A study of mixed-mode dynamic fracture in advanced particulate
composites by optical interferometry, digital image
correlation and finite element methods
Except where reference is made to the work of others, the work described in this
dissertation is my own or was done in collaboration with my advisory committee. This
dissertation does not include proprietary or classified information.
Madhusudhana S. Kirugulige
Certificate of Approval:
Thomas S. Denney Jr.
Professor
Electrical and Computer Engineering
Hareesh V. Tippur, Chair
Alumni Professor
Mechanical Engineering
Jeffrey C. Suhling
Quina Distinguished Professor
Mechanical Engineering
Winfred A. Foster
Professor
Aerospace Engineering
Joe F. Pittman
Interim Dean, Graduate School
A study of mixed-mode dynamic fracture in advanced particulate
composites by optical interferometry, digital image
correlation and finite element methods
Madhusudhana S. Kirugulige
A Dissertation
Submitted to
the Graduate Faculty of
Auburn University
in Partial Fulfillment of the
Requirements for the
Degree of
Doctor of Philosophy
Auburn, Alabama
August 04, 2007
A study of mixed-mode dynamic fracture in advanced particulate
composites by optical interferometry, digital image
correlation and finite element methods
Madhusudhana S. Kirugulige
Permission is granted to Auburn University to make copies of this dissertation at its
discretion, upon the request of individuals or institutions and at
their expense. The author reserves all publication rights.
Signature of Author
Date of Graduation
iii
Vita
Madhusudhana Kirugulige was born on January 25, 1975 in Shimoga, India. He grad-
uated with Bachelor of Engineering degree in Mechanical Engineering major from M. S.
Ramaiah Institute of Technology, Bangalore University, Bangalore, India in 1997. In 1998,
he joined Indian Institute of Science, Bangalore, India for masters program in Mechanical
Engineering. He was awarded Master of Science degree for his thesis entitled ?Experimental
and Numerical Investigations of Stable Crack Propagation in Adhesively Bonded Joints ?in
2001. Following graduation, he worked in two private sector companies, Turbotech Preci-
sion Engineering Private Limited, Bangalore and General Electric Global Research Center
in Bangalore. In the former he was involved in design, analysis and testing of steam tur-
bines, gas turbines and centrifugal compressors. In the latter, he worked on finite element
modeling of entire gas turbine engine casing in order to evaluate running clearance between
stator and rotor. He joined Auburn University in spring 2003 as a Ph.D. candidate in
Mechanical Engineering where he conducted research on mixed-mode dynamic fracture of
novel materials using optical interferometry, digital image correlation and finite element
methods.
iv
Dissertation Abstract
A study of mixed-mode dynamic fracture in advanced particulate
composites by optical interferometry, digital image
correlation and finite element methods
Madhusudhana S. Kirugulige
Doctor of Philosophy, August 04, 2007
(M.S., Indian Institute of Science, Bangalore, India, 2001)
(B.E., Bangalore University, Bangalore, India, 1997)
204 Typed Pages
Directed by Hareesh V. Tippur
Understanding the fracture mechanics of materials under stress wave loading is essen-
tial for impact resistant design of structures. In this context, mixed-mode dynamic fracture
behavior of two-phase composites - a functionally graded material (FGM) and a syntactic
structural foam are investigated experimentally and numerically. FGMs are macroscopi-
cally nonhomogeneous engineered materials with spatially varying volume fraction of the
constituents. They are used as thermal barrier coatings in high temperature components, as
core materials in sandwich structures, as inter layers in micro-electronic packages, to name
a few. Syntactic foams are homogeneous buoyant materials used in naval/marine applica-
tions as well as for energy dissipation in military and industrial environments. Catastrophic
failure in these materials is often observed to occur in a mixed-mode fashion involving a
combination of tensile and shear fractures. Real-time and full-field measurement of crack
tip deformations in these circumstances is rather challenging because the events typically
last only a couple of hundred microseconds, and need optical tools coupled with ultra-high
v
speed imaging devices to understand the associated failure mechanisms. To date very few
methods are available for performing direct measurements of crack tip fields. This dis-
sertation aims to address these by studying the dynamic fracture behavior of such novel
materials by developing suitable measurement and modeling tools.
The first part of this research extends the optical method of Coherent Gradient Sensing
(CGS) to the study of mixed-mode dynamic fracture behavior of functionally graded ma-
terials. FGMs studied are the ones with a continuously varying volume fraction of ceramic
filler particles in a polymer matrix having edge cracks initially along the property gradient
and subjected to impact loading. The mixed-mode loading is generated by loading samples
eccentrically relative to the initial crack plane. CGS and high-speed photography are used
to map transient crack tip deformations. Two configurations, one with a crack on the stiffer
side of a graded sheet and the second with a crack on the compliant side, are examined
experimentally. Differences in both pre- and post-crack initiation behaviors are observed in
terms of crack initiation time, crack path, crack speed and stress intensity factor histories.
A crack kinks by a much larger angle when it originates from the stiffer side of the FGM
compared to the compliant side. Crack speeds, however, are higher in the latter configura-
tion by nearly 100 m/sec. Prevailing crack tip field descriptions do successfully predict the
observed crack path differences.
In the second part of this work, the method of digital image correlation is developed to
study transient deformations associated with rapid mixed-mode crack growth in materials.
Edge cracked polymer beams and syntactic foam samples are studied under low-velocity
impact loading conditions. Decorated random speckle patterns in the crack tip vicinity
are recorded using an ultra high-speed CCD camera at framing rates of 200,000 frames
vi
per second. A three-step digital image correlation technique is developed and implemented
in a MATLAB environment for evaluating crack opening/sliding displacements and the
associated strains. Using this approach, the entire crack tip deformation history, from
the time of impact to complete fracture, is mapped successfully. Over-deterministic least-
squares analyses of crack tip displacements are performed to extract dynamic stress intensity
factor (SIF) histories. The current work being the first of its kind using a rotating mirror
type multi-channel high-speed digital camera system, calibration tests and procedures are
established by carrying out a series of benchmark experiments. The accuracy of measured
displacements is in the range 2 to 6% of a pixel (0.6 to 1.8 ?m) and that of the dominant
strain is about 150-300 micro strains.
In the last part, finite element modeling of mixed-mode dynamic crack growth in FGM
using cohesive element formulations is performed. The formation of new surfaces is ac-
complished by using bilinear tensile and shear traction-separation laws. A user-defined
subroutine is developed and linked with ABAQUS implicit procedure. The spatial varia-
tions of material properties are incorporated into the continuum elements by performing a
thermal analysis first and then by applying temperature dependant material properties to
the model. Measured mode-I crack initiation toughness data from homogeneous samples
of various volume fractions of the filler are used to introduce spatial variation of cohesive
element properties to the model. The simulated crack paths are in agreement with the ex-
perimental ones. The computed results prior to crack initiation show the presence of larger
negative constraining stresses (T-stresses) near the crack tip when the crack is situated on
the compliant side of the FGM. The simulations reveal that more energy is dissipated when
the crack is situated on the compliant side of the sample compared to when it is on the
vii
stiffer side. This is in consistent with the higher crack speeds observed when the crack
initiates from the complaint side.
viii
Acknowledgments
I would like to thank my research supervisor, Dr. Hareesh V. Tippur for his valuable
guidance, constant encouragement and advice extended throughout the research work of
this thesis. I am grateful to him for providing excellent experimental and computational
facilities to work on and encouraging me to present the research work at various forums.
Special thanks to Dr. Thomas S. Denney for his valuable help and guidance with image
processing issues of this research work. Thanks to Dr. Jeffrey C. Suhling and Dr. Winfred
A. Foster for having kindly agreed to serve on my committee. My sincere thanks to thank
Dr. Michael J. Stallings for agreeing to become the external reader for my thesis. Thanks
are also due to Dr. Michael E. Miller of Auburn University Research Instrumentation
Facility (AURIF) department for allowing me to use his vacuum evaporator. I would like
to thank US Army Research Office (grants # W911NF-04-10257, DAAD19-02-1-0126 and
DAAD19-01-1-0745) and for extending financial and equipment support for this work.
I would like to acknowledge my colleagues and friends in Auburn for the memorable
and enjoyable time I spent with them. I had very good time in the lab with Rajesh, Mike,
Piyush, Taylor, Dong and Rahul. Espeicially the lively atmosphere of the lab in the presence
of Piyush, Rajesh and Taylor was really enjoyable.
Finally, the support, encouragement and love I had from my wife Asha Dixit is some-
thing invaluable. I dedicate this work to her.
ix
Style manual or journal used Journal of Approximation Theory (together with the style
known as ?aums?). Bibliograpy follows van Leunen?s A Handbook for Scholars.
Computer software used The document preparation package TEX (specifically LATEX)
together with the departmental style-file aums.sty.
x
Table of Contents
List of Figures xiv
List of Tables xxi
1 Introduction 1
1.1 Motivation and Literature Review . . . . . . . . . . . . . . . . . . . . . . . 5
1.1.1 Review of failure characterization of FGM . . . . . . . . . . . . . . . 7
1.1.2 Review of optical methods to study fracture . . . . . . . . . . . . . . 8
1.1.3 Review of numerical methods to simulate fracture . . . . . . . . . . 10
1.2 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.3 Organization of Dissertation . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2 Experimental method of CGS to study dynamic fracture 17
2.1 Optical set-up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.2 Working principle of CGS . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.3 Experimental procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.4 Extraction of stress intensity factors from interferograms . . . . . . . . . . . 24
2.4.1 Pre-crack initiation period . . . . . . . . . . . . . . . . . . . . . . . . 25
2.4.2 Post-crack initiation period . . . . . . . . . . . . . . . . . . . . . . . 26
2.5 Computation of crack speed . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
2.6 Crack tip fields in FGM with linear material property variation . . . . . . . 30
2.6.1 Crack along the direction of property gradation . . . . . . . . . . . . 30
2.6.2 Crack inclined to the direction of property gradation . . . . . . . . . 32
2.7 Extraction of SIFs with difference formulation . . . . . . . . . . . . . . . . . 33
3 Mixed-mode dynamic fracture of FGM using CGS 35
3.1 Material Preparation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
3.2 Material characterization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
3.3 Specimen surface preparation . . . . . . . . . . . . . . . . . . . . . . . . . . 38
3.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
3.4.1 Experimental details . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
3.4.2 Crack growth and crack speed histories . . . . . . . . . . . . . . . . 43
3.4.3 Mixed-mode stress intensity factor histories . . . . . . . . . . . . . . 44
3.4.4 Initial crack path prediction . . . . . . . . . . . . . . . . . . . . . . . 50
3.5 Mixed-mode SIF history from FGM crack tip fields . . . . . . . . . . . . . . 52
xi
4 The method of Digital Image Correlation 56
4.1 The approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
4.1.1 Initial estimation of displacements (Step-1 . . . . . . . . . . . . . . . 57
4.1.2 Refining displacements (Step-2) . . . . . . . . . . . . . . . . . . . . . 59
4.1.3 Smoothing of displacements and estimation of strains (Step-3) . . . 60
4.2 Static experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
4.3 Dynamic experimental set-up . . . . . . . . . . . . . . . . . . . . . . . . . . 67
4.4 High-speed camera calibration . . . . . . . . . . . . . . . . . . . . . . . . . . 69
4.5 Benchmark experiments for high-speed camera . . . . . . . . . . . . . . . . 74
4.5.1 Intensity variability test . . . . . . . . . . . . . . . . . . . . . . . . . 76
4.5.2 Translation test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
4.5.3 Rotation test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
4.6 Flash lamp light characteristics . . . . . . . . . . . . . . . . . . . . . . . . . 87
5 Dynamic fracture studies using DIC method 91
5.1 Sample preparation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
5.2 Experimental procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
5.3 Finite element simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
5.3.1 Mixed-mode fracture of syntactic foam . . . . . . . . . . . . . . . . . 94
5.3.2 Mode-I fracture of epoxy . . . . . . . . . . . . . . . . . . . . . . . . 96
5.4 Results - Mixed-mode dynamic fracture of syntactic foam . . . . . . . . . . 97
5.4.1 Extraction of stress intensity factors . . . . . . . . . . . . . . . . . . 99
5.4.2 Estimation of strains . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
5.5 Results - Mode-I dynamic fracture of epoxy . . . . . . . . . . . . . . . . . . 107
5.5.1 Extraction of stress intensity factors . . . . . . . . . . . . . . . . . . 109
5.5.2 Estimation of strains . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
6 Numerical procedures for modeling dynamic fracture in FGM 116
6.1 Elastodynamic governing equations . . . . . . . . . . . . . . . . . . . . . . . 116
6.2 Implicit integration of dynamic equations in ABAQUS . . . . . . . . . . . . 119
6.3 Formulation of an element in ABAQUS . . . . . . . . . . . . . . . . . . . . 120
6.4 Cohesive element formulation . . . . . . . . . . . . . . . . . . . . . . . . . . 121
6.4.1 Exponential traction-separation law . . . . . . . . . . . . . . . . . . 123
6.4.2 Bilinear traction-separation law . . . . . . . . . . . . . . . . . . . . . 126
6.5 Implementation details . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
6.6 Implicit dynamic scheme and time step control . . . . . . . . . . . . . . . . 130
7 Numerical simulation of mode-I and mixed-mode dynamic fracture in
FGM 133
7.1 Modeling aspects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
7.2 Application graded material properties to continuum elements . . . . . . . . 136
7.3 Application of material properties to cohesive elements . . . . . . . . . . . . 137
xii
7.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
7.4.1 Energy computations . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
7.4.2 Effect of the initial slope of traction-separation law . . . . . . . . . . 141
7.4.3 Crack path history . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
7.4.4 T-stress history . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148
8 Conclusions 150
8.1 Future Directions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154
Bibliography 156
Appendices 164
A A note on accuracy of strains and time resolved displacements 164
A.1 A note on accuracy of strains . . . . . . . . . . . . . . . . . . . . . . . . . . 164
A.2 Time resolved displacements . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
B Computation of stiffness coefficients in tration-separation laws 170
B.1 Exponential traction-separation law . . . . . . . . . . . . . . . . . . . . . . 170
B.2 Bilinear traction-separation law . . . . . . . . . . . . . . . . . . . . . . . . . 173
C Finite element simulation of Mode-I dynamic fracture of FGM 175
C.0.1 Material preparation and characterization . . . . . . . . . . . . . . . 175
C.0.2 Experimental results . . . . . . . . . . . . . . . . . . . . . . . . . . . 177
C.0.3 Modeling details . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180
C.0.4 Finite element results . . . . . . . . . . . . . . . . . . . . . . . . . . 180
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List of Figures
1.1 Schematic of a functionally graded material showing variation in elastic, fail-
ure and fracture properties when constituent volume fraction is varied. . . . 2
1.2 Applications of syntactic foams, (a) deepwater insulated oil and gas pipelines
(Courtesy: Cuming corporation), (b) buoyancy foam for deep underwater
floatation (Courtesy: Syntech materials, Inc.) and (c) Impact resistant sand-
wich structures (Courtesy: Goodrich Corporation) . . . . . . . . . . . . . . 3
1.3 Schematic illustrating modes of fracture . . . . . . . . . . . . . . . . . . . . 4
1.4 Mixed-mode dynamic fracture evidences, (a) environmentally assisted crack
propagation from leading edge to the inside cooling surface of a gas tur-
bine blade (Courtesy: Gas Turbine technology) and (b) Concrete damage
after missile impact (Courtesy: Dept. of Civil and Structural Engineering,
University of Sheffield, UK.) . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.1 Schematic for reflection-mode CGS set-up with a high-speed camera . . . . 18
2.2 Working principle of CGS, (a) Diffraction of a collimated beam though two
parallel Ronchi gratings, (b) undeformed object wave front and (c) deformed
object wave front. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.3 Global and local crack tip coordinate systems for (a) stationary crack and
(b) propagating crack . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.4 (a) Schematic of FGM sample with linear material property variation, (b)
elastic modulus variation in graded samples (broken line denotes the crack
tip location) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
3.1 Schematic of FGM sample used in experiments . . . . . . . . . . . . . . . . 36
3.2 Variation of longitudinal and shear wave speeds along the width of the sample 37
3.3 Variation of elastic modulus and mass density along the width of the sample 39
3.4 Variation of dynamic initiation toughness (impact velocity = 5.4 m/sec) with
Elastic modulus. (Broken line is a trend line) . . . . . . . . . . . . . . . . . 40
xiv
3.5 Selected CGS interferograms representing contours of ?w/?x in FGM and
homogeneous samples. (The vertical line is at 10 mm from the crack). (a)
crack on the compliant side and (b) crack on the stiffer side (c) homogeneous
(Plexiglas) sample. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
3.6 Crack growth behavior in FGM samples under mixed-mode dynamic loading.
(a) Crack growth history, (b) normalized crack speed history. (VR: local
Rayleigh wave speed) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
3.7 Mixed-mode dynamic stress intensity factor histories (impact velocity=5.2
m/sec). (Circles: E1 E2). (Time base is altered such
that t?ti = 0 corresponds to crack initiation) . . . . . . . . . . . . . . . . . 45
3.8 Photographs showing multiple fractured specimens (right half) demonstrat-
ing experimental repeatability (a) FGM with a crack on the compliant side
(E1 E2). . . . . . . . 47
3.9 Photographs showing fractured specimens for (a) FGM with a crack on the
compliant side (E1 E2)
and (c) a homogeneous specimen. Impact point is indicated by letter ?I? and
initial crack tip by letter ?C? . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
3.10 Crack growth behavior in FGM samples under mixed-mode dynamic loading.
(a) Crack growth history, (b) normalized crack speed history. (VR: local
Rayleigh wave speed) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
3.11 Stress intensity factors extracted from CGS interferograms by performing
over-deterministic least-squares analysis on difference formulation of CGS
governing equation formulated by using crack tip stress fields obtained for
FGM with linear elastic modulus variation. . . . . . . . . . . . . . . . . . . 54
3.12 The quality of least-squares fit (plots of synthetic contours generated from
Eq. 2.34 superimposed on collected data points) for (a)E1 E2 (t?ti = ?20 ?s). . . . . . . . . . . . . . . . . . . . . 55
4.1 (a)Undeformed and deformed sub-images chosen from images before and
after deformation, respectively and (b) typical plot of impulse response
G?p(kx ?u,ky ?v) generated from cross-correlation between two sub-images. 58
4.2 (a) Schematic of the experimental set-up for static experiment, (b) specimen
and loading details . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
4.3 Photograph of the static experimental set-up . . . . . . . . . . . . . . . . . 64
xv
4.4 In-plane displacements obtained from Step-1, 2 and 3 of the image correlation
process. The interval between contours is 7 ?m. . . . . . . . . . . . . . . . . 65
4.5 Static experimental results. (a) and (c) u-displacement (mm) from DIC
and FEA, (b) and (d) ?xx (?-strain) from DIC and FEA, (e) and (f) u-
displacement and ?xx-strain at section AA and BB. Rigid body displace-
ments have been subtracted out both in (a) and (c) . . . . . . . . . . . . . 66
4.6 Schematic of the dynamic experimental set-up . . . . . . . . . . . . . . . . . 67
4.7 Photograph of the dynamic experimental set-up . . . . . . . . . . . . . . . . 68
4.8 Optical schematic of cordin-550 camera: M1,M2,M3,M4,M5 are mirrors;R1
and R2 are relay lenses; r1,r2,???r32 are relay lenses for CCDs; c1,c2,???c32
are CCD sensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
4.9 (a) Image of the 5?5 dot pattern template used for calibration experiment
and (b) Inverted binary image of the template in order to find the control
points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
4.10 Mean and standard deviations of intensity values of images acquired in total
darkness (with lens cap on). Images were recorded at 50,000 frames per
second in experiment 1 and at 200,000 frames per second in experiment 2. . 77
4.11 Experimental set-up for conducting ranslation test for high-speed digital camera 79
4.12 Translation test results for D=400 mm and 200 mm (see Fig. 4.8. (a) mean
and (b) standard deviations of u- and v- displacement fields for X- and Y-
translations of ? 60 ?2?m (c) mean and (d) standard deviations of u- and
v-displacement fields for X- and Y- translations of 300 ?2?m. Magnification
= 35.6 ?m/pixel for D=400 mm and 27 ?m/pixel for D=200 mm. . . . . . 82
4.13 Translation test results for D = 400 mm (see Fig. 4.8) and out-of-plane
displacement (w) =30 ?m. (a) mean and (b) standard deviation of u- and
v-displacement field. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
4.14 Estimated full-field quantity ?xy from one pair of the images taken from
camera # 1 in a rotation experiment (Imposed rotation = 0.0056?0.00035
radians). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
4.15 Results from rotation test (applied rotation = 0.0056 ? 0.00035 radians).
(a) mean and (c) standard deviation of rotation field estimated from image
correlation. (b) mean (d) standard deviations of in-plane strains estimated
(ideally these strains need to be zeros). . . . . . . . . . . . . . . . . . . . . . 88
xvi
4.16 Photo detector output proportional to flash lamp light intensity, A1, A2 and
B1, B2 are two repeated acquisitions when photodiode was placed one inch
away in the plane perpendicular to optical axis of the camera. . . . . . . . . 89
5.1 Specimen configuration for (a) mixed-mode test of syntactic foam and (b)
mode-I test of epoxy. Impactor force history and support reaction histories
recorded by Instron Dynatup 9250 HV drop tower for (c) mixed-mode exper-
iment and (d) mode-I experiment. The sample dimensions are a = 10 mm,
W = 50 mm, S = 25.4 mm, L = 200 mm, B = 8.75 mm, Impact velocities,
V1 = 4.5 m/sec and V2 = 4.0 m/sec. . . . . . . . . . . . . . . . . . . . . . . 92
5.2 Finite element mesh used for elsto-dynamic finite element analysis of (a)
mixed-mode problem and (b) mode-I problem. . . . . . . . . . . . . . . . . 95
5.3 Acquired speckle images of 31 ? 31 mm2 region at various times instants.
(Crack tip location is shown by an arrow.) . . . . . . . . . . . . . . . . . . . 98
5.4 Crack growth behavior in syntactic foam sample under mixed-mode dynamic
loading. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
5.5 Crack opening and sliding displacements (in mm ) for pre- and post-crack
initiation instants. (a) v-displacement and (c) u-displacement before crack
initiation (at t=150 ?s); (b) v-displacement and (d) u-displacement after
crack initiation (t=220 ?s). Crack initiation time ? 175 ?s. (A large rigid
body displacement can be seen in (c) an (d) due to movement of the sample. 100
5.6 Stress intensity factors extracted from displacement fields obtained from im-
age correlation. SIF history obtained from finite element simulation up to
crack initiation is also shown. . . . . . . . . . . . . . . . . . . . . . . . . . 104
5.7 The mode-mixity,?obtained from experiments and finite element simulation.
The broken line corresponds to crack initiation time. . . . . . . . . . . . . . 105
5.8 Crack tip normal strains (in micro strains) at t = 150 ?s. (a) from experi-
ment and (b) From FEA. Crack initiation time = 175 ?s. . . . . . . . . . . 107
5.9 Acquired speckle images of 31 ? 31 mm2 region at various times instances.
Current crack tip location is shown by an arrow. . . . . . . . . . . . . . . . 108
5.10 Crack growth behavior in epoxy sample under mode-I dynamic loading.
Crack length history and crack speed history . . . . . . . . . . . . . . . . . 109
xvii
5.11 Crack opening and sliding displacements (in ?m ) for pre- and post-crack
initiation instants. (a) v-displacement and (c) u-displacement before crack
initiation (at t = 120 ?s); (b) v-displacement and (d) u-displacement after
crack initiation (t = 151 ?s). Crack initiation time ? 133 ?s. . . . . . . . . 110
5.12 Examples showing quality of least-squares fit of displacement data; Crack
opening displacement field (?m) obtained from DIC and synthetic contours
for (a) t = 124 ?s (before crack initiation) and (b) t = 151 ?s (after crack
initiation). Crack initiation time = 133 ?s . . . . . . . . . . . . . . . . . . . 112
5.13 Stress intensity factors extracted from displacement field obtained fromimage
correlation. SIF history obtained from finite element simulation up to crack
initiation is also shown. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
5.14 Crack tip in-plane constraint, ? obtained from experiments and finite element
simulation. The broken line corresponds to crack initiation time. . . . . . . 114
5.15 Crack tip normal strains (in micro strains) for pre- and post-crack initiation.
Normal strain ?yy at (a) t = 120 ?s and (b) at t = 151 ?s and (c) ?yy from
finite element analysis at t = 120 ?s. Crack initiation time = 133 ?s. . . . . 115
6.1 (a) Undeformed and (b) deformed finite element mesh near a notch tip, (c)
Schematic showing separation of nodes in a cohesive element and (d) local
and global coordinate system used for a cohesive element. . . . . . . . . . . 122
6.2 Exponential traction-separation law showing uncoupled loading: variations
of (a) pure normal traction with normal separation and (b) pure tangential
traction with tangential separation. . . . . . . . . . . . . . . . . . . . . . . . 125
6.3 Exponential traction-separation law showing coupled loading: variations of
(a) normal traction and (b) tangential traction. . . . . . . . . . . . . . . . 126
6.4 Prescribed bilinear traction-separation law for (a) pure normal traction ver-
sus normal separation and (b) pure tangential traction versus tangential
separation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
6.5 Steps involved in implementing a cohesive element as user-defined element in
ABAQUS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
7.1 (a) Overall view of the finite element mesh used for the analysis (b) Mag-
nified view of mesh showing region 1 and region 2 (c) Enlarged view of the
mesh at the interface where the elements from region 1 and region 2 meet. . 134
xviii
7.2 (a) Nodal temperature results from thermal analysis, (b) magnified view of
the cohesive element region. . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
7.3 Mixed mode dynamic fracture of plxiglas sample. (a) Crack path observed
in experiments and (b) initial crack path from finite element simulations . . 139
7.4 Evolution of different energy components in dynamic simulation for both
FGM configurations: (a) kinetic energy and strain energy and (b) energy
dissipated by cohesive elements . . . . . . . . . . . . . . . . . . . . . . . . . 142
7.5 Effect of initial slope of the traction-separation law on (a) displacement and
(b) on stress results in elastodynamic simulation . . . . . . . . . . . . . . . 144
7.6 Snapshots of ?yy stress field at two different time instants, (a) 120 ?s and
(b) 150 ?s for E1 E2 (crack initiation time = 130 ?s). . . . . . . . . . . . 146
7.7 Snapshots of uy displacement field at two different time instants, (a) 120 ?s
and (b) 150 ?s for E1 E2 (crack initiation time = 130 ?s). . . . . . . . . 147
7.8 Crack growth behavior in FGM sample under mixed-mode loading. Absolute
crack length history from (a) experiments and (b) finite element simulations,
ti is crack initiation time (ti = 155 ?s for E1 E2
in experiments, ti ? 130 ?s for both E1 E2 in simulations). 148
7.9 (a) Variation of apparent T-stress with crack length at certain time instant
before crack initiation (b)T-stress history up to crack initiation for E1 E2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
A.1 Results of benchmark experiment conducted to estimate the accuracy of dis-
placements and strains. (a) full-field u-displacement between image 1 and
image 2 before deforming image 2 (ideally u-displacement shoud be zero).
(b) u-displacement after applying a constant strain to image 2 but before
smoothing, (c) u-displacement after smoothing and (d) normal strain after
stretching image 2 uniformly. . . . . . . . . . . . . . . . . . . . . . . . . . . 166
A.2 Time resolved crack opening displacements for image # 1 to 12. Time at
which each image was acquired after impact, is indicated above each figure.
The interval between each contour is 3.5 ?m. . . . . . . . . . . . . . . . . . 167
A.3 Time resolved crack opening displacements for image # 13 to 24. Time at
which each image was acquired after impact, is indicated above each figure.
The interval between each contour is 3.5 ?m. . . . . . . . . . . . . . . . . . 168
xix
A.4 Time resolved crack opening displacements for mixed-mode dynamic test,
image # 25 to 32. Time at which each image was acquired after the impact,
is indicated above each figure. The interval between each contour is 3.5 ?m. 169
B.1 Reversible and irreversible unloading . . . . . . . . . . . . . . . . . . . . . . 171
C.1 (a) Schematic of the FGM specimen, (b) Material properties variation along
the width of the sample and (c) Variation of dynamic crack initiation tough-
ness along the width of the sample. . . . . . . . . . . . . . . . . . . . . . . . 178
C.2 Selected CGS interferograms representing contours of ?w/?x in functionally
graded epoxy syntactic foam sheet impact loaded on the edge opposing the
crack tip. (The vertical line is at a distance of 10 mm from the crack.) (a)
Crack on the compliant side E1 E2.
Fringe sensitivity ? 0.015o /fringe. . . . . . . . . . . . . . . . . . . . . . . . 179
C.3 Finite element mesh used for the analysis . . . . . . . . . . . . . . . . . . . 181
C.4 Snapshots of ?yy stress field at two different time instants, (a) 85 ?s and (b)
125 ?s for E1 E2 (crack initiation time = 127 ?s). . . . . . . . . . . . . . 182
C.5 Crack growth behavior in syntactic foam FGM samples undermode-I loading.
absolute crack length history from (a) experiments and (b) finite element
simulations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183
C.6 Evolution of various energies in mode-I dynamic simulation for both FGM
configurations: (a) kinetic energy and strain energy and (b) energy dissi-
pated by cohesive elements . . . . . . . . . . . . . . . . . . . . . . . . . . . 183
xx
List of Tables
3.1 Nominal bulk properties of the constituent materials . . . . . . . . . . . . . 35
3.2 Predicted crack kink angle based on estimated SIF data from CGS interfer-
ogrmas before crack initiation . . . . . . . . . . . . . . . . . . . . . . . . . . 51
3.3 Observed crack kink angle from three CGS interferograms just after crack
initiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
4.1 Alignment differences between individualoptical channels of Cordin-550 cam-
era; Stretch, rotation and translations of different images with respect to the
image taken by camera # 09 . . . . . . . . . . . . . . . . . . . . . . . . . . 75
4.2 Gray scale values at a particular pixel in five repeated sets of images of
speckle pattern acquired at 200,000 frames second. Note the repeatability of
the gray scale values between different sets of images. . . . . . . . . . . . . 78
4.3 Details of translation tests: Six sets of 32 images were recorded in each
configuration. In Configuration-2, the camera was kept twice as close as in
Configuration-1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
4.4 Mean and standard deviations of in-plane strain fields estimated from mea-
sured displacements in translation test . . . . . . . . . . . . . . . . . . . . . 85
xxi
Chapter 1
Introduction
Over the last two decades, there has been an increasing demand for stiffer, stronger,
light-weight, energy absorbing materials. Aerospace applications have placed some of the
stringent requirements on the performance of the materials used in space planes and re-entry
vehicles. One of them is to withstand large stress and/or thermal gradients over a small
spatial dimension. For example, large temperature gradients in re-entry vehicles generate
enormous amount of thermal stresses. This is true with parts exposed to high temperatures
in gas turbines and IC engines as well. They are traditionally made by plasma spraying parts
[1] made of nickel based alloys using thermal barrier coatings (TBC). These conventional
TBCs suffer in terms of durability due to their poor bond strength, oxidation/corrosion
resistance, and delamination or spallation. Therefore a new class of materials called func-
tionally graded materials (FGM) having gradual compositional variation from heat resistant
ceramic to fracture resistant metals have emerged as potential replacements for discretely
layered conventional TBCs. They are manufactured by continuously varying the volume
fraction of constituent phases along a spatial direction. Other applications of FGM include
surface hardened tribological surfaces, impact resistant structures for armors and ballistics,
interlayers in microelectronic and optoelectronic components, heat shields in rockets, to cite
a few. Manufacturing methods for FGM have also been evolving over the recent years. The
processing techniques such as chemical and physical vapor deposition, powder processing,
infiltration techniques, buoyancy assisted casting, and diffusion are commonly used for pro-
ducing FGM. The study of dynamic failure of FGM becomes important because many of
1
the aforementioned applications involve dynamic loading including mechanical shock and
impact. Figure 1.1 shows schematic of the FGM sample prepared using gravity assisted
casting.
Figure 1.1: Schematic of a functionally graded material showing variation in elastic, failure
and fracture properties when constituent volume fraction is varied.
There has been steady increase in the usage of sandwich composites in aerospace,
marine, transportation and packaging industries. The sandwich composites are made by
attaching two thin plates called skin or face sheets to a thick and light-weight material called
the core. Syntactic foams (polymers filled with thin walled hollow microballoons) have
gained popularity as core materials in sandwich structures due their high energy absorption
capability, compressive strength and low moisture absorption. Syntactic foams are also
used in a variety of other applications such as buoyancy modules in boat hulls, structural
components in helicopters and airplanes, antenna assemblies, thermal insulators in oil and
gas industries, to name a few (see Fig. 1.2). Syntactic foams are particulate composites
manufactured by dispersing prefabricated microballoons in a matrix material. The porosity
in these materials results in lower density and superior thermal, dielectric, fire resistant,
hygroscopic properties and sometimes radar or sonar transparency. Syntactic foams can
be tailored to suit a particular application by selecting a wide range of microballoons of
2
different sizes (hollow glass micro spheres, carbon or polymer microballoons, cenoshepres,
etc.) along with any metallic, polymeric or ceramic matrix material. Since one of the main
functions of a syntactic foam is to absorb energy to withstand impact and shock loading, it
is important to study the dynamic failure behavior of these materials.
Figure 1.2: Applications of syntactic foams, (a) deepwater insulated oil and gas pipelines
(Courtesy: Cuming corporation), (b) buoyancy foam for deep underwater floatation (Cour-
tesy: Syntech materials, Inc.) and (c) Impact resistant sandwich structures (Courtesy:
Goodrich Corporation)
The crack initiation and propagation under transient dynamic loading occurs in many
engineering applications. Pressure induced shocks in reactor vessels, failure of metallic
armor by projectile impact, blast loading in an aircraft are few examples. Although quasi-
static fracture is fairly well understood theoretically as well as experimentally [2, 3], many
issues still remain unresolved in the area of dynamic fracture of materials in general and
heterogeneous materials in particular. A dynamic fracture event can be classified into
mode-I or mixed-mode type depending on whether a crack propagates in the direction of
initial crack orientation or not. Figure 1.3 shows a schematic of different modes of fracture.
Depending on the type of loading and the way in which crack flanks move with respect
to each other, three different fracture modes can be identified. They are mode-I (opening
mode), mode-II (sliding mode or in-plane shearing mode) and mode-III (tearing mode or
3
out-of-plane shearing mode). A combination of any of these modes is referred to as a
mixed-mode problem. In the current work, the problems involve a combination of mode-I
Figure 1.3: Schematic illustrating modes of fracture
and mode-II loading of a crack. Therefore the word ?mixed-mode? henceforth refers to a
combination of mode-I and mode-II fractures.
Asfar asmode-I fractureis concerned, it isgenerally accepted that rapid crack initiation
and propagation are governed by an equality between the dynamically evaluated crack
driving force and the resistance of a material for crack extension [2, 4]. Consequently, the
mode-I dynamic fracture criterion is expresses as
KdI(a(t),t,P) = KD(v), (1.1)
where a(t) is the time dependent crack length and v is the crack speed. The dynamic stress
intensity factor KdI measures the strength of the near tip fields which drive crack propaga-
tion. The right hand side of the equation, KD is the so-called dynamic fracture toughness
which is identified as a material property. This forms the basis for mode-I dynamic fracture
4
mechanics. Extensive research has been conducted over the past two decades in order to
experimentally measure KD and to determine whether it indeed is a material property.
Practical problems, however, often belong to a mixed-mode type because advanced
materials often fail in a mixed-mode fashion. Physical mechanisms governing the dy-
namic mixed-mode fracture are not fully understood, especially regarding crack curving
and branching. Based on observations from quasi-static mixed-mode fracture in materials,
it is assumed that under mixed-mode loading, crack tends to grow according to the lo-
cal mode-I conditions (KII = 0 criterion or Maximum Tangential Stress (MTS) criterion).
Some common examples of mixed-mode dynamic fractures are shown in Fig. 1.4.
Figure 1.4: Mixed-mode dynamic fracture evidences, (a) environmentally assisted crack
propagation from leading edge to the inside cooling surface of a gas turbine blade (Courtesy:
Gas Turbine technology) and (b) Concrete damage after missile impact (Courtesy: Dept.
of Civil and Structural Engineering, University of Sheffield, UK.)
1.1 Motivation and Literature Review
A great deal of experimental and numerical research has been reported on mode-I
dynamic fractureof homogeneous and functionally graded materials. However, as mentioned
earlier, practical problems often belong to a mixed-mode type. Further, a mixed-mode
5
dynamic crack propagation in FGM is more complex than in a homogeneous case because
mode-mixity in the former arises not only from geometric and loading conditions, but also
from mechanical property gradients. These introduce both normal and shear tractions
ahead of the crack tip. Therefore mixed-mode dynamic fracture behavior of FGM needs to
be studied. The full-field measurement of crack tip deformations for mixed-mode dynamic
fracture studies is rather demanding due to a combination of spatial and temporal resolution
challenges involved. Consequently, there is hardly any reported work in the literature about
mixed-mode dynamic failure of FGM and structural foams.
In the current work, mixed-mode dynamic crack propagation is studied using two dif-
ferent experimental techniques. An optical interferometer called Coherent Gradient Sensing
(CGS) is used to study mixed-mode dynamic failure of FGM. A digital image correlation
method with high-speed digital imaging technology is developed to study mixed-mode fail-
ure of syntactic foams. Experiments are complemented by finite element simulations of
mixed-mode dynamic failure in FGM. Here a cohesive element formulation is implemented
to study the formation of new surfaces in nonhomogeneous materials.
In the following, the literature review for the current research is provided in three parts.
In the first part, works on failure characterization of FGM is reviewed. In the second part,
development of various full-field optical methods particularly the digital image correlation
method to study fracture are reviewed. In the third part, numerical methods to simulate
mixed-mode crack propagation under dynamic loading are reviewed.
6
1.1.1 Review of failure characterization of FGM
Delale and Erdogan [5] and Eischen [6] have shown that stress intensity factors in non-
homogeneous materials such as FGM are affected by compositional gradients even though
the inverse ?r singularity near the crack tip is preserved as in homogeneous materials. Is-
sues pertaining to fracture mechanics of FGM under static loading have been addressed in
the recent literature. Jin and Batra [7], Gu and Asaro [8] provide quasi-static stress inten-
sity factors for cracks in FGM for different geometry and loading conditions. Konda and
Erdogan [9] have provided expressions for stress intensity factors (SIFs) of a mixed-mode
fracture problem in a FGM. Abanto-Bueno and Lambros [10] have conducted experiments to
study quasi-static mixed-mode crack initiation and growth in FGM. Shukla and coworkers
[11, 12, 13] have reported crack tip stress fields for dynamically growing cracks in function-
ally graded materials for mode-I and mixed-mode loading conditions. They have derived
asymptotic expansions for stresses and displacements in FGM with linear and exponential
variations of elastic modulus. Tippur and his coworkers [14, 15, 16, 17] have addressed
several issues related to mode-I dynamic fracture of FGM. Among the numerical studies re-
lated to FGM, Wang and Nakamura [18] have simulated crack propagation in elastic-plastic
functionally graded materials using cohesive elements. Kim and Paulino [19] have addressed
issues pertaining to crack path trajectories in FGM under mixed-mode and non-proportional
loading conditions.
Ramaswamy et al.[20] have successfully used Coherent Gradient Sensing (CGS) to
study mixed-mode crack tip deformations under static loading using a modified flexural
specimen geometry. The problem of a crack located in a homogeneous material but close
to an interface between two dissimilar linear elastic materials is examined by Lee and
7
Krishnaswamy [21]. Mason et al.[22] have used CGS to map mode-I and mode-II stress
intensity factors in homogeneous polymer sheets under dynamic loading conditions. Prabhu
and Lambros [23] have studied mode-I and mixed-mode crack tip fields in homogeneous
materials by using the methods of CGS and caustics simultaneously.
Among the experimental investigations on mixed-mode fracture of FGM, works of
Butcher et al.[24], Rousseau and Tippur [25] and Marur and Tippur [26] are noteworthy. In
Ref. [24], feasibility of processing glass-filled epoxy beams for mixed-mode static fracture
studies using optical interferometry is demonstrated. The role of material gradation on
crack kinking under static loading conditions is presented in Ref. [25]. The possibility of
using optimally positioned strain-gages near a crack tip undergoing mixed-mode loading to
obtain SIF histories during impact loading is demonstrated in Ref. [26].
1.1.2 Review of optical methods to study fracture
Measuring surface deformations and stresses in real-time duringa transient failure event
such as dynamic crack initiation and growth in opaque materials is quite challenging due to
a combination of spatial and temporal resolution demands involved. One of the very early
efforts in this regard dates back to the work of de Graaf [27]. In this work, photoelastic
measurement was attempted to witness stress waves around a dynamically growing crack in
steel. This method continues to be popular in the study of fast fracture events [28, 29]. In
recent years, a lateral shearing interferometer called Coherent Gradient Sensing (CGS) has
become a tool of choice for studying dynamic fracture problems of opaque solids because of
its robustness and insensitivity to rigid body motions and vibrations [16, 17, 30, 31]. Moir?e
interferometry has also been used in the past to measure in-plane displacement fields in
8
dynamic fracture experiments [32]. The electronic speckle pattern interferometry (ESPI)
and digital speckle photography [33, 34, 35] methods have been found suitable for measuring
both in-plane as well as out-of-plane deformations in opaque solids. The former uses speckle
intensity patterns formed by the illumination of an optically rough surface with coherent
light. An interferogram containing a fringe pattern is formed when speckle images, recorded
before and after deformation, are subtracted digitally or when a photographic filmis exposed
twice is optically filtered. In the latter, the intensity of speckle images is compared digitally
to determine local translations of speckles. This method does not require any reference wave
and incoherent light is sufficient making it simple and easy to use. Using these methods,
vibration measurements have been attempted by employing a high-speed camera in Refs.
[36, 37]. Duffy?s double aperture imaging scheme [39] has been modified by Sirohi et al.[38]
to measure displacement-derivatives using ESPI. Chao, et al.[40] have studied deformations
around a propagating crack using digital image correlation method with the aid of a Cranz-
Schardin filmcamera. In this work, theyhave scanned film recordsobtained fromthe camera
to perform correlation operations between successive images to estimate displacements.
It should be noted here that photoelasticity and interferometric techniques can all
measure surface deformations in real time but they require somewhat elaborate surface
preparation (transferring of gratings in case of moir?e interferometry and preparing a spec-
ularly reflective surface in case of CGS, birefringent coatings in reflection photoelasticity,
etc.). For cellular materials (syntactic foams, polymer metal foams, cellulosic materials,
etc.) such surface preparations are rather challenging and in some cases may not be feasible
at all. In those instances, digital image correlation method with white light illumination
is a very useful tool due to the relative simplicity in this regard. It involves decorating a
9
surface with alternate mists of black and white paints. Recent advances in image process-
ing methodologies and ubiquitous computational capabilities have made it possible to apply
this technique to a variety of applications - in bio-mechanics to measure displacements of
arterial tissues [41, 42], in metal forming to measure deformations during cold rolling [43],
to measure displacements and strains in C/C composites [44], - just to name a few. Early
contributors to the development of the method include Peters and Ranson, [45] and Sutton
and his coworkers [46, 47]. Chen and Chiang [48] have subsequently developed a spectral
domain approach to measure displacements of digitized speckle patterns. A stereo-vision
methodology for measuring 3D displacement fields has also been introduced in recent times
[49, 50].
With the advent of digital high-speed imaging technologies, imaging rates as high as
several millions frames per second can be achieved at a relatively high spatial resolution.
This has opened the possibility of extending digital image correlation (DIC) method to esti-
mate surface displacements and strains for extracting dynamic fracture/damage parameters.
In the current work, the DIC technique is extended to mode-I and mixed-mode dynamic
fracture studies under stress wave loading conditions.
1.1.3 Review of numerical methods to simulate fracture
Numerical modeling of crack growth in a mixed-mode dynamic fracture event is very
challenging. Material nonhomogeneity adds to the complexity in case of FGM. In order to
predict the crack kinking direction in a FGM sample, the numerical scheme should have
the following features built-in. The model should be able to represent continuous spatial
10
variation of material properties and the evolution of crack path must be a natural out-
come of the analysis. There are mainly three types of approaches within the framework
of finite element method to simulate the current problem of mixed-mode crack growth in
nonhomogeneous medium. The first one is automatic moving finite element method with
local re-meshing along the crack path. This approach requires that a user defined crack
increment be provided and relies on one of the mixed-mode fracture criteria for determin-
ing crack growth direction. Ingraffea and co-workers [51] and Nishioka [52] have used this
approach to simulate mixed-mode crack propagation in homogeneous materials. Nishioka
et al.[53] were able to predict crack path of a mixed-mode dynamic fracture experiment us-
ing moving singular finite element method based on Delaunay automatic mesh generation.
Kim and Paulino [19] have used local re-meshing technique to predict the crack path of
the mixed-mode fracture tests conducted by Rousseau and Tippur [25] under static loading
conditions. Recently, Tilbrook et al.[54] have simulated crack propagation in functionally
graded materials under flexural loading. The limitations of this approach are that it requires
robust automatic re-meshing algorithm, elaborate book-keeping system of node numbering
to re-adjust the mesh pattern periodically and a mesh re-zoning procedure for mapping
of the solution fields in the previous mesh onto those in the current mesh. The second
approach is to use cohesive elements whose idea dates back to the work of Dugdale [55]
and Barenblatt [56]. There are two basic types of cohesive zone models in the literature -
intrinsic and extrinsic. The former is characterized by its increasing (hardening) and de-
creasing (softening) portions of a traction-separation law (TSL) whereas the latter has only
the decreasing portion. The intrinsic cohesive formulation was first proposed by Needleman
11
[57]. Numerous other investigators have used this intrinsic type of formulation with differ-
ent shapes of TSL. They are exponential [57, 18, 58], bilinear [59, 60, 61], and trapezoidal
[62, 63] types. Xu and Needleman [58] have conducted mixed-mode dynamic crack growth
simulation in brittle solids using such formulations. Wang and Nakamura [18] have used
exponential TSL to simulate dynamic crack propagation in elastic-plastic graded materials.
Zhang and Paulino [61] have conducted mode-I and mixed-mode dynamic fracture simula-
tions in FGM samples. Madhusudhana and Narasimhan [63] have used trapezoidal TSL to
simulate mixed-mode crack growth in ductile adhesive joints. The extrinsic type of formu-
lation is also used by many researchers [64, 65, 66, 67]. Using extrinsic formulation, Ruiz
et al.[67] have simulated mixed-mode dynamic fracture experiments of Guo and Kobayashi
[32] and captured the experimentally observed crack path and displacement fields. Recently,
Belytschko and co-workers [68, 69] have proposed a new method called extended finite ele-
ment method (X-FEM) to model arbitrary discontinuities in finite element meshes. They
added discontinuous enrichment functions to the finite element approximation to account
for the presence of a crack while preserving the classical displacement variational setting.
This flexibility enables the method to simulate crack growth without re-meshing. Using
this method, Rabczuk et al.[70] have predicted crack path in a notched beam subjected to
asymmetric four-point bending.
Physical mechanisms governing dynamic crack propagation under mixed-mode loading
in FGM are not clearly understood. Observations based on mixed-mode quasi-static fracture
indicate that under mixed-mode loading, cracks tends to grow according to a local mode-I
condition (KII = 0 criterion or MTS criterion). Extension of these methods to mixed-
mode dynamic fracture of FGM requires evaluation of one of the fracture criterion and
12
local re-meshing. However, cohesive elements allow crack initiation and kinking to occur
spontaneously without defining the crack path a priori. Therefore in the current work,
intrinsic cohesive element method with bilinear traction-separation law is used to model
the mixed-mode dynamic crack growth in FGM.
1.2 Objectives
The literature review suggests that there is a need for experimental and computational
techniques to study mixed-mode dynamic fracture of novel heterogeneous material systems.
Since most of the practical problems belong to the mixed-mode type, fracture of such
materials is studied experimentally and numerically in this work. The following are the
primary objectives of the present research:
? Investigate mixed-mode dynamic fracture behavior of FGM under impact loading
conditions using optical interferometry and high-speed photography.
? Extract mixed-mode dynamic stress intensity factor histories using CGS interfero-
grams.
? Study the effects of material gradation on crack path, crack speed and stress intensity
factor histories.
? Develop an experimental method based on digital image correlation (DIC) and high-
speed photography to measure transient surface deformations such as the one associ-
ated with a rapid growth of cracks in materials.
13
? Study mixed-mode dynamic fracture of syntactic foam specimen under impact loading
conditions using the method of DIC with high-speed photography, map full-field in-
plane displacements, strain and extract mixed-mode SIF histories.
? Perform finite element simulations of mixed-mode dynamic crack growth in FGM
using cohesive element formulation.
? Compare the simulation results with experimental observations and explain the failure
process.
1.3 Organization of Dissertation
This dissertation is organized into eight chapters including Introduction. In Chapter
2, the optical interferometric method of CGS is explained. The CGS experimental set-up,
governing equations and experimental procedure are described. The asymptotic expres-
sions for stresses in FGM considering linear variation elastic modulus are presented. The
implementation of these equations into CGS governing equations to extract mixed-mode
SIF from CGS interferograms is described.
In Chapter 3, mixed-mode dynamic fracture of glass-filled epoxy beam samples is ex-
perimentally studied using CGS. Preparation and characterization of FGM samples are
described. The experimental results namely, crack length and crack speed histories and SIF
histories are presented. Using the MTS fracture criterion, the initial crack kink angles are
predicted and compared with the experimentally observed ones.
In Chapter 4, development of DIC method to measure surface deformations and strains
is explained. A three-step digital image correlation technique is formulated for evaluating
crack opening displacements and strains. The calibration and benchmark experiments and
14
their results for a rotating mirror type ultra high-speed digital camera system are presented.
Using this method, surface deformations and strains of a specimen loaded in three-point
bend configuration under static loading conditions are evaluated and compared with the
corresponding finite element simulations.
In Chapter 5, mode-I dynamic fracture of a polymeric sample and mixed-mode dy-
namic fracture of syntactic foam samples are studied using digital image correlation and
high-speed photography. The sample preparation and the experimental procedure are ex-
plained in detail. The crack opening and sliding displacements and crack tip dominant
strain histories are computed from the speckle images. Dynamic stress intensity factors
are extracted by performing over-deterministic least-squares analysis on crack opening and
sliding displacements.
In Chapter 6, the details on finite element modeling of mixed-mode dynamic crack
propagation in FGM is explained. The cohesive element formulation and its implementation
in ABAQUSTM commercial finite element software package under the option of user defined
element (UEL) is described. Using this option, exponential and bilinear type of traction
separation laws are implemented.
Chapter 7 deals with the simulation of mixed-mode dynamic crack growth in function-
ally graded materials. Modeling aspects and application of graded material properties to
the finite element model are explained. Simulations are carried out for two configurations,
crack on the compliant side and crack on the stiffer side of a sample as explained in Chapter
2. The evolution of strain energy, kinetic energy as well as cohesive energies and theT-stress
histories up to crack initiation are also computed.
15
Finally, the main conclusions of this dissertation are summarized in Chapter 8. Few
potential topics for future research are highlighted.
16
Chapter 2
Experimental method of CGS to study dynamic fracture
Inthis chapter, theoptical interferometric method ofCoherent Gradient Sensing(CGS),
used to study mixed-mode dynamic fracture of functionally graded materials, is discussed.
The attractive features of this technique are realtive ease of implementation in conjunction
with high-speed camera and insensitivity to rigid body motions of the sample during a test.
The crack tip fields for a dynamically loaded stationary crack as well as for a propagating
crack are also explained in this chapter. The asymptotic expressions for crack tip stresses
in FGM considering linear variation of elastic modulus are presented. The implementation
of these equations into CGS governing equations to extract mixed-mode SIF from interfer-
ograms is described.
2.1 Optical set-up
CGS measures in-plane gradients of out-of-plane surface displacements (surface slopes)
when used to study opaque solids. A schematic of the optical set-up [30, 71] is shown in Fig.
2.1. The measurement system consisted of an impactor, pulse-laser, CGS interferometer
and a continuous access high-speed camera. The light beam was processed using a CGS
interferometer comprising of a pair of Ronchi gratings (chrome-on-glass gratings) and a
Fourier filtering/imaging lens. An argon-ion laser beam (wavelength ? = 514 nm) was
expanded and collimated into a 50 mm diameter beam and made to illuminate the opaque
specimen with a specularly reflective surface. The reflected object wave front propagates
through two Ronchi gratings separated by a distance ? and undergoes diffraction in several
17
Figure 2.1: Schematic for reflection-mode CGS set-up with a high-speed camera
discrete directions as shown in Fig. 2.1. In this experiment, the principal direction of the
gratings is along X1 direction (in the direction of the crack). This causes the diffraction
to occur in X1-X3 plane resulting in surface slopes ?u3/?X1, where u3(X1,X2) denotes
the out-of-plane displacement component. The filtering lens L spatially filters the field
distribution emerging from the G2 plane and the associated spatial frequency content is
displayed on its back focal plane (filter plane in the schematic). By locating an aperture
around either the ?1-diffraction orders, the information corresponding to the displacement
gradient is obtained on the image plane of the lens.
2.2 Working principle of CGS
The working principle of CGS interferometer is shown schematically in Fig. 2.2. Con-
sider a plane wave reflected-off of the specimen surface. The incident wave is diffracted into
18
Figure 2.2: Working principle of CGS, (a) Diffraction of a collimated beam though two
parallel Ronchi gratings, (b) undeformed object wave front and (c) deformed object wave
front.
several diffraction orders 0, ?1, ?2 ... by the first grating G1. The corresponding complex
amplitude distribution of the diffracted waves are denoted by Eo, E1, E?1 ... Consider only
three diffracted wave fronts E0 and E?1 for the simplicity analysis. That is the gratings are
assumed to have a sinusoidal transmission function. The magnitude of the angle between
the propagation direction of Eo and E?1 is given by grating equation ? = sin?1(?/p), where
? is the wave length of light used and p is the grating pitch. Upon incidence on the second
grating G2, the wave fronts are further diffracted into E0,0, E0,+1, E0,?1, E+1,0, E+1,?1
19
etc. The second subscript in each of these represents the diffraction order after leaving the
grating G2. These wave fronts propagating in distinctly different directions are brought
into focus at spatially separated spots on the back focal plane of the filtering lens L. The
spacing between these diffraction spots is directly proportional to diffraction angle ? or
inversely proportional to grating pitch p. Note from the Fig. 2.2 that the diffracted waves
E0,?1 and E?1,0 are propagating in the same direction (parallel lines) but are displaced (or
sheared) laterally in the X1-direction. These two overlapping but laterally displaced light
beams produce ?1 diffraction spots. Using a filtering aperture placed at the focal plane
of the filtering lens, the diffraction order either +1 or ?1 is allowed to pass through while
blocking all others.
Consider the complex amplitudes of two plane wave fronts E0 = A0eikl1 and E?1 =
A?1eikl2 for analysis. Here A0 and A?1 are amplitudes, l1 and l2 are optical path lengths
of E0 and E?1, respectively, and k = 2?/? is the wave number. The light intensity I for
the undeformed object (see Fig. 2.2(b)) is proportional to,
I = (E0 +E?1)(E0 +E?1)? = A+B cos{k(l1 ?l2)}. (2.1)
The light intensity reaches its maximum value when the following condition for constructive
interference is satisfied:
k(l1 ?l2) = 2N? (N = ?1,?2,...). (2.2)
20
It can be seen from Fig. 2.2(b) that
(l1 ?l2) = ?[1?(cos?)?1], (2.3)
where ? is the diffraction angle of the grating. For small angles, (sin? ? ? = ?/p). Now
expanding the right hand side of Eq. 2.3 using the binomial expansion, we get
(l1 ?l2) = ?[1?(cos?)?1] = ?
bracketleftBig
1?
parenleftBig
1? ?
2
2 +
?4
8 ?...
parenrightBigbracketrightBig
? ?
parenleftBig?2
2
parenrightBig
. (2.4)
Combining Eqs. 2.2 and 2.4, and noting that k = 2?/?, we get
??
2
2 = N?. (2.5)
Equation 2.5 suggests that N is a constant since ?, p and ? are all constants for an un-
deformed object. Therefore, initially the interferometer produces a uniform fringe in the
entire field of observation.
When the specimen deforms, the collimation of the object wave is perturbed or light
rays are deflected relative to the optical axis. For simplicity, consider light ray deflections
only in the X1 ?X3 plane and ? be the angular deflection of a generic ray. Let l?1 and l?2
be the optical paths of Eo and E+1 after deformation (see Fig. 2.2(c)). Now the intensity
of light is given by,
I? = A? +B?cos[k(l?1 ?l?2)], (2.6)
21
where,
(l?1 ?l?2) = ?
bracketleftBig
(cos?)?1 ?(cos(???))?1
bracketrightBig
= ?
bracketleftbiggbraceleftbigg
1? ?
2
2 +???
bracerightbigg
?
braceleftbigg
1? (???)
2
2 +???
bracerightbiggbracketrightbigg
? ?
bracketleftbigg
???+ ?
2
2
bracketrightbigg
. (2.7)
Again, the intensity of light I attains a maximum value when condition for constructive
interference namely,
k(l?1 ?l?2) = 2M? (M = ?1,?2,...), (2.8)
is met. By combining Eqs. 2.2, 2.7 and 2.8, we can get
???? = (M ?N)? (2.9)
or,
?= (N ?M) ??? = m ???/p = mp?, m = 0,?1,?2,... (2.10)
where m = (N ?M). For reflection mode CGS, ? = ?(?s)/?X1, where ?s = 2w is the
optical path change due to the applied stress with w being the out-of-plane displacement
component. For plane stress conditions, we have
w ? ??B2E (?x +?y). (2.11)
22
Therefore, the governing equation for reflection-mode CGS with plane stress assumption
becomes
?w
?X1 ?
?
?X1
bracketleftbigg??B
2E
parenleftBig
?x +?y
parenrightBigbracketrightbigg
= m p2?. (2.12)
2.3 Experimental procedure
The specimen was initially rested on two soft blocks of putty to preclude any interaction
from the supports (anvils) while the specimen undergoes stress wave loading. The top
edge of the specimen was taped with a thin adhesively backed copper strip. During the
experiment, a pneumatically operated cylindrical steel hammer with a hemispherical tip
was launched towards the specimen (velocity 5 m/sec). During its descent, a velocity flag
(of width ? 6.4 mm), attached to the impactor, first triggered a photo-detector to open a
capping shutter located in front of the high-speed camera allowing light to reach its internal
cavity. When the impactor-head contacted the adhesive backed copper tape affixed to the
top edge of the specimen, it closed an electrical circuit initiating a series of laser pulses for
a duration corresponding to a single sweep of the laser beam on a stationary photographic
film (Kodak TMAX-400) track. The light entering the camera was reflected-off of a spinning
three-facet mirror mounted on a turbine shaft driven using compressed air. The reflected
light beam was swept on the film held in the film track as discrete images. At the end
of that period, the capping shutter was closed to prevent over-writing on the film. In the
current experiments, the laser pulse was repeated every 5 ?s (200,000 frames per second)
with a pulse width (exposure time) of 40 ns and a total recording duration of approximately
320 ?s. With these settings, roughly 70 images were recorded on the photographic film.
23
The light beam reflected off the specimen surface carries information about local surface
deformations. The resulting fringes represent surface slopes in the initial crack direction.
2.4 Extraction of stress intensity factors from interferograms
In Eq. 2.12, ?x and ?y are the Cartesian stress components. By using chain rule of
differentiation, one can transform quantities from X1 ?X2 coordinates to r?? coordinate
system. The quantity ?x +?y is an invariant under coordinate transformation. Hence,
?w
?X1 =
?w
?r
?r
?X1 +
?w
??
??
?X1, (2.13)
where r2 = X21 +X22, ? = tan?1(X2/X1), X1 = r cos?,X2 = r sin?. Further, ?r/?X1 =
X1/r = cos?, ??/?X1 = ?X2/r2 = ?sin?/r. By expressing (?x + ?y) in Eq. 2.12
using asymptotic expansion for crack tip stresses, in-plane gradients of the out-of-plane
displacement become [30, 72]
?w(t)
?X1 =
??B
2E
bracketleftBigg ?summationdisplay
N=1
CN(t)
parenleftbiggN
2 ?1
parenrightbigg
r(N2 ?2) cos
bracketleftbiggparenleftbiggN
2 ?2
parenrightbigg
?
bracketrightbiggbracketrightBigg
= Mp2?, (2.14)
?w(t)
?X2 =
??B
2E
bracketleftBigg ?summationdisplay
N=1
CN(t)
parenleftbiggN
2 ?1
parenrightbigg
r(N2 ?2) sin
bracketleftbiggparenleftbiggN
2 ?2
parenrightbigg
?
bracketrightbiggbracketrightBigg
= Mp2?, (2.15)
for the crack opening mode (mode-I) and
?w(t)
?X1 =
??B
2E
bracketleftBigg ?summationdisplay
N=1
DN(t)
parenleftbiggN
2 ?1
parenrightbigg
r(N2 ?2) sin
bracketleftbiggparenleftbiggN
2 ?2
parenrightbigg
?
bracketrightbiggbracketrightBigg
= Mp2?, (2.16)
?w(t)
?X2 =
??B
2E
bracketleftBigg ?summationdisplay
N=1
DN(t)
parenleftbiggN
2 ?1
parenrightbigg
r(N2 ?2) cos
bracketleftbiggparenleftbiggN
2 ?2
parenrightbigg
?
bracketrightbiggbracketrightBigg
= Mp2?, (2.17)
24
for the in-plane shear mode (mode-II). In the above (r, ?) are the polar coordinates centered
at the crack tip. In Eqs. (2.14)-(2.17), CN and DN are coefficients for mode-I and mode-II,
respectively. The above equations can be used for a dynamically loaded stationary crack
by making an implicit assumption that inertial effects enter the coefficients while retaining
the functional form of the quasi-static counterpart. Accordingly, the coefficients CN and
DN are represented as functions of time t.
2.4.1 Pre-crack initiation period
Figure 2.3: Global and local crack tip coordinate systems for (a) stationary crack and (b)
propagating crack
Figures. 2.3(a) and (b) show the crack tip coordinate systems followed in this work for
digitizing CGS fringes and to extract mode-I and mode-II stress intensity factors. Consider
the situation prior to crack initiation as shown in Fig. 2.3(a). Here X1 ?X2 is the global
25
coordinate system which is aligned with the local (crack tip) coordinate system. Since the
crack is aligned with the principal direction of the gratings, CGS governing equation for
mixed-mode stress intensity factors is obtained by superposing mode-I and mode-II fields
from Eqs. 2.14 and 2.16:
?w(t)
?X1 =
?B
2E
bracketleftBiggbraceleftbigg ?summationdisplay
N=1
CN(t)
parenleftbiggN
2 ?1
parenrightbigg
r(N2 ?1) cos
bracketleftbiggparenleftbiggN
2 ?2
parenrightbigg
?
bracketrightbiggbracerightbigg
+
braceleftbigg ?summationdisplay
N=1
DN(t)
parenleftbiggN
2 ?1
parenrightbigg
r(N2 ?1) sin
bracketleftbiggparenleftbiggN
2 ?2
parenrightbigg
?
bracketrightbiggbracerightbiggbracketrightBigg
= Mp2?. (2.18)
The coefficients of the terms (with N = 1) in the asymptotic series are related to mode-I
and mode-II stress intensity factors KID(t) and KIID(t), respectively, as
C1(t) = KID(t)
radicalbigg2
?, D1(t) = KIID(t)
radicalbigg2
?. (2.19)
2.4.2 Post-crack initiation period
Since the crack path during growth can be along an arbitrary direction, it is convenient
to define a local (variable) coordinate system (X?1,X?2) which is instantaneously aligned with
the current crack path as shown in Fig. 2.3(b). While digitizing fringes in the post-crack
initiation period, coordinates of the digitized points are transformed from the global system
(X1,X2) to the local rotated system (X?1,X?2). The CGS fringes represent surface slopes
as indicated by Eq. 2.12. Since the crack is at an arbitrary angle, surface slope has to be
resolved along the local coordinates as,
?w
?X1 =
?w
?X?1 cos?+
?w
?X?2 sin?, (2.20)
26
where ? is the crack kink angle as shown in Fig. 2.3(b). By using modified versions of Eqs.
2.14-2.17 and 2.20, for a steadily propagating crack, deformations can be related to optical
measurements as, [74, 12]
??B
2E
bracketleftBigg
? 12r?3/2l
braceleftbigg
f(V;CL,CS)C1(t)cos
parenleftBig3?l
2 +?
parenrightBig
+
g(V;CL,CS)D1(t)sin
parenleftBig3?l
2 +?
parenrightBigbracerightbigg
+
?summationdisplay
N=2
CN(t)
parenleftBigN
2 ?1
parenrightBig
r(
N
2 ?2)l cos
braceleftBig
?+
parenleftBig
2? N2
parenrightBig
?l
bracerightBig
+
?summationdisplay
N=2
DN(t)
parenleftBigN
2 ?1
parenrightBig
r(
N
2 ?2)
l sin
braceleftBig
?+
parenleftBig
2? N2
parenrightBig
?l
bracerightBigbracketrightBigg
= Mp2?. (2.21)
Here f and g are functions associated with the instantaneous crack velocity and (rl,?l)
denote the crack tip polar coordinates associated with a growing crack,
rl =
braceleftBig
(X?1)2 +?2L(X?2)2
bracerightBig1/2
, ?l = tan?1 ?LX
?2
X?1 . (2.22)
For plane stress, the functions f and g are given by [12],
f(V;CL,CS) =
parenleftbigg1+?
1??
parenrightbigg (1 +?2
S)(1??2L)
4?L?S ?(1 +?2S)2, (2.23)
g(V;CL,CS) =
parenleftbigg1+?
1??
parenrightbigg 2?
S(1??2L)
4?L?S ?(1 +?2S)2, (2.24)
where,
?L =
bracketleftbigg
1? ?(1??)2? V2
bracketrightbigg1/2
, ?S =
bracketleftbigg
1? ??V2
bracketrightbigg1/2
. (2.25)
HereV is the crack speed,?and?are the local shear modulusand mass density, respectively.
27
Analyzing fringes in FGM using Eqs. 2.18 and 2.21 implicitly assumes a locally ho-
mogeneous material behavior in the crack tip vicinity. This needs justification since crack
tip stress fields for steadily and transiently growing cracks for nonhomogeneous material
have been made available by Shukla and coworkers [11, 12] in recent years. For a relatively
shallow elastic gradients such as the ones used in the current study, it is shown in an earlier
work [15] that results would not be greatly affected (differences being less than 5 %) by
such an assumption. Further, there are also difficulties associated with utilizing existing
FGM crack tip fields [11] to analyze optical interferograms in the current work. Specifically,
theoretical derivations [11, 12] use spatial variation of elastic modulus and mass density
to have a single nonhomogeneity parameter for an exponential type description or assume
a constant mass density while varying the modulus. However, particulate composites in
general and glass-filled epoxy FGM prepared for this work in particular have significantly
different elastic modulus and mass density variations (2.5 fold (4.0 GPa to 10 GPa) over a
width of 43mm where as mass density variation was 1.5 fold (1175 kg/m3 to 1700 kg/m3)
over the same length) which limit the usage of those reported equations.
The role of crack tip transients namely the rate of change of stress intensity factors
and crack accelerations/decelerations are described in Ref. [12]. Based on the experimental
results (to be described later), these effects were found to be negligible for the current
work. For example, it is noted in Ref. [12] that the rate of change of SIF on out-of-plane
displacements becomes relatively insignificant ifdKID(t)/dtis within1.0?105 MPa m1/2/sec
and in the current work dKID(t)/dt values were an order of magnitude less than this value.
Also out-of-plane displacements are said to be minimally affected if accelerations are less
than 1.0 ? 107 m/sec2. Again, maximum acceleration recorded during this work was an
28
order of magnitude lower. Thus extracting SIF from fringes using steady-state assumptions
is quite reasonable.
While digitizing interferograms, the current crack tip location was identified and the
crack kink angle was evaluated. Simultaneously, around the crack tip, the fringe location
r,? and the fringe order M were also recorded. The collected data was used to perform an
over-deterministic least-squares analysis [73, 30] on Eq. 2.18 or 2.21 depending on whether
the interferogram belonged to pre-initiaiton or post-initiation period to extract KI andKII.
To maintain the accuracy of the digitized data points as well as to exclude the region where
3D-effects [30] dominate, data points in the range (0.3 E2.
Except for this change all other conditions are same for type-(a) and (b) experiments.
Specimens were subjected to mixed-mode loading by impacting at an offset distance (S
40
= 25.4 mm) relative to the initial crack plane. Since specimen dimensions and all other
experimental conditions are same, any difference in stress intensity factor histories, crack
speed histories, and crack path are directly attributable to the compositional gradients.
A few representative fringes are shown in Fig. 3.5 for experiments conducted on two
types of FGM samples as well as a homogeneous sample. In all the Figs. 3.5(a)-(c), the
first two interferograms correspond to pre-initiation period, third image at a time instant
when the crack was about to initiate and the fourth one is in the post-initiation period. The
legends correspond to the time instant at which the image was recorded after impact. The
impact point was located outside the window of observation and hence cannot be seen in
these images. Once the impact occurs, the compressive stress waves originate at the impact
point and travel through the specimen. They reflect from the bottom edge and sides of
the sample as tensile waves and load the crack tip. At about 60 ?s after impact, the crack
tip experiences stresses large enough to exhibit fringes after which a monotonic increase in
the number of fringes around the crack tip was seen. Two important observations can be
made from these images. Firstly, fringes are seen for the case E1 E2 indicating more deformation around the crack tip when the crack is located on
the compliant side. On the other hand, fewer fringes around the crack tip are seen when
it is situated on the stiffer side. Secondly, the crack tip loading is essentially of the mixed-
mode type. That is, one can see clearly tilting of fringe lobes towards the left of the initial
crack indicating the presence of a negative shear component in the beginning. However,
just before crack initiation, fringes tend to become symmetric with respect to the crack
suggesting initiation under a predominantly mode-I condition accompanied by a vanishing
KII.
41
Figure 3.5: Selected CGS interferograms representing contours of ?w/?x in FGM and ho-
mogeneous samples. (The vertical line is at 10 mm from the crack). (a) crack on the
compliant side and (b) crack on the stiffer side (c) homogeneous (Plexiglas) sample.
42
3.4.2 Crack growth and crack speed histories
A number of experiments (typically 3-4) were conducted for both configurations E1 <
E2 and E1 > E2 to ensure repeatability. In the following, results are presented for a
representative experiment of each configuration. Figure 3.6(a) shows crack growth histories
for both experiments. The crack initiates at about 145 ?s when it is situated on the stiffer
side and at about 160 ?s when on the compliant side. This is consistent with the fact that
more crack tip deformation occurs for the case E1 < E2 than E1 > E2 as evidenced by a
relatively large number of fringes in Fig. 3.6(a) than in Fig. 3.6(b). Also, the slope of the
crack growth history curve is steeper for E1 E2. This shows that crack speeds are generally higher when the crack is on the
compliant side of the sample. Once the crack initiates, the normalized crack speed remains
nearly constant until it propagates through the lower half of the specimen. Subsequently,
the crack speed history shows a decreasing trend for both the configurations as the crack
tip approaches the impact point.
3.4.3 Mixed-mode stress intensity factor histories
The stress intensity factors for both cases were extracted as explained previously and
are shown in Fig. 3.7. In this plot, the crack initiation time is identified as (t = 0 ?ti) so
that the positive values correspond to the post-initiation period and the negative ones to
the pre-initiation period. The stress intensity factors were initially computed by considering
the K-dominant term (N = 1 in Eq. 2.18 or 2.21) and up to four higher order terms (N
= 6) sequentially. After analyzing a few experiments, it was found that the K-dominant
solution (N = 1) was inadequate to capture the mixed-mode stress intensity factor histories
throughout the experiment. A two-term (N = 3 in Eq. 2.18) or a three-term (N = 4)
solution was found to be stable and capture the overall fracture behavior. In Fig. 3.7(a)
the mode-I stress intensity factor monotonically increases up to crack initiation. The rate
of increase of in the stress intensity factor in the early stages of crack tip loading is about
3 ? 104 MPa m1/2/s. Following crack initiation, a small dip in KI is seen suggesting a
44
Figure 3.7: Mixed-mode dynamic stress intensity factor histories (impact velocity=5.2
m/sec). (Circles: E1 E2). (Time base is altered such that t?ti = 0
corresponds to crack initiation)
45
sudden release of stored energy from the initial notch tip. After this drop, KI values show
a modest increase in case of E1 E2,
after initiation KI gradually decreases in the observation window. The mode-II SIFs (Fig.
3.7(b)) for both FGM are initially negative and once initiation occurs, KII continues to
be at a small but negative value for E1 < E2 whereas it attains a small positive value for
E1 >E2 within the observation window.
As mentioned earlier, a number of experiments (typically 3-4) were conducted for both
configurations E1 < E2 and E1 > E2 to ensure repeatability. Four fractured samples
from each configuration are shown in Fig. 3.8. A distinctly different crack path can be
seen for these two configurations from Figs. 3.8(a) and (b). Also the repeatability of
crack paths in each configuration for all four specimens can be readily noted. Figure 3.9
shows photographs of the fractured specimens for one representative experiment in each
configuration. The impact point is located on the top edge of each image and the initial
crack tip is at the bottom edge as indicated. The reflective area on each specimen surface is
the region of interest where surface deformations are monitored during experimentation. A
vertical line (on the right side of the crack for FGM samples and on the left side of the crack
for homogeneous sample) seen in these figures is located 10 mm away from the crack tip to
help establish the scale. In Fig. 3.9(a), the crack is on the compliant side (E1 E2) and in Fig. 3.9(c), fractured
Plexiglas specimen is shown. The striking feature in these images is the differences in crack
paths in the lower half of the specimen. For the case E1 E2).
whereas for the case E1 >E2, the crack growth occurs at an initial kink angle of ?? 16o,
(see Fig. 3.9(b)). Subsequent crack growth (say, ? 4 mm beyond the initial growth) in
E1 E2 the crack growth is essentially self-similar following initiation
with a continued growth at an angle of ? 16o with respect to the X1-axis. In the upper half
of the sample, the crack growth is affected by a combination of free-edge and impact point
effects. All the parameters (specimen dimensions, impact velocity, etc.) are same for these
two experiments except for the reversal of compositional grading. Hence, the differences
in the two crack paths are attributable directly to the respective compositional gradations
(elastic as well as fracture toughness gradients). Having seen distinctly different crack paths
for the above two configurations, homogeneous specimens made of Plexiglas were also tested
under similar conditions and the resulting crack path is shown in Fig. 3.9(c). The crack
shows an initial kink angle ? ? 10o which is bounded by the ones observed in case of the
two FGM configurations.
47
Figure 3.9: Photographs showing fractured specimens for (a) FGM with a crack on the
compliant side (E1 < E2), (b) FGM with a crack on the stiffer side (E1 > E2) and (c)
a homogeneous specimen. Impact point is indicated by letter ?I? and initial crack tip by
letter ?C?
48
Figure 3.10: Crack growth behavior in FGM samples under mixed-mode dynamic loading.
(a) Crack growth history, (b) normalized crack speed history. (VR: local Rayleigh wave
speed)
The ratio of in-plane shear stress to normal stress near a crack tip can be quantified
by the mode mixity ? = tan?1(KII/KI). The mode mixity histories for all the three cases
are plotted in Fig. 3.10. A large negative value of ? can be seen at the initial stages after
impact indicating the presence of significant negative in-plane shear component at the crack
tip. But just before crack initiation, ? approaches zero. This suggests that crack initiated
under dominant mode-I conditions in all the three cases. Once the crack initiates, ? remains
slightly negative for the case E1 E2 as shown in Fig.
3.10. Interestingly, for the case of a homogeneous sample, mode mixity is essentially zero
(oscillaions about zero) during propagation suggesting the possibility of crack growth in
FGM occurring under conditions of nonzero KII.
49
3.4.4 Initial crack path prediction
The maximum tangential stress (MTS) criteria introduced by Erdogan and Sih [78] is
found to predict crack kink angles in FGM reasonably well under static loading conditions
for glass-filled epoxy [25] and hence its validity in the current dynamic experiments is
considered next. The MTS criteria states that the crack kinks in the direction of maximum
tangential stress. Thus the crack kink angle ? can be computed uniquely by solving the
equation
KI sin?+KII(3cos??1) = 0. (3.2)
Here KI and KII are mode-I and mode-II stress intensity factors and the kink angle ? is
positive in the counter-clockwise direction in the usual crack tip coordinate system (see
Fig. 2.3(b)). In this work, the crack kink angle at initiation was predicted based on SIF
histories 10 ?s (2 frames) prior to initiation. Thus predicted kink angle was verified by
the observed angle near the initial crack tip. The crack kink angle in experiments was
determined from interferograms using MATLABTM as follows. The images corresponding
to post-crack initiation regime were loaded into the software environment. First, a point
corresponding to the current crack tip was located. Then, a second point was located on the
crack establishing a tangent to the current crack path. By using these two points, horizontal
and vertical components of crack extension were identified and ? was calculated. A third
point was marked on the initial crack tip to continuously track the current crack tip with
respect to the initial crack tip. The crack kink angle thus calculated was also verified by
post-mortem examination of the fractured specimens.
Table 3.2 lists crack kink angles predicted by the MTS criteria using estimated SIFs
from the two interferograms just before crack initiation. The crack initiation time is? 160?s
50
E1 E2
Time (?s) ? Time (?s) ?
150 7.9o 135 17.6o
155 5.8o 140 18.3o
Average 6.8o 17.9o
Table 3.2: Predicted crack kink angle based on estimated SIF data from CGS interferogrmas
before crack initiation
E1 E2
Time (?s) ? Time (?s) ?
165 4.3o 150 16.8o
170 5.0o 155 16.4o
Average 4.6o 16.6o
Table 3.3: Observed crack kink angle from three CGS interferograms just after crack initi-
ation
for E1 E2. It can be seen from Table 3.2 that the average kink
angle is greater for E1 > E2 compared to E1 < E2. This indicates that the crack would
kink more when it is situated on the stiffer side compared to the compliant side of the FGM.
This can be readily verified from Fig. 3.9. The crack propagation in fractured specimens
showing an initial crack growth (? 10 mm) are shown as insets in Figs. 3.9(a) and (b) for
E1 E2, respectively. The observed kink angles are listed in Table 3.4.4 for 5
and 10 ?s after crack initiation. It can be seen that the average crack kink angles predicted
by the MTS criteria agrees reasonably well with the observed ones at the early stages of
crack growth.
51
3.5 Mixed-mode SIF history from FGM crack tip fields
The SIF histories presented in Fig. 3.7 were computed using crack tip stress fields
derived for homogeneous materials. However, in this section, SIFs computed by considering
a 4-term expansion (terms associated with r?1/2,r0,r1/2,r1) of the asymptotic series for
stresses which incorporates the variation of elastic modulus in the sample. The stress
intensity factors, extracted as explained in Section 2.7 for both configurations, are shown
in Fig. 3.11. In this plot, the crack initiation time (ti) is identified as t = 0 so that positive
values correspond to the post-initiation period and negative ones to the pre-initiation period.
It should be noted here that SIF trends are somewhat different compared to the ones in
Fig. 3.7 since they are affected by the non-homogeneity terms ?f and ?f (see Eq. 2.32).
In Fig. 3.11(a), KI increases monotonically up to crack initiation for both configurations.
The value of KI at crack initiation is roughly 1.5 MPa m1/2 for both configurations. After
crack initiation, KI values show an increase in the case of E1 < E2 since the crack grows
into a region with an increasingly higher volume fraction of the filler. However, for the case
of E1 >E2 FGM after initiation, KI gradually decreases in the observation window. This
difference in KI history in the post-initiation region is also confirmed by the finite element
simulations (to be discussed in Chapter 7) where higher energy is absorbed when the crack
is situated on the compliant side than on the stiffer side. The KII (see Fig. 3.11(b)) values
for both FGM configurations are initially negative and once initiation occurs, KII continue
to be a small but negative value for E1 E2 similar to the results reported in Fig. 3.7.
The faithfulness of Eq. 2.34 to represent surface slopes observed in experiments is
also tested. Thus, the synthetic contours generated from Eq. 2.34 are superimposed on
52
CGS interferograms obtained from experiments and are shown in Fig. 3.12. One image
from the pre-initiation and one from the post-initiation time period is considered for both
FGM configurations. It should be noted here that only the lobes behind the crack tip
were digitized while performing over-deterministic analyses. Accordingly, the synthetic
contours (orderN = ?1,?1.5 and ?2) are superimposed on the back lobes of the respective
interferograms. The least-square fit considering a 4-term FGM solution for the crack tip
field shows a reasonably good fit with the optical data.
53
Figure 3.11: Stress intensity factors extracted from CGS interferograms by performing over-
deterministic least-squares analysis on difference formulation of CGS governing equation
formulated by using crack tip stress fields obtained for FGM with linear elastic modulus
variation.
54
Figure 3.12: The quality of least-squares fit (plots of synthetic contours generated from
Eq. 2.34 superimposed on collected data points) for (a) E1 E2 (t?ti = ?20 ?s).
55
Chapter 4
The method of Digital Image Correlation
In this chapter, measurement of surface deformations and strains using the method
of digital image correlation (DIC) is explained. A three-step approach is developed during
this research for evaluating crack-tip displacements and strains from random speckle images.
In the first step, a 2D cross-correlation coefficient is computed to obtain initial estimates
of full-field in-plane displacements. In the second step, an iterative technique based on
nonlinear least-squares minimization is implemented to refine the estimated displacements
from the first step. In the third step, displacements are smoothed and strains are computed.
Next, the experimental set-up and test procedures for measuring transient surface de-
formations near a rapidly growing crack using a rotating mirror type ultra high-speed digital
camera is detailed. Since the current work is the first of its kind using a newly introduced
multi-channel high-speed digital camera system, calibration tests are conducted to estimate
and correct misalignments between different optical channels. A series of benchmark ex-
periments including intensity variability test, translation test and rotation tests are also
conducted and the accuracy of measured displacements and strains are reported.
4.1 The approach
In the digital image correlation technique, random speckle patterns on specimen surface
are monitored during a fracture event. These patterns, one before and one after the defor-
mation, are acquired, digitized, and stored. Then a sub-image in the undeformed image is
chosen and its location in the deformed image is sought. Once the location of a sub-image
56
in the deformed image is found, the local displacements can be readily quantified. In the
current work, a three-step approach is developed in a MATLABTM [79] environment to
estimate planar displacements and hence strains.
4.1.1 Initial estimation of displacements (Step-1
In the first step, 2D cross-correlation is performed between two selected sub-images.
The peak of the correlation function was detected to a sub-pixel accuracy (1/16th of a
pixel) by bicubic interpolation. This process is repeated for the entire image to get full-field
in-plane displacements. The method is briefly explained in the following. Consider two sub-
images, f(x,y) from undeformed image and g(x,y) from the deformed image (Fig. 4.1(a)).
Note that g(x,y) can be approximated as a shifted copy of f(x,y) with some random noise
?(x,y). That is, g(x,y) = f(x?u,y?v) +?(x,y), where u and v denote displacements.
The cross-correlation can now be performed in the frequency domain as [48],
P(?x,?y) = F(?x,?y)G
?(?x,?y)
|F(?x,?y)G(?x,?y)|1??p ? |F(?x,?y)|
2?pe{j2pi(u?x+v?y)} (4.1)
where F and G are Fourier transforms of f(x,y) and g(x,y), respectively, (?x,?y) denote
the frequency domain variables and ?p is a constant which can be varied from 0 to 1.
By performing Fourier transform of the function P(?x,?y), a distinct peak in the second
Fourier domain can be obtained as
G(kx,ky) =
integraldisplayintegraldisplay
P(?x,?y)e[?j2pi(?xkx+?yky)]d?xd?y
=
integraldisplayintegraldisplay
|F(?x,?y)|2?pe[?j2pi{?x(kx?u)+?y(ky?v)}]d?xd?y
= G?p(kx ?u,ky ?v), (4.2)
57
where (kx,ky) are frequency domain coordinates of the second Fourier domain. Determining
u and v displacements of a sub-image then reduces to detecting the spatial location of the
peak of the impulse function G?p accurately (Fig. 4.1(b)). The quality of the signal peak
(impulse function) depends on the chosen value of the exponent ?p. If ?p = 0 is chosen, it
represents an ideal case (existence of zero noise in the images), and the spectrum relation
of Eq. (4.1) becomes a pure phase field and the response will degenerate into a Dirac-delta
function located at (u,v). But in reality, due to the presence of noise, the signal peak is often
suppressed. Therefore, it is necessary to make P(?x,?y) a halo-weighted complex spectrum
rather than a pure phase field by choosing ?p to be greater than zero. A systematic study
about the selection of ?p and its effects on measured displacements can be found in [48]. In
the current work, ?p = 0.25 is adopted. This choice ensures good signal-to-noise-ratio and
the probability of a distinct peak appearing at (u,v) is maximized.
Figure 4.1: (a)Undeformed and deformed sub-images chosen from images before and after
deformation, respectively and (b) typical plot of impulse response G?p(kx ? u,ky ? v)
generated from cross-correlation between two sub-images.
58
4.1.2 Refining displacements (Step-2)
In this step, an iterative approach is used to minimize the 2D correlation coefficient by
using a nonlinear optimization technique. Theu and v displacements obtained in step-1 are
used as initial guess values for the iterative scheme. The correlation coefficient is defined as
[47],
s
parenleftbigg
u,v,?u?x,?u?y,?v?x,?v?y
parenrightbigg
= 1?
summationdisplay
i,j
bracketleftBig
F(xi,yi)? ?F
bracketrightBigbracketleftBig
G(x?i,y?i )? ?G
bracketrightBig
summationdisplay
i,j
bracketleftBig
(F(xi,yi)? ?F)2
bracketrightBigsummationdisplay
i,j
bracketleftBig
(G(x?i,y?i)? ?G)2
bracketrightBig. (4.3)
Here F(xi,yi) is the pixel intensity or the gray scale value at a point (xi,yi) in the unde-
formed image andG(x?i,y?i) is the gray scale value at a point (x?i,y?i ) in the deformed image.
The symbols ?F and ?G are the mean values of intensity matrices F and G, respectively (Fig.
4.1). The coordinates or grid points (xi,yi) and (x?i,y?i ) are related by deformation between
the two images. If the motion occurs in a plane perpendicular to the optical axis of the
camera, then the relation between (xi,yi) and (x?i,y?i) can be approximated by a 2D affine
transformation,
x? =x+u+ ?u?x?x+ ?u?y?y,
y? = y+v+ ?v?x?x+ ?v?y?y. (4.4)
Here u and v are translations of the center of the sub-image in X- and Y- directions,
respectively. The distances from the center of the sub-image to a generic point (x,y) are
denoted by ?x and ?y. Thus, the correlation coefficient s is a function of displacement
components (u,v) and displacement gradients (?u/?x, ?u/?y, ?v/?x, ?v/?y). Therefore,
59
a search has to be performed for optimum values of displacements and their gradients
such that s is minimized. In the current work, the Newton-Raphson method which uses
line search and BFGS (Broyden, Fletcher, Goldfarb and Shanno) algorithm to update an
inverse Hessian matrix is employed [80]. This method is applied in two phases. In the
first phase, minimization is done in only a two variable (u,v) space by using the initial
estimates from Step-1. In the second phase, minimization is carried out in a six variable
space (displacements and displacement gradients) by using values for (u,v) from Step-1
and zeros for all the gradients. It should be noted here that estimation of displacements is
accurate if the minimization is done in a six variable space rather than in a two variable
space. However, the gradients obtained are quite noisy (especially when gradients are small
as in the present work).
4.1.3 Smoothing of displacements and estimation of strains (Step-3)
The displacement gradients obtained during the correlation process represent average
values for each subsetand they tend to benoisy. Therefore it isnecessary to applysmoothing
algorithms to (u,v) fields in order to extract strains. There are a number of methods
available in the literature to smooth the data [81, 82]. The one employed here uses an
unbiased optimum smoothing parameter based on the noise level present in the displacement
field. It should be noted here that displacements are discontinuous across the crack. A
generic smoothing method tends to smooth displacements across the crack faces and hence
strain concentration effects near the crack tip will be interpreted inaccurately. Therefore a
smoothing method which allows discontinuity of displacements across the crack faces was
introduced. A regularized restoration filter [83] with a second order fit was employed for
60
this purpose. This method minimizes the functional,
?(f) = bardblg?Hfbardbl2 +?sbardblLfbardbl2, (4.5)
wheref is the displacement field to be restored andgis the noisy displacement field obtained
from DIC method, both arranged in a single column format. In Eq. (4.5), H is a Point
Spread Function (PSF) of a degradation model. The objective in the current work was to
remove the random noise in order to restore/smooth displacement fields. Therefore, H was
assumed as an identity matrix. The Laplacian operator (?2/?x2 +?2/?y2) is denoted by L
in Eq. (4.5). Here ?s is a smoothing parameter selected on the basis of the noise present
in the displacement filed. The operation bardbl.bardbl denotes l2-norm of a vector. Now, Eq. (4.5)
can be written as,
?(f) = (g?Hf)T(g?Hf)+?sfTLTLf. (4.6)
The above functional is minimized by differentiating ?(f) with respect to f and equating
the result to zero. Upon simplification we get,
f = (HTH +?sLTSL)?1Hg, (4.7)
whereS is a diagonal matrix needed if a different amount of smoothing is desired in different
parts of the image. In the current work, however, an identity matrix was used for S with
appropriate diagonal elements set to zero to turn off smoothing across the crack. The
Laplacian (?2/?x2+?2/?y2) = ?2f was found by defining a 3 x 3 discrete Laplacian kernel
61
as,
L(m,n) =
?
??
??
??
?
0 1 0
1 ?4 1
0 1 0
?
??
??
??
?
. (4.8)
The data points corresponding to the crack faces were excluded from this operation so that
displacement discontinuity was preserved along the crack faces. The smoothing parameter
?s was so chosen that data infidelity satisfies the condition [81, 82],
1
n
nsummationdisplay
j=1
[f(j)?g(j)]2 = ?2, (4.9)
with n being the total number of data points. The quantity ?2 is the variance of the
noise present in the displacement data to be estimated by a calibration process. Once the
displacements are smoothed, strains are obtained by numerical differentiation.
4.2 Static experiments
Before applying DIC technique to study dynamic fracture, a static experiment was con-
ducted so that 2D in-plane displacements and strains evaluated from DIC can be compared
with the ones obtained from finite element simulations. Figure 4.2(a) shows the experimen-
tal set-up used. The specimen coated with a random speckle pattern was illuminated by two
light sources. A Nikon D100 digital camera with an objective lens (Nikkor 28-300 mm) was
used to image the specimen surface. A bellows extension with sliding arrangement was used
in between the camera and the lens in order to control the optical zoom. The specimen was
loaded in three point-bend configuration (Fig. 4.2(b)) in a INSTRON 4465 loading frame.
The actual photograph of the set-up is shown in Fig. 4.3. A central region of 45?30 mm of
62
Figure 4.2: (a) Schematic of the experimental set-up for static experiment, (b) specimen
and loading details
the specimen surface was imaged. The images recorded at load levels of 50 lbs and 300 lbs
were correlated. The selection of the image at 50 lbs as reference image helped to preclude
some of the initial rigid body displacements close to zero loads entering the analysis. The
resolution of the image was 3000 ?2000 pixels. The sub-image size chosen for correlation
was 32?32 pixels. The magnification used was such that one pixel represented 15 ?m on
the specimen surface.
The in-plane displacement results obtained from step-1, 2 and 3 are shown in Fig. 4.4.
The sub-image size chosen was 50?50 pixels. The displacements obtained from step-1 look
quite noisy in the figure, nevertheless, they serve as good guess values for step-2. Figure
4.5 shows few representative results from the static test. The full-filed u-displacement and
63
Figure 4.3: Photograph of the static experimental set-up
?xx-strain from DIC are shown in Figs. 4.5(a) and (b), whereas same results from FEA
are shown in Figs. 4.5(c) and (d). The qualitative similarity between experimental and
finite element results can be noted from these figures. The u-displacement and ?xx-strain
values along the section AA and BBwere collected and plotted in Figs. 4.5(e) and (f). A
close agreement (within ? 7.2%) between DIC and finite element results can be seen for
u-displacement in Fig. 4.5(e). The ?xx values obtained from experiments also agree with
the ones from FEA but with a greater uncertainty (within 15%). This is expected since
generally strains computed from DIC are less accurate when compared to displacements
as a result of numerical differentiation. In these static experiments, displacements were
resolved to an accuracy of 3 % of a pixel (0.45 ?m) and the strain accuracy was about 108
??.
64
X (mm)
Y (mm)
Step?1, ux
0 10 20 30 400
5
10
15
20
25
X (mm)
Y (mm)
Step?1, uy
0 10 20 30 400
5
10
15
20
25
X (mm)
Y (mm)
Step?2, ux
0 10 20 30 400
5
10
15
20
25
X (mm)
Y (mm)
Step?2, uy
0 10 20 30 400
5
10
15
20
25
X (mm)
Y (mm)
Step?3, ux
0 10 20 30 400
5
10
15
20
25
X (mm)
Y (mm)
Step?3, uy
0 10 20 30 400
5
10
15
20
25
Figure 4.4: In-plane displacements obtained from Step-1, 2 and 3 of the image correlation
process. The interval between contours is 7 ?m.
65
Figure 4.5: Static experimental results. (a) and (c) u-displacement (mm) from DIC and
FEA, (b) and (d) ?xx (?-strain) from DIC and FEA, (e) and (f) u-displacement and ?xx-
strain at section AA and BB. Rigid body displacements have been subtracted out both in
(a) and (c)
66
4.3 Dynamic experimental set-up
A schematic of the experimental set-up used in this study is shown in Fig. 4.6. It
Figure 4.6: Schematic of the dynamic experimental set-up
consisted of a Instron-Dynatup 9250-HV drop-tower for impact loading the specimen and
a Cordin 550 ultra-high-speed digital camera (with a 28-300 mm macro lens) for capturing
the images in real-time. The drop-tower had an instrumented tup for recording the impact
force history and a pair of anvils for recording support reaction histories. The set-up also
consisted of a delay/pulse generator to generate a trigger pulse when the tup contacts the
specimen. Since all the images were recorded during the event lasting over a hundred micro
seconds, the set-up used two high-energy flash lamps, triggered by the camera, to illuminate
the specimen. The set-up also utilized two computers, one to record the tup force and anvil
67
reaction histories (5 MHz acquisition rate) and the other to record the images. The actual
photograph of the dynamic set-up is shown in Fig. 4.7.
Figure 4.7: Photograph of the dynamic experimental set-up
The high-speed camera uses a combination of CCD based imaging technology and
high-speed rotating mirror optical system. It can capture images up to 2 million frames
per second at a resolution of 1K x 1K pixels per image. It has 32 independent CCD image
sensors positioned radially around a rotating mirror which sweeps light over these sensors
(Fig. 4.8). Each sensor is illuminated by a separate optical relay. Thus small misalignments
between images are to be expected. (The effect of these parameters on displacement results
are discussed in Sections 4.4 and 4.5). These misalignments preclude the possibility of
image correlation between two images recorded by different CCD sensors. However, the
above artifacts are absent between two images if they are captured by the same CCD
sensor at two different time instants close to one another.
68
Therefore, the following approach was adopted. Prior to impacting the specimen, a set
of 32 images of the specimen were recorded at a desired framing rate (200,000 frames per
second in this work). While keeping all the camera settings (CCD gain, flash lamp duration,
framing rate, trigger delay, etc.) same, next set of images, this time triggered by the impact
event, were captured. For every image in the deformed set, there is a corresponding image
in the undeformed set. That is, if an image in the deformed set was recorded by say sensor
#10, then the image recorded by the same sensor #10 in the undeformed set was chosen
for image correlation. By doing this, the optical path could be maintained same for the two
images under consideration and the only source of error now becomes the CCD noise which
is in the range of 4 to 6 gray levels in an 8-bit intensity image (Section 4.5.1). In order
to get meaningful results, it is essential that no extraneous camera movements occur while
recording a set of images and during the time-interval between the two sets of images. This
was achieved by triggering the camera electronically.
4.4 High-speed camera calibration
As noted earlier, in the high-speed digital camera, different geometrical distortions are
present in the images. This is because light travels through different optical paths (relays)
before reaching individual CCD sensors, as shown in Fig. 4.8. In this work, the specimen
was located at approximately 270 mm away from the objective lens. The field lens was
about 620 mm away from the specimen. The image at the field lens was then relayed
through various optical elements before being recorded by a sensor. For the camera system,
four main types of misalignments in the images can be identified.
69
Figure 4.8: Optical schematic of cordin-550 camera: M1,M2,M3,M4,M5 are mirrors; R1
andR2 are relay lenses;r1,r2,???r32 are relay lenses for CCDs;c1,c2,???c32 are CCD sensors
70
? Focusing errors between frames: This aberration can be minimized by careful align-
ment of each optical path but cannot be entirely eliminated. However, since two
images one before deformation and one after deformation, recorded from the same
camera were correlated in the current work, this error does not affect measurements.
? Translation between two images: The images could have an in-plane (X- and Y- di-
rections) relative translation of 5-7 pixels (out of 1000?1000 pixel image). Since the
evaluation of fracture parameters depends primarily on locating the crack tip, trans-
lation of the whole image is not detrimental to the accuracy of results. However, these
translations between frames were estimated accurately by calibration. Subsequently,
the images were aligned to get good registration of one frame relative to the next.
? In-plane rotation between two images: The individual camera images could also have
relative rotation (a maximum of 0.18o between the frames). This rotation was esti-
mated accurately in the calibration experiment and then the frames were aligned with
respect to each other.
? Perspective effect: In order to minimize errors due to perspective effects, camera needs
to be located sufficiently far away from the specimen and the images must be recorded
using higher F# numbers. In the current experiment, the field lens was situated 620
mm away from the specimen and with an F# of 5.6.
A calibration experiment was performed to quantify the above imperfections. The objective
here was to estimate the correction parameters to be used later on for aligning each optical
channel relative to a reference. A template with 5 x 5 array of targets (dark circles) was
printed on a white glossy background and affixed to a flat surface. The camera was focused
71
on the template and flash lamps were adjusted to illuminate the targets uniformly. A set
of 32 pictures of the template were recorded at 200,000 frames per second. Figure 4.9
shows a corresponding image recorded by one of the sensors. It should be noted here that
calibration of the camera is needed only once before the actual fracture experiments. The
various recording parameters (distance between the lens and the specimen, framing rate,
magnification, etc) subsequently need to be maintained same between the calibration and
real experiments.
Figure 4.9: (a) Image of the 5 ? 5 dot pattern template used for calibration experiment
and (b) Inverted binary image of the template in order to find the control points
In the calibration experiment, distortions were corrected using a two-step process. In
the first step, various correction factors were estimated between the images. To do this, one
image was chosen as the base image and the distortions of all other 31 images (called input
images) with respect to the base image were estimated. The base image and one of the
input images were considered and histogram equalization was carried out on them. Then
these images were converted into binary images by performing a thresholding operation
72
followed by an intensity inversion operation to get white circles with a black background
(see Fig. 4.9(b)). The location of the center of each circle was estimated for both the base
and input images. These center locations (also called control points) were further fine-tuned
for the input image by performing normalized cross-correlation operation locally [84]. This
operation matches the template in the neighborhood of a control point in the base image
with that of an input image and fine-tunes the location of the control point of the input
image. This process is repeated for all other 31 images and the control points for every base
image-input image pair were stored.
In the second step, correction was applied to the real images recorded in an experiment
based on the transform inferred from the control points. Here it should be emphasized that
no histogram equalization and thresholding operations were performed on real images. The
input image was transformed with respect to the base image by a linear conformal mapping
transformation. In this transformation, shape of the input image was unchanged, but the
image was deformed by some combination of translation, rotation, and scaling. That is,
by doing this, straight lines remained straight and parallel lines remained parallel. The
transformation used is given by,
?
??
?
xi
yi
?
??
? =
?
??
?
a1 b1
?b1 a1
?
??
?
?
??
?
xb
yb
?
??
?+
?
??
?
ao
bo
?
??
?, (4.10)
where (xi,yi) and (xb,yb) are coordinates of the input image and the base image, respec-
tively. Also, (ao,bo) represent translations in the X- and Y- directions, respectively. The
stretch and rotation are denoted by a1 and b1. Since there are four unknowns in Eq. (4.10),
two pairs of control points are sufficient to find these unknowns. Since there are 25 pairs
73
of control points (5?5 array of circles), the unknowns were determined in this work in an
over-deterministic least-squares sense. The transformation structure generated from each
pair of control points is listed in Table 4.1. Here the image from camera # 09 was chosen as
the base image and misalignments of all other images with respect to this image are listed.
It can be noted from this table that there is horizontal and vertical misalignment between
the successive images. The horizontal movement is in the range of 0 to 9 pixels where as the
vertical movement is in the range of 0 to 4 pixels. Rotation between the frames is within
0.003 radians. As noted earlier, for every experiment, two sets of images were recorded,
one set before impact loading and another set after. The above transformation was applied
for both the sets. That is, if a control points pair came from analyzing images of sensors
1 and 10 (1 being the base image and 10 being the input image) of the reference image
set, then the transformation was also applied to the images captured by sensors 1 and 10
in the undeformed set as well as the deformed set. The two images from the same camera
were then correlated (that is, between the images captured by sensor 10 of the undeformed
set and the deformed set). Again it should be noted here that since the same transforma-
tions are applied to both the undeformed and deformed images, they would not influence
measured deformations but improve the quality of sequential displaying (or animation) of
images helpful in visualizing the failure process.
4.5 Benchmark experiments for high-speed camera
In view of the presence of distortions/misalignments in the camera system, it becomes
important to assess the camera performance to measure transient deformations in a dynamic
74
a1 b1 (radians) ao (pixels) bo (pixels)
Camera 00 0.9966 0.0003 1.3468 1.9695
Camera 01 0.9997 0.0004 -0.4052 -1.0066
Camera 02 0.9986 0.0003 0.7136 0.8484
Camera 03 1.0006 0.0002 -1.7187 0.0524
Camera 04 0.9986 -0.0009 0.5622 -0.1669
Camera 05 1.0001 -0.0003 -1.1137 -0.3886
Camera 06 1.0002 0.0007 -2.2279 -0.3266
Camera 07 1.004 0.0029 -6.5381 2.656
Camera 08 1.007 0.0056 -3.7658 -1.05
Camera 09 1.0000 0.0000 0.0000 0.0000
Camera 10 0.9983 -0.0019 0.0058 -0.9471
Camera 11 0.9995 -0.0012 1.6056 -1.9433
Camera 12 0.9946 -0.0003 8.1177 -0.6218
Camera 13 1.0003 0.0002 0.318 -0.2026
Camera 14 0.998 -0.0015 2.2551 -0.7557
Camera 15 0.9961 0.0005 2.6524 3.1431
Camera 16 0.9939 -0.0017 9.4083 1.9072
Camera 17 0.9977 -0.0005 4.6258 -0.7484
Camera 18 0.9984 -0.0003 5.4465 -1.8866
Camera 19 0.9969 -0.0013 4.5083 -1.7703
Camera 20 0.999 -0.0008 3.8725 -1.873
Camera 21 1.0000 -0.0009 3.2238 -3.2697
Camera 22 0.9993 -0.0011 4.8201 -2.9866
Camera 23 1.0018 -0.0008 3.2343 -3.8285
Camera 24 0.9998 0.0006 3.7571 -1.9882
Camera 25 1.0017 -0.0012 1.4998 -4.764
Camera 26 0.999 0.0001 1.7903 -1.4227
Camera 27 0.9976 -0.0012 4.97 -0.3825
Camera 28 0.9999 -0.0007 4.1851 -2.8842
Camera 29 0.9984 -0.0014 4.5328 -2.8868
Camera 30 1.0002 -0.0019 4.1973 -3.5948
Camera 31 0.9995 -0.0016 3.2673 -2.8311
Table 4.1: Alignment differences between individual optical channels of Cordin-550 camera;
Stretch, rotation and translations of different images with respect to the image taken by
camera # 09
75
test. Therefore certain benchmark tests - image intensity variability test, translation text
and rotation tests were conducted.
4.5.1 Intensity variability test
In the current work, 8-bit (0 to 255 gray levels) images were captured and analyzed.
The CCD noise in these images was first estimated. The CCD noise in an acquired image
depends on the value of CCD gain that can be pre-set in the camera on a scale of 0 to 1000
before conducting an experiment. For all the experiments reported in this work, this value
was set in the range of 500 to 550. A value of 700 becomes an upper limit since it results
in saturation of a few pixels in the acquired images and hence was avoided. Thus, two
sets of 32 images were acquired at framing rates of 200,000 and 50,000 frames per second
in total darkness (with the lens cap on). All the images in these two sets had their pixels
representing the gray scale values in the range of 0 to 8. Thus, it can be said that lower 3
bits in an 8-bit image represents CCD noise and the intensity represented by the remaining
5 bits can be faithfully measured. Figure 4.10 shows the mean and standard deviations of
intensity values of all pixels (1 million pixels in a 1000x1000 pixel image) in various images
captured in darkness. It can be seen from this figure that all the images have their mean
intensity values in the range 6 to 8 with a relatively narrow spread (standard deviation in
the range 2 to 4).
As already mentioned, in the current work, transient deformations are estimated by
performing image correlation between two images acquired from the same CCD sensor,
one before the impact and another after the impact. Therefore it is important to know
the intensity variations between two images recorded by the same CCD sensor at different
76
Figure 4.10: Mean and standard deviations of intensity values of images acquired in total
darkness(with lens cap on). Images were recorded at 50,000 frames persecond in experiment
1 and at 200,000 frames per second in experiment 2.
times. To this end, five sets of 32 images of a stationary sample, decorated with random
speckle pattern, were acquired at 200,000 frames per second. The gray scale values at a
few randomly chosen pixels were stored (same set of random pixels were chosen from all
the images). The intensity values at a particular pixel from all the five images acquired
by the same CCD sensor was examined. This variation is shown in Table 4.2 for all the
32 CCD sensors. Significant difference in intensity value at a pixel is observed between
images acquired by different CCD sensors. More importantly, a very small variation in
gray scale value at a pixel from images acquired by the same CCD sensor can be seen.
Thus, the standard deviation values are in the range 2 to 6 gray levels for most of the CCD
sensors (apparently this is in the same range as the standard deviation values observed
for the images recorded in total darkness, see Fig. 4.10). This demonstrates that between
an undeformed and deformed image registered during an actual experiment, there are no
77
Camera # Set 1 Set 2 Set 3 Set 4 Set 5 Mean St. dev
00 121 116 123 120 122 120.4 2.7
01 106 102 104 108 102 104.4 2.61
02 119 120 125 116 120 120 3.24
03 93 95 111 109 102 102 8.06
04 122 125 126 116 120 121.8 4.02
05 97 102 106 105 105 103 3.67
06 106 97 108 112 103 105.2 5.63
07 79 80 74 82 74 77.8 3.63
08 84 81 84 89 84 84.4 2.88
09 123 118 128 129 129 125.4 4.83
10 111 105 110 114 116 111.2 4.21
11 118 112 111 110 117 113.6 3.65
12 82 88 76 82 82 82 4.24
13 117 115 115 114 117 115.6 1.34
14 88 98 93 87 93 91.8 4.44
15 94 96 93 92 91 93.2 1.92
16 77 73 73 77 72 74.4 2.41
17 63 60 59 65 63 62 2.45
18 97 93 92 93 98 94.6 2.7
19 76 66 71 72 72 71.4 3.58
20 69 70 60 71 67 67.4 4.39
21 87 86 93 95 85 89.2 4.49
22 114 109 113 115 110 112.2 2.59
23 82 80 79 82 76 79.8 2.49
24 92 89 93 96 84 90.8 4.55
25 124 119 120 130 122 123 4.36
26 122 120 126 117 123 121.6 3.36
27 78 83 86 79 78 80.8 3.56
28 76 74 73 77 70 74 2.74
29 95 93 92 91 96 93.4 2.07
30 73 71 70 71 69 70.8 1.48
31 95 91 95 96 88 93 3.39
Table 4.2: Gray scale values at a particular pixel in five repeated sets of images of speckle
pattern acquired at 200,000 frames second. Note the repeatability of the gray scale values
between different sets of images.
78
abrupt light intensity variations apart from the random CCD noise. This is a rather subtle
but important point to note. Also it is unique to this type of high-speed camera system
where one can perform image correlation between two images acquired from the same CCD
sensor to obtain highly accurate displacements.
4.5.2 Translation test
In this experiment, a specimen (decorated with random b/w speckles) was mounted
on a 3D-translation stage, as shown in Fig. 4.11. A series of known displacements were
Figure 4.11: Experimental set-up for conducting ranslation test for high-speed digital cam-
era
imposed in the X- and Y- directions separately and the images were captured. The mean
and standard deviations of the displacement fields were computed and compared with the
applied displacements. Also, a small out-of-plane (Z-direction) displacement of 30 ?2?m
79
1 was applied to the sample and a set of images were captured. The objectives of these
translation tests were as follows:
? To estimate noise levels in the measured in-plane displacement fields (or, to determine
the smallest in-plane displacement that can be measured reliably from the camera
system).
? To compare displacement fields obtained by the 32 individual cameras when they are
used to measure the same applied displacement.
? To determine the effect of out-of-plane displacement on the accuracy of measured
in-plane displacements (or, to address the issue of whether the accuracy of in-plane
displacements is affected if the sample undergoes a small out-of-plane deformation
during an experiment).
? To determine the effect of variation in the working distance (D) distance between the
sample and the objective lens (see Fig. 4.11), on the quality of the measured in-plane
displacements.
The details of translation tests are given in Table 4.3. Totally six sets of 32 images were
recorded in each configuration. In Configuration-1, the objective lens of the camera was
400 mm away from the sample. The first set of 32 images of the undeformed sample makes
Set-1. In Set-2 and Set-3, images were recorded after applying 60 ?2?m of translation in
the X-direction and 60 ?2?m translation in the Y-direction, respectively. This is typically
1This is typically the amount of out-of-plane displacement that occurs in the vicinity of a crack tip in
an experiment conducted in this work. For example, in Ref. [17] one can see roughly 7-9 interferometric
(CGS) fringes near the crack tip over a distance of ?10 mm. Since these fringes represent surface slopes and
the resolution of the set-up was ? 0.015o/fringe, one can estimate the out-of-plane displacement around the
crack tip to be ?23 ?m.
80
the deformation level observed in the current tests. In Set-4 and Set-5, specimen was
translated by 300 ?2?m in the X- and 300 ?2?m the Y-direction, respectively. These
represent the amount of rigid body displacements expected in the dynamic tests. In Set-
6, images were recorded after applying 30 ?m translation in the Z-direction. The same
exercise was repeated for Configuration-2 where the camera was kept twice as close as in
Configuration-1. It should be noted here that all these translations were applied manually
by micrometers in a xyz-translation stage.
Configuration 1 Configuration 2
Working distance (D) = 400 mm Working distance (D) = 200 mm
Magnification = 35.6 ?m/pixel Magnification = 27 ?m/pixel
Set 1 Undeformed Set 1 Deformed
Set 2 X-translation = 60?2?m Set 2 X-translation = 60?2?m
Set 3 Y-translation = 60?2?m Set 3 Y-translation = 60?2?m
Set 4 X-translation = 300?2?m Set 4 X-translation = 300?2?m
Set 5 Y-translation = 300?2?m Set 5 Y-translation = 300?2?m
Set 6 Z-translation = 30?2?m Set 6 Z-translation = 30?2?m
Table 4.3: Details of translation tests: Six sets of 32 images were recorded in each configu-
ration. In Configuration-2, the camera was kept twice as close as in Configuration-1.
The full-field displacements were computed for all these tests. The sub-image size
chosen was 30 ? 30 pixels which gave 32 ? 32 = 1024 data points for a 1K ? 1K image.
The results shown in Figs. 4.12(a) and (b) are: (i) the mean and standard deviations of
u-displacement (between images of Set-1 and Set-2 of Configuration-1, ?solid circle?), (ii)
the mean and standard deviations of v-displacement (between images of Set-2 and Set-3
of Configuration-1, ?solid square?), (iii) mean and standard deviations of u-displacement
(between images of Set-1 and Set-2 of Configuration-2, ?solid triangle?) and (iv) mean and
81
standard deviations of v-displacement (between images of Set-2 and Set-3 of Configuration-
2, ?solid diamond?). Tests were conducted in Configuration-2 to examine the effect of the
Figure 4.12: Translation test results for D=400 mm and 200 mm (see Fig. 4.8. (a) mean
and (b) standard deviations of u- and v- displacement fields for X- and Y- translations of
? 60 ?2?m (c) mean and (d) standard deviations of u- and v-displacement fields for X-
and Y- translations of 300 ?2?m. Magnification = 35.6 ?m/pixel for D=400 mm and 27
?m/pixel for D=200 mm.
working distance ?D ?on the accuracy of the measured displacements. It should be noted
that magnification in Configuration-1 was 35.6 ?m/pixel and in Configuration-2 it was 27
?m/pixel on the image plane. In view of this, a constant value of ?60 ?m (1.6 pixels)
was expected for ?solid circles ?and ?solid squares ?of Figs. 4.12(a). Similarly a value of
82
?2.2 pixels was expected for ?solid triangles?and ?solid diamonds?of Fig. 4.12(a). It can be
seen from Fig. 4.12(a), that a constant displacement is being measured in all 32 cameras
within a scatter band of 0.2 pixels. Also the error levels in the displacements measured by
each of the cameras fall within the range of 2 to 6% of a pixel (see Fig. 4.12(b)). Similar
results for X- and Y- translations of 300 ?m (between images of Sets-1 and -4 and Sets-4
and -5 of Configurations-1 and 2) are presented in Figs. 4.12(c) and (d). By comparing
the results in Figs. 4.12(b) and (d), it is evident that there is no significant difference in
standard deviations of the measured displacement fields. This implies that displacements
of 8 to 10 pixels can be measured rather easily to an accuracy of less than 6% of a pixel.
By comparing the values of ?solid triangles?and ?solid circles?in Fig. 4.12(b) or (d), it is
clear that the quality of measured displacements is not affected significantly if the working
distance D is reduced by a factor of 2. Finally, Fig. 4.13 shows the effect of the imposed
uniform out-of-plane displacement on the measured in-plane displacements. The mean and
standard deviations of u- and v-displacements that occurred between the images of Set-5
and Set-6 (Z-translation of 30 ?m) are shown in Figs. 4.13(a) and (b). Again, both u- and
v-displacements are within 0.1 pixels in Fig. 4.13(a). The standard deviation of u- and v-
displacements is in the range 1 to 6% of a pixel as evident from Fig. 4.13(b).
It is instructive to study in-plane strain fields estimated from measured displacements
in these translation tests. To this end, the displacements were smoothed by the restoration
method explained in Section 4.1.3 and strains were obtained by performing numerical dif-
ferentiation. The mean and standard deviations of ?xx and ?yy strains are presented for two
tests in Table 4.4. These tests correspond to X- and Y-translations of 60?2?m (between
images of Set-1 and Set-3 of configuration 1) and of 300?2?m (between images of Set-1 and
83
Figure 4.13: Translation test results for D = 400 mm (see Fig. 4.8) and out-of-plane
displacement (w) =30 ?m. (a) mean and (b) standard deviation of u- and v-displacement
field.
Set-5 of configuration 1). Since the applied displacement was a rigid translation, ideally,
zero strains for all the images are expected. However, numerical differentiation amplifies the
noise in the displacements which manifests itself in the estimated strains. The mean values
of strains were in the range 0 to 150 ?? in both the experiments. The standard deviations
of strains were in the range 0 to 300 ?? for various individual cameras. Interestingly, the
mean and standard deviations were not affected by the amount of translation imposed. The
implication of this on an actual experiment is that a relatively large rigid body motion can
be accommodated without sacrificing accuracy in the measured displacements and strains.
84
Xtrans, Ytrans = 60?2?m Xtrans, Ytrans = 60?2?m
?xx (??) ?yy (??) ?xx (??) ?yy (??)
Camera # Mean Std Mean Std Mean std Mean Std
00 -20.7 203.1 259.7 217.1 0.9 209 2.2 229.1
01 -65 88.8 190.6 258.6 -35.3 240.6 15.4 244.6
02 -200.4 92.6 124.7 200.3 0.8 248.7 78.8 253.7
03 -210.1 139.9 30.2 209.4 13.8 341 106.4 230.6
04 -119.2 69.9 153.6 184.8 1.6 250 -9.7 241.2
05 -83.8 102.9 53.9 101.8 -52.1 167.8 10.6 335.6
06 -68.3 147 68.5 125.2 -178.7 159.5 51.6 254
07 -129.6 186 27.4 134.3 -65.5 196.4 85 277.9
08 0.4 94.6 100.3 157.4 -19.7 211.4 79.1 195.7
09 -44.1 118.4 71.8 153.8 11.2 219.7 117.1 206.1
10 -27.8 112.3 106.7 159.3 -0.7 215.2 45.4 237.1
11 -2.4 121.5 52.2 204.2 -1.5 219.5 83.5 214.2
12 -14.6 128.3 125.2 203 -42.3 201.2 -5.2 215.4
13 -7.5 140.6 36.8 284.6 -13.8 193.2 79.4 195.7
14 11.4 135.3 73.5 211.2 -50.5 235.9 3.8 202.2
15 11.5 84.6 14.9 315.3 -16.7 216.6 102.1 58.5
16 -6.5 103 168.8 308.1 -75.7 247 50.7 92.3
17 -15 113.3 3.7 319.3 -8 239.4 16.8 168.6
18 55.8 164.8 60.4 258.6 7.2 222.4 88.5 243.9
19 -7.7 126.4 192 215.1 -2.5 242.2 -33.5 242.8
20 32.4 117.1 88.8 242.2 -69.5 214.9 22.3 248.6
21 15.2 104.8 188.7 148.3 -17.3 288.1 55.5 234.6
22 28 150.3 195.5 167.1 -39.9 246.3 -34.1 247.7
23 4.5 161.9 175.8 142.5 -25.1 210.8 62.4 253.9
24 -171.4 107.7 42 217.1 15.3 224 137.2 311.6
25 -153.3 99.8 42.5 167.1 43.4 231.5 -4.9 287.1
26 -93.7 133.1 25.8 218.2 -59.5 230.3 4.1 187.9
27 -161.2 123.7 -24.1 197 -18.7 218.9 22.5 320.9
28 -133.6 92.5 162.1 126.6 10.8 237 93.1 248.6
29 -174.7 84 133.2 210.5 -16.4 203.2 -69.3 167.4
30 -93.2 149.4 -17.3 191.9 -15.5 194.5 72.2 219.8
31 -10.1 123.7 255.4 214 -3.2 200.9 53.7 221.6
Table 4.4: Mean and standard deviations of in-plane strain fields estimated from measured
displacements in translation test
85
4.5.3 Rotation test
The objectives of the rotation test were (a) to estimate the accuracy with which a
pure rotation can be measured using the high-speed camera system, (b) to compare the
performance of different individual cameras when they are used to measure same applied
rotation and (c) to examine whether the applied rotation produces any spurious strain. In
the rotation test, a specimen decorated with random b/w speckle pattern was mounted on
a rotation stage. Two sets of 32 images were recorded at 200,000 frames per second, one set
before, and another set after imposing a rotation of 0.32o (0.0056 radians). The full-field
displacements were computed between these two sets of images. The sub-image size chosen
was 30?30 pixels so that the displacements were available on a grid of 32?32 = 1024 points.
These displacements were smoothed by the restoration method explained in Section 4.1.3.
The cross derivative terms ?u/?y and ?v/?x were computed by numerical differentiation.
The rotation ?xy was then evaluated as,
?xy = 12
parenleftbigg?u
?y ?
?v
?x
parenrightbigg
. (4.11)
Figure 4.14 shows a representative full-field plot of ?xy from one pair of the images.
The estimated values are close to the applied value of rotation everywhere in the image
except near the boundaries. This is expected because the errors displacement derivatives
(strains and rotations) get magnified near the boundaries due to the so-called edge effects.
Next, mean and standard deviations of?xy were computed for each image (while computing
these quantities for a 32 ? 32 matrix, three rows and three columns of data points were
excluded near the border of the image in view of the presence of larger errors at these
86
Figure 4.14: Estimated full-field quantity?xy from one pair of the images taken from camera
# 1 in a rotation experiment (Imposed rotation = 0.0056?0.00035 radians).
locations). Figure 4.15 shows the mean and standard deviation of estimated rotations and
strains from this test. It can be seen from Fig. 4.15(a) and (c) that an applied rotation of
0.0056 ?0.00035 radians is measured by all the individual cameras with in an error band
of ?5?10?4 radians (10 % error). A rigid rotation imposed to the sample will not produce
any strains. Consequently, zero strains are expected from this test. However, the mean
values of strain fields obtained were within 100 ?? and standard deviations were up to 300
??. Thus, it can be said that in an actual experiment involving some rigid rotation of the
sample, spurious values of in-plane strains in the range to 100 to 300 ?? can be expected.
4.6 Flash lamp light characteristics
One of the main assumptions while performing image correlation between two images
recorded by a high-speed digital camera system is that illumination of the specimen is
87
Figure 4.15: Results from rotation test (applied rotation = 0.0056?0.00035 radians). (a)
mean and (c) standard deviation of rotation field estimated from image correlation. (b)
mean (d) standard deviations of in-plane strains estimated (ideally these strains need to
be zeros).
uniform and stable (spatially as well as temporally) and also repeatable. Spatial stability
means the light intensity needs to be uniform in the region of interest (in the current work
31x31 mm2 area). By temporal stability, we expect the light intensity to remain constant
during the event of interest (? 150 ?s in this work during which all the 32 images were
acquired). The repeatability is also important since the light intensity need to remain same
between any two successive experiments. To be more specific, undeformed set of 32 images
and the deformed set of 32 images need to be exposed by the same light intensity.
88
The light intensity from the flash lamps ramps up initially, dwells for a while and then
decreases. In the current work, dwell time was set to 9 ms. In order to test flash lamp
characteristics, a photo detector having 1 mm2 sensing area was placed at a location where
the sample was placed in the real experiment. The voltage signal proportional to the light
intensity was recorded with time using a high-speed data acquisition system at a sampling
rate of 1 MHz. This exercise was repeated twice in order to check for the repeatability of the
flash lamp characteristics. Subsequently, the photo detector was moved to a new location
in the plane perpendicular to the optical axis of the camera by 25 mm inch and the output
was again recorded twice. The voltage signal registered by the photo detector is plotted in
Fig. 4(b). An excellent repeatability in the light intensity can be seen. A dwell time of 9
Figure 4.16: Photo detector output proportional to flash lamp light intensity, A1, A2 and
B1, B2 are two repeated acquisitions when photodiode was placed one inch away in the
plane perpendicular to optical axis of the camera.
ms can be seen from all these plots. While conducting the real experiment, all the images
89
were captured during this dwell time by appropriately triggering the event. Thus, the flash
lamps were found to be stable and no hot spots were found in the captured images.
90
Chapter 5
Dynamic fracture studies using DIC method
In this chapter, the method of DIC to study dynamic fracture in polymeric materi-
als is discussed. Mode-I and mixed-mode dynamic fracture of epoxy and syntactic foam
samples are examined using this method. The sample preparation and the experimental
procedure are explained in detail. The crack opening and sliding displacements and domi-
nant strain histories are computed from speckle images. The dynamic stress intensity factor
histories are extracted by performing over-deterministic least-squares analysis on estimated
displacements.
5.1 Sample preparation
Edge cracked epoxy and syntactic foam samples were prepared for conducting mode-I
and mixed-mode dynamic fracture experiments, respectively, using DIC method. Epoxy
samples were made from bisphenol-A resin and an amine based hardener in the ratio 100:38
for mode-I experiments. For mixed-mode tests, samples were made by mixing 25% (by
volume) of hollow microballoons in the epoxy matrix. The microballoons used in this study
were commercially available hollow glass spheres of mean diameter of ? 60 ?m and wall
thickness ? 600 nm. The microballoons were carefully stirred into the epoxy resin while
avoiding air bubbles and agglomeration. Stirring was continued until the mixture showed a
tendency to gel and then poured into molds. This helped to eliminate any buoyancy induced
floatation of microballoons during the cure cycle. The elastic modulus and Poisson?s ratio,
measured ultrasonically, were 4.1 GPa and 0.34 for epoxy samples and 3.02 GPa and 0.34
91
for syntactic foam samples, respectively [24]. While casting the mixture, a sharp razor
blade was inserted into the mold. After the sample was cured, a sharp ?edge notch ?was left
behind in the specimen. Further details about this method of introducing a sharp crack
into specimens can be found in Ref. [77]. Finally, the specimen was machined into beams
of height 50 mm with a crack of 10 mm length (a/W = 0.2) as shown in Fig. 5.1(a) and
(b). A random speckle pattern was created on the specimen surface by spray-painting with
black and white paints alternatively.
Figure 5.1: Specimen configuration for (a) mixed-mode test of syntactic foam and (b)
mode-I test of epoxy. Impactor force history and support reaction histories recorded by
Instron Dynatup 9250 HV drop tower for (c) mixed-mode experiment and (d) mode-I
experiment. The sample dimensions are a = 10 mm, W = 50 mm, S = 25.4 mm, L = 200
mm, B = 8.75 mm, Impact velocities, V1 = 4.5 m/sec and V2 = 4.0 m/sec.
92
5.2 Experimental procedure
Since the event to be captured is highly transient in nature, the total recording time
is rather small and hence the high-speed camera was synchronized with the event. The
sequence of events in a typical experiment was as follows: The specimen was initially rested
on two instrumented supports/anvils of the drop tower. The camera, anchored firmly to the
ground, was focused on 31?31 mm2 region of the sample in the crack tip vicinity (see Fig.
5.1 (a) and (b)). A set of 32 pictures of the stationary sample were recorded at the desired
framing rate (225,000 frames per second in mode-I and 200,000 frames per second in mixed-
mode experiment). Next, an impactor was launched at a desired velocity (?4.5 m/sec for
mode-I ?4.0 m/sec for mixed-mode) towards the sample. As soon as the tup contacted
an adhesive backed copper tape affixed to the top of the specimen, a signal was generated
by a pulse/delay generator to trigger the camera. The camera also sent a separate trigger
signal to the high intensity flash lamps. A trigger delay was pre-set in the camera system to
capture images ? 75 ?s after the initial impact. This time delay provides sufficient time for
the high intensity flash lamps to ramp up to their full intensity level and provide uniform
illumination during recording. Since the measurable deformations around the crack tip for
the first 85 ?s are relatively small, there was no significant loss of information during this
period. A total of 32 images were recorded with 4.44 ?s (5 ?s in mixed-mode tests) interval
between images for a total duration of 142 ?s (160 ?s in mixed-mode tests). Once the
experiment was complete, the captured images were stored in the computer. Just before
the impact occurs, the velocity of the tup was recorded by the drop-tower system. Also
recorded were the tup force and support reaction histories.
93
The tup force and support reaction histories are shown in Fig. 5.1(c) and (d) for
asymmetric impact of syntactic foam specimen and symmetric impact of epoxy specimen,
respectively. In these plots, multiple contacts between the specimen and the tup can be
evidenced by the occurance of more than one peaks. Since complete fracture of the specimen
had taken place before 240 ?s in both the experiments, only the first peak of the impact
force history is of relevance here. From Fig. 5.1(d) it can be seen that supports feel the
impact force only after ? 400 ?s for the mode-I test. In case of the mixed-mode test, since
left support is closer to the impact point, the impact force records start earlier than the
one for the right support (see Fig. 5.1(c)). Also, it should be noted that anvils register a
noticeable impact force after 220 ?s by which time the crack had propagated through half
the sample width (see Fig. 5.4). Thus, the reactions from the anvils do not play any role
in the crack initiation and initial growth. Accordingly, for both the tests, the samples were
subsequently modeled as a free-free beam in finite element simulations with impact resisted
by the specimen inertia.
5.3 Finite element simulations
5.3.1 Mixed-mode fracture of syntactic foam
Elasto-dynamic finite element simulations were conducted up to crack initiation under
plane stress conditions. The finite element mesh used is shown in Fig. 5.2(a) along with
the force boundary conditions at the impact point. Experimentally determined material
properties (elastic modulus = 3.1 GPa, Poisson?s ratio 0.34 and mass density = 870 kg/m3)
were used as inputs for the finite element analysis. The numerical model was loaded using
the force history recorded by the instrumented tup (only the first peak of the tup force
94
Figure 5.2: Finite element mesh used for elsto-dynamic finite element analysis of (a) mixed-
mode problem and (b) mode-I problem.
from Fig. 5.1(c) was used). (Before applying, the force history data was interpolated and
smoothed for the following two reasons: (a) The time step of the force history measurement
was larger than the one used in the simulations and (b) The force history recorded by
the tup had experimental noise. Therefore smoothed cubic splines were fitted to the force
history data before applying to the model.) The implicit time integration scheme of the
Newmark ? method with parameters ? = 0.25 and ? = 0.5 and 0.5% damping was adopted
in the simulations.
Instantaneous crack opening and sliding displacements along the crack flanks were used
for extracting mode-I and mode-II stress intensity factors (SIF). Using Williams? asymptotic
expansion for crack opening and sliding displacements, apparent stress intensity factors can
95
be expressed as
[KI(t)]app =
?2?Eu
y(t)|?=?pi
8?r ?KI(t) +C1r,
[KII(t)]app =
?2?Eu
x(t)|?=?pi
8?r ?KII(t)+C2r. (5.1)
Here uy and ux are the crack opening and sliding displacements, respectively, KI and KII
are mode-I and mode-II stress intensity factors, E is the elastic modulus of the material,
C1 and C2 are higher order coefficients. For each time instant, the values of [KI(t)]app and
[KII(t)]app are plotted as a function of r and the extrapolated values KI = limr?0[KI]app
and KII = limr?0[KII]app were identified as the mode-I and mode-II instantaneous stress
intensity factors.
5.3.2 Mode-I fracture of epoxy
The finite element simulation procedures for the mode-I test are same as the mixed-
mode test except the numerical model was loaded using one-half of the force history recorded
by the instrumented tup due to symmetry of the model. Finite element mesh of the half-
model used is shown in Fig. 5.2(b) along with the boundary conditions. Experimentally
determined material properties for this case are (elastic modulus = 4.1 GPa, Poisson?s ratio
0.34 and mass density = 1175 kg/m3). As explained in the previous section, the mode-I
SIF was calculated by regression of crack opening displacements by the formula
[KI(t)]app =
?2?Eu
y(t)|?=?pi
4?r ?KI(t) +Cr. (5.2)
96
The T-stress was also determined using a modified stress difference method [77]. Instanta-
neous normal stress difference (?x ??y) ahead of the crack tip along X1-axis was used to
calculate the non-singular T-stress as
(?x ??y)|r,?=0o ?T +Dr, (5.3)
where D is a higher order coefficient of the asymptotic expansion of crack tip stresses.
5.4 Results - Mixed-mode dynamic fracture of syntactic foam
From each experiment 64 images were available, 32 each from the undeformed and the
deformed sets, each having a resolution of 1000?1000 pixels. Figure 5.3 shows four selected
speckle pattern images from the deformed set of 32 images. The time instant at which the
images were recorded after impact is shown below each image and the current crack tip
location is indicated by an arrow. The position of the crack tip is plotted against time in
Fig. 5.4. It can be seen from this figure that crack initiates at about 175?s. Upon initiation,
the crack rapidly accelerates and subsequently attains at a relatively steady velocity of ?
270 m/s. The magnification used in this experiment was such that the size of a pixel was
equal to 31 ?m on the specimen. A sub-image size of 26?26 pixels was chosen for image
correlation. The 2D in-plane displacements were estimated for all the 32 image-pairs. The
crack opening displacement, uy, and sliding displacement, ux, for the sample images (one
before crack initiation and one after) are shown in Fig. 5.5.
Figures 5.5(a) and (c) show uy- and ux-displacements at 150 ?s after impact and
Figs.5.5(b) and (d) show the corresponding displacement components at t = 220 ?s after
impact. These are smoothed values of displacements (aspects of smoothing were discussed
97
Figure 5.3: Acquired speckle images of 31?31 mm2 region at various times instants. (Crack
tip location is shown by an arrow.)
98
Figure 5.4: Crack growth behavior in syntactic foam sample under mixed-mode dynamic
loading.
in Chapter 4). A significant amount of rigid body displacement component can be seen in
the ux-field (Figs. 5.5(c) and (d)). In the current work, the displacements were resolved to
an accuracy of 2 to 6% of a pixel or 0.6 to 1.8 ?m.
5.4.1 Extraction of stress intensity factors
Both crack opening and sliding displacement fields were used to extract dynamic stress
intensity factors in the current work. The asymptotic expressions for crack tip displacement
fields for a dynamically loaded stationary crack are given by [72],
ux =
Nsummationdisplay
n=1
(KI)n
2?
rn/2?
2?
braceleftbigg
? cos n2?? n2 cos
parenleftBign
2 ?2
parenrightBig
?+
braceleftBign
2 +(?1)
nbracerightBigcos n
2?
bracerightbigg
+
Nsummationdisplay
n=1
(KII)n
2?
rn/2?
2?
braceleftbigg
? sin n2?? n2 sin
parenleftBign
2 ?2
parenrightBig
?+
braceleftBign
2 ?(?1)
nbracerightBigsin n
2?
bracerightbigg
, (5.4)
99
Figure 5.5: Crack opening and sliding displacements (in mm ) for pre- and post-crack
initiation instants. (a) v-displacement and (c) u-displacement before crack initiation (at
t=150 ?s); (b) v-displacement and (d) u-displacement after crack initiation (t=220 ?s).
Crack initiation time ? 175 ?s. (A large rigid body displacement can be seen in (c) an (d)
due to movement of the sample.
100
uy =
Nsummationdisplay
n=1
(KI)n
2?
rn/2?
2?
braceleftbigg
? sin n2?+ n2 sin
parenleftBign
2 ?2
parenrightBig
??
braceleftBign
2 +(?1)
nbracerightBigsinn
2?
bracerightbigg
+
Nsummationdisplay
n=1
(KII)n
2?
rn/2?
2?
braceleftbigg
?? cos n2?? n2 cos
parenleftBign
2 ?2
parenrightBig
?+
braceleftBign
2 ?(?1)
nbracerightBigcos n
2?
bracerightbigg
. (5.5)
In the above equations, ux and uy are crack sliding and opening displacements, (r, ?)
are crack tip polar coordinates, ? is (3 ??)/(1 + ?) for plane stress where ? and ? are
shear modulus and Poisson?s ratio, respectively. The coefficients (KI)n and (KII)n of the
leading terms (n = 1) are the mode-I and mode-II dynamic stress intensity factors (SIF),
respectively. Equations (5.4-5.5) implicitly assume that inertial effects enter the coefficients
while retaining the functional form of the quasi-static crack tip equation. However, once
the crack initiates, asymptotic expressions for crack sliding and opening displacement fields
for a steadily propagating crack are used [85]:
ux =
Nsummationdisplay
n=1
(KI)nBI(C)
2?
radicalbigg2
?(n+1)
braceleftbigg
rn/21 cos n2?1 ?h(n)rn/22 cos n2?2
bracerightbigg
+
Nsummationdisplay
n=1
(KII)nBII(C)
2?
radicalbigg2
?(n+1)
braceleftbigg
rn/21 sinn2?1 +h(?n)rn/22 sin n2?2
bracerightbigg
, (5.6)
uy =
Nsummationdisplay
n=1
(KI)nBI(C)
2?
radicalbigg2
?(n+1)
braceleftbigg
??1rn/21 sin n2?1 + h(n)?
2
rn/22 sin n2?2
bracerightbigg
+
Nsummationdisplay
n=1
(KII)nBII(C)
2?
radicalbigg2
?(n+1)
braceleftbigg
?1rn/21 cos n2?1 + h(?n)?
2
rn/22 cos n2?2
bracerightbigg
, (5.7)
where
rm =
radicalBig
X2 +?2mY2, ?m = tan?1
parenleftbigg?
mY
X
parenrightbigg
m = 1,2
101
?1 =
radicalBigg
1?
parenleftbigg c
CL
parenrightbigg2
, ?2 =
radicalBigg
1?
parenleftbigg c
CS
parenrightbigg2
CL =
radicalBigg
(?+1)?
(??1)?, CS =
radicalbigg?
?, ? =
3??
1 +? for plane stress
h(n) =
?
???
???
2?1?2
1+?22 for odd n
1+?22
2 for even n
h(?n) = h(n+1)
BI(C) = 1 +?
22
D , BII(C) =
2?2
D , D = 4?1?2 ?
parenleftBig
1 +?22
parenrightBig2
. (5.8)
Here (X, Y) and (r, ?) are crack the tip Cartesian and polar coordinates instantaneously
aligned with the current crack tip (see, Fig. 5.5(b)) and c is the speed of the propagating
crack tip, CL and CS are dilatational and shear wave speeds in the material, ? and ? are
shear modulus and Poisson?s ratio, respectively. Again, coefficients (KI)n and (KII)n of
the leading terms are the mode-I and mode-II dynamic stress intensity factors, respectively.
For a mode-I problem uy is the dominant in-plane displacement and hence used for
extracting mode-I SIF history. However, in a mixed-mode problem, both ux and uy dis-
placements can be equally important. It can be thought that crack opening displacement
uy as having mode-I rich information whereas sliding displacement ux as having mode-II
rich information. Therefore, one can use uy to extract KI and ux to extract KII accurately.
On the other hand, one can use either radial (ur) or tangential (u?) displacements (com-
puted by transforming ux and uy) to extract both KI and KII together more accurately
compared to using ux and uy alone. Following Yoneyama et al.[86] who have demonstrated
this recently, in the current work the radial displacement component ur was used to extract
both KI and KII histories.
102
For extracting SIF from displacement data, the current crack tip location was identified
and the Cartesian and polar coordinate systems (X?Y and r??) were established at the
crack tip. A number of data points (usually 100 to 120) were collected in the region around
the crack tip (0.3 < r/B < 1.6) and (?145o E2 case since stiffer material is
located near the impact point. After about 90 ?s, for E1 < E2 (120 ?s for E1 > E2), the
stored strain energy is gradually converted into fracture energy. The energy absorbed by
the cohesive elements is shown in Fig. 7.4(b). Initially a small portion of the total energy
gets stored in the cohesive elements which cause a slow increase of UCE up to 120 ?s. A
sudden change in the slope of UCE curves at about 125 ?s indicates the crack initiation
event after which the fracture energy becomes a major portion ofUCE. More importantly, it
can be observed from this plot that more energy is absorbed throughout the loading history
by the cohesive elements for the case of E1 E2) are shown in Figs. 7.6(b) and (d).
The crack initiation times in simulations are nearly same for both the configurations (129?s
forE1 E2). The similarity in crack paths between the experiments
and simulations can be seen for the initial ? 9 mm of crack growth by comparing figures
7.6(b) with 3.9(a) and 7.6(d) with 3.9(b). When the crack is situated on the compliant side
(E1 E2 and they tend to decrease after initiation
since crack grows into a progressively compliant region. The opposite trend is observed
for the other configuration where lower stresses are seen before crack initiation and they
increase after the initiation. Figure 7.7 shows snapshots of crack opening displacements
at two time instants, one before and one after the crack initiation. Typical crack opening
displacement fields for a mixed-mode problem can be seen from Figs. 7.7(a) and (c). As
expected, prior to crack initiation, larger displacements are seen for the case of E1 < E2
compared to the one withE1 >E2. Upon comparing Figs 7.7(b) and (d), it can be said that
displacements rapidly increase for E1 > E2 configuration when compared to the E1 < E2
configuration since the crack grows into a progressively compliant material in the former.
145
Figure 7.6: Snapshots of ?yy stress field at two different time instants, (a) 120 ?s and (b)
150 ?s for E1 < E2 (crack initiation time = 129 ?s), and (c) 120 ?s and (d) 168 ?s for
E1 >E2 (crack initiation time = 130 ?s).
The crack length history from experiments and simulations are plotted in Fig. 7.8(a)
and (b) against time t?ti, where ti is the time at crack initiation. In simulations, the
crack initiation takes place at approximately 130 ?s for both configurations. This is in
contrast to the experimental results shown in Fig. 7.8(a) where crack initiation time is in
the range 145 ?s to 155 ?s. This difference is attributed to the fact that in experiments, the
initial crack had a finite root radius of ? 150 ?m whereas in finite element simulations, it
was modeled as a sharp crack with zero thickness. Therefore, in experiments considerable
amount of energy had to accumulate at the notch tip before the crack initiated. Further, the
146
Figure 7.7: Snapshots of uy displacement field at two different time instants, (a) 120 ?s
and (b) 150 ?s for E1 E2 (crack initiation time = 130 ?s).
crack propagates at higher speed when it initiates from the compliant side of the sample.
This agrees well with the experiments (higher slope for E1 < E2 in Figs. 7.8(a) and Fig.
7.8(b)). The higher crack speeds lead to greater roughness of the fracture surfaces due to
the formation of micro cracks at the main crack tip and hence greater energy dissipation.
147
Figure 7.8: Crack growth behavior in FGM sample under mixed-mode loading. Absolute
crack length history from (a) experiments and (b) finite element simulations, ti is crack
initiation time (ti = 155 ?s for E1 E2 in experiments, ti ? 130
?s for both E1 E2 in simulations).
7.4.4 T-stress history
In order to understand the marked difference in crack paths for the two configurations,
a measure of in-plane crack tip constraint, T-stress was computed up to crack initiation.
A modified stress difference method [77] was employed where regression of normal stress
difference (?x ??y) ahead of the crack tip was used to find the instantaneous T-stress as
(?x ??y)?=0 = T +Dr, (7.5)
where D is the higher order coefficient associated with r1 term in the asymptotic expansion
of (?x ??y). It can be seen from Fig. 7.9(a) that (?x ??y) has an excellent linearity in
the range where straight line is fit to the computed data. This process was repeated for all
the time steps to get T-stress history of each FGM configurations. The computed T-stress
history is plotted in Fig. 7.9(b) up to crack initiation for both configurations. A larger
148
negative T-stress is observed for the case of E1 < E2. This indicates that crack is likely
to grow in this case in its original direction and has less tendency to kink compared to the
other configuration.
Figure 7.9: (a) Variation of apparent T-stress with crack length at certain time instant
before crack initiation (b) T-stress history up to crack initiation for E1 E2.
149
Chapter 8
Conclusions
Mixed-mode dynamic crack propagation in particle filled composites is investigated ex-
perimentally and numerically. A Coherent Gradient Sensing (CGS) optical interferometer
was used to study mixed-mode dynamic failure of functionally graded materials (FGM).
In this technique, the surface slopes (in-plane gradients of out-of-plane displacement) were
measured in real-time and fracture parameters were extracted subsequently. A digital image
correlation method with high-speed digital imaging technology was also developed to study
mixed-mode dynamic failure of syntactic foams. Here, in-plane displacements were mea-
sured and strains and fracture parameters were estimated from the measured displacements.
Experiments were complemented with finite element simulations of mixed-mode dynamic
failure in FGM. Here, a cohesive element formulation was implemented to study formation
of new surfaces in nonhomogeneous materials.
In the first part, the optical method of Coherent Gradient Sensing (CGS) was used to
investigate the mixed-mode fracture behavior of functionally graded materials. The FGM
samples studied were the ones with a continuously varying volume fraction of ceramic filler
particles in a polymer matrix having edge cracks initially oriented along the gradient and
subjected to impact loading. Mixed-mode loading of the crack was generated by impacting
the samples eccentrically relative to the crack plane. The optical method of CGS and
high-speed photography were used to map transient crack tip deformations before and
after crack initiation. Two configurations, one with a crack on the compliant side of a
graded sheet (E1 < E2) and the second with a crack on the stiff side (E1 > E2), were
150
examined experimentally. The differences in both pre- and post-crack initiation behaviors
were observed in terms of crack path, crack speed and stress intensity factor histories.
Following conclusions were drawn from the study:
? The crack initiates earlier for the case of a crack on the compliant side of the beam
(E1 < E2) compared to the one with a crack on the stiffer side (E1 > E2). Higher
crack speeds were observed in the latter case compared to the former.
? The crack initiation in both the FGM configurations occurred when KII approached
values close to zero. Yet, during crack growth KII remained at a small negative value
when the crack was on the compliant side but maintained a small positive value when
it was on the stiffer side of the FGM sheet. This raises the possibility of a non-zero
KII during mixed-mode dynamic crack growth in FGM.
? The crack paths differed significantly for the two FGM configurations studied. That
is, the crack kinked less when situated on the compliant side compared to the stiffer
side of the FGM sample.
? The initial crack kink angle was predicted for both the configurations using MTS
criteria based on the SIF values just prior to crack initiation. Thus predicted crack
kink angles agree reasonably well with the observed ones during the early stages of
crack growth.
In the second part of this work, the method of digital image correlation was developed to
the study of transient deformations such as the one associated with a rapid growth of cracks
in materials. Edge cracked polymer beams and syntactic foam samples were studied under
low-velocity impact loading conditions. Decorated random speckle patterns in the crack tip
151
vicinity were recorded using an ultra high-speed digital camera at framing rates of 200,000
frames per second. Two sets of images were recorded, one set before impact and another
set after impact. A three-step digital image correlation technique was developed and imple-
mented in a MATLABTM environment for evaluating crack opening/sliding displacements
and the associated strains. In the first step, a 2D cross-correlation coefficient was com-
puted to obtain initial estimates of full-field in-plane displacements. In the second step, an
iterative technique based on nonlinear least-squares minimization was carried out to refine
the estimated displacements from the first step. In the third-step, a regularized restoration
smoothing technique, which smoothes the displacements while allowing for discontinuity of
displacements across the crack faces was developed and strains were computred.
The current work being the first of its kind using a rotating mirror type multi-channel
high-speed digital camera system, calibration tests and procedures were established. A
series of benchmark experiments such as intensity variability test, rigid translation and
rotation tests were conducted and the accuracy of measured displacements and strains are
reported. The accuracy of the measured displacements is in the range 2 to 6 % of a pixel
(0.6 to 1.8 ?m) and that of dominant strain is about 150 to 300 micro strain.
Using the developed methodology, mode-I dynamic fracture of epoxy and mixed-mode
dynamic fracture of syntactic foam samples were studied. The crack opening and sliding
displacements and crack tip dominant strain histories from the time of impact upto com-
plete fracture were computed from the speckle images. The crack length and crack speed
histories were evaluated. The dynamic stress intensity factors were extracted by performing
over-deterministic least-squares analyses on crack opening displacements (in case of mode-I
dynamic test) and radial displacement component (in case of mixed-mode dynmaic test).
152
The accuracy of estimated displacements and strains are reported. The mode-mixity his-
tory in case of mixed-mode dynamic fracture test and the crack tip T-stress history in
case of mode-I dynamic fracture test are also evaluated. The measurements were in very
good agreement with companion finite element results. The current approach seems to be
a powerful method to investigate dynamic failure events in real time.
In order to understand the marked crack path and other observed differences in fracture
parameters between the two FGM configurations from the first part of this research, finite
element simulations were undertaken in the last part of this dissertation. An intrinsic cohe-
sive element method with bilinear traction-separation laws were used to model mixed-mode
dynamic crack growth. A user subroutine was developed and augmented with ABAQUSTM
(Version 6.5) under the ?user-defined element ? (UEL) option to implement the cohesive
elements. The spatial variation of material properties in continuum elements were incor-
porated by conducting a thermal analysis and then applying material properties (elastic
properties, density and crack initiation toughness) as temperature dependant quantities.
The pre-initiation T-stresses were also computed by a modified stress difference method.
The finite element simulations have successfully captured the dominant characteristics
of crack kinking under mixed-mode impact loading conditions. The simulated crack paths
show a greater kink angle when the crack is on the stiffer side of the FGM. The computed
T-stress values prior to crack initiation are more negative when the crack is situated on the
compliant side of the sample indicating a greater likelihood of a crack to grow in its original
direction and has a lower tendency to kink. Also, as in the experiments, higher crack
speeds occur when the crack initiates from the compliant side of the FGM. The computed
energy histories reveal greater energy dissipation throughout the observation window by the
153
cohesive elements for the case of a crack on the compliant side of the FGM. Since higher
crack speeds are accompanied by greater fracture surface roughness due to micro-cracking
in dynamic fracture events, this observation supports experimental observation of higher
crack speed when a crack initiates from the compliant side of the sample.
8.1 Future Directions
The FGM samples used in this research have a shallow gradient (variation in elastic
modulus was ?2.5 fold 43 mm). The specimens with large material gradients need to be
tested experimentally. They are likely to absorb more energy and may delay crack initiation
if the initial crack is oriented appropriately with respect to the property gradients). In this
work, fracture parameters were extracted for mixed-mode dynamic fracture experiments by
measuring transient surface deformations using CGS interferometer. This approach can be
extended to a bimaterial systems where the crack is situated close to an interface or the
crack running into an interface at an arbitrary angle.
The digital image correlation technique combined with ultra high-speed imaging tech-
nology developed in this work promises to be a powerful tool for measuring transient de-
formations. With the rotating miror type high-speed digital camera system, displacements
as small as 2 to 6% of a pixel have been measured. This can be applied to a number
of applications. For example, to understand the dynamic failure characteristics of fiber
reinforced composites, cellular materials subjected to stress wave loading, for developing
fundamental understanding of nonlinear deformation in rubber like materials subjected to
transient loading, to study the damage casued by blast, detonation and shock wave loading
in military applications, to name a few.
154
As a first step, the cohesive element modeling implemented with ABAQUS can be
modified to simulate some interesting problems such as interaction of crack with a cylindri-
cal/spherical inclusions. This has great practical applications in understanding toughening
mechanisms in particle reinforced composites. As a next step, the cohesive element model
can be incorporated with representative volume elements (RVE) for multi-scale modeling of
the failure of novel materials such as syntactic foams. The simulation of crack propagation
using the current implementation of cohesive element model is computationally intensive.
In order to overcome this, an extended finite element method (X-FEM) (which enriches
the regular elements selectively with discontinuous shape functions) can be implemented in
ABAQUS environment.
155
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163
Appendix A
A note on accuracy of strains and time resolved displacements
A.1 A note on accuracy of strains
Since the materials tested using DIC in the current work are epoxy and syntactic foam
made from glass microballoons with epoxy matrix, strains observed in the crack tip vicinity
were relatively small and elastic. The strains presented here were obtained by differentiating
displacements estimated from DIC. Therefore greater errors in the strains when compared
to displacements are to be expected. It has been shown in Section 4.5.2 that the accuracy of
displacements is in the range of 2 to 6% of a pixel. The accuracy of strains can be estimated
by conducting benchmark tests at different stages/levels, as discussed below. The central
requirement of these tests is to generate a known constant strain for the entire image and
then try to estimate it using the computational methodology:
? Generate a synthetic image mathematically (speckle image can be generated by adding
random noise of zero mean and a constant variance to a uniform image). Then apply
a known strain to this image mathematically and measure the same using the image
correlation program.
? Acquire an image of a speckle pattern, then deform it mathematically by applying
uniform strain and estimate the same.
? Acquire two images back to back without any deformation between them and then
deform one of the images mathematically and estimate the strains between them.
164
? Acquire an image, subject the sample to a uniform mechanical load to impart a
constant strain and then estimate that strain.
Out of these four possible approaches, the results of (a) will be highly accurate but may
not be realistic as it cannot be achieved in practice. Even the accuracy of results in (b)
is seldom achieved because the random errors associated while acquiring an image, twice
by a CCD camera, are not modeled in such an exercise. The exercise of type (d) takes
all experimental errors into account but the results will depend on the characteristics of
the mechanical device used as well as inherent experimental complexities due to rotation of
the sample, slip in the grips, etc. Considering all the above, benchmark test of type (c) is
proposed to assess the accuracy of strains in the current work.
Two images of a random speckle pattern were recorded back to back without any
deformation. The full-field (horizontal) displacement data between these two images is
presented in Fig. A.1(a) which has a mean of 0.039 pixels and a standard deviation of 0.0015
pixels (Ideally, these values should be all zeros). One of these images is mathematically
stretched by imposing a strain of 2500 ?? in the horizontal direction and the displacements
were extracted. The resulting linearly varying u-displacement field can be seen from Fig.
A.1(b). This displacement was smoothed using the restoration method explained in Section
4.1.3 and normal strain ?xx was computed. The smoothed displacement field is shown in
Fig. A.1(c) and the strain plot is shown in Fig. A.1(d). It can be seen from Fig. A.1(d)
that ?xx oscillates about 2500 ?? and has a standard deviation of about 142 ??. Thus it can
be said that the strains estimated in this work have errors approximately equal to 142 ??.
A.2 Time resolved displacements
165
Figure A.1: Results of benchmark experiment conducted to estimate the accuracy of dis-
placements and strains. (a) full-field u-displacement between image 1 and image 2 before
deforming image 2 (ideally u-displacement shoud be zero). (b) u-displacement after apply-
ing a constant strain to image 2 but before smoothing, (c) u-displacement after smoothing
and (d) normal strain after stretching image 2 uniformly.
166
t = 85 ?s t = 90 ?s t = 95 ?s
t = 100 ?s t = 105 ?s t = 110 ?s
t = 115 ?s t = 120 ?s t = 125 ?s
t = 130 ?s t = 135 ?s t = 140 ?s
Figure A.2: Time resolved crack opening displacements for image # 1 to 12. Time at which
each image was acquired after impact, is indicated above each figure. The interval between
each contour is 3.5 ?m. 167
t = 145 ?s t = 150 ?s t = 155 ?s
t = 160 ?s t = 165 ?s t = 170 ?s
t = 175 ?s t = 180 ?s t = 185 ?s
t = 190 ?s t = 195 ?s t = 200 ?s
Figure A.3: Time resolved crack opening displacements for image # 13 to 24. Time at
which each image was acquired after impact, is indicated above each figure. The interval
between each contour is 3.5 ?m.
168
t = 205 ?s t = 210 ?s
t = 215 ?s t = 220 ?s
t = 225 ?s t = 230 ?s
t = 235 ?s t = 240 ?s
Figure A.4: Time resolved crack opening displacements for mixed-mode dynamic test, image
# 25 to 32. Time at which each image was acquired after the impact, is indicated above
each figure. The interval between each contour is 3.5 ?m.
169
Appendix B
Computation of stiffness coefficients in tration-separation laws
B.1 Exponential traction-separation law
While deriving expressions for stiffness coefficients, the history dependency needs to
be taken into account. Refering to Fig. B.1, if the history dependancy is not taken into
account and if there is an unloading at point ?C?, then the same traction curve is followed
as during the loading. This implies that to achieve unloading, traction need to be increased
which is not realistic. Thefore unloading path should be linear leading to the origin as
shown in broken line in Fig. B.1. This means that the stiffness matrix need to be different
for loading and unloading part of the traction-separation curve. Thus defining the effective
separation parameter, ? as,
? =
radicalBiggparenleftbigg
?n
?n
parenrightbigg2
+?2c
parenleftbigg?
t
?t
parenrightbigg2
(B.1)
with ?c = (0,?). In the current work, history dependency is taken care by defining a single
history dependent damage parameter ?max as,
?max = max{?(?)|0 ?? ?t} (B.2)
Loading is said to occur when ? = ?max and ?? ? 0, and unloading/reloading when
? ? ?max. Stiffness coefficients for loading are obtained by diferentiating Eqs. 6.25 and
6.26 with respect to ?n and ?t
SS11 = ?Tt??
t
= 2?n?2
t
e(??n?n )
bracketleftbigg
q+
braceleftbiggr?q
r?1
bracerightbigg?
n
?n
bracketrightbiggbracketleftbigg
1? 2?
2t
?2t
bracketrightbigg
e(?
?2t
?2t ), (B.3)
170
Figure B.1: Reversible and irreversible unloading
SS12 = ?Tt??
n
= 2?n?2
t
?t
?n e
(??n?n )e(?
?2t
?2t )
bracketleftbiggbraceleftbiggr?q
r?1
bracerightbiggbraceleftbigg
1? ?n?
n
bracerightbigg
?q
bracketrightbigg
, (B.4)
SS21 = ?Tn??
t
= 2?n?2
t
?t
?n e
(??n?n )e(?
?2t
?2t )
bracketleftbiggbraceleftbigg1?q
r?1
bracerightbiggbraceleftbigg
1? ?n?
n
bracerightbigg
? ?n?
n
bracketrightbigg
. (B.5)
SS22 = ?Tn??
n
= ?n?
n
e(??n?n )
bracketleftbiggbraceleftbigg 1
?n ?
?n
?2n
bracerightbigg
e(?
?2t
?2t ) +
braceleftbigg1?q
r?1
bracerightbiggbraceleftbigg
1?e(?
?2t
?2t )
bracerightbiggbraceleftbigg?
n
?2n ?
1
?n ?
r
?n
bracerightbiggbracketrightbigg
.
(B.6)
The unloading stiffness matrix is calculated as follows. The separations ?n and ?t are
first scaled by a factor ?max/?, the tractions associated with these scaled separations are
computed, and these tractions are scaled back by multiplying them by ?/?max [92]. Thus,
Tun = ??
max
Tn
parenleftbigg?
max
? ?n,
?max
? ?t
parenrightbigg
= ??
max
Tn(??n,??t), (B.7)
Tut = ??
max
Tt
parenleftbigg?
max
? ?n,
?max
? ?t
parenrightbigg
= ??
max
Tt(??n,??t). (B.8)
171
where
??n = ?max? ?n, ??t = ?max? ?t. (B.9)
In the above, Tun and Tut are unloading tractions in normal and tangential directions re-
spectively. The variation of ? is required in order to differentiate Tun and Tut which is given
as,
?2 =
parenleftbigg?
n
?n
parenrightbigg2
+?2c
parenleftbigg?
t
?t
parenrightbigg2
2??? = 2?n?2
n
??n + 2?
2c?t
?2t ??t
?? = ?n??2
n
??n + ?
2c?t
??2t ??t (B.10)
The variation of traction in normal direction is given by,
?(Tun) = 1?
max
Tn(??n,??t)??+ ??
max
?[Tn(??n,??t)]
= 1?
max
Tn??+ ??
max
parenleftbigg?T
n
???n??
?
n +
?Tn
???t ??
?
t
parenrightbigg
= 1?
max
Tn??+ ??
max
bracketleftbigg?T
n
???n
braceleftbigg?
max
? ??n ?
?max?n
?2 ??
bracerightbiggbracketrightbigg
+ ??
max
bracketleftbigg?T
n
???t
braceleftbigg?
max
? ??t ?
?max?t
?2 ??
bracerightbiggbracketrightbigg
. (B.11)
Substituting for ?? and simplifying we get,
?(Tun) =
bracketleftbigg T
n?n
?max??2n +
parenleftbigg
1? ?
2n
?2?2n
parenrightbigg?T
n
???n ?
?n?t
?2?2n
?Tn
???t
bracketrightbigg
??n
+
bracketleftbigg?2
cTn?t
?max??2t +
parenleftbigg
1? ?
2c?2t
?2?2t
parenrightbigg?T
n
???n ?
?2c?n?t
?2?2t
?Tn
???t
bracketrightbigg
??t. (B.12)
172
Similarly the variation of traction in tangential direction can be simplified to,
?(Tut ) =
bracketleftbigg T
t?n
?max??2n +
parenleftbigg
1? ?
2n
?2?2n
parenrightbigg ?T
t
???n ?
?n?t
?2?2n
?Tt
???t
bracketrightbigg
??n
+
bracketleftbigg ?2
cTt?t
?max??2t +
parenleftbigg
1? ?
2c?2t
?2?2t
parenrightbigg?T
t
???t ?
?2c?n?t
?2?2t
?Tt
???n
bracketrightbigg
??t. (B.13)
Finally the stiffness coefficients for unloading are given by [92],
SSu11 = ?T
ut
??t =
bracketleftbigg ?2
cTt?t
?max??2t +
parenleftbigg
1? ?
2c?2t
?2?2t
parenrightbigg?T
t
???t ?
?2c?n?t
?2?2t
?Tt
???n
bracketrightbigg
, (B.14)
SSu12 = ?T
ut
??n =
bracketleftbigg T
t?n
?max??2n +
parenleftbigg
1? ?
2n
?2?2n
parenrightbigg ?T
t
???n ?
?n?t
?2?2n
?Tt
???t
bracketrightbigg
, (B.15)
SSu21 = ?T
un
??t =
bracketleftbigg?2
cTn?t
?max??2t +
parenleftbigg
1? ?
2c?2t
?2?2t
parenrightbigg?T
n
???n ?
?2c?n?t
?2?2t
?Tn
???t
bracketrightbigg
, (B.16)
SSu22 = ?T
un
??n =
bracketleftbigg T
n?n
?max??2n +
parenleftbigg
1? ?
2n
?2?2n
parenrightbigg?T
n
???n ?
?n?t
?2?2n
?Tn
???t
bracketrightbigg
. (B.17)
B.2 Bilinear traction-separation law
Stiffness coefficients can be calculated by differentiating the tractions with respect to
separations from Eq. 6.30 through 6.33. For loading/unloading in the range 0 ????cr,
S11 = ?Tt?u
t
= ?c?
t
Tmax
?cr , S22 =
?Tn
?un =
1
?n
Tmax
?cr , S12 = S21 = 0. (B.18)
for loading in the range ?cr E1.
The Coherent Gradient Sensing (CGS) method was used in conjunction with high-speed
photography [30] in this study to perform real-time measurements of instantaneous surface
deformations around the crack tip. The details about the method are explained in Chapter
2.
The resulting interference fringes for the two cases are shown in Fig. C.2(a) and (b).
For each case, the representative interferograms corresponding to pre- and post-initiation
time instants are included. The crack initiation occured at t = 115 ?s after impact for
177
Figure C.1: (a) Schematic of the FGM specimen, (b) Material properties variation along
the width of the sample and (c) Variation of dynamic crack initiation toughness along the
width of the sample. 178
E1 < E2 and t = 135 ?s after impact for E1 > E2. The legends correspond to the time
instant at which the image was recorded after impact. At earlier times severe concentration
of interference fringes are seen at the impact location (near the top edge) while only a few
fringes are seen at the crack tip (near the bottom edge). With the passage of time crack
tip deformations increase, as evidenced by an increasing number of fringes at the crack tip,
followed by crack initiation and growth. The fringe pattern in each case is symmetric on
either side of the crack, indicating mode-I crack tip deformations.
Figure C.2: Selected CGS interferograms representing contours of ?w/?x in functionally
graded epoxy syntactic foam sheet impact loaded on the edge opposing the crack tip. (The
vertical line is at a distance of 10 mm from the crack.) (a) Crack on the compliant side
E1 E2. Fringe sensitivity ? 0.015o /fringe.
179
C.0.3 Modeling details
The overall view of the finite element mesh is shown in Fig. C.3(a). The dimensions
of the notch (150 ?m root radius) was also modeled in the simulations as can be seen from
Fig. C.3(b). The cohesive elements were inserted along a line in which the crack is allowed
to propagate can be seen from this figure. The smallest element size used in the mesh was
less than the characteristic cohesive length scale (see Section 7.1). The sample was modeled
as a free-free beam and a velocity of 5 m/sec was imposed on the node located at the
impact point. The variations of elastic modulus, E (4.2 GPa to 2.1 GPa) and mass density
? (1175 kg/m3 to 690 kg/m3) were approximated by linear functions and applied to the
model as explained in Section 7.2. The variation of mode-I fracture energy GI computed
from experimentally obtained variation of crack initiation toughness was applied to cohesive
elements as detailed in Section 7.3. The peak stress Tmax was assumed as E(X)/100, and
the values for ?c and ?c are chosen as 1.0.
C.0.4 Finite element results
Snapshots of crack tip normal stresses before and after crack-initiation are shown in
Figs. C.4(a) and (c) for the case of a crack on the compliant side (E1 E2 since the crack is
growing into a microballoon-rich region in this case. The opposite trend is observed in the
other configuration. Instantaneous crack length histories for both cases of monotonically
graded foam sheets are shown in Figs. C.5(a) and (b). The crack initiation occurs earlier
180
Figure C.3: Finite element mesh used for the analysis
in case of the specimen with a crack on the compliant side (E1 < E2) when compared to
the one with a crack on the stiffer side (E1 >E2). The crack initiation times in simulations
are 106 ?s and 127 ?s for E1 < E2 and E1 > E2, respectively against 115 ?s and 135 ?s
observed experimentally. Higher crack speeds were observed for the case of E1 E2, respectively. The corresponding values in simulations are 385 m/s and 278 m/s.
Thus, the trends observed in crack speeds are preserved in simulations.
181
Figure C.4: Snapshots of ?yy stress field at two different time instants, (a) 85 ?s and (b)
125 ?s for E1 < E2 (crack initiation time = 106 ?s), and (c) 105 ?s and (d) 145 ?s for
E1 >E2 (crack initiation time = 127 ?s).
The evolutions of various energy components was also studied. The evolution of strain
energy (USE) and kinetic energy (UKE) are shown in Fig. C.6(a). Similar to the results
presented in Section 7.4.1, both USE and UKE increase rapidly for the case of E1 < E2.
The energy dissipated by the cohesive elements (UCE) is shown in Fig. C.6(b). A rapid
change in the slope of UCE curve at t ? 106 ?s for E1 < E2 (t ? 127 ?s for E1 > E2)
182
Figure C.5: Crack growth behavior in syntactic foam FGM samples under mode-I loading.
absolute crack length history from (a) experiments and (b) finite element simulations.
signifies crack initiation. Also as noted earlier in the Section 7.4.1, the higher crack speed
for the case of E1