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ADVANCES IN STRUCTURAL DAMAGE ASSESSMENT USING
STRAIN MEASUREMENTS AND INVARIANT SHAPE DESCRIPTORS

By

Amol Suhas Patki

A DISSERTATION
Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of
Doctor of Philosophy

Mechanical Engineering

2010

ABSTRACT

ADVANCES IN STRUCTURAL DAMAGE ASSESSMENT USING
STRAIN MEASUREMENTS AND INVARIANT SHAPE DESCRIPTORS

By
Amol Suhas Patki

Energy conservation has become one of the most important topic of engineering
research over the last couple of decades all around the world and implies reduced energy
consumption in order to preserve rapidly depleting natural resources. Along with
development of fuel-efficient power plants and technology utilizing alternate fuel to
traditional fossil fuels, the design and manufacturing of light-weight energy-efficient
structures plays a major role in energy conservation. However this reduction in material
and/or weight cannot be achieved at the expense of safety. Thus it is essential to either
increase the confidence in the analysis of mechanics of traditional isotropic materials to
reduce safety factors or develop new structural materials, such as fiber-reinforced (FRP)
polymer matrix composites, which tend to have a higher strength to weight ratio. This
doctoral research work will focus on two problems faced by the structural mechanics
community viz. effects of closure and overloads on fatigue cracks and structural health
monitoring of composites.

Fatigue life prediction is largely empirical which in recent years has been shown
to be a conservative design model. Investigation of crack growth mechanisms, such as
crack closure can lead to design optimization. However, the lack of understanding and
accepted theories introduces a degree of uncertainty in such models. Many of the

complexity and uncertainty arise from the lack of an experimental technique to quantify

crack closure. In this context, this research work offers the most compelling evidence to
date of the effects of overload retardation and a confirmation of the Wheeler model using
direct experimental observations of the stress field and crack tip plastic zone with the aid
of thermoelastic stress analysis.

On the other hand, the uncertainties in the post-damage behavior of energy saving
FRP-composite materials increase their capital cost and maintenance cost. Damage in
isotropic materials tends to be local to the area surrounding the damage, while damage in
orthotropic materials tends to have more global repercussions. This calls for analysis of
full-field strain distributions adding to the complexity of post-damage life estimation.
This study explores shape descriptors used in the field of medical imagery, military
targeting and biometric recognition for obtaining a qualitative and quantitative
comparison between full-field strain data recorded from damaged composite panels using
sophisticated experimental techniques. These descriptors are capable of decomposing
images with 103 to 106 pixels into a feature vector with only a few hundred elements.
This ability of shape descriptors to achieve enormous reduction in strain data, while
providing unique representation, makes them a practical choice for the purpose of
structural damage assessment. Consequently, it is relatively easy to statistically compare
the shape descriptors of the full-field strain maps using similarity measures rather than
the strain maps themselves. However, the wide range of geometric and design features in
engineering components pose difficulties in the application of traditional shape
description techniques. Thus a new shape descriptor is developed which is applicable to a
wide range of specimen geometries. This work also illustrates how shape description

techniques can be applied to full-field finite element model validations and updating.

ACKNOWLEDGEMENTS

I would like to take this opportunity to express my sincere and heartfelt thanks to my
advisor Dr. Eann Patterson (Professor, department of mechanical engineering and
department of chemical engineering and material Science, Michigan State University,
East Lansing, Michigan and Director, Composite Vehicles Research Center) for trusting
me and believing in me to complete this doctoral dissertation. I would like to thank him
for offering me a research assistantship under his guidance which is nothing less than an
honor while his mentoring and constructive criticism has helped me to successfully reach
my professional and personal goals. I also thank him for providing me with the much
needed financial support for completing my graduate studies at Michigan State
University (MSU). I have learnt a lot about composite materials, experimental solid
mechanics, fracture mechanics and related topics by working under his guidance on this

dissertation. No written words can express my gratitude and respect for him.

I would also like to thank Prof. Dahsin Liu, Prof. Carl Boehlert and Prof. Ronald Averill
for agreeing to be a part of my doctoral dissertation committee and provide valuable
inputs and suggestions. I like to say a special thank to Dr. Dahsin Liu for providing me
with the fiber-reinforced polymer composite specimen for this study and he also happens
to be one of my favorite teachers. I would also like to thank all the faculty and
researchers working at the composite vehicle research center for their support and
encouragement. I would finally like to thank Michael McLean for his help in the machine

shop.

iv

The support from my family has always been comforting throughout my life for which I
would like to thank my parents Shobha Patki and Suhas Patki, my brother Satyajit Patki
and my sister-in-law Dipti Patki. I thank all my friends from Michigan for their support

and for making my stay at MSU enjoyable.

Finally, I would like to thank the love of my life, my fiance Stefanie Nowak for her

companionship, affection and support.

TABLE OF CONTENTS

LIST OF TABLES ........................................................................................................... viii
LIST OF FIGURES ........................................................................................................... ix
NOMENCLATURE: PART-A ...................................................................................... xviii
NOMENCLATURE: PART -B ........................................................................................ xix
1 Introduction ................................................................................................................ 1
2 Part - A: Thermoelastic stress analysis of fatigue cracks subject to overloads ......... 7
2.1 Summary ............................................................................................................. 7
2.2 Literature review ................................................................................................. 8
2.3 Thermoelastic stress analysis - principles & equipment ................................... 11
2.4 Thermoelastic stress analysis - data processing for crack analysis ................... 14
2.5 Experimental details and results ....................................................................... 21
2.6 Discussion ......................................................................................................... 27
2.7 Conclusions ....................................................................................................... 38
2.8 Future work ....................................................................................................... 39

3 Part B: Structural damage assessment in fiber-reinforced composite using shape
analysis techniques .................................................................................................. 42
3.1 Summary ........................................................................................................... 42
3.2 Literature review ............................................................................................... 42
3.3 Experimental technique selection ..................................................................... 59
3.3.1 In-plane Moiré .......................................................................................... 60
3.3.2 Thermo—Photoelasticity ............................................................................. 62
3.3.3 Digital Image Correlation ......................................................................... 64
3.3.4 Rational decision making model ............................................................... 69
3.4 Specimens ......................................................................................................... 72
3.5 Experimental set-up and method ...................................................................... 76
3.6 Shape analysis ................................................................................................... 80
3.7 Zemike moments .............................................................................................. 83
3.7.1 Technique .................................................................................................. 83
3.7.2 Rotational invariance ................................................................................ 89
3.7.3 Mapping a polygon to a circle .................................................................. 90
3.7.4 Results & discussion ................................................................................. 92
3.7.5 Advantages, limitations and solutions .................................................... 103
3.8 Fourier descriptors .......................................................................................... 105
3.8. 1 Technique ................................................................................................ 105
3.8.2 Elliptical shape descriptor and clustering ............................................... 108
3.8.3 Results & discussion ............................................................................... 116

vi

3.8.4 Advantages, limitations and solutions .................................................... 119

3.9 Fourier — Zemike moments ............................................................................. 120
3.9.1 Technique ................................................................................................ 121

3.9.2 Results & discussion ............................................................................... 122

3.9.3 Advantages, limitations and solutions .................................................... 126

3.10 Discussion ....................................................................................................... 130
3.13 Conclusions ..................................................................................................... 167
3.14 Future work ..................................................................................................... 171

4 Concluding remarks ............................................................................................... 177
BIBLIOGRAPHY ........................................................................................................... 196
APPENDIX—A: Source code for ImPaCT ...................................................................... 185

vii

LIST OF TABLES
Table 1: Test Parameters and applied overloads ............................................................. 21
Table 2: Essential Attributes for Technique Selection ...................................................... 70
Table 3: Soft attributesfor technique selection ................................................................ 71
Table 4: F iber—reinforced polymer composite specimen .................................................. 73
Table 5: Finite element -model mesh parameters ........................................................... 170

viii

LIST OF FIGURES

“Images in this dissertation are presented in color. ”

Figure 1: Phase difference, A between signal measured by the thermoelastic stress
analysis system and the forcing signal, as a map (bottom) around a crack
growing from left to right together with a section (top) through the map
along a horizontal line through the crack tip. The regions A, B and C
indicate the elastic field, the cyclic plastic zone and the crack wake
respectively. The approximate pixel resolution is 14 pixels/mm. .................... 15

Figure 2: A map of the thermoelastic signal, S (DT units) obtained from the
thermoelastic stress analysis system around a crack growing from left to
right; with a superimposed plot of (I/Smax)2 as a function of the
perpendicular distance from the crack tip, y. The regions P, S and N
indicate the cyclic plastic zone, the singularity dominated elastic zone and
non-singularity dominated elastic zone respectively. The approximate
pixel resolution is 14 pixels/mm. ...................................................................... 17

Figure 3: three-dimensional representation of the phase difference data from figure
I. The approximate pixel resolution is 14 pixels/mm. ..................................... 18

Figure 4: Crack tip plastic zone obtained by applying a binary filter to the phase
difference data to identify the negative values; the white area in the binary
image indicates the region with cyclic crack tip plasticity. The
approximate pixel resolution is 14 pixels/mm. ................................................ 19

Figure 5: Specimen geometry showing the strain gage rosette which was bonded
15mm to the right of the notch and 15mm above the crack line (all units
are in ‘mm’). .................................................................................................... 20

Figure 6: Stress intensity factors and plastic zone area as a function of crack length
for a typical specimen (# 7 in Table 1) subject to a constant amplitude
loading. ............................................................................................................ 24

Figure 7: Stress intensity factors and plastic zone area as a function of crack length
for a typical specimen (#4 in Table 1) subject to a 50% overload for ten
cycles at a crack length of 25 mm. ................................................................... 25

Figure 8: Radius and area of plastic zone as a fimction of crack length for

specimen#4 which was subjected to a 50% overload for 10 cycles at a
crack length of 25 mm. ..................................................................................... 29

ix

Figure 9: Crack growth curves for typical specimens subjected to a 50% overload for
a single cycle (specimen#I) and for 10 cycles (specimen#4) at a crack
length of 25mm; with data before and after the application of the overload
fitted with trend lines to highlight crack growth retardation. ......................... 30

Figure 10: Stress intensity factor at 200 cycles after the application of overload as a
fimction of percentage overload. ..................................................................... 31

Figure 11: Crack tip plastic zone size at 200 cycles after the application of overload
as a function of percentage overload. .............................................................. 32

Figure 12: Wheeler exponent, m estimated from thermoelastic stress analysis data
using expression (10) and shown as a function of percentage overload. ........ 33

Figure 13: Experimental crack growth rate curve for a typical specimens subjected to
a 50% overload for a single cycle (specimen#l) superimposed with the
corresponding Wheeler model estimation of crack growth rate obtained
using data from thermoelastic stress analysis. Some data points (solid
symbols) from the experiment are numbered and some from the model
lettered to indicate the direction of crack growth. ........................................... 36

Figure 14: Experimental crack growth rate curve for a typical specimens subjected to
a 50% overload for ten cycles (specimen#4) superimposed with the
corresponding Wheeler model estimation of crack growth rate obtained
using data from thermoelastic stress analysis. Some data points (solid
symbols) from the experiment are numbered and some from the model
lettered to indicate the direction of crack growth. ........................................... 37

Figure 15: Schematic of experimental setup for in-plane Moiré interferometry with

one pair of coherent beams. ............................................................................. 59
Figure 16: Experimental setup for Thermo-Photoelasticity. ............................................. 62
Figure I 7: Experimental setup for two-dimensional digital image correlation. ............... 67
Figure 18: Experimental setup for three-dimensional digital image correlation. ............ 68

Figure 19: Technique Performance plotted against soft attributes. The soft attribute
of low capital cost was excluded for this analysis since all three
techniques were readily available at the MS U. ............................................... 72

Figure 20: Incrementally damaged composite specimens using a drop-weight testing
machine with increasing impact energy arranged from left-to-right along
with a composite specimens with an impacted hole (top right most) and
with a machined hole (bottom right most). ...................................................... 74

Figure 21 : C -scan image and time of flight maps obtained from ultrasonic evaluation
of the composite specimens with a machined hole (top left and bottom left
respectively) and with a impacted hole (top right and bottom right
respectively). .................................................................................................... 75

Figure 22: Raw digital image correlation images of composite specimen with impact
damage without speckle represented in two color schemes; grey (left) and
Istra (right) ....................................................................................................... 76

Figure 23: Raw digital image correlation images of composite specimen with impact
damage with speckle represented in two color schemes; grey (left image)
and Istra (right image). .................................................................................... 78

Figure 24: Mapping a polygon to a circle ......................................................................... 91

Figure 25: Original image representing the maximum principal strain map obtained
from digital image correlation of a virgin composite specimen without any
damage with a speckle pattern. ........................................................................ 92

Figure 26: Zemike moments evaluated for the maximum principal strain map
obtained from digital image correlation of a virgin composite specimen
without any damage with a painted speckle pattern, shown in figure 25. ....... 94

Figure 27: Reconstructed maximum principal strain map using Zemike moments in
figure 26. .......................................................................................................... 95

Figure 28: Zemike moment descriptor (middle) and reconstructed image (right) of
the original image (left) representing a simulated maximum principal
strain map with uniform strain of 879 ,ustrain. ................................................ 96

Figure 29: Zemike moment descriptor (middle) and reconstructed image (right) of
the original image (left) representing a simulated maximum principal
strain map with uniform strain of 1 ustrain. .................................................... 96

Figure 30: Original image representing the maximum principal strain map obtained
by subtraction of the average strain of the strain map in figure 25 from the

strain map in figure 25. .................................................................................... 97

Figure 31: Zemike moments evaluated for the maximum principal strain map in
figure 30. .......................................................................................................... 97

Figure 32: Reconstructed maximum principal strain map using Zemike moments in
figure 31 ........................................................................................................... 98

xi

Figure 33: Original image representing the maximum principal strain map obtained
from digital image correlation of a virgin composite specimen without any
damage without a speckle pattern. ................................................................... 99

Figure 34: Zemike moments evaluated for the maximum principal strain map
obtained from digital image correlation of a virgin composite specimen
without any damage without a speckle pattern. ............................................. 100

Figure 35: Reconstructed maximum principal strain map using the Zemike moments
in figure 24. .................................................................................................... 101

Figure 36: Original image representing the maximum principal strain map obtained
from digital image correlation of a composite specimen with a machined
hole ................................................................................................................. 102

Figure 37: Reconstructed maximum principal strain map using Zemike moments
obtained for the maximum principal strain map in figure 36 with
maximum order of Zemike moments of Nmax=8. ............................................ 102

Figure 38: Reconstructed maximum principal strain map using Zemike moments
obtained for the maximum principal strain map in figure 36 with
maximum order of Zemike moments of Nmax=20. .......................................... 103

Figure 39: Convergence curves for Zemike moment shape descriptors evaluated for a
composite specimen with a machined hole. ................................................... 104

Figure 40: Two-dimensional representations of the Fourier descriptors in terms of
the phase value map (top right), map of absolute value of the discrete
Fourier transform (bottom left) and a map of the logarithm of the absolute
value of the discrete Fourier transform (bottom right) of the original
image (top lefi) representing a maximum principal strain map obtained
from digital image correlation of a virgin composite specimen without any
damage. .......................................................................................................... 108

Figure 41: ' Three-dimensional representations of the Fourier descriptors in terms of
the phase value map (top right), map of absolute value of the discrete
Fourier transform (bottom left) and a map of the logarithm of the absolute
value of the discrete Fourier transform (bottom right) of the original
image (top left) representing a maximum principal strain map obtained
from digital image correlation of a virgin composite specimen without any
damage. .......................................................................................................... 109

Figure 42: Two-dimensional representations of the Fourier descriptors in terms of
the phase value map (top right), map of absolute value of the discrete
Fourier transform (bottom left) and a map of the logarithm of the absolute
value of the discrete Fourier transform (bottom right) of the original

xii

image (top left) representing a maximum principal strain map obtained
from digital image correlation of a composite specimen with a machined
hole. ................................................................................................................ 1 10

Figure 43: Three-dimensional representations of the Fourier descriptors in terms of
the phase value map (top right), map of absolute value of the discrete
Fourier transform (bottom left) and a map of the logarithm of the absolute
value of the discrete Fourier transform (bottom right) of the original
image (top left) representing a maximum principal strain map obtained
from digital image correlation of a composite specimen with a machined
hole. ................................................................................................................ 11 1

Figure 44: Absolute values of low frequency components of the two-dimensional
discrete Fourier transforms and its elliptical descriptors, evaluated for the
strain distribution in composite specimen with a machined hole. ................. 112

Figure 45: The maximum principal strain distributions in the incrementally damaged
composite specimens under a tensile load of 4000N. .................................... 113

Figure 46: Absolute values of low frequency components of the two-dimensional
discrete Fourier transforms and their corresponding elliptical descriptors,
evaluated for the strain distribution in composite specimens with varying
amount of impact damage. ............................................................................. l 14

Figure 47: Variation in the elliptical descriptor with varying amount of damage in
composite specimen. Specimens I to 4 are the incrementally damaged
composite specimens with increasing impact-energy. Specimen 5 is a
virgin specimen with no damage while specimen 6 and 7 are composite
specimens with an impact hole and machined hole respectively. .................. 115

Figure 48: Fourier Zemike moments with a maximum order of Nmax=20, of the
original image representing a map of maximum principal strain obtained
from digital image correlation of a composite specimen with a machined
hole. The strain map is reconstructed using the reconstructed logarithm
magnitude map and original phase value map of the discrete Fourier
transform. ....................................................................................................... 120

Figure 49: Comparison of reconstructed strain distribution from Zemike moments
(center) and Fourier-Zemike moments (right) evaluated for the maximum
principal strain distribution (left) in the composite specimen with a
machined hole, under a tensile load with a maximum order of Zemike
moments, Nmm = 8 . ....................................................................................... 122

Figure 50: Fourier Zemike moments with a maximum order of Nmax=20, of the

original image representing a map of maximum principal strain obtained
from digital image correlation of a virgin composite specimen. The strain

xiii

map is reconstructed using the reconstructed logarithm magnitude map
and original phase value map of the discrete Fourier transform. ................. 123

Figure 51 : Fourier Zemike moments with a maximum order of Nmax=8, of the
original image representing a map of maximum principal strain obtained
from digital image correlation of a virgin composite specimen. The strain
map is reconstructed using the reconstructed logarithm magnitude map
and original phase value map of the discrete Fourier transform. ................. 124

Figure 52: Comparison of reconstructed strain distribution from Zemike moments
(center) and Fourier-Zemike moments (right) evaluated for the maximum
principal strain distribution (left) in the virgin composite specimen under
a tensile load with a maximum order of Zemike moments, Nmax = 8 . .......... 125

Figure 53: Convergence curves for F ourier-Zemike moment shape descriptors
evaluated for a composite specimen with a machined hole. .......................... 127

Figure 54: Original maximum principal strain distribution in a composite
specimen#I under tensile load of 8000N obtained using digital image
correlation. The composite specimen was damaged in a drop weight
testing machine with an impact-energy of 37.3 joules. .................................. 128

Figure 55: Plot of Fourier-Zemike moments (left) and filtered Fourier-Zemike
moments (right) evaluated for the maximum principal strain distribution
for composite specimen#I illustrated in figure 50. ........................................ 129

Figure 56: Reconstructed maximum principal strain distribution using Fourier-
Zemike moments (left) and those using filtered F ourier-Zemike moments
(right) from figure 50. .................................................................................... 131

Figure 57: Plot of Fourier-Zemike moments evaluated for the maximum principal
strain distribution for composite specimen#I subjected to a drop weight
impact with an impact-energy of 3 7.3 joules, as a fimction of tensile loads
varying from 4501b-force to 18001b-force in 150 lb-force increments. ....... 133

Figure 58: Plot of Fourier-Zemike moments evaluated for the maximum principal
strain distribution for composite specimen#2 subjected to a drop weight
impact with impact-energy of 3 7.3 joules, as a function of tensile loads
varying from 450 lb-force to 18001b-force with 150 lb-force increments. ...134

Figure 59: Plot of Fourier-Zemike moments evaluated for the maximum principal
strain distribution for composite specimen#3 subjected to a drop weight
impact with impact-energy of 3 7.3 joules, as a fimction of tensile loads
varying from 450 lb-force to 1800 lb-force with 150 lb-force increments. ...135

xiv

Figure 60: Plot of Fourier-Zemike moments evaluated for the maximum principal
strain distribution for composite specimen#4 subjected to a drop weight
impact with impact-energy of 37.3 joules, as a function of tensile loads
varying from 4501b-force to 1800 lb-force with 150 lb-force increments. ...136

Figure 61: Plot of Fourier-Zemike moments evaluated for a maximum principal
strain distribution for a virgin composite specimen, as a function of tensile

loads varying from 450 lb-force to 1800 lb-force with 150 lb-force
increments. ..................................................................................................... 139

Figure 62: Plot of Fourier-Zemike moments evaluated for the maximum principal
strain distribution for a composite specimen subject to an impact load
with impact-energy enough for the tup to penetrate through the specimen
forming an impacted hole, as a function of tensile loads varying from 450
lb-force to 1800 lb-force with 1501b-force increments. ................................ 141

Figure 63: Plot of Fourier-Zemike moments evaluated for the maximum principal
strain distribution for a composite specimen with a machined hole, as a
function of tensile loads varying from 450 lb-force to 1800 lb-force with
1501b-force increments. ................................................................................ 142

Figure 64: Plot of Pearson ’s correlation coefi‘icient to measure the similarity between
the F ourier-Zemike moment shape descriptors of the principal strain
distribution in seven composite specimens under tensile loading and those
for a virgin composite specimen with no damage (left). Chart of right plots
the same figure on left excluding the virgin composite specimen and the
composite specimens with an impact and machined holes. ........................... 143

Figure 65: Euclidean distance measured between the F ourier-Zemike moment shape
descriptors of the principal strain distribution of five incrementally
damaged composite specimens under tensile loading and those for a virgin
composite specimen with no damage (left). Chart of right plots the same
figure on left excluding the virgin composite specimen and the composite
specimens with an impact and machined holes. ............................................ 145

Figure 66: Cosine similarity measured between the Fourier-Zemike moment shape
descriptors of the principal strain distribution in five incrementally
damaged composite specimens under tensile loading and those for a virgin
composite specimen with no damage (left). Chart of right plots the same
figure on left excluding the virgin composite specimen and the composite
specimens with an impact and machined holes. ............................................ 146

Figure 67: Similarity measures between the F ourier-Zernike moments evaluated for
the maximum principal strain distribution in the incrementally damaged
composite specimens and those evaluated for the maximum principal
strain distribution in the virgin composite specimen in terms of the

XV

Pearson’s correlation coefiicient, cosine similarity and Euclidean distance
plotted as a function of impact-energy. .......................................................... 147

Figure 68: Plot of Zemike moments evaluated for the maximum principal strain
distribution for composite specimen#I subjected to a drop weight impact
with impact-energy of 3 7.3 joules, as a function of tensile loads varying
from 4501b-force to 1800 lb-force with I501b-force increments. ................ 148

Figure 69: Plot of Zemike moments (excluding the first Zemike moment) evaluated
for the maximum principal strain distribution for composite specimen#I
subjected to a drop weight impact with impact-energy of 3 7.3 joules, as a
function of tensile loads varying from 450 lb-force to 1800 lb-force with
I501b-force increments. ................................................................................ 149

Figure 70: Plot of Zemike moments evaluated for the maximum principal strain
distribution for composite specimen#Z subjected to a drop weight impact
with impact-energy of 3 7.3 joules, as a function of tensile loads varying
from 450 lb-force to 1800 lb-force with I501b-force increments. ................ 150

Figure 71: Plot of Zemike moments evaluated for the maximum principal strain
distribution for composite specimen#3 subjected to a drop weight impact
with impact-energy of 37.3 joules, as a function of tensile loads varying
from 4501b-force to 18001b-force with I501b-force increments. ................ 152

Figure 72: Plot of Zemike moments evaluated for the maximum principal strain
distribution for composite specimen#4 subjected to a drop weight impact
with impact-energy of 37.3 joules, as a function of tensile loads varying
from 450 lb-force to 18001b-force with 1501b-force increments. ................ 153

Figure 73: Pearson ’s correlation coefficient to measure the similarity between the
Zemike moment shape descriptors of the principal strain distribution in
five incrementally damaged composite specimens under tensile loading
and those for a virgin composite specimen with no damage (left). The
chart on the right shows the same figure on left excluding the virgin
composite specimen and specimen#l. ............................................................ 154

Figure 74: Euclidean distance measured between the Zemike moment shape
descriptors of the principal strain distribution in four incrementally
damaged composite specimens under tensile loading and those for a virgin
composite specimen with no damage (left). The chart on the right shows
the same figure on the left excluding the virgin composite specimen and
specimen#I. .................................................................................................... 155

Figure 75: Cosine similarity measured between the Fourier-Zemike moment shape

descriptors of the principal strain distribution in four incrementally
damaged composite specimens under tensile loading and those for a virgin

xvi

composite specimen with no damage (left). The chart of right shows the
same figure on left excluding the virgin composite specimen and
specimen#l. .................................................................................................... 157

Figure 76: Similarity measures between the Zemike moments evaluated for the
maximum principal strain distribution in the incrementally damaged
composite specimens and those evaluated for the maximum principal
strain distribution in the virgin composite specimen in terms of the
Pearson ’s correlation coefficient, cosine similarity and Euclidean distance
plotted as a function of impact-energy. .......................................................... 158

Figure 77: C -scan images obtained from ultrasonic evaluation of non-impact surface
which is also the bottom surface in the ultrasonic test configuration for
composite specimen#I (top left), specimen#2 (top right), specimen#3
(bottom felt) and specimen#4 (bottom right) with varying impact energies. .159

Figure 78: C-scan images obtained from ultrasonic evaluation of impact surface
which is also the top surface in the ultrasonic test configuration for
composite specimen#I (top left), specimen#2 (top right), specimen#3
(bottom felt) and specimen#4 (bottom right) with varying impact energies. .160

Figure 79: Time of flight images obtained from ultrasonic evaluation of composite
specimen#l (top left), specimen#2 (top right), specimen#3 (bottom felt)
and specimen#4 (bottom right) with varying impact energies. ...................... 161

Figure 80: A comparison between damage assessment in incrementally damaged
composites using ultrasonic non-destructive evaluation, stress
concentration factors and shape description similarity measures as a
fimction of impact energy. .............................................................................. 163

Figure 81: A comparison between damage assessment in incrementally damaged
composites using stress concentration factors and shape description
similarity measures as a function of impact energy (left) and size of
delamination (right) obtained using ultrasonic non-destructive evaluation. 166

Figure 82: Similarity measures between the Fourier-Zemike moments evaluated for
maximum principal strain distribution in the finite element model with
increasing mesh density of an aluminum plate with a hole subjected to
tensile loads and those evaluated for the corresponding experimental
maximum principal strain distribution in an aluminum plate with a hole in
terms of the Pearson’s correlation coefi‘icient, cosine similarity and
Euclidean distance plotted as a function of mesh density factor. .................. 175

xvii

(32s.;

m6

{AC/J

max

"I

j

NOMENCLATURE: PART-A

Crack length
Surface emissivity
Wheeler empirical exponent

Paris empirical exponent

Radius of crack tip plastic region

Thermoelastic calibration constant
Paris empirical constant

Specific heat at constant strain

Specific heat at constant pressure

Mode I stress intensity factor
Mode 11 stress intensity factor

Number of cycles

Heat Input

Ratio of 6m," to (rmx

Thermoelastic signal

Maximum thermoelastic signal per row of pixels
Absolute temperature

Linear coefficient of thermal expansion

Strain tensor

xviii

mm
unitless
unitless

unitless

mm

MPa/T S

unitless
J/Kg-K
J/Kg-K
MPa-Vm
MPa-vm

cycles

Cal.

unitless
TS
TS
°C
K-l

u-strain

81&82

AT

A¢

Principal strains

Wheeler retardation parameter
Density

Stress tensor

Principal stress

Yield stress

Poisson Ratio

Phase angle

Temperature change

Change in phase angle

NOMENCLATURE: PART-B

Error function
Pitch of grating

Elliptical descriptor

Normalized elliptical descriptor

Mean value

complex number

order of Zemike polynomial

Thickness of photoelastic coating

xix

u-strain
unitless
kg/m3
MPa
MPa

MPa

unitless

degrees
°C

degrees

N/A
lines/mm

N/A
N/A

N/A

N/A

unitless

mm

u = Displacement in X-direction mm

v = Displacement in Y-direction mm

x, y = Cartesian coordinates mm

A = Thermoelastic calibration constant MPa/T S
D F ( f ) = Fourier transform N/A

D F (4:) = Discrete Fourier transform N/A

D F (u, v) = Two-dimensional discrete Fourier transform N/A
Dn = Photoelastic data MPa
Dw = Wavelet transform N/A

I = Original strain map u-strain
i = Reconstructed strain map u-strain
T = Two-dimensional Fourier transforms of image, I l/mm2
[or = Rotated image N/A
IDIC = Reference image N/A

I j) [C = Current image N/A

K = Photoelastic constant l/MPa
N XM = Size of digital image correlation raw image Pixel2
N x = Fringe order in X—direction unitless
N y = Fringe order in Y-direction unitless
N 9 = Size of feature vector unitless

XX

PxQ

11,171

R,F

Size of strain map
Radial orthogonal polynomial

Radial coordinates in mapped strain map

Thermoelastic signal

Thermoelastic signal

Real valued even Zemike polynomial

Real valued odd Zemike polynomial
Zemike polynomials
Cartesian coordinates in reference image

Zemike moments

Maximum shear strain

Kronecker delta

Strain in X-direction

Strain in Y-direction

Wavelength of light source
radial coordinates
Standard deviation

Standard deviations

Principal stress

Angle of rotation

xxi

 

mm

N/A

mm, degrees

TS

TS

N/A

N/A

N/A

N/A

degrees

unitless

u-strain

u-strain

nm
mm, degrees
N/A

N/A

MPa

degrees

AT
4(1)

Phase angle

Wavelet

Temperature change
Change in phase angle

Pearson’s correlation coefficient

xxii

degrees
N/A

°C
degrees

unitless

1 Introduction

The depletion of reserves of fossil fuels has made it a priority to find ways to either
replace fossil fuels as a source of energy by renewable energy resources or ensure
efficient usage of fossil fuels. According to the Annual Energy Review 2008 (www.cia.
doe.gov/aer/pdf/aer.pdf) by the Energy Information Administration only 3% of the
transportation sector utilizes renewable energy sources while 97% relies on petroleum
products. Also 228% of the world’s produced-energy is utilized by the transportation
sector. The Annual Energy Outlook 2010 (www.eia.doe.gOV/oiam
pdf/0383(2010).pdfi by the Energy Information Administration also suggests that not
much is going to change in the next twenty years with an estimation of only 10% of the
transportation sector utilizing renewable energy with 90% continuing to rely on fossil
fuels. Thus it is necessary to find ways to utilize the available resources of fossil fuels
more efficiently. Some of these ways include hybrid engines, smaller engines that tend
burn lesser fuel, smarter engines that ensure complete combustion of fuels minimizing
any scavenging and lighter vehicles reducing the pay load. In fact, lighter vehicles also
lead to the possibility of using even smaller engines. Thus many of the mechanical
engineers and researchers around the world are working on methods to make automobiles
and aircraft lighter while adhering to safety regulations. These methods include reducing
the factor-of-safety or using fiber-reinforced polymer composites with higher strength to

weight ratios.

Engineering designs include a specified factor-of—safety which varies from 1.1 to 4 or

sometimes even higher depending on factors such as,

1) knowledge of loads, strength, wear and environmental conditions

2) type of expected failure,

3) cost of over—engineering, and

4) quality of/ confidence in engineering analysis
Thus the factor-of-safety is supposed to account for any uncertainties in the
aforementioned factors. Buildings are designed with a factor—of—safety of 2 since the
loads are well studied and known, while pressure vessels are designed with a factor-of-
safety of 3.5 — 4 due to the chance of potentially dangerous catastrophic failures. In the
transportation sector automobile structures are designed with a factor-of—safety of 3 while
aircrafts and spacecrafts are built with a comparatively lower factor-of-safety of 1.5 — 3
since the cost associated with structural weight are quite higher. On the other hand, the
limited use of fiber-reinforced polymer matrix composites is associated with, among
other things the uncertainty of their post-damage load carrying capacity and life
estimation. Thus the standard practice is to replace the damaged composite component at
the slightest detection of damage. This is not a cost effective solution based on material
cost as well as losses associated with down-time. This issue can be addressed by
developing techniques for,

1) developing materials with higher strength and damage resistance,

2) repairing damaged fiber-reinforced polymer composites, and

3) developing techniques for their post-damage life estimation.
This doctoral research concentrates on using shape description techniques to obtain shape
features that uniquely represent full-field strain distributions recorded using state-of—the-

art techniques such as thermoelastic stress analysis and digital image correlation to

provide a basis for full-field comparisons of strain maps. It is proposed that most of the
afromentioned uncertainties in the design of engineering components can be eliminated
or substantially reduced with such comparisons. This will be illustrated by addressing
two well-defined problems faced by the structural mechanics community viz. effects of
closure and overloads on fatigue cracks and structural health monitoring of fiber-

reinforced polymer composites.

Fatigue of engineering materials has been studied by engineers and scientists since the
early 19th century while the phenomenon of crack closure was discovered in the early
1970’s and shown to have a significant impact on crack growth calculations under
random loading. Researchers have shown through their experimental results that the
presence of crack closure, which implies that the crack tip is shielded from the complete
loading cycle, leads to reduced stress intensity values and hence improved fatigue life.
However, the lack of accepted theories makes the inclusion of crack closure in fatigue
life calculations difficult. Among the many causes of crack tip shielding, the size of the
crack tip plastic zone as a result of the stress singularity at the crack tip is one of the most
prominent. The difficulty in studying this phenomenon arises from the fact that there are
no experimental techniques available to measure the size of the crack tip plastic zone and
to study its shape. In this work thermoelastic stress analysis will be used to record the
full-field distribution of the first invariant of stresses surrounding the crack tip in an
Aluminum compact tension specimen. This stress distribution will be represented by a
single factor called the stress intensity factor, K which is a function of specimen

geometry, size and the location of the crack and loading configuration. This stress

intensity factor can be considered an invariant descriptor of the stress distribution if
normalized with the load. This work will also introduce a technique to measure the actual
crack tip plastic zone size in the specimen using phase information of the recorded
thermoelastic stress analysis data. This is understood to be the first time the crack tip
plastic zone size has been measured experimentally. Data recorded at different crack
lengths shows a typical trend in the stress intensity factor as well as the crack tip plastic
zone size. The study of the effect of crack closure and applied overloads shows an
increase in the crack tip plastic zone size which is associated with a corresponding
reduction in the stress intensity factor and crack growth rate. Thus this technique can be
used for a thorough quantitative assessment of the effect of crack closure in different
loading conditions as well as different materials to gain confidence and improved

understanding of the phenomenon.

On similar lines, the full-field strain distribution in damaged composites can provide
useful information about the load carrying capacity of the damaged composite. This is
based on an understanding that the slightest amount of damage, which might be invisible
to the naked eye, will still cause a change in the strain distribution in the composite.
Optical techniques such as thermoelastic stress analysis and digital image correlation
which are capable of recording high resolution full-field strain data are sensitive enough
to detect these changes. However, the ability to not only detect damage but to be able to
differentiate between damage caused due to different types of impact and impact energies
requires comparison of full-field strain maps of composite panels with different level of

damage amongst themselves. A pixel-to-pixel comparison of two such high resolution

strain maps is not practical and is computationally expensive. A solution to this problem
is to employ a technique which will form a basis of comparison between these high
resolution strain maps. The techniques of shape analysis and description were explored
for this puropse. Shape description techniques are capable of uniquely representing two-
dimensional as well as three-dimensional shapes with only a few invariant shape features.
They have been used for feature recognition, medical imagery, military targeting and
biometric recognition for over a decade. In this work it is proposed that shape description
techniques can be used to uniquely represent strain distributions in composites recorded
using digital image correlation. Traditional shape descriptors such as Zemike moments
and Fourier descriptors were investigated and shown to be incapable of representing
generic engineering structural components which might contain discontinuities such as
through holes or rivets. Thus a new shape descriptor viz. the Fourier-Zemike descriptor
was developed by combining Fourier transforms with Zemike moments. This new shape
descriptor was capable of accurate and unique description of strain distribution in
composite specimens with varying geometries. Shape descriptors evaluated for an
incrementally damaged composite specimen with same type of impact damage with
varying impact energies were compared with each other in terms of three different
similarity measures viz. Euclidean distance, the Pearson’s correlation coefficient and
cosine similarity. The sensitivity of these shape descriptors to the change in strain
distributions in the composite specimen with the slightest difference in the intensity of
damage was studied. This shape description technique can be used to uniquely represent
the strain distributions while providing a considerable reduction in data to be analyzed

making their comparisons computationally efficient.

Along with damage detection and quantification, residual life estimation of the structural
component made from fiber-reinforced polymer composite is important. In case of
fracture mechanics the stress intensity factor, K not only represents the strain distribution
in the vicinity of a crack tip but also provides a measure of residual fatigue life.
Similarly, the stress concentration factor in case of stress raisers such as holes, slots, etc.
also provide a measure of the residual life. The similarity parameters discussed above
were studied in view of being able to predict the residual life of the corresponding
composite structural component, similar to the stress intensity and stress concentration

factors.

This doctoral research work introduces computationally efficient novel shape description
schemes to study and compare full-field strain distributions recorded using thermoelastic
stress analysis or digital image correlation without discarding any useful data. These
schemes can be used to study problems such as crack closure effects and post-damage life
predictions of composite structural components as discussed in this work to increase
confidence limits in the application of these theories and/or materials to attain optimal

design of structural components in terms of minimal energy consumption.

2 Part - A: Thermoelastic stress analysis of fatigue cracks subject to

overloads

2.1 Summary

In this study the effects of plasticity-induced crack closure and overloads were
investigated using thermoelastic stress analysis (TSA) in aluminum alloy 2024. The
amplitude of the stress intensity factors were evaluated during crack growth by fitting a
Muskhelishvili-type description of the crack tip stress fields to the thermoelastic stress
analysis data using the multi-point over-deterministic method. A new method to directly
measure the extent of the crack tip plastic zone is proposed based on the phase difference
between the measured temperature and the forcing signal. The presence of crack tip
plasticity was clearly identified and correlated with changes in the stress intensity factor
values obtained from the thermoelastic stress analysis data both during constant
amplitude loading as well as single and multiple overloads. Immediately post-overload
the plastic zone increased in size by up to 50% while the amplitude of the stress intensity
factor and the crack growth rate decreased until the crack grew through the plastic region
created by the overload when the pre-overload value of the crack growth was re-attained.
The experimental data of mode 1 stress intensity factor, crack growth rate and radius of
crack tip plastic zone obtained from thermoelastic data in the region affected by the
overload were used to estimate the exponent in the Wheeler crack growth model which

was found to be a function of percentage overload and the number of cycles of overload.

2.2 Literature review

Crack closure is known to cause retardation of crack growth which implies there is a
potential opportunity to extract more life from a mechanical component with a crack
when crack closure is present during a significant portion of the fatigue life. However
this cannot be at the expense of safety. The phenomenon of crack closure was first
studied and documented by Elber [1, 2] in the 19705 when he studied the effects of crack
tip plasticity on a fatigue crack in an Al 2024-T3 sheet under cyclic tensile loading.
Since then various methods including direct observation, measurement of compliance,
and indirect (non-compliance) thickness averaging techniques have been developed to
study the phenomenon of crack closure [3]. Stanley and Chan [4] were probably the first
researchers to consider using thermoelastic stress analysis (thermoelastic stress analysis)
for studying fatigue cracks. They used Westergaard’s equations [5] to represent the first
stress invariant, which is proportional to the temperature measured in thermoelastic stress
analysis, around the crack tip [4, 6] to evaluate the mode-I and mode-II stress intensity
factors. In thermoelastic stress analysis, stresses are evaluated from minute temperature
changes recorded at an array of points on the object surface instead of using the applied
load or overall deformation of the object. This makes thermoelastic stress analysis a
powerful technique to study local effects such as stress concentrations. This attribute
along with the capacity to capture and process real-time temperature data makes
thermoelastic stress analysis an ideal choice to study fatigue cracks with crack closure.
Stanley and Dulieu-Smith [7] attempted to calculate stress intensity factors from contours
of constant first stress invariant around the crack tip. Their results were in good

agreement with theoretical values for mode 1 cracks but showed only a moderate level of

agreement with theoretical values for mixed mode cases where mode 11 behavior was
dominant. Similar issues existed in the photoelastic characterization of fatigue cracks
until Nurse and Patterson [8] fitted Muskhelishvili-type stress descriptors [9] to
photoelastic data using the multi-point over—deterministic method (MPODM) developed
by Sanford and Dally [10]. Good correlation was obtained between the stress intensity
factors evaluated with this method and those computed using analytical methods for
mode I as well as mixed mode cracks. Tomlinson et al. [11] applied this concept to

thermoelasticity using a system with a single, scanning infrared detector.

The effect of closure was first witnessed using thermoelastic stress analysis by Leaity et
a1. [12] and subsequently by Fulton et a1. [13] who found the evaluated stress intensity
factors to be considerably lower than the values expected from theory. Tomlinson et al.
[14] performed further experiments with a thermoelastic stress analysis system
incorporating a focal-plane array camera and were able to conclude that some form of
closure was detected which tended to decrease the values of AK. when compared with
those calculated using analytical methods. Subsequently Dulieu-Barton et a1. [15]
improved their computational techniques for calculating stress intensity factors from
contours of constant first stress invariant around the crack tip which was done manually
in their earlier work [7]. These improvements gave better agreement between their
experimental and theoretical values including mixed mode cracks. They were also able
to detect the effects of closure with as much as a 12% reduction in the value of AK]. Diaz
et a1. [16] studied the map of phase difference between the applied load and specimen’s

surface temperature and concluded that these maps contained important information

which could be used to identify plasticity in the specimen. They made use of phase
information obtained from thermoelastic stress analysis signals to locate the crack tip in
recorded thermal images of the specimen. Their results showed a reduction in the
difference between the experimental and theoretical stress intensity values with increased
load ratios (R-ratio) demonstrating the sensitivity of thermoelastic stress analysis to the

presence and effects of crack closure.

Following some initial attempts in the 19605 to predict crack growth associated with
overloads and underloads, Elber [2] and, then Wheeler [17], introduced considerably
more elegant growth prediction models based on the Paris law. Wheeler assumed that,
following an overload, a retardation process occurs as the crack propagates through the
plastic zone produced by the overload. Matsuoka et a1. [18] developed a model based on
crack closure concepts including crack blunting and claimed it was different from

Wheeler’s approach. In a similar manner to Elber, Lu et a1. [19] developed a model by

modifying the range of mode-I stress intensity factor, AK] with an empirical constant

which was based on the crack length and the size of the crack tip plastic zone so that the

model has the potential to account for some of the effects on AK] of closure induced by

plasticity. Subsequently, most of the researchers in the field have employed either
Elber’s or Wheeler’s model, or a modifying version thereof, fitted to their experimental
data using the retardation parameters [20, 21]. This process requires an evaluation of the
stress intensity factor and then derived from it, the radius of the crack tip plastic zone. A
number of models can be used to evaluate the stress intensity factor and the plastic zone

size however thermoelastic stress analysis has some significant advantages in this field

10

because, as has been demonstrated previously, the stress intensity factor can be evaluated
directly; and, as will be demonstrated, the radius of the plastic zone ahead of the crack tip
can be measured directly. These two quantities can be obtained from experiments with a
high level of reliability thus allowing the Wheeler exponent to be evaluated more directly

than has been possible before.

In this study fatigue cracks were propagated in A1 2024 CT specimens to study the effects
of crack closure on stress intensity factors and crack growth rate. The methodology
proposed by Diaz et a1. [16] using phase data to locate the crack tip was explored further
by Patki and Patterson [22] to define the extent of plasticity so that the plastic zone size
as well as the stress intensity factor can be calculated as a function of crack length.
Single and multiple cycle overloads with a range of magnitudes were applied to
specimens at the same nominal crack length to study the effect of the overloads on the

size of the crack tip plastic zone and the stress intensity factor [22].

2.3 Thermoelastic stress analysis - principles & equipment

Thermoelastic stress analysis is a non—contact method which uses the thermoelastic effect
to evaluate stresses in a material. In 1853 Lord Kelvin proposed that every material in
nature that undergoes a change in shape and volume as a consequence of mechanical
stress experiences a change in its temperature [23]. When compressed, a material
experiences an increase in temperature while under tension its temperature decreases.

This process of converting mechanical energy to thermal energy and vice-versa can be

11

 

reversible only if the object is stressed within its elastic range and when there is no

significant heat transfer during the loading and unloading.

When reversible and adiabatic conditions are maintained in the object the change in
temperature of its surface is proportional to the first stress invariant, and the relationship

is given by [24]:

 

T 80,-]- Q
ATz— e~ —— (1)
pop: at '1 peg

where, AT is the temperature change, T is absolute temperature of the material, p is the

density, C], is the specific heat at constant pressure, 03-]- the stress tensor, 8,-1- the strain

tensor, Q is the heat input and C8 is the specific heat at constant strain. By applying

cyclic loads of suitable frequencies it can be ensured that no heat conduction takes place
within the object in which case the second term on the right hand side of expression (1)

can be neglected. Then equation (1) can be rewritten as [24]:

AT = "ET—M01 + 02) (2)
pC

P

where, a is the linear coefficient of thermal expansion and 0', & 02 are the principal

stresses. Expression (2) is valid only when the frequency of loading is high enough to
maintain adiabatic and reversible conditions in the object. This limiting frequency of
loading varies with the thermal conductivity of the material as well as the stress and
temperature gradients within the object. The variation in temperature of the surface can

be recorded by using a suitable infrared detector in terms of a voltage output, S.

12

The spectral radiant photon emittance recorded for an object depends on its surface
emissivity e, the absolute temperature of the object T and the type of infrared detector
used. The emissivity of the surface can be maintained constant with an appropriate
surface preparation, such as a thin and uniform coat of flat black paint. The black paint
also provides low surface reflectivity, reducing unwanted reflections into the infrared
camera from the test specimen. Under reversible and adiabatic conditions and uniform
surface emissivity, equation (2) takes the form of the typical thermoelastic expression:
A(O'1+0'2)= AS (3)
where, S is the signal recorded by the infrared detector and A is the thermoelastic constant
which depends on the material properties of the object including, the coefficient of
thermal expansion, density, specific heat capacity under constant pressure and the surface
temperature as well as the detector parameters. Various calibration techniques to obtain
the calibration constant A in expression (3) are described by Dulieu-Smith [25]. The
calibration constant A depends on the radiometric properties of the detector, the material

properties, absolute temperature of the specimen and the emissivity of the surface.

In this study a DeltaTherm®1550 system, produced by Stress Photonics Inc (Madison,
WI) was used. This system has a focal plane array camera capable of capturing 256x320
resolution full-field infrared images using an array of 81,920 Indium-Antimonide infrared
detectors. The camera is capable of taking 400 images per second. These images were
integrated over a specified time to obtain cumulative data captured over a few loading
cycles. The use of an infrared zoom lens provided a maximum spatial resolution of 25pm

with a thermal resolution of 2mK [26]. To relate the temperature changes detected by the

13

camera to change in stress using expression (1) it is essential to correlate the infrared
signal with a signal that is representative of the applied load. This signal is known as the
reference signal and was acquired from the fatigue test machine [26]. The data captured
using the thermoelastic stress analysis system can be represented as a vector where the
magnitude represents the thermoelastic signal which is proportional to the temperature
changes in the object while the angle is the phase shift between the thermoelastic and
reference signals. The phase angle is usually uniform over the object surface unless

adiabatic conditions are lost due to heat conduction and, or generation.

2.4 Thermoelastic stress analysis - data processing for crack analysis

The Muskhelishvili type complex stress descriptors are derived from the mathematical
theory of elasticity for isotropic materials [9]. In this analysis MPODM is used to fit such
mathematical models to the stress field around the crack tip which can be considered to
be divided into three zones: immediately surrounding the crack tip is the crack tip plastic
zone then beyond it is the singularity dominated elastic zone and eventually the far-field
or non-singularity dominated zone is reached. The thermoelastic expression given by
equation 3 is not valid in the plastic region due to loss of adiabatic conditions and
combined with the requirement that Muskhelishvili stress descriptors are applicable only
to elastic behavior imply that it is essential to reliably identify the extent of the plastic
zone around the crack tip. It is also crucial to estimate the maximum extent of the
singularity dominated zone so that the analysis can be restricted to the region in which it
is valid. The co-ordinate system used in the analysis had its origin at the crack tip and

thus locating the crack tip was also important.

14

Qw-

Signal Profile in X-direction

r T T T fl T

Crack Tip . :. .

 

    
 
  

A

A Phase Angle
A
Q

 

1 i

50 100 150 260 250 3&3

 

 

 

 

  
     

 

o 8
A Phase Angle

—L

-10

50100 150 200 250
Pixel

Figure 1: Phase difference, A between signal measured by the thermoelastic stress
analysis system and the forcing signal, as a map (bottom) around a crack growing from
left to right together with a section (top) through the map along a horizontal line through
the crack tip. The regions A, B and C indicate the elastic field, the cyclic plastic zone and

the crack wake respectively. The approximate pixel resolution is 14 pixels/mm

15

The method proposed by Diaz et a1. [16] was used to locate the crack tip in the specimen
using the phase angle data. This method can be better understood by reference to figure
1, the lower image of which shows the distribution of phase angle, A in the vicinity of a
crack tip. Phase angle values along a line through the crack path (y = 0) shown by the
arrow in the lower image, are plotted in the graph above. It can be observed from the
image in figure 1 that the phase angle is reasonably uniform in the far-field (on the right
in the image) and has been set to zero, as can be seen in the plot along y = 0, for
160<x<320 pixels. In the plot, as the crack is approached from the far-field on the right
and proceeding left, the phase angle becomes negative reaching a minimum at B (x z 120
pixels) and then increasing back to zero, at x z 110 pixels. This pattern was observed in
the phase maps immediately in front of the crack from all seven specimens. The change
in phase angle, A indicates a loss of adiabatic conditions in this region due to heat
generation associated with plastic work [27]. To the left of this region (x < 110 pixels)
the phase angle values become positive. These positive values of phase angle, A are
believed to be associated with a loss of adiabatic conditions caused by contact between
the residual plastic regions in the crack wake. Contact between the crack flanks will
generate heat during the unloading part of the fatigue cycle; whereas dislocation
movement in the plastic region ahead of the crack will generate heat during the loaded
part of the cycle. Hence it would be expected, as observed, that these zones would
exhibit phase shifts in the temperature signal of opposite signs relative to the adiabatic,
synchronous temperature field. The point at which the phase angle changes from
negative to positive was taken to be a first approximation of crack tip location by Diaz et

a1. [16] as shown in the plot in figure 1.

16

X102

 

1 20
’3‘
9
_ 60
o
.5
9;
.9- g;
35. '5
L“ 0 l—
o D
E
2
a)
8
9'60
.92
D

 

-120

0 100 200 300 400 500 600
Inverse square of the maximum thermoelastic signal (1 X10-101/DTunitsz)

Figure 2: A map of the thermoelastic signal, S (DT units) obtained from the thermoelastic

stress analysis system around a crack growing from left to right; with a superimposed
plot of (I/Smax)2 as a function of the perpendicular distance from the crack tip, y. The

regions P, S and N indicate the cyclic plastic zone, the singularity dominated elastic zone
and non-singularity dominated elastic zone respectively. The approximate pixel

resolution is 14 pixels/mm.

17

 
 
 

Crack Flank
Residual Plasticity

    
 

    
 

 

 

   

 

 

*‘60
60—» ........
-50
401' l 140
g 100 150 200 250 300
§20i Pixels
or
(D
m
& o~
<1
'ZOT Crack Tip
Plasticity
Agdamm “-4—-
2 200 *\\V M 300
150 100 50 0 0 MET-T00 - 200
Pixels Pixels

Figure 3: three-dimensional representation of the phase difference data from figure I.

The approximate pixel resolution is 14 pixels/mm.

The extent of the singularity-dominated elastic zone was identified using a technique
developed by Stanley and Chan [4] who proposed a relationship between the square of
the inverse of the maximum thermoelastic signal along a line of constant perpendicular

distance from the crack, y:

_ 375K} 1 4
477A Smax

 

This relationship is linear only in the singularity-dominated elastic field [4] which defines
the region over which data can be collected. Figure 2 shows a plot of (I/Smax 2 as a

function of the perpendicular distance from the crack tip, y (open black circles) super-

imposed on the corresponding thermoelastic stress analysis image. A pair of white

18

dashed lines are superimposed on the mid-section of these data plots to identify the linear
region of the plots which corresponds to the singularity—dominated elastic field. The y
co-ordinates of the included points at the extremes define the inner and outer radii of an
annulus from which data is sampled for the calculation of the stress intensity factor. To
eliminate the plastic zone in the crack wake, a small sector is removed using a manually-

defined angle, 0.

 

 

 

 

   

'75
5 I ‘ "- a: "i‘ at «re ”a ...- -
. .j "t" ' a c ‘3' - é ‘ ‘:~ *1 * "=1 ‘ g - 31'- .. .1 ‘ 4" "5' - a0
0 P ‘ ?-'- ' - W .‘a 45 r’ ‘
2 -5 r 9
8’ Plastic Zone In front of crack tip . -5 E)
< -10 ~ E
o. '15 —1 0 CL
4 Q
-20 ~
-25 ~ -15
-30 am m_w
50 “HTTP—“fin “um,._ _ ._-

Pixel

Pixel
Figure 4: Crack tip plastic zone obtained by applying a binary filter to the phase
difference data to identify the negative values; the white area in the binary image
indicates the region with cyclic crack tip plasticity. The approximate pixel resolution is

I 4 pixels/mm.

Figure 3 shows a three-dimensional representation of the phase data from figure 1, in

which the specimen surface is plotted in the X Y—plane and the phase angle, A is plotted on

the Z axis. It can be observed that the phase is uniform and zero except in two regions:

19

an area of negative phase angle forming an inverted cone in front of the crack tip which is
associated with crack tip plasticity; and an area of positive phase angle forming a cone
which is associated with the residual plasticity in the crack wake region. This phase
information was used to obtain the size of the crack tip plastic zone. A binary image was
created as shown in figure 4 which distinguishes the negative region (white) from the rest
of the phase angle data (black). The white region of the binary image corresponds to the

plastic zone at the crack tip and was used to calculate the area of the plastic zone and an
experimental value for the radius of plastic zone, rp which was taken as the distance

across the crack tip plastic zone from the crack tip in the direction of crack growth.

 

 

 

 

 

 

 

 

 

' Gages -:
‘ I4 , _._-.-_._ 33.5
15.5 - "i 1
‘ l

 

 

 

 

 

 

 

 

 

 

155 i i
l -- - 335
i \ 014.0 -‘
l _
£150 ‘
= 90.0 = __, 2.0
Front View Side View

Figure 5: Specimen geometry showing the strain gage rosette which was bonded 15mm

to the right of the notch and 15mm above the crack line (all units are in ‘mm ’).

2O

Table I : Test Parameters and applied overloads

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

21

the thickness was only 2 mm to avoid through thickness effects (figure 5).

Location of

t: . Overload Overload in

0

§ Loading L03.“ Load1ng (% of Overload terms of

0 Ratio Frequency .

a (N) (R) (Hz) loading Type Crack

VJ amplitude) Length ‘a’

(mm)

Set-point = 1000 Single

1 Amplitude = 500 1/3 10 50% cycle 25 mm
Set-point = 1000 Single

2 Amplitude = 500 “3 10 25% cycle 25 mm
Set-point = 1000 Single

3 Amplitude = 500 U3 10 10% cycle 25 mm
Set-point = 1000 0 10

4 Amplitude = 500 1/3 10 50 /0 cycles 25 mm
Set-point = 1000 10

5 Amplitude = 500 “3 10 25% cycles 25 mm
Set-point = 1000 10

6 Amplitude = 500 U3 10 10% cycles 25 mm

7 Set-point = 1000 1/3 10 No No overload was

Amplitude = 500 overload applied
2.5 Experimental details and results

CT specimens of extra-strength aircraft-grade aluminum alloy 2024-T3 (yield stress =

290MPa) were manufactured based on ASTM E399 with H = 67mm and W = 72mm and

specimens were loaded in a uniaxial 50,000N MTS servo—hydraulic loading frame. A
cyclic load with a peak-to-peak amplitude and mean of 1000N each was applied at a

frequency of 10 Hz causing the initiation and growth of a fatigue crack in each of the

 

The

specimens. Data were collected at approximately every millimeter of growth of the crack
using the focal plane array camera to observe an area 18mm x 23mm. Each data map
was integrated over a 20 second period. Seven different specimens were tested using the
parameters provided in Table 1. The overloads were applied at a nominal crack length of
25mm and a load ratio R of 0.33 was maintained during the constant amplitude cycle as
well as during the overloads. A range of single as well as multiple cycle overloads were
applied as listed in Table 1. The thermoelastic stress analysis data were calibrated using
an orthogonal strain gage rosette bonded to the reverse face of the specimen in a region of
uniform stress, where the thermoelastic signal was constant. As illustrated in figure 5 the
strain gage rosette was located 15mm to the right of the notch and 15mm above the crack
line for all the specimens. From the rosette the first strain invariant was obtained and
converted to stress using Hooke’s law. The calibration constant, A was calculated by
correlating the thermoelastic signal from the strain gage location with the signal from the

strain gage rosette using [14]:

A: E Aiex + 8),) (5)
1+1) S

 

The full—field stress maps around the crack tip were processed using a freely available
software package, FATCAT (www.experimentalstress.com) which implements the
MPODM using the methodology of Diaz et al. [16] which is based on the formulation
developed by Tomlinson et a1. [11] to evaluate the stress intensity factor and crack tip
location from thermoelastic data using Muskhelishvili’s approach to describe the stress
field in terms of a pair of complex Fourier series. Five terms were used in the Fourier
series based on a convergence analysis of the evaluated stress intensity values with

respect to the number of terms [11]. The crack tip plastic zone size was estimated from

22

the phase information as described above using a specially written MATLAB® program.
The experimental results from a typical specimen subject to constant amplitude loading
are shown in figure 6 and for a typical overload case in figure 7 together with 95%
confidence limits calculated using [8]:

(1+,u—2s)KS SKS(1+y+2s)KS (6)
where, ,u and s are the mean and variance of the least square difference between the stress
field equations and the experimental data and K, is solution from the MPODM. For a
constant amplitude case, the AK, value was calculated from theory using a plane stress
empirical formulation [28] and from a finite element model. ANSYS® was used to
develop and solve a plane stress/strain FE model for the compact tension specimen and
the stress intensity factor was calculated using the energy (J -integral) method under plane
stress conditions [29]. The variation of the radius and area of the crack tip plastic zone

were plotted as a function of crack length in figure 8. The theoretical radius of the crack
tip plastic zone, rp was calculated using [28]:

2
lAKI

r :— 7
p 27: 20,, U

The crack growth curves for two typical overload cases, Le. a single overload
(specimen#l) and multiple overloads (specimen#4) are plotted in figure 9. The AK,
values and area of the crack tip plastic zone were plotted as functions of the magnitude of

the overload in figures 10 and 11 respectively.

23

 

 

 

 

 

 

 

 

 

<> AKI obtained from FE analysis
¢_ -TheoreticalAKl
A Experimental Data for AKI (Specimen 7)
13 Experimental Data for AKll (Specimen 7)
0 Experimental Radius of Plastic Zone
— — Theoretical Radius of Plastic Zone
20 1 I ' : 4
15 — ~- 3
’E‘
E
A of
A; 10 r “‘ 2 g
m N
‘5 i e
:: . 8
x ’ _
<1 _ i “F a.
5 : 1 .5
' (I)
: I 3
, . B
. . m
I i I : n:
0‘ EEDEIUEUDDEDUEIEEI “0
-5 4 L S i -1
15 20 25 30 35 40

Crack Length ‘a' (mm)

Figure 6: Stress intensity factors and plastic zone area as a function of crack length for a

typical specimen (# 7 in Table I ) subject to a constant amplitude loading.

24

 

 

 

 

 

 

 

 

 

 

 

A Experimental Data for AKI
El Experimental Data for AKII
0 Experimental Area of Plastic zone
16 a . . . . 20
I I 'o I I I
r : £3 '0 ' 1
l o .1) I I
: g a ' : A:
I as 0- : I I
12 - : g g I : é : 3- 16
I I .o I I I A
. : r91 : 4: : ‘é‘
: M1} 2 s e
E 8 - ' [EA . ‘5 E i _. 12 8
z, . : 1 : ,9,
o. : : : o
2 , I . : ‘5
E . i 00 : ' : B
Q 4 - I ,. I I — I I —_ 3 O-
' o ' CI): : “5
893859 “539 :o o; m
I I I I 9
: : : : : <
o—~ mugmqgnmmtblfl r3. -— 4
-4 I i i i i o
15 20 25 3O 35 40 45

Crack Length 'a' (mm)

Figure 7: Stress intensity factors and plastic zone area as a function of crack length for a
typical specimen (#4 in Table 1) subject to a 50% overload for ten cycles at a crack

length of 25 mm.

As discussed earlier, thermoelastic stress analysis is capable of providing experimental

estimates for AK, and the radius of plastic zone in front of the crack tip. A direct

application of these would be to evaluate the exponent in the Wheeler crack growth

25

model and study its dependence on the type of overload. The Wheeler model is obtained
by modifying the Paris law [17]:

31% = C(AK )” (8)

where C and n are empirical coefficients, and da/dN and AK are the crack growth rate and
range of stress intensity factor respectively. The thermoelastic data for da/dN and AK for
the specimen subject only to constant amplitude loading (#7 in Table 1) was used to
estimate the values of the empirical coefficients, C and n by performing a linear

regression for ln(da/dN) plotted as a function of ln(AK). Wheeler’s model [17] is a

modified version of equation (8):

 

da
where,
’ m
rpi <
...... a'+r ~_a +r
§=< a0L+rp.o,.—a.- ' ’“ 0L “0" (10>
(1..............................a,-+er-2a0L+rp,0L

 

where, cf and m are the empirical coefficients known as the ‘Wheeler retardation
parameter’, the ‘Wheeler empirical exponent’ respectively; a and rp are the crack length
and radius of the crack tip plastic zone respectively with subscripts i and OL indicating
current and overload values respectively. For each specimen subjected to an overload the
value of the Wheeler retardation parameter, 4' was estimated during the period after
overload until the crack propagated through the overload induced plastic zone., i.e. in the

range am > a,- > rpm using values of AK, and da/dN obtained from the thermoelastic data

26

together with the values of C and n obtained in the constant amplitude case. Subsequently
the Wheeler empirical exponent, m was estimated using the values of (f and the values for
aOL, rag, and no, obtained from the thermoelastic data. For each specimen the average
value of m was calculated over the period no, > a, > rm), and is shown in figure 12. In
order to give an indication of the reliability of Wheeler’s model, these average values of
m have been used to calculate the crack growth rate, da/dN for a single overload case
(specimen#l in Table 1) and a ten-cycle overload (specimen#4 in Table 1) in figures 13
and 14 respectively using the values for AK measured using thermoelastic stress analysis
in equation (9) and in equation (7) to find no. Thus the data in figures 13 and 14 represent
the prediction of crack growth rate that would be obtained with only experimental data
for the stress intensity factor as might be the case if only compliance measurements were

available.

2.6 Discussion

Tomlinson et a1 [11] have examined the reliability of the method used in this work for
estimating stress intensity factors from thermoelastic data and found that the average
difference between data from experiment and theory was 3.4% thus providing a high
level of confidence in the method. The effect of closure is evident in the experimental
results as the AK, values shown in figure 6 are lower than those predicted from theory and
the numerical model. Christopher at al [30] have argued that plasticity at the crack tip
and along the flanks shields the crack tip from the complete loading cycle which means
that it experiences only a part of the peak-to-peak stress amplitude. The presence of

plasticity in these specimens has been confirmed by the loss of adiabatic conditions

27

manifested as a phase difference measured between the temperature signal from the
material and the forcing signal (figures 1 and 3). Thus, since such effects were not
accounted for in the theory or finite element model, it is reasonable to attribute the
difference between values for the amplitude of the stress intensity factor, AK obtained
from thermoelastic stress analysis and theory, and between thermoelastic stress analysis
and finite element analysis, to the presence of plastic shielding of the crack tip in a
behavior associated with crack closure. This behavior confirms the work of Fulton et al.
[13], Tomlinson et al. [14] and Diaz et al. [16]. AK,, values are also plotted in figures 6
and 7 and are very close to zero suggesting the presence of pure mode I displacements
which would be expected for this specimen geometry and loading. The 95% confidence
limits shown in figures 6 and 7 are small suggesting that the accuracy of the data obtained
from the thermoelastic stress analysis is high. The maps of the phase difference between
the thermoelastic stress analysis and forcing signals have been used to identify the crack
tip and the associated plastic zone following Diaz at al [16]. The proposed procedure
measuring the radius of the crack tip plastic zone was straightforward to apply and the
results in figure 6 for constant amplitude loading show good agreement with those

predicted by equation (7 ), though with some scatter.

Chona et a1. [31] provide an alternate method to estimate the inner radius of the
singularity dominated zone; however the method described here which was used by Diaz
et a1. [16] is comparatively faster and reasonably accurate. By way of comparison, for the
same geometry Chona et a1 [31] found rmin/a : 0.04 at W = 0.5 compared to rmin/a z

0.05 in this study.

28

 

 

<> Radius of Plastic zone (from phase data)

— Theoretical radius of plastic zone for constant amplitude loading

 

0 Area of Plastic zone (for phase data)

 

 

 

 

 

 

 

 

4 ; . . 17
‘ 5‘8 ~
I 8%
. ~05
3 E i; g?) 15
‘r _ (DU
1 Q)
I §§
A i 1 NCO a
E - E
E 2] ~ % ~ 13g
(1) G)
C C
191 191
-.‘=’ 11 ~~11-f.—’
3 3
E o.
“5 “5
.3 0" “”9 E3
g <
c:
-1~ 7
-2 I . i i I 5
15 20 25 30 35 4O 45

Crack Length 'a' (mm)

Figure 8: Radius and area of plastic zone as a function of crack length for specimen#4

which was subjected to a 50% overload for 10 cycles at a crack length of 25 mm.

29

 

43

4 4

o 50% single cycle overload

   
  
      
  

O 50% ten cycles overload

 

l
" 1
38 . ,
l I.
O
b
.1
.1

O l
1

(mm)

 

Zone affected
by applied

Crack Length, a

 

 

l
|
|
I
l
l
I

 

0 50000 1 00000 1 50000 200000 250000
Number of Cycles, n

Figure 9: Crack growth curves for typical specimens subjected to a 50% overload for a
single cycle (specimen#I ) and for 10 cycles (specimen#4) at a crack length of 25mm;
with data before and after the application of the overload fitted with trend lines to

highlight crack growth retardation.

The experimental data in figure 7 was obtained from specimen#4 in Table 1 to which a

50% overload was applied for ten cycles at a crack length of a=25.3 mm which

represents the most aggressive overload. Immediately after the application of this

30

overload the values of AK, dropped by 20% while the area of the crack tip plastic zone
increased by 40%. Crack tip plasticity has been established as a cause of crack closure
[31] and an increase in the crack tip plastic zone size due to the application of an
overload would intensify the effect of plasticity-induced shielding of the crack tip and
crack closure, thus causing a reduction in the amplitude of the stress intensity factor, AK,.
The crack growth curve for specimen#4 in figure 9 demonstrates a simultaneous

retardation in crack growth.

 

 

 

 

 

 

 

 

8.00 . I
7.50 é *1 ........ l ........ I _______
SAIOO i -------- 7 ******** I“ ------ ‘1 *******
E : : , . :
(U l 1 1 l
a. 1 ; 1 : 5
E . ' 1 :
5650—- ........ ................ 5 ........
6°00 <>Sing|eCycIeOverIoad ?
O 10 Cycle Overload j
A No overloads(a=25 mm) I
5.50 I I I I I I
-2 6 14 22 3O 38 46 54
°/oOverload

Figure 10: Stress intensity factor at 200 cycles after the application of overload as a

function of percentage overload.

31

 

 

 

 

 

 

1 1 . .
0 Single Cycle Overload ' O
O 10 Cycle Overload I ‘ '
A No overloads (a=25 mm) I
A 10 ~ . _. _______ . |_ _
NE I i i i i :
7; * : ; 3 3 0
c I I ‘ I l
0 ~ . .
N I i
.9 9 - I ‘.
TB
53 I
n- I
B I
8 :
a 8 - J
E . :
3 1
.§ 0
“’6 o
2 7 g 0
E O
A : i :
6 I I i L ;
-2 6 14 22 30 38 46 54

% Overload

Figure 11: Crack tip plastic zone size at 200 cycles after the application of overload as a

function of percentage overload.

A straight line was fitted to the portion of the crack growth curve immediately prior to the
application of the overload as shown in figure 9; and a parallel line was fitted to the post-
overload data in order to identify the cycle (£130,000) at which the crack growth rate
returned to its pre-event value. The gap between these parallel lines indicated the period
of crack retardation which corresponds to about 3.4mm of crack growth (z90,000 cycles).

Similar behavior is evident from the data related to the size of the plastic zone in figure 8.

32

 

 

 

 

 

 

 

 

0 Single Cycle Overload i i

3.5 O 10 Cycle Overload ~ ------ I ‘

A No overloads (a=25 mm) I 1

E a

g. 2.5 - ,
C
Q)
C

8 I l

x 2 “

”J

E :

83 1.5 ~ 1

S I

3 I

1 r . .

| O .

0.5 « ; 3 .

5 O : 5 ? i I

0 J—A——+—<> I I I I I

-2 6 1 4 22 30 38 46 54
% Overload
Figure 12: Wheeler exponent, m estimated from thermoelastic stress analysis data using

expression (10) and shown as a function of percentage overload.

The values from experiment for both radius and area of the plastic zone exhibit a large
step when the overload is applied; with the radius reaching 3.2mm compared to a
predicted value of 2.6mm. The values for both the radius and area return to the
approximate pre-event values after about 3.4mm of additional growth. Similar features
are evident in the data for specimen#l when a single 50% overload is applied. In other

words the crack retardation after overload extends over the length of the plastic zone

33

created by the overload. This behavior was observed some time ago in 2124 T351
aluminum alloy based on calculated plastic zone size [32] and more recently in 7075-
T651 aluminum alloy [21]; however this is believed to be the first confirmation of such

behavior via direct measurements of the plastic zone.

In figures 10 and 11 it can be observed that, with an increase in the amplitude of
overload, the crack tip plastic zones are larger and the corresponding AK, values lower
and that these effects are larger for a single cycle overload than for 10 cycles of constant
overload. The exact mechanism underlying the observed behavior is not evident however
it is surmised that the first cycle of the overload induces the same increase in plastic zone
size as observed in the single overload event; and that this enlarged plastic zone shields
the crack tip from a portion of the subsequent overloads thus avoiding any further
enlargement of the crack tip plastic zone. In the subsequent overload cycles, the crack
grows through the enlarged plastic zone at a faster rate than in the corresponding cycles
after the single overload event due to the magnitude of the overload cycles.
Consequently, at the end of the ten-cycle overload event, the crack is longer than at 10
cycles after the single overload event resulting in a faster return towards the constant
amplitude state as observed 200 cycles after the event in the data in figures 10 and 11.
This explanation is also supported by the crack growth curves in figure 9 which show that
following the 50% single overload event approximately 117,000 cycles are required to
return to the pre-event growth rate compared to 90,000 cycles for the corresponding ten
cycle overload, i.e. about 30% longer. In effect, the data from experiment presented in

figures 13 and 14 presents the same evidence; the initial impact of both overload regimes

34

on the crack growth rate is essentially identical but by the first measurement of stress
intensity factor, AK the single overload case shows a large change in AK relative to the
pre-event level. In this case the crack growth rate, da/dN remains low for a longer period
giving a small gradient between points 3 and 4 in figure 13 and then accelerates as the
crack tip emerges from the overload-induced plastic zone, between points 5 and 6 in
figure 13 causing the plot to trace out a figure of eight pattern. By comparison, the
multiple overloads lead to faster recovery of the crack grthh rate, da/dN so that the
‘cross-over’ in the figure of eight in the experimental data occurs very early after the

initial overload and is not simulated by Wheeler’s model (figure 14).

The data in figure 12 clearly shows that the Wheeler empirical exponent, m is a function
of the applied overload both in terms of its magnitude and the number of cycles. The
level of agreement in figures 13 and 14 between the values from experiment for crack
growth rate, (1de as a function of the stress intensity factor, AK and Wheeler’s model is
to be expected since the model is purely empirical. It is accepted that the Wheeler
empirical exponent is not a material property [17] and previously has been found to vary
between 1 and 3 with R-ratio, size of overload and crack length and hence the Wheeler
model has very limited predictive capability [21]. The data in figure 12 shows values of
the Wheeler exponent, m of less than one for overloads of 25% and less; however there
are clear trends in the data with percentage overload at constant R-ratio (r = 0.33) and

crack length at overload (aOL = 25 mm).

35

ln(AK,)

 

 

 

 

 

 

 

1.7 1.8 1.9 2.0 2.1 2.2 2.3
-7.5 m . I I I
-8.0
-8.5
E -9.0
8
'0
E
-9.5
-10.0 ,
-10.5 ~
-11.0
0 Experimental data - Before overload
0 Experimental data - After overload
—Whee|er model - using experimental AKI

 

Figure 13: Experimental crack growth rate curve for a typical specimens subjected to a
50% overload for a single cycle (specimen#I ) superimposed with the corresponding
Wheeler model estimation of crack growth rate obtained using data from thermoelastic
stress analysis. Some data points (solid symbols) from the experiment are numbered and

some from the model lettered to indicate the direction of crack growth.

36

 

 

 

 

 

1.8 1.9 2.0 2.1 2.2 2.3
-8.0 ‘ I ‘ f
-8.5 ~
g -9.0
R
‘D
‘5’
-9.5 -
-10.0 -
-10.5
0 Experimental data - Before overload
0 Experimental data - After overload
—Whee|er model - using experimental AKI

 

 

Figure 14: Experimental crack growth rate curve for a typical specimens subjected to a
50% overload for ten cycles (specimen#4) superimposed with the corresponding Wheeler
model estimation of crack growth rate obtained using data from thermoelastic stress
analysis. Some data points (solid symbols) from the experiment are numbered and some

from the model lettered to indicate the direction of crack growth.

37

2. 7 Conclusions

In this study thermoelastic stress analysis was combined with the multi-point over-
deterministic method to evaluate stress intensity factors for fatigue cracks in compact
tension specimens of aluminum alloy 2024 in order to study the effects of overloads on
fatigue crack growth in the presence of crack closure. It was found that the values from
experiment for the amplitude of the stress intensity factor, AK, were lower than the
corresponding values obtained from both finite element analysis and theory in which
crack closure was not modeled. These results are consistent with the presence of closure

in the specimen and confirmed the trend in the values of AK, observed in previous studies

[13,14, 15 &16].

The concept of using the map of the phase difference between the measured signal from
thermoelastic stress analysis and the forcing signal to obtain the location of the crack tip
proposed by Diaz et al. [16] has been developed further in this study to locate the plastic
zone ahead of the crack. A technique has been developed to measure the radius of the
plastic zone, rp directly from this phase information; and for constant amplitude loading

gave results consistent with estimates from theory.

Overloads were applied to a series of specimens while the plastic zone size and stress
intensity factor were monitored using the data from thermoelastic stress analysis. After
an overload had been applied, the plastic zone size was larger, AK, smaller and the
growth was slowed. It was observed that these changes were proportional to the applied

overloads. As the crack propagated through the enlarged plastic zone, the area of crack

38

tip plastic zone reduced while the AK, values and the growth rate increased and returned
to approximately the pre—event values when the post-overload growth equaled the plastic

radius at the overload.

Wheeler’s model for fatigue crack growth rate following an overload was employed and
the coefficients found using the thermoelastic stress analysis data for stress intensity
factor, crack length, and plastic zone size. The Wheeler exponent was found to be a
function of both the magnitude of the overload and the number of cycles. Wheeler’s
model followed the fatigue growth behavior with a reasonable degree of fidelity when the

appropriate values of the coefficients were used.

It is believed that this is the first time the effects of overloads have been documented
experimentally and was possible because of the capacity of thermoelastic stress analysis

to capture this local phenomenon.

2.8 Future work

In this work the dependence of crack closure on crack tip plasticity has been discussed.
Thermoelastic stress analysis proved to be an ideal technique to measure the extent of
crack tip plasticity in front of a fatigue crack due to it high sensitivity to local change in
temperatures in the specimen caused due to loading conditions as well as the high spatial
resolution obtained with the use of a two position infrared zoom lens. The phase data in
front of the crack tip was used as any deviation in the phase angle value from its uniform

value in the far field of the cracked specimen was associated with loss of adiabatic

39

 

conditions caused due to plastic work in the corresponding region of the specimen.
Although according to the theory of thermoelastic stress analysis it was impossible to
quantify the amount of plastic strain experienced by the specimen this phenomenon was
capable of providing an accurate estimate of the size of the crack tip plastic zone. The
thermoelastic stress analysis signal recorded from the specimen was calibrated in terms of
stresses and combined with a multi—point over-detenninistic method to evaluate the

experimental stress intensities.

Digital image correlation could be used instead of thermoelastic stress analysis to not
only obtain the size of the crack tip plastic zone but also to quantify the plastic strain in
this region. Since the measurements with digital image correlations are in terms of
displacements and not temperature changes as in the case of thermoelastic stress analysis
it is a more direct measure of plasticity and might tend to be more accurate. On the other
hand digital image correlation requires some assumption about the yield strain and
application of an appropriate yield criterion, all of which are prone to errors that are hard
to assess, particularly in the presence of strain hardening; while thermoelastic stress
analysis identifies the plastic zone since the material behaves differently i.e. non-
adiabatically. The limitations of digital image correlation in terms of spatial resolution as
well as requirement of a very fine speckle pattern at higher magnifications tend pose
restrictions to its application in the field of fracture mechanics. As a consequence the
displacement and strain maps tend to be noisy causing errors in both the measurement of
crack tip plastic zone size and evaluation of the stress intensities using multi-point over-

deterministic method. Work needs to be done in identifying a method to obtain a very

40

 

fine speckle pattern or surface texture on the fracture specimen to improve the strain
resolution in the considerably small singularity dominated zone and even smaller fracture

process zone.

Other than crack tip plasticity, crack closure is also affected by crack wake plasticity. The
thermoelastic stress analysis technique can be used to measure the extent of crack wake
plasticity and a similar study could be performed to study the effect of crack wake
plasticity on crack closure. The crack wake region is adjacent to the crack flanks or
surfaces which are continuously in motion due to the opening and closing of the crack
under cyclic loads. Care needs to be taken while analyzing data in this region since time
integration of this data might produce erroneous results due to inadequate motion-

compensation.

41

3 Part B: Structural damage assessment in fiber-reinforced composite

using shape analysis techniques

3.1 Summary

Shape analysis techniques such as geometric moment descriptors, Fourier descriptors,
wavelet descriptors etc. have been used commercially in the fields of biometrics for
finger print matching, iris matching and facial recognition for almost over a decade.
Initial test results suggest that these techniques can be used to assess the type and extent
of damage in composite panels by comparing the key geometric features in the full-field
displacement, strain or stress maps of damaged composites with those in the
corresponding maps of undamaged composites. In this study Zemike moment descriptors
and Fourier descriptors will be used to represent full-field stress and strain data obtained
from digital image correlation for composite tensile specimens with different levels of
damage and their performance will be evaluated and rated against one another. The
advantages and short-comings of these shape descriptors will be discussed together with
the possibility of combining Fourier decomposition with Zemike moments to provide a

simple index of damage.

3.2 Literature review

Different methods adopted for damage assessment of structural composites over the years
include non-destructive testing, vibration characteristics and strain measurement. These
techniques can be classified based on the level of damage information provided by them

[33].

42

Level 1: Damage detection

Level 2: Level 1 plus determination of location of damage

Level 3: Level 2 plus determination of the extent of damage

Level 4: Level 3 plus post-damage life estimation
Non—destructive techniques including ultrasonic methods, magnetic field methods,
radiography, eddy current methods and thermal field methods are capable of providing
level-1 and level-2 damage information. However, these techniques require easy access
to the component. Methods like ultrasound require the component to be submerged
underwater to obtain a liquid coupling. Thus in recent studies active and passive piezo-
electric detectors have been used which are based on the concept of Lamb waves [34].
Due to the uncertainty in the post-damage behavior of fiber-reinforced polymer
composites the common practice is to replace the component at the onset of damage. This
implies that non-destructive evaluation focuses on simply finding damage with relatively
little attention paid to quantifying its extent and none to quantifying its effect. The cost
implications associated with down-time and replacement of damaged components hinders
the application of these energy-conserving composite materials in structural applications.
Thus it is necessary to be able to accurately predict post-damage life and load carrying
capacity of damaged composites. Different strain measurement techniques including
smart composites with inbuilt fiber optic strain gages and full-field strain measurement
techniques such as digital image correlation are capable of providing Level 3
information. Work is in progress on enhancing these techniques to perform Level 4
damage assessment. Thus these strain-based techniques provide the opportunity to

discriminate between inconsequential damage that does not alter the strain pattern and

43

damage that weakens the structure by raising the strains. They also provide the

opportunity to assess residual life based on the measured strain.

Zhou and Sim [35] and Kuang and Cantwell [36] have provided thorough reviews on the
use of fiber optic strain gages for structural damage assessment. Steenkiste [37] reviews
the different types of fiber optic sensors and their applications. They benefit from their
immunity to electromagnetic interference, easy temperature correction, reduced weight
and cost effective application [38]. They don’t need an external power supply and the
required light source and monitoring system is comparatively compact. Udd [38] has
given a review of different setups for the application of fiber optic sensors. Multiple
gratings can be incorporated in a single optical fiber to record strain signals at multiple
points along its length with reasonable accuracy. It is also possible to perform a multi-
dimensional strain measurement using polarization-preserving fiber optic grating strain
sensors as demonstrated by Udd et al. [39]. Optical fibers can be used in a multiplexed
configuration and signals from each can be monitored using a single monitoring unit.
NASA Langley has in fact developed a novel grating based system capable of
multiplexing thousands of Bragg gratings in a single optical fiber. This system is based
on optical frequency domain reflectometry (OFDR) as explained by Foggatt and Moore
[40]. The application of this system was later illustrated by Childers et al. [41] by
simultaneously reading 3000 strain data points in four eight meter long optical fibers.
One concern with the application of fiber optics sensors is their fragility outside the
structure which can be addressed by advanced fiber optics technology developed by

telecommunications and optical-electronics industry. Also the optical fibers tend to be

44

about 100 to 300 pm thick, which is at least an order of a magnitude bigger than the
reinforcing fibers and hence are expected to compromise the mechanical properties of the

composite [36].

An alternative to using embedded fiber optic strain gages is using full-field strain
measuring techniques such as Electronic Speckle Pattern Interferometry (ESPI),
Reflection Photoelasticity, Thermoelastic Stress Analysis (TSA) and Digital Image
Correlation (DIC). The only advantage that fiber optic strain gages might have over full-
‘field strain measurement techniques is that they can be embedded between different
laminates of the composite and provide strain data at locations within the composite,
while full-field strain measurements are capable of recording only strain of the surface of
the composite specimen. Richardson et a1. [42] were able to detect internal delamination
in composite panels using both intensity fringes and the phase map recorded using ESPI.
Although ESPI was unable to record the exact location of the delamination within the
composite the effect of this internal delamination on the strain distribution in the surface
laminate makes its detection possible. They did not attempt to use this strain information
to obtain a measure of the residual life of the composite but explored the possibility of
real-time non-destructive testing for damage detection in composites using a full-field
strain measurement technique. On similar lines Horn et a1. [43] used thermoelastic stress
analysis for detection of damage to obtain a measure of the intensity of damage and
residual fatigue life of the damaged composite. However, they fail to utilize all the full-
field recorded data and obtain a stress concentration factor using the thermoelastic stress

signal in the vicinity of the damage normalized by that in the far-field. They assume that

45

the damage stress concentration factor to be a direct measure of the level of damage as
well as the residual life of the composite. Thus this technique is no different from the one
proposed by Caprino and Tecchio [44] who evaluate the residual strength as a ratio of the
failure load of a damaged specimen and an undamaged virgin specimen. The only
difference being that the latter evaluate the residual strength under monotonic tensile and

compressive loads while Horn et al. evaluate the residual fatigue life of the specimen.

This doctoral research concentrates on utilizing the entire pixel data-set of high resolution
strain distributions which can be obtained using any full-field strain measuring technique.
Comparison of these full-field strain distributions recorded from damaged composites
with those recorded from undamaged virgin composites has a potential to provide
necessary information for level 4 diagnosis of damage; i.e. the extent of damage as well
as post—damage life predictions. However, a pixel-to-pixel correlation of these high
resolution images is not computationally efficient leading to very high computational
costs. As a solution shape description techniques will be explored as discussed below to
obtain a unique representation of the full-field strain distribution while attaining a
considerable reduction in the data to be analyzed. These shape descriptors provide a

basis to characterize damage in a manageable decision making process.

An image or scene can be composed of one or more objects and in order to understand
the contents of an image it is essential to identify these objects. These objects can be
represented by their shape, e.g. a round ball or a square shaped box. In many imaging

applications analysis and identification of these shapes is a critical step. Some examples

46

would include organs in medical images, clouds in storm-tracking, aircrafts and/or
buildings in military targeting and machine parts in mechanical diagnostics. In these
cases image analysis is usually reduced to the analysis of shapes. The field of shape
analysis has been extensively researched since the early 19705 with availability of
cheaper computers and dedicated hardware. A number of review papers [45 — 48] and

books [49 - 54] have been written on the topic since then.

Shape analysis techniques can be classified based on different criteria such as topography
of shape, type of outcome of analysis and information preservation and retrieval. Palvadis
[45 and 46] proposed a classification based on topography which differentiates between
shape boundary information techniques and shape body information techniques. He
specified that shape analysis techniques, which deal solely with the boundary of the shape
are boundary information techniques, while those which dealing with the boundary as
well as the interior body of the shape are shape body information techniques or global
information techniques. Boundary information techniques are based on concepts of
bending energy [55], auto regressive models [56], time series [57], shape matrices [58
and 59], Fourier transforms of planar curves [60 - 64] and medial axis transforms [65, 66
and 67]. The next classification of shape analysis techniques depends on whether the
outcome of the analysis is numeric or not. Some techniques which produce images and
matrices are called non-numeric space-domain techniques while others which produce
scalars and vectors are called numeric or scalar transform techniques. Two-Dimensional
Fourier transforms [68 and 69] are example of space-domain techniques, while moment-

based techniques [70 - 76] fall under the category of scalar transform techniques. The

47

third type of classification differentiates between shape-analysis techniques based on
their ability to accurately reconstruct the original shape. Techniques which allow full
reconstruction are called information preserving while those capable of only partial
reconstruction are called information non-preserving. The elliptical description technique
for two-dimensional Fourier transforms developed by Wang et al. [68 and 69] is an
information non-preserving technique while Zemike moment description techniques [68,

69 and 74] are information preserving.

Shape analysis methods are capable of shape representation as well as shape description.
The non-numeric space domain techniques mentioned above are also sometimes know as
shape representative techniques while numeric, scalar transform techniques are shape
descriptive [48]. The idea behind shape description is to condense three-dimensional
shape information from surfaces and their boundaries into one or more one-dimensional
shapes. Feature recognition involves shape matching and discrimination where known
objects are compared to unknown objects detected in an image. The comparisons can be
simplified and made faster by comparing the corresponding representative shape features
of the objects using an appropriate metric as opposed to comparing the objects

themselves.

In this research work the objective was to be able to use shape descriptors to represent
full-field experimental strain maps which are computationally more efficient to compare
than comparing the high resolution strain maps themselves. It is envisioned that such a

representation of experimental strain maps could be a novel approach to efficient

48

structural damage assessment, structural health monitoring as well as experimental
validation of finite element models. Some common examples of simple global shape
descriptors are the centroid distance function which is formed from the distance of all the
boundary points of the shape from its centroid, perimeter of the surface boundary, surface
area, circularity (perimeterZ/area), eccentricity and bending energy. Peura and Iivarinen
[75] discuss other simple global shape descriptors like convexity and circular & elliptical
variance and their performance in describing experimental shapes. Though
computationally efficient, these simple shape descriptors are only capable of
discrimination between shapes with large differences. Some other techniques included in
contour-based shape representation classification are elastic matching, stochastic
methods, spectral transforms, chain code representation, polygon decomposition, smooth

curve decomposition and shape invariant techniques.

Elastic matching is also called the method of deformed templates. It is a feature
recognition technique that evaluates the strain energy and bending energy required to
deform a template into the shape of the object under consideration. These are evaluated
by optimizing the similarity between the bent template and object. The complexity of the
object shape and curvature are also considered along with the strain energy, bend energy
and the error function to represent the original object [76]. The method of elastic
matching can tend to be computationally inefficient depending on the complexity of the
shape. Stochastic methods on the other hand make use of time series and autoregressive
modeling which is a random process that predicts the current value of a function based on

previous observations [79, 80, 81, 82, 83 and 84]. The stochastic model representing the

49

closed boundary is invariant to transformations like scaling, translation, choice of starting
point, and rotation over angles that are multiples of 27r/N, where N is the number of

observations [56]. These invariant properties make these stochastic models viable for

shape representation.

Most of the contour-based global shape descriptors mentioned above tend to be sensitive
to noise in the original image. This problem can be alleviated by working in a frequency
domain as opposed to the spatial domain. This is achieved by taking spectral transforms
such as Fourier and wavelet transforms of one-dimensional vector representing the
contour of the object. Fourier descriptors were first introduced by Cosgriff [83] in the
1960’s and are one of the most widely used descriptors in the fields of biometrics,
medical imagery and topographical recognition. Fourier Descriptors can be used to
represent not only closed curves but also partial shapes as demonstrated by Lin et al. [84]
and Mitchell et a1. [85]. Arbiter et al. [86] and Grandlund [87] discuss the invariant
properties of Fourier descriptors which make them an effective shape descriptor. Short
term Fourier descriptors were introduced by Eichmann et al. [88] which are able to
represent local shape features effectively as compared to traditional Fourier descriptors
but at the same time have been shown to be out performed by the traditional Fourier
descriptors in shape retrieval by Zhang and Lu [89] due to a lack of global integration.
Wavelet transforms are a recent introduction to the field of shape description however
complicated matching schemes and their dyadic nature make them an inappropriate

choice for representing shape boundaries [90, 91 and 92].

50

Chain code representation of boundaries, also called the unit vector method, was
introduced by Freeman in 1961 [93 and 94]. In this technique an arbitrary curved
boundary of an object is represented by a sequence of unit vectors and a limited set of
possible directions. This technique tends to have higher dimensions for complex
geometries and is sensitive to boundary noise. Though it is an outdated technique for
shape description it can still be effectively used as an input to more recent techniques.
Groskey et al. [95] used a similar approach in breaking down the object boundary into
line segments and vertices using polygon decomposition. Each vertex is then described
using a four element vector composed of the internal angle, distance to next vertex, and
its coordinates. Berretti et al. [96] extended this method of polygon decomposition to
generic objects calling it smooth curve decomposition. They use the curvature of the
zero-crossing points of a Gaussian smoothed boundary to obtain primitives called tokens,
rather than using the vertices. Each token was represented by its curvature and orientation
and a weighted Euclidean distance was used for comparison. These techniques are not
invariant to scaling and rotation which is considered as a major drawback in shape

description.

The theory of invariance was formed as a mathematical discipline in the mid-19’h century
and was developed to address problems in number theory, algebra and geometry.
Mathematically invariance is a property of a mathematical object which remains
unchanged under geometric transformations of certain kind. A number of shape

invariants have been developed since including:

51

(i) geometric invariants such as cross-ratio, length ratio, distance ratio, angle,
area [97], triangle [98]
(ii) algebraic invariants such as determinants, Eigen values [99] and trace of a
tensor
(iii) differential invariants such as curvature, torsion and Gaussian curvature.
Geometric and algebraic invariants are suitable for shapes with boundaries which can be
represented algebraically while differential invariants are suitable for more random
geometries. Structural components have varying geometries and loading conditions and
thus strain distributions recorded from such structural components tend to have different
location, orientation as well as spatial resolution depending on the optical equipment used
for recording purposes. A shape descriptor which is invariant to affine transformations
such as translation, rotation and scaling eliminates the requirement of preprocessing of
the strain maps by spatial correlation, data point synchronization and orientation
synchronization prior to assessment. Thus invariant properties are highly desirable for

shape representation in damage assessment and low noise sensitivity.

The contour-based shape descriptors discussed above are of the boundary information
type as discussed previously. While studying the response of an engineering component
subject to static and/or dynamic loads the strain distribution throughout the specimen is
more critical than the shape of the specimen restricting the application of boundary
information shape descriptors as stand alone descriptors in the field of experimental
mechanics. However, these boundary information shape descriptors are still useful as

filters or are sometimes combined with other shape descriptors for shape representation

52

and discrimination. Shape description techniques in which all the pixels within the
boundary of the object are taken into account are called region-based shape descriptors

and can be divided into Global methods and structural methods.

Geometric moment invariants which were first applied to two-dimensional pattern
recognition applications by Hu [70] are global region-based shape descriptors. Here a
non-linear combination of lower order moments form invariant moments which are
invariant to rotation, translation and scaling. However, a normalization process like z-
score normalization [100] becomes necessary for their implementation. The problem with
application of geometric moments is that only a few low order geometric moments are
not sufficient to accurately represent complex shapes and higher order moments are
difficult to handle. Zhang and Lu [89] performed an elaborate study of geometric moment
invariants and discovered that their performance with respect to affine transformations of
simple shapes is much better compared to scaled, perceptively transformed or subjective
test shapes. Algebraic moment invariants, introduced by Taubin and Cooper [101 and
102] differ from geometric moment invariants in that they are not restricted to lower
order moments and at the same time are invariant to affine transformations. However,
results from the work done by Scassellati et al. [103] prove that the algebraic moment

invariants tend to be inconsistent in shape description.

Teague [71] was the first to use Zemike polynomials and Legendre polynomials to form

Zemike moments and Legendre moments respectively. The orthogonal properties of the

Legendre and Zemike polynomials ensure accurate reconstruction of the described shape

53

making optimal utilization of the shape information thus classifying Zemike moments
and Legendre moments as truly orthogonal moments. Pseudo-Zemike moments which
are obtained by using the real-valued radial components of the Zemike polynomials are
also orthogonal moments. Teh and Chin [104] have studied the effectiveness of non-
orthogonal as well as orthogonal moments in describing a wide variety of shapes and
have concluded that Zemike moments and pseudo-Zemike moments are the least affected
by noise. They conclude that Zemike moments are preferable over the rest of the
orthogonal shape descriptors. Liao and Pawlak [105] have extended Teh and Chins work
in studying accuracy of moments under different image resolutions and introducing
techniques to increase the accuracy and efficiency of moments. Thus the more concise,
robust and computationally efficient, orthogonal moment descriptors prove to be very

promising in the field of shape description.

Zhang and Lu [106] proposed that the normalized coefficients of the two—dimensional
Fourier transforms of the shape image can be used as shape descriptors. Compared with
Zemike moment descriptors they tend to be easier to compute and have better retrieval
performance due to multi-resolution representation in both the spatial and spectral
domains. Through their follow-up work, Zhang and Lu [107] have shown that Fourier
descriptors obtained from two-dimensional Fourier transforms outperform contour-based
shape descriptors and most of the region-based shape descriptors. Fourier descriptors
have been recently implemented for mode shape recognition and numerical model

updating by Wang et al [68 and 69].

54

The grid based method proposed by Lu and Sajjanhar [108] and the use of a shape matrix
proposed by Goshtasby [58] are also global region-based techniques that have been used
for shape representation [109-114] but are not able to match up to the performance in

shape retrieval shown by orthogonal moment descriptors and Fourier descriptors.

For representation of full—field strain maps of damaged composite materials orthogonal
Zemike moment descriptors and Fourier descriptors were selected based on some of the
essential attributes discussed earlier. Dudhani et al. [113] explored the possibility of using
invariant moment descriptors for aircraft identification with minimal success in the
19705. However over a decade later Bhanu et al. [114 and 115] recognized these shape
descriptors to have a potential in the field of automatic target recognition. In a similar
application, Pizarro et al. [116] have used Zemike moments for underwater
archaeological excavations. In the field of medical imagery these invariant moment
descriptors are used for tumor detection [117] and polyp detection [118]. Grandison et al.
[119] have used these techniques in an interesting application for studying molecular
shapes and shape changes of proteins to understand their function. In the field of
biometric recognition these shape descriptors are used for facial feature recognitions
[120, 121 and 122], hand print analysis [123] and lip reading [124]. Invariant moments
also find applications in agronomics in sorting grains [125] and flowers [126]. Similarly,
Fourier descriptors also find applications in the field of military targeting and aircraft
recognition [127], medical imagery [128 and 129], biometrics [130, 131 and 132], lip
reading [133] and agronomics [134]. These techniques have recently been used by Wang

et al. [68, 69 and 74] in vibration mode-shape recognition and finite element model

55

updating however, they have never been applied in the field of experimental mechanics
for detection and assessment of damage or for the experimental validation of finite

element models.

A disturbance or irregularity in the strain distribution of a structural component under
known loading conditions can be associated with damage which can be recorded using
experimental techniques such as thermoelastic stress analysis and digital image
correlation. In this PhD research, for the first time, Zemike moment descriptors and
Fourier descriptors are used to represent full-field strain distributions in structural fiber-
reinforced polymer matrix composites obtained using digital image correlation for the
purpose of damage detection and structural life assessment. Also, the possibility of
combining these shape descriptors to obtain a more robust shape descriptor that will be
able to uniquely represent a wide range of displacement, strain or stress maps will be
explored. Patki and Patterson [135] have justified the combination of Fourier transforms
and Zemike polynomials to obtain a new hybrid shape descriptor useful for representing

strain maps of damaged composite panels.

The shape descriptors used in this work are mathematically capable of unique
representation of shapes. However, none of the literatures cited above applying these
shape descriptors, provide reconstructed data to validate the ability of these shape
descriptors to provide unique representation and most of the time this property is taken
for granted. This work stresses on the importance of such a validation and thus the in-

house developed software by the author provides reconstructions of the strain

56

distributions for validation purposes and each shape descriptor is evaluated based on its

ability to reconstruct back to the original strain distribution.

It is well known that finite element predictions need to be validated using experimental
results, since the finite element model ought to deform exactly like the actual physical
material under applied loads. Thus optimizing the finite element model or in other’words
finite element model updating is a crucial step in the development of numerical models.
Mottershead & Friswell [136] and Friswell et al. [137] discuss the importance and
complexities of finite element model updating using experimental data in detail. Recently
Patki et al [138] and Wang et al [139] have discussed the importance and advantages of
performing full-field experimental validations of finite element model for composites and
they have explored the possibility of using shape description techniques to achieve such a
full-field validation. The same scheme used for structural damage assessment will be
applied to experimental validation of finite element models using full-field experimental
strain data. Numerical models used in stress analysis for engineering design are efficient
and effective. However, they need to be validated against analytical models and more
importantly experimental data. This validation is usually done by comparing the stresses
from the numerical models to those from analytical models or experiments at single
critical locations in the specimen. The experimental data is usually collected using strain
gages. Although this technique is suitable for traditional isotropic materials it is not good
practice to apply the same to anisotropic materials like fiber-reinforced composites.
Experimental techniques such as digital image correlation, thermoelasticity and reflection

photoelasticity which are capable of recording full-field displacement, strain and stress

57

maps in test specimens or engineering components can be used for full-field experimental

validation of finite element models.

Patki et al. [135 and 138] used these shape descriptors for structural damage assessment
as well as full-field FEA validation using full-field experimental data obtained from
digital image correlation. Feligiotti et al. [140] have enumerated various different
methodologies employed by researchers all over the world to obtain full-field numerical
model validations and they have acknowledged the innovation and promise delivered by
the strain distribution representation technique using invariant descriptors which was

developed as a part of this doctoral research.

Both these schemes mentioned above viz. structural damage assessment using full-field
strain data and full-field experimental validation of finite element models require
comparison of high resolution full-field data maps either obtained from full-field
experimental measurement techniques or finite element models. However, there are no
tools available to achieve such comparisons efficiently. The aim of this PhD research
project is to devise such a technique using shape descriptors to obtain a considerable
reduction in strain data without sacrificing any shape information from the strain
distributions. These shape descriptors in turn provide quantitative as well as qualitative

basis for comparison of the strain distributions that they uniquely represent.

58

Mirrors

 

 

 

 

 

 

 

 

 

 

 

Spadal
Filter ,\
l 1’ Specimen
I i with grating
Light (i
Source Objective Collimator
:>
CCD Camera

 

 

A
\
Figure 15: Schematic of experimental setup for in-plane Moire’ interferometry with one

pair of coherent beams.

3.3 Experimental technique selection

Multiple techniques are available today for full-field displacement, strain and stress
measurements. Selection of the appropriate experimental technique is often based on its
sensitivity to the change in the desired quantity with satisfactory accuracy under a wide
range of loading conditions. Eleven such techniques were identified for studying full-
field stress or strain maps in damaged composite panels based on their capability and
availability. A rational decision making model [141] was employed to select the
appropriate experimental mechanics technique for this work. Out of the eleven techniques

three were short-listed based on their possession of essential attributes such as their

59

ability to record full-field in-plane displacements / strains during loading, their ability to
either separate principal strains or record separated strains and the availability of the
technique. The assessment of these eleven techniques against the essential attributes is
shown in table 2. The three short listed techniques were in-plane moire’, digital image

correlation and thenno-photoelasticity.

3.3.1 In-plane Moire’

In-plane Moiré is a very effective full-field technique for two-dimensional measurements
of displacements and strains. Different variants of the technique exist depending on the
setup used and are known as geometric moiré, shadow moire, moiré interferometry,
Fourier transform moiré, etc. The most common and easy-to-use moire technique for in-
plane strain measurements is geometric or mechanical moire. In this technique two
gratings are required, a specimen grating and a master grating. As the specimen grating
deforms along with the specimen it moves relative to the stationary or master grating and
an interference pattern forms fringes which are related to the relative displacement of the
gratings. The specimen grating can be obtained by projecting a shadow of a grating on to
the specimen. In this case the geometric moiré technique is called shadow moiré.
However, these techniques are limited by the frequency of the master grating with a
typical upper limit of 40 lines/mm requiring at least 0.025mm deformation to form one
fringe. Moiré interferometry or optical moire on the other hand employs the principal of
light interference and diffraction which provides high enough sensitivity to measure
small elastic strains. Figure 15 shows a schematic of a two beam moiré interferometry
setup capable of measuring in-plane displacement only in one direction. A number of

pairs of such coherent light beams can be used in different orientations to resolve strains

60

in that direction, e. g. in case of four beam moire interferometry setup, the U—displacement
field would be generated by the pair of beams in the horizontal plane while the V-
displacement field would be generated by the pairs of beams in the vertical plane. For
two-dimensional measurements of displacement either two line specimen gratings or dot

array patterns need to be imprinted on the specimen surface. The displacements; U and V

and the strains; 8x, 8y and 7,0, can be calculated from the fringe map using the following

 

 

expressions,
Nx
u=— (11)
f
N,
v=——} (12)
f
Bu lde
8x:—:— (13)
6x f Bx
av IaNy
3y f 3y
Bu av 1 3N, aNy
7xy=—+—=— + (15)
By ax f 6y 6x

The surface preparation plays an important role as a high quality grating can produce
accurate and low noise moire’ fringes. Different methods of producing high density
diffraction gratings on the specimen have been developed [142-148] and selection of the
appropriate grating depends on the spatial resolution and strain resolution requirements of

the measurements as well as the geometry of the specimen.

61

    

Light Source

Poleidoscope

Specimen with
Photoelastic coating

Figure I 6: Experimental setup for Thermo—Photoelasticity.

3.3.2 Thermo-Photoelasticity

Thermoelasticity is a full-field experimental technique for measuring stresses which
yields a temperature map proportional to the sum of principal strains while
photoelasticity provides a full—field intensity map proportional to the difference between
the principal strains. Thus it is possible to combine both of these methods to separate the
principal stresses. This technique is called thermo-photoelasticity. The thermoelastic

signal obtained in plane stress is related to principal strain as [26],
A(O'1+O'2)= AS (16)
where S is the thermoelastic stress analysis signal, A is the calibration constant and a, and

0’2 are the principal stresses. Similarly for photoelasticity,

62

xi
21K

C

 

Ymax : 01— 02 : Dn : N =Nf (17)

If Sn and D" are the corresponding thermoelastic and photoelastic data, after spatial
correlation and data point synchronization of the two images, one obtained from
thermoelasticity and the other from photoelasticity the principal strains can be separated

as follows [149and 150]:

Sn+Dn
a =—— 18
1 2 ( )
—D
022—5" 2 " (19)

Figure 16 illustrates a schematic diagram of the experimental setup used for recording
simultaneous thermoelastic and photoelastic data. This setup and technique were
developed by Green et al. [149] to facilitate full-field separation of principal stresses
recorded from the surface of a component. An ordinary CCD camera with a photoelastic
poleidoscope was used to record photoelastic fringe patterns while a DeltaTherm®1550
system was used to record data for thermoelastic stress analysis. The specimen surface
preparation included gluing a reflection photoelastic coating to its surface. These coatings
are made of polycarbonate material which is opaque or black in the infrared spectrum.
Thus this reflection photoelasticity coating acts as a strain witness not only for
photoelasticity but also for thermoelastic stress analysis without the need for a surface
coating of matt black paint [149, 151 and 152]. As shown in figure 16 the thermoelastic
stress analysis camera and the poleidoscope were mounted side-by-side and they
observed the same field of view since they observe the specimen along the same optical

axis via the beam splitter. A beam-splitter was used that was specially designed to reflect

63

radiation in the visible spectrum into the poleidoscope and to transmit the infrared
radiation into the infrared camera for thermoelastic stress analysis. The specimen was
subsequently illuminated with a circularly polarized light source for the photoelastic

measurements.

3.3.3 Digital Image Correlation

The technique of digital image correlation was developed in the University of South
Carolina in the 19805 shortly after Peters and Ranson proposed the analysis of digital
ultrasound images of solids subjected to two-dimensional loads to obtain in—plane
displacements [153]. Over the next ten years the concept was modified for use in the
visible spectrum to obtain in-plane as well as out-of—plane surface displacements. The
displacement field for an object is obtained at different locations by choosing facets from
the initial image and searching for an optimal match in a second image [154]. The facet
size and facet step for the computations can be adjusted before the analysis depending on
the quality of the pattern and expected strain values [155]. If we assume that for every
pixel denoted by X, Y, dx and dy are the corresponding displacements after deformation
of the specimen and u and v are the displacement vectors the new position of any pixel
can be written as x=X+u and y: Y +v. The recorded intensity pattern after deformation

can be related to the one before deformation by,

['DIC (x, y) = IDIC (X +u(X, Y), Y + V(X, Y» (20)

then,

64

1}),C(x+dx,y+dy))=1 IDIC[X+u(X, Y)+(l+g—u—]dX+ 8“ —dY,

x x
v(X,Y)+{l+a—)dY+§KdX (21)
By x

To find the values of u, v, 6u/(3x, 6v/6x, 6u/6y and 6v/6y a normalized correlation

function given by,

 

1—C1[u v— Bu 6_u 8v iv] (22)
6x d—y 6x 8y
is used where,
N M '
XXIIDmiXthl'IDICiXi”(Xi’leYj‘LAXz-lell
i=1 '=1
c1: 1 l
. 2
gill ch(X i1 Yj) IDIC (Xi+“(Xi1Yj)’Yj+V(Xi9Yj)):l2
I F

The function given by equation (22) above is optimized using Newton-Raphson method

Bu 6_u_ 6v av

to determine the six components of the strain tensor including it, v,— —an nd —

6x 6y 6x 3y

which include the displacements of the corresponding pixels [156, 157 and 158]. Thus for
every pixel in the intensity pattern after an initial estimate of displacements u’ and v’ is
made all six parameters are obtained by minimizing the correlation parameter given by
equation (22). This process is repeated until data is obtained for the whole image [159].
Fast Fourier transforms can be used as an alternative where the in-plane strains and rigid
body motion in the test specimen are relatively small [159 and 160]. Two-dimensional

digital image correlation requires that the surface of the test specimen have a pattern that

65

produces a varying intensity of diffusely reflected light from the surface. This pattern
may be applied to the object or it may occur naturally. It is also important to make sure
that the sensor plane of the CCD is parallel to the plane of the planar specimen [154].
Two—dimensional digital image correlation however has its own limitations [154]; a
change in magnification due to out-of—plane displacements can introduce significant
errors and the two-dimensional digital image correlation method is incapable of
measuring these out-of-plane displacements which might be equally important as the in-
plane displacements depending on the loading conditions. Even with its limitations two-
dimensional digital image correlation is still successfully used in fracture mechanics and
in applications involving concrete where out-of—plane deformations are negligible as well
as in wood, paper and forest products. After applying different techniques to minimize
the errors in two-dimensional digital image correlation, finally the concept of measuring
full three-dimensional displacements using digital image correlation was developed as a
solution [154]. Figure 17 shows a typical two-dimensional digital image correlation setup
used for recording full-field displacement maps in composite panels under tensile loads.
The camera was located in accordance to the requirements discussed above and a ring
light concentric to the camera lens was used to illuminate the composite specimen. The
first three-dimensional digital image correlation measurements were performed by Sutton
et al. [154] in 1988 and were obtained by using one horizontal traversing camera to

obtain two views of the object.

66

El CWLI‘. war-r1“! ____J .
~.-‘-

0..—
..o-
.—-‘.. ...

.\ 1‘

‘ O

’

 

‘1'-
M... ‘.’ 9:».—

Figure 17: Experimental setup for two-dimensional digital image correlation.

67

The two camera version of three-dimensional digital image correlation was developed in
1991 by Luo et. al. [160] which provided stereo vision images of the test specimen which
they used to analyze fracture problems. Since the position of the two cameras is known
and the position of every pixel on the test specimen surface with respect to each camera is
known along with all imaging parameters it is possible to exactly define the location of
every point on the specimen surface with three-dimensional coordinates [161]. This two
camera three-dimensional version was further refined by Helm et al. [162 and 163]. A
three-dimensional digital image correlation setup is shown in figure 18 is capable of
eliminating rigid body movements and recording in-plane as well as out-of-plane
displacements.

W)
t

I
1
.1
V.‘
.1
‘
\

".
I“:
l'.
"I

It
1.

1 '.

~l

Ill

 

 

Figure 18: Experimental setup for three-dimensional digital image correlation.

68

3. 3.4 Rational decision making model

The three techniques discussed in the previous section for recording full-field
displacement, strain and stress data from composite specimen were rated against each
other based on a set of soft attributes including setup and process cost, ease of use,
limitations, etc. Seventeen engineers participated in a survey in which each of them were
asked to rate the importance of every soft attribute on a scale of 1 to 5 while they had to
rate the techniques against each soft attributes on a scale of 1 to 10. Every participant was
also asked to rate their own experience and expertise with each technique. Based on their
replies a weighted average was taken and the scores of all three techniques were

compared against one another as shown in table 3.

It was observed that both thermo-photoelasticity and digital image correlation scored low
on the attribute regarding their initial capital costs. This attribute was ignored in the final
decision making since both the techniques were readily available. Thermo-photoelasticity
and In-plane moire both had low scores in regards to intensive surface preparation
required for both techniques. Digital image correlation scored a fraction better on most of
the soft attributes compared to In-plane moire’ techniques and also edged out thermo-
photoelasticity on simplicity of application. The scores of every technique were plotted
against the soft attribute as shown in figure 19. Digital image correlation had the highest
score, followed by thermo-photoelasticity and in—plane moire’ in that order. Based on this
study digital image correlation was selected from amongst 11 different techniques as the
best suited experimental technique to record full-field, in-plane displacements in

components made of composite materials.

69

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a O
5 O
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O Thermo-Photoelasticity (TEPE)
0 Digital Image Correlation (DIC)
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—- Weighted-average score (TEPE)
— Weighted-average score (DIC)
2 l T T t l
O 2 4 6 8 10 12

Soft Attributes

Figure I 9: Technique Performance plotted against soft attributes. The soft attribute of

low capital cost was excluded for this analysis since all three techniques were readily

available at the MS U.

3.4 Specimens

A glass fiber-reinforced polymer matrix composite CYCOM® 1004 manufactured by
CYTEC Industries Ltd, New Jersey was used in this study. The composite material is

made of nine cross-plies in the [O/90/0/90/O/90/0/90/O]° configuration. The thickness of

72

the composite material sheet was 3.5mm. A tensile test was carried out on two specimens
with dimensions 177.8mm x 25.4mm to determine the material properties. One specimen
had its principal material direction aligned with the longitudinal direction which was the
loading direction while the other had the same aligned across the principal direction. It
was observed that the material properties did not change significantly between the two
specimens. Seven specimens of dimensions 240mm x 60mm were machined using a
dimond edge saw.

Table 4: Details of F iber—reinforced polymer composite specimens

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Dead Height of Impact-
. Damage Type Impactor .
Spec1men T e Weight release energy
yp (Kg) (cm) (Joules)
Impact Spherical
1 (Drop Weight) Tup 5 76 37'3
Impact Spherical
2 (Drop Weight) Tup 5 71 34'8
Impact Spherical
3 (Drop Weight) Tup 5 66 32'4
Impact Spherical
4 (Drop Weight) Tup 5 61 30
Virgin N/A N/A N/A N/A N/A
Impact Impact Spherical Enough for the tup to penetrate
Hole (Drop Weight) Tup through the specimen.
Machined . Machined Wllh. N / A N / A N / A N / A
Hole minimal delamination

 

 

One specimen was impacted with a drop-weight testing machine (Dynatup®3500) with
enough energy for the tup to penetrate through the specimen leaving it with a central

impact hole. The second specimen had a machined hole, which was approximately the

73

same size of the hole formed due to impact damage in the first specimen. The third
specimen was a virgin specimen without any damage. The remaining four specimens
were incrementally damaged in a drop weight testing machine. The impact-energy was
kept low enough to ensure that the tup did not penetrate any of these four composite
specimens. The weight of the drop weight was kept constant for all four composite

specimens and the height was varied thus in-turn varying the impact-energy.

 

Figure 20: Incrementally damaged composite specimens using a drop-weight testing

machine with increasing impact energy arranged from left-to-right along with a
composite specimens with an intpacted hole (top right most) and with a machined hole

(bottom rightmost).

74

 

D 0.5 1 1 .5 2 0 0.5 1 1 .5 2

Figure 21: C -scan image and time of flight maps obtained from ultrasonic evaluation of
the composite specimens with a machined hole (top left and bottom left respectively) and

with a impacted hole (top right and bottom right respectively).

Aluminum tabs were glued to each side of the composite specimens at the ends to
provide a location for gripping the specimen. These metal tabs protect the composite
specimens from failure due to excessive crushing forces in the hydraulic loading grips of

the servo-hydraulic test frame, and provided a uniform and gradual load transfer from the

75

hydraulic gn'ps to the specimen. These seven specimens are shown in figure 21 arranged
from left to right in increasing amount of damage due to increased impact-energy as
described in table 4. In the specimen with the machined hole, the hole was machined
carefully to avoid any delamination during machining. C-scan and time of flight images

of these specimens obtained from ultrasonic evaluation in figure 21 confirm that, the

specimen with the machined hole had no delamination surrounding the hole.

 

Figure 22: Raw digital image correlation images of composite specimen with impact

damage without speckle represented in two color schemes; grey (left) and Istra (right)

3.5 Experimental set-up and method

The specimens were loaded in a uniaxial 50,000N MTS servo-hydraulic loading frame.
Two-dimensional digital image correlation was used to record the data at each loading.
step. A Digital Image Correlation System (Q-400 — Dantec Dynamics, Denmark) was

used for the image correlation. The digital image correlation system comprising of two

76

used for the image correlation. The digital image correlation system comprising of two
high resolution (2048 X 2048 pixels) digital cameras and a HILIS cold light system for

uniform illumination of the specimen surface was used for strain analysis.

The specimen was located in the servo-hydraulic loading frame and was subjected to
tensile loads. The cameras were focused on the specimen to give a clear and sharp image.
The digital image correlation system was located at half a meter distance from the
specimen. The specimen was removed and without moving the cameras a calibration was
performed using a 4mm X 4mm glass calibration plate supplied with the instrument. The
calibration was carried out using both the cameras for the three—dimensional system.
After the calibration was performed the specimen was repositioned in the loading frame
and the data was collected at every load increment. Care was taken so that none of the
digital image correlation equipment was moved from its original position as that would
have disrupted the calibration and introduced errors in the final results. Three-
dimensional digital image correlation was performed on the composite material surface
using only the surface texture of the composite. Figure 22 plots the representation of raw
digital image correlation image of a composite specimen without a speckle pattern in two
different color schemes for easier comparison. A special illumination technique was used
to emphasize the surface texture of the composite specimen for image correlation using
two LED light sources. The light sources and the cameras were positioned such that a
substantial amount of glare was present in the images. The intensity of the light source
was adjusted to obtain a distinctive glare pattern formed by the surface irregularities of

the composite which was sufficient for image correlation.

77

k‘!

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Figure 23: Raw digital image correlation images of composite specimen with impact
damage with speckle represented in two color schemes; grey (left image) and Istra (right

image).

Subsequently the specimens were painted with a black and white speckle pattern and a
single light source was adjusted to obtain a clear pattern of varying intensity recorded in
the CCD camera. Again, data was recorded at every load step. Figure 23 shows
corresponding raw digital image correlation images of the same composite specimen with
a black and white speckle pattern. The experiments were repeated to get six sets of data

for statistical comparison.

During evaluation of the data the default facet size of 17 and grid spacing of 12 were

used so that none of the data was lost and a full-field data set of displacements was

78

obtained for every specimen. These displacement datasets were differentiated to obtain
full-field maps of the maximum principal strain component. These strain maps evaluated
from displacement data recorded by digital image correlation for specimen with and
without a speckle pattern were compared to explore the viability of the non-speckle

digital image correlation technique.

Three different shape descriptors including the more traditional Zemike moments and
Fourier descriptors along with the newly developed Fourier-Zemike moments were
evaluated for all the specimens using a stand alone Matlab® code, ImPaCT (Image
Pattern Comparison Technique) developed by the author. The source code for ImPaCT is
provided at the back of this dissertation in Appendix—A. ImPaCT is capable of importing
strain maps in different formats including ASCII “.dtl format” for thermoelastic stress
analysis by Stress Potonics Inc., “.HDF format” for digital image correlation by Dantec
Dynamics and all different image formats including “TIFF”, “JPEG”, “BMP”, etc.
Masking operation involves setting up a square mask to extract relevant strain data from
the imported strain map. The square mask is generated with user defined center and size.
Following the masking operation, the two-dimensional Fourier magnitude map and the
Fourier phase map are evaluated for the strain distribution. The user is prompted to
choose to evaluate the Zemike moments, Fourier descriptors or the Fourier-Zemike
moments for the imported strain distribution. In case of Fourier descriptors ImPaCT plots
the maps of the absolute value of the two-dimensional discrete Fourier transform, the
logarithm of the absolute value of the two-dimensional discrete Fourier transform and the

unwrapped phase of the two-dimensional discrete Fourier transform. For the Zemike

79

moments or the Fourier-Zemik moments ImPaCT evaluates the Zemike moments for the
strain distribution or the logarithm of the absolute value of the two-dimensional discrete
Fourier transform of the strain distribution respectively. As discussed earlier it is essential
to validate the unique representation of the imported strain distribution which is achieved
by reconstructing the strain distribution with just the shape descriptors. After evaluating
the Zemike / Fourier-Zemike moments ImPaCT also provides an automatic
reconstruction of the strain maps. All the results are plotted as two-dimensional maps as

well as their three-dimensional renditions.

The shape descriptors of the full-field strain maps obtained from the speckle and non-
speckle data were compared for the composite specimens to examine the effectiveness of

the shape analysis techniques proposed in this work.

3.6 Shape analysis

Digital image correlation is capable of recording full-field displacement maps in
structural components as discussed in the previous sections. These displacement maps
can be used to obtain the full-field strain distribution under different loading conditions.
Along with loading conditions these full—field strain distributions also vary with the
structural integrity of the component. Thus a full-field strain distribution obtained by
performing digital image correlation can be used to not only detect the onset of damage
but also to quantify the extent of damage. However, such an assessment of damage would
necessitate comparisons of these full-field strain distributions obtained from components

with different levels and types of damage with the strain distribution in undamaged

8O

components and amongst themselves. Digital image correlation evaluates full-field strain
map from high resolution raw data captured with CCD cameras. These strain maps could
contain 104 to 106 pixels of strain data and a pixel-to-pixels comparison of two such data
sets is computationally expensive. Shape analysis employs shape description techniques
to condense all this strain information into only a few feature vectors making these
comparisons easier and computationally acceptable. Such techniques have been used in
the field of medical imagery, storm tracking and biometric recognition. Initial test results
suggest that these techniques can be successfully applied to structural damage assessment
and other experimental mechanics applications such as experimental validation of finite
element models. However varying size, geometries and design features of structural
components, their orientation and varying loading conditions pose challenges in the
application of such shape description techniques to the field of experimental mechanics.
Thus, the selection of the appropriate shape descriptors or the introduction of new ones to
suit the application is essential. Some of the desired properties of shape descriptors used
for structural damage assessment will be discussed in this section to render the selection
process easier. The first step to shape analysis is image acquisition and processing as

explained below.

Pre-processing of the image under consideration is an important step in its shape

recognition. This evolves the following steps:
a) Image acquisition and storage: This step includes different techniques to capture
the images of the object and saving them in a particular format that is easy to

handle. In this work, full-field techniques of recording displacements and stresses

81

in composite specimen such as digital image correlation and thermoelastic stress
analysis will be used. Both these techniques have their unique format to store the
recorded data.

b) Noise Filtering: In case the recorded data is corrupted with unnecessary noise,
suitable image filters can be used for noise reduction. This is a very crucial step as
care needs to be taken so that important information regarding the measured
quantity is not lost during filtering.

c) Shape processing: Every strain map can be considered to have distinct shape
features. This step includes conditioning of the image including steps such a
masking, scaling, and normalization of the measured quantity so that comparisons

of two or more shapes make sense.

After preparing the images for analysis, different shape description and pattern
recognition techniques can be applied to condense all the shape information in the image
into a set of feature vectors called shape descriptors. These shape descriptors are
supposed to have a certain set of desirable properties that qualify them as reliable shape
descriptors, i.e.

a) The descriptors must be computable efficiently.

b) The descriptors should be invariant under affine transformations. This is required

since these transformations, by definition, do not change the shape of the object.
c) The shape representation by the descriptors should be unique for every shape

pattern.

82

d) Derived from c) above one should be able to easily retrieve the shape information

from the shape descriptors by reconstruction of the image.

There are many different approaches to shape description based on template matching,
statistical methods, syntactic approach and artificial neural networks that possess the
above properties qualifying them as reliable shape descriptors. Some of these shape
descriptors have been explored during this study and their performance will be discussed

systematically in the following sections.

3. 7 Zemike moments

Zemike moments are a type of orthogonal geometric moments capable of decomposing

three-dimensional shapes with pixels of the order 103 to 105 into only a few hundred

feature vectors and at the same time retaining all the original shape information. They are
capable of unique shape representation and are robust to noisy data making them a viable
candidate for representation of full-field experimental displacement, strain or stress data.
They are invariant to rotation, location and size of the data and thus can be potentially
used for comparison of experimental data from different full-field experimental

techniques and experimental validation of finite element models.

3. 7.1 Technique

Geometric moments are formed using a set of monomials basis. Geometric moment
descriptors have been shown to be invariant to rotation, scaling, translation, etc. however;

they are not orthogonal moments and are capable of containing high degree of redundant

83

information. In other words, these polynomials increase rapidly in range as the order
increases, producing highly correlated descriptions. This can result in important
descriptive information being contained within small differences between moments,
which lead to the need for high computational precision. Moments developed using a
orthogonal basis set has the advantage of needing lower precision to represent differences
to the same accuracy as the monomials. Two functions are said to be orthogonal if their
inner product. equals zero. Thus the orthogonality condition simplifies the reconstruction
of the original function using the generated moments by providing a unique
representation with only a few representative moments. A set of orthogonal complex
polynomials defined over a circular domain of unit radius introduced by Zemike [164]

are given as:

Vn,m(x’ y) : Vn,m(p96) : Rn,meim6 (23)

where, n is a non-negative integer which represents the order of the radial polynomial, m

is an integer subject to the constraint n-I ml is even and [ml 5 n, (x, y) are Cartesian

coordinates while (0,6) are the polar coordinates in the complex plane and Ram is a radial

polynomial defined as:

rl-lml

2 —S . _
lemme): z (1)8 1" l' p02 2.) (24)

5:0 s! ("HM—s]! {n—Iml_s]!
2 2

which is valued and satisfy the orthogonality relation [165] i.e. the inner product yields;

 

l
1
<Rp,q9Rn,m> : IRp,q(p)’Rn,m (,0)de = m n,p (25)
O

84

where, 5,”, is the kronecker delta. Also these polynomials have the property,

Rn,—m (,0) 2 Rn,m (,0) (26)

The inner product of the Zemike polynomials can be written as.

<VPCI’V'1m>: JIVp,q(p6),[Vnm(6p’)r/)dpd6 (27)

271'

1
: lIRP,q(p)elq6Rn.m(p)e-lm6pdpd6
0 0

2n 1
0 0

Substituting from equation (25) we can modify equation (28) as;

 

1 2n _( )6?
_ l q—m _
<Vp,ann,m> _ 2M 2 Ie d6 6,”, (29)
0
also,

27: .

I e‘(‘1"")‘9 d6 = 2mg”, (30)
0

thus, using equations (29) and (30) we get,

(Vp,q,Vn,m> = L6 5 (31)

n, m,
n+1 p q

Equation (31) proves the orthogonal property of Zemike polynomials. The Zemike

moment descriptor of an image [(x, y) is defined as its representation using a complex

85

orthogonal Zemike polynomial of order n and with m repetitions and is mathematically

given as,

<I(x, Y), Vn,m>

<Vp.q’vn.m>

Using result in equation (31) we get,

Zn,m :

 

(32)

me = n +1 ”Rx, y)[Vn,m(x, y)] *dxdy (33)

71'
x2+yZSl

 

Equation (33) can be expressed in a polar coordinate system as,

27[ 1 *
6)an,m(p.6)l pdpdé? (34)

             

00

127i'1 .
9)Rn,m(p)e_'m6p (1,0516l

             

00

2a 1
e) Rmmgo) [cos(me) — isin(m6’)] p dp d6

             

00

27:1

             

6) Rmm (p) cos(m6) ,o dp d6
0 0

 

n +12”1
—z' j 1 1mm Rmmtp) sin<m6>pdp d6
7’ 0 0

n+2”
El

12 l
—i";1[ Ila/3,6) ”UK/961MB]
O 0

 

l

lime) 6U}? pdpde

O (35)
fl"

 

86

Where,

6U}? = Rmm cos(m 6) (36)
are the real-valued even Zemike polynomials, and

0U,'," = Rmm sin(m (9) (37)

are the real-valued odd Zemike polynomials and these are related to the complex Zemike

polynomials by the following relation,
e m .o m
Vn,m: Un +1 Un (38)
Equation (35) along with equations (36) and (37) are used to decompose a given shape in

the form of a two-dimensional image, [(0,6) into the corresponding Zemike moments,

Zn’m.

Zn,m =6sz —i0Z,’.f' (39)

As discussed in previous sections, one of the most important requirements of an ideal
shape descriptor is its ability to represent the shape uniquely. This uniqueness implies
reliable reconstruction of the original shape using the corresponding set of feature
vectors. The reconstructed image of size P x Q pixels can be mathematically represented

as,

1(1), 6)= Z ZZm Vn,m(p.6) (40)

n=0 m
This reconstruction of the image using an infinitely large number of Zemike moments is

not computationally efficient and thus this series expansion is truncated at a finite order

NW.

87

N max

“1096): Z ZZanVan (,0,9) (41)

n=0 m

Using equations (18) and (39) we can write equation (41) as,

N max

11m): 2 2(622" _.- 02th.... W
n=0 m

A Nmax

ma): 2 21%?ng cos(mt9)+ 02;," sin(m6)] (42)
n=0 m

This maximum order of Zemike moments is derived by optimizing the closeness of the
original image, [(0, 6’) and the reconstructed images represented by equation (41) based

on a pre-defined threshold on a quantity like a RMS error as shown below,

 

 

 

1
( 2 )5
P Q Nmax eZ m cos(m 6) +
Z 2 [exp (p,t9)— Z Zan I; m .
p=o (1:0 ":0 m Zn s1n(m 9)
err/n = P x Q (43)
x l

The optimal values of the coefficients 8Z3, and 02;," were evaluated by minimizing

the error function given in equation (43). The complex Zemike moments were
represented by equation (39), rather than evaluating the inner products in equation (32)
the magnitudes of which form the feature vector. The Zemike moments are arranged in
this feature vector in ascending order of the radial polynomials, n and the ascending order

of their repetition, m.

88

3. 7.2 Rotational invariance

Invariance of Zemike moments to rotation can be easily demonstrated [74]. If [a is the

rotated image through an angle, a with respect to the Z-axis, while I is the original image

such that,

Ia(p,6)=l(p,t9—a) (44)
The Zemike moments of the original image and the rotated image can be written using

equation (34) as.

 

 

27:1
+1 _-
2mm = " j [1(p, 0)Rn,m(p)e 'mgpdpdfi (45)
7‘ 0 0
n+1”.1 '
Z“ = I[Ia(p,6—a)Rnm(p)e_’m9pdpd6 (46)
"’m E 00 ,

respectively. Equation (46) can be rewritten as follows,

nm—
0

 

l
75 [1am e — a)Rn,m( p)e“""(9‘“+a) p dp d(6— a+ a)
’ 0
Substituting 9a = 6 — a we get,

Za _n+12]-z
n,m
7’ 0

 

1 . o
l 1am9a)Rn,m(p)e"’m(‘9a)e"""“p dp Ma) W)
0

sz = Z,,,me“""a (48)
-. Z“ = Z (49)
n,m n,m

 

 

 

 

89

Equation (49) suggests that the magnitudes of Zemike moments which form the feature
vectors are equal for both the original and rotated images thus proving rotational
invariance of Zemike moments. The orientation angle, a between the original and rotated
images can be evaluated using,

arg(Z )— arg(Za )
n,m [1,)”

m

 

Invariance of Zemike moments to translation and scaling can be demonstrated in a
similar manner. These geometric properties and their ability to uniquely define a given
image make them an ideal choice to represent full-field strain maps obtained from digital

image correlation.

3. 7.3 Mapping a polygon to a circle

As mentioned earlier, Zemike moments are orthogonal only over a circular domain of
unit radius. This requirement makes it necessary to map a rectangular image onto a circle
of unit radius while retaining the shape features of the original image. This mapping is

illustrated in figure 24 and can be performed mathematically by using,

 

p_(-|y3 yak -(xB-xA)y|R (51)
,0, XAYB xBYA

9" U)" M y l(FB _FA)+ FA (52)
I ya yAlx -(xB-xA)y|

 

where, (p, 6’) are the unknown radial coordinates of a general point P’ in the mapped

geometry and (x, y) are the Cartesian coordinates of the corresponding point P in the

original geometry. (xA, yA) and (x3, y3) are Cartesian coordinates of points A and B in

90

the original geometry while (R, FA) and (R, F3) are the radial coordinates of the

corresponding points A’ and B‘ respectively in the mapped geometry.

    

l
.W A
— - _

7"A'

     

 

 

 

 

 

A'(R,FA)

 

 

x 9

Figure 24: Mapping a polygon to a circle

To summarize the process, the first step requires mapping of the centroid of the polygon
to the centroid of the circle (top right image in figure 24) which happens to be its center.
The polygon is then divided into triangles as shown in the top left image of figure 24. The
circle is divided in the same number of sectors as the number of triangles for the polygon
and the angle of each sector is decided by the area percentage of the corresponding

triangle in the polygon. The rest of the points in the triangle (the bottom left image in

91

figure 24) are mapped to the corresponding points in the sector (bottom right comer of
figure 24) using mapping functions given by equations (51) and equation (52). For
simplicity points within the polygon are represented by a Cartesian coordinates with the
origin lying at the centroid of the polygon while the mapped points are represented by
radial coordinates with the origin at the center of the circle. This mapping was performed
automatically and was coded as an integral part of the stand alone Shape analysis
Matlab® code, ImPaCT. Zemike moments were evaluated for all the composite

specimens using ImPaCT and the results are discussed in the following section.

 

ustrain
950
900 950
7:” .
.g 900‘. ......
850 3 850. ..........
s:
g 8004 ......
300 m 750:
100 .......

 

~60

 

'40

 

50 20
20 40
Width (mm) Length (mm) 0 Width (mm)

Figure 25: Original image representing the maximum principal strain map obtained from
digital image correlation of a virgin composite specimen without any damage with a

speckle pattern.

3. 7.4 Results & discussion

Figure 25 shows a two-dimensional as well as three-dimensional representation of a map
of the principal strain distribution in a virgin composite specimen with a painted speckle

pattern under tensile loads. Though the strain distribution is not uniform it varies only

92

over 200 p-strain which is 20% of the average strain in the specimen. Zemike moments

were evaluated using ImPaCT with a maximum order of Nmm. = 8 and are plotted as a bar

diagram in figure 26. The inserted plot in figure 26 has the zero order moment removed
to reveal the details of the higher order moments. This technique requires only 45 Zemike
moments, as shown in figure 26, to uniquely represent the strain map in figure 25 which
has more than 2500 data points (pixels). This claim was validated by a reconstruction of
the strain map using just the Zemike moments. The reconstruction was obtained using
ImPaCT as well and the reconstructed strain maps are plotted in figure 27. The original
and reconstructed strain maps from figure 25 and figure 27 can be compared and with an
initial visual comparison it can be concluded that the Zemike moments are capable of not
only representing the shape features of the strain map but also the average level of strain

with a high level of accuracy.

The zero order moment in figure 26 was observed to have a value of 879 )u-strain which
was the same as the average strain across the strain map in figure 25. To substantiate this
observation which suggests that the zero order Zemike moment represents the average
strain value of the distribution which is related to the applied load, a similar exercise was
performed with simulated data with a maximum principal strain map of uniform strain
equal to the average strain across the experimental map in figure 25. The simulated strain
map is plotted in figure 28 (right) and the corresponding Zemike moments are plotted in
the same figure in the middle. To confirm unique representation of the strain map using
Zemike moments, image reconstruction was performed and the reconstructed image

plotted in figure 28 (left) was compared to the original strain map. It was observed that

93

only the zero order Zernike moment was relevant with a value equal to the value of
uniform strain, while the rest of the moments were unchanged with a value of zero. The
same analysis was performed for a simulated strain map with uniform strain of unity and
the results are plotted in figure 29. The same trend was observed with the zero order
moment having a magnitude of unity which is equal to the value of uniform strain while
the rest of the moments were unchanged with a value of zero. This ensures that the zero

order Zemike moment is equal to the average value of the strain distributions.

 

 

 

900 35 1 I . , , T , .j
800) 30_ 4
700L 25_ , -
600 g 20_ 4
é 500’ §> 15~ ‘
-—- 2
Eu
«s 400~ _ —
2 10
soot 5- 1. -
. 1 1 .
: . l :
200 3 i 5 5 l l 1 l l ‘ . j _
l 00 10 20 3O 40
100 Zemike moments q

.L

 

 

 

J_ _L_

0 5 10 15 20 25 30 35— 40 45

Zemike Moments

Figure 26: Zemike moments evaluated for the maximum principal strain map obtained
from digital image correlation of a virgin composite specimen without any damage with a

painted speckle pattern, shown in figure 25.

94

ustrain

 

 

 

950
900 ’5 950‘
.y. '- 00~~
g 9
850 3 850“"
E 800«
U)
800 750-
100
60
750 20
20 40 60 Length (mm) 0 Width (mm)

Width (mm)

Figure 27: Reconstructed maximum principal strain map using Zemike moments in

figure 26.

To further illustrate this claim the average strain value was subtracted from the strain
distribution in figure 25 and is plotted in figure 30. Thus the average strain value of the
distribution represented by figure 30 is zero. Zemike moments were evaluated for this
strain distribution in figure 30 and are plotted as a bar diagram in figure 31. It can be
observed that the zero order Zemike moment has a value of “zero” and upon close
examination the Zemike moments in figure 31 are identical to the Zemike moments the
embedded plot of figure 26 which excludes the zero order Zemike moment for. These
observations confirm the claim that the zero order Zemike moment is equal to the
average strain of the strain distribution that the corresponding set of Zemike moments
uniquely represents, while the rest of the moments contribute towards the shape of the
strain map. Thus a set of Zemike moments evaluated for a strain map represent not only
the shape of the strain distribution but also the level of average strain in the composite

specimen. Figure 32 shows the reconstructed strain distribution obtained using the

95

Zemike moments in figure 31. This reconstructed strain map is identical to the original

strain distribution in figure 30.

ustrain pstram

10 i
Ezo 600 EM 600
E30 400 a L”400 . °°
3 4o

50

60 - ~

00 20 4o 29 40 60
Width (mm) Zemike Moments Width (mm)

 

800

600

400

200

 

 

O

 

Figure 28: Zemike moment descriptor (middle) and reconstructed image (right) of the
original image (left) representing a simulated maximum principal strain map with

uniform strain of 879 ,ustrain.

 

 

 

 

nstrain “strain
10 . . . - 10
1

A 10 8 g 10 8
E 20 30-3 E20
5 6 -— a 6
E30 E015 .2530
= 40 4 0.4 233040 4
.3 ._1

50 2 0,2 50 2

60 0 - - . - 60 0

. 60 0o 10 20 30 4o 20 4o 60
Width (mm) Zemike momnts Width (mm)

Figure 29: Zemike moment descriptor (middle) and reconstructed image (right) of the
original image (left) representing a simulated maximum principal strain map with

uniform strain of 1 ,ustrain.

96

ustrain

 

 

 

 

100
50
h 0
-50 '
.. (I)
-100 50
20 40 60 h 0 .
Width (mm) Lengt (mm) Width (mm)

Figure 30: Original image representing the maximum principal strain map obtained by
subtraction of the average strain of the strain map in figure 25 from the strain map in

figure 25.

 

35 I I I I I I I I I

30* -

 

 

0 5 10 1 5 2O 25 30 3 5 40 45
Zemike moments

Figure 31: Zemike moments evaluated for the maximum principal strain map in figure

30.

97

pstrain

 

 

 

100
10
50 A
E 20 .5
S 0 g
.s: 30 3.
E040 50 -§
.3 i:
U)
50 -100 f
20 40 60
Width (mm) Length (mu!) 0 Width (mm)

Figure 32: Reconstructed maximum principal strain map using Zemike moments in

figure 31

To illustrate the validity of performing digital image correlation on a composite specimen
without any surface preparation or speckle pattern imprinting, strain maps were obtained
for the same virgin specimen using the non-speckle method introduced in the
Experimental set-up and methods chapter. The same tensile load was applied as in the
speckled case. The map of maximum principal strain for this non-speckle specimen is
plotted in figure 33. It can be observed that even though the strain distribution in the
specimen is different from the one seen in figure 25 for its speckled counterpart, the level
of strain is approximately the same. In the speckle case the coat of paint is acting as a
strain witness while in the non-speckle case the strains are evaluated using displacement
maps recorded directly of the specimen surface. The quality of the glare affects the digital
image correlation measurements locally producing variations which results in the
observed discrepancy in the strain distribution of the two cases. These observations

combined with the previous argument is enough to deduce that to validate the non-

98

speckle digital image correlation method the strain levels are more relevant that the strain
distribution. Comparing the zero order Zemike moments from figure 26 and figure 34 we
can conclude that the surface texture of the composite specimen combined with special
illumination can be used for digital image correlation eliminating surface preparation and

dependence of strain distribution on the quality of the speckle pattern.

 

  

 

 

ustrain
900
10
A :p 850 E 900~
E 20 E
E30 800 :55 800~............~
t: " .5
3 40 r :3
750 7) 700
50 100 ...____';:_-....-.-_'_'_'." .
20 4o 60 700 50 0 20 _
Width (mm) Length (mm) Width (mm)

Figure 33: Original image representing the maximum principal strain map obtained from
digital image correlation of a virgin composite specimen without any damage without a

speckle pattern.

Even though Zemike moments look promising in fulfilling the objective of being a
unique shape descriptor for full-field strain maps, their performance in handling
geometric and design features in the structural component still needed to be assessed.
Usually damage initiates in critical regions of the structural components which contain
specific design features like holes and rivets which act as stress raisers. The boundaries of
such geometric features manifest themselves in the strain maps as sharp discontinuities

and they also sometimes lead to high level of stress concentrations. To verify the

99

effectiveness of Zemike moments in representing more realistic strain maps with such
sharp discontinuities, two composite specimens were used in form of a plate with a hole.
One specimen had an impact hole caused by a drop weight projectile penetrating the
composite panel while the other specimen had a machined hole. The map of maximum
principal strains obtained using digital image correlation for the machined hole has been
plotted in figure 36 showing a typical strain distribution for a plate with a hole with stress
concentrations at the hole boundary along the horizontal axis and a corresponding stress
depression along the vertical loading axis. Zemike moments were evaluated for the strain
map in figure 36 with maximum order of NW = 8 and the corresponding reconstruction

is plotted in figure 37.

 

 

 

800 — I I I I I I I I I
35-
700" 30_ r
600* 25 r
500‘ -
o
“O
£3400
8
2
300* -
200" . r
0 l l . t
0 10 20 30 40
100 ’ Zemike moments ‘

_L .1. _l _ L

20 25 30 35 40 45
Zemike moments

 

 

 

Figure 34: Zemike moments evaluated for the maximum principal strain map obtained
from digital image correlation of a virgin composite specimen without any damage

without a speckle pattern

100

ustrain

 

  

 

 

900
10 .-
E 20 .5
E g
30 800 g. , """
g) . E“ 800
3’ 4° I E
:4 750 m
700> ......
50 100 _
60 .. § ........ 40 60
20 40 60 700 50 20
Width (mm) Length (mm) 0 Length (mm)

Figure 35: Reconstructed maximum principal strain map using the Zemike moments in

figure 24.

It can be observed from figures 27 and 28 that Zemike moments completely fail to
record the shape features in the strain map, one solution to which is to increasing the
maximum order of Zemike moments. Thus Zemike moments corresponding to NM = 20
were evaluated and the corresponding reconstructed strain map is represented in figure 37
as well. Although the shape features become more prominent, the representation is still

not satisfactory.

Convergence curves for the composite specimen with a machined hole are plotted in
figure 39 for the root mean squared error between the original and reconstructed images
and the computational time plotted as a function of the maximum order of Zemike
moments. It can be observed that as the maximum order of Zemike moments increases
the root mean squared error reduces and the reduction between the NM = 8 and NM = 20

cases is 235% while the corresponding increase in the computation time is =900%.

101

N
O
O
O

Strain (ustrain)
S
O
O

 

 

100’

50
20 40 60
Width (mm) Length (mm) 0 Width (mm)

 

Figure 36: Original image representing the maximum principal strain map obtained from

digital image correlation of a composite specimen with a machined hole

pstrain

N
O
O
O

Strain (ustrain)
S
O
O

 

 

100‘

 

Length (mm) 0 Width (mm)

 

Figure 3 7: Reconstructed maximum principal strain map using Zemike moments

obtained for the maximum principal strain map in figure 36 with maximum order of

Zemike moments of Nmax=8.

102

ustrain

   

 

 

     

{:2000 2500~ . t
"E 2°°°~"""” ; -- .
I. 1000 3 10004 .
W 500. =
100 ~60
500 40
20 40 60 50 20
Width (mm) Length (mm) 0 Width (mm)

Figure 38: Reconstructed maximum principal strain map using Zemike moments

obtained for the maximum principal strain map in figure 36 with maximum order of

Zemike moments of Nmax=20.

3.7.5 Advantages, limitations and solutions

It can be concluded from the previous results section that Zemike moment descriptors are
robust shape descriptors capable of unique shape representation of high resolution strain
maps with less than a hundred Zemike moments. Also, their unique properties of
rotational, translational and scaling invariance make them a desirable shape descriptor for
representing full-field strain maps in the field of experimental mechanics. However,
Zemike moment descriptors are not able to cope with the discontinuities in the strain field

due to geometric and design features such as through holes, impact damage holes, etc.

103

 

 

 

 

400.0 , , . r. : 2400
O RMS error—Strain map f , 5
350.0 _- ITime """" '7' 2100
300.0 —————————————————— ------- = ------- : ------- moo
E 250.0 ~ ------ 9 ““““ : ““““ TW“1500A
a a a 3 El ' E:
:: 200.0 it ------ j ---- 3 """"" L “““ T ““““ E """" T """" '1200 0
g : : J : : I ,§
0 , . . . t—
m 150.0 4I————--~-} ------- i --- :"'“““: ””””” i ‘‘‘‘‘‘ ' ’’’’’’ .__9m
2 ; I ' i i i
9‘ i 3 3 . : :
1000 ~~~~~~~~ 1 """"""" : ******* r ------ 1 “““ r “““““ 600
50.0 -------------- a ------- i ------ e ——————— ----- —~ 300
: I 1 — E E
0.0 - ~ ' i i l 't i 0

 

 

0 3 6 9 12 15 18 21
Maximum order of Zemike moments

Figure 39: Convergence curves for Zernike moment shape descriptors evaluated for a

composite specimen with a machined hole.

One solution to this is to increase the maximum order of the Zemike moments and thus
was explored and discussed in the previous results section. Higher values of N mean high
computational cost as depicted by the convergence curve in figure 39 making this an
impractical solution. Another solution to this problem is to use the Gram-Schmidt
approach to obtain arb-Zernike polynomials [139, 166 and 167]. Here only those Zemike
polynomials relevant to the annular region of the unit circle excluding the central hole are

evaluated. This technique can be extended to different geometries including circular

104

plates with rectangular holes, etc. Although an effective solution, identifying the
appropriate Zemike moments can be a time consuming and tedious proposition for
application in the field of structural experimental mechanics due to the variation in
component geometries and design features. The Gram- Schmidt approach is not in the
scope of this work and attention is moved to Fourier descriptors and wavelet descriptors
to study their advantages and disadvantages as suitable shape descriptors for full-field

strain maps of damaged composite materials.

3.8 Fourier descriptors

Fourier descriptors are amongst some of the most widely used shape descriptors in the
field of pattern recognition. These are mainly obtained by taking Fourier transforms of
signals or three—dimensional shapes. The frequency component as well as the phase
component of the Fourier transform can both be used for the purpose of shape
description. Fourier descriptors are invariant to location but are not invariant to rotation

and size as will be discussed in the following section.

3. 8. 1 Technique

Fourier descriptors are obtained by taking the Fourier transforms of a signal. This can be

obtained by [68 and 69],
DF (f) = [um e‘iw’ dt (53)

105

The amplitude as well as the phase obtained from the above transform contains useful
information for shape description. The above equation (53) can be represented in discrete
form as,

.27t

_l____

N—l 1
DF(§)= Z u(t)e N (54)
n=0

This discrete Fourier transform (DFT) has some useful properties which are relevant to it
being a good shape descriptor. The discrete Fourier transform is invariant in translation

while rotation in the spatial domain can be easily accounted for by multiplying each

. . . . 'a . .
coordinate 1n the frequency domain. With a term 8/ where a is the angle of rotation.

Thus, rotation affects only the phase of the discrete Fourier transform and rotational
invariance can be achieved by ignoring the phase information and using only the
magnitudes of the transform coefficients [168 and 169]. The phase information is
sometimes used in the cases of finger print and iris matching. Scaling of the original
signal requires scaling of the discrete Fourier transform with just a simple scalar. Also,
the most significant attribute of Fourier descriptors is that the low frequency components
reflect the global configuration of shape while high frequency components store the
shape details. Thus based on these attributes it is necessary to normalize the Fourier
descriptors to eliminate their dependence on the position, size, orientation and starting
point of the signal. The reconstruction of the original signal can be easily obtained by

taking an inverse Fourier transform of the Fourier descriptor.

106

Similarly, the concept of Fourier Descriptors can be extended to two-dimensional images
by taking multi-dimensional Fourier transforms. The expression for a two-dimensional
Fourier transform is given as,

+oo+oo
DF(u,v)= I Ie—12”(“x+vy)l(x,y)dxdy (55)

—OO —00

which can be written in its discrete form as,

DF (u,v)— —

1 K— 1L—1 i—27:[5—k +77%)
k—Mz 2e K I(k,l) (56)

where, f=0,....,K-I and n=0,.....,L-I.

Alternatively, Fourier descriptors such as frequency-weighted FD can also be defined by
dividing the Fourier components with their corresponding frequencies to reduce the

sensitivity to high-frequency noise.

Since the strain data is real valued, the conjugate symmetry of the Fourier transform
given by equation (57) below suggests that only half of the frequency plane is uniquely

determined by the real valued image [68].
~ ~*
1 (u, V) = 1 (-u.-V) (57)

where, * denotes the complex conjugate and T is given by,

+oo+oo
T(u,v) = I Jl(x, y)e—2m(xu+yv)dxdy (58)

—oo —oo

which is the two-dimensional Fourier transform of the image.

107

ustrain radians

950

ii900 ’7
' E

800

 

20 40 6O 2 20 40
Width (mm)

Vthh(nnn) 14““!

X 104 log(1/mm2)

Length (mm)
VI A L» N
O O O O

 

O\
O

20 40 6O 20_ 40
Width (mm) Width (mm)

Figure 40: Two-dimensional representations of the Fourier descriptors in terms of the
phase value map (top right), map of absolute value of the discrete Fourier transform
(bottom left) and a map of the logarithm of the absolute value of the discrete Fourier
transform (bottom right) of the original image (top left) representing a maximum
principal strain map obtained from digital image correlation of a virgin composite

specimen without any damage.

108

radians

  

  

 

 

 

   
  

 
  

 

 

 

 

   

[—
‘5 5~
900~-~“ 95
.. Q)
0
700" E _5’ V‘
50 50
Length (mm) 0 Width (mm) Length (mm) 0 Width (mm)
l/mm2 4 t—
A x 10 u. log(l/mn12)
"E: “ Q
Q—t
\ O
c 10 E 2
E 5.5 -
“a 5..., '8 g ..
o (H 4°10
E S, g
g o. 3 04
3100 100
<
50 50
Length (mm) 0 Width (mm) Length (mm) 0 Width (mm)

Figure 4]: Three-dimensional representations of the Fourier descriptors in terms of the
phase value map (top right), map of absolute value of the discrete Fourier transform
(bottom left) and a map of the logarithm of the absolute value of the discrete Fourier

transform (bottom right) of the original image (top left) representing a maximum
principal strain map obtained from digital image correlation of a virgin composite

specimen without any damage.

109

ustrain radians

 

 

20 4o 60
. 20 40 60
Width (mm) 2 Width (mm)

20 40 60 20 40 60
Width (mm) Width (mm)

Figure 42: Two-dimensional representations of the Fourier descriptors in terms of the
phase value map (top right), map of absolute value of the discrete Fourier transform
(bottom left) and a map of the logarithm of the absolute value of the discrete Fourier

transform (bottom right) of the original image (top left) representing a maximum
principal strain map obtained from digital image correlation of a composite specimen

with a machined hole.

3.8.2 Elliptical shape descriptor and clustering

The pattern distribution of the Fourier descriptors thus obtained can be further described

using a normalized elliptical vector, feu.

110

r . w
centrald, u
centroid, v

= < orientation (59)
ell

eccentricity
(spread

 

 

 

 

   

 

 

 

 

 

 

radians
[—
LL.
Q 5
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:3 m
> "-1
0 1:
a 3"; -
u: -51 ‘
50
Length (mm)
[—
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E D
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c .3 A
i: 3 "E
D 53 g 5
“a e a
.15 “5 a“
a no :3
> 3 0.
£31 100
< 50
Length (mm) 0 Width (mm) Length (mm)

Figure 43: Three-dimensional representations of the Fourier descriptors in terms of the
phase value map (top right), map of absolute value of the discrete Fourier transform
(bottom left) and a map of the logarithm of the absolute value of the discrete Fourier

transform (bottom right) of the original image (top left) representing a maximum
principal strain map obtained from digital image correlation of a composite specimen

with a machined hole.

111

Composite Specimen - Machined Hole l/Pixel2

x 106
10_ ......................................... DDDDDDDDDD
9_ ......................................... DD . .DDDDD...
8L ......................................... D ENDS
7. ........................................ C’DD.DDD. . DE]...
6' ........................................ DUNE-JESSE
5_ ....................................... DD.DDE:H:H:H:I
4t ....................................... DDDDDDDD

CHIC!)

 

 

 

 

 

Pixel

 

 

 

. IDDD'D
1. ........................................ ...-DQDD

 

 

 

 

ii34séis910
Pixel

Figure 44: Absolute values of low frequency components of the two-dimensional discrete
Fourier transforms and its elliptical descriptors, evaluated for the strain distribution in

composite specimen with a machined hole.

Figures 40 and 42 show two-dimensional representations of the absolute value of the
frequency component (bottom left) and the phase component of the discrete Fourier
transform (tOp right) of the strain distribution (top left) for a virgin composite specimen
and a composite specimen with a machined hole respectively. Figures 41 and 43 show the

corresponding three-dimensional representations. It can be observed that only a few low

112

frequency components are significant and thus only the first 10x10 frequency

components in the first quadrant are chosen for shape description as shown in figure 44.

     

   

. Specimen #4 . Specimen #3 _
Virgin Specimen p-stram (Impact energy = 30J) u-stratn (Impact energy = 32.51) u-stram
3000 00 3000
30 30 30
€25 A25 E25
520 2000 520 2000 v20 2000
{3015 at 5 "£301 5
‘3 t: t:
310 1000310 1000310 1000
5 5 5
0
10 20 30 0 10 20 30 O 10_ 20 30
Width (mm) Width (mm) Width (mm)
Specimen #2 Specimen #1 ll' strain Specimen with '
(Impact energy = 351) p-strain (Impact energy = 37.51) impact hole p—stram
5000
3000 3 0 4000 30
A25 3000 E25 4000
2000 520 520 3000
E 2000 -5
2°15 ”’1 5 2000
1000 310 £10
1000 1000
5 5
0
10 20 30 0 10 20 30 0 10. 20 30
Width (mm) Width (mm) Width (mm)

Figure 45: The maximum principal strain distributions in the incrementally damaged

composite specimens under a tensile load of 4000N.

Figure 44 shows only the first 10x10 frequency components obtained by taking a two-
dimensional discreet Fourier transform of the maximum principal strain distribution for a
specimen with a machined hole. The location of the weighted centroid was evaluated
based on the geometry and was weighted using the magnitude of each frequency
component in the 10x10 grid. The major axis, minor axis, eccentricity and orientation

were obtained for an ellipse which had the same normalized second central moment of

113

superimposed on the map of absolute value of the frequency component and was centered
at the weighted centroid of the region as shown in figure 44. The parameters describing
the shape, size and location of the shape representative ellipses are combined to form the
corresponding elliptical shape descriptor given by equation (59). Since the components of
the elliptical shape vector have different units they are normalized to dimensionless units

using the mean, u and standard deviations, o [68].

~ f -#
felt = end (60)

_ . 2 5 . 2 5 . 2 5
Virgin SpeCimen up "$910 Specimen#4 VP 1x£110 Specimen #3 UP 1x£110

 

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Figure 46: Absolute values of low frequency components of the two-dimensional discrete
Fourier transforms and their corresponding elliptical descriptors, evaluated for the

strain distribution in composite specimens with varying amount of impact damage.

114

 

 

 

 

 

 

<> Weighted Centroid: X coordinate
El Weighted Centroid: Y coordinate
A Eccentricity
Cl Major axis of elliptical descriptor
0 Minor axis of elliptical descriptor
A Orientation (Secondary Axis)
15 I I I I 100
E:3 [II 5 n
12 E] : : [3 : E3 80 33‘
f : : : a
A x i 4: e”
g 9Z3 : ‘ “25560 E
I ' on
g g C? Q C? Q E
6g : a + 2 5 ~- 40 .2
: : : 2
l 1 i I 0)
1 [j] : [j [J 5'
32> g2   Ea ; 2520
E : O O Q
i A A A A
0 ' ‘ ' ' A 0
5 <1- m N ; 2
.§ i: 4: 4: g E
O Q) 0) G) H
a .e e E E a
m 8 a 8 8 2*
'5 E
'3
m

Figure 4 7: Variation in the elliptical descriptor with varying amount of damage in
composite specimen. Specimens I to 4 are the incrementally damaged composite
specimens with increasing impact-energy. Specimen 5 is a virgin specimen with no
damage while specimen 6 and 7 are composite specimens with an impact hole and

machined hole respectively.

115

Figure 45 shows distribution of the maximum principal strain in six incrementally
damaged composites. Elliptical vectors were evaluated for each of the absolute values of
the frequency components obtained for these strain distributions. The frequency
components of the two-dimensional discrete Fourier transforms for the maximum
principal strain distributions in the six incrementally damaged composite specimen are
plotted in figure 46 superimposed by the ellipses represented by the corresponding
elliptical vectors. It can be observed that the weighted centroid moves vertically up along
the y—axis with increased level of impact damage. Also, with increased impact damage
the spread of the elliptical descriptor in terms of the minor axis increases until it becomes

equal to the major axis for the composite specimen with an impacted hole.

A hierarchal clustering algorithm can be applied to a set of such elliptical descriptors,
which based on a predefined threshold segregates the data based on the similarity of the

normalized elliptical shape descriptors [68].

3.8.3 Results & discussion

The two-dimensional Discrete Fourier transforms (Fast Fourier transforms) of the full-
field strain maps were evaluated using equation (56). The corresponding absolute value
of the frequency components and the distribution of the phase component of the discrete
Fourier transforms were studied as illustrated by figures 40 & 41 and figures 42 & 43 for
a virgin composite specimen and a composite specimen with a machined hole
respectively. The logarithm of the absolute value of the frequency components of the
discrete Fourier transforms were calculated and are plotted in the bottom right corner of

figures 40 to 43. All three of these data sets can be used as unique shape descriptors.

116

These Fourier descriptors are of limited value on their own since they are images which
are of the same size as the original strain map and hence provide no reduction in the
quantity of data. Various techniques have been employed to extract useful shape
information from these Fourier descriptors one of which was discussed in the previous
section. The elliptical shape vectors are plotted along with the corresponding Fourier
descriptors in figure 46 for composite specimens with the same type of impact damaged

caused by a drop weight testing machine but with different impact energies.

The variations in the components of the elliptical shape vector with respect to six
different strain distributions are plotted in figure 47. In the specimen with through holes
the strain distribution is influenced by the presence of the hole giving a completely
different strain distribution compared to the incrementally damaged composite specimen,
which had varying amounts of matrix cracking and delamination due to impact. The
incrementally damaged composite specimen had no through holes as the impact-energy
was maintained at a low enough level to keep the impactor from penetrating the
specimen. Thus it can be observed that the elliptical vectors are capable of distinguishing
between the incrementally damaged specimen and the composite specimen with impact
hole. With increased impact damaged the orientation angle and the Y-coordinate of the
weighted centroid increase while the X-coordinate of the weighted centroid seems to
decrease. The spread and eccentricity of the ellipse show no definite trend with increasing
level of damage. However, these variations were very small concluding that the elliptical
shape vector evaluated from the discrete Fourier transforms of the strain distribution are

not sensitive enough to be applied to the field of structural damage assessment. Besides

117

this drawback there is no method to obtain a reconstruction of the strain distribution from

these elliptical descriptors, making there unique shape representation questionable.

The discrete Fourier transform of the strain distribution in figure 42 and figure 43 shows
no discontinuity due to the presence of a hole in the specimen. Thus it is proposed that
the Zemike moments descriptors can be obtained for the Fourier descriptors of the
original strain map of the specimen. Since both the Fourier descriptors and the Zemike
moment descriptors have been proven to represent every shape uniquely the Zemike
moment descriptor of the Fourier descriptor of a strain map in theory should be a unique
shape descriptor. However, the Fourier descriptor in consideration which is the
magnitude of the discrete Fourier transform of the strain map, has a peak in the central
portion with a large magnitude causing a steep gradient in its shape. Though the Fourier
descriptor is continuous, it is unlikely that this peak can be modeled using Zemike
moments descriptors with a lower order. To overcome this anomaly a new Fourier
descriptor is introduced which is simply the natural log of the magnitude of the discrete
Fourier transform of the strain map denoted by ln(DF T). This Fourier descriptor is the
same size as the original image, while it contains more high quality shape information
with lower gradients which can be easily represented using lower order Zemike moment
descriptors. However, it still remains to be proven that this proposed combined Fourier-

Zemike descriptor is capable of uniquely representing the shape of the full-field strain

field.

118

Even with the difficulty of interpreting and comparing Fourier descriptors it was
observed that they were able to cope with the sharp discontinuities in strain maps as
illustrated in figure 42 and figure 43. In this case a simple inverse Fourier transform
yielded the original strain map while neither the frequency component nor the phase
component of the discrete Fourier transform of the strain distribution of the composite
specimen with a machined hole possessed any sharp discontinuities due to the presence

of the hole.

3. 8.4 Advantages, limitations and solutions

It can be concluded from this section that Fourier descriptors are difficult to interpret and
provide no reduction in the quantity of data unless combined with different techniques to
extract useful shape features from the Fourier descriptors. One such technique was
discussed in detail in this section where the Fourier descriptor was described by a
representative elliptical vector. However, these elliptical vectors were ruled out as an
application in structural damage assessment based on their relative insensitivity to
differentiate between specimens with the same type but different levels of damage. Also,
they are information non-preserving shape descriptors incapable of reconstruction of the

strain map making their ability of unique representation questionable.

Even with all of its drawbacks Fourier descriptors are easy to apply and were observed to
effectively represent sharp discontinuities in strain maps due to geometric features in the
specimen. This property gives them an edge over the geometric moment descriptors
discussed in the previous chapter with respect to applications in the field of experimental

mechanics.

119

Original strain map Map of log of as. value of DFT Fourier - Zemike moments

 

 

  
   

 

 

 

 

 

ii-strain 8
2500 , / ln( 1 "5‘5“ )
' " " ‘ ‘ 0.4
g 2000A i2 0 6
E 20 520 'o 0.3
2’ 1500 5 '0 ’3 4 0 2
ED fit 8 g) .
3 40 1000 540 2 0-1
,_1 6 2 0 i
500 0 100 200
Wdth 20 40 60 0 100 200
‘ (mm) Width (mm) Zemike moments
Reconstructed strain map . Map of phase value of DF T Reconstructed logarithm magnituczle
H'Stram radians map ln( l/mm )
0 12
g 2000 g ’5‘
£20 E20 3 20 IO
_. -50
ft. .. 1500‘s?) it
540 540 5 40
,_1 ..1 t—l 8
1000
60 60 -100 60
20 40 60 20 40 6O 20 40 60
Width (mm) Width (mm) Width (mm)

Figure 48: Fourier Zemike moments with a maximum order of Nmax=20, of the original

image representing a map of maximum principal strain obtained from digital image
correlation of a composite specimen with a machined hole. The strain map is
reconstructed using the reconstructed logarithm magnitude map and original phase value

map of the discrete Fourier transform.

3.9 Fourier - Zemike moments

The objective of this study is to perform shape decomposition which is simple to apply as
well as interpret. It is easy to decompose a given image with data points of the order of at
least 103 in terms of only a few hundred Zemike moments which are easy to interpret.
However, they are incapable of handling sharp discontinuities in the image. On the other

hand, two-dimensional discrete Fourier transforms provide us with the magnitude of the

120

Fourier coefficients and the phase information both of which can be used as shape
descriptors. These Fourier descriptors are of limited value on their own since they are
images of the same size as the original strain map and hence provide no reduction in the
quantity of data. In fact, Fourier transform of a strain map results in a magnitude map and
a phase map of the frequency component, each being the same size as the original strain
map leading to twice the amount of useable data. Various techniques have been used to
extract useful shape information from these Fourier descriptors [68, 69 and 74]; one of
which was discussed in the previous section. However, it was observed that the logarithm
magnitude map obtained by taking a logarithm of the absolute value of the discrete
Fourier transform of the original strain map with a sharp discontinuity was continuous.
The logarithm reduced the dynamic range of the absolute value of the discrete Fourier
transform. Many of these shortcomings can be alleviated by combining Fourier
transforms and Zemike moments as will be illustrated through examples in the following

sections.

3. 9. 1 Technique

Figure 48 describes the method of obtaining Zemike moment descriptors. Since this
shape descriptor is supposed to be able to uniquely represent strain distributions with
discontinuities its was initially evaluated for the map of logarithm of the absolute value of
the discrete Fourier transform of the strain map for a composite specimen with a
machined hole. The ability of this shape descriptor to uniquely represent the shape of
each full-field strain map was verified by attempting to reconstruct the strain map from
the corresponding shape descriptors. First the Fourier descriptors were evaluated for the

strain map (top left) in figure 48 by taking the natural logarithm of the absolute value of

121

the discrete Fourier transform, ln(DIC) which is plotted in the top center of figure 48. The
phase value map of the discrete Fourier transform is stored and plotted bottom middle in
figure 48. The Zemike moment descriptors were then evaluated for the logarithm
magnitude map. It is proposed that these Zemike moments can be used as a unique shape
descriptor for the original strain map in figure 48. Since the discrete Fourier transform of
the original strain map is a unique shape descriptor of the latter and the Zemike moments
are capable of uniquely representing the logarithm magnitude map, then the Zemike
moments must be a unique representations of the original strain map. This is the first time
Zemike moments of the Fourier descriptor have been used as an integral shape descriptor
and applied in the field of experimental mechanics. This combined shape descriptor is
named as the Fourier-Zemike moment descriptor based on the order in which the two

techniques are applied.

     

ii-strain u-strain it-strain
2500
A 2000 A
3520 2000 E20 52 2000
E E E
V V 1500 4:
a 1500 .2: 8'0 1500
= G40
5:40 1000 340 1°00 .3
500 . 1000
500 ,
60 60
60 20 40 60 20 40 60 20 40 60
Width (mm) Width (mm) Width (mm)

Figure 49: Comparison of reconstructed strain distribution from Zemike moments
(center) and F ourier-Zernike moments (right) evaluated for the maximum principal strain
distribution (left) in the composite specimen with a machined hole, under a tensile load

with a maximum order of Zemike moments, NM = 8.

122

 

 

 

 

      

 

 

         

Original strain map , Map Of 3133- value 0f DFT 2 Fourier Zemike moments
ti-strain ln(l/mm ) 5 7 - . - .‘
950 10 4 0.4 t .
2 %
900 v O 8 g 3 0 2
850 '51; Q 2 °
5 6 2
800 -‘ i 0
60 ' * 0
20 40 60 29 40 60 O 50 100 150 200
Width (mm) Wldth (mm) Zernike moments
Reconstructed strain map Map of phase value of DFT Reconstructed logarithm
ir-strain radians magnitude map ln( l/mmz)
10
950 2 A
E E
900 £20 0 £20 8
"i3: "3)
850 540 -2 540 6
800 '4 "I
4 4
60 60
20 40 60 20 40 6O 20 40 60
Width (mm) Width (mm) Width (mm)

Figure 50: Fourier Zemike moments with a maximum order of Nmax=20, of the original
image representing a map of maximum principal strain obtained from digital image
correlation of a virgin composite specimen. The strain map is reconstructed using the
reconstructed logarithm magnitude map and original phase value map of the discrete

Fourier transform.

3.9.2 Results & discussion

To study the ability of Fourier-Zemike moments to uniquely represent the corresponding
original strain maps an attempt was made to reconstruct the strain distribution starting
with just the Fourier-Zemike moments. The logarithm magnitude map, ln(DIC) was
reconstructed from the Zemike moments and is plotted bottom right of figure 48. The

absolute value of discrete Fourier transform were retrieved from this reconstructed

123

logarithm magnitude map which was combined with the original phase information
plotted bottom center in figure 48 to reconstruct the full-field maximum principal strain
map. After visual comparison of the reconstructed strain distributions using the Zemike
moments and Fourier-Zemike moments with the original strain distributions as shown in
figure 49, it was concluded that the Fourier-Zemike moments are indeed capable of
capturing the correct shape features of the full-field strain distribution with a

discontinuity due to the presence of a hole.

 

 

    

 

 

 

    

   

Original strain map . Map'of log of abs. value of DFT 5 Fourier ' Zemike moments
ln(l/mmz) ' ' - ' ;
10 o 4
”5’2 B 3
v 8 .—
34 6 1
4
60 0
20 40 20 40 60 0 IO .20 30 40
Width (mm) Width (mm) Zemike moments
d , M f h 1 fDFT Reconstrcted lograthm
Reconstructe strain mfiystrain ap o p ase va ue o radians mamtude map ln(l/mmz)
950 2 8
”£2 €20 520
v 900 z 0 4:
g» E0 E” 6
:34 850 340 -2 .340
“4 4
60 800 60 60
20 40 6O 20 40 60 29 4O 60
Width (mm) Width (mm) Wldth (mm)

Figure 51: Fourier Zemike moments with a maximum order of Nmax=8, of the original

image representing a map of maximum principal strain obtained from digital image
correlation of a virgin composite specimen. The strain map is reconstructed using the
reconstructed logarithm magnitude map and original phase value map of the discrete

Fourier transform.

124

Fourier-Zemike descriptors were evaluated for the strain distribution in figure 25 for a

virgin composite specimen under tensile loads. The same procedure discussed above was

employed with a maximum order of Zemike moments of Nmax = 8 and NW = 20 and the

corresponding Fourier—Zemike moments and reconstructions are plotted in figure 50 and

figure 51 respectively. It can be observed that the case with me = 20 provides a better

representation of the original strain distribution. The reconstructed strain maps using the

Zemike moments and Fourier-Zemike moments for a strain distribution in the virgin

specimen under tension are plotted along with the original strain distribution in figure 52.

It can be concluded that the Zemike moment descriptors are more effective and efficient

compared to the Fourier-Zemike descriptors for strain maps without discontinuities.

 

 

u-strain u-strain u-strain
950 A 950 A 950
E20 900 E 20 900 E20
E I z 900
"3 850 85° ...
E040 $5040 §40 850
3 800 -—l 800 ...:
800
60 750 60
6O 20 40 60 20 4O 60 20 4O 60
Width (mm) Width (mm) Width (mm)

Figure 52: Comparison of reconstructed strain distribution from Zemike

moments (center) and F ourier-Zemike moments (right) evaluated for the maximum

principal strain distribution (left) in the virgin composite specimen under a tensile load

with a maximum order of Zemike moments, NW = 8 .

As for the Zemike moments, convergence curves were plotted for Fourier-Zemike

moment descriptors as shown in figure 53. The root mean squared error was calculated

between the original and reconstructed strain maps with increasing values of the

125

maximum order of the Zemike moments. These root mean squared errors along with the

corresponding computational times were plotted as a function of the maximum order of

Zemike moments. It can be observed that for the Nmm. = 20 case the computational time

for evaluating Fourier-Zemike moment descriptors is 2900 seconds which is less than
half the time required to evaluate the Zemike moment shape descriptors for the same
strain map. It can also be observed that the root mean square errors are much bigger in
the Fourier-Zemike moments case than the Zemike moments case thus suggesting that
the Zemike moment descriptors are more accurate representation of the strain map than

the Fourier-Zemike moments.

3. 9.3 Advantages, limitations and solutions

Fourier-Zemike moments that are obtained by combining Fourier description techniques
and Zemike moment descriptors possess all the desirable properties of its parent shape
descriptors. Zemike moments are invariant to rotation, translation and scaling while these
transformations can be easily accounted for in Fourier description techniques. They form
a robust shape description technique that provides a unique representation with only a
few low order Zemike polynomials. They are easy to implement and computationally
efficient to evaluate. At the same time they are capable of eliminating the limitations of
Fourier descriptors and Zemike moment descriptors. They are able to cope with sharp
discontinuities in the strain map unlike Zemike moments and simultaneously are easier to
interpret unlike Fourier descriptors. All these properties make Fourier-Zemike moments

the preferred choice for the application of shape analysis techniques to structural damage

126

assessment, especially in anisotropic materials such as fiber-reinforced polymer

 

 

 

composites.
650.0 , . . ; 7 r 1000
O RMS error-Strain map
. : : : I
I Time 1 1 I
600.0 5 ’ I f ’ ’ i E E 800
A Q.
.5 2 i I l i
a 550.0 — : : : :+ : 600 3
1 I i I I I 8
:j t : : : ' 3
o O : : : I E
E 00 O - : : : : 400 F
m 5 . ' : : 1
2 1 I I I
m i i i
: 1 ' I :
450.0 s " 200
g a 2 o
4000 — i i i i 0

 

 

 

 

0 3 6 9 12 15 18 21
Maximum order of Zemike moments

Figure 53: Convergence curves for F ourier-Zernike moment shape descriptors evaluated

for a composite specimen with a machined hole.

In case of virgin composite specimens, it was observed that the quality of the
reconstructions obtained from Fourier-Zemike moments was worse than that for the more
traditional Zemike moments. According to the convergence curves in figures 39 and 53

the root mean square values suggest that the Zemike moments are a better choice even

127

for representation of strain distributions with discontinuities however the visual
comparison in figure 49 suggests otherwise. Thus Fourier-Zemike moments are a good
compromise between unique and accurate shape description, especially for strain

distributions with discontinuities.

gF6000

"35000

 

Length (mm)

 

IO 20 3O 4O 50
Width (mm)

Figure 54: Original maximum principal strain distribution in a composite specimen#l
under tensile load of 8000N obtained using digital image correlation. The composite
specimen was damaged in a drop weight testing machine with an impact-energy of 3 7.3

joules.

128

In this newly introduced shape descriptor the Zemike moments are evaluated for the
logarithm of the absolute values of the discrete Fourier transforms. Thus the Fourier —
Zemike moments evaluated for composite specimens with gradually increasing damage
tend to be very similar with subtle difference, e.g. two specimens subject to impacts at
different energies would tend to have similar strain distributions with different
magnitudes of strain. Such specimens would tend to have similar logarithm magnitude
maps with small differences. Care needs to be taken in the selection of the appropriate
technique for comparing two sets of Fourier—Zemike moments obtained from specimens
with similar damage patterns especially when assessing the level of damage. In the
following chapters the set of six composite specimens with incremental impact damage
are examined and different correlation coefficients and closeness models will be

investigated as means for comparing the corresponding sets of Fourier-Zemike moments.

3 . . 3 -

 

 

Magnitude
N

y—A

 

 

 

 

 

 

0 0in ll. ll

0 100 200 0 100 200
Fourier-Zemike moments Fourier-Zemike moments

Figure 55: Plot of F ourier-Zernike moments (left) and filtered F ourier—Zemike moments
( right) evaluated for the maximum principal strain distribution for composite specimen#l

illustrated in figure 50.

129

3.10 Discussion

As discussed earlier, direct comparisons of high-resolution full-field strain distributions
are computationally expensive making the approach impractical. The above sections also
discuss how shape description techniques using Zemike moment descriptors and Fourier-
Zernike moment descriptors are capable of uniquely representing these strain maps using
feature vectors that provide a considerable reduction in the size of data. The comparisons
of these sets of Zemike and Fourier-Zemike moments which uniquely represent their
corresponding strain maps in-turn facilitate the comparison of the strain maps themselves.
These feature vectors evaluated for the strain distributions using the specially written
ImPaCT software package can set up in the form of a vector with N,, number of elements.
Thus they can be compared using similarity metrics such as the nearest-neighbor distance
as illustrated by Khotanzad and Hong [170]. They measured the distance between two
feature vectors, using the Euclidean distance which assumes that the two feature vectors
each with N,, number of elements represent two points in Np-dimensional space. Then the
feature vectors with the minimum Euclidean distance are labeled as the nearest-neighbors

to one another.

The classical techniques mentioned above for comparison of feature vectors consider
only the magnitude of the complex Zemike moments. Revaud et al. [171 and 172] argue
that although loosing the phase information from the complex Zemike moments is
associated with the advantage of rotational invariance, it also can cause erroneous results
in symmetrical patterns. Thus they have introduced a similarity measure that incorporates

the magnitude as well as phase information of the Zemike moments making the

130

comparison technique comparatively more robust. However, the strain distribution in
structural components is dependent on the loading conditions. Thus each strain map is
associated with its own frame of reference which depends on the direction of loading. As
long as the feature vector representing these strain distributions are invariant to rotation
and translation the orientation of the components becomes irrelevant. Thus it is argued
that for applications in experimental mechanics the inclusion of phase information in the
similarity measure used for the comparisons of shape vectors is not a requirement.

u-strain u-strain

 

8000
7000
6000
5000
10 20 30 40 50 10 20 30 40 50
Width (mm) Wldth (mm)

Figure 56: Reconstructed maximum principal strain distribution using F ourier-Zemike

moments (left) and those using filtered F ourier-Zernike moments (right) from figure 50.

Some other state-of—the-art methods that have been developed in the last decade for
matching shape features are geometric hashing [173] and deformation tolerant
generalized Hough transforms [174 and 175]. However, both the application of these
techniques is quite complicated and they are also slow compared to the techniques
discussed previously. Revaud et al. [172] have compared the performance of all these

techniques and concluded that both geometric hashing and deformation tolerant

131

generalized Hough transforms are 104 times slower than the technique of nearest-
neighbor. This study will concentrate on using the simplest technique for comparison of
shape features which includes the technique of nearest-neighbor using Euclidean
distance. In addition, for comparison two more similarity metrics viz. the Pearson’s
correlation coefficient and the cosine similarity will be used in addition to the Euclidean

distance.

Pearson’s correlation, p(X, Y) between two series or vectors, X & Y with Ng number of

elements each can be written as,

 

No
2061' — 7000' — y)
52 X,Y = i=1
( ) (N9 —1)0'x0'y (60)

where, f & y are the means of vectors X & Y and O' x & 0' y are the standard

deviations. On the other hand cosine similarity between the two vectors X & Y is
evaluated as the cosine of the angle between them and can be evaluated using the

expression for the dot product between the two vectors.

cos 6 = —X-—Y— (61)
llX ||||Yl|
No
X xm

cos (9 2 151—— (62)
"X llllYll

132

Both the Pearson’s correlation coefficient and cosine similarity have values between zero

and unity; unity implies that X and Y are the same while a value of zero implies no

 

 

    
 
   
   
 
   
  

 

 

 

correlation.
30000 . , T
27500 DFZM#1 '
[:JFZM#5 , . .
‘a‘ 25000 » DFZM#11&15 g
E 22500-. AFZM#13 5 3 ., 9. ___.EJ
ii 20000 AFZM#25 ‘ : ;
fig AFZM#41&91 ~ - E '
[3 17500 + OFZM#37&45 ‘ j : .
E 15000 OFZM#79 — e ******* ' ' - ‘ ————— E]
o OFZM#137&153 3 '
E; 12500~ . w , —————————————
O
o i
3 10000
So
2

 

 

 

 

450 600 750 900 1050 1200 1350 1500 1650 1800
Load (lb-force)

Figure 5 7: Plot of F ourier-Zemike moments evaluated for the maximum principal strain
distribution for composite specimen#I subjected to a drop weight impact with an impact-
energy of 3 7.3 joules, as a function of tensile loads varying from 4501b-force to 18001b-

force in 1501b-force increments.

133

 

    
  
  
  
 
   

 

 

 

 

22500 f . j g 1:]
DFZM#1 ; ' 5 .
20000” EJFZM#5 """"" ------ ——————
‘a' EIFZM#11&15 i E E : l:3
«v 17500 —- ~~~~~ ,,. ...... .......... , _____ ._
g AFZM#13 : : 1 1
E 1 : [3 .
g 15000-~ AFZM#25 ..... . ...... . ......
E OFZM#37&45 ‘ ' U
N 12500 + . . .......................................
.5 l '
E 10000~ '
“-4
o I
l” _ :
B 7500 :
a) A
(6 —» we
2 5000 :
2500E~__, t... . . ~__ _,_>
: "' -... ’- : A
0 : l i i 5 E i E

 

 

 

450 600 750 900 1050 1200 1350 1500 1650 1800
Load (lb-force)

Figure 58: Plot of F ourier-Zernike moments evaluated for the maximum principal strain
distribution for composite specimen#2 subjected to a drop weight impact with impact-
energy of 3 7.3 joules, as a function of tensile loads varying from 4501b-force to 18001b-

force with 1501b-force increments.

The third measure of similarity between two feature vectors is the Euclidean distance.
This method assumes that each feature vector with Ng elements actually represents the
coordinates of a point in an Ng -dimensional space and evaluates the distance between

two such points, X & Y using the least-distance formula given by,

134

 

 

 

 
  
  
  
  
   

E = 209-002 (63)
i=1
27500 . . . r . . :
DFZM“ : : : E 3
25000” “‘*t ‘‘‘‘‘ 7: ------ Ir —————— :--47,_|T _____
22500“ DFZM#11&15 **** g ------------------
: r : : . 1:]
20000 —~ AFZM#13 —————— ...... U _____
17500—- AFZW” ..... ..... ______
OFZM#37&45 5 3 ' '

 

 

 

15000 ” . ---|* ------ : ----- — __________% ______
12500

10000 ‘*

Magnitude of Fourier—Zemike moment

 

 

 

 

 

450 600 750 900 1050 1200 1350 1500 1650 1800
Load (lb-force)

Figure 59: Plot of F ourier-Zernike moments evaluated for the maximum principal strain
distribution for composite specimen#3 subjected to a drop weight impact with impact—
energy of 3 7.3 joules, as a function of tensile loads varying from 450 lb-force to 1800 lb-

force with 150 lb-force increments.

135

 

27500

 

   
 
     

 

 

 

ClFZM#1 i 5 5 i :
25000” DFZM#5 """" g ------ 1 ------ 5 ------ t3
22500” ElFZM#11&15 ______ _____ _____
20000 a AFZM#25 ----- r ****** ------ ----- [j ------
OFZ #7 4 ' 1 '
17500-. ,M3.& 5
15000~*

12500 7* '

10000 7

Magnitude of Fourier-Zemike moment

 

 

 

 

 

450 600 750 900 1050 1200 1350 1500 1650 1800
Load (lb-force)

Figure 60: Plot of F ourier-Zernike moments evaluated for the maximum principal strain
distribution for composite specimen#4 subjected to a drop weight impact with impact-
energy of 3 7.3 joules, as a function of tensile loads varying from 4501b-force to 18001b-

force with 1501b-force increments.

In using these similarity measures for qualitative and quantitative comparisons of the
feature-vectors, the feature vector corresponding to the maximum principal strain
distribution in the virgin composite specimen under tensile loads will be considered as a

master feature vector. The feature vectors of the damaged composite specimen were

1.36

compared with each other in terms of their closeness to the master feature vector of the
virgin composite specimen. This closeness was evaluated using the closeness coefficients

discussed above and their effectiveness will be explored.

In order to study the suitability of the Zemike moment and Fourier-Zemike moment
descriptors for structural damage assessment their sensitivity to the level of applied load
and degree of impact damage was investigated. Maximum principal strain maps were
obtained for all seven composite specimen listed in table 4 using digital image correlation
on both sides of the specimens at loads of 450 lb—force to 1800 lb-force in 150 lb-force
increments. The opposite face to the impacted face of the specimens was chosen for
analysis since this face showed the maximum delamination and higher strains due to the
strain concentration caused by this delamination. Both Zemike moments as well as
Fourier-Zemike moments were evaluated for the four incrementally damaged composite
specimens and the virgin composite specimen while only Fourier-Zemike moments were
evaluated for the remaining two composite specimens; one with an impact hole and the

other with a machined hole.

The Zemike moments and Fourier-Zemike moments were evaluated for the strain

distributions in all seven composite specimens with the maximum order of Zemike-

moments Nmax = 20 which corresponds to 230 moments. However, as explained in the

previous chapters, only a few of these moments are substantial while the others are either
zero or close to zero and do not contribute to the three-dimensional shape of the

corresponding strain distribution. Thus a filter was employed where only those moments

137

greater than 10% of the maximum moment corresponding to the highest load level of
1800 lb-force or 8000N were considered while the rest were eliminated. This reduces the
size of the feature vector even further making it more efficient to store and analyze. The

left plot in figure 55 shows the Fourier-Zemike moments evaluated for the strain map

illustrated in figure 54, with a maximum order of Zemike moments, Nmax = 20 while the

plot on the right shows the same set of Fourier-Zemike moments after applying the 10%-
filter. This reduces the number of Fourier—Zemike moments from 230 to only 13 relevant
Fourier-Zemike moments. To illustrate that these 13 filtered Fourier Zemike moments
are capable of representing the strain distribution in figure 54 the strain maps were
reconstructed with both the sets of Fourier-Zemike moments shown in figure 55. The
reconstructed strain maps are shown in figure 56. The plot on the left is the reconstructed
strain map using all 230 Fourier-Zemike moments while that on the right is the
corresponding reconstructed strain map using just the 13 filtered Fourier-Zemike
moments. It can be observed that both the strain distributions in figure 56 are identical.
The 10% filter was applied to all the feature vectors. For the qualitative and quantitative
comparison of the feature vectors using the similarity coefficients, the strain distributions
were normalized using the load providing 10 feature vectors for each specimen enough to

perform a valid statistical argument.

These filtered Fourier-Zemike moments evaluated for the maximum principal strain
distribution in the four incrementally damaged specimens subject to tension have been
plotted as a function of the applied load in figures 57 to 60 respectively and for the virgin

specimen in figure 61. In all these figures the magnitudes of all the relevant Fourier-

138

Zemike moments tend to increase with increasing load, suggesting that this shape
description technique can not only be used to represent the shape of the strain distribution

but also the level of strains experienced by the specimens due to the applied loads.

 

 

     

 

 

 

60000 ‘ ' ‘. : I I . ;
E3F234#1 E 5 E 5 : 3
, mm : 5 2 2 2 5
50000" AFZM#13 """" 1 """" T """" ““““““““ E3
AFZM#25

40000— —————— J: ,
30000+

20000 7

Magnitude of Fourier-Zemike moment

 

 

 

450 600 750 900 1050 1200 1350 1500 1650 1800
Load (lb-force)

Figure 6]: Plot of F ourier-Zernike moments evaluated for a maximum principal strain

distribution for a virgin composite specimen, as a function of tensile loads varying from

4501b-force to 18001b-force with 1501b-force increments.

Another observation that can be inferred from these plots is that for all the specimens the

same Fourier-Zemike moments are relevant. However, as the degree of damage reduces

139

the higher Fourier-Zemike moments stop contributing towards the shape of the strain
distributions, e.g. some of the relevant Fourier-Zemike moments for the strain
distribution in specimen#l including 41, 91, 79, 137 and 153 do not pass the 10%
criterion in any of the other specimens with lower levels of impact damage. The first part
of this observation can be attributed to the same type of impact damage present in all of
incrementally damaged composite specimens giving rise to the same type of strain
distribution. At the same time it is important to remember that for the Fourier-Zemike
moments, the Zemike moments are evaluated for the logarithm of the absolute value of
the two-dimensional discrete Fourier transform of the strain distribution, which tends to
have the same generic shape with only very subtle differences. While the second part of
the observation is explained by studying the level of damage in each specimen and the
corresponding level of strain concentration induced in the specimen. Specimen#1 was
impacted with the maximum impact energy making it the specimen with the highest
degree of damage and resulting strain concentration. This high level of strain
concentration makes the strain distribution surrounding the damage intense compared to
the rest of the specimens. Thus higher order Zemike polynomial terms are required to
capture the strain distribution in specimen#l. As the degree of impact damage decreases
in subsequent composite specimens the effect of the strain concentration due to the
damage reduces making it easier to represent the resulting strain distribution with fewer
and fewer high order Zemike polynomial terms. The strain distribution of the Virgin
composite specimen without any damage and strain concentrations is represented by only
four Fourier-Zemike moments viz, 1, 5 13 and 25. The above observations suggest that

the Fourier-Zemike moments are capable of representing the strain concentrations caused

140

due to the impact damage which is a direct indication of the level of damage in the

structural component and can be used as a measure of post damage residual life [43].

 

 

 

 

 

 

 

 

 

7000 1 I . r
. E S , o 450 lb force
6000 aw ________ ’ ° 600lbforce
a E E E 5 o 7501b force
3 . 1 1 1
1:3 ‘ ° 9001b force
5000* ““““““ """""" “““““ A10501bforce '
t on ; 2 5 i A 12001bforce
A I A l3501bforce
o 4000 “A w T .
3 AA DD 7 i D ISOOIbel'CC
S0 A AA . , D 16501b force
2 3000’OM* E118001bforce
3 AA I i l
2000 .. 3‘2 . 5 i
:00 A A C; D I I 1
’m B , 212 ii a : t
1000 “its? 383 a B s A D D ‘ ”D
9% .t::.: age gas 9 0 ago; i 2‘ i ;
a, n n 9 ° 013°? 1
ea % §§§ ° ° 333: 53% i :
0 + ; i I :
0 40 80 120 160 200 240
Fourier-Zemike Moments

Figure 62: Plot of F ourier-Zernike moments evaluated for the maximum principal strain
distribution for a composite specimen subject to an impact load with impact-energy
enough for the tup to penetrate through the specimen forming an impacted hole, as a

function of tensile loads varying from 4501b-force to 1800 lb-force with 150 lb-force

increments.

141

 

6000

 

o ' 1 °4501bforce
a 3 _ ; °6001bforce
“ " °7501bforce
°9001bforce
4000__ . 10501bforce,

CID

D

 

 

 

 

 

 

 

 

DD A 12001b force
.8 A 1 1 A 1350 lb force
8 AA _
'E 3000 “56 ______ 1 1 1 - D 15001b force
go AA : ; : :
2 3 M i 0 1650lbforce
30 (53> D: n i I D 1800 lb fOI'CC
2000 :3" “‘0’ “““““““ 1 1
oo ODD DD , i i
“36 ago AA Ci 0 J :
OOAOCID AA g g i .
ma) AiA a; Cl D g 2 Cl Cl1 :
1% pm: A i A A g a? i u D
> g% g 3883 a a 9 “b: : B a
$3 W§§§§H3§
as 33 . . ; g g = 8
0 1 1 1 l l
0 40 80 120 160 200 240
F ourier-Zemike Moments

Figure 63: Plot of F ourier-Zernike moments evaluated for the maximum principal strain
distribution for a composite specimen with a machined hole, as a fimction of tensile loads

varying from 4501b-force to 18001b-force with 150 lb-force increments.

Figures 62 and 63 show plots of the Fourier-Zemike moments for the strain distribution
in the specimen with an impacted hole and one with a machined hole respectively under

tensile loads plotted as a function of applied load. It was observed that the strain

142

distributions in these specimens were complex, requiring more Fourier-Zemike moments
for their representation compared to the incrementally damaged specimens. This
complexity arises from the presence of a physical hole in the specimen, which acts as a
sharp discontinuity in the strain map making the higher frequency terms relevant along
with the low frequency ones in the corresponding Fourier representation. To
accommodate all of the qualifying Zemike moments using the 10% filter and still study
their trend with respect to changing loads, they have been plotted differently than the
previous examples. It was observed that the magnitude of the Fourier-Zemike moments

increased with increasing loads similar to the case of incrementally damaged composites.

 

 

 

 

 

 

 

 

 

 

 

I Virgin specimen I Specimen#4 _ .
#
I SPecimen#3 I Specimen#2 I Spec1men 4 . Specunen#3
I Specimen#l I Impact hole _ .
I Machined hole I Spearmint2 I Specnmenttl
1-2 0.97
G G
'2 ’2 0 96
° .2 . 1
a: a:
Q) 4)
o o
O U
.5 g 0.95 ~
E a
0 0
g E 0.94 —
U U
.2 ’2
o o
E g 0.93 ~
31’ a
0.92 ~
Specimens Specimens

Figure 64: Plot of Pearson ’s correlation coefficient to measure the similarity between the
F ourier-Zernike moment shape descriptors of the principal strain distribution in seven
composite specimens under tensile loading and those for a virgin composite specimen
with no damage (left). Chart of right plots the same figure on left excluding the virgin

composite specimen and the composite specimens with an impact and machined holes.

143

As mentioned earlier, the Fourier—Zemike moments evaluated for the normalized strain
distribution of all seven specimens were compared using similarity coefficients. The
Pearson’s correlation coefficient, cosine similarity and the Euclidean distance was
calculated in order to compare the Fourier-Zemike moments of each damaged specimen
and the corresponding Fourier-Zemike moments for the virgin composite specimen for
ten load values per specimen. These similarity coefficients are plotted in figure 64, figure
65 and figure 66 as bar charts for comparison. In all these figures the plot on the left
includes all seven composite specimens while the ones on the right take a closer look at
the variation in the similarity coefficients for the four incrementally damaged composites.
Also, all three similarity coefficients were plotted on one graph for the four incrementally
damaged composites for comparison as shown in figure 67. Error bars plotted for each
specimen were based on a 95% confidence limit evaluated based on the ten datasets per

specimen.

The same trend can be observed for the plots of the Pearson’s correlation coefficient and
cosine similarity in figure 64 and figure 66. Both these similarity coefficients are unity
for the virgin specimen as is expected for the correlation coefficient of a vector with itself
is always unity. As the degree of damage increases, the strain distribution deviates more
and more from that of the virgin specimen as is depicted by the gradual drop off in the
values of similarity coefficient from unity. On the other hand, the Euclidean distance
between the feature vectors of the virgin specimen with itself is zero and this distance
gradually increases with increase in the level of damage. Thus it can be observed from

figure 65 that the Euclidean distance similarity coefficient shows a reverse trend to the

144

Pearson’s correlation coefficient plot in figure 64 and cosine similarity plot in figure 66.
However, it is still capable of distinguishing between the feature vectors representing

maximum principal strain maps in composite specimen with different levels of damage.

 

 

 

 

 

 

 

 

 

 

   

I Virgin specimen I Specimen#4 _ .
#
I SPecimen#3 I Specimen#2 I Spec1men 4 . Specunen#3
I Specimen#l I Impact hole _ .
I Machined hole I Spec1men#2 I 3138011116114“
6 .
3 .
Q.) d)
8 g 2.3 ~
3 a
'"5 ”—6
5 g 2.6
Q) Q)
’3 .12
m m 2.4 .
2.2 ~
Specimens Specimens

Figure 65: Euclidean distance measured between the F ourier—Zernike moment shape
descriptors of the principal strain distribution of five incrementally damaged composite
specimens under tensile loading and those for a virgin composite specimen with no
damage (left). Chart of right plots the same figure on left excluding the virgin composite

specimen and the composite specimens with an impact and machined holes.

Figure 67 provides a comparison between the three similarity coefficients when applied
to the incrementally damaged composite specimen. It can be observed that the Euclidean
distance varies non-linearly with respect to level of damage, while the Pearson’s

correlation coefficient and the cosine similarity both follow a fairly linear trend with a

145

residual least square parameter of 20. 94. Thus the slope of these trend lines represent the
sensitivity of both these similarity coefficients to change in the level of damage which
was observed to be -0.0032 joule" for the Pearson’s correlation coefficient and -0.003
joule'l for cosine similarity. However, the sensitivity of Euclidean distance to change in
level of damage varies due to its non-linear behavior. The Euclidean distance tends to be
more sensitive when the variation between the two feature vectors is less corresponding
to lower levels of impact in this case, while its sensitivity decreases with an increase in

the level of damage.

 

 

 

 

I Virgin specimen I Specimen#4 I Specimen#4 I Specimen#3
I Specimen#3 I Specimen#2

I Specimen#l I Impact hole Specimen#2 I Specimen# l
I Machined hole 0.96 4

 

1.2

 

0.95

.O
00

Cosine similarity
o
O

Cosine similarity
C
so
A
l

 

 

 

 

 

0.4
0.93 -
0.2
0 - 0.92 -
Specimens Specimens

Figure 66: Cosine similarity measured between the Fourier-Zemike moment shape
descriptors of the principal strain distribution in five incrementally damaged composite
specimens under tensile loading and those for a virgin composite specimen with no
damage (left). Chart of right plots the same figure on left excluding the virgin composite

specimen and the composite specimens with an impact and machined holes.

146

 

 

 

 

 

 

  

 

 

 

 

 

 

 

A Pearson's correlation coefficient
I Cosine similarity
0 Euclidean Distance (Secondary Axis)
0.97 ; 1 ; .- 3
E 0.96 7 _ : l 1: i 7* 2.8 8
.12 I 1 1 1=
.2 i ‘ i .. i S
8:: .- 1 ' i .2
18’ : - a a - 9:
390.955 i- 7726 8
E a : a g
E 5 E 5 [1:3
"’ 0.94 ~ 1 § : :1 -— 2.4
0.93 i i l l ' 2.2
29 31 33 35 37 39

Impact Energy (joules)

Figure 67: Similarity measures between the F ourier-Zemike moments evaluated for the
maximum principal strain distribution in the incrementally damaged composite
specimens and those evaluated for the maximum principal strain distribution in the virgin
composite specimen in terms of the Pearson’s correlation coefficient, cosine similarity

and Euclidean distance plotted as a function of impact-energy.

147

 

 

 

 

  

 

    

 

 
 

 

 

 

 

 

 

A Pearson's correlation coefficient
I Cosine similarity
O Euclidean Distance (Secondary Axis)
0.97 7 ; ; .- 3
.5 0.96 7 : 77 2.8 8
.12 . . c
.2 I 1 3
a: ' i .52
s 5 53
3‘ 0.95 7 77 2 6 3
E a §
.§ i 5 113
m 0.94 .. e ~ 1 . f —— 24
0.93 l l l l ' 2.2
29 31 33 35 37 39

Impact Energy (joules)

Figure 67 : Similarity measures between the F ourier-Zernike moments evaluated for the
maximum principal strain distribution in the incrementally damaged composite
specimens and those evaluated for the maximum principal strain distribution in the virgin
composite specimen in terms of the Pearson ’s correlation coefficient, cosine similarity

and Euclidean distance plotted as a function of impact-energy.

147

 

El ZM#l Cl ZM#5

 

 

 

 

Cl ZM#l2 & 14 A ZM#13
A ZM#24 & 26 A ZM#25
O ZM#40, 42, 60& 62 O ZM#41&61
O ZM#84 & 86 O ZM#85
O ZM#112, 114 & 145 O ZM#113
— Linear (ZM#12 & l4) — Linear (ZM#5)
— Linear (ZM#I)
3500 . . . I I I :
3000 7 I |
E 2500 -
E
C
E
E3 2000 ~
E
Q
N
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,, 1500 .
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fl _., , ,
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us-
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3

450 600 750 900 1050 1200 1350 1500 1650 1800
Load (lb-force)

Figure 68: Plot of Zemike moments evaluated for the maximum principal strain
distribution for composite specimen#l subjected to a drop weight impact with impact-
energy of 3 7.3 joules, as a function of tensile loads varying from 4501b-force to 18001b-

force with 150 lb-force increments.

148

 

 

 

 

 

 

1:1 ZM#5 1:1 ZM#12 & 14
A ZM#13 A ZM#24 & 26
A ZM#25 ' o ZM#40, 42, 60 & 62
o ZM#41& 61 o ZM#84& 86
<> ZM#85 <> ZM#112, 114 & 145
0 ZM#113 --Linear (ZM#12 & 14)
—Linear(ZM#5)
900 1 ,
: : ' : é?
800—~ E . : 3 3 i 9
' ' 1 E ' 8 E A
700 A 1 t 1 '1— _._._ n _____ a

‘5‘, . . . E E 5 i 1.;

g 60,. e 2 s a = a i a  4 ~11

0 E 1 3 i 9 E i @‘ g;

“a 5009 g ; g e: —~ .4: A

a a a o ' 8 g

”5 4007'“ 5 CD *°‘ A” g *7? “““ A

18 : 0 II @ : 6 .

a ' t! 3 ' ' '3 1:1

- 1 : o -. - -1 _. U ......

g 300 E 8 @ 5 Q :

2 Q t a " ' 1
200‘“ a “ a
100,3:1‘ 5 i

0 I I 4 : 1 4 1 : i

 

450 600 750 900 1050 1200 1350 1500 1650 1800
Load (lb-force)

Figure 69: Plot of Zemike moments (excluding the first Zemike moment) evaluated for
the maximum principal strain distribution for composite specimen#I subjected to a drop
weight impact with impact-energy of 3 7.3 joules, as a function of tensile loads varying

from 450 lb-force to 1800 lb-force with 150 lb-force increments.

149

 

 

     
   

 

 

 

 

 

 

 

3000 : I I I .
[12mA#I : : I I I VJ
C] # i i i i
2500 e 234 5 ————:~ I ----- ; ————— ————
tjznu#118:15 I :
E i ‘ * '
E I I I ‘
o 2000—~————I-—————*——_———-———————_« ———————-————I -----
E . . —- ,
a) I I I
f! I
E ; I
§ 1500 I j I ”””””” T """
Q— l | '
o l l l l
a) I I I I
“O I l I I
a l I I I
“a lOOO-x---~% --------- %-~**‘-—--I ----- . ----- I"--I -----
01) I I I | l I
w | l l l | l I
2 r : : I : : : :
“ : : : : I : : I
500 —————— I —————————— ; ————— I ————————— . ————————— fl
- . m S $ 3: 9.3 § §
0 'T' I I I I I I I

 

 

450 600 750 900 1050 1200 1350 1500 1650 1800
Load (lb-force)

Figure 70: Plot of Zernike moments evaluated for the maximum principal strain
distribution for composite specimen#2 subjected to a drop weight impact with impact-
energy of 3 7.3 joules, as a function of tensile loads varying from 450 lb—force to I8001b-

force with 1501b-f0rce increments.

Zemike moments were evaluated for the strain distributions in the incrementally
damaged composite specimens subject to tensile loads, since these specimens are intact
with no sharp geometric discontinuities. The only disadvantage of using Zemike
moments is that they cannot be used as a basis of comparison between the incrementally
damaged composite specimen and the specimens with holes. Nonetheless their
effectiveness in differentiating between the strain distributions in composite specimens

with varying levels of damage can be assessed. A similar analysis was performed for

150

Zemike moments as discussed above for Fourier-Zemike moments; the only difference
being that the Zemike moments were evaluated for the strain distributions directly while
in the case of Fourier-Zemike moments the Zemike moments were evaluated for the
absolute values of the two-dimensional discrete Fourier transform of the strain
distributions. Zemike moments were evaluated for each specimen at ten different load
steps with maximum order of, Nmax = 20 resulting into 230 Zemike moments each. The
10% filter was applied to each set of Zemike moments to short-list only those moments
which are responsible for the representation of the corresponding strain distributions.
Zemike moment descriptors for the un-normalized strain distributions were used to study
the effect of applied loads, while those obtained from strain maps which were normalized
with the applied loads were used to perform a statistical analysis to study the effect of

level of damage using similarity coefficients.

Similar to plots of Fourier-Zemike moments in figures 57 to 60, the Zemike moments
evaluated for the maximum principal strain distribution in the four incrementally
damaged composite specimens are plotted in figures 68 to 72. Figure 68 plots the relevant
Zemike moments for the strain distribution in specimen#l as a function of increasing
load. The specimen with the highest degree of impact damage has 21 relevant Zemike
moments where the first moment, ZM#I is proportional to the level of applied load. Thus
figure 69 illustrates the same plot in figure 68 but without the first Zemike moment to
take a closer look at the variation of the higher order Zemike moments with respect to
applied loads. It was observed from both these figures that the Zemike moments

increased linearly with increase in load, unlike the case of Fourier-Zemike moments

151

where this relationship was non-linear. A similar linear trend was observed for the
remaining three composite specimens as illustrated in figures 70 to 72. Similar to the
Fourier-Zemike analysis, it was observed that with reduced intensity of damage the
number of relevant Zemike moments representing the strain distribution are reduced to
only a few lower order moments until one reaches the virgin undamaged specimen where
only the first Zemike moment is relevant, which is proportional to the applied load or the

average uniform strain in the specimen as was demonstrated in chapter 3.7.4.

 

 

 

 

 

   

 

 

 

 

3000 T ' I . *T I I I J
r12m4#1 I : : I I I
2500 1:1 ZM#2 & 3 I : I I I .
CI ZM#5 I I I I I I
E I I I I I + I I
g : I I I I . I :
o 2000 ~---—I ----- : ~~~~~~ I---—I ————— ' ——-—I---—I——-—I -----
E : : I : . I I I
a" I I I I I | I
i I I | I I
E I : : : I : I I
IS 1500“ """" 7 ***** : """ “ ’I‘T""’T ccccc : “““ T”"‘"T """"
‘4— I I I I I I I
o I l | I | l I I
0) '1— I I I I I I
"O I I I I I I I
a I I I I I I I I
'a 1000" ----- I ———I ----- I ----- I ----- : ————— I ————— I-——-I -----
DO I I I I I I I I
a I I | I I I I I
55 : I : I I I I I :
500 ------ I ————— : ————— : ————— I—--—I ————— I ————— I————I .....
I I ', I* . 3
fl I I TI I I J
0 I“ “r "IJ :1. T I I I I

 

 

 

450 600 750 900 1050 1200 1350 1500 1650 I800
Load (lb-force)

Figure 71 .° Plot of Zemike moments evaluated for the maximum principal strain
distribution for composite specimen#3 subjected to a drop weight impact with impact-
energy of 3 7.3 joules, as a function of tensile loads varying from 450 lb-force to 18001b-

force with I501b-force increments.

152

 

3000

 

      
   

III ZM#I

 

 

2500 ‘“ DZM#2&3 "

 

_ ____._r_.__——.—__I

_L_ ._ ——L_~—
l
l
I
|
._.______L________
I
I
I
|
|

  
 
   

 

 

J.‘
I
I
E I I I
Q) I I
E I I '
E 2000—I~~~~I ~~~~~~~~~~~~ ~——————————— ——————I ————— I —————
I - I I
I I I
G)
x ; I I
E I I I
o _+____;____c_c_--t__ _ _2__ ----_---__L____' ______
N 1500 I I j
“5 I I |
I-Qo) I I I I
a I ' I I I
'a 1000— ————— I : . I : I ----- I -----
go I I I I I I I
2 . I I I : I I I
r" I I I I I I I I
L I I I I I I I I
I I I I I I I I
500* ----- 'I ————— I ----- r—’—*1 ————— I """ I “““ T"—"1 ““““
I : I : ' : I - I
: I I I I I
I" l | I I
L I I I I I I I
O i 41 ? I I i I 'T

 

450 600 750 900 1050 1200 1350 1500 1650 1800
Load (lb-force)

Figure 72: Plot of Zemike moments evaluated for the maximum principal strain
distribution for composite specimen#4 subjected to a drop weight impact with impact-
energy of 3 7.3 joules, as a function of tensile loads varying from 4501b-force to I8001b-

force with 150 lb-force increments.

Similarity coefficients in terms of the Pearson’s correlation coefficient, cosine similarity
and the Euclidean distance were evaluated to compare the Zemike moment vectors for all
the incrementally damaged composite specimens and for the virgin composite specimen
and are plotted as bar diagrams in figures 73 to 75. Also, all three similarity coefficients
were plotted on one graph for the four incrementally damaged composites for comparison
as shown in figure 76. Error bars plotted for each specimen were based on a 95 percent

confidence limit evaluated based on the ten datasets per specimen.

153

 

 

I virgin I Specimen#4 . .
I Spec1men#4 I Spec1men#3
I Specimen#3 I Specimen#2

 

 

 

 

 

 

 

 

 

 

 

I Specimen#l El Specimen#2

1.05 0.98
E 1 . S
.2 '8
“~= E
8 a)
o 0-95 ’ 8 0.97 I
‘3 I:
.9 .2
E 0.9 «3
f: 0.85 f; 0.96 I
c: t:
8 a
‘5 0.8 Is
fl II:

0.75 ‘ 0.95

Specimens Specimens

Figure 73: Pearson’s correlation coefi‘icient to measure the similarity between the
Zemike moment shape descriptors of the principal strain distribution in five
incrementally damaged composite specimens under tensile loading and those for a virgin
composite specimen with no damage (left). The chart on the right shows the same figure

on left excluding the virgin composite specimen and specimen#I.

Figures 73, 74 and 75 show similar trends in the variation of the similarity factors with
respect to varying level of damage as those for Fourier-Zemike moments discussed
previously. The Pearson’s correlation coefficient and the cosine similarity have a value of
unity for the virgin specimen as expected and then the values decrease gradually with the
increasing level of impact damage, suggesting that the strain distribution deviates further

away from the corresponding strain distribution in the virgin composite specimen with

154

increasing level of damage. The Euclidean distance measured between the feature vectors
is zero for the virgin composite specimen and gradually increases with the increasing
level of damage as would be expected. Thus Zemike moments, like Fourier-Zemike
moments, also demonstrate the ability of representing the type and level of damage as

well as identifying the level of applied loads.

 

 

 

 

 

 

 

 

 

 

 

 

   

I virgin I Specimen#4 , ,
I Spec1men#4 I Spec1men#3
I Specimen#3 I Specimen#2
I Specimen#l Specimen#2
0.45 0.27
0.4
0.35 0.25 .
8 0.3 “ 8
C'- I:
g 0.25 . g 0.23 a
‘6 '6'
g 0.2 5
rag). 0.15 " .40) 0.21 _.
3 0.1 ~ '3
5 :3
0.05 4 0.19 r
0 4
-005 4 0.17 r
Specimens Specimens

Figure 74: Euclidean distance measured between the Zemike moment shape descriptors
of the principal strain distribution in four incrementally damaged composite specimens
under tensile loading and those for a virgin composite specimen with no damage (left).
The chart on the right shows the same figure on the left excluding the virgin composite

specimen and specimen#I.

For Zemike moments all three similarity coefficients tend to vary non-linearly as a

function of impact energy or degree of damage as shown in figure 76. Previously, for

155

Fourier-Zemike moments the Pearson’s correlation coefficient and the cosine similarity
showed a linear relationship with the degree of damage. From figure 76 all three
similarity coefficients seem to have a low sensitivity at lower levels of impact damage
which increases with the level of damage. This contradicts the behavior of the Euclidean
distance similarity coefficient observed previously for Fourier—Zemike moments where
the Euclidean distance was less sensitive at higher degree of damage. This discrepancy
can be attributed to the principal difference in the two shape descriptors. To reiterate for
Zemike moment descriptors, the Zemike moments are evaluated directly for the strain
distribution while for Fourier-Zemike moments, Zemike moments are evaluated for the
absolute value of the two-dimensional discrete Fourier transform of the strain
distribution. Thus in the frequency or Fourier domain the strain distribution for higher
degrees of impact damage tend to be similar, while on the contrary in the real
displacement domain the strain distributions for lower level of impact damage tend to be
similar. Since all four composite specimen have the same damage pattern caused by an
impacter in a drop weight testing machine only with different levels of impact energy the
frequency information of the strain distribution, which is captured by the discrete Fourier
transform, tends to be similar and the effect of the magnitude of strains is not reflected in
the frequency domain. However, for simple Zemike moments the strain distributions in
the specimen domain are highly influenced by the level of stress concentration, which is

more intense at higher levels of impact damage.

It was observed that none of the three similarity coefficients was able to differentiate

between specimen#2 and specimen#3. The lowest sensitivity for the Pearson’s correlation

156

coefficient, cosine similarity and Euclidean distance for the Zemike moment descriptors
was measured between specimen#2 and specimen#3 with values of -0.0005 joule', 0.0005
joule'l and -0.0002 joule" respectively compared to -0.003 joule’l, -0.003 joule'1 and 0.03
joule" respectively for Fourier-Zemike moments. Thus the Fourier-Zemike moments are

more effective in damage assessment compared to simple Zemike moments.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

I virgin I Specimen#4 . .
I Spec1men#4 I Spec1men#3
I Specimen#3 I Specimen#2
I Specimen#l I Specimen#2 I Specimen#l
1.15 1
0.95 - 0.95 a
E‘ 0.75 a g; 0.9 s
.2 ‘a‘
g o 55 E 0.85 .
.i‘e’ :2
e 0.35 — '5 -
8 8 0.8
0.15 ~ 0.75 ~
-0.05 e 0.7
Specimens Specimens

Figure 75: Cosine similarity measured between the F ourier-Zemike moment shape
descriptors of the principal strain distribution in four incrementally damaged composite
specimens under tensile loading and those for a virgin composite specimen with no
damage (left). The chart of right shows the same figure on left excluding the virgin

composite specimen and specimen#I.

157

 

A Pearson's correlation coefficient
I Cosine similarity

O Euclidean Distance (Secondary axis)
; . , . 0.45

 

 

  

   

0.95 i

l

i + —— 0.4

 

0.9 -'

l
l
c
U.)
Euclidean Distance

l
.O
N
U:

Similarity coefficient
0
00
LII

 

 

l

0.75

 

 

IL
.o
N

 

 

i
l
_q__-_-_—____——_______——
I

I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
l

-—I——--—-—-———— -—-_-

I
I
I
I
I
I
I
l

 

0.7 0.15

29 3 1 33 35 37 39
Impact Energy (joules)

Figure 76: Similarity measures between the Zemike moments evaluated for the maximum
principal strain distribution in the incrementally damaged composite specimens and
those evaluated for the maximum principal strain distribution in the virgin composite

specimen in terms of the Pearson ’s correlation coefiicient, cosine similarity and

Euclidean distance plotted as a function of impact-energy.

158

2.: P ~ 2" 7'3" T‘ 7:12,: *._r-.-*'::‘»";23"
I " .

Impact Energy: 37. 5 J

   

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Figure 77: C -scan images obtained from ultrasonic evaluation of non-impact surface
which is also the bottom surface in the ultrasonic test configuration for composite
specimen#I (top left), specimen#2( top right), specimen#3( bottom felt) and

specimen#4( bottom right) with varying impact energies.
Figure 77, figure 78 and figure 79 show the C-scan images and time of flight images for

the incrementally damaged specimens obtained using an ultrasonic setup ( UltrapacII®).

Figure 77 shows C-scan images fromthe opposite face from the impacted face which was

159

also on the bottom in the ultrasonic setup, thus showing delamination between all the
layers overlapped one on top of the other. Figure 78 shows C-scan images of the impact

surface which was the top surface in the ultrasonic setup showing delamination only

between the surface laminate and the laminate immediately below it.

2-1

imam}: ' ' as ..52
Impact Energy: 37. 5 J " '5- Impact Energy: 35 J

1.6.; ,0f1;' _ . ' I ‘ . , l . ’ .- _.
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Figure 78: C -scan images obtained from ultrasonic evaluation of impact surface which is
also the top surface in the ultrasonic test configuration for composite specimen#1(top
left), specimen#2( top right), specimen#3( bottom felt) and specimen#4( bottom right) with

varying impact energies.

160

-‘

1 . Impact Energy = 35 J

a
8 .

 

Figure 79: Time of flight images obtained from ultrasonic evaluation of composite
specimen#l (top left), Specimen#2(top right), specimen#3( bottom felt) and

specimen#4( bottom right) with varying impact energies.

161

The time of flight images shown in figure 79 were used to estimate the amount of
delamination between each laminate of the composite. The lighter the shade of blue the
further away from the ultrasonic detector is the delaminated surface which reflects back
the sound wave. Damage evaluation from the shape description technique was compared
to that from the non-destructive ultrasonic technique. Since the strain maps used for the
shape description technique were recorded using digital image correlation from opposite
face to the impact surface the size of delamination between this surface laminate and the
laminate adjacent to it was measured from the time of flight images in figure 79. The
length and width of delamination were measured as shown in figure 79. The area of
delamination was also measured using a MATLAB® code specially developed for this
purpose which masks the damaged region to form a binary image of the specimen such
that the region of the specimen without delamination is black and that with delamination
is white. The area of the white region in the specimen can be easily evaluated. Since the
size of delamination grows with increasing impact energy it has a similar trend as the
Euclidean distance similarity coefficient. The length, width and area of the delamination
obtained from the ultrasonic time of flight images and the Euclidean distance similarity
coefficients for both the Zemike moment descriptors and Fourier-Zemike moment
descriptors were plotted on the same graph as a function of increasing impact energy as
shown in figure 80. These damaged parameters were normalized using their
corresponding mean and standard deviations to facilitate the comparison with the same

scale as shown in figure 80.

162

 

A Area of Delamination: Ultrasonic

A Length of Delamination: Ultrasonic

A Width of Delamination: Ultrasonic

O Euclidean Distance: Zemike Moments

0 Euclidean Distance: Fourier-Zemike Moments

0 Inverse of Pearson correlation coefficient: Zemike moments

<> Inverse of Pearson correlation coefficient: Fourier-Zemike moments
D Inverse of cosine similarity: Zemike Moments

Ll Inverse of cosine similarity: Fourier-Zemike Moments

X Stress concentration factor

2.5

 

 

 

 

1.5

Damage - Normalized Quantity

0.5

 

 

 

 

 

30 32 34 36 38
Energy of Impact (J)

Figure 80: A comparison between damage assessment in incrementally damaged
composites using ultrasonic non-destructive evaluation, stress concentration factors and

shape description similarity measures as a function of impact energy.

163

Figure 80 shows that, the size of the delamination between the opposite face to the
impact face laminate and the laminate beneath it increases with increase in impact energy
which is in accordance to previous work on ultrasonic evaluation of impact damage in
fiber-reinforced polymer composites [34]. The Euclidean similarity coefficient between
the Fourier-Zemike moments evaluated for the strain distribution in the damaged
composite specimens and the virgin composite specimen follow a similar trend to the
ultrasonic of damage parameters while the Pearson correlation coefficient and cosine
similarity decrease with increasing impact damage. Thus the inverse of the Pearson
correlation coefficient and the cosine similarity have been plotted in figure 80 for
comparison purposes. However, other than the qualitative similarity parameter for the
Fourier-Zemike moments the moments themselves represent the strain field in the
damaged composite which is a measure of the load carrying capacity of the damaged
composite. Thus the Fourier-Zemike moments are capable of a qualitative damage
analysis in composites similar to non-destructive ultrasonic evaluation as well as
quantitative damage evaluation in terms of the load carrying capacity of the damaged
composite. Stress concentration factors have been used and considered as one of the
easiest ways of structural damage characterization [43]. Thus the stress concentration
factors were evaluated from the maximum principal strain distributions for the
incrementally damaged composite specimens shown in figure 45 by normalizing the
maximum strain in the strain distribution with the uniform strain value in the far field.
These stress concentration factors are also plotted in figure 80 with respect to increasing

impact energy.

164

It can be observed that the length and the area of the delamination follow very similar
trends as a function of impact energy while the behavior of the width of the delamination
is very peculiar. The variation in the width of the delamination suggests that it increased
with increasing impact energy until it reaches a point where it remains constant with any
further increase in impact energy. This behavior depends on the size and type of the
impactor used in the drop weight testing machine. The other hand the Euclidean distance
similarity feature evaluated for Fourier-Zemike moments shows an identical trend as the
width of the delamination as a function of the energy of impact. Thus both these
parameters are not appropriate measures of damage since they portray a wrong notion of
the composite material being able to sustain high energy impacts without any
considerable deterioration of their ability to perform as structural materials. The loading
axis was perpendicular to the length of the delamination caused due to a drop weight
impact making it the more critical measure of damage and will be used for comparison

with shape description techniques.

The similarity features associated with the Zemike moment descriptors illustrate the low
sensitivity of these descriptors to impact damage with relatively low level of impact
energy. This behavior suggests that impact damages caused by a drop weight with impact
energy of 34 J or less do not affect the strain distribution and the complete structural
integrity of the composite is retained. On the other hand the Pearson correlation
coefficient and cosine similarity evaluated for the Fourier-Zemike moments show a linear

behavior as a function of the energy of impact similar to the stress concentration factor.

165

 

6 Pearson correlation coefficient: Fourier-Zemike moments 0 Pearson correlation coefficient: Fourier-Zemike moments

 

 

 

 

 

 

 

A Cosine similarity: Fourier-Zemike moments A Cosine similarity: Fourier-Zemike moments
-1 Stress concentration factor ' Stress concentration factor
1.075 5 1.075

I
l
|
l
l

A

 
 
 

1.065 1

i
DJ

Stress concentration factor

1.055 1 ' 1.055 1

i
N

 

   

1.045 1 1.045 1

Stress concentration factor

I
|
l
l
l
|

Similarity feature: Shape description
Similarity feature: Shape description

 

 

 

I
I
l
l

 

 

 

1.035 I 0 1.035

30 32 34 36 38 5 10 15 20 25 30
Energy of Impact (.1) Length of delamination: Ultrasound (mm)

Figure 81: A comparison between damage assessment in incrementally damaged
composites using stress concentration factors and shape description similarity measures
as a function of impact energy (lefi) and size of delamination (right) obtained using

ultrasonic non-destructive evaluation.

Figure 81 plots the inverse of the Pearson correlation coefficient and the cosine similarity
evaluated for the Fourier-Zemike moments which were evaluated for full-field strain
distributions in incrementally damaged composites under tensile loads along with the
corresponding stress concentration factors as a function of impact energy (right plot) and
length of delamination (left plot). These plots illustrate the ability of Fourier-Zemike
moment descriptors to as a factor of damage characterization in composites and unlike
the stress concentration factor they Fourier-Zemike moments represent the whole strain
distribution in the composite. Thus both of these similarity measures canbe used for

characterizing damage in composites.

166

3.13 Conclusions

The application of shape description to structural damage assessment in fiber-reinforced
polymer composites was explored in this doctoral research. Over the last decade these
shape description techniques have been successfully applied commercially to the fields of
biometric recognition, medical imagery, military targeting, storm tracking and recently in
vibration mode shape recognition. Shape descriptors were carefully chosen based on their
ability to attain a significant amount of data reduction during shape analysis of high
resolution full—field strain maps while retaining all the corresponding contour
information. Along with considerable data reduction, these shape descriptors possess
many other desirable properties, such as low computational cost and invariance to
rotation, translation and size of the full-field strain data eliminating the requirement of
complicated and time consuming image processing. These properties make shape
description techniques viable for comparison of full-field stress, strain or displacement
data recorded from composite panels with different degrees of damage using
sophisticated experimental techniques such as digital image correlation, thermoelastic
stress analysis, photoelasticity, etc which provided the necessary motivation for this

research project.

A rational decision making model was developed and used in the selection of an
appropriate full-field experimental technique for this work. Based on some logistical and
technical limitations that formed a set of essential attributes three techniques viz. digital
image correlation, thermo-photoelasticity and in-plane Moiré were shortlisted. These

three techniques were then rated against one another in a survey based on a set of

167

desirable (soft) attributes. Seventeen researchers and scientists in the field of
experimental mechanics with different levels of expertise and knowledge in these
competing techniques participated in this survey. Digital image correlation was rated as
the most versatile technique for structural damage assessment of fibre reinforced polymer
composites through this exercise and was used to record full-field strain data from
composite specimens in this study. The possibility of performing digital image
correlation on images of composite specimens without any surface preparation or the
application of a speckle pattern was explored. The surface texture of the composite
specimen was enhanced with special illumination either using multiple LED light arrays
or a LED ring-light fitted onto the camera lens. The correlation results with and without a
speckle pattern were compared using the shape description technique developed in this

study and the results validated the surface texture-based digital image correlation method.

Along with traditional shape descriptors viz. Zemike moments and Fourier transforms, a
new shape descriptor was developed to address complications arising from the shape
description of full-field strain maps. It was necessary to devise a generic shape
description scheme which would be applicable to a wide variety of structural
components. This included composite panels with randomly oriented holes of different
shape and sizes and other geometric as well as structural design features such as rivets
and threaded joints. The shape description technique also needed to be sensitive enough
to detect the smallest change in the strain contour caused by damage to the corresponding
composite panel which might be small enough to be invisible to the naked eye. The

Fourier-Zemike descriptor was developed to address these practical problems. This

168

descriptor combined the Fourier transformation of the full-field strain maps with Zemike
moments and was shown to inherit all the desirable properties of both of its parents while

eliminating their disadvantages.

The relevant Zemike moments as well as Fourier-Zemike moments for a series of
incrementally damaged composites were compared and similarity coefficients such
Pearson’s correlation coefficient, cosine similarity and Euclidean distance were employed
to assess the difference between the corresponding feature vectors for the damaged
specimens and a virgin undamaged specimen. It was observed that the Pearson
correlation coefficient and the cosine similarity both decreased below unity with
increasing damage caused by increasing impact energy while the Euclidean distance
increased with increasing damage or impact energy. It was observed that the Pearson’s
correlation coefficient which provides an element-to-element correlation unlike the
cosine similarity and Euclidean distance was the preferred similarity feature due to its
higher sensitivity to damage. The Euclidean distance provides a more classical trend of
increasing damage with increasing energy of impact [34] and can be readily compared to
similar results obtained from non—destructive testing while the inverse of the Pearson
correlation coefficient and cosine similarity were considered for comparison purposes. It
was concluded that the Pearson correlation coefficient and cosine similarity coupled with
the Fourier-Zemike moment descriptors were the preferred similarity coefficients for

shape description applied to structural damage assessment.

169

Zemike moment descriptors were able to provide accurate distinction between the strain
distributions in the incrementally damaged composite specimens under tensile loads in
terms of similarity coefficients. However, since Zemike moments are not capable of
representing strain distributions with discontinuities, they were not evaluated for the
specimen with an impact hole and they can not be used to compare the incrementally
damaged specimens with the one with an impact hole. On the other hand, the Fourier-
Zemike moments were able to differentiate between the incrementally damaged
specimens and it was possible to compare them to the corresponding feature vector for
the specimens with a through hole caused by a high-energy impact. The accuracy of
Fourier-Zemike moments descriptors in terms of unique representation was lower
compared to that for Zemike moment descriptors while their ability to handle
discontinuities in the strain distribution due to geometric features make them more suited

for applications in structural damage assessment.

Table 5: Finite element —model mesh arameters

 

 

 

 

 

 

Mesh Type of Number of
Name
Parameter element elements

CPS4

PEA-2 2 (4 node quads) 6“
CPS4

PEA—4 4 (4 node quads) 1221
CPS4

PEA—6 6 (4 node quads) 1832
CPS4

FEA’S 8 (4 node quads) 2442
CPS4

FEA_16 16 (4 node quads) 4884

 

 

 

 

 

 

170

This study provides a basis for qualitative as well as quantitative comparison of full-field
data recorded using different experimental methods and predictions from finite element
models using shape description techniques; thus laying down the foundation for structural
damage assessment in fiber-reinforced polymer composites as well as full-field

experimental validation of finite element models.

3.14 Future work

The viability of shape descriptors other than the ones applied in this study can be
explored for structural damage assessment in composites as well as experimental

validation of finite element models.

Wavelets are one such mathematical tool that are capable of extracting important shape
information from different kinds of data including and certainly not restricted only to
audio signals and images. Wavelet transforms have been shown to have improved time
frequency resolution over Fourier transforms. A wavelet is a mathematical function that
is capable of dividing a given signal into different scaled components. A continuous
wavelet transform involves taking an inner product of the signal under consideration with

the selected wavelet which results in a representation of the signal in terms of the wavelet

and is given by [176],
DW(a9bx9by) : <V/bx,by,a ’ [(JC, 20> (61)
+00 +00 *
= I IV/bx,by,a (x, Y) ' [(95, Y) dx dy (62)

171

where, w is the mother wavelet and I is the signal. Since the computational cost of
calculating continuous wavelet transforms is very high the discretisation of the wavelet
parameters is performed. These discrete wavelet transforms can be applied to two-
dimensional images to extract important shape information. Wavelets are known to
handle discontinuities in the shape better than Fourier descriptors and when combined
with Zemike moment descriptors could prove to be more accurate than the Fourier-
Zemike shape descriptors. A through study of different wavelets will be required and
their effectiveness in unique representation of full-field displacement, strain or stress
maps could be rated against one another. This will form the basis of a selection process

for the most suitable wavelet for the application to the field of experimental mechanics.

The application of shape description technique to full-field validation of finite element
models can be explored. Finite element models developed with meshes with varying
coarseness and types of elements were validated against corresponding experimental data
using the shape description scheme introduced in this work. In case of traditional
isotropic materials this critical step is often neglected or reduced to placing a single strain
gage at the predicted hot-spot of stress. Even though modern techniques of optical
analysis offer full-field maps of displacement, strain or stress to be obtained from real
components with relative ease and at modest cost, validations continue to be performed
only at predicted and, or observed hot-spots and most of the wealth of data is ignored.
This ignored data however becomes more critical for orthotropic materials such as fiber-
reinforced polymer composites which were extensively studied in this work. With

variations in fiber layup, weaves and manufacturing techniques prediction of stress

172

distribution in fiber-reinforced polymer composites becomes progressively difficult for
the simplest of loading conditions. The shape description technique developed in this
work was capable of fast, full-field comparisons of displacement, strain or stress maps

obtained from experimentation to those obtained from finite element models.

A very simple example of an isotropic plate with a hole was considered to illustrate the
effectiveness of the shape description technique in full-field finite element validations.
.Thus, this technique can be integrated with finite element model updating schemes to
obtain optimal numerical models. An aluminum specimen with dimensions of 50mm X
100mm with a central hole of diameter of 25mm was loaded under tension in a MTS
servo-hydraulic test frame. The maximum principal strain distribution was recorded using

digital image correlation, forming the reference set of experimental data.

A corresponding finite element model was developed using Abaqus-CAE. The mesh
parameter was varied from 2 to 16 with 2 representing a sparse mesh with the lowest
number of elements, while 16 represents a most dense mesh with the highest number of
elements. The details of the five sets of finite element data considered are provided in
Table 5. Since the specimen had a hole causing a sharp discontinuity in the strain
distribution only Fourier-Zemike moments were evaluated for the strain distributions
obtained from the finite element models with varying mesh densities as well as for the
experimental data obtained earlier. The closeness parameters evaluated between the
shape features of the numerical and experimental strain maps were able to track the

effects of the coarseness of the mesh viz. with increased coarseness of the mesh the

173

Euclidean distance between the feature vectors of the numerical and experimental strain
maps increased while the Pearson’s correlation coefficient and the cosine similarity

decreases below unity.

The variation of all three similarity coefficients viz., Pearson’s correlation coefficient,
cosine similarity and Euclidean distance were plotted as a function of increasing mesh
parameter as shown in figures 82. The Pearson’s correlations coefficient and the cosine
similarity both increase towards unity with increase in mesh density until they reach a
value of 0.992 and 0.9908 respectively for mesh factor, 16. The Euclidean distance tends
to decrease with increasing mesh density implying that the finite element solution is
converging to the experimental solution with increasing mesh density. It is observed that
for the example considered in this section of an isotropic plate with a hole, the sensitivity
of all three similarity coefficients is less for denser meshes and more for rarer meshes.
This work could be advanced further by using shape descriptors to optimize of more
complex finite element models and the selection of material models using corresponding

experimental full-field data for anisotropic materials such as fiber reinforces composites.

Finally the influence of individual elements of shape vectors on the reconstruction of the
strain distribution can provide valuable information regarding the quantification of
damage in the engineering components using nothing but the corresponding shape

descriptors.

174

 

 

 

 

A Pearson's correlation coefficient
I Cosine similarity
O Euclidean Distance (Secondary axis)
0.993 , , , ; ; 1.75
0.992 — ' g f ‘ i —— 1.6
*a l 5 A 3 i i 8
8 a a I a s E
1%. 0.991 1 11 11 1.45 E
8 : ' : . : g
>» 1 i I . I 1 <1)
3:: : A : : : T2
5 0.99 1* 5 1' 1 E 3 11 1.3 7:)
E E o E : E m
m = . s = s a
0.989 1 A *1” 1.15
0.988 i I 5 i I 1

 

 

0 3 6 9 12 15 18
Mesh density factor

Figure 82: Similarity measures between the F ourier-Zernike moments evaluated for
maximum principal strain distribution in the finite element model with increasing mesh
density of an aluminum plate with a hole subjected to tensile loads and those evaluated

for the corresponding experimental maximum principal strain distribution in an
aluminum plate with a hole in terms of the Pearson’s correlation coefi‘icient, cosine

similarity and Euclidean distance plotted as a function of mesh density factor.

175

These shape analysis techniques can be used to devise a structural health management
scheme for diagnosis and prognosis of damage in composite structures. An active and/or
passive sensory monitoring including a grid of fiber-optic strain sensors incorporated into
the composite during manufacturing would be able to detect the onset of damage in the
structure. The signal recorded from this grid of sensors and existing diagnostic techniques
can be combined with shape analysis for post-damage prognosis and life assessment of

the structure.

176

4 Concluding remarks

This work introduces a technique for utilizing full-field strain data for structural analysis
including crack propagation in metals and impact damage in a fiber-reinforced
composite. The aim of this research study was to develop better tools for understanding
the structural behavior of engineering materials which will lead to the elimination of
uncertainties in their life estimation and to reduced safety factors and optimal energy
saving structural engineering designs. The author believes that using advanced techniques
capable of recording full-field strain distributions can provide an added advantage over
traditional strain measurement techniques in assessing damage mechanisms and
evaluating accurate life predictions in isotropic as well as orthotropic engineering
materials. However, the computational cost associated with analysis of high resolution
full-field strain distributions is discouraging. Thus shape description techniques have
been investigated which are capable of condensing these high-resolution strain maps into
a feature vector with a considerably smaller number of elements capable of uniquely
representing the strain distribution. The shape descriptor selected for these applications
possess properties such as invariance to rotation, translation and scaling which provide
added advantages in assessing structural components with a wide variety of geometries,

shapes and sizes.

Thermoelastic stress analysis is capable of recording the full-field distribution of the first
invariant of stress in a specimen. The raw data recorded by thermoelastic stress analysis
surrounding a crack tip in aluminum compact tension specimen was studied and found to

be useful in studying local effects due to crack closure and applied overloads on fatigue

177

cracks. The stress intensity factor was evaluated using the full-field stress distribution
combined with multi-point over-deterministic method and a Muskhelishvili type
description of stresses, a technique which was developed by Nurse et al. [8]. Since this
stress intensity factor is evaluated using the experimental stress distribution surrounding
the crack tip, it is the actual or effective stress intensity factor which captures the effects
of crack closure. It was observed that the experimental stress intensity factors were lower
than those predicted by theory or finite element analysis for the same specimen geometry
and loading conditions. This observation was consistent with the presence of closure in
the specimen and confirmed the trend in the values of the stress intensity factor observed

in previous studies [13-16].

The phase data recorded by the thermoelastic stress analysis system, which has been
previously used for locating the crack tip by Diaz et al. [16], was further studied in this
work to provide an accurate measure of the crack tip plastic zone. This is understood to
be the first time the crack tip plastic zone size has been measured experimentally as a
direct measurement without the use of any empirical relationships. In specimens
exhibiting the effects of crack closure, it was observed that the experimentally measured
crack tip plastic radius was higher than the Irwin’s theoretical crack tip plastic radius.
Thus it was concluded that the increase in the size of the crack tip plastic radius causes
increased shielding of the crack tip from the complete applied loading cycle which in turn
causes a drop in the amplitude of the stress intensity factors. On the application of single
cycle overloads it was observed that the size of the crack tip plastic zone immediately

after the application of overload was higher, which causes a drop in the stress intensity

178

and further retards the crack growth. On the other hand, in case of a multi-cycle
overloads, the first few overload cycles form an increased crack tip plastic zone while the
remaining overload cycles assist the crack in growing faster through this plastic zone.
Thus the crack growth retardation effect in case of multi-cycle overloads was found to be

diminished.

The stress intensity factor, the radius of the crack tip plastic zone and the crack length
obtained using the thermoelastic stress analysis data was used to optimize the so called
empirical constants in the Wheeler model including the shaping exponent, m. This
Wheeler’s shaping exponent was previously used to modify the wheeler model to best fit
the corresponding experimental data [19, 20 and 21] and was shown to have values
between 1 and 3. This is the reason why the Wheeler model was considered to have
limited predictive capability. However, with the optimized empirical constants obtained
in this work, it was shown that the Wheeler’s exponent can have values less than 1 and
the model was capable of accurately predicting the effects of crack closure and applied

overloads on the crack growth rate.

This work provides the most compelling evidence to date of the effect of crack tip
shielding and overload retardation with a conformation of the Wheeler’s model using
experimental stress intensity factor and radius of crack tip plastic zone obtained using
thermoelastic stress analysis. The stress intensity factor can be considered as a feature
descriptor for the strain distribution in the vicinity of the crack tip and depends on the

geometry of the specimen and the loading conditions. The geometric stress intensity

179

factor, g can be considered to be invariant of the loading condition and is an invariant

descriptor of the stress distribution surrounding a crack tip.

This work also demonstrates the application of shape description techniques to structural
damage assessment in fiber-reinforced composites. Traditional shape descriptors such as
Zemike moment descriptors and Fourier descriptors which have been used in the fields of
medical imagery, military targeting and biometrics were explored. It was observed that
Zemike moment descriptors were very efficient for complex shape description but were
not able to handle discontinuities, while Fourier descriptors were more difficult to
interpret because they provide no reduction in the size of data that needs to be analyzed.
Thus a new shape descriptor was developed by combining Zemike moments with Fourier
transforms. This was able to uniquely represent strain distributions with discontinuities
and was as easy to interpret as Zemike moments while still being able to preserve all the
shape information for reconstruction. This new shape descriptor can be used to represent
strain distributions from a range of structural components with varying shape, size and

geometries.

A special illumination technique was developed which allowed using the surface texture
of the composite specimen for digital image correlation without the application of a
speckle pattern. The experimental strain maps, obtained by performing digital image
correlation on composites with just the surface pattern were validated using experimental
strain maps obtained by performing digital image correlation on composites with a spray

painted speckle pattern. This full-field validation was performed using the shape

180

description technique introduced in this doctoral dissertation. Digital image correlation
was selected for full-field structural damage assessment out of eleven different
techniques based on its versatility and simplicity. The non-speckle technique further
simplifies the application of digital image correlation by eliminating the requirement of

generating a high quality black and white speckle pattern on the specimen surface.

Incrementally damaged composites subjected to impact with increasing levels of energy
were tested under tension and the strain maps obtained using digital image correlation
were compared using Zemike moments as well and Fourier-Zemike moments. Similarity
parameters such as Pearson’s correlation coefficient, cosine similarity and Euclidean
distance between the strain distributions in the damaged composites and the virgin
undamaged specimen were investigated. The results show a trend in the similarity
coefficients with increasing impact energy, thus being able to distinguish between the
levels of damage in each specimen. The shape description technique introduced in this
doctoral dissertation, which makes use of the newly developed Fourier-Zemike moment
descriptors, is capable of full-field structural damage assessment in a variety of

engineering components with varying shape, size, location and geometry.

Stress concentration factors have been previously used as a measure of damage and
residual life [43 and 44]. The results in this dissertation prove that the Fourier-Zemike
moments evaluated for full-field strain distributions show the same evolution as the stress
concentration factor with increasing damage in fiber reinforced polymer composites.

Thus it was concluded that these shape descriptors can indeed be used as a measure of

181

damage and residual life. Unlike stress intensity factors these shape descriptors are a
unique representation of the full-field strain distribution in the component and thus posses
an added advantage over the traditional stress intensity factor in term of information
retrieval. These shape descriptors provided a basis for quantitative comparison between
different strain distributions. The properties, such as orthogonality and invariance to
rotation, translation and scaling while providing an immense reduction in data, make
them a practical choice for representation and full-field comparisons of high-resolution

strain maps.

This work explores shape description techniques as a tool for full-field structural damage
assessment which is especially crucial in the case of anisotropic composite materials
because of their uncertain post-damage behavior. This required introduction of a new
generic shape descriptor which can be applied to a variety of structural engineering

components.

This doctoral dissertation provides the following original contributions in the field of
fracture mechanics and structural health assessment of composites,
1. A method for experimental measurement of crack tip plastic zone size using
thermoelastic stress analysis—phase data [22].
2. A comprehensive study of the effects of crack closure and overloads on fatigue
cracks in terms of the experimental stress intensity factors, radius of crack tip

plastic zones and crack growth rates [22].

182

3. The development of a shape description technique and a new shape descriptor for
structural damage assessment of composites [135].

4. A validation of the non-speckle technique for digital image correlation which uses
a special illumination scheme to facilitate the use of surface texture of the
composite for digital image correlation.

5. A new measure of damage and life prediction using shape features which
represent a full-field strain distribution in the whole component instead of using
just the maximum strain level.

6. A technique for full-field experimental validation of finite element models [138].

This work provides a better understanding of the phenomenon of crack closure and
overload retardation. The technique utilizing thermoelastic stress analysis can be used to
study different fracture scenarios and materials to increase the confidence in the effects of
crack closure on fatigue crack growth rate. This will help develop better post-fracture life
prediction models than can be used for less conservative, energy saving engineering

designs.

The study involving the use of shape description techniques and Fourier-Zemike
moments provides a basis for post-damage life assessment of structural composites. The
evaluation of damage in composites and the measurement of their residual load carrying
capacity, in terms of shape features, is equivalent to application of the stress
concentration / intensity factors used in the case of isotropic materials. Since the shape

features represent the full-field strain distribution in composites, they represent the global

183

effect of damage on the post-damage load carrying capacity of these anisotropic
materials. Thus this technique can be used to estimate the residual life of the damage
composite and reduce the replacement of damaged composites on the slightest onset of
damage. This facilitates a wider application of these energy saving, light-weight
structural materials in every-day—life engineering designs. A pilot was performed where
the same shape descriptors were used for experimental validation and updating of finite
element models using full-field displacement, strain or stress distributions measured
using techniques such as thermoelastic stress analysis and digital image correlation.
These shape description techniques can be used as a link between current smart structures
with diagnostic capabilities and current prognostic techniques to develop an integrated

real-time structural health monitoring system.

This doctoral dissertation provides new and better tools for understanding the structural
behavior of engineering materials reducing the uncertainties in their life estimation and
eventually leading to reduced safety factors and optimal energy saving structural

engineering designs.

184

 

APPENDIX—A: Source code for ImPaCT

clear all
clc
close all
tic;

% User Inputs
PixPerMM=input('Pixels/mm = ');
L = input('Normalize Strain with Load?(yes/no): ','s');
if strcmpi(‘yes',L)==l
Load=input('Load in Newtons = ');

else

Load =1;
end
strl = '.hdf5';
Str2 = '.dtl';

fprintf('Select data for shape description...\n')
[name b.path]=uigetfile('c:\programas\*.*','Select file');
[pathstr, titel, ext, versn] = fileparts(name);
if strcmpi(strl ,ext)==l
strain_pl = hdf5read(name,’/strains/strain_p 1 ');
strain_pl = double(strain_p1); %strain in "mm/m"
strain_pl = transpose(strain_p 1 );
CalFactor= 1000;
strain_pl = strain_p] *CalFactor; %strain in micro-strain
elseif strcmpi(str2,ext)==
CalFactor = input('Calibration factor (MPa/DTunits) = ');
if name==0
return
else
fid=fopen([b.path,name],'r’);
vari="; % Auxiliar variable
while~strncmp(vari,"'Data0 l " ,3 20,256,0,5',8);
vari=fgetl(f1d);
end
if strcmp(vari,"’Data01",320,256,0,5')==l;
var=l;
Ximage=fscanf(fid,'%f,[320,256]); % Read the x matrix from the original file
b.Ximage=Ximage';
while~strcmp(vari,"'Data0 l " ,320,256, 1 ,5');
vari=fscanf(fid,'%s', l );
end
Yimage=fscanf(fid,'%f.[320,256]); % Read the Y matrix from the original file.
b.Yimage=Yimage';
fclose(fid);
end
Rimage=(b.Ximage."2+b.Yimage."2)."0.5: % Calculate r from x and y
b.Rimage=Rimage;
Phase=atan2(b.Yimage,b.Rimage)* l 80/pi;
b.Phase=Phase;
strain_p l =b.Rimage*CalFactor;
end
else

185

A = imread(name);
[mm,nn,oo]=size(A);
CalFactor = input('Calibration factor = ');
ifoo==l
strain_p I =double(A):
strain_p l =CalFactor*strain_p l ;
elseif oo==3
AA=double(A)/255;
ind=rgb2ind(AA,jet,'nodither'):
strain_p l =double(ind);
end

end

strain_p l =strain_p 1 /Load;

% Data Pre-processing & masking
str3='yes';
YON = input('lmport saved mask?(yes/no): ',‘s');
if strcmpi(str3,YON)==l
[nameofmask b.path]=uigetfile('c:\programas\*.*','Select file');
read_mask=dlmread(nameofmask);
rect1=read_mask( l ,2):
rect2=read_mask(2,:);
X=read_mask(3, l );
Y=read_mas k(4, l );
A=read_mask(5 , l );
manscale=read_mask(6, 1 );
S = imcrop(strain_p1,rectl );
S=imresize(S,manscale);
[mm] = size(S);
x=( l : l :n);
=(1: 12m);
if A==l
SIG=0; NUM=0;
for k=1: 1 :m
for j=l :l:n
Svalue=S(k,j);
if Svalue~=0
SIG=SIG+Svalue;
NUM=NUM+1;
end
end
end
ang=SIG/NUM;
for k=l:l:m
for j=l:l:n
Sig=S(k.j);
if sig==
S(k,j)=ang;
end
end
end
else
ang=mean(mean(S));
end
Cvect=[m n X Y];

186

C = min(Cvect):
C = round((2*C-8)/2);
o=X-C;
p=Y-C:
Sl= imcrop(S, rect2):
strain_map=S 1 *Load;
Sl=Sl-ang;
else
fprintf('Crop region of interest...\n')
figure(); imagesc(strain_pl);axis image;
title('ACTION REQUIRED: Crop region of interest')
[S rectl] = imcrop;
close
[m,n] = size(S);
minSize=min(m,n);
if minSize>=750
scfact=0.3;
elseif minSize>=500 && minSize<750
scfact=0.4;
elseif minSize>=250 && minSize<500
scfact=0.5;
else
scfact=l;
end
fprintf('Size of the input image is %d X %d. \n',m,n):
fprintf('The recomended scaling factor is %f. \n',scfact);
manscale=input('Scaling factor = ');
S=imresize(S,manscale);
[m,n] = size(S);
x=( l : l :n);
y=( l : l :m);
answer = input('Are internal geometric features present(yes/no): ','s');
if strcmpi(strB,answer)==l
A=l;
SIG=0; NUM=0;
for k=1 : l :m
for j=1 : l :n
Svalue=S(k,j);
if Svalue~=0
SIG=SIG+Svalue;
NUM=NUM+];
end
end
end
ang=SIG/NUM;
for k=l : l :m
for j=l : 1 :n
sig=S(k,j);
if sig=—
S(k,_j)=ang;
end
end
end
else
A=0;
ang=mean(mean(S));

187

end

fprintf('Select a pixel as the approximate center of the region of interest...\n')
figure(); imagesc(S);axis image;

title('ACTION REQUIRED: Select a pixel as the approximate center of the region of interest')
[X Y vals]=impixel;

close

Cvect=[m n X Y];

C = min(Cvect);

C = round((2*C-8)/2);

o=X-C;

p=Y-C;

rect2=[o p 2*C 2*C];

Sl= imcrop(S, rect2);

strain_map=S 1 *Load;

Sl=Sl-ang;

%Save mask as ASCII file
mask_name = input('Save mask as...(type name of file with extension ".txt"): ','s');
dlmwrite(mask_name, rectl );
dlmwrite(mask_name, rect2, '-append');
dlmwrite(mask_name, X, '-append');
dlmwrite(mask_name, Y, '-append');
dlmwrite(mask_name, A, '-append');
dlmwrite(mask_name, manscale, '-append');
end

% Fourier Descriptor
FF=fft2(S l );
FFF=Iog(abs(FF));

FF(1,1)=real(FF(1,1))+(li*real(FF(l,l )))/1000;
FFreal=log(real(FF));

FFimag=log(imag(FF));

RFFreal=real(FFreal);

IFFreal=imag(FFreal);
RFFimag=real(FFimag);
IFFimag=imag(FFimag);

mag=abs(FF);

logmag=log(mag);

phase=angle(FF);

phi=cunwrap(phase); % Cunwrap: Copyright (c) 2009, Bruno Luong
diffph=phi~phase;

factph=diffph/(2*pi);

roundph=round(factph);
corphi=phase+2*pi.*roundph;

% Selecting the type of Shape Descriptor
descriptor = menu('Choose a descriptor','Zernike Moment Descriptor','Fourier Descriptor','Fourier-Zemike
Descriptor','Wavelet Descriptor');
if descriptor ==
Image=S l +ang;

188

elseif descriptor ==
Image=S l;
elseif descriptor == 3
parameter = menu('Choose a Fourier descriptor','Logarithm of magnitude','Phase','Logarithm of Real
part of Fourier transform','Logarithm of Imaginary part of Fourier transform');
if parameter ==
Image=logmag;
P=cos(phase)+ l i*sin(phase);
elseif parameter ==
Imagezcorphi;
elseif parameter ==
Image=RFFreal;
elseif parameter ==4
Image=RFFimag;
end
elseif descriptor == 4
fprintf('This option is not yet available, select another option \n');
descriptor = menu('Choose a descriptor','Zemike Moment Descriptor','Fourier Descriptor','Fourier-
Zemike Descriptor','Wavelet Descriptor');
if descriptor ==
Image=S l +ang;
elseif descriptor == 2
Image=S l;
elseif descriptor ==
parameter = menu('Choose a Fourier descriptor','Natural log of magnitude',‘Phase','Real part of Fourier
transform','lmaginary part of Fourier transform');
if parameter == 1
Image=logmag;
P=cos(phase)+ 1 i*sin(phase);
elseif parameter ==2
Image=corphi;
elseif parameter ==
Image=RFFreal;
elseif parameter ==4
Image=RFFimag;
end
end
end
[m,n] = size(Image);
CentY=C+2/2;
CentX=C+2/2;
x=(l :1 :n);
y=( l : l :m);
xx=zeros(m,n);
yy=zeros(m,n);
for i=1 : l :m
for j=l:1:n
xx(i,j)=x(j)-CentX;
yy(i,j)=y(i)-CentY;
end
end
XXXX=(xx+CentX)/PixPerMM;
YYYY=(y+CentY)/PixPerMM;

189

if descriptor ~=2
% Mapping the specimen onto a unit circle
[th,r] = cart2pol(xx,yy);
for ii=l : l :n
for j=l : l :n
t=th(ii,j);
if t>0
th(ii,j)=t;
else
th(ii,j)=2*pi+t;
end
end
end
th 1 =th* l 80/pi;
a=th( l ,n);
b=th( 1 , 1 );
c=th(n, l );
d=th(n,n);
e=(360*pi/ l 80)-th(CentY,n);
f=th(CentY,n);
g=th(CentY,1);
impangle=[a b c d e f g];
impangle=impangle* l 80/pi;
xa=xx( l ,n);
xb=xx( l ,l );
xc=xx(n,l);
xd=xx(n,n);
ya=yy( l ,n):
yb=yy( 1,1);
yc=yy(n. l ):
yd=yy(n.n);
xef=xx(CentY,n);
yef=yy(CentY,n);
xg=xx(CentY, l );
yg=yy(CentY.l );
rho=zeros(m,n);
theta=zeros(m,n);
for ii=l : l :m
for j=l : 1 :n
xcord=xx(ii,j);
ycord=yy(ii,j);
angle=th(ii,j):
if (angle>=d) && (angle<=c)
rho(ii,j)=abs((((yc-yd)*xcord)—((xc-xd)*ycord))/(xd*yc-xc*yd)):
theta(ii,j)=d+(c—d)*abs((xcord*yd-xd*ycord)/(((yc-yd)*xcord)-((xc-xd)*ycord)));
elseif (angle>=e) && (angle<=d)
rho(ii,j)=abs((((yd-yef)*xcord)—((xd-xei)*ycord))/(xef*yd-xd*yef));
theta(ii,j)=e+(d-e)*abs((xcord*yef—xef*ycord)/(((yd-yef)*xcord)-((xd—xet)*ycord)));
elseif (angle>=a) && (angle<=t)
rho(ii,j)=abs((((yef-ya)*xcord)-((xef-xa)*ycord))/(xa*yef-xef*ya));
theta(ii,j)=a+(f-a)*abs((xcord*ya-xa*ycord)/(((yef-ya)*xcord)-((xef-xa)*ycord)));
elseif (angle>=b) && (angle<=a)
rho(ii,j)=abs((((ya-yb)*xcord)-((xa-xb)*ycord))/(xb*ya-xa*yb));
theta(ii,j)=b+(a-b)*abs((xcord*yb-xb*ycord)/(((ya-yb)*xcord)-((xa-xb)*ycord)));
elseif (angle>=e) && (angle<=b)
rho(ii,j)=abs((((yb-yc)*xcord)-((xb-xc)*ycord))/(xc*yb-xb*yc));

190

theta(ii,j )=c+(b-c)*abs((xcord*yc-xc*ycord)/(((yb-yc)*xcord)-((xb-xc)*ycord))):
end
end
end
for ii=l : l :n
if ii>(n+l )/2
theta((m+] )/2,ii)=2*pi;
else
theta((m+ l )/2.ii)=pi:
end
end
theta l =theta;
[XX,YY] = pol2cart(theta,rho);

% Setting up the Zemike moments paramenters for optimization
N = input('lnput maximum order of Zemike moments = ');
nnumber=N+l;
Nvalues=(0: l :N);
mz=N+l;
sizeN=size(Nvalues);
sizeN=sizeN( l ,2);
Mvalues=zeros(nnumber,nnumber);
for j=l : l :sizeN
l=l;
valuen=Nvalues( l ,j ):
if valuen==0
Mvalues(j,l)=0;
l=l+l;
else
for ii=-valuen: l :valuen
a=valuen-abs(ii);
b=a/2;
c=a-2*round(b);
if c==0
Mvalues(j,l)=ii;
l=l+l;
end
end
end
end
[w,ww] = size(Mvalues);
zs=sum(Nvalues+l);
20=zeros( 1 ,2*zs);
h=1 ;hh=l;
sizes 1 =size(S l );
R=zeros(sizeS l ( 1,1),sizeS l( l ,2),zs);
Theta=zeros(sizeS l( 1,1 ),sizeS l( 1 ,2),zs);
Mmatrix=zeros(sizeS 1( l , l ),sizeS l ( l ,2).zs):
Nmatrixzzeros(sizeSl(l,l),sizeSl(1,2),zs);
x=sizeS l(1,2);
for ii=l:1:w
for j=l : l :ww
if j<(ii+l)
=Nvalues( 1 ,ii):
m=Mvalues(ii.j);

191

S=(n-abs(m))/2;
RR=0;
for 3:0: 1 :S
RRR=(((- l )"s)*(factorial(n-s)/(( factorial(s))*(factorial(((n+abs(m))/2)-s))*(factorial(((n-
abs(m))/2)-s)))).*(rho."(n-2*s)));
RR=RR+RRR;
end
for k=l : l :x
for 1:] : l :x
R(k,l,h)=RR(k,l):
Theta( :,:,h)=theta;
Mmatrix(:,:,h)=m;
Nmatrix(:,:,h)=n;
end
end
h=h+l;
end
end
hh=hh+l ;
end
CosMatrix=cos(Mmatrix.*Theta);
SinMatrix=sin(Mmatrix.*Theta);

%Optimization using 'Tminunc " - "errfn " is a pointer to the function being optimized

options=optimset( 'MaxFunEvals',( 10"8),'MaxIter',( 10"6),'To|Fun',( 10"(-
8)),IargeScale','off,'Display','iter');

[z,E,exitflag,output]=fminunc(@errfn,zO,options,R,Image,CosMatrix.SinMatrix);

%Calculating Zemike moments

sizez=size(z);

sizez=sizez( l ,2);

11:1 ;

Zdirection=zeros( l ,zs);

ma meoment=zeros( l ,zs);

forj=l :2:sizez
Zmoment=sqrt(((z( l ,j))"2)+((z( l ,j+] ))"2));
Zdirection( l ,ll)=( l 80/pi)*atan2(z( l ,j+ 1 ),z( 1 .j));
mameoment( l ,ll)=abs(Zmoment);
ll=ll+l ;

end

% Reconstructing the image using Zernike moments

u=l; F=O;

l=l;

sizel=size(zO);

sizel=sizel( l ,2)/2;

for i=1 :1 :sizel
f=R(:,:,i).*(z(u).*CosMatrix(:,:,i)+z(u+l).*SinMatrix(:,:,i)):
F=F+fi
u=u+2;

end

192

% Correlation Coefficient
F_bar=sum(sum(F))/3. l4;
Image_bar=sum(sum(Image))/3. 14;

N l =F-F_bar;

N 2=Image-Image_bar;

N=sum(sum(N 1 .*N2));

D l =sum(sum((F—F_bar)."2));
D2=sum(sum((Image-Image_bar)."2));
D=sqrt(Dl *D2);

corrcoff=N/D;

%Reconstruction of images in case of F ourier-Zernike descriptors
if descriptor == 3

if parameter ==
ilogmag=exp(F);
G=ilogmag.*P;
iG=real(ifft2(G));
iG=(iG+ang)*Load;
dl=(strain_map-iG)."2;
ggl =mean(mean(d l ));
gl=sqrt(ggl);
F_bar=sum(sum(iG))l3. 14;
Image_bar=sum(sum(strain_map))/3. l4;
N1=iG-F_bar;
N2=Image—Image_bar;
N=sum(sum(N 1 .*N2));
D1=sum(sum((iG-F_bar)."2));
D2=sum(sum((strain_map-Image_bar).“2));

=sqrt(D 1 *D2);

corrcoff_strain=N/D;

elseif parameter ==2
P=(cos(F))+l i.*(sin(F)):
G=mag.*P;
iG=real(ifft2(G));
iG=(iG+ang)*Load:
d l =(strain_map-iG)."2;
gg l =mean(mean(d l ));
gl=sqrt(ggl);
F_bar=sum(sum(iG))l3. l4;
Image_bar=sum(sum(strain_map))/3. l4;
Nl=iG-F_bar;
N2=Image-Image_bar;
N=sum(sum(N 1 .*N2));
D l =sum(sum((iG-F_bar)."2));
D2=sum(sum((strain_map-Image_bar)."2));
D=sqrt(Dl *D2);
corrcoff_strain=N/D;

elseif parameter ==3
G=exp(F+l i.*IFFreal)+l i.*exp(FFimag);
iG=real(ifft2(G));
iG=(iG+ang)*Load;
d1=(strain_map-iG)."2;
gg l =mean(mean(d l ));
gl=sqrt(ggl);
F_bar=sum(sum(iG))l3. 14;

193

Image_bar=sum(sum(strain_map))/3. 14;
N l =iG-F_bar;
N2=Image-Image_bar;
N=sum(sum(N 1 .*N2));
D1=sum(sum((iG-F_bar)."2)):
D2=sum(sum((strain_map-lmage_bar)."2));
D=sqrt(Dl *D2);
corrcoff_strain=N/D;

elseif parameter =-
G=exp(FFreal)+ l i.*(exp(F+ l i. *IFFimag)):
iG=real(ifft2(G));
iG=(iG+ang)*Load;
d l =(strain_map-iG)."2;
gg l =mean(mean(d1 ));
g|=sqrt(ggl );
F_bar=sum(sum(iG))/3. l4;
Image_bar=sum(sum(strain_map))/3. 14;
N l =iG-F_bar;
N 2=Image-Image__bar;
N=sum(sum(N] .*N2));
D1 =sum(sum((iG-F_bar)."2));
D2=sum(sum((strain_map-Image_bar)."2)):
D=sqrt(Dl *D2);
corrcoff_strain=N/D;

end

end
end

% Plotting Results
[mm nn]=size(S l );
vectX=( l : 1 2mm);
vectY=( l : 1 mm);
XXX=vectXlPixPerMM;
YYY=vectY/PixPerMM;
if descriptor ==
figure( 1 );subplot( l ,2, l ); surf(XXXX,YYYY,strain_map,'lines','none');title('Original Strain map')
figure(2);subplot( l ,2, l ); imagesc(XXX,YYY,strain_map);axis image; colorbar;title('Original Strain
map');set(gca,'YDir','normaI')
figure( 1 );subplot( 1,2,2); surf(XXXX,YYYY,F,'lines','none'):title('Reconstructed Strain map')
figure(2);subplot( l ,2,2); imagesc(XXX,YYY,F);axis image; colorbar;title('Reconstructed
map'):set(gca,'YDir','norma1')
figure(5):bar(mameoment):title('Zernike Moments')
elseif descriptor ==
figure(2); subplot(2,3, l ); surf(XXXX.YYYY,S1,'1ines','none'):colorbar;title('lmage')
figure(2): subplot(2,3,2); surf(XXXX,YYYY,mag,'lines','none'):colorbar;title('Fourier transform')
figure(2); subplot(2,3,3); surf(XXXX,YYYY,corphi,'lines’,'none');colorbar;title('Phase angle')
figure(2); subplot(2,3,4); surf(XXXX,YYYY,logmag,'lines','none');colorbar;title('Logarithm of Fourier
transform')
figure(2); subplot(2,3,5); surf(XXXX,YYYY,RFFreaI,'lines','none'):colorbar;title('Logarithm of Real part
of Fourier transform')
figure(2); subplot(2,3,6): surf(XXXX,YYYY,RFFimag,'lines','none'):colorbar;title('Logarithm of
Imaginary part of Fourier transform')
figure(3); subplot(2,3,l ); imagesc(XXX,YYY,S 1 );colorbar; axis
image:colorbar;title('lmage');set(gca.'YDir','normal')
figure(3); subplot(2,3,2); imagesc(XXX,YYY,mag);colorbar: axis image:title('Fourier
transform');set(gca,'YDir','normal')

194

figure(3); subplot(2,3,3); imagesc(XXX,YYY,corphi);colorbar; axis image;title('Phase
angle'):set(gca,'YDir','normal')

figure(3); subplot(2,3,4); imagesc(XXX,YYY,logmag);colorbar; axis image;title('Logarithm of Fourier
transform');set(gca,'YDir','normal')

figure(3); subplot(2,3,5); imagesc(XXX,YYY,RFFreal);colorbar;axis image;title('Logarithm of Real part
of Fourier transform');set(gca,'YDir','normal')

figure(3); subplot(2,3,6); imagesc(XXX,YYY,RFFimag);colorbar;axis image;title('Logarithm of
Imaginary part of Fourier transform');set(gca,'YDir','normal')
elseif descriptor ==

figure( 1 );subplot( l ,2, 1); surf(XXXX,YYYY,strain_map,'lines','none'):title('Original Strain map')

figure(2);subplot( l ,2, l ); imagesc(XXX,YYY,strain_map):axis image; colorbar;title('Original Strain
map');set(gca,'YDir','normal')

figure( I );subplot( 1,2,2); surf(XXXX,YYYY,iG,'lines','none');title('Reconstructed Strain map')

figure(2);subplot( l ,2,2); imagesc(XXX,YYY,iG);axis image; colorbar;title('Reconstructed Strain
map');set(gca,'YDir','normal')

figure(3);subplot( 1,2,1); surf(XXXX,YYYY,Image,'lines','none');title('Original Fourier descriptor')

figure(4);subplot( 1,2,1); imagesc(XXX,YYY,Image);axis image; colorbar;title('Original Fourier
descriptor');set(gca,'YDir','normal')

figure(3);subplot(1,2,2); surf(XXXX,YYYY,F,'1ines','none');title('Reconstructed Fourier descriptor')

figure(4);subplot( l ,2,2); imagesc(XXX,YYY,F);axis image; colorbar;title('Reconstructed Fourier
descriptor'):set(gca,'YDir','normal')

figure(5);bar(mameoment);title('Zernike Moments')
end

time=toc:

195

 

10.

11.

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