y = 6 mm in order to keep the Q correction factor accurate to within 0.5%. The

cursors indicate the maximum strain E, and the strain at y = 0.1 mm. For x = 2 mm. E~~

is 993 x 106 at y = 2.1 mm. A similar graph of strain at z = 4 mm is shown in Fig.3-3.

The strains are smaller as expected and the peak strain has shifted away from the crack from

y = 2.1 rnnz to y = 4-1 mm.

A 2-dimensional plot of the straia distribution in a 4.5 mm2 area in the first quadrant in

front of the crack is shown in Fig.3-4.

Figure 3-4: Theoretical 2-d strain field near crack tip

The distribution compares favorably to isochromatic fringe patterns obtained using pho-

toelastic techniques shown in Fig.3-5 [14]. The two patterns are very similar even though

the crack in the photoelastic sarnple is relatively wide. This confirrns the suggestion that the

crack width does not overly affect the strain distribution a t the crack tip as long as the tip is

relatively t hin [24].

3.1.4 Case 1 Analysis

A method for deterrnining crack length independent of load magnitude normalizes the peak

strain in the y- direction (perpendicular tu the crack). This is done by using the plate's far-

field strain in the load direction. Graphs of crack length versus the normalized peak strain

for a 2000 iv and 5000 N static load are shown in Fig.3-6 and 3-7. The raw data used to

create these graphs can be found in Appendix B.1. The relationship between crack length

and normaiized peak strain seems unaffectecl by load magnitude. The relationship seems to

be a polynomial of 4th order. Since the relationship is invariant with the load magnitude, it

could be used to determine crack length. However, a second sensor wouid be needed to measure

the far-field strain. Another limitation of this method is its increasing loss of resolution with

increased crack length. As the crack grows, it becornes harder to differentiate the magnitude

of the normalized peak strain.

Another method that could be used to detennine crack length uses the location of the peak

strain along the y- direction. Graphs of crack length versus the y- location of the peak strain

for a 2000 N and 5000 N static load are shown in Fig.3-8 and 3-9. The raw data used to create

these graphs can be found in Appendix B.2. The relationship is linear and irrespective of load.

The location of the peak strain is only dependent on the length of the crack and thus provides

an elegant method of determining crack length. By knowing only the properties of the plate

and determining the peak strain location along a Bragg grating sensor located perpendicular

to the crack, we can determine how long a crack is.

Figure 3-5: Isochromatic fringe pattern: a / W = 0.4

Crack Iength vs. Normrliwd peak stnin

Figure 3-6: Crack length versus normalized peak strain (peak strain/far-field strain) for 2000 iV static load

Cmck lsngth vs. Nomulkrd peak stmin

Figure 3-7: Crack length versus normalized peak strain (peak strain/far-field strain) for 5000 !Y static load

C r u k kngth vs. P.rk siriln b e d o n rlong y-uls

Figure 3-8: Crack lengtk versus peak strain location along y- axis for 2000 X static load

crack longth vs. Peak *train location along y-ails

Figure 3-9: Crack length versus peak strain location along .y- axis for 5000 :V static load

I a MOON [ - ~inear fa j

40E-02

2 OOE-02

180E-02

1 60E-02

1 40E-02

Ê 120E-02 a

f UI

1 00E-02 - .x U

8 ME-03

6.00E-03

4 00E-03 .-

2.00E-03 -. 0 WEc00 -

Peak grrrln W o n along y -u t8 (p.rp.ndculw 10 u ick te) [ml

*

-

- -

.-

.-

,- - ,

- -

- -

O 00€+00 2 00E-03 4 00E-03 6.00E-03 8.00E.03 1 .WE-02 1.20E-02 1

3.2 Case 2: Patched Plate/Static Load

A 2 stage analysis using Rose's method found in [6] was used to determine the stress and strain

distribution near the crack tip:

a Plate and patch are both isotropic and have the sarne u

a 'io residual thermal stress due to bonding

a -411 deformations are linearly elastic

a Plate and patch are in plane stress, thus no variation across thickness

a Adhesive layer acts as a shear spring

Stage 1

The plate is assumed to be uncracked and reinforced. The reinforced region is viewed as an

inclusion of liigher stifiess which means that there can be no relative displacement between

plate and reinforcement. This is valid when the width of the load transfer region around the

boundary of the reinforced area is smali compared with in-plane dimensions of plate. The

redistribution of stress in the uncracked plate due to the reinforcement is deterrnined by the

1-d theory of bonded joints. This alIows for comparatively simple calculations of uo in the

plate.

Stage 2

A cut is made in the plate, allowing a0 = O. The stress intensity factor K for the reinforced plate

is then determined by assuming that the plate and reinforcement are infinite when compared to

the load transfer region. Wit h an infinite reinforcement, K will not increase indefinitely wit h

increasing crack Iength and instead, will approach an asymptotic value. This asymptotic value

for K can allow us to apply the stress intensity approach to determine the crack tip strain field.

3.2.2 Case 2 Input and Results

So simulations have been performed for this case because the adhesive has yet to be chosen.

3.3 Case 3: Dynamic Load

For dynarnic loading, crack growth can be predicted using the range of stress intensity:

AK = Km - Km;, = (ama - ami,) Q&

3.3.1 Assumptions

0 Sinusoidal stress cycles as shown in Fig.3-10

Linear crack propagation accumulation hypothesis - the crack grows with eveq stress

cycle

For high-strength aerospace aluminum alloys, Forman's Law is used for predicting crack

propagation:

tvhere CI. rn and Kf are evperimentally determined constants different for each material

as shown in Fig.3-11 [II and R = %.

Forman's Law is only applicable when:

Ah' Km= = -

1 - R < Kf

3.3.2 Case 3 Input

0 Specimen geometry same as in Case 1

0 Sinusoidal load varying with time

R = 0.1 for al1 runs

- n (number ot cycles)

Figure 3-10: Crack behaviour due to cyclic loading

mm c y c l e

CRACK P R O P A G A T I O N R A T E S A P P R O X I M A T I O N W I T H F O R M A N ' S L A W

B E S T D A T A F I T C U R V E S O F M E A N VALUES

Material : 7075-T6 s h e e t D a t a t a k e n from : R e p o r t T N M 2 f 1 1 r N L R I A m s t e r d a m

Figure 3-12: Crack growth due to 2000 N cyclic load

3.3.3 Case 3 Results

For a niaximum load of 2000 N, it took 2721 cycles for the crack to grow fiom a Iength of 40.0

nzrrz to 58.0 mm. The simulation was ended at this point because Forman's Law was no longer

valid. The finite correction factor Q was also not accurate to within 0.5% because the crack

length a had grown to be a significant portion of the width W. The effect of the increasing

crack Imgth wodd be to increase the tension in the remaining material in the plate sipificantly

more than could be accounted for by the Q factor. The crack would thus probably take fewer

cycles to reach the 58.0 mm mark than predicted. A larger sample would be more appropriate

for tracking crack growth. This case was studied as a follow-up to Case 1. The existing Case 1

sample can be loaded Further and behaviour compared to the computer results. Fig.3-12. 3-13

and 3-14 show several runs utilizing different maximum loads.

3.3.4 Case 3 Analysis

For al1 three load cases, the cracks appear to grow exponentially. Even though Forman's

law and thus the computer program is invalid for longer crack lengths, the plate would have

Figure 3-13: Crack growth due to 3000 N cyclic load

Figure 3-14: Crack growth due to 4000 N cyclic load

65

presumably failed by this point. The initial stages of crack growth are also the most important

in determining the life of the structure. By using the methods described in Section 3.1.4 to

determine crack length, the structure can De monitored and a warning given before the crack

grows to a critical size, Le. before the crack 'rounds the corner' in the exponential cunre.

Chapter 4

Experiment al Result s

4.1 Unpatched Single-Edge Cracked Specimen

A specirnen similar to the computer simulation was created to measure the strain field near a

crack tip. A notch was made to simulate a crack in the aiuminum. The initial cut was 0.5 mm

wide and 3.5 cm long and done using a standard disc saw. The last 0.5 cm of the crack was cut

by hand using a jeweller's saw. A 7/0 gauge blade, which corresponds to a width of 0.18 mm,

was used. The total length of the crack is 40 n m . A very thin crack tip was desirable since the

computer mode1 is based on an infinitely thin crack tip. However, under loading, thin cracks

may emanate from stress concentrations in these notches. These cracks are as thin as c m be

practicalIy made and thus the initial width cf the notch is not as important.

4.1.1 Bragg Grating Sensors

Currently, only two non-chirped gratings, 6 mm and 9 mm long are available. They both have

reflectivities under 30% and reflect wavelengths near 1543 nnz as shown in Fig.4-1 and42.

The specimen d l 1 be fitted with Bragg grating sensors running dong the y direction as shown

in Fig.4-3. Since the strain field should be symmetric across the x-axis, the gratings will only

measure the strain in the positive y-direction. The start of the grating will be at y = -1 mnt

because of uncertainty in the location of the crack tip. Bond lines should be approximately

1 mm nide. Care will be taken to ensure that bond lines are as thin as possible and do not

overlap.

SPAN 4 .812 ni j ST 388 i s e c i

1

Figure 4-1: Characteristics of grating #l

CENTER 1 5 W . 9 9 1 nm SPRN % H l 2 nm 430 0.88 nia U0 200 Hz S ST 300 i s e c

Figure 4 2 : Characteristics of grating #2

Crack

Sensor 1 :

Sensor x = 4mm y= -0.1 to

Figure 4-3: Bragg grating sensor locations

4.1.2 Static Load Strain Profiles at the Crack Tip

Loading will be the same as in Case I and 2 of the computer model. The load will be 1000 .V

which will cause negligible crack growth: 155 nm for 1 cycle of 1000 N.

1-d Measurements

By placing two sensors dong x = 2 m and x = 4 mm, results can be compared to Case 1

of the computer model. The strains are al1 in the hundreds of p~ which are easily detected by

thesc Bragg gratings which can measure in the tens of pz.

2-d Measurements

LYith more sensors: a grid of sensors c m be placed ahead of the crack tip. By interpolating

results. a 2-d strain field can be constructed.

Cornparison

Comparisons ndl be made to the Case f and Case 2 computer models for both the 1-d and 2-d

measurements.

4.1.3 Dynamic Load and Crack Growth

The specirnen will be further loaded. The loading will now be cyclic as in the Case 3 computer

model.

1-d and 2-d Measurements

Crack growth can be tracked by using one fiber sensor or several. As the crack grows tonyard the

sensor, the change in the 1 or 2-d strain field can be detected. The location of peak strain can

be tracked! giving an indication of crack length. This can be done with only 1 Bragg grating

sensor positioned perpendicular to the crack. Comparisons c m be made to the computer model

Case 3 as well as to visual measurements of the crack length. The crack can also be allowed to

grow past the optic fiber sensor. As long as the crack width remains small enough not to break

the sensor, useful data may stiil be obtained. For multiple sensors, cracks can be ailowed to

grow past several fibers to the point where some fibers break. These fiber sensors then act as

light/no-light sensors and the point at which they break can be correlated to the distributed

strain field ahead of the crack.

Chapter 5

Conclusions

Bragg grating sensors can measure strain within several tens of Knowing this, and using

the computer program, it has been determined theoretically that a strain perpendicular to a

crack 4 cm long in an aluminum plate of dimensions s h o m in Fig.3-1 can best be measured by

a sensor located perpendicular to the crack and 2 mm from the crack tip. At this distance, a

difference of several hundred p~ exists between the peak strain and the strain locâted at y = O

rrrn2. The peak can therefore be found and tracked in a short 5 cm long Bragg grating, which

minimises cost. The program illustrated that tracking this peak strain was the best met hod for

determining crack length. The relationship is l inea and offers an elegant method of tracking

crack gron-th over time. The sensor can even act as an intensiometric sensor in the extreme

case of the crack growing to a size that would break the sensor. In this case, the loss of light

reflected back to the demodulation system wodd indicate the break. This scenario also satisfies

the limitations of the LEFM equations used in the program as discussed in the Assumptions

sections of Case 1 through 3.

The obstacle to verifying this experimentally has been the inability of the Bragg grating

sensing system to differentiate strain in the plate from interference in the system. The location

of the peak strain cannot be determined accurately. With better demodulation techniques to

filter out this interference, this system should be viable.

Bibliography

Il] Material Data of High-Strength Aluminium Allogs /or Dumbility Eualvation of Structures.

Aluminium-Verlag, Dusseldorf, 1986.

131 R. D. Adams and N. A. Peppiatt. J. Strain Anal., 83134, 1973.

131 R. D. Adams and N. A. Peppiatt. J. Strain Anal., 9:185, 1974.

141 Robert D. Adams and William C. Wake. Structural Adhesive Joints in Engineeràng. Else-

vier Applied Science Publishers, New York, 1984.

[5] D. J. Allrnan. A theory for elastic stresses in adhesive bonded lap joints. Quarterly J.

Mechanics Appl . Maths., 30:415, 1977.

[ G ] A. -4. Baker and R. .Jones. Bonded Repair of Aircraft Structures. Martinus Sij hoff Pub-

lishers, Boston, 1988.

[7! 3. K. Benson. Influence of stress distribution on joint st rengt h. Adhesion-Fundamentals

and Practice.

181 Jason Bigue. Development of a novel serially multiplexed fiber bragg grating sensor system

using fourier analysis. Master's thesis, University of Toronto, 1997.

[9] hl. R. Bowditch. The durability of adhesive joints in the presence of water. Int. J. Adhesion

and A dhesives, 16:73, 1996.

j l O ] C.D. Butter and G.B. Hocker. Fiber optics strain gauge. Applied Optics, 17(18):2867.

1978.

jll] J e r l J.S. Chwang. Smart patch sensing racl. Tedinical report, University of Toronto

Institute for Aerospace Studies.

[I2] A. D. Crocombe and R. D. Adams. Journal of Adhesion, 12(2):127, 1981.

[13] A. D. Crocombe and R. D. Adams. Journal of Adhesian, 13(2):141, 1981.

[14] J.R. Dixon, J.S. Strannigan, and J. McGregor. Stress distribution in a tension specimen

notched on one edge. Journal of Stmin Analysis, 4:27, 1969.

! 151 Graham Duck. Bragg Gmting Sensors: Implementations and Applications. PhD thesis,

University of Toronto, to be completed in 1999.

il61 M. Goland and E. Reissner. The stresses in cemented joints. J. Appl. Mech., Ran. , ll:A17,

1944.

[lï] L. .I. Hart-Smith. Stress analysis: A continuum mechanics approach. In A. J. Kinloch,

editor, De.velopments in Adhesives, volume 2, chapter 1.

[18] S. Huang, M. LeBlanc, M. Lowery, R. Maaskant. and R.M. Measures. Distributed fiber

optic strain sensing for anchorages and other applications, 1996.

j19] G.R. Invin. Analysis of stresses and strains near the end of a crack traversing a plate. J.

Appl. Mech., Tkan., 24:361, 1957.

[?O] D. Kutscha and K. E. Hofer. Feasibility of joining advanced composite flight vehicle struc-

tures. Technical Report AD 690 616, ITT Research Inst., 1969.

[21] Michel LeBlanc. A prototype fibre optic damage assessrnent system for an aircraft com-

posite leading edge. M aster's t hesis, University of Toronto, 1990.

[Z] Murray Lowery. A study on fiber optic bond integrity measurements. Master's thesis,

University of Toronto, 1996.

[23] Raymond M. Measures. Fiber optic sensing for composite smart structures. Composites

Engineering, 3:715, 1993.

[24] S. A. Meguid. Engineering Fracture Mechanics. Elsevier Applied Science, New York, 1989.

[25] Ian Munm. Adhesive bond inspection using nondestructive test ing. Materials World,

3:527. 1995.

[26] Y. Murakami, editor. Stress zntemzty factors handbook* volume 1 and 2. Pergamon, New

York, 1987.

[27] X.I. Muskhelishvili. Some B u i c Pmblems of the ~lfathematical Theory of Elasticity. Noo-

ordhoff. Leiden, 1953.

i28j Paul C. Paris and George C. Sih. Stress analysis of cracks. Fracture Toughness Testing

and Its Applications.

;29] 3. A. Peppiatt. Stress analysis in adhesive joints. PhD thesis, University of Bristol, 1974.

[30] W. J. Renton and J. R. Vinson. The analysis and design of composite material bonded

joints under static and fatigue loadings. J. Adhesion, 7.

i31j Bahaa E. A, Saleh and Malvin Car1 Teich. Fundamentals of Photonics. John Wiley &

Sons, Inc., Toronto: 1991.

13-] E. W. Thrall. The primaq adhesively bonded structure technology program. page 1 ? 1980.

[33] 0. Volkersen. Die nietkraftverteilung in zugbeanspruchten nietverbindungen mit knon-

stanten laschenquerschnitten. Luftfahrtforschung, 15:4, 1938.

j34] HM. Westergaard. Bearing pressures and cracks. J. Appl . Mech., 6:49, 1939.

Appendix A

Comput er Program

The cornputer prograrn used to analyse the theoretical strain field in cracked plates. both

patched and unpatched and subject to both static and dynamic loads was written using Lab-

WEW 4. LabVIEW is a graphical computing program which provides a graphical user interface

(GUI) for controlling input variables as well as being able to display results in the f o m of graphs

and tables. Both the interface and the graphs are created as a matter of course during the pro-

grarnrning process.

A. 1 Graphical User Interface

Crack . vi Last rnodified on 1/26/99 at 12:17 PM

. OOOE.0 rack Excenslon F Crack EXLenalcn f o r c e 4

:rd? , B@ ] ave s r r a a n ave crack va normall r v c crack vr

6-D Scrain F l c l d r & d Crack ?in

Crack . vi L a s t modified on 1/26/99 at 3:54 AM

- - - ---- - --A ---- - - -- -- -

Connector Pane)

Y

Crack. vi r a y e

Controls and Indicators

Plate Material

ta [ml tP Iml tr tml A [ml B [ml Biaxial ai [ml Stop Button Ratio of M i n J M a x .

ae [ml Width [ml Crack Geometry Max. r [*IOA-4mJ

Stress

Max. deg fram x-wis Reinf orcement Material Reinforced? Cyclic ~oading? Coordinates ~ax.x-coordinate [lOA-4m] Max. y-coordinate [IOA-4m] x [IOA-&ml Load [NI Increment Number for calculations Location of fiber eensor [ml r [*IOA-4m] Save strain profile? Save crack vs. normalized peak? Save crack va. peak location? b [mA-11 s

Crack. vi Last modified on

D

sinf [Pa]

Load Lap Joint ~ransfer Length

Ga [Pa] Crack Extension C r a c k Extension

Force Force

Characteristic Crack Length L [ml

Khax < Kf? Use Forman's Law?

a+ [ml a [ml number of cycles' C r a c k Length vs. m e r of Sinusoidal Loading, R=lO%

r a y e E$ C r a c k . vi Last modified on 1/26/99 at 3:54 AM

Correction factor accurate to O.5%?

Strain Matrix2 1 1DBL-j 1 Strain y Strain Matrixl

Degtees froan x-axis

Sttain in y-Direction Constant Axial Location x

r/a less than 16%?: stresses accurate to approximately 10%

2-D Strain Field Near Crack Tip Static Load lOOON

KI Stress ~ntensity Factor K

sO/sy<0.8? Elastic Approximation Valid?

normalized strain location of mrry strain Er [Pa]

A.2 Computer Code

Crack. vi Last modified on 1/26/99 at 3:54 AM

Block Diagram

Last modified on 1/26/99 at 3:54 AM

:rack .vi ~ a s t modified on 1/26/99 at 3:54 AM Last I

- rack Lemch v s . inusordsl toadinq. R=

I

C r a c k . vi L a s t modified on 1/26/99 at 3:54 AM

'"Y'

C r a c k . vi L a s t modified on 1/26/99 at 3 :54 AM

P o - , & - - - r - 1 ---

I vat rd?

- rack EZtanslln Farce ,I a)

crack. vi L a s t modified on 1/26/99 at 3 : 5 4 AM

pl t r ror

- - -- - - - - - - - - - -- - - -- - -- - - - -

Position in Hierarchy

L a s t rnodified on 1/26/99 at 3 : 5 4 AM

cerisclc Crack k n g - r. c [p. -

rack. vi ast rnodified on 1/26/99 at 3:54 AM

C r a c k . vi Last modified on 1/26/99 at 3:54 AM

.

L i s t of SubVIs

t : : . ~ Beep . vi 161 C : \LABVIEW\vI. 1ib\Utility\syçtem. lIb\~eep.vi

-- History

"Crack.vi History" Current Revision:

NOTE TO USERS

Page(s) not included in the original manuscript are unavailable from the author or univemity. The

manuscript was microfilmed as received.

This reproduction is the best copy available.

UMI

Appendix B

Raw Data Output

This data is obtained hom the computer program when the user chooses to Save the raw data.

The data is saved as columns of numbers separated by tabs. These fiIes are then imported into

Microsoft Excel 95 for analysis.

B.l Crack length versus N o r d s e d peak strain

2000N Crack length Nomalised peak strain

1.00E+00 1 -0ûE-02 1 -01 E +O0 1 .O1 E-02 1.02€+00 1.02E-02 1.02€+00 1.03E-02 1.03E+00 1.04E-02 1.04E+00 1 .OSE02 1.04E+00 1.06E-02 1.05E+00 1 -07E-02 1,06E+00 1.08E-02 1.07€+00 1.09E-02 1.07E+00 1.1 0502 1.08€+00 1.1 1E-02 1.09E+00 1.1 2E-02 1.1 0E+00 1.1 3E42 1 -1 0E+00 1.1 4E-02 1.1 1 €+O0 1.1 5E-02 1 -1 2Ei-00 1.1 6E-02 1.1 3E+00 1.1 7E-02 1.1 4E+00 1.1 BE42 1 -1 4E+00 1.1 9E-02 1 -1 5E+00 1.20E-02 i -1 6E+00 1.21 €42 1.1 7 E 4 0 1 22E-02 1.1 8E+00 1.23E-02 1.19E+00 1.24E-02 1.20€+00 1.25E-02 1 -21 E+00 1.26E-02 1.21 E+00 1.27E-02 1.22EdIO 1.28E-02 1.23Ei00 1.29E-02 1.24E+ûO 1.30E-02 1.25€+00 1.31 E-02 1.26E+00 1.32E-02 1.27€+00 1.33E-02 1.28€+00 1 34E-02 1.29E+00 1.35E-02 1.30E+00 1.36E-02 1.31 €+O0 1.37E-02 1.32€+00 1.38E-02 1.34€+00 1.39E-02 1.35€+00 1.40E-02 1.36E+OO 1.41 €42 1.37€+00 1.42E-02 1.38€+00 1 A3E-02 1.39E+00 1 A4E-02 1.41 E+00 1.45E-02 1.42E+00 1.46E-02 1.43E+ûO 1.47E-02 1.44EMO 1.48E-02 1.46€+00 1.49E-02

5000N Crack length Nonnalised p

1 .oQE+00 1 .O1 E+OO 1.02E+ûû 1.02E+00 1.03€+00 1 .04E+ûû 1.04€+00 1 .OSE+00 1.06€+00 1.07E+00 1.07E+00 1.08E+ûû 1.09E+00 1.1 OEm 1.1 0 E m 1.11E+00 1.1 2E+00 1.1 3E+00 1 .14E+00 1 .14E+00 1.1 SE- 1.1 6E+00 1.1 7E+00 1.1 8E+ûû 1.1 9E+00 1.20€+00 1.21 €+O0 1.21 E+OO 1.22E+00 1.23E+ûû 1.24€+00 1.25E+ûû 1.26E- 1.27E+ûû 1.28E+00 1.29E+00 1.30E+ûû 1.31 E+ûû 1.32E+00 1.34E+00 1.35E+ûû 1.36E+ûû 1.37€+00 1.38€+00 1.39E+00 1.41 E+00 1.42E+ûû 1.43E+ûû 1.44€+00 1.46€+00

ieak strain 1.00E-02 1.01 E-02 1.02E-02 1 -03E-02 1.04E-02 1 -05E-02 1.06E-02 1.07E-02 1.08E-W 1 -09E-02 1.10E-02 1.11E-02 1 -12E-02 1 .l3E-O2 1 -14E-02 1.1 5E-02 1 -16E-02 1.1 7E-02 1 -18E-02 1.1 9E-02 1.20E-02 1 -21 E-02 1 -22E-02 1 -23E-02 1 -24E-02 1 -25E-02 1 -26E-02 1 -27E-02 1 -28E-02 1 -29E-02 1.30E-02 1 -31 E-02 1.32E-02 1 -33E-02 1.34642 1 -35E-02 1.36E-02 1 -37E-02 1.38E-02 1.396-02 1.40E-02 1 -41 €42 1.42E-02 1 -43E-02 1 .44E42 1 -45E-02 1.46E42 1.47E-02 1 M E 4 2 1.49E-02

B.2 Crack length versus Peak strain location dong y- axis

The graphs for these cases only utilised every third data point in order to illustrate the trend

lines more clearly.

2000N Crack Length y-location of peak strain

1.25E-O2 1.00E-02 1.24E-02 1 .O1 €42 1.23E-02 1.02E-02 1.22E-02 1 -03E-02 1.21 E-02 1 .ME-02 .20E-02 1.05E-02

1.1 9E-02 1 -06E-02 1 -1 8E-02 1.07E-02 1.1 7E-02 1.08E-02 1 -1 6E-02 1 -09E-02 1.1 SE-02 1.10E-02 1.1 4E-02 1.1 1E-02 1.13E-02 1.1 2E-02 1.1 2E-O2 1 -1 3E-02 1.1 1 E-02 1 .14E-02 1.1OE-O2 1 -1 5E-02 1.09E-02 1.16E-02 1.08E-02 1 -17E-02 1.06E-02 1.18E-02 1 .OSE-02 1 -19E-02 1.04E-02 1.20E-02 1.03E-02 1.21 E-02 1.02E-O2 1 22E-02 1.01 E-02 1.23E-02 1 .OOE-O2 1.24E-02 9.90E-03 1.25E-02 9.80E-03 1.26E-02 9.70E-03 1.27E-02 9.60E-03 1 -28E-02 9.50E-03 1.29E-02 9.40E-03 1.30E-02 9.30E-03 1 -31 €102 9.20E-03 1.32E-02 9.1 O€-03 1.33642 9.00E-03 1 -34E-02 8.90E-03 1.35E-02 8.80E-03 1 -36E-02 8.70E-03 1.37E-O2 8.60E-03 1 -38E-02 8.50E-03 1 -39E-02 8.40E-03 1.40E-02 8.20E-03 1.41 € 4 2 8.1 OE-03 1.42E-02 8.00E-03 1 M E 4 2 7.90E-03 1 -44E-02 7.80E-03 1 -45E-02 7.70E-03 1.46E-02 7.60E-03 1.47E-02 7.5OE-03 1 A8E-02 7.40E-03 1 -49E-02

5000N Crack Length y-location of peak strain

1.25E-02 1.00E-02 1.24E-02 1 -01 E-02 1.23E-02 1.02E-02 1 22E-02 1.03E-02 1.21 E-02 1.04E-02 1.20E-02 1.05E-02 1 -19E-02 1 -06E-02 1 -18E-02 1 -07E-02 1.17E42 1.08E-02 1 -16E-02 1.09E-02 1.1 SE42 1 -1 0E-02 1.1 4E-02 1 .11E-02 1 .13E-02 1.1 2E-02 1.1 2E-02 1 -1 3E-02 l.llE-02 1.1 4E-02 1 .1 0E-02 1 -1 SE42 1.09E-02 1 .16E-02 1 -08E-02 1 -17E-02 1 ME-02 1.18E-02 1.05E-02 1.19E-02 1.04E102 1.20E-02 1.03E-02 1.21 E-02 1.02E-02 1.22E-02 1.01 E-02 1 -23E-02 1.00E-02 1.246-02 9.90E-03 1.2SE-02 9.80E-03 1.26E-02 9.70E-03 1.27E-02 9.60E-03 1 -28E-02 9.50E-03 1 -29E-02 9.40E-03 1 -30E-02 9.30E-03 1.31 E-02 9.20E-03 1.32E-02 9.10E-03 1 -33E-02 9.ûûE43 1 34E-02 8.9ûE43 1.35E-02 8 &O€-Q3 1 -36E-02 8.70E-03 1 -37E-02 8.60E-03 1 38E-02 8.SOE43 1.39E-02 8.40E43 1.40E-02 8.20Ed3 1 -41 €42 8.10E-03 1.42E-02 B.OOE-03 1 A3E-02 7.90E-03 1 .ME42 7.80E-03 1.45E-02 7.70E-03 1 -46E-02 7.60E-03 1.47E-02 7.SOEQ3 1 .48E42 7.40E-03 1 -49E-02