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On the dynamic fracture behavior of polymeric materials subjected to extreme conditions
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On the dynamic fracture behavior of polymeric materials subjected to extreme conditions
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Content
On the Dynamic Fracture Behavior of
Polymeric Materials Subjected to Extreme
Conditions
Orlando Delpino Gonzales
Dissertation
submitted to
Faculty of the USC Graduate School
University of Southern California
in partial fulfillment for the award of the degree of
Doctor of Philosophy
in Aerospace Engineering
August, 2016
Dedication
to my family
Acknowledgements
I would like to thank my advisor Prof. Veronica Eliasson for giving me the opportunity
to work on this project. I will always be grateful for her patience, guidance and support.
Many thanks to Prof. Steven Nutt for providing helpful suggestions and allowing me access
to his facilities to run experiments, in addition to serving as my thesis committee member.
I would also like to acknowledge Prof. Vincent Lee for being on my thesis committee. My
sincere gratitude to Prof. Andrea Hodge and Samantha Graves for guiding me in the initial
stages of my PhD, I really appreciate all your patience and help.
I am grateful to Behrad Koohbor and Prof. Addis Kidane of the University of South
Carolina for their immense help. Their guidance and tips were crucial in the progress of my
project. Also,abigthankyoutoProf.LeslieLambersonofDrexelUniversityforallheruseful
suggestions. ThankyoutoDougLoupoftheCarderockDivisionoftheNavalSurfaceWarfare
Center for providing carbon-fiber/vinyl ester samples. I would like to thank Yunpeng Zhang
and Theresa Juarez for taking so muchof their time to help me runexperiments and prepare
samples. A big thank you to all the awesome people working at the USC machine shop,
they helped me meet impossible deadlines many times. The National Science Foundation
and Office of Naval Research are sincerely acknowledged for funding this work.
To my shockwave-lab family: Dr. Chuanxi Wang, Dr. Gauri Khanolkar, Stelios Koumlis,
Shi Qiu, Hongjoo Jeon, Qian Wan and Jack Gross - working with you has been great and
I wish you all the very best. Particularly, thank you to Chuanxi and Stelios, whom I learnt
so much from. Additionally, I was lucky to have worked with a great team of undergraduate
students. ThankyouforyourhelpCatherinaTicsay,EricSiryj,AntonSchuetze,KimLoung,
Heidi Homma, Austin Nicassio, Austin Simons and Michael Werner.
2
A mis padres, todo es gracias a ustedes. Gracias por siempre darme apoyo incondicional.
A mi hermana Alejandra, a pesar de la distancia, nuestra comunicacion constante ha sido y
es un gran soporte para m´ ı, eres una gran hermana y amiga. A mi hermana Paola, gracias
por ayudarme en mis inicios en USA, siempre te estar´ e muy agradecido. A mis abuelos, a
pesar de que tres de ustedes ya no est´ en con nosotros, todos tuvieron una gran influencia en
m´ ı. Espero que mi trabajo y esfuerzo los llene de orgullo.
Finally, to Katie - I cannot thank you enough for all your help and encouragement. You
believedinmeeverydayandyouweresupportiveduringmygoodandbaddays. Thiswould
not be possible without you... We did it together.
3
Contents
1 Introduction 13
I. Outline and Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2 Background 20
I. Polymeric Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
II. Fiber Reinforced Polymeric Materials . . . . . . . . . . . . . . . . . . . . . . 21
III. Dynamic Fracture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
A. Dynamic Crack Initiation . . . . . . . . . . . . . . . . . . . . . . . . 26
B. Dynamic Crack Growth . . . . . . . . . . . . . . . . . . . . . . . . . 29
Brittle and Ductile Fracture . . . . . . . . . . . . . . . . . . . . . . . 30
C. Dynamic Fracture of FRPs . . . . . . . . . . . . . . . . . . . . . . . . 31
IV. Environmental Stress Cracking . . . . . . . . . . . . . . . . . . . . . . . . . 35
A. Effect of Water Content on Fracture . . . . . . . . . . . . . . . . . . 36
PMMA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
Vinyl Ester Resin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
Fiber Reinforced Polymers . . . . . . . . . . . . . . . . . . . . . . . . 39
B. Effect of Surrounding Liquids on Dynamic Fracture . . . . . . . . . . 40
3 Methods 43
I. Experimental Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
A. Dynamic Loading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
B. Strain Gauges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
II. Optical Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
A. Methods of Caustics . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
4
B. Digital Image Correlation . . . . . . . . . . . . . . . . . . . . . . . . 53
C. Interferometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
4 Influence of Water Uptake on Dynamic Fracture Behavior of PMMA 59
I. Experimental Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
A. Material and Conditioning . . . . . . . . . . . . . . . . . . . . . . . . 60
II. Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
III. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
5 Effect of Water Content on Dynamic Fracture Initiation of Vinyl Ester 71
I. Experimental Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
A. Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
B. Sample Conditioning . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
II. Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
III. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
6 EvaluationoftheEffectofWaterContentontheStressOpticalCoefficient
in PMMA 81
I. Material and Conditioning . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
II. Experimental Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
III. Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
IV. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
7 Experimental Investigation of Dynamic Fracture Initiation in PMMA
Submerged in Water 93
I. Methods and Material . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
A. Experimental Procedure . . . . . . . . . . . . . . . . . . . . . . . . . 94
B. Method of Caustics in Liquid Media . . . . . . . . . . . . . . . . . . 94
II. Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
III. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
5
8 EffectofWaterContentonDynamicFractureInitiationofCarbon-fiber/vinyl
ester 103
I. Analytical Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
A. Extraction of the Stress Intensity Factor . . . . . . . . . . . . . . . . 104
Isotropic Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
Orthotropic Materials . . . . . . . . . . . . . . . . . . . . . . . . . . 109
B. Alternate Method to Extract Mode-I SIF for Orthotropic Materials . 110
II. Material and Conditioning . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
III. Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
IV. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
9 Future Direction 123
I. Estimation of Dynamic Mechanical Behavior using Displacement Fields . . . 125
Bibliography 130
6
List of Figures
2.1 Fracture modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.2 Fracture toughness threshold . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
2.3 Comparison of fracture toughness . . . . . . . . . . . . . . . . . . . . . . . . 29
2.4 Interlaminar and intralaminar fracture in FRPs. . . . . . . . . . . . . . . . . 33
3.1 Experimental setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
3.2 Schematic of impact setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
3.3 Velocity profile for sabot passing a velocity sensor. . . . . . . . . . . . . . . . 45
3.4 Alignment schematic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
3.5 Test section . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
3.6 Exploded view of test section . . . . . . . . . . . . . . . . . . . . . . . . . . 48
3.7 Alternative alignment setup . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
3.8 Schematic of method of caustics . . . . . . . . . . . . . . . . . . . . . . . . . 52
3.9 Top view of experimental setup . . . . . . . . . . . . . . . . . . . . . . . . . 52
3.10 Caustic curve schematic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
3.11 DIC Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
3.12 Experimental setup for DIC . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
3.13 Interferometer setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
4.1 Drawing of notched PMMA sample . . . . . . . . . . . . . . . . . . . . . . . 60
4.2 Average weight change of PMMA samples as a function of time . . . . . . . 62
4.3 Crack propagation sequence for a PMMA sample conditioned at 11% RH . . 65
4.4 Strain gauges response for 10 repeated experiments . . . . . . . . . . . . . . 66
4.5 Results for the RH cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
7
4.6 Crack propagation comparisons with PMMA samples exposed to various en-
vironments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
4.7 Stress intensity factor for PMMA samples with different water contents . . . 69
4.8 Comparison of the critical SIF variation with water content and strain rate . 70
5.1 Drawing of notched vinyl ester specimen . . . . . . . . . . . . . . . . . . . . 73
5.2 Average weight change as a function of time . . . . . . . . . . . . . . . . . . 75
5.3 Strain gauge response for 12 experiments . . . . . . . . . . . . . . . . . . . . 76
5.4 Crack propagation sequence for an impacted VE sample . . . . . . . . . . . 77
5.5 Vinyl ester crack-tip speed . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
5.6 Stress intensity factor for VE samples . . . . . . . . . . . . . . . . . . . . . . 79
5.7 Stress intensity factor variation with crack-tip speed . . . . . . . . . . . . . . 79
6.1 Drawing of tensile test sample . . . . . . . . . . . . . . . . . . . . . . . . . . 82
6.2 Weight increment due to water sorption on PMMA tensile test samples . . . 83
6.3 Field of view and fringe appearance . . . . . . . . . . . . . . . . . . . . . . . 84
6.4 Intensity plot showing fringe time history . . . . . . . . . . . . . . . . . . . . 85
6.5 Comparison of response of extensometer and fringe count . . . . . . . . . . . 86
6.6 Elastic modulus variation of PMMA samples as a function of water content
for two different strain rates . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
6.7 Stress optical coefficient comparison . . . . . . . . . . . . . . . . . . . . . . . 89
6.8 Optical and mechanical contributions to SOC . . . . . . . . . . . . . . . . . 91
7.1 Sample geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
7.2 Comparison of caustic size for samples surrounded by air and water . . . . . 96
7.3 Crack propagation sequence for a sample immersed in water . . . . . . . . . 98
7.4 Average strain response comparison for four samples per case. . . . . . . . . 99
7.5 Crack-tip speed corresponding to four PMMA samples exposed to air . . . . 100
7.6 Stress intensity factor corresponding to four PMMA samples exposed to air . 101
8
8.1 Area around crack-tip used for SIF analysis . . . . . . . . . . . . . . . . . . 105
8.2 Weight increase due to water sorption on CFVE samples. . . . . . . . . . . . 114
8.3 Schematic of DIC calibration experiment . . . . . . . . . . . . . . . . . . . . 114
8.4 Strain measurement comparison between gauge and DIC . . . . . . . . . . . 115
8.5 Comparison of strain response for repeatability. . . . . . . . . . . . . . . . . 116
8.6 Inherent noise obtained . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
8.7 Longitudinal wave speed calculation . . . . . . . . . . . . . . . . . . . . . . . 117
8.8 Full-field strain maps around a crack-tip for a PMMA sample . . . . . . . . 118
8.9 Impact schematic for CFVE sample with fiber orientation . . . . . . . . . . . 119
8.10 CFVE SIF measurements and full-field strain maps . . . . . . . . . . . . . . 120
8.11 Critical SIF comparison for 3 dry and 3 soaked CFVE samples. . . . . . . . 121
1 Schematic of the sample impacted . . . . . . . . . . . . . . . . . . . . . . . . 127
2 Full-field acceleration maps for a PMMA sample . . . . . . . . . . . . . . . . 129
3 Average acceleration measurement from full field visualization . . . . . . . . 129
9
List of Tables
4.1 Composition of saturated salt solutions . . . . . . . . . . . . . . . . . . . . . 61
4.2 Average water absorption for experimental samples . . . . . . . . . . . . . . 63
4.3 Values used for the method of caustics . . . . . . . . . . . . . . . . . . . . . 63
4.4 Summary of high-speed cameras settings . . . . . . . . . . . . . . . . . . . . 64
4.5 Average critical stress intensity factor for 40 days of conditioning . . . . . . . 69
5.1 Composition of saturated salt solution . . . . . . . . . . . . . . . . . . . . . 74
5.2 Average weight change due to water sorption on experimental samples . . . . 74
5.3 Values used for the method of caustics . . . . . . . . . . . . . . . . . . . . . 76
6.1 Averagevariationfromdryconditionsofopticalandmechanicalcontributions
due to water sorption . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
7.1 Material properties used for the method of caustics . . . . . . . . . . . . . . 97
8.1 Properties of unidirectional CFVE samples with a fiber orientation of 90
◦
. . 113
8.2 Comparison of the SIF calculation using two methods . . . . . . . . . . . . . 119
8.3 ComparisonofstressintensityfactorcalculationofCFVEcompositeandvinyl
ester resin matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
10
Abstract
The useful properties that polymeric materials offer make them good candidates to be
considered in naval and aircraft applications as the matrix constituent in a composite ma-
terial, such as fiber reinforced polymers (FRP). Naturally, material properties are prone
to degenerate as a result of exposure to their environment and the characteristics of the
material. In order to confidently use these materials in outdoor environments in which,
for instance, humidity levels are varied, it is important to understand how environmental
conditions may affect their performance under different loading conditions. In this work,
an experimental study was completed regarding the effects of varied humidity levels and
sorbed water amounts on dynamic crack initiation and propagation of three polymeric ma-
terials, Poly(Methyl Methacrylate) (PMMA), vinyl ester neat resin and carbon-fiber/vinyl
ester (CFVE) composite, subjected to stress pulses created by an gas-gun. Edge-on impact
experiments were performed on samples conditioned in differentenvironments, including dry
specimens, specimens exposed to different relative humidity environments, and distilled wa-
ter saturated specimens. Experiments varied by immersion time, and similar loading rates
were applied to all sample groups. The optical, mechanical and fracture properties were in-
vestigated under different water contents. Additionally, the effect of water as a surrounding
medium was studied on the dynamic fracture behavior of PMMA. High-speed photography
combined with three different non-invasive visualization techniques, namely the method of
transmitted caustics, a Fizeau interferometer, and digital image correlation, and simultane-
ous strain gauges were utilized to obtain quantitative data from the experiments depending
on the material and measurements required. Among the properties measured during these
experiments were the stress intensity factor, stress optical coefficient, and crack-tip speed.
Results yielded repeatable responses within the same materials studied. The fracture be-
havior of these polymeric materials did not show a significant change due to water content
or surrounding environment. This was attributed to the effect of high strain rate, which
overcame the effect of water. On the other hand, the fracture behavior of CFVE was sig-
11
nificantly affected by water content, in that the fracture toughness of the material degraded
by 30%. This dissertation is part of an ongoing effort in which the experimental techniques
implemented and the results obtained will be used as a baseline to study the change of dy-
namic fracture behavior of fiber-reinforced polymers subjected to similar conditioning and
loading conditions.
12
Chapter 1
Introduction
Navalandaerospaceindustriesarecontinuouslysearchingforthelatestandmostefficient
technology, while simultaneously trying to maintain a low overall cost of its fleet. In recent
years, the improved design and manufacturing of lightweight materials, such as polymeric
composites and sandwich structures, have led to an increased interest in replacing existing
metallic parts on naval vessels with such materials [1]. The main reasons are that fiber
composites have a low maintenance cost and high strength-to-weight ratio [2], which are
properties that are beneficial for applications that require lighter, faster, stronger and more
durable materials.
Structures are designed based on the loads they are expected to experience while in
service. Largestressconcentrationsareavoided,andareasonablemarginofsafetyisimposed
to assure maximum loads are never reached during operation. As industries progress, safety
margins have decreased due to higher requirements for energy and material conservation.
Additionally, when a defect is detected, it does not make the part instantaneously unusable;
instead,itmaystillhaveusefullifeuntilitneedstobereplaced. Thisismostlyapplicablefor
expensive materials or structures that would be inconvenient to replace. Regardless of the
source of the flaws, they are common, and history contains many examples of catastrophic
failures in structures. Furthermore, materials tend to deteriorate as a result of exposure to
their respective environments. Due to their wide range of useful properties, polymers and
composites are being used in naval and aircraft applications, meaning they are exposed to
an outdoor environment, in which climatic conditions may vary significantly and could lead
to changes in properties of the material.
There already exist a few examples of naval structures, both above and below the wa-
terline, that have been changed to composite materials. Perhaps the largest naval metallic
13
structure to be replaced with a lightweight option was the superstructure on the DDG-1000
Zumwalt class. The DDG-1000 is a multi-mission destroyer with an emphasis on naval
surface fire support, which is designed to operate in near-shore waters. The composite su-
perstructure measures 39.6 m by 18.3 m by 12.2 m, and is divided into four levels made of
balsa-cored glass and carbon-fiber/vinyl ester (CFVE) sandwich panels [3]. Research and
development of composite propellers are also being completed in order to replace the most
commonly used metal alloys (nickel-aluminum-bronze). An example of these devices is the
QinetiQ (Rosyth, UK) 5-blade composite propeller [4], which is 2.9 m in diameter. The
lighter composite material allowed the propeller blades to be thicker to improve cavitation
performance and, thus, reduce vibrations and underwater signatures. In Europe, several
marine vessels made out of composite materials have been assembled. One example is the
Norwegian stealth missile craft KMS Skjold, the fastest armed craft in the world to date .
KMS Skjold, a catamaran-type hull vessel, is made out of glass and carbon-fiber composite
materials.
Marine structures are often subjected to extreme loading scenarios in which high loading
rates occur as a result of, e.g., wave slamming and underwater explosions [5]. In addition,
these structures are exposed to cyclic UV-radiation, which can generate micro-cracks even
in coated polymeric composites. Therefore, if these materials contain micro-cracks and are
subjected to long-term exposure to seawater, it is necessary to understand their dynamic
fracture behavior with varying levels of water content. When materials are subjected to ex-
tremeconditions,itisalmostimpossibletopreventfractureinitiation,andfractureinitiation
and subsequent crack propagation can occur at multiple sites. In order to confidently use
thesematerialsinsuchanenvironment,thestudyoffractureinitiationinmaterialssubjected
to dynamic loading is necessary to estimate the allowable stresses that structures can with-
stand at high loading rates. Furthermore, it is important to understand how environmental
conditions may affect their performance under different loading conditions.
The main challenge consists of connecting the high loading rate experiments with prop-
erly conditioned samples and creating a controllable and highly repetitive realistic shock or
14
impact environment. Unfortunately, the relationship between the applied loading conditions
– especially for high loading rates – and environmental effects leading to water-uptake and
the following crack initiation and propagation are still not clear. Therefore, in order to im-
prove the current knowledge of the dynamic response of polymeric materials, it is necessary
to understand, quantitatively, the correlation between high loading rates, water uptake and
resulting crack propagation.
This dissertation adopted an experimental multi-disciplinary approach in which solid
mechanics, fracture dynamics and fluid mechanics were applied with the purpose of investi-
gatingthedynamicresponseandfailureofpolymericmaterialsinextremeenvironmentswith
afocusonsampleswithvaryingwatercontentsandsurroundingmediums. Theexperimental
approach consisted of impact experiments in which a projectile launched from a gas-gun was
usedtogeneratehighstrainrateloadingconditionsonpre-conditionedsamples. Thefirstset
of experiments was performed on a well-characterized material, Poly(Methyl Methacrylate)
(PMMA),inordertodeveloptheexperimentalsetupandvisualization techniques, aswell as
to characterize the dynamic behavior of PMMA at a different loading rate. Next, materials
relevant to marine applications, vinyl ester and CFVE composites, were subjected to high
strain rate loading and exposed to long-term water exposure. The effects of these conditions
were studied to determine the dynamic fracture behavior of these materials and to gain a
better understanding of the positive or negative effects that the environmental conditions
haveonthedynamicfracturebehaviorofthesematerialswhensubjectedtodynamicloading.
I. Outline and Motivation
A description of the content and motivation of each chapter in this dissertation is provided:
Chapter 2: An overview of the relevant literature and previous results on the topic of
this dissertation are presented to illustrate the contributions that were added with this work
to the field of dynamic fracture.
Chapter 3: This chapter presents the basic concepts of the experimental and visualiza-
15
tion techniques used in this work. A series of experiments using state-of-the-art high-speed
photography with simultaneous strain measurements were conducted to enhance the current
understanding of the dynamic response and fracture mechanisms of polymeric materials in
extremeenvironments. Theexperimentalsetuphasbeendesignedtogeneratehighlyrepeat-
able edge-on impacts onto solid structures. Since each polymeric sample is impacted only
once and then compared to other samples, it is crucial to have full control of the external
loadingconditionsthatareappliedtothesampleintermsofpulseduration, pulseamplitude
and maximum strain. Additionally, three non-invasive visualization techniques, utilized to
obtain quantitative measurements, were implemented and are described in this chapter.
Chapter4: Astudyontheeffectsofdifferenthumiditylevelsandsorbedwateramounts
on crack initiation and propagation in PMMA subjected to stress pulses created from a gas-
gun was completed. Impact testing was performed on six sets of samples conditioned in
different environments: dry specimens, specimens exposed to three different relative humid-
ity environments (11 %, 54 %, 98 %) using three saturated salt solutions (Lithium Chloride,
Magnesium Nitrate, and Potassium Sulfate, respectively), distilled water and seawater sat-
urated specimens. Experiments varied by immersion time and loading rates in order to
generate similar strain rate (10
2
s
−1
) on the sample groups. The method of transmitted
caustics was used to obtain quantitative measurements from these experiments. The main
objective of this study was to determine the behavior of crack propagation in PMMA un-
der dynamic loading as humidity levels varied. Experimental techniques and conditioning
procedures used for the duration of this thesis work were established performing this study.
Chapter 5: Fiber-reinforced polymer (FRP) matrix composite materials are commonly
based on organic thermoset resins, such as vinyl ester. Vinyl ester resins are widely used as
matrixmaterialsinFRPsbecausetheyconstitutealow-costoptionthatprovideslightweight,
effective corrosion resistance to seawater environment and a number of stealth properties,
including low thermal, magnetic signatures and good noise-dampening properties [6]. On
the other hand, it has been reported that they suffer from low-impact resistance, which, due
to the nature of a marine environment, could lead to catastrophic consequences if subjected
16
to high loading rate, i.e., wave slamming or shock loading [7]. Even though FRPs are the
materials to be used in marine applications, any changes in subjected loading amplitude and
loadingratecanaffectthepropertiesofthepolymermatrix,whichcouldinturninfluencethe
fracturetoughnessofthecomposite. Thus,themechanicalpropertiesandtheratesensitivity
of the polymer matrix properties directly determine the rate sensitivities of the polymeric
composite[8]. Thisstudyexaminedtheeffectsofwatercontentonmode-Icrackpropagation
invinylesterneatresinspecimens. Thegoalwastounderstandandcharacterizethefracture
behavior of vinyl ester resins under dynamic loading conditions. Previous studies on vinyl
ester have not included the effects of water content on its dynamic fracture initiation when
subjected to strain rates at or above 10
2
s
−1
. In this work, samples were subjected to equal
amplitude tensile stress pulses by edge-on ballistic loading reaching strain rates of 10
2
s
−1
.
The relationship between water content and dynamic fracture initiation is discussed in this
chapter.
Chapter 6: This chapter constitutes the study of optical properties on PMMA to vali-
date the properties utilized in the analysis of dynamic fracture. The experimental determi-
nation of stresses within a body under tension using non-contact visualization techniques in
transmission requires measurements of the variations of the optical paths along the principal
stress directions. These optical path differences are related to the applied stresses through
the stress optical coefficient (SOC). The SOC depends on the variation of mechanical prop-
erties and the refractive index of a polymer, both of which are known to be affected by water
[9, 10]. It is relevant to study these properties of PMMA because the applications for which
thismaterialcanbeutilizedareprincipallybasedonitstransparencyanddurability, making
it a good candidate for use as a component in aerospace and naval structures (e.g. windows,
lenses,canopyandcovers). Asaresult,theseapplicationsoftenrequirethematerialtobeex-
posedtoanoutdoorenvironmentwhereitwillbesubjectedtoenvironmentaleffects. Among
these effects, humidity or water exposure are conditions that cause aging in the material and
are known to gradually degrade its useful properties. The motivation for this study resides
in the need to obtain precise values of the SOC under different environmental conditions.
17
These values could be used for studies of deformation and fracture in various visualization
techniques such as interferometry, classical photoelasticity and the method of caustics. For
this reason, the SOC of PMMA samples with different water contents was measured using a
Fizeau interferometer during tensile testing. Additionally, the SOC behavior of PMMA was
separated into optical and mechanical properties to understand the effect that sorbed water
has on the optical and mechanical contributors of the SOC of PMMA.
Chapter 7: PMMA samples were subjected to impact loading as they were surrounded
by water to study the effect of their surrounding medium on dynamic crack propagation.
This chapter can be considered a continuation of Chapter 4 with a much shorter water
immersion duration, limited to seconds, and with surrounding water still present as the
fracture event occurred. Underwater dynamic loading events constitute situations involv-
ing fluid-solid interactions, which could threaten the integrity of structures and potentially
lead to catastrophic failure. For instance, coastal buildings, ship hulls subjected to wave
slamming, or submarines subjected to underwater explosions are scenarios in which catas-
trophic failure can occur. Therefore, it is necessary to understand the fracture behavior of
structures in underwater situations when subjected to dynamic loading conditions in order
to improve future design considerations and minimize damage. An experimental study of
structures subjected to underwater dynamic loading, reaching strain rates of 10
2
s
−1
, was
conducted and in-situ measurements were recorded with the aid of visualization techniques
and high-speed photography. This work was particularly challenging since the application of
visualizationtechniquesandloadcharacterizationmethodscanbenon-trivialforunderwater
studies compared to air environment experiments. Results obtained for underwater crack
propagation were compared to the results reported in Chapter 4.
Chapter 8: CFVE composites and PMMA samples were studied under similar condi-
tions as in Chapter 4. Due to the opaque nature of the composites samples, Digital Image
Correlation (DIC) was implemented for this investigation, different from the visualization
technique used in Chapter 4, Chapter 5 and Chapter 7. The results obtained for the PMMA
samples were compared to the results obtained in Chapter 4 to calibrate and confirm that
18
the visualization technique was implemented correctly. DIC offers multiple advantages be-
cause it provides full-field measurements, which allows for analytical methods to be used to
enable extraction of stress intensity factors using an overdeterministic approach. This work
describes the initial results of the dynamic fracture behavior of CFVE composite samples
subjected to long-term exposure in water. The results from this chapter were compared to
the results presented in Chapter 5 to perform a matrix-composite comparison.
The overall goal of this dissertation was to develop an understanding in order to predict
the dynamic response and mechanisms behind failure of polymeric materials subjected to
highlytransientloadingsubjectedtolongtermexposureorimmediateexposuretowater. In
particular, this thesis aimed to investigate how cracks initiate and propagate in pre-soaked
polymers as the level of water content was varied, so in the future, these polymeric materials
can be confidently used in different humidity environments with quantitatively determined
allowable dynamic loads.
19
Chapter 2
Background
I. Polymeric Materials
Polymeric materials are a common subject of research and development since they represent
a solution to the need of saving fuel in the aircraft, naval and automotive industries. For
instance, a car is made out of more than 300 pounds of plastics, without including paints,
rubber in tires and upholstery. Also, new aircrafts include increasing amounts of polymers
and fiber reinforced polymers for insulation, windows and structural components. The ap-
plication of polymers in other industries is still growing (for example, in piping, thermal
insulation, paints, and bottles), such that the use of polymers exceeds that of metals on a
mass basis.
Polymers are products of the petrochemical industry, which have five major applications:
protective coatings, plastics, adhesives, rubbers and fibers. In this work, we will focus on
the application of polymers as plastics. A plastic is normally thought of as being the type
of polymer that possesses a degree of structural rigidity. The molecular requirement for
a polymer to be used in a plastic form is the polymer must be below its glass transition
temperature. The main distinction between types of polymers is based on their reaction to
heating and cooling, classifying as thermoplastics or thermosets.
• Thermoplastics: Certain polymers soften upon heating and could then be made to
flowwhenastresswasapplied. Whencooledagain, theywouldregain theirsolidform.
Thesepolymersareknownasthermoplastics. Continuedheatingofthermoplasticswill
ultimately lead to degradation, but they will generally soften at temperatures below
their degradation point. By analogy, ice and solder, though not polymers, behave as
20
thermoplastics.
• Thermosets: Although they might be heated to the point where they would soften and
flow under stress, the process would be irreversible. This means that heating causes
them to undergo a curing reaction. Further heating of thermosets would ultimately
lead to degradation instead of softening and flow under stress. By analogy, eggs and
concrete behave as thermosets [11].
In general, the behavior of materials of low relative molecular mass, such as polymers,
is usually discussed in terms of two types of materials: elastic solids and viscous liquids.
Elastic solids have a defined shape and, when deformed by external forces, they achieve a
newequilibriumshape. Furthermore,uponremovaloftheseforces,thematerialreturnstoits
original form. The solid stores all the energy that it obtains from the external forces during
the deformation, and this energy is available to restore the original shape when the forces
are removed. On the other hand, a viscous liquid has no definite shape and flows irreversibly
under the action of external forces. One of the most interesting features of polymers is
that a given polymer can display all the intermediate range of properties between an elastic
solid and a viscous liquid depending on the temperature and the experimentally chosen
time-scale. Thus, the mechanical behavior of these materials depends on strain rate and
temperature. In general, higher strain rates and lower temperatures yield higher moduli and
smaller elongations [12].
II. Fiber Reinforced Polymeric Materials
Composite materials refer to the type of materials that result from bonding two or more
homogeneous materials with different properties to obtain a final result with certain desired
material,mechanicalpropertiesandcontrolledanisotropy. Compositematerialsaregenerally
usedbecausetheyhavedesirablepropertiesthatcannotbeachievedbyanyofitsconstituent
materials independently. FRPs are a category of composite materials that specifically use
fiber materials to mechanically enhance the strength and elasticity of polymers. In FRPs,
21
the original polymeric material without fiber reinforcement is known as the matrix compo-
nent. The matrix is a tough but relatively weak plastic that is reinforced by stronger and
stiffer fibers and simultaneously serves to protect the fibers from external damage and en-
vironmental attacks. Reinforcement of a matrix component occurs when the FRP material
exhibits increased strength or elasticity relative to the strength and elasticity of the matrix
alone. The strengthening magnitude depends on the mechanical properties of both fiber and
matrix, their volume relative to each other, and the fiber length and orientation within the
matrix. In general, FRPs possess a higher strength-to-weight ratio (specific strength) and a
higher stiffness-to-weight ratio (or specific stiffness) compared to bulk materials [13, 14].
The independent usage of fibers allow extremely high tensile strength and stiffness of
a material; however, in fiber form they are not stiff in compression and their transverse
mechanicalpropertiesareoftennotaseffectiveasthecorrespondinglongitudinalones. Thus,
fibersaregenerallyuselessasstructuralmaterialsunlesstheyareheldtogetherinastructural
unitwithamatrixmaterial. Mostcompositepropertiesaredependentonthecombinationof
the matrix and fiber reinforcement together. However, in some cases, the fibers will largely
determine the mechanical behavior of a structure with little contribution from the matrix,
while in others, the matrix will dominate. In a single ply, the tensile strength parallel to
the fibers is largely influenced by the fiber properties since their elastic modulus is typically
one order of magnitude larger than that of the matrix. In contrast, the elastic and strength
properties perpendicular to the fibers and shear properties of the ply are all dominated by
the matrix, since the load must be transferred by the matrix through the fiber diameter
and are, therefore, much lower than the properties parallel to the fibers. As expected, when
the reinforcing fibers are combined with a matrix material to form a composite, there is a
reduction in the high strengths and elastic modulus of the fibers alone, but the composites
still have a significant advantage with respect to these properties.
The most efficient composites have most of their fibers oriented in the primary load di-
rection and sufficient fibers oriented in the other directions to carry secondary loads, provide
transverse reinforcement and hold the structure together. Efficiency means both low weight
22
and low cost, so any fibers that do not carry much load could likely be removed. Because
moststructuresarenotloadedinasingledirection,eventhoughonedirectionmaydominate,
it is necessary to orient fibers in multiple directions. This is accomplished by stacking multi-
ple plies together. Such a stack is called a laminate. One of the critical factors in composite
technology is the ability to integrate design and manufacturing with the main performance
requirements to produce a component in which the reinforcing fibers are arranged within the
optimum fiber architecture. There are several architectural variables including arrangement
of fibers, fiber length and volume fraction of fibers to take into account for optimization
[15, 16].
One attribute in which FRPs are usually not as effective as conventional bulk materi-
als is their ultimate strain, or strain to failure. Many advanced fibers exhibit nearly linear
stress-strain curves to failure, which is considered to be characteristic of brittle materials.
Fortunately, because fiber orientation can be modified, a composite laminate can be con-
structed in such a way that its overall behavior is similar to that of ductile metals, even
though the reinforcing fibers exhibit brittle behavior [14]. Specifically, in this work, we will
focus on CFVE because of the advantages (higher modulus, higher strength and lower den-
sity)thatcarbon-fiberhasovermorecommonlyusedfibers, e.g., glassfiberandconventional
bulk materials [17]. CFVE is a desirable material for naval structures due to the superior
resistance to water degradation that carbon-fiber and vinyl ester exhibit.
III. Dynamic Fracture
In the early 17th Century, strength and fracture studies started with observations from
Galileo, who was the first to associate the changes in strength response with the presence of
defects within the structure. Nevertheless, fracture mechanics was truly developed through
investigations by Griffith [18] and Irwin [19], who suggested an effective method of equi-
librium for a cracked elastic domain. Griffith suggested the following criterion for crack
initiation: “The energy required to create new surfaces must be equal to that released by the
23
solid in deforming to the new configuration.” Irwin [19] suggested that fracture will occur
when the critical toughness value, K
Ic
, is reached for the material. K is denominated the
stress intensity factor (SIF), and it quantifies the stress state near the tip of a crack. The
SIF depends on load applied and geometry, and is used to establish failure criteria due to
fracture. The reason the SIF is used in fracture mechanics is because the stress values near
a crack-tip are always very high, approaching infinity at the crack-tip. The strength of ma-
terials approach dictates the material will fail when the stress applied exceeds some critical
value (ultimate or yield stress). Thus, the strength of materials approach cannot be used in
fracture studies. When a small load is applied to a cracked plate, the stress field near the
crack-tip increases significantly, but the plate does not fail. However, as the applied load
increases to some critical value, the plate fails. In fracture mechanics, instead of comparing
the maximum stress value with a critical stress value, the material failure is predicted by
comparing the stress intensity factors, K with some critical value K
c
. The critical value
is called the critical stress intensity factor, or the fracture toughness of the material. In
other words, stress is related to strength as the stress intensity factor is related to fracture
toughness. Note that K
c
is a material property, as the ultimate stress and yield stress are,
while K depends on the problem geometry and applied loads [20, 21].
From these contributions, the Theory of Linear Elastic Fracture Mechanics (LEFM) was
developed. This theory constitutes the fundamentals of fracture analysis. Although LEFM
was formulated for slowly moving cracks (crack speed < 1 m/s), its basic principles were
assumed to be applicable for dynamic crack propagation. However, several basic experi-
mental results could not be explained by LEFM [22]. These limitations will be discussed
in the following sections. LEFM separates the conditions at which crack propagation oc-
curs according to the different ways a force is applied to a structure. Irwin [19] proposed a
classification corresponding to the three situations represented in Figure 2.1.
In mode-I, or opening mode, the body is loaded by tensile forces such that the crack
surfaces are pulled apart in the y-direction. In mode-II, or sliding mode, the body is loaded
by shear forces parallel to the crack surfaces, which slide over each other in the x-direction.
24
Figure 2.1: Fracture modes. Image from [23].
In mode-III, or tearing mode, the crack surfaces move relative to one another and parallel to
the leading edge of the crack. Mode-I fracture is the most common occurring in engineering
designbutnotnecessarilythemostcommoninnature[23]. Theideabehindthisclassification
systemisthatoneneedstodecidetheloadingconditionstodetermineunderwhichscenarios
and direction the crack will propagate.
The main difference between quasi-static fracture and dynamic fracture is the presence
of stress waves on the latter. These waves arise due to either the stresses released from
the crack-tip at fracture or externally applied loads [24]. Dynamic loading is a widely used
techniquetodeterminethedynamicpropertiesofamaterial. Thereareseveralmethodsused
togeneratedynamicloading[25]. Onemethod,usedforthiswork,isbyimpactofaprojectile
launchedfromthebarrelofagunontotheedgeofasample,whichgeneratesstresswavesthat
travel through the solid. Initially, the generated stress wave travels as a compressive pulse.
Oncethecompressivepulseimpingesthefreerearsurfaceofthespecimen,ararefactionwave
is created and reflected as a tensile pulse that returns the specimen to ambient pressure [26].
The amplitude of the tensile pulse depends on the impedance mismatch at the interface of
the free end and the fluid to which it is exposed [27].
As previously mentioned, the criteria given by Griffith [18] and Irwin [19] are not fully
applicable to dynamic loading. One explanation is that the energy required to create a new
surface may depend on the rate of the load applied, which in turn influences the crack speed
25
[28]. For fast propagating cracks, the governing equation under elastodynamic conditions
can be stated in a simple form as K
I
(t)=K
I
D
(v), where K
I
(t) is the instantaneous dynamic
stress intensity factor and K
I
D
(v) is the material resistance, a function of crack speed [29].
After decades of research attempting to find a unique criterion to analyze dynamic loading,
there are still scattered results that either support or contradict each other.
A. Dynamic Crack Initiation
Dynamicinitiationtoughness,K
I
d
,isdefinedastheminimumstressintensityfactorrequired
to initiate crack propagation. At high loading rates, K
I
d
is higher than the quasi-static
fracture toughness. The dynamic initiation toughness depends on the loading rate and is
described by Equation (2.1):
K
dyn
I
(t
f
) = K
I
d
(T,
˙
K
dyn
I
) (2.1)
where t
f
is time to fracture, T is temperature, and
˙
K
dyn
I
is the dynamic loading rate [25].
Shockey and Curran [30] completed an experiment in which a specimen was impacted
at three different speeds (14, 32, 57 m/s) with a stress pulse duration of 2.8 μs and stress
amplitude of 27, 60 and 94 MPa, respectively. They found that none of the cracks grew
at the lower impact speeds, but, at the highest speed, all cracks grew. This analysis was
based on the inspection of samples after the experiments. Shockey et al. [31] interpreted
the loading rate dependence of the dynamic initiation toughness by adding, while it was
essential for the stress intensity factor to reach a critical value, it was also necessary for it to
be maintained over a certain time period for the fracture processes to develop completely.
Ravi-Chandar and Knauss [28] performed the first direct experimental measurement of
the loading rate dependence for dynamic mode-I crack initiation. Their results showed that
there was an increase in dynamic crack initiation toughness with increasing loading rate
in Homalite 100 (a brittle polyester). In order to explain their fracture initiation results
satisfactorily, they included microscopic phenomena related to the fracture process. The
26
reasoninggivenwasthatthereisanintrinsictimeassociatedwiththenucleationandgrowth
of cracking processes. Thus, if the rate of loading is smaller than the rate of microprocesses,
quasi-static results would be obtained and the stress intensity factor would remain constant.
Conversely, if the loading rate is greater, the microscopic processes would not be completely
developed, leading to a tolerance for higher loads. In other words, when crack-tips are
subjected to dynamic loading rates or shorter stress pulses, the stress intensity factor of the
material increases monotonically, allowing the material to withstand higher loads before the
crack propagates.
As a summary of these results, Figure 2.2 shows an example of the findings. Assume
that the crack is loaded such that the stress intensity factor increases up to 0.75 and then
decreases to zero along the loading path indicated by the solid black line. Since the loading
never exceeds the fracture threshold, crack growth will not be initiated. On the other hand,
if one decreases the loading rate to cause the loading along the path indicated by the dashed
line, while maintaining the same peak load, the crack will initiate at the point indicated by
the dot. This shows that when crack-tips are subjected to higher loading rates or shorter
stress pulses, the stress level needed for a crack to initiate increases [28].
ThemechanismdescribedinFigure2.2appliestoPMMA,whichisanamorphouspolymer
that exhibits a viscoelastic nature with a high dependence on strain rate (increment of
5.2 MPa per decade of strain rate) [32], a relevant condition in dynamic loading experiments
[33]. ItiswellunderstoodthatthetensilestrengthofPMMAwillincreasewithhigherstrain
rate [32, 34, 35] because the loading rate is faster than the time that it takes microcracks
and void mechanisms to form before crack initiation. This allows the material to withstand
higher levels of stress before fracture occurs [29].
However, Figure 2.2 does not entirely apply to vinyl ester resin whose fracture toughness
does not seem to have significant loading rate sensitivity for strain rates of 10
−3
− 10
2
s
−1
[36, 37]. The mode-I critical stress intensity factor, K
Ic
, of vinyl ester has been previously
studiedinexperimentsconductedatastrainrateof 10
−3
s
−1
onsingle-edge-notchbendtests
[38–41]. The values obtained are within the range of K
Ic
= 0.7−1.7 MPa
√
m.
27
Figure2.2: Fracturetoughnessthreshold. Highloadingrates(blacksolidcurve)donotinitiate
fracture due to higher fracture toughness at those rates. Lower strain rates (dashed curve)
initiate crack propagation. Image from [28].
The variation in the value of K
Ic
is a result of the various compositions of the vinyl ester
resinutilizedbydifferentresearchgroups. Also, DelpinoGonzalesandEliasson[37]reported
values for the mode-I critical stress intensity factor of 1.5 MPa
√
m for vinyl ester samples
subjected to a strain rate of 10
2
s
−1
.
Liu et al. [42] used inertial constraints to provide an alternative explanation to dynamic
crack initiation and loading rate dependence. Rather than applying the fracture criterion
based on the stress intensity factor reaching a critical value, Liu et al. [42] proposed that the
crack will grow when the normal stress component reaches a critical value. This criterion
was based on a brittle material. It was proposed that nucleation, growth and coalescence
of microcracks around the crack-tip process zone require a certain critical stress level to be
reached. Results demonstrated that inertial effects drive the fracture rate dependence of
brittle materials. In the case of metals, Kalthoff [43] found that there was an increase in the
stressintensityfactorwithstrainrateforsteelbutfoundnoratedependenceforAralditeB(a
28
brittleepoxy). Figure2.3showstheresultsfortheloadingdependenceofthecrackinitiation
toughness for Homalite-100 and 2024-T3 aluminum. It demonstrates that a similarity exists
in the rate dependence of these two materials even though their fracture mechanisms are
significantly different (brittle and ductile); hence, the similarity must originate from other
common conditions, for instance, inertial effects as described by Liu et al. [42]. This theory
remains to be fully proved due to the difficulty of obtaining in-situ measurements of the
normal stress component in the vicinity of a crack-tip.
I
d
Figure2.3: Comparison of fracture toughness between Homalite-100 and Aluminum 2024-T3.
Image from Ravi-Chandar [25].
B. Dynamic Crack Growth
Dynamiccrackgrowthinitiatesduetotheconditionssetbythedynamicinitiationtoughness;
however, crack growth is determined by a different set of criteria. The dynamic stress field
near a running crack is characterized by the dynamic stress intensity factor, K
I
D
, where
the subscript D indicates dynamic crack growth. Now, this is a function of crack position,
29
loading rate,
˙
K
dyn
I
, and crack-tip speed, v, as seen in Equation (2.2):
K
dyn
I
(t,v) = K
I
D
(v,
˙
K
dyn
I
). (2.2)
It must be recognized that crack initiation is not a characteristic point on the dynamic
crack growth criterion curve. That is, it does not represent the value of the stress intensity
factor when crack speed approximates to zero, as shown in Equation (2.3). This may be
a consequence of various factors, such as rate dependence, initial crack-tip bluntness, or
loading conditions,
K
I
D
(v→ 0,
˙
K
dyn
I
,T)6= K
I
d
(
˙
K
dyn
I
,T). (2.3)
A direct consequence of this phenomenon is that cracks will accelerate to high velocities im-
mediately after initiation. Dynamic crack propagation is controlled and influenced by stress
waves impacting either a stationary or running crack-tip by injecting energy into its tip to
generate or continue propagation [44]. It is fairly complex to establish the stress intensity
factor at the tip of a running crack. Some experimental efforts to determine K
I
D
have re-
sultedinconflictingconclusionsbecausevariationinthemeasurementsfordifferentspecimen
geometries, loading and unloading conditions, and crack-tip acceleration have influenced the
interpretation of the outcome. Some materials show an increased K
I
D
at high loading rates,
whereas others exhibit a decrease in fracture toughness [24, 45, 46].
Brittle and Ductile Fracture
Depending on the fracture characteristics of the material, there are two fracture behaviors:
ductile and brittle. In ductile fracture, the toughness of the material is determined by the
plastic deformation around the growing voids. Ductile solids flow or bend before they break
and are more resilient. On the other hand, brittle fracture does not exhibit large plastic
deformation. Brittle solids break easily and catastrophically. In general, dynamic loading
experiments show higher toughness for ductile materials than for brittle ones [24, 47]. Some
of the most relevant findings on crack growth are described in this section.
30
Dally [48] showed an experimental characterization of the dynamic crack growth tough-
ness. The most relevant finding was that the crack speed changes from quasi-static to about
300 m/s for a 20% increase in K
I
D
. After this, for a crack speed between 300 and 400 m/s,
K
I
D
more than doubles. An opposite trend was found by Arakawa and Takahashi [49], who
introduced the K-v relationship, which depends on whether the crack was accelerating or
decelerating.
Kalthoff [50] conducted studies on Araldite B, which initially showed a large scatter in
the dynamic K-v relationship. In order to sort this out, three sets of experiments were
completed. The same batch of the specimen material was used to make the samples, but
their geometries were varied. Since different results were obtained for each configuration,
it was demonstrated that there was a clear influence of specimen geometry on the K-v
relationship.
Ravi-Chandar and Knauss [44] performed experiments eliminating the geometry depen-
dence. They found that, even though the dynamic stress intensity factor continued to vary
subsequently to crack initiation, the crack speed remained constant. Unlike polymeric mate-
rials, experimental characterizations of the dynamic crack growth toughness done by Zehn-
der and Rosakis [51] have resulted in suggestions that a unique relationship between the
SIF and crack-tip speed, K-v, is possible in metallic materials. The contrast in response
between brittle materials and ductile materials in the K-v relationship must be attributed
to the deformation and failure mechanisms that occur in the fracture process zone, such as
the dissipation shown in ductile materials due to plasticity.
C. Dynamic Fracture of FRPs
Composite materials derive many of their advantages from their anisotropy; yet, anisotropy
alsogivesrisetocomplicatedthreedimensionalstatesofstressinthepresenceofgeometrical
discontinuities, such as cutouts, notches or ruptures. In FRPs, multiple local or small scale
damage phenomena can occur, including delamination, debonding, fiber pullout, matrix
crackingandvariousformsoffiberfracture, makingthestudyoffracturemechanicsofFRPs
31
very complex [52]. For FRPs, the use of LEFM is not quite so straightforward because this
theory can only be rigorously applied for homogenous materials. Nevertheless, a pragmatic
application to continuous fiber reinforced plastics has demonstrated that LEFM can be
helpful in defining the fracture anisotropy but also in simply measuring the toughness. The
nature of the anisotropy of the continuous fiber dominates the fracture characteristics. In
order to analyze and obtain the stress intensity factor, the first theory and equations were
derived by Sih et al. [53].
Fracture modes in continuous fiber reinforced composites can be divided into three basic
fracture types. These involve translaminar fracture (breaking fibers), intralaminar fracture
(a crack running between fibers) and interlaminar fracture or delamination (a crack running
between the plies that make up the composite). Translaminar fractures are those oriented
transverse to the laminated plane in which conditions of fiber fracture are generated. When
considered in microscale, interlaminar and intralaminar fracture types can be similarly de-
scribed. In both cases, fracture occurs on a plane parallel to that of the fiber reinforcement.
In a similar manner to that described for bulk materials, fracture of either type can occur
under mode-I tension, mode-II, mode-III or any combination. Since this work focuses on
in-plane loading, more details will be provided for interlaminar and intralaminar fracture.
Interlaminar fracture (delamination) of FRPs has received considerable attention in the
researchcommunityindifferentmodesforseveralyears,suchthattherearealreadystandard-
izedproceduresformeasurementofmode-I,mode-IIandmixedmode-I/IIfracturetoughness.
There exists state-of-the-art testing methods of interlaminar fracture toughness which have
been comprehensively reviewed by several authors over the past years [54]. Delamination is
definedastheseparationoflayersfromeachother. AlthoughFRPsarecurrentlyextensively
used, their potential for delamination, or separation of the laminate, is still a major prob-
lem because the interlaminar strength is matrix-dominated. Delamination is classified as a
critical failure in composite structures, not necessarily because it will cause the structure to
break into two or more pieces, but because it can degrade the laminate to a such degree that
it will be useless. The interfacial separation caused by delamination may lead to premature
32
buckling of the laminate, excessive vibration, intrusion of moisture, stiffness degradation
and loss of fatigue life. Delamination may lead to redistribution of stresses which could
eventually promote total failure [55, 56].
More recently, advances in fracture mechanics have enabled studies of additional frac-
ture mechanisms, such as intralaminar fracture. The difference between interlaminar and
intralaminar is shown in Figure 2.4. It can be seen that delamination is defined as a dis-
continuity in the x-y plane between two adjacent plies of a laminate. An intralaminar crack
is a discontinuity in the y-z plane that, when subjected to tensile loading, propagates longi-
tudinally along the fiber direction. Frequently, delaminations and intralaminar cracks arise
concurrently within a composite structure and intralaminar cracks often act as delamination
migration pathways between adjacent interfaces or as boundaries that constrain delamina-
tion growth. The fracture toughness parameters involving crack growth parallel to the fibers
(intralaminar) and between layers of aligned fiber in a laminate (interlaminar) are strongly
matrix-dependent. Thedifferentelasticpropertiesofthefiberandmatrixresultinsignificant
stress concentrations in the matrix between the fibers. The measured fracture toughness of
intralaminar failure modes are comparable to the interlaminar ones [16, 57, 58].
Interlaminar
cracking
σ
σ
z
y
x
Intralaminar
cracking σ
σ
Figure 2.4: Interlaminar and intralaminar fracture in FRPs.
Additionally, there are secondary mechanisms in FRPs that are responsible for a multi-
tude of local or small-scale damage phenomena including tearing, debonding (separation of
33
fibers and matrix), crazing, cavitation, voiding, yielding and matrix cracking. Matrix crack-
ing and debonding are considered the two more relevant small-scale damage phenomena for
this thesis, so they will be briefly described.
Matrix microcracking is often the first form of damage in FRP laminates. They are
intralaminar cracks that cross the thickness of the ply and run parallel to the fibers of
the ply. The most common observable microcracking is cracking in the 90
◦
plies during
axial loading applied in the 0
◦
direction. These microcracks are transverse to the loading
direction and are often termed as transverse cracks. Microcracks can form in any plies
but predominantly they are found in plies off-axis to the loading axis. When the part is
mechanically loaded over a certain limit for the first time, additional tiny cracks occur.
As long as these cracks are non-visible and limited to a fiber/matrix-scale they are called
microcracks. Both length and the number of cracks grow if the stress level in the matrix is
increased. Eventually, a macroscopic crack is formed, which runs through the thickness of a
laminate. This macro-mechanical damage is called matrix cracking or Inter-Fiber Fracture.
Once matrix cracking occurs, other forms of damage can occur such as delamination, fiber
breakage, or the provision of pathways for the entrance of corrosive liquids. Such damage
modes may subsequently lead to laminate failure. Fiber debonding occurs due to weak
bonding between the matrix and fibers. In FRPs, during crack propagation, the stress near
the crack-tip usually causes the fibers to debond from the matrix; if the fibers do not break,
a bridge or gap remains between the fiber and matrix [59, 60].
Despite the current importance of FRPs, dynamic fracture mechanics studies and ex-
perimental data for understanding the mode of failure and energy absorption mechanism
of laminated composite plates are very limited. Static or quasi-static fracture behavior of
composites has been extensively studied by many authors and it is well established in the
examination of unidirectional fiber-reinforced composite behavior. By contrast, dynamic
fracture toughness in composites has not received as much attention. Until now, there has
been no completely adequate method to characterize and measure dynamic fracture tough-
ness of solid materials due to the difficulties in dynamic fracture theory and experimental
34
techniques. Thus, there exists no proper understanding of the behavior of FRPs at higher
loading rates. The work is still limited due to imaging capabilities; thus, most of the studies
are focused on the feasibility of studying impulse loading induced crack initiation and rapid
crack growth in fiber-reinforced composites using new visualization techniques.
A more limited volume of work exists in the study of fracture of FRPs, mostly unidi-
rectional laminates, as a result of stress wave loading, from which arose intralaminar crack
propagation. Liu et al. [61] and Yu et al. [62] conducted some of the first studies on dynamic
fracture of composite materials, studying dynamic fracture behavior on unidirectional FRPs
using interferometric techniques. As DIC, a relatively new visualization technique, has pro-
gressed and anisotropic constitutive relationships have been applied to displacement fields
obtained from DIC, more robust dynamic fracture toughness measurements have been possi-
ble. Leeetal.[63]studiedthefracturebehaviorofmultilayeredunidirectionalgraphite/epoxy
composite materials under impact loading conditions using DIC. Studies were conducted in
mode-I or mixed-mode fracture. Rubio-Gonz´ alez et al. [64] studied the strain rate effect on
fracture toughness of unidirectional FRPs. Results showed that dynamic fracture toughness
to be greater than the quasi-static one. Mallon et al. [65] performed the first study of wo-
ven composites in which stress intensity factors and crack direction were investigated as a
function of fiber orientation using DIC.
Specifically,mechanicalbehaviormeasurementsontheFRPstudiedinthisthesis,namely
carbon-fiber/vinyl ester, have been previously completed to characterize its behavior at
different fiber orientation for quasi-static loading conditions [66], fluid-solid interaction [67]
and blast loads [68]; however, no quantification studies on its dynamic fracture toughness
have been conducted yet.
IV. Environmental Stress Cracking
The effect of the environment in material fracture is referred to as environmental stress
cracking, which is one of the most common causes of unexpected brittle failure of polymers.
35
An overview of the relevant studies done in the field of environmental stress cracking of
PMMA, vinyl ester neat resin and CFVE will be described.
A. Effect of Water Content on Fracture
Some of the relevant studies found in the literature regarding the effect of water content of
the fracture behavior of the three polymeric materials studied in this work are described as
follows.
PMMA
The variation of the mechanical properties of PMMA due to water sorption has been investi-
gatedpreviouslyfordifferentloadingconditions,temperatures,geometriesandcompositions.
These studies include deformation, fracture strength and crack propagation [9, 69–71, 71–
78]. Experimental results show that for strain rates of 10
−3
−10
−8
s
−1
, the fracture behavior
of PMMA changes as water uptake increases. For each of these studies, the maximum water
content level reached at saturation is less than 2 wt % because PMMA and water do not
have high chemical compatibility. It is well understood that the process of water sorption
in PMMA follows a Fickian diffusion law [73]. In general, it is accepted that water acts
as a mild plasticizer for concentration values up to 1%. Initially, it was observed that the
deformation behavior of PMMA changes from brittle to ductile mode as water is absorbed
[72, 74]. However, for water concentrations higher than 1%, the deformation reverses from
ductile to brittle. This indicates that, at a certain point of absorption, water embrittles the
material and ceases to act as a plasticizer. In other words, as water concentration increases,
the strain to fracture initially increases (i.e. increased ductility), peaks at the threshold of
1%, and then decreases at higher water concentrations [9].
Shenetal.[9]performedexperimentsonsamplessubjectedtoacceleratedagingbycondi-
tioningtestspecimensinwaterfor22daysat60
◦
C.ItwasfoundthatinPMMAtheductility
initially increases with increasing water content but then reduces at higher concentrations
above 1.1 wt %, such that it appears that the brittleness increases for water contents higher
36
than 1.1 wt %. This phenomena was attributed to the change in the mechanism by which
water is accommodated in the PMMA microstructure as water concentration increases. It
was suggested that, at water concentrations above 1.1 wt %, the behavior changes as water
molecules begin to cluster and act more as filler particles than as a plasticizer, leading to
stress concentration effects and early fracture. Thus, water acts predominantly as a mild
plasticizer for PMMA at water contents of about 1.1 wt %, and, at higher water contents,
clustering occurs, which yields significant changes in the deformational response and a drop
in the tensile strength [9].
Arnold [73] completed quasi-static tensile tests at a strain rate of 10
−4
s
−1
on PMMA
samplespre-conditionedinwaterandotherliquidsforperiodsoftimerangingfrom1minute
to 14 days. Results showed that ductility increases with immersion time. It must be noted
that all of their samples exhibited a water content lower than 1 wt % after conditioning, so
thefindingsshownbyShenetal.[9]werenotobserved. Nevertheless,theseresultssupported
the concept that plasticization occurs when PMMA samples contain water concentrations
lower than 1 wt %.
Hamouda [74] studied the tensile deformation and fracture of PMMA on samples that
were exposed to different humidity levels and immersed in water to achieve 0.8 wt % of
water uptake. The experiments occurred at a loading rate of 1 mm/min. Properties such
as crazing, tensile fracture stress, strain-to-fracture and fracture surface morphology were
investigated. It appears that this study was an extension of the work done by Shen et al.
[9] since more cases were analyzed, which added more data points on the material behavior
as water content increased. Hamouda [74] found that water sorption leads to not only the
improvement of flexibility, ductility, and fracture toughness of PMMA, but also a reduction
in hardness and stiffness for tensile tests in samples with levels of water absorption of up to
0.8%. The transition from brittle to ductile mode was clearly observed as the water content
increased from 0 to 0.8 wt %, which indicated that the material toughness increases with
watercontentuptoaconcentrationof0.8wt%. Ingeneral,inagreementwithShenetal.[9],
it was concluded that ductility increases with water content up to 0.8 wt %. Furthermore,
37
the tensile strength for PMMA reduces with the increasing water content.
Ishiyama and Higo [69] performed tensile tests for three different humidity conditions
at three different strain rates (10
−3
s
−1
, 10
−4
s
−1
, 10
−5
s
−1
) in an effort to study the effect
of humidity levels on the Young’s modulus of PMMA samples exposed to their respective
environments for 60 days. Results showed that strain rate and water content affected the
Young’s modulus of PMMA, such that it decreases with increasing water content.
All of the previously mentioned studies describe changes in the mechanical behavior
of PMMA when subjected to quasi-static testing with varying amounts of sorbed water.
Thus, it is apparent that the mechanical response of PMMA after water exposure is well
understood for strain rates of 10
−5
− 10
−3
s
−1
. As water absorption increases and higher
strain rates are applied, the tensile strength of PMMA will be influenced by two competing
effects: a reduction due to water absorption; and an increase due to higher strain rates
applied. Therefore, results for lower loading rates cannot be generalized to higher loading
rates without previous experimentation.
Vinyl Ester Resin
The mechanics of water sorption in vinyl ester are well understood for samples subjected to
accelerated aging tests (e.g. hygrothermal aging) or natural aging tests (e.g. room temper-
ature water immersion) [79–81], and it is commonly accepted that the low water sorption
capability of vinyl ester resin – with values ranging from 0.2 wt % to 0.8 wt %, depending
on the conditioning temperature and duration – is due to few ester groups available in its
structureforhydrolysistooccur[82–84]. Theeffectofwatercontentonthemechanicalprop-
erties of vinyl ester neat resin is an active research area, and it has shown greater retention
of mechanical properties over commonly used resins for marine applications (e.g., polyester).
These studies also included the interaction of vinyl ester as a matrix in FRP with various
water contents for different loading conditions [37, 84–89].
38
Fiber Reinforced Polymers
The sorption of moisture in a FRP can cause plasticization of the matrix. Plasticization
mainly affects materials negatively by reducing their stiffness and strength properties. If the
matrix material is a significant factor in both of these properties, then the properties of the
matrix are likely to be critical in how FRPs respond to moisture sorption. Furthermore,
the presence of moisture at the fiber/matrix interface can modify their interfacial bonding,
resulting in an alteration of the mechanical properties of FRPs. The structural integrity and
life time performance of FRPs exposed to different humidity environments are strongly de-
pendent on the quality of the fiber/polymer interfacial region and the hydrolysis capabilities
of the constituent matrix material [14, 90].
Environmental studies for different loading conditions and long-term water exposure on
CFVE have been previously conducted by a few research groups. Results characterizing the
effect of water content on the mechanical behavior of CFVE samples vary between differ-
ent researchers. This variation can be attributed to the inconsistency in the manufacturing
procedure of FRPs among research groups. Since the composition and post-curing proce-
dures yield different mechanical properties in composite materials, it is difficult to draw
comparisons among different results encountered in the literature.
Ramirez et al. [91] studied the degradation of the mechanical properties of single fiber
CFVE samples after water immersion by monitoring their tensile strength, flexural strength,
and fiber/matrix interface strength. The tensile and flexure strengths of single fiber CFVE
samples were reduced by 40 and 45%, respectively. The reduction in tensile strength was
attributed to the degradation in the fiber/matrix interface. Siriruk and Penumadu [85]
studied water degradation in fatigue experiments on CFVE samples. Results showed that
fatigue life was shortened by up to 85% for samples immersed for at least 6 months at 40
◦
C
in seawater, showing a very significant loss of mechanical strength under cyclic loading.
Additionally, Siriruk and Penumadu [92] performed tensile tests at a strain rate of 10
−6
s
−1
to study degradation on the failure strength on CFVE samples due to long-term exposure to
water (16 weeks). A reduction of 6% in the failure strength of CFVE samples was identified
39
duetowaterdegradation. Siriruketal.[93]studiedtheeffectofseawateronmode-Ifracture
toughnessofinterfacialdelaminationofCFVEsandwichstructuresunderquasi-staticloading
(2 mm/min). Results showed that CFVE sandwich samples, exposed to water for 90 days,
experienced a 30% degradation in delamination fracture toughness.
On the other hand, Kootsookos and Mouritz [94] evaluated the effect of seawater immer-
siononthedurabilityofthefiber/matrixinterphaseofthecompositesbymode-Iinterlaminar
fracture toughness testing at a loading rate of 10 mm/min. Results exhibited considerable
scatter in the interlaminar fracture toughness values, and no significant variation was ob-
servedforsampleswithdifferentwatercontents. Thefracturesurfacesofthemode-Isamples
tested were examined for visible signs of fiber/resin degradation using SEM. There was not
noticeable difference between fracture surfaces, supporting the observation that the tough-
ness of fiber/matrix interphase to the composites was not affected by seawater immersion.
While a few studies for different loading conditions and water contents have been con-
ducted for CFVE composite materials, there is still a lack of fundamental understanding
on how CFVE laminates react as they are exposed to different humidity environments and
subjectedtodynamicloading, whichcanprovideahighlydynamicresponseresultinginhigh
strain rates within a composite structure.
B. Effect of Surrounding Liquids on Dynamic Fracture
It is well-known that the surrounding environment also has an important role in the fracture
behavior of a material [95]. Specifically, the effects of liquids surrounding a propagating
crack have been studied for many types of materials and liquid environments. Some of the
most relevant studies and their findings will be briefly described.
One of the first studies on the effect of liquid environments on fracture was performed
on PMMA and inorganic glass by Williams and Marshall [96]. A fracture mechanics ap-
proach was used in conjunction with time-dependent material parameters to describe crack
propagation in both air and liquid environments. A fluid flow model was inserted into crack
propagation analysis to describe how failure processes were developed under different sur-
40
rounding environments. This model was applied for crack speeds of up to 10
−1
m/s and,
among their results, it was stated that water causes plasticization of crack fronts.
Mai[97]performedanexperimentalstudytoexaminecontinuousslowcrackingofPMMA
in the presence of a variety of liquid environments. Single-edge notched specimens were
subjectedtouniaxialloadingatstrainratesontheorderof10
−3
−10
−2
s
−1
. Itwasdiscovered
that there is an increase in the energy required for fracture to occur in the presence of
liquids, such as water, oils and alcohols, thus, plasticizing the crack-tip and increasing the
toughness of the material. These results complement the findings by Williams and Marshall
[96]. Additionally, it was discovered that the effect of water on fracture toughness is rate
sensitive, which means that the effect of water on crack propagation was dependent on the
crack-tip speed. Michalske and Frechette [98] conducted further studies to analyze the rate
sensitivity of fracture toughness and proposed that for crack-tip speeds higher than 0.1 m/s,
an accelerating crack is able to completely escape all effects of water. This behavior was
attributed to the viscosity of the liquid, which does not allow the liquid to flow into the
crack-tip fast enough. In other words, the moment the crack-tip speed is fast enough for it
to escape the effect of the surrounding liquid, the fracture behavior of the material is the
same as in air.
Additional studies have yielded similar results, which can be summarized in that, at low
crack-tip speeds, fracture properties are drastically changed when a crack propagates while
immersedinwater. Atlowstrainrates, thefracturetoughnessofPMMAincreasestoalmost
double that of the material in air due to the plasticization effect of water. The reason for
this behavior is that the duration of the fracture event is longer, which allows more time
for water to flow into the crack tip and promote plasticization. On the other hand, as the
strain rate is increased, water does not seem to affect the fracture behavior of the material
[99, 100]
Alternative mechanisms have been suggested to explain the effect of liquids on low-
speed crack propagation. Among some of those mechanisms are surface tension, hydrogen
bonding breakage or chemical interactions between the surrounding liquid and the polymer,
41
and the absorption of some of the stored elastic strain energy by the liquid [95, 101–103].
Nevertheless, it seems that plasticization occurring on the crack-tip at strain rates lower
than 10
−2
s
−1
[102] or crack propagation speeds lower than 10
−1
m/s [98] prevails as the
mostacceptedexplanationforthevariationoffracturebehaviorofPMMAinthepresenceof
water. Allthepreviouslymentionedexperimentshaveshownresultsforspecimenssubjected
to quasi-static loading and resulting crack-tip speeds in the range of 3×10
−5
to 2×10
−1
m/s.
Moreover, no studies were found in the literature for materials subjected to dynamic loading
when exposed to a liquid environment.
There exists a general understanding of the effect of surrounding water on polymer frac-
ture for quasi-static loading and low-speed crack propagation. Still, most of the studies
have used simple fracture tests to quantify the effects of surrounding liquids, and real-time
crack-tip behavior under extreme environmental conditions has been significantly less stud-
ied [100]. An extension of these studies to higher strain rates could possibly identify other
mechanisms that may affect the dynamic fracture behavior of a material since it is well
known that the mechanical response of PMMA changes when it is subjected to dynamic
loading. For this reason, crack initiation in PMMA immersed in water when subjected to
an impact were investigated in this work. These experiments were executed to observe if
there is any sort of coupling between the surrounding fluid and the solid during the impact
event that would affect the dynamic behavior of cracks propagating underwater, e.g., surface
energy dissipation into the surrounding water may arise due to higher strain rate. To the
best of the authors’ knowledge, there have been no previous experimental studies utilizing
high-speed visualization to measure in-situ underwater fracture initiation when a polymeric
material has been subjected to strain rates in the orders of 10
2
s
−1
.
42
Chapter 3
Methods
The type of work done in this thesis was predominantly in the area of experimental solid
mechanics. Thus, several techniques, experimental and optical, had to be developed. In this
section, the design and principles of the experimental techniques are described.
I. Experimental Setup
A challenge of dynamic loading experiments is to obtain repeatable loading conditions, so
the results from consecutive experiments can be compared. Since each polymeric sample is
impacted only once and then compared to other samples, it is crucial to have full control of
the external loading conditions that are applied to the sample in terms of pulse duration,
pulse amplitude and maximum strain. The experimental setup presented was designed to
generate repeatable impacts onto solid structures under dry or wet conditions.
A. Dynamic Loading
The setup, shown in Figure 3.1, consisted of a pressurized gas-gun, a visualization system
with a high-speed camera, a catcher box to contain the experimental samples during and
after impact, and a test section or sample holders placed inside the catcher box. The optical
section of the experimental setup shown in Figure 3.1 consisted of a series of two flat mirrors
and two concave mirrors used to transmit collimated light through the test sample. The
visualization system was set up to allow for different types of visualization techniques (e.g.
schlieren, shadowgraph), which justifies the Z-folded mirror system in Figure 3.1.
Thevelocityoftheprojectilewasdependentonthepressureinthegas-gunreservoiratthe
time of launch. This pressure was monitored by a GE Druck DPI 104 digital pressure gauge.
43
Figure 3.1: Experimental setup consisting of an (1) gas-gun, a (2) catcher box, a (3-6)
visualization system, and a test section placed inside of the catcher box.
For all the dynamic loading experiments in this work, a 10 mm diameter steel sphere was
loaded into a sabot, a 25.4 mm long and 50.7 mm diameter acrylonitrile butadiene styrene
(ABS) cylinder. The center of the sabot contained a bored hole to carry the projectile down
the 2.1 m long and 50.8 mm inner diameter gun barrel during the launch. As the sabot
exited the gun barrel, it was stopped mechanically by a metal plate with a hole large enough
to allow only the spherical projectile to continue its trajectory and impact onto the edge of
the sample, see Figure 3.2. To monitor the speed of the projectile, and trigger the camera
and strain gauge equipment, a laser beam (650 nm - 5 mW), denoted (1) in Figure 3.2,
was aimed through the barrel 100 mm ahead of the gun barrel exit. Placed opposite to
the laser beam was a photodiode receiver (HFBR-25x5AZ, AVAGO Technologies), see (3) in
Figure 3.2. When the laser light was interrupted by the sabot, the output signal from the
photodiode was used to trigger all necessary equipment. Additionally, the projectile speed
was measured using the output signal from the photodiode receiver. As the projectile passed
44
Figure 3.2: Schematic of impact setup (shown to scale). Impact occurs from left to right. (1)
Laser beam, (2) Laser, (3) Photodiode receiver, (4) Gun barrel, (5) 10 mm diameter steel
projectile carried inside sabot (6) ABS sabot, (7) Metal stopper, (8) 10 mm diameter steel
projectile before impact, (9) Sample, (10) Area of interest upon which camera is focused.
by this sensor, it generated a square wave shown in Figure 3.3. The pulse duration of the
square wave, Δt, represents the time it took the sabot to travel through the sensor; thus, Δt
and the length of the sabot were utilized to calculate the impact speed. A Phantom V711
0 0.5 1 1.5 2 2.5
0
1
2
3
4
5
6
Voltage [V]
Time [ms]
Δt
Figure 3.3: Velocity profile for sabot passing a velocity sensor.
high-speed camera was synchronized with a pulsed laser (SONY SLD1223V 670-nm laser
diode combined with PicoLAS LDP-V drive module) used as a light source. More details
on the synchronization system are presented in [104]. The pulsed laser reduced blur in the
images since the duration of the laser pulse is 70 ns, which is shorter than the minimum
45
exposure time capabilities of the high-speed camera (0.3μs). Thus, the duration of the laser
pulse functioned as the exposure time for these experiments.
One of the main challenges due to alignment issues was to obtain repeatable central
edge-on impacts onto the samples to compare the results from consecutive experiments. For
this reason, an alignment setup, contained inside the catcher box, was designed such that
samples were aligned with the longitudinal axis of the gun barrel (the direction of impact).
A schematic drawing of the apparatus utilized to generate dynamic loading conditions is
featuredinFigure3.4. Thesetupconsistedoftwoaluminumstands,two0.6mlongalignment
pins, and a stainless-steel test section. The front aluminum stand, which was clamped onto
the end of the gun barrel, and the rear stand (not shown in Figure 3.4), were both bolted
onto an optical table. The stands included two holes located (vertically) above and below
of the gun barrel, aligned with the barrel’s centerline. The two alignment pins were inserted
into the holes to create a rail system used to properly align the test section in front of the
exit of the gun, as shown in Figure 3.4.
Thetestsectionwasdesignedtoensurecentraledge-onimpactsontothesamplesinorder
to obtain reproducible successive experiments. The test section consisted of a 150 mm long
chamber with a thickness of 12 mm. The windows were made of 6.35 mm thick optically
clear PMMA sheets. The test section allows visualization techniques in transmission, and
it is capable of containing liquids without clamping the sample when the impact occurs,
see Figure 3.5. An exploded view of the test section is shown in Figure 3.6 to illustrate its
features in detail. The test section comprises of an assembly of a stainless steel frame and
twoacrylicsidewindowstoallowforlighttransmission. Additionally,thetestsectionhasthe
capability of containing liquids in the volume encapsulated by the windows and the frame.
O-rings were included as part of the assembly to create a seal between components of the
testsectionandavoidwaterleakageattheinterfaces. Thewatersupplywasfedintothetest
section through a small orifice located on its top surface. The water was supplied through
a 2 mm diameter hose that connected the test section to the top of the catcher box, where
a small water-filled reservoir was placed. The experimental samples were inserted into the
46
1
2
3
4
5
6
Figure3.4: Alignment schematic. The setup consists of: (1) Gun barrel, (2) Front aluminum
stand, (3) Sabot utilized to carry projectile inside of gun barrel during launch, and the sabot
was mechanically stopped as it exited the barrel (mechanism not shown in this schematic),
(4) 10 mm diameter stainless steel sphere projectile, (5) Alignment pin, (6) Stainless-steel
test section positioning sample in place.
Figure3.5: Test section designed to maintain the pre-notched area of the samples surrounded
by liquid or air during impact and allowing visualization techniques relying on transmitted
light to capture fracture events. Direction of impact is indicated by sphere and arrow.
47
test section as shown in Figure 3.4 allowing the middle part of the specimen, containing the
notch, to be immersed in water. Note that during the experiments, samples were allowed to
move in the longitudinal direction to avoid introducing complex boundary conditions. For
this to occur, it was required that there was no tight seal in the gap through which the
samples were inserted into the frame. Thus, water was allowed to leak through the interface
between the test section and the specimens. In order to maintain the test section filled with
water for the duration of the experiment, a constant feed of water was provided from the
reservoir. Once the samples were placed in the test section and immersed in water, the
specimens were surrounded by a 5 mm thick water sheet on each of its sides.
1
2
3
4
5
1
Figure 3.6: Exploded view of test section: (1) Window, (2) O-ring to prevent water leakage,
(3) Frame metal insert to secure sample, (4) Sample, (5) Holder frame.
Alternatively, the experiments described in Chapter 5 did not require the use of the
stainless steel test section because these experiments occurred only in an air environment,
so sample holders were designed to ensure central edge-on impact onto the sample. These
sample holders were made out of acrylic and were used to position the sample in place as
shown in Figure 3.7.
48
Figure 3.7: Alignment setup consisting of two holders used to align the samples with the gun
barrel by the use two steel pins.
B. Strain Gauges
A 3.5 mm long strain gauge (Micro-Measurements EA-13-125BZ-350/LE) was attached to
each sample to record the impulse loading and to ensure repeatability of loading conditions
betweensuccessiveexperiments. Thestraingaugeswereinstalledontoacleansamplesurface
following the instruction bulletin (Micro-Measurements B-127-14). Once the electrodes of
the gauges were soldered to an electrical wire, the gauges were covered with a protective
layer of silicon rubber (Micro-Measurements M-coat C). The strain gauges were connected
to a quarter Wheatstone bridge from a signal conditioning amplifier (Vishay 2310B). The
excitationvoltageusedis2.7Vandagainof49. Theoutputfromtheamplifierwasrecorded
byanoscilloscope(LeCroyWaveSurfer24Xs-A)withasamplingrateofupto500MHz. The
data obtained was post-processed using Equation (3.1):
ǫ =
4V
out
V
ext
×G×K
, (3.1)
49
where V
out
is the voltage recorded by the oscilloscope, V
ext
is the excitation voltage, G is the
gain applied and K is the gauge factor.
II. Optical Methods
Non-invasive visualization diagnostics were used simultaneously with strain gauges. The
data gathered during an experiment was used to understand the material’s response during
loading. The visualization techniques had various purposes; first, to quantitatively mea-
sure the response of the sample without introducing any disturbances to the sample itself,
and second, to understand the loading condition that is applied to the sample. A total of
three different visualization techniques were implemented during this study: the method
of caustics, Digital Image Correlation and a Fizeau Interferometer. The principles of the
visualization techniques utilized and their respective applications in this work are described
in this section.
A. Methods of Caustics
The method of caustics was used to obtain qualitative and quantitative dynamic fracture
data from the experiments performed on transparent samples. The method of caustics was
applied in compliance with the guidelines outlined in [105]. Figure 3.8 shows the physical
principle of the method of caustics.
A point light source is utilized to generate collimated light and illuminate a sample
containing a crack which is loaded by a tensile load. The stress concentration in the area
surrounding the crack-tip due to the tensile load causes a reduction of the thickness of the
specimen. As a result, the area around the crack-tip acts similar to a divergent lens, the
light transmitted through the specimen is deflected outwards. This light is projected onto a
parallel plane (the reference plane) at a distance z
0
from the specimen. The location of the
reference plane in the experimental setup used for this work can be seen in Figure 3.9, and it
represents the plane where the lens of the high-speed camera was focused. On the reference
50
plane, the crack-tip appears surrounded by a dark shadow. The shadow is surrounded by an
intense light concentration called a caustic. The light is concentrated because the angular
deflection of the light rays decreases with increasing radial distance from the crack-tip. As
an example of its appearance, Figure 3.10 shows a caustic obtained from one experiment on
PMMA. The size of the caustic is directly related to the stress-strain conditions in the area
around the crack-tip and the stress intensity factor. This occurs because the magnitude of
the deflections of the light rays is correlated to the magnitude of the stress concentration at
the crack-tip. The stress intensity factor can be calculated using this method by applying
Equation (3.2) [106]:
K
d
I
=
2
√
2πF(v)
3m
3/2
ctz
0
D
3.17
5/2
, (3.2)
where,
F(v) =
4α
1
α
2
−(1+α
2
2
)
2
(α
2
1
−α
2
2
)(1+α
2
2
)
,
α
1
=
1−
v
2
c
2
L
1
2
,
α
2
=
1−
v
2
c
2
T
1
2
.
The SIF can be expressed in terms of the transverse diameter of the caustic, D, the
thickness of the sample, t, and the crack-tip speed, v. Figure 3.10 shows one of the caustic
images obtained in this study next to a schematic of a caustic curve to exemplify how the
transversediameterandthelocationofthecrack-tipwereobtained. Themeasurementswere
taken at the points of the caustic curve where the highest intensity pixels were located, since
these represent the highest concentration of light [107].
Themethodofcausticswasutilizedfortransparentmaterialsandforsamplessurrounded
by water. The analysis performed to adapt this method to experiments with water is shown
in Chapter 7. Further improvements can be applied to the method of caustics to better
quantifythedynamicfracturebehaviorofmaterialsbyincludingtransienteffectsasdescribed
by [108]. Transient effects were not included in this work because not enough temporal and
spatial resolution were available to accurately solve the equations shown in [108]. However,
a correction factor, F(v), was utilized to adjust for crack-tip speed.
51
D
Reference
Plane
Sample
Incident collimated
light rays
σ
σ
x
x
2
3
d
z
o
Crack
Figure 3.8: Schematic of caustic formed by light rays passing through a transparent sample.
The reference plane, located at a distance z
o
from the specimen, is shown in the image on
the left.
Figure 3.9: Top view of experimental setup showing distance to focus plane, z
o
, used for the
method of caustics.
52
Figure 3.10: Caustic curve image (left) and schematic (right) indicating the caustic trans-
verse diameter and crack-tip location. This measurement procedure was obtained from [107].
B. Digital Image Correlation
Digital Image Correlation (DIC) has been extensively applied in the field of experimental
solid mechanics, as it is a powerful tool to obtain quantitative data from events in which
displacement measurements are required. DIC is a full-field non-contact optical technique
that is used to measure displacement fields for both transparent and opaque samples of var-
ious materials in any shape. DIC relies on comparing images that represent the specimen in
an undeformed state with subsequent images representing the sample in a deformed state.
A schematic of the basic principle of this optical method is shown is Figure 3.11. A subset
of certain pixel size centered at point P in the undeformed subset is used to find its loca-
tion in the deformed subset according to a correlation function. When the subset is most
similar between images before and after deformation, the correlation coefficient maximum is
detected. The variables, u
x
and u
y
, are the displacement components for point P in the x
and y directions, respectively [109–112].
The setup, shown in Figure 3.12, used for experiments is the same as the one previously
described. The visualization system was modified to obtain images from the reflection of
53
P
Q
Q’
P’
Δx
Δy
u
u
x
y
Undeformed subset
Deformed subset
x
y
Figure3.11: Digital Image Correlation principle showing subset prior and after deformation.
high-intensity LEDs onto the speckle pattern surface of the sample. As part of the exper-
imental procedure to use DIC, a speckle pattern was applied to the surface of the samples
prior to the experiments. A layer of white paint was first applied to the specimen surface.
Then, a black pattern was applied on top of the white paint using spray paint, resulting in a
random speckle pattern. The light intensity of the speckle patterns was recorded before and
after deformation. The random speckles on the samples were then monitored using high-
speed photography during a fracture event. Deformed images corresponding to the fracture
event were paired with the ones prior to loading. Images taken during the impulse loading
were then exported to the commercial software VIC-2D (digital image correlation software
by Correlated Solutions Inc.), in which a 2D cross-correlation between the two selected sub-
sets was applied. This process was repeated for the entire image until full-field in-plane
displacements were obtained over an array of grid points to extract the displacement and
strain fields. In this software, subset and step sizes of 17 and 3 pixels, respectively, were
used to calculate the displacements [65, 113].
ThereasonDICwasusedforthisexperimentsisbecauseithasprovenasuccessfulmethod
to study dynamic response in a variety of materials including mode-I and mixed-mode frac-
54
(1)
(2)
(2)
(3)
(4)
Figure 3.12: Sketch of experimental setup for DIC showing: (1) Phantom camera, (2) high-
intensity LED lights, (3) sample placement, and (4) gun barrel.
ture[113–116]. Itwasofparticularinteresttoapplythedisplacementmeasurementsobtained
from DIC to extract the SIF from crack propagation experiments.
C. Interferometry
A Fizeau interferometer was employed to quantify the effect of water on the stress optical
coefficientofPMMAsamplesintheworkdescribedinChapter7. Thistechniquewasapplied
because it is capable of relating the stress applied to the optical path variation. The Fizeau
interferometer was introduced by Theocaris and Gdoutos [117] as an experimental method
for solving plane-stress elastic problems. A Fizeau interferometer is based on individual
measurements of the absolute retardation of a monochromatic light. This method uses the
specimen as the interferometer, and the interference fringes obtained are of the Haidinger
type. One of the main advantages of this technique is that the samples do not need to be
optically flat provided that the deviations from flatness are not abrupt. Additionally, it is
55
oneofthemoststraightforwardmethodsbasedonmeasurementsoftheabsolutevariationsof
the optical paths for the solution of elasticity problems because it does not use any external
elements for creating interference, and it has an uncomplicated optical setup. The main
disadvantageofthemethodisthattheprincipalstressdirectionsmustbeknownbeforehand,
but this is easily determined in uniaxial tensile tests. Raftopoulos et al. [118] applied this
technique to study the static and dynamic SOC of dry PMMA specimens. The authors
provided an in-depth description of their experimental setup, which was utilized to build the
experimental setup used in this current study.
The experimental setup is shown in Figure 3.13. A He-Ne laser with a wavelength of
λ=632 nm, similar to the one utilized in [118], and a 10 mm beam diameter, generated
by using a 15× beam expander, was used to impinge light onto the sample. The light
is normally incident on a thin plate under generalized plane-stress conditions to obtain a
reflection from the front and rear faces of the sample. The absolute variations of the two
reflections from the front and rear faces of the plate results in a full field map containing
information about the thickness and refractive index of the samples in their unloaded state
[117]. A Phantom V711 high-speed camera was used to record the fringes as the samples
were loaded, allowing full-field visualization, which yielded multiple points for analysis.
The interference created from the reflection of each surface creates well-defined fringes.
Once a load is applied on the specimen, a displacement of the interference fringes along the
field is created due to the change in thickness of the sample. By counting the number of
fringes passing across each point, the SOC can be calculated. The reason this technique
is applied is not only because it is straightforward, but it was also shown that a Fizeau
interferometer is significantly more sensitive and accurate than any other interferometric
method using an external interferometer or classical photoelasticity to perform coefficient
measurements [117].
The theoretical analysis of stress-optical relationships that result in the equations used
for the Fizeau interferometer are not described here, since the theoretical derivation has ap-
peared in previous work, see references [117–119]. Only the final formulation of the previous
56
Figure3.13: Interferometersetup: (1)PhantomV711high-speedcamera, (2)PMMAsample,
(3) 15X beam expander, (4) He-Ne laser with λ=632 nm.
analysis is shown. The assumptions made to arrive at these equations are that the material
utilized, PMMA, is an optically isotropic material, also called optically inert, which means
that its optical properties are the same in all directions, and that the fringe count will be
the same in both principal stress directions [117, 118].
The equation used to calculate the SOC in transmission, C
t
, is given by
C
t
=
α
t
+β
t
2
, (3.3)
where α
t
and β
t
are the stress optical coefficients in transmission for each principal direction
1 and 2, respectively. Equation (3.3) is derived from the Maxwell-Neumann stress-optical
law. Thisequationaccountsforthechangeinthicknessofthematerialandhowitaffectsthe
interaction of light rays reflected from the front and rear face of the sample. Additionally,
mechanical properties, such as Young’s modulus, E, and Poisson’s ratio, ν, are included in
the analysis.
The equations used to relate the number of fringes counted along the two principal
57
stress directions, N
1
and N
2
, during loading and the stress optical constants describing the
interference caused by reflected rays, α
∗
and β
∗
, for uniaxial tension test (σ
2
= 0) are
α
∗
=
N
1
2bσ
, (3.4)
β
∗
=
N
2
2bσ
, (3.5)
where b is the thickness of the samples and σ is the load applied. To relate the coefficients
α
t
andβ
t
and the constantsα
∗
andβ
∗
, the relationsα
t
=−λα
∗
+ν/E andβ
t
=−λβ
∗
+ν/E
can be utilized. Since PMMA is an optically isotropic material and presents very weak
birefringence, it has been previously proven that the fringe count in the principal directions
are equal, i.e. N
1
= N
2
, which means that α
∗
= β
∗
[118]. By combining Equation (3.3)
and (3.4), the SOC can be explicitly expressed as
C
t
=−λ
N
1
2bσ
+
ν
E
. (3.6)
Equation (3.6) shows a direct contribution from both mechanical and optical properties.
This was utilized to further analyze the contribution of water content on the variation of the
optical or mechanical properties of the material.
58
Chapter 4
Influence of Water Uptake on Dynamic
Fracture Behavior of Poly(Methyl
Methacrylate)
A study was completed to assess the effects of various humidity levels and amount of
sorbed water on the fracture behavior of notched PMMA samples subjected to stress pulses
generated by the impact of a projectile launched from the dynamic loading setup described
in Chapter 3. Impact experiments were performed on six sets of samples conditioned in
different environments: dry samples; samples exposed to three different relative humidity
environments(11%,60%,and98%)usingsaturatedsaltsolutions(LithiumChloride,Sodium
Bromide, and Potassium Sulfate, respectively); and distilled water- and seawater-exposed
samples. Experiments varied by immersion time and water content, while loading conditions
were kept constant. The main goal of this study was to understand the effects of sorbed
water on the fracture behavior of PMMA when subjected to high strain rate impacts. It
was observed that when PMMA is subjected to strain rates of 10
2
s
−1
, the effect of water
content is not a dominant mechanism on the crack initiation and crack-tip speed of PMMA.
This work was published in the journal of the Society for Experimental Mechanics under
Influence of Water Uptake on Dynamic Fracture Behavior of Poly(Methyl Methacrylate) by
Delpino Gonzales and Eliasson [78].
59
I. Experimental Procedure
Rectangular PMMA samples were impacted on their short edge by a spherical projectile
launched from a gas-gun. Loading conditions were monitored using strain gauges, and the
crack initiation and propagation were visualized using the method of caustics. Details of the
experimentalsetupandenvironmentalconditioningparametersareexplainedinthefollowing
two subsections.
A. Material and Conditioning
The material used for the experimental samples was commercially available PMMA sheets
with the same geometry as used in [120]. Samples with a thickness of 3.125±0.2 mm were
laser-cut to rectangular shapes with dimensions 300 mm × 40 mm. At half-length of each
sample, a 10 mm long and 0.152 mm wide notch was sawed. This type of geometry, shown
in Figure 4.1, allowed for a pure mode-I crack to occur when a tensile pulse was applied.
Due to its cross-sectional aspect ratio (12:1), three-dimensional effects could be ignored, and
it could be assumed that the wave traveling through the sample due to impact was a plane-
stress wave [120]. A total of 61 samples were divided into three different sets corresponding
300 mm
40 mm
(1)
Figure 4.1: Drawing of notched PMMA sample with dimensions 300×40×3.125 mm
3
. The
location of the strain gauge is indicated by (1).
to their respective environment exposure: (1) relative humidity (RH); (2) liquid; and (3)
dry. Prior to environmental conditioning, all samples were heated for 24 hours at 76
◦
C to
obtain dry samples and remove residual stresses.
60
The first set of samples investigated the effects on fracture behavior due to exposure to
three different RH environments. These three RH environments, at 11%, 60%, and 98%,
were obtained by using three saturated salt solutions (Lithium Chloride, Sodium Bromide,
andPotassiumSulfate, respectively)at 25
◦
C[74]. Theamountsofsaltusedandthesolution
intensities are shown in Table 4.1. The saturated solutions were placed in air-tight plastic
containerswiththesamplespositionedinholders40mmabovethesurfaceoftheliquidsolu-
tions. The relative humidity values in the closed containers, generated by the vapor pressure
from the solutions, were frequently monitored using a thermo-hygrometer and matched the
results from Greenspan [121]. Samples presented in this manuscript were conditioned in
their various environments for up to 40 days, each at 25
◦
C.
Table 4.1: Composition of saturated salt solutions
Solution RH [%] Used amount [g] Volume of water used [ml] Intensity [g/ml]
LiCl 11 2505 1000 0.835
NaCl 60 2199 1000 0.733
K
2
SO
4
98 333 1000 0.111
The second set of samples were used to investigate the effects on dynamic fracture be-
havior due to conditioning in two liquid environments: distilled water and seawater. The
seawater was obtained from a simulated sea salt mix (Lake Products Company LLC) that
meets the ASTM D1141-52 standard as a substitute for seawater. Samples were exposed to
liquid environments for 3, 10, 20 and 40 days. The third set of samples were the dry samples
used for comparison purposes.
The weight change in the samples during their exposure time was frequently monitored
using a weight scale (ML4002E/03, Mettler Toledo), with a readability of 0.01 g, to obtain
thewatercontentasafunctionoftime. Theweightchangewascalculatedusingthefollowing
equation:
C =
W
f
−W
i
W
i
×100 (4.1)
whereC (wt%)isthewatercontent,W
i
istheweightofasamplebeforeimmersion,andW
f
is
theweightofthesamplecontainingsorbedwater. Figure4.2showstheweightmeasurements
61
as a function of time. It could be observed that the water content of each sample increased
with time, but the rate of absorption decreased with time, in agreement with [70]. Also, the
higher the relative humidity environment, the higher the water content in PMMA, as shown
in Figure 4.2(a). As expected, the water absorption was higher for the liquid environments
compared to the two lower RH environments of 11% and 60%, see Table 4.2. However, the
samples exposed to the 98% RH environment showed a similar rate of absorption as the
samples exposed to liquid environments, see Figure 4.2(c).
0 5 10 15 20 25 30 35 40
0
0.2
0.4
0.6
0.8
1
1.2
Water content [%]
Time [days]
98% RH
60% RH
11% RH
(a) 3 RH environments
0 5 10 15 20 25 30 35 40
0
0.2
0.4
0.6
0.8
1
1.2
Water content [%]
Time [days]
Seawater
Distilled Water
(b) Distilled water and seawater
0 5 10 15 20 25 30 35 40
0
0.2
0.4
0.6
0.8
1
1.2
Water content [%]
Time [days]
Seawater
Distilled Water
98%
(c) Comparison
Figure 4.2: Average weight change as a function of time due to water absorption for samples
exposedto(a)threedifferentRHenvironments(11%, 60%, and98%)for40days, (b)distilled
water and seawater up to 40 days, (c) comparison of weight absorption for samples exposed
to 98% RH, seawater, and distilled water. See details in Table 4.2.
The visualization technique utilized to extract quantitative data from these experiments
was the method of caustics. The material properties and experimental constants used for
the analysis are listed in Table 4.3. Experiments were performed in order to determine
62
Table 4.2: Average water absorption for experimental samples
Environment
Weight change (%)
3 Days 10 Days 20 Days 40 Days
11% 0.03±0.03 0.04±0.03 0.05±0.03 0.06±0.02
60% 0.12±0.03 0.41±0.02 0.55±0.03 0.63±0.06
98% 0.19±0.01 0.57±0.02 0.83±0.05 1.08±0.08
Distilled water 0.14±0.07 0.55±0.02 0.81±0.08 1.12±0.16
Seawater 0.13±0.03 0.50±0.03 0.78±0.03 1.06±0.06
if the longitudinal wave speed, c
L
, an inherent material property, was affected by water
absorption. Longitudinal wave speed measurements were performed under constant impact
loading conditions for a total of six PMMA samples: three dry samples and three samples
that had been immersed in distilled water for 40 days. For these six experiments only, an
additional strain gauge to the one shown in Figure 4.1 was glued to the sample such that
bothstraingaugeswereplaced100mmapart(50mminfrontofandbehindthenotch)along
the centerline of the sample. The samples were impacted, and the longitudinal wave speed
was calculated by measuring the time of arrival at each strain gauge. However, no difference
could be found between the longitudinal wave speeds for samples with high water contents
(∼1.12 wt %) and dry samples. The average wave speed measured for all six samples was
c
L
= 2,272± 16 m/s, which is similar to c
L
= 2,290 m/s, the longitudinal wave speed
(plane-stress) reported in [120]. Additionally, an ultrasonic thickness gauge (Olympus 38DL
PLUS) was used on six dry PMMA samples to determine their transverse wave speed, c
T
.
The value obtained is shown in Table 4.3.
Table 4.3: Values used for the method of caustics
Notation Value Unit
Distance to reference plane z
0
1.4 m
Scale factor m 1.01 -
Longitudinal wave speed c
L
2,272 m/s
Transverse wave speed c
T
1,501 m/s
Stress optical coefficient† c −5.5×10
−11
m
2
/N
† Value obtained from [120]
63
Two high-speed cameras were utilized for imaging crack propagation: (1) a Kirana high-
speed camera with a 400 mm lens and a frame rate of 500,000 fps used for critical SIF
measurements; and (2) a Phantom V711 with a 200 mm or a 50 mm lens for critical SIF
measurementsandcrack-tipspeedmeasurements,respectively. SeemoredetailsinTable4.4.
Table 4.4: Summary of high-speed cameras settings
Camera
Lens focal length Frame rate Time interval Exposure time Resolution Scale factor
[mm] [fps] [µ s] [ns] [pix] [µ m/pix]
Kirana 400 500,000 2.00 100 924×768 58
Phantom V711 200 80,000 12.62 90* 256×256 78
Phantom V711 50 170,000 5.85 30* 224×112 350
*Obtained by synchronizing camera with pulsed laser diode to reduce effective exposure time. See [122] for more details
Figure 4.3 shows a sequence of caustic images obtained with the Phantom V711 high-
speed camera for samples that were conditioned at 11% for 40 days. To obtain quantitative
data from these images, the post-processing procedure first removes any optical distortions
that have been introduced in the system. For this, a virtual grid with known points was
mapped onto the recorded images. Then, the transverse diameter, D, and the crack-tip
location were measured as demonstrated in Figure 3.10. It must be noted that the image
sequence in Figure 4.3 shows certain curviness that could appear as mode-mixity. How-
ever, mode-I was the dominant failure mode. For instance, the initial notch appears to be
nonlinear (or not perpendicular to the sample geometry). Nevertheless, in reality, it was
perpendicular to the cross section of the sample. This is due to a result of the caustics tech-
nique where samples are purposefully imaged out of the focused plane of the sample. This
creates additional optical distortions, which appear to exhibit a mixed-mode crack path.
64
(a) 0 μs (b) 17.55 μs (c) 35.10 μs
(d) 52.60 μs (e) 70.10 μs (f) 87.60 μs
(g) 105.1 μs (h) 122.6 μs (i) 140.1 μs
Figure 4.3: Crack propagation sequence for a sample conditioned at 11% RH. Images show
area of interest as indicated in Figure 3.2.
65
II. Results and Discussion
The repeatability of the impact experiment was measured by performing multiple experi-
mentsandrecordingtheresponsefromastraingaugeattachedtothesampleatthecenterline
50 mm behind the notch, denoted (1) in Figure 4.1. Results of the strain signals are shown
in Figure 4.4. The strain gauge signals correspond to an impact velocity of 37.8±0.40 m/s,
which generated a maximum strain of ǫ
max
= 2480± 255 µ strain on all samples. For the
0 50 100 150 200 250
−4000
−3000
−2000
−1000
0
1000
2000
3000
Strain [με]
Time [μs]
Average Response
Figure 4.4: Strain gauges response for 10 repeated experiments on PMMA samples with
impact velocity 37.8±0.40 m/s.
experiments in this study, the projectile speed was 37.83± 0.82 m/s. This impact speed
generated stress waves in the samples with a pulse duration of 70 μs. The results obtained
for crack-tip displacement for samples conditioned in the three different RH environments
are shown in Figure 4.5(a). A slight difference in crack propagation can be observed; cracks
propagated faster as the water content increased. However, Figure 4.5(b) shows that the
crack-tip speed for all three cases converged to the same value as the crack continued to
propagate. The main difference in the crack-tip speed was observed only in the initial seg-
ment of the crack propagation. The strain plot shown in Figure 4.5(c) confirms that the
applied loading conditions were the same in all cases.
No difference was observed in the crack propagation behavior of samples exposed to
66
0 20 40 60 80 100 120
0
5
10
15
20
25
Crack length [mm]
Time [μs]
11% RH
60% RH
98 %RH
(a)
0 5 10 15 20 25
0
50
100
150
200
250
300
350
Crack speed [m/s]
Crack length [mm]
11% RH
60% RH
98% RH
(b)
0 50 100 150 200 250
−4000
−3000
−2000
−1000
0
1000
2000
3000
Strain [με]
Time [μs]
(c)
Figure 4.5: Results for PMMA samples exposed to different RH cases; (a) crack-tip displace-
ment, (b) crack-tip speed. The loading conditions were similar for all cases, as seen in (c),
in which the strain responses are plotted.
distilled water and seawater, similar to the results obtained by [72]. This result could be
attributed to the fact that the water absorption levels of both types of samples were the
same throughout the 40 days of conditioning. In addition, soaked samples were compared
to the samples exposed to 98% RH, see Figure 4.6(a). Their behavior did not show any
significant difference, as expected since the water uptake was almost identical. The same
behavior was confirmed for all the cases that had a similar water concentration at the time
of impact: 11% RH and 3-day liquid immersion, or 60% RH and 10-day liquid immersion,
see Figure 4.6(b) and (c), respectively. Finally, comparisons were made between the SIF for
samplesconditionedinthreedifferentenvironments: 11%RH,60%RHanddistilledwaterfor
40 days. Note that in this study, the main focus was to quantify the effects of water content
on the fracture toughness of PMMA; thus, only the initial stages of crack propagation were
67
0 20 40 60 80 100 120
0
5
10
15
20
25
Crack length [mm]
Time [μs]
Distilled water
Seawater
98% RH
(a)
0 20 40 60 80 100 120
0
5
10
15
20
25
Crack length [mm]
Time [μs]
Distilled water
11% RH
Dry
(b)
0 20 40 60 80 100 120
0
5
10
15
20
25
Crack length [mm]
Time [μs]
Distilled water
Seawater
60% RH
(c)
Figure 4.6: Crack propagation comparisons with samples exposed to various environments
resulting in similar water uptake: (a) 40 day exposure in seawater, distilled water and 98%
RH; (b) 11% RH for 40 days, dry samples and samples exposed to distilled water for 3 days;
(c) 60% RH for 40 days and samples exposed to seawater and distilled water for 10 days.
consideredwhencalculatingtheSIF.Resultsshowthatthefracturetoughnessofthematerial
doesnotvarysignificantlywithwatercontent. TheaveragecriticalSIFforeachcaseislisted
inTable4.5. Additionally, Figure4.7showstheSIFattheinitialstagesofcrackpropagation
(up to 15 mm of crack extension). Evidently, at these loading rates, water content has no
significanteffectonthefracturetoughnessofPMMA.Asitcanbeseen, allthreecasesfollow
the same trend and show a clearly defined critical SIF (∼2 MPa
√
m), identified in Figure 4.7
as the stage where the crack begins propagating. Figure 4.8 shows a comparison of the
variation of the critical SIF with water content between the results obtained in this study
with the results obtained by Bokoi et al. [70]. The latter reported critical SIF values for
cracks that propagated at speeds on the order of 10
−7
−10
−8
m/s under static tensile testing
68
(no strain rate value was reported). The effect of water content can be easily identified from
Figure 4.8 for lower strain rates: water acts as a mild plasticizer; thus, the toughness of the
material increases with water content. However, the results reported in this study show that
when the strain rate is increased, the toughening effect of water content does not seem to
prevail, and the effects of the strain rate dominate the fracture behavior of PMMA.
Table 4.5: Average critical stress intensity factor for 40 days of conditioning
Environment 11% RH 60% RH Distilled water
K
d
Ic
[MPa
√
m] 2.02±0.04 2.00±0.03 2.01±0.04
0 5 10 15
0
1
2
3
4
Stress Intensity Factor [MPa √m]
Crack Length [mm]
Distilled Water
60% RH
11% RH
Figure 4.7: Stress intensity factor for PMMA samples with different water contents. Uncer-
tainty for each point is represented by marker size.
These findings are opposite to the results by [9, 70, 72–74], who observed differences in
thefracturebehaviorofsampleswhoseimmersiontimesvariedfromafewminutestoseveral
months when subjected to strain rates of 10
−3
− 10
−8
s
−1
. Neither was there a significant
variation between samples with large water content differentials, such as samples exposed to
Lithium chloride (0.06 wt %) and distilled water (1.12 wt %). Thus, results obtained in this
study showed that water content is not a dominant mechanism at strain rates on the order
of 10
2
s
−1
.
69
0 0.5 1 1.5
0
0.5
1
1.5
2
2.5
Stress Intensity Factor [MPa √m]
Water content [wt. %]
Figure 4.8: Comparison of the critical SIF variation with water content between the results
obtained for static tensile testing by Bokoi et al. [70] ( ) with the results obtained in this
study ( ) for PMMA.
III. Conclusions
ThedynamicfracturebehaviorofPMMAsamplesconditionedinvariousenvironmentsyield-
ing different water uptake in the samples was investigated. The results can be concluded as
follows:
• A new experimental setup was designed, and it has been shown to yield repeatable
resultsforsamplesimpactedinadryenvironment. Impactvelocitieshadavariationof
±0.82 m/s for all 61 experiments performed, and strain responses showed that loading
conditions were similar with an average maximum strain of ǫ
max
= 2480±255 µ strain
and pulse duration of 70 μs for all samples tested.
• The mode-I critical SIF value obtained at ∼10
2
s
−1
was similar for different water
contents. This differs from the results presented by Bokoi et al. [70], who showed a
significant variation in the fracture toughness of the material. Additionally, the crack
propagation and speed of PMMA for different water contents had a similar behavior,
regardless of the conditioning procedure. Contrary to previous studies performed at
lower strain rates [9, 70, 74], results obtained in this study showed that water content
is not a dominant mechanism at strain rates on the order of 10
2
s
−1
.
70
Chapter 5
Effect of Water Content on Dynamic Fracture
Initiation of Vinyl Ester
The effects of water content on the dynamic fracture initiation of notched vinyl ester
neat resin samples were examined. The samples were subjected to stress pulses generated by
the impact of a projectile launched from the dynamic loading setup described in Chapter 3.
Two sets of samples of samples were used: the first set was conditioned in an 11% rela-
tive humidity (RH) environment using a saturated salt solution (Lithium Chloride), and the
second set was immersed in distilled water. Both sets were kept in their respective environ-
ments for 43 days. The dynamic loading conditions were kept constant to analyze the effect
of water content on the dynamic fracture initiation of both sample sets. It was observed
that the fracture toughness and crack-tip speed showed no significant difference despite a
water content differential of 0.49 wt% between the sample sets. This work was published
in the journal of the Society for Experimental Mechanics under Effect of Water Content on
Dynamic Fracture Initiation of Vinyl Ester by Delpino Gonzales and Eliasson [37].
I. Experimental Procedure
A spherical steel projectile launched from an gas-gun was utilized to impact the short edge
of rectangular vinyl ester samples. Loading conditions were monitored using strain gauges,
and fracture initiation was visualized using the method of caustics. Details of the material
preparation, experimental setup and environmental conditioning parameters are explained
in the following three subsections.
71
A. Materials
Vinyl ester resin sheets were provided by collaborators at the NSWC Carderock Division.
The sheets were formed from Ashland Derakane 510A vinyl ester resin, mixed with a Cobalt
Naphthenate (CoNap) promoter, N,N-Dimethylacetoacetamide (DMAA) accelerator, 2, 4-
Pentanedione(2, 4-P)gel timeretarder, andTrigonox (239A) initiator. Theresinwas mixed
at ambient temperature conditions with the following ratios to achieve an approximate two-
hour gel time: CoNap 0.2 wt%; DMAA 0.05 wt%; 2,4P 0.1 wt%; and 239A 1.5 wt%. The
mixed resin was agitated in a vacuum chamber for approximately 30 minutes for degassing.
The mold used to form the resin sheets consisted of release coated plates with spacers in
between to form a rectangular cavity of 3.3 mm thick× 280 mm wide× 360 mm long and
open at both ends. The mold was enclosed in a vacuum bag with plastic tubing arranged
to form a resin inlet port at one end of the cavity and a vacuum port at the other end.
The mold was arranged at an angle with the resin inlet port approximately 150 mm lower
than the vacuum outlet port. The vacuum bag and mold cavity were evacuated to a vacuum
pressure of approximately 0.5 atm. The resin port was opened, and resin was allowed to
flow into the cavity. Once the cavity was filled and the resin had begun to enter the vacuum
tubing, the vacuum pressure was reduced until the resin stopped flowing into the mold. The
pressure was held at that level, with the outlet end of the mold elevated until the resin
cured, for approximately 12 hours. The resin sheets were subsequently post-cured at 60
◦
C
for 6 hours, prior to being machined to the final specimen dimensions. It is known that
vinyl ester resin will undergo further crosslinking to reach full strength if exposed to even
higher temperatures during post-cure. However, the objective was to establish the material
performance developed at room temperature. This is because elevated temperatures are
not used during the curing processes of full-scale ship structures. Thus, to replicate the
ship structure performance achieved after many months of room temperature curing, an
accelerated, slightly elevated temperature post-cure cycle was used.
The samples used in this study had the same geometry as used in [120]. Specimens with
a thickness of 3.3±0.09 mm were machined to rectangular shapes with dimensions 300 mm
72
× 40 mm. At half-length of the long edge of each specimen, a 10 mm long and 0.152 mm
wide notch was sawn. This type of geometry, shown in Figure 5.1, allowed for a pure mode-I
cracktooccurwhenatensilepulsewasapplied. Duetoitscross-sectionalaspectratio(12:1),
three-dimensional effects could be ignored, and it could be assumed that the wave traveling
through the specimen due to impact was a plane-stress wave [120].
Figure5.1: Drawing of notched vinyl ester specimen with dimensions 300×40×3.3 mm
3
. The
location of the strain gauge is indicated by (1).
B. Sample Conditioning
A total of 20 specimens were divided into two different sets corresponding to their respective
environment exposure: (1) 11% relative humidity (RH) and (2) distilled water. The first
set of samples investigated the effects on fracture behavior due to exposure to an 11% RH
environment. ThisenvironmentwasobtainedbyusingaLithiumChloridesaturatedsolution
at 25
◦
C [74]. The amount of salt usedand the solution intensityare shownin Table 5.1. The
saturated solution was placed in air-tight plastic containers with the specimens positioned
in holders 40 mm above the surface of the liquid solution. The relative humidity value
in the closed container, generated by the vapor pressure from the solution, was frequently
monitored using a thermo-hygrometer (PTH 8707). The second set of samples were used
to investigate the effects on dynamic fracture behavior due to immersion in distilled water.
The results presented in this study correspond to samples conditioned in these environments
for 43 days at 25
◦
C because water sorption rates significantly decreased after this. The
rationale in selecting 11% RH and distilled water environments was to simulate the opposite
humidity level conditions to which the material could be exposed. Note that after the
73
material was machined to its final sample dimensions, following the post-curing cycle, their
water content was not equilibrated by heat treatment prior to conditioning them in their
assigned environments. This was done to avoid any further cross-linking in the resin.
Table 5.1: Composition of saturated salt solution
Solution RH [%] Used amount [g] Volume of water used [ml] Intensity [g/ml]
LiCl 11 2505 1000 0.835
The weight change in the samples during their exposure time was frequently monitored
using a weight scale (ML4002E/03, Mettler Toledo), with a readability of 0.01 g, to obtain
thewatercontentasafunctionoftime. Theweightchangewascalculatedusingthefollowing
equation:
C =
W
f
−W
i
W
i
×100 (5.1)
whereC (wt%)isthewatercontent,W
i
istheweightofasamplebeforeimmersion,andW
f
is
theweightofthesamplecontainingsorbedwater. Figure5.2showstheweightmeasurements
as a function of conditioning time. It was observed that the water content of the specimens
changed and the rate of change decreased with time. As expected, the water content was
higher for samples immersed in distilled water compared to the ones exposed to 11% RH,
see Table 5.2. There was an average water content differential of 0.49 wt% (0.26 g) between
samples conditioned at different environments. Since the samples were not heat treated
prior to conditioning and the relative humidity level at room conditions was higher than
11%, the samples exposed to the 11% RH environment underwent desorption until they
reached equilibrium with their environment, as shown in Figure 5.2.
Table 5.2: Average weight change due to water sorption on experimental samples
Environment
Weight change (g)
2 Days 8 Days 26 Days 43 Days
11% -0.02 -0.04 -0.04 -0.05
Distilled water 0.06 0.11 0.19 0.21
The visualization technique utilized to extract quantitative data from these experiments
was the method of caustics. The Phantom V711 high-speed camera was used to record
74
0 5 10 15 20 25 30 35 40 45
−0.1
0
0.1
0.2
0.3
0.4
Water content [%]
Time [days]
Figure 5.2: Average weight change as a function of time for ( ) vinyl ester samples exposed
to distilled water ( ) vinyl ester samples exposed to 11% RH, both conditioned for 43 days.
See details in Table 5.2.
caustic image sequences with a frame rate of 80,000 frames per second, which resulted in a
timeintervalof12.62µ sbetweenconsecutiveimagesandaresolutionof256×256pixels. The
fieldofviewofthecameracorrespondedtoa20×20mm
2
sectionofthesample,resultingina
scaling factor of 78 µ m per pixel. The experimental setup constants and material properties
utilized are shown in Table 5.3. Additionally, measurements were performed in order to
determine if the longitudinal and transverse wave speed, inherent material properties, were
affected by water content. The wave speeds of ten vinyl ester samples (five samples from
eachenvironment)weremeasuredusinganultrasonicthicknessgage(Olympus38DLPLUS).
Both sets of samples showed similar results with an average longitudinal wave speed of
c
L
= 2,194±14 m/s and transverse wave speed of c
T
= 1,114±16 m/s.
Noavailableinformationwasfoundintheliteratureforthestressopticalcoefficient(SOC)
ofvinyl ester resin, thus, thevalue utilizedfor theSOC,c, was determinedbyusing theratio
−ν/E as shown by [106]. The elastic constant, E, was calculated applying Equation (5.2),
E =
c
2
L
ρ(1−ν)
(1+ν)(1−2ν)
. (5.2)
This yielded in a value of E = 3.4 GPa, which is similar to the value obtained by [87]. The
75
values used are shown in Table 5.3.
Table 5.3: Values used for the method of caustics
Notation Value Units
Distance to reference plane z
0
1.4 m
Scale factor m 1.01 -
Longitudinal wave speed† c
L
2,194 m/s
Transverse wave speed† c
T
1,117 m/s
Elastic modulus E 3.4 GPa
Density ρ 1,320 kg/m
3
Stress Optical Coefficient c −1.12×10
−10
m
2
/N
† Value obtained from ultrasonic gage measurement
II. Results and Discussion
The repeatability of the impact experiments was measured by recording the strain response
from a strain gauge attached to the sample at the centerline 50 mm behind the notch of all
the samples impacted, denoted (1) in Figure 5.1. Results of the strain signals are shown in
Figure 5.3. The strain gauge responses correspond to an impact velocity of 31.9±0.40 m/s
and an average maximum strain of ǫ
max
= 4106±145 µ strain for all samples. The projectile
0 50 100 150 200 250 300 350
−6000
−4000
−2000
0
2000
4000
6000
Strain [με]
Time [μs]
Figure 5.3: Strain gauges response for 12 experiments on vinyl ester samples with impact
velocity 31.9±0.40 m/s.
76
speed used for these experiments was 31.93± 0.40 m/s. This impact speed caused stress
waves to travel through the samples with a pulse duration of 95 µ s. The method of caustics
was then used to quantify the dynamic fracture initiation and initial propagation stages for
the various samples. The SIF was calculated by measuring the transverse diameter of the
caustic curves as the crack propagated. Figure 5.4 shows a sequence of caustic images. To
post-process these images, any optical distortions that had been introduced in the system
were removed. For this, a virtual grid with known points was mapped onto the recorded
images. Then, the location and the diameter of each caustic were measured at the points of
the curve where the highest intensity pixels were located, since these represent the highest
concentrationoflight. Thelocationofthecrack-tipwasassumedtobeatadistanceof0.527
timesthetransversediameterofthecausticfromthemostforwardpointofthecausticcurve
[107]. An average of 6 frames were analyzed for each experiment.
(a) 0 μs (b) 12.62 μs (c) 25.24 μs
(d) 37.86 μs (e) 50.48 μs (f) 63.10 μs
Figure5.4: Crack propagation sequence for an impacted sample. Images show area of interest
(20×20 mm
2
) as indicated in Figure 3.2.
77
Theresultsobtainedforcrackpropagationforthreesamplesexposedto11%RHandtwo
samples immersed in distilled water are shown in Figure 5.5. Even though samples had been
conditioned in opposite humidity level environments for 43 days, no significant difference
was found in the initial stages of crack propagation. Crack-tip speed was calculated from
the crack-tip displacement, and velocities up to 454±16 m/s were recorded.
0 2 4 6 8 10 12 14 16 18
0
100
200
300
400
500
Crack Speed [m/s]
Crack Length [mm]
Figure 5.5: Crack-tip speed corresponding to ( ) vinyl ester samples exposed to distilled water
( ) vinyl ester samples exposed to 11% RH. Uncertainty for each data point is represented
by marker size.
Figures 5.6 and 5.7 show the SIF variation with crack length and speed for the initial
stages of crack propagation, respectively. Again, it can be seen in Figure 5.6 that the critical
SIF is similar for both sets of samples with an approximate value of K
Ic
=1.5 MPa
√
m. This
valuefallswithintherangeobtainedatlowerstrainrates(∼ 10
−3
s
−1
),whichagreeswiththe
low loading rate sensitivity of the mode-I fracture toughness of vinyl ester resin reported by
[36]. Additionally, this shows that water has no significant effect on the fracture toughness
of vinyl ester resin for strain rates of 10
2
s
−1
. Also, both sets show similar SIF for the initial
stages of crack propagation. Note in Figure 5.7, as a general trend for the initial stages of
crack propagation, the SIF increases with increasing crack-tip speed.
78
0 2 4 6 8 10 12 14 16 18
0
0.5
1
1.5
2
2.5
3
3.5
Stress Intensity Factor [MPa √m]
Crack Length [mm]
Figure 5.6: Stress intensity factor corresponding to ( ) samples exposed to distilled water ( )
samples exposed to 11% RH. Uncertainty for each data point is represented by marker size.
0 100 200 300 400 500
0
0.5
1
1.5
2
2.5
3
3.5
Stress Intensity Factor [MPa √m]
Crack Speed [m/s]
Figure 5.7: Stress intensity factor variation with crack-tip speed for ( ) vinyl ester samples
exposed to distilled water ( ) vinyl ester samples exposed to 11% RH. Uncertainty for each
data point is represented by marker size.
79
III. Conclusions
In this study, the dynamic fracture initiation of vinyl ester resin samples conditioned in two
differentenvironments, yielding ina water contentdifferentialof 0.49 wt%, was investigated.
The results can be concluded as follows:
• The value obtained for the mode-I critical SIF, K
Ic
=1.5 MPa
√
m, agrees with what
was reported in previous studies, K
Ic
=0.7-1.7 MPa
√
m, at strain rates on the order of
∼10
−3
s
−1
. Also, it implies that fracture toughness of the material does not seem to
be rate sensitive up to a strain rate of∼10
2
s
−1
.
• The crack propagation behavior of vinyl ester neat resin samples exhibited no signif-
icant variation with different water content. Both sets of samples showed a similar
behavior in the initial stages of crack propagation (up to 16 mm extension) with an
average maximum crack-tip speed of 454±16 m/s.
• For the strain rate at which these tests were conducted (∼10
2
s
−1
), water content had
a negligible effect on the fracture toughness of vinyl ester neat resin. Comparisons
were made by calculating the SIF on both sets of samples and bothsets showed similar
behavior in the initial stage of crack propagation.
These results show that water content has an insignificant effect on the dynamic fracture
initiation of vinyl ester resin, agreeing with previous studies regarding the resistance to
degradation of vinyl ester resin upon humidity exposure. This is advantageous because it
confirms the benefits of using vinyl ester resin as the matrix component of FRP composites
in applications that require high humidity exposure. To further understand the dynamic
behaviorofvinylesterresin,theanalysisofthecausticsmustbeextendedtoincludetransient
effects due to fast crack propagation as described by [108]. Additionally, these results will
be used as a baseline for future studies on the dynamic fracture behavior of FRPs subjected
to similar conditioning and loading conditions.
80
Chapter 6
Evaluation of the Effect of Water Content on
the Stress Optical Coefficient in PMMA
A series of experiments was completed to study the effect of water content on the stress
optical coefficient of PMMA samples under uniaxial tension. The stress optical coefficient
is an inherent material property that is utilized in interferometric techniques, classical pho-
toelasticity and the method of caustics to obtain deformation measurements and fracture
values. The stress optical coefficient depends on the variation of the mechanical properties
and the refractive index of the polymer, both of which are affected by water uptake. The
effect of water sorption on the stress optical coefficient of PMMA is not yet clear. The most
relevant research on this subject was done by Abo-El Ezz et al. [10], who studied the vari-
ation of the optical constants of PMMA when immersed in a liquid environment. However,
this study did not include the effects of water sorption after pre-conditioning the samples
for certain periods of time, but rather the immediate effect of the surrounding liquid during
the experiments.
The main objective of this study was to quantify the variation of the stress optical
coefficient with water uptake, so that it can be utilized in the analysis of visualization
techniques and applied to deformation and fracture studies in humid or liquid environments.
Experiments were performed at two different strain rates, 5×10
−4
−5×10
−3
s
−1
, for both
dry and pre-conditioned samples. The preconditioned samples were submerged in distilled
water for either 8 or 40 days, allowing samples to reach water contents of 0.8 wt % and
1.4 wt %, respectively. Measurements were completed utilizing a Fizeau interferometer and
high-speed photography. Results showed a variation on the stress optical coefficient of up to
11% for a water content of 0.8 wt % and a strain rate of 5×10
−4
s
−1
. This variation was no
81
longer observed whenthestrainrate was increased byan orderofmagnitude. This workwas
publishedinthePolymerTestingjournalunderEvaluation of the Effect of Water Content on
the Stress Optical Coefficient in PMMA by Delpino Gonzales, Nicassio and Eliasson [123].
I. Material and Conditioning
Commercially available PMMA (from Emco Industrial Plastics, Inc) was used for these
experiments. All samples met the ASTM D638-10 (Type II) standards with dimensions of
3.125mmthickness, 183mmoveralllengthand19mmoverallwidth, asshowninFigure6.1.
Experiments were performed using samples that were submerged in distilled water for 8 and
40 days, as well as dry samples for comparison purposes. These pre-conditioning durations
were determined such that water content was lower than 1 wt % for the samples exposed to
water for 8 days and higher than 1 wt % for samples immersed in water for 40 days. Prior
to placing the samples in their respective environments, they were dried in an air oven at
70
◦
C for 24 hours, in compliance with the method described in [69]. The weight change
in the samples during their exposure time was periodically monitored using a weight scale
(ML4002E/03, Mettler Toledo), with a readability of 0.01 g, to obtain the water content as
a function of time. Water absorption percentage was calculated using Equation (6.1)
Figure6.1: Drawing of PMMA tensile test sample. This shape corresponds to Type II ASTM
D638-10 with a thickness of 3.125 mm.
M =
W
f
−W
i
W
i
×100, (6.1)
82
where M (wt %) is the water content, W
i
is the weight of a sample before immersion,
and W
f
is the weight of the sample containing sorbed water. Figure 6.2 shows the weight
measurements as a function of time. It was observed that the water content of each sample
increased with time, but the rate of absorption decreased with time, in agreement with [70].
As it can be seen in Figure 6.2, there was a weight increment of 0.69 wt % due to water
sorption after the first five days of conditioning. This sorption rate decreased with time such
that in the last ten days of conditioning, only a 0.17 wt % weight increment was observed.
Theresultantwatercontentforthesamplesexposedtowaterfor8and40dayswas0.8wt%
and 1.4 wt %, respectively.
0 5 10 15 20 25 30 35 40
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
Water content [% ]
Time [days]
Figure 6.2: Weight increment due to water sorption on PMMA samples for an immersion
time of up to 40 days.
II. Experimental Procedure
A series of experiments under uniaxial tension were completed using a 5567 Instron machine
with a 5 kN load cell. Experiments for each conditioning set were performed at strain rates
of 5×10
−4
s
−1
and 5×10
−3
s
−1
with loads increasing from zero up to fracture at a constant
temperature of 25
◦
C. A small pre-load of 40 N was applied to all samples prior to each
83
experiment to ensure alignment. A Fizeau interferometer, which principles are described in
Chapter 3, was utilized in these experiments. Figure 6.3 shows a representative image of
the appearance of the well-defined fringes captured by the high-speed camera. The image
of the specimen located next to the fringes in Figure 6.3 shows the field of view observed
during the experiments. The high-speed camera was set to a resolution of 750×300 pixels
and an exposure time of 30μs. A 200 mm lens (AF Micro-Nikkor, f/4D) was used, resulting
in a scale of 70 μm/pixel. The high-speed camera was set to record at 24 and 300 fps
for the experiments performed at strain rates of 5× 10
−4
and 5× 10
−3
s
−1
, respectively.
These frame rates were carefully selected to minimize measurement errors by recording and
averaging results from 8–10 frames per experiment.
4 mm
Figure 6.3: Field of view (22×9 mm
2
) and fringe appearance (Resolution: 70 µm /pixel).
Measurements such as stress and strain were obtained directly from the load cell in the
Instronmachineinadditiontoaclip-onextensometeraddedontoeachsample. Thesampling
rate of the Instron machine was 0.1 seconds, which yielded an average of 100 and 1000 data
84
pointsforthesamplesloadedatthestrainratesof 5×10
−3
s
−1
and 5×10
−4
s
−1
, respectively.
Byusingthestressmeasuredbytheloadcellandthefringecountobtainedfromtherecorded
images, the SOC was calculated as the load increased. The correlation between the images
recorded and the stress/strain data was performed by shifting the time data so that the
instant fracture occurred was set as a common point in time.
Thepost-processingprocedurewascompletedusingMATLABtoobtaintheelasticmod-
ulus and to count the pixel intensity at multiple sample locations. To do this, two points
within the field of view of the sample were selected, which represented the endpoints of a
line along which the intensity of every pixel was measured as the load was increased. This
yielded a time history of the intensity of each pixel. Data from several pixels was obtained
for analysis, contrary to the method employed by Raftopoulos et al. [118] who only mea-
sured a single point. Figure 6.4 shows a digitized image of the fringe count obtained for the
intensity of a single pixel as a function of time. To facilitate and automate the fringe count,
a zero-phase digital filter was used to smooth the data. Each peak was considered a fringe,
thus, once the data was smoothed out, fringe count was easily performed. Once the fringes
(or peaks) were computed, the fringe count was grouped in intervals of applied stress level.
Lastly, Equation (3.6) was used to calculate SOC values as the load was increased.
0 5 10 15 20 25 30 35
0
400
800
1200
1600
2000
Time [sec]
Pixel Intensity , ( )
0
100
200
300
400
500
Load [N] , ( )
Loading Starts
Fracture
Figure 6.4: Intensity plot showing fringe time history for a single pixel and its simultaneous
loading history.
85
III. Results and Discussion
As it was previously mentioned, the values that represent the mechanical response of the
material were directly measured from the load cell and clip-on extensometer used during
the experiments, and the values for the optical response were measured with the Fizeau
interferometer and high-speed photography. The results from both sources of measurements
were compared to confirm their agreement; thus, the fringe count, which represents the
change in thickness of the material, was normalized by the maximum number of counted
fringes and compared with the normalized strain obtained from the clip-on extensometer.
These two terms are proportional to the change in thickness of the material as it is loaded.
Figure 6.5 shows an example of the results when they are compared. Their agreement
confirms that the fringe count is a measurement that accurately represents the sample’s
behavior. This type of comparison was performed during the analysis of every sample to
ensure the fidelity of the measurements obtained from the Fizeau interferometer.
0 0.2 0.4 0.6 0.8 1
0
0.2
0.4
0.6
0.8
1
N
1
/N
max
ε /ε
max
σ/σ
max
Extensometer: ε/ε
Fringe Count: N / N
and
1 max
max
Figure 6.5: Comparison of response of extensometer and fringe count as a function of stress
applied. Excellent agreement is obtained between the measurements.
The variation of the mechanical response of PMMA was measured by investigating the
86
effect of water content on its elastic modulus. Figure 6.6 shows the variation of the elastic
modulus measured at two different strain rates as a function of water content. The elastic
modulus decreased with increasing water content for both strain rates, as well as it increased
with increasing strain rate. The effect of strain rate on PMMA is well understood as it
is known that it is highly rate dependent, such that it experiences an incremental increase
on the elastic modulus as strain rate increases. This can be attributed to the fact that as
the strain rate increases, the molecular mobility of the polymer chains will be decreased,
thus, making the chains stiffer, and a stiffer material has a higher elastic modulus [124, 125].
Additionally, the effect of water content on the mechanical response of PMMA agrees with
the findings by [69, 72]. The effect of water is known to act as a mild plasticizer and, thus,
results in a decrease in the elastic modulus. In this study, the analysis of the mechanical
response of PMMA is focused on the variation of its elastic modulus because, as seen in
Equation (3.6), it is a critical term for the calculation of the SOC. Another mechanical
property involved in the analysis of the SOC is Poisson’s ratio. However, it was reported by
Theocaris [119] that the Poisson’s ratio does not vary significantly in PMMA for different
loading rates. For this reason, in this study, Poisson’s ratio wasassumed to remain constant,
ν=0.36.
Figures 6.7(a) and 6.7(b) show the variation of the SOC for the three different condition-
ing scenarios at both strain rates studied. Each of these curves feature results for at least
5samplespercase. Notethatonlyafewpointsfromtheentiredatasetareshownforeaseof
reportingtheresults. However,thetrendsandvaluesarewellrepresentedintheseplots. The
values calculated for the SOC immediately prior to fracture agree with the results reported
in the literature by Katsamanis et al. [126] who reported values of c
t
=1.21×10
−10
m
2
/N and
Theocaris [119] who reported values of c
t
=1.1 to 1.7×10
−10
m
2
/N. It has also been reported
that the SOC is strain rate dependent. In this study, a reduction of approximately 10± 4%
was observed as the strain rate was increased by an order of magnitude. This behavior has
been observed in comparisons between dynamic and static loading for which reductions of
up to 40% the SOC values have been reported [118, 126].
87
0 0.5 1 1.5
0
500
1000
1500
2000
2500
3000
3500
4000
Water content [wt %]
Elastic Modulus [MPa]
5 x 10
−3
s
−1
5 x 10
−4
s
−1
Figure 6.6: Elastic modulus variation as a function of water content for two different strain
rates (5×10
−3
s
−1
and 5×10
−4
s
−1
).
Overall, it is apparent that the variation of the SOC due to water content is not very sig-
nificant. For both strain rates applied, the behavior of the SOC in samples pre-conditioned
in water is similar regardless of the duration of time that they were immersed in water.
Nevertheless, there was a small variation between the SOC in the dry samples and the ones
exposed to water for the lower strain rate scenario (5×10
−4
s
−1
). To further understand
the effect of water content on the SOC, the approach adopted was to study the contribu-
tion from the optical and mechanical properties of the material by independently analyzing
the response from the first and second term in Equation (3.6), respectively. Figure 6.8(a)
and 6.8(b) show the contributions from each term (λα∗ and ν/E) as the load increased for
samples subjected to strain rates of 5×10
−3
s
−1
and 5×10
−4
s
−1
, respectively.
The comparison analysis was performed by studying the variation of the results of the
samples pre-conditioned in a water environment against the results obtained for the dry
samples. From Figure 6.8(a) and 6.8(b) it is apparent that for all cases the dominant
contribution is the term representing the optical properties. Figure 6.8(a) shows that a
similar effect was experienced by both mechanical (ν/E) and optical (λα∗) contributions for
samples that were immersed in water for the same time duration.
88
0 5 10 15 20 25 30 35 40
0
0.5
1
1.5
x 10
−10
Stress [MPa]
c
t
[m
2
N
−1
]
1.4 wt %
0.8 wt %
0 wt %
(a) 5×10
−3
s
−1
0 5 10 15 20 25 30 35 40
0
0.5
1
1.5
x 10
−10
Stress [MPa]
c
t
[m
2
N
−1
]
1.4 wt %
0.8 wt %
0 wt %
(b) 5×10
−4
s
−1
Figure6.7: Stress optical coefficient comparison for PMMA samples subjected to two different
strain rates and with varying water contents.
89
Furthermore, the magnitude of variation due to water content of both mechanical and
optical contributions was higher for the samples with a water content of 1.4 wt %. On the
other hand, Figure 6.8(b) shows that the variation in the optical properties were similar for
samples pre-conditioned in a water environment, yet their mechanical properties showed a
dissimilar behavior. The samples with a water content of 1.4 wt % experienced a significant
variation in their mechanical response compared to the samples with a water content of 0.8
wt %. A change in the response of the optical properties seems to occur as water content is
increased in the samples tested at a strain rate of 5×10
−4
s
−1
. However, this behavior was
not observed for experiments performed at a strain rate of 5×10
−3
s
−1
. Further studies must
be completed to understand the mechanism causing this response.
Tosummarizetheseresults, Table6.1showsthevariationexperiencedbybothcontribut-
ing terms for each of the cases studied. It must be noted that the values shown represent an
average variation from the dry samples for the duration of the experiment. Figures 6.8(a)
and 6.8(b) show a complete trend and distribution of the results.
Table 6.1: Average variation from dry conditions of optical (λα*) and mechanical (ν/E)
contributions due to water sorption
Case ǫ ν/E λα* C
t
0.8 wt % 5×10
−3
s
−1
3.6±4% 0.5±3% 3.1±7%
0.8 wt % 5×10
−4
s
−1
2.8±5% 14.1±2% 11.3±7%
1.4 wt % 5×10
−3
s
−1
20.7±2% 17.8±3% 2.9±5%
1.4 wt % 5×10
−4
s
−1
19.9±1% 12.7±3% 7.2±4%
Table6.1showsthatwatercontenthadthegreatestinfluence,atbothstrainratesapplied,
on the mechanical properties in the cases where the samples had a water content of 1.4 wt
%, reaching values of up to 20% variation. However, the changein the optical properties was
notsignificantlylower,exceptforthesampleswithawatercontentof0.8wt%andsubjected
to a strain rate of 5×10
−4
s
−1
, as previously discussed. The last column of Table 6.1 shows
the variation of the SOC as a result of water content. Due to the nature of Equation (3.6),
the condition that determines the overall effect of water content on the SOC is the difference
between the contributions of the mechanical properties over the optical ones.
90
0 5 10 15 20 25 30 35 40
0
0.5
1
1.5
2
2.5
x 10
−10
Stress [MPa]
ν/E & λα* [m
2
N
−1
]
1.4 wt %
0.8 wt %
0 wt %
ν/E
λα*
(a) 5×10
−3
s
−1
0 5 10 15 20 25 30 35 40
0
0.5
1
1.5
2
2.5
x 10
−10
Stress [MPa]
ν/E & λα* [m
2
N
−1
]
1.4 wt %
0.8 wt %
0 wt %
ν/E
λα*
(b) 5×10
−4
s
−1
Figure6.8: Optical and mechanical contributions to SOC for PMMA samples at two different
strain rates.
91
As it can be seen, the average values of the SOC are a good representation of what is
displayed in Figure 6.7. A small effect of water content on the SOC for samples subjected
to a strain rate of 5×10
−4
s
−1
is observed, and this effect was reduced with increasing strain
rate. Hence, there was no significant difference in the SOC between dry samples and pre-
conditioned samples for the strain rate of 5×10
−3
s
−1
.
IV. Conclusions
The effect of water content on the SOC of PMMA when subjected to two different strain
rates under quasi-static loading was investigated. The results can be concluded as follows:
• Water content had an effect on the mechanical contribution, ν/E, of the SOC by
causing a variation of up to 20% from dry conditions. The effect of water on the
optical contributions, λα*, of the SOC was marginally lower, reaching values of up to
18% variation from dry conditions.
• Since the effect of water on the optical and mechanical contributors was similar, this
yielded a variation in the SOC of up to 11% for samples subjected to a strain rate
of 5×10
−4
s
−1
. This effect was reduced with increasing strain rate, such that there
was no significant difference in the values of the SOC between dry and pre-conditioned
samplesat the strain rate of 5×10
−3
s
−1
.
92
Chapter 7
Experimental Investigation of Dynamic
Fracture Initiation in PMMA Submerged in
Water
The mode-I dynamic fracture response of notched PMMA plates submerged in water
was investigated. The experimental setup utilized was designed to study underwater crack
propagation with the aid of visualization techniques to obtain quantitative and qualitative
results. High-speed imaging and the method of transmitted caustics were used to complete
the measurements. The main objective of this study was to determine the effect of sur-
rounding water on the dynamic fracture behavior of samples as they were impacted. The
properties measured to assess the effect of water were the stress intensity factor and crack-
tip speed in the initial stages of crack propagation. Experiments were also completed for
samples surrounded by air for comparison purposes. Results showed that the presence of
water had no effect on the fracture behavior of PMMA at strain rates of the order of 10
2
s
−1
.
ThisworkisunderreviewinthejournalofDynamicBehaviorofMaterialsunderExperimen-
tal Investigation of Dynamic Fracture Initiation in PMMA Submerged in Water by Delpino
Gonzales, Luong, Homma and Eliasson.
I. Methods and Material
To simulate dynamic loading conditions on structures immersed in a water environment,
an impact was generated using a 10 mm diameter spherical steel projectile launched onto
rectangular PMMA notched samples with mid-sections immersed in water.
93
A. Experimental Procedure
Commercially available PMMA (from Emco Industrial Plastics, Inc) was used to make the
samples for these experiments. The geometry of the samples, shown in Figure 7.1, was
the same as the ones described in Chapter 4, chosen to obtain mode-I crack propagation
during impact. Prior to each experiment, one strain gauge was installed on each sample.
The setup utilized to generate impacts and obtain quantitative measurements was described
in Chapter 3. The high-speed camera was set to record at a frame rate of 80,000 frames
per second, which resulted in a time interval of 12.62 μs between consecutive images and
a resolution of 256 × 256 pixels. Figure 7.1 shows the field of view of the camera that
corresponded to a 20× 20 mm
2
section of the sample, resulting in a scaling factor of 78μm
per pixel. Optical distortions resulting from the visualization system were removed using
the same procedure mentioned in Chapter 4.
300 mm
5 mm
50 mm
Strain gauge
Figure 7.1: Sample geometry. The field of view of the camera is indicated and the sample’s
appearance through the camera is shown (Resolution: 78 µ m/pixel).
B. Method of Caustics in Liquid Media
The method of caustics is suitable for these experiments because it allows visualization of
crack propagation in transparent media, such as air or water, which was used to quantify
the SIF and crack-tip speed of PMMA. Equation (3.2) is applied to calculate the SIF, as
94
described in Chapter 3. The SIF, K
I
, is expressed as a function of the transverse diameter
of the caustic, D, the sample thickness, t, the crack-tip speed, v, the longitudinal and
transverse wave speed, c
L
and c
T
, respectively, as well as other experimental setup constants
and material properties shown in Table 7.1.
The presence of water will cause an optical effect that reduces the size of the caustic
curve. Abo-El-Ezz et al. [127] and Takahashi et al. [128] were the first ones to observe
this optical effect when the method of caustics was applied to investigate the crack growth
of PMMA submerged in water and methanol while subjected to uniaxial loads. In these
studies, a change in the diameter of the caustic curves surrounded by water was observed
whencomparedtocausticcurvesinairsubjectedtothesameloadingconditions. Thechange
in diameter occurs because the refractive index of water, n=1.33, is in between that of air,
n=1, and of PMMA, n=1.49. Therefore, when a propagating crack is immersed in water,
the light ray deviation angle is smaller than in air after refraction, which generates a smaller
caustic curve [129].
Figure7.2showsimagesobtainedinthisstudyforsamplessubjectedtothesamedynamic
loading conditions in air and water. The images are taken at the moment prior to crack
growth in each experiment; thus, it is assumed that the transverse diameter of the caustic
should be of the equal. However, as it can be seen, the transverse diameter of the caustic
in water is smaller than the one in air. For this reason, an adjustment was applied to the
analysis of the SIF. The variation in the length of the transverse diameter of the caustic was
taken into consideration by estimating a different value of the stress optical coefficient of the
sample to account for the surrounding water. Equation (7.1) was used to estimate the SOC
of PMMA submerged in water [130]
c =
1
E
[(1−2ν)(1+νk)b+(n−n
0
)ν(k−1)], (7.1)
where b is a material constant (b=−0.557 for PMMA [130]), n is the index of refraction of
the material, n
0
is the index of refraction of the liquid medium surrounding the specimen
(water with n
0
=1.33), and ν is Poisson’s ratio. The k value is a measurement of the three-
95
D
air
(a) Air environment (D
air
= 7.55 mm)
D
water
(b) Water environment (D
water
= 6.72 mm)
Figure 7.2: Comparison of caustic size for samples surrounded by air and water using the
same optical setup.
dimensionality of the state of stress applied with a range between the values of 0 and 1 for
plane stress and plane strain, respectively. Here, the loading condition is considered to be
plane stress, so k=0. By making this assumption, Equation (7.1) simplifies to
c =
1
E
[b(1−2ν)−ν(n−n
0
)]. (7.2)
Equation (7.3) was used to quantify the dynamic elastic modulus of PMMA to be used
in the calculation of its SOC
E =
c
2
L
ρ(1−ν)
(1+ν)(1−2ν)
, (7.3)
The value obtained for the dynamic elastic modulus of PMMA was E = 5.56 GPa, which
is similar to the value, E = 5.60 GPa, reported by Rosakis [131]. Finally, the experimental
constants and material properties used for the analysis of the results from the method of
caustics are shown in Table 4.3 and some additional properties utilized to adjust for the
presenceofsurroundingwaterareshowninTable7.1. Notethatduetothemanyassumptions
taken to obtain Equation (7.2), the SOC values utilized in the present work are estimates
only.
96
Table 7.1: Material properties used for the method of caustics
Notation Value Unit
Stress optical coefficient in water c −4.01×10
−11
m
2
/N
PMMA constant
∗
b −0.557 -
∗
Value obtained from [130]
II. Results and Discussion
The results reported next concern fracture initiation and early stages of crack propagation
in dry PMMA surrounded by air and water. Prior to the impact experiments, all samples
were treated for 24 hours at 76
◦
C to remove any moisture content and establish equal initial
conditions. All results regarding crack propagation in air correspond to data reproduced
from [78] in which the same experimental setup was used. Figure 7.3 shows a sequence of
caustic images exemplifying the footage obtained from these experiments. Each of these
images were post-processed to measure the transverse diameter of the caustic and to obtain
the crack-tip location. An average of 5 frames were analyzed for each experiment.
The repeatability of the loading conditions was interpreted by comparing the impact
speed for each experiment, and the strain response obtained from the strain gauges attached
to each sample. The projectile speed used for these experiments was 37.83±0.4 m/s and
38.83±1.1 m/s for experiments in air and water, respectively. The average strain response
obtained for a total of eight samples exposed to either environments is shown in Figure 7.4.
From these results it was calculated that the samples were subjected to a strain rate of
approximately 10
2
s
−1
. Overall, the strain responses for both cases are similar and follow
thesamebehavior, acompressionpulsefollowedbyatensionpulse. Still, asitcanbeseenin
Figure 7.4, it appears that the addition of water on the surroundings of the specimens had a
dampening effect on the amplitude of the reflected tensile pulse. The strain response of the
samples exposed to air exhibit a stronger nonlinear behavior, which could be attributed to
high-frequency vibrations within the structure picked up by the strain gauge. Contrastingly,
these vibrations may not occur when water is present as the density of water, which is
97
(a) 0 μs (D = 0 mm) (b) 12.62 μs (D = 5.01 mm) (c) 25.24 μs (D = 6.55 mm)
(d) 37.86 μs (D = 7.32 mm) (e) 50.48 μs (D = 7.47 mm) (f) 63.10 μs (D = 7.16 mm)
Figure 7.3: Crack propagation sequence for a sample immersed in water. Images show area
of interest (20 ×20 mm
2
) as indicated in Fig. 7.1 and corresponding caustic diameters, D,
are presented.
98
significantly higher than the density of air, might not allow it.
0 50 100 150 200
−4000
−3000
−2000
−1000
0
1000
2000
3000
Strain [ μstrain]
Time [ μs]
Air
W a ter
Figure 7.4: Average strain response comparison for four samples per case.
Figure 7.5 shows a comparison of the crack-tip speed of a total of eight samples that
weresubjectedtodynamicloadingwiththeirnotchsurroundedbyeitherairorwater. These
resultsrevealthatsamplessurroundedbywatershownosignificanteffectontheinitialstages
of the crack-tip speed of PMMA. Figure 7.6 shows a comparison of the SIF measurements
obtained for cracks propagating in air and water. It is clear that the presence of water
has no effect on the fracture toughness or SIF of this material, as both scenarios show a
clearly defined critical SIF (∼2 MPa
√
m), interpreted as the instant when the crack starts
propagating. Additionally, it can be seen that the SIF for propagating cracks submerged
in water was 10% higher than the SIF for cracks propagating in air. This variation in the
SIF can be attributed to the use of Equation (7.2), as it only calculates an estimate value
of the SOC, which could have introduced a rough approximation of the SIF. Nevertheless,
if Equation (7.2) was not utilized in the analysis of this experiments there would be a 35%
difference in the SIF values between cracks propagating in air and water. In any case, it
is likely that the 10% increment in the SIF was not caused due to plasticization generated
from water exposure. The reason for this suggestion is because of the results presented in
Figure7.5, whichshowednodifferenceinthecrack-tipspeedbetweenbothcases. Thecrack-
99
tip speed measurements were not affected by an estimated value and were directly measured
from the footage obtained during the experiments. Thus, if the crack-tip speeds showed
the same behavior and the SIF presented similar values for both cases, it is proposed that
surrounding water does not significantly alter the dynamic fracture behavior of PMMA.
0 1 2 3 4 5 6
0
50
100
150
200
250
300
Crack Speed [m/s]
Crack Len gth [mm]
Air
W a ter
Figure 7.5: Crack-tip speed corresponding to four samples exposed to air ( ) [78] and four
samples exposed to water ( ). Uncertainty for each data point is represented by marker size.
Inthiswork,itwasinitiallyproposedthatathighloadingrates,acouplingeffectbetween
the structure and surrounding liquid may result in a variation of dynamic fracture of the
material. However, eventhoughasmalldifferenceinthestrainresponsewascapturedbythe
strain gauges, namely a dampening effect on the amplitude of the tensile pulse exemplified
in Figure 7.4, no significant effect on the fracture behavior of the samples was observed.
These findings partially agree with previously reported results in which surrounding water
showed no effect on cracks propagating faster than 10
−1
m/s. These studies indicated that
at some point the crack-tip speed was too fast for the liquid to diffuse into the crack-tip and
plasticize it. On the other hand, the inability of a liquid to diffuse into a propagating crack
does not explain why no plasticizing effect was observed on the notch before crack initiation
occurred. Prior to the crack initiation stage, there was sufficient time for surrounding water
to plasticize the crack-tip, yet the fracture behavior was the same as in air. Therefore,
100
0 1 2 3 4 5 6
0
1
2
3
4
Stress Intensity Factor [MPa
Crack Length [mm]
Air
W a ter
√m]
Figure 7.6: Stress intensity factor corresponding to four samples exposed to air ( ) [78] and
four samples exposed to water ( ) for the early stages of crack growth. Uncertainty for each
data point is represented by marker size.
it is suggested that plasticization did occur prior to the fracture event, but the effect of
the strain rate, which also increases the fracture toughness of the material, dominated the
fracture behavior of the material, resulting in similar results for both environments studied.
III. Conclusions
An experimental investigation on underwater dynamic fracture initiation of PMMA was
completed. To the best of the authors’ knowledge, this constitutes the first study in which
dynamic underwater crack propagation is observed and quantitative data is obtained using
visualization techniques. The stress intensity factor and crack-tip speeds were measured
using the method of caustics coupled with high-speed photography. Results showed that
water media has no significant effect on the dynamic fracture initiation of the material when
it is subjected to an impact resulting in strain rates of 10
2
s
−1
. It was initially proposed
that other influencing mechanisms on underwater crack propagation, such as surface energy
dissipation into the surrounding water, may arise due to higher strain rate. However, no
significant variation was observed in the analysis of the crack-tip speed or SIF. Results
101
also imply that the mechanism that prevented water from having an effect on the crack
propagationofPMMAsubjectedtolowstrainratesprevailsatthehigherstrainratesstudied
here (up to 10
2
s
−1
). Moreover, since the notch of the samples was static prior to crack
initiation, it was proposed that water does plasticize the crack-tip before fracture occurs,
but strain rate effects overcome any effect that water induced. Thus, strain rate seems
to dominate the fracture event by overcoming the plasticizing mechanism prior to crack
initiation and by generating fast crack-tip speeds that do not to allow water to diffuse into
the propagating crack.
Clearly, the aim of this study has been to demonstrate the experimental capabilities
available to determine the dynamic fracture of transparent materials underwater. The main
limitation in the present work are the assumptions taken to overcome the optical effect
that water causes. More accurate results could be obtained if a more robust visualization
technique, such as digital image correlation (DIC), could be implemented to quantify the
dynamic fracture of materials submerged in water. The use of DIC will require higher
temporal and spatial resolution, which can be resolved by using ultra high-speed cameras.
While beyond the scope of this work, the aforementioned discussion is part of an ongoing
effort to continue the study of dynamic fracture in extreme environments and to contribute
to the experimental mechanics community.
102
Chapter 8
Effect of Water Content on Dynamic Fracture
Initiation of Carbon-fiber/vinyl ester
This chapter describes the implementation and progress of a study performed using DIC
and its application on dynamic fracture on CFVE composite materials. This effort repre-
sents a starting point for future studies on dynamic fracture using full-field analysis in our
laboratory. Themotivationforthisworkwastobeabletoobtainquantitativemeasurements
from dynamic loading experiments done on CFVE composite samples. Since CFVE samples
are opaque, the method of transmitted caustics could not be utilized for this work, so a dif-
ferent visualization technique had to be implemented. The approach followed for this work
consisted in developing DIC and analytical tools to quantify dynamic fracture. After that,
the setup shown in Chapter 3 was calibrated by performing dynamic loading experiments on
PMMA and correlated the deformation images recorded with a Phantom V711 high-speed
camera using DIC. The measurements were then compared to the results described in Chap-
ter4. Next,dynamicloadingexperimentswereperformedonunidirectionalCFVEcomposite
samples obtained from collaborators at the Naval Sea Warfare Center (NSWC) Carderock
Division. The implemented analytical tools and their assumptions are described in detail
for isotropic and orthotropic materials. The work presented in this chapter represents the
first analytical steps and initial experimental results of a study describing the degradation
effects on the dynamic fracture behavior of CFVE composite samples immersed in water for
43 days and subjected to dynamic loading.
103
I. Analytical Methods
The procedure followed to calculate mode-I SIFs around a crack-tip in isotropic and or-
thotropicmaterialssubjectedtodynamicloadingisdescribedinthissection. Thedescription
oftheanalysisusedforeachtypeofmaterial, isotropicandorthotropic, isshownbecausethe
analysis for isotropic analysis was used for PMMA for calibration purposes and the analysis
for orthotropic materials was applied for the CFVE samples.
A. Extraction of the Stress Intensity Factor
The coordinates of a crack-tip can be expressed in polar coordinates as:
r =
p
(x
k
−x
0
)
2
+(y
k
−y
0
)
2
,
θ = tan
−1
y
k
−y
0
x
k
−x
0
, (8.1)
where (x
0
,y
0
) are the location of the crack-tip relative to an arbitrary Cartesian coordinate
system and (x
k
,y
k
) is the location of the of the point analyzed within the displacement field
under consideration. The SIF was determined using the displacement fields calculated with
DIC coupled with an overdeterministic approach and a least-square optimization method
[132]. The area around the crack-tip considered to calculate the SIF (see Figure 8.1) was
sampledfromanannularareawithaninnerradiusR
1
andouterradiusR
2
. Theinnerradius,
R
1
, was determined for it to avoid the plastic region around the crack-tip, where there is the
presence of plasticity and three-dimensional effects; thus, R
1
was chosen to be greater than
half the thickness of the sample (2 mm). The outer radius, R
2
, was chosen such that more
than 150 displacement data points were included in the analysis to increase the accuracy of
the SIF estimate [133, 134]. In general, the displacement data points analyzed around the
crack-tip were bounded by 0.5≤ r/B≤ 1.5, where B is the thickness of the sample.
104
R
R
1
2
Figure 8.1: Area around crack-tip used for SIF analysis. More than 150 displacement data
points within the area bounded by R
1
and R
2
were used for SIF analysis to avoid the plastic
region around the crack-tip.
Isotropic Materials
The general equations to describe the displacement field around a crack-tip for an isotropic
material are [134]:
u =
∞
X
n=1
A
n
2µ r
n/2
[(χ+
n
2
+(−1)
n
cos
n
2
θ−
n
2
cos(
n
2
−2)θ)]−
∞
X
n=1
B
n
2µ r
n/2
[(χ+
n
2
−(−1)
n
sin
n
2
θ−
n
2
sin(
n
2
−2)θ)],
v =
∞
X
n=1
A
n
2µ r
n/2
[(χ−
n
2
+(−1)
n
sin
n
2
θ +
n
2
sin(
n
2
−2)θ)]−
∞
X
n=1
B
n
2µ r
n/2
[(−χ+
n
2
−(−1)
n
cos
n
2
θ−
n
2
cos(
n
2
−2)θ)],
(8.2)
105
where u and v are the horizontal and vertical displacement components, respectively, µ is
the shear modulus and χ = (3− ν)/(1 + ν) for plane stress and χ = (3− 4ν) for plane
strain, ν is Poisson’s ratio, r andθ are described by Equation (8.1), A
n
andB
n
are constants
that correspond to the mode-I and mode-II SIFs, respectively. These equations allow for the
estimation of the SIF for mode-I and mode-II fracture. However, only the mode-I SIF was
consideredinthisstudy. Thedisplacementcomponents,uandv,wereextractedusingaDIC
commercial code, VIC-2D
TM
. Furthermore, the displacement fields for isotropic materials
can also be described as:
u =
N
X
n=1
A
n
f
n,m
(r,θ)−
N
X
n=1
B
n
g
n,m
(r,θ),
v =
N
X
n=1
A
n
h
n,m
(r,θ)−
N
X
n=1
B
n
l
n,m
(r,θ), (8.3)
where N is the number of terms of the expansion of the displacement field. The number of
terms in the series was chosen as the value at which the SIF estimate converged, which was
N = 5 for this work. The subscript m represents the total number of displacement points.
Since more than 150 displacement data points (m≥ 150) were used to solve for the SIF, i.e.,
thenumberofdisplacementdatapointsutilizedismuchlargerthanthenumberofunknowns
in Equation (8.3), a least-square fit method was used to solve the system of equations. To
allow for rigid body translation and rotation during the experiments, additional terms were
included to Equation (8.3):
u =
N
X
n=1
A
n
f
n,m
(r,θ)−
N
X
n=1
B
n
g
n,m
(r,θ)+T
x
−rsin(θ)R,
v =
N
X
n=1
A
n
h
n,m
(r,θ)−
N
X
n=1
B
n
l
n,m
(r,θ)+T
y
+rcos(θ)R, (8.4)
where T
x
and T
y
express the rigid body translation for the x- and y-directions and R is the
rigid body rotation. The terms f
n,m
and h
n,m
are used for mode-I calculations and g
n,m
and
106
l
n,m
can be used for mode-II. Equation (8.4) can be expressed in matrix form as:
u
1
.
.
.
u
m
v
1
.
.
.
v
m
=
f
1,1
··· f
n,1
0 g
1,1
··· g
n,1
1 0 −r
1
sin(θ
1
)
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
f
1,m
··· f
n,m
0 g
1,m
··· g
n,m
1 0 −r
m
sin(θ
m
)
h
1,1
··· h
n,1
0 l
1,1
··· l
n,1
0 1 r
1
cos(θ
1
)
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
h
1,m
··· h
n,m
0 l
1,m
··· l
n,m
0 1 r
m
cos(θ
m
)
A
1
.
.
.
A
m
B
1
.
.
.
B
m
T
x
T
y
R
. (8.5)
To solve the parameters A
n
, B
n
, T
x
, T
y
and R, an iterative procedure was utilized [114,
132]. To do this, Equation (8.5) was rewritten as:
h
x
=
N
X
n=1
A
n
f
n,m
(r,θ)−
N
X
n=1
B
n
g
n,m
(r,θ)+T
x
−rsin(θ)R−u,
h
y
=
N
X
n=1
A
n
h
n,m
(r,θ)−
N
X
n=1
B
n
l
n,m
(r,θ)+T
y
+rcos(θ)R−v, (8.6)
Subsequently, using a Taylor series expansion on Equation (8.6):
(h
x
)
i+1
= (h
x
)
i
+
δh
x
δA
1
i
ΔA
1
+···+
δh
x
δB
1
i
ΔB
1
+···+
δh
x
δT
x
i
ΔT
x
+
δh
x
δR
i
ΔR,
(h
y
)
i+1
= (h
y
)
i
+
δh
y
δA
1
i
ΔA
1
+···+
δh
y
δB
1
i
ΔB
1
+···+
δh
y
δT
y
i
ΔT
y
+
δh
y
δR
i
ΔR. (8.7)
Each subscript i represents the ith iteration step and Δ are corrections to the previous
estimation of A
n
, B
n
, T
x
, T
y
and R. The end goal of this iterative process is for (h
x
)
i+1
=
107
(h
y
)
i+1
= 0, which yields,
−(h
x
)
i
=
δh
x
δA
1
i
ΔA
1
+···+
δh
x
δB
1
i
ΔB
1
+···+
δh
x
δT
x
i
ΔT
x
+
δh
x
δR
i
ΔR,
−(h
y
)
i
=
δh
y
δA
1
i
ΔA
1
+···+
δh
y
δB
1
i
ΔB
1
+···+
δh
y
δT
y
i
ΔT
y
+
δh
y
δR
i
ΔR. (8.8)
These terms can be expressed in matrix form as:
h = bΔ. (8.9)
The solution to Equation (8.9) is solved using a least-square fit by inserting h calculated
using Equation (8.6) and b is the matrix shown in Equation (8.5):
Δ =
b
T
b
−1
b
T
h. (8.10)
This procedure was followed from [115, 134, 135]. The given solution is a result of the
iterative correction to the coefficients A
n
, B
n
and constants T
x
, T
y
and R. Thus, the best-fit
set of coefficient was obtained by:
(A
1
)
i+1
= (A
1
)
i
+ΔA
1
,
.
.
.
(B
1
)
i+1
= (B
1
)
i
+ΔB
1
,
.
.
.
(T
x
)
i+1
= (T
x
)
i
+ΔT
x
,
(T
y
)
i+1
= (T
y
)
i
+ΔT
y
,
(R)
i+1
= (R)
i
+ΔR.
(8.11)
Usually only two or three iterations were necessary to obtain an acceptably small correction
Δ. The method described allowed for estimation of SIFs from in-plane displacement fields.
108
Finally, the mode-I stress, K
I
, can be calculated by using:
K
I
= A
1
√
2π. (8.12)
Orthotropic Materials
Thegeneralequationsfordisplacementfieldfunctionsaroundcrack-tipsinorthotropicbodies
are derived using a complex variable approach. The stress intensity factors (SIFs) can be
directly evaluated from these functions. The same overdeterministic approach and a least-
square optimization method used to calculate SIFs for isotropic materials were used for the
composite materials. To dothis, Equation (8.2) was replacedbythedisplacementfield(u,v)
given by [53, 113].
u = A
1
Re
1
γ
2
−γ
1
(p
1
γ
2
z
1
−p
2
γ
1
z
2
)
√
r,
v = A
1
Re
1
γ
2
−γ
1
(q
1
γ
2
z
1
−q
2
γ
1
z
2
)
√
r, (8.13)
where the quantities p
j
, q
j
and z
j
(j = 1,2) are defined as:
p
j
= γ
2
j
s
11
+s
12
−γ
j
s
16
,
q
j
= γ
j
s
12
+
s
22
γ
j
−s
26
,
z
j
=
p
cosθ +γsinθ,
where the variables γ
j
(j = 1,2) are the roots of the characteristic equation, shown in
Equation (8.14), with positive imaginary part and s
ij
(i,j = 1,...,6) are the elements of the
compliance matrix of the material:
s
11
γ
4
−2s
16
γ
3
+(2s
12
+s
66
)γ
2
−2s
26
γ +s
22
= 0. (8.14)
In this work, a planar state of stress and an orthotropic material system aligned with the
primary fiber directions was assumed. Thus, the compliance matrix of the material can be
109
expressed as [136, 137]:
s
11
= 1/E
1
s
12
=−ν
12
/E
1
0
s
21
=−ν
12
/E
1
s
22
= 1/E
2
0
0 0 s
66
= 1/G
12
.
For this case, the only terms considered were s
ij
(i,j = 1,2,6). Then, the quantities p
j
,
q
j
and z
j
(j=1,2) were redefined as:
p
j
= γ
2
j
s
11
+s
12
,
q
j
= γ
j
s
12
+
s
22
γ
j
,
z
j
=
p
cosθ +γsinθ,
and the characteristic equation can be expressed as:
s
11
γ
4
+(2s
12
+s
66
)γ
2
+s
22
= 0. (8.15)
Once the roots have been calculated, the displacement components u and v were used
to evaluate the term A
1
following the overdeterministic least-square method previously de-
scribed. Finally, the SIF for orthotropic materials can be calculated by:
K
I
= A
1
r
π
2
. (8.16)
B. Alternate Method to Extract Mode-I SIF for Orthotropic Ma-
terials
A second method was used to evaluate the SIFs for orthotropic materials to validate the
SIFs calculated using the overdeterministic approach previously described. In this method,
the SIF can be estimated using the linear relation between the displacement field (u,v) and
110
the SIF, K
I
, shown in Equation (8.13).
To do this, the displacement component described in Equation (8.13), u
2
, was con-
structed assuming a SIF of K
I
= 1 MPa
√
m. The terms of the characteristic equation,
Equation (8.15), and the displacement field coordinates (r,θ) used for u
2
were the same as
the ones used in the analysis using the displacement component, u, calculated using DIC.
Note that the subscript 2 was assigned to the analytically calculated displacement compo-
nent, u
2
, to differentiate it from the displacement component, u, obtained using DIC. Then,
the ratio of u and u
2
yields:
u
u
2
=
K
I
p
π/2Re[(p
1
γ
2
z
1
−p
2
γ
1
z
2
)/γ
2
−γ
1
]
√
r
(1)
p
π/2Re[(p
1
γ
2
z
1
−p
2
γ
1
z
2
)/γ
2
−γ
1
]
√
r
.
The SIF can be estimated from the ratio of the displacement component, u
2
, calculated
assumingK
I
= 1MPa
√
mandthedisplacementcomponent,u,obtainedfromDIC,resulting
in K
I
= u/u
2
. Note that only the horizontal displacement component, u, was used for this
method as it was assumed that the load was uniaxial, contrary to the least-square method
in which both displacement components were included in the analysis. This method is a
simple way to check the SIF values calculated using the overdeterministic approach.
Finally,itmustbeclarifiedthattheSIFwascalculatedonlyuptowhenfractureoccurred.
Once a crack starts propagating, the exact location of the moving crack-tip is necessary
to calculate the correct SIF. Unfortunately, one of the main drawbacks of using DIC for
dynamic fracture studies is the difficulty of locating the crack-tip due to spatial resolution
requirements. Currently, the spatial resolution available in our laboratory setup does not
allow to confidently locate the moving crack-tip, so the results presented in this work will
be focused in reporting the critical SIF. However, once the spatial resolution is increased,
there are linear and non-linear methods [114, 138] available to calculate the exact location
of a moving crack-tip. The spatial resolution reported by previous research groups that were
able to locate the crack-tip to calculate the SIF as the crack propagated varied between
12–31 μ/pixel [113, 114, 134].
111
II. Material and Conditioning
The CFVE laminates were provided by collaborators at the Naval Surface Warfare Center
CarderockDivision. Thesampleswerefabricatedfromunidirectionalnonwovencarbonfabric
(12K tows) and 510A vinyl ester resin (same resin material used as samples in Chapter 5).
The tows were held together with a thin hot-melt polyester strand applied across the tows
on an approximately 10 mm spacing. The panel was 11 plies thick and was fabricated in
3 separate infusions of 4 plies, 4 plies, and 3 plies. The first 4 plies were infused with the
bottom ply placed directly against the mold surface. A heat treated nylon fabric release ply
was placed on the upper ply of the 4-ply stack, between the dry uni-carbon fabric and the
infusion resin distribution media. The nylon release fabric was used to create a clean and
bond-ready surface which required no additional preparation for good bond strength with
subsequent infusions. Following the infusion of the 4 plies, the release fabric was removed
anda4additionalplieswereinfuseddirectlyontopoftheinitialstack. Thefinal3plieswere
placed directly on the mold surface and the previously infused 8-ply laminate was flipped
over onto the 3-ply stack. For the last infusion step, the resin was infused into the dry
fabric between the mold surface and the previously infused laminate. The final laminate was
post-cured at 60
◦
C to replicate the same degree of cure as the neat resin panels used for
the samples described in Chapter 5 and to replicate the standard procedure followed by the
Navy. The completed laminate had a desired smooth molded finish on both top and bottom
surfaces for ease of use with visualization techniques.
The geometry of the samples used in these experiments was the same as the ones used
in Chapter 5. The samples were cut from the laminates using a waterjet machine such that
the fiber orientation of the samples was 90
◦
. Then, a diamond blade saw was used to cut
9 mm long and 0.4 mm wide notches at half length of the samples, and the tip of the notches
were gently hit with a blade to create a natural crack. The notches were parallel to the fiber
orientation. The material properties for the CFVE samples are listed in Table 8.1. These
estimate values were provided by the manufacturer at the NSWC Carderock Division.
112
Table 8.1: Properties of unidirectional CFVE samples with a fiber orientation of 90
◦
Property Value
E
1
55 GPa
E
2
7 GPa
ν
12
0.05
G
12
26 GPa
V
f
0.6
ρ 1,600 kg/m
3
B 3.3 mm
The conditioning procedure consisted of dividing a total of 8 specimens into two sets
correspondingtotheirrespectiveenvironmentexposure: (1)dryand(2)distilledwater. The
first set of samples was dried in an oven for 24 hours at 70
◦
C to remove any water content.
The second set of samples were used to investigate the effects on dynamic fracture initiation
due to immersion in distilled water for 43 days at 25
◦
C. The weight change in the samples
duringtheirconditioningtimewasmonitoredusingaweightscalewithareadabilityof0.01g.
ThewatercontentwascalculatedusingthesameprocedureasinChapter5. Figure8.2shows
the weight measurements as a function of time. It can be seen that water content increased
with time. However, the rate of sorption and final water content of this material was less
than half that of the vinyl ester resin samples (shown in Chapter 5) immersed in water for
the same amount of time. This behavior is characteristics of CFVE and exemplifies why it
is an attractive option for marine applications due to its low hydrolysis capabilities.
As mentioned in Chapter 3, DIC requires the surface of the samples to be covered by
a random speckle pattern that deforms together with the object. The speckle pattern was
appliedusingblackandwhiteRustoleumspraypaint. Forthesamplesthatwereconditioned
in water, the speckle pattern was applied post-conditioning and prior to impact.
113
0 5 10 15 20 25 30 35 40 45
0
0.05
0.1
0.15
0.2
Water content [% ]
Time [days]
Figure 8.2: Weight increase due to water sorption on CFVE samples.
III. Results
Experiments were performed with the purpose of calibrating the system and to prove that
the implementation in the laboratory setup yields satisfactory measurements. Figure 8.3
shows the field of view for these experiments.
Strain gauge
Figure 8.3: Schematic of DIC calibration experiment. Field of view showing speckle pattern
and strain gauge location (on opposite side).
A PMMA plate was impacted using the same procedure as in Chapter 4 [78]. In this
experiment, astraingaugewasplacedonthesideofthesamplesoppositetothesidecovered
with a speckle pattern, as shown in Figure 8.3. Once the sample was impacted with a speed
114
of 40 m/s, the response of the strain gauge was compared to the strain measured with DIC
at the same location on the opposite face on which the gauge was installed. The response
of the gauge was recorded with an oscilloscope, and the speckle pattern was recorded using
the high-speed camera at a frame rate of 100,000 fps and a resolution of 500×90 pixels.
The results plotted in Figure 8.4 show agreement between DIC measurements and the strain
gauge response. Therefore, these results confirm that the speckle pattern applied, the in-
house built high-intensity LED system, and the DIC post-processing procedure are suitable
to measure strain for these experiments.
0 20 40 60 80 100 120
−3000
−2000
−1000
0
1000
2000
3000
Strain [μstrain]
Time [μs]
DIC
Strain Gauge
Figure 8.4: Strain measurement comparison for impact experiment on PMMA between the
response of a strain gauge and the calculated strain obtained from DIC.
A repeatability study was also conducted by subjecting two additional PMMA samples
to the same loading condition as shown in Figure 8.3. As part of the repeatability study,
two different Nikon lenses with focal lengths (FL) of 200 mm and 105 mm were used with
the high-speed camera for two different experiments to study the effect of resolution on the
strain measurements. A resolution of 71 μm/pixel and 166 μm/pixel were obtained with
the 200 mm and 105 mm lenses, respectively. Figure 8.5 shows the strain response measured
25 mm away from the free-end of the samples. Results show good repeatability in amplitude
andsimilaroveralltrend. Theuseofdifferentlensesdoesnotseemtohaveasignificanteffect
on the measurements. Any differences can be attributed to the response of the material as
similar variations in the strain response have been observed when using strain gauges for the
115
work done in Chapter 4.
0 20 40 60 80 100 120 140 160
−5000
0
5000
Strain [μstrain]
Time [μs]
200 mm FL
105 mm FL
105 mm FL
Figure 8.5: Comparison of strain response for repeatability. Uncertainty for each data point
is represented by marker size.
Finally,theinherentnoisefromthemeasurementswasquantifiedbycalculatingthestrain
of nine consecutive images of the static sample. The measurements shown in Figure 8.6 were
averagedoverafieldofviewof35mmby5mm. ComparedtotheresultsshowninFigure8.6,
the maximum noise measured represents less than 5% of the maximum strain recorded from
the response of the material to an impact.
0 2 4 6 8 10
−300
−200
−100
0
100
200
300
Strain [μstrain]
Image #
Figure 8.6: Inherent noise showing less than 5% of the maximum strain recorded from the
response of the material to an impact.
Using the same data obtained from the experiment shown in Figure 8.3, additional mea-
116
surements, other than strain, can be performed to characterize the dynamic mechanical
behavior of materials. One of these measurements is the longitudinal wave speed. The wave
speed was calculated using DIC by measuring the time of arrival of the wave at two differ-
ent locations within the area of interest of the sample, designated A and B in Figure 8.7.
The time of arrival at each point can be clearly recognized by analyzing the plot shown in
Figure 8.7, from which Δt was calculated. The wave speed measured was 2,590±400 m/s,
which is within the range of the reported values of the longitudinal wave speed for PMMA.
However, the temporal resolution needs to be increased to reduce the uncertainty of this
measurement. The principles and preliminary results of additional measurements that can
be completed with DIC to characterize the dynamic mechanical behavior of materials are
described in Appendix 9.
B A
0 5 10 15 20
−3
−2
−1
0
1
Wave Speed [m/s]
Image #
Point A
Point B
Δt
Figure 8.7: Longitudinal wave speed calculation. The time of arrival of the longitudinal wave
was recorded for Point A and B to calculate Δt, which was used to compute the wave speed.
Experiments were performed to study the dynamic fracture initiation of PMMA using
DIC to compare to the results obtained in Chapter 4 and to assure that the implementation
of the overdeterministic approach algorithm was performed correctly. Figure 8.8 shows an
example of a sequence obtained for dynamic fracture experiments on PMMA. Even though
117
the results in this work are limited to calculating the critical SIF, it is worth mentioning
that the main advantage of DIC in fracture studies is that it provides a full-field strain mea-
surement around the crack-tip, showing and quantifying the development of the deformation
around the crack as the stress concentration increases. Full-field measurements, such as
the one shown in Figure 8.8, were used to calculate the critical SIF of two PMMA samples
subjected to an impact of 40 m/s.
(a) 0 μs (b) 7 μs (c) 14 μs (d) 21 μs (e) 28 μs
(f) 42 μs (g) 63 μs (h) 77 μs (i) 119 μs (j) 126μs
Figure 8.8: Full-field strain maps around a crack-tip for a PMMA sample subjected to dy-
namic loading.
TheresultsobtainedforthecriticalSIF,K
Ic
, ofPMMAusingDICcomparedtothecriti-
calSIFmeasuredusingthemethodofcausticsarepresentedinTable8.2. Bothvisualization
techniquesyieldedsimilarresults, within10%difference. Thisresultwasaconfirmationthat
the implementation of the overdeterministic approach algorithm was completed adequately.
Thus, it could confidently be utilized for other materials.
118
Table 8.2: Comparison of the critical SIF obtained using DIC and the method of caustics for
an impact of 40 m/s
Method K
Ic
[MPa
√
m]
Method of Caustics 2
Digital Image Correlation 2.2
Next, a total of six CFVE samples conditioned in two different environments were sub-
jected to a 66 m/s impact. Figure 8.9 shows a schematic of the impact experiment and field
of view of the high-speed camera. The 90
◦
fiber orientation of the CFVE samples is also pre-
sented. This fiber orientation represents the weakest configuration of a FRP for this loading
scenario since the fibers are parallel to the notch and perpendicular to the load direction.
x
y
β
Figure 8.9: Impact schematic for CFVE sample showing its 90
◦
fiber orientation, parallel to
the notch.
The plot displayed in Figure 8.10 correlates the SIF and the mechanical response of the
CFVE samples. The full-field images shown represent the strain field around the crack-tip
during crack initiation. The sequence shown on the right side of Figure 8.10 illustrates a
runningcrack-tipcapturedwithDIC.Clearly, thehigherstrainisattainedasthecrackprop-
agates when the crack faces are furthest apart. As expected, mode-I fracture was observed
at the notch location parallel to the fiber orientation. After impact, samples lost most of
their structural rigidity; however, they were not split into two pieces because they remained
held together by the thin hot-melt polyester strand applied across the tows. Longitudinal
intralaminar matrix cracking, leading to fiber bridging were identified as the main failure
mechanisms.
Figure 8.11 shows a comparison of the SIF measurements for CFVE samples. Certainly,
there was a significant degradation on the fracture toughness of the material, approximately
30%, after immersion in water for 43 days. The large scatter observed on the dry samples
119
0 2000 4000 6000 8000 10000
0
1
2
3
4
5
6
Stress Intensity Factor [MPa √m]
Strain [με]
Fracture
Figure 8.10: CFVE SIF measurements and full-field strain maps displaying crack propaga-
tion.
120
could be attributed to the drying procedure, which might have required a longer period of
time to fully remove the water content in the samples prior to impact.
Dry Soaked
0
2
4
6
8
Stress Intensity Factor [MPa √m]
Figure 8.11: Critical SIF comparison for 3 dry and 3 soaked CFVE samples.
Table 8.3 shows a matrix-composite comparison featuring the average SIF for dry and
soaked CFVE composite and vinyl ester neat resin. There is only a single value displayed
for vinyl ester neat matrix because, as shown in Chapter 5, the SIF for dry and soaked
specimens remained unchanged.
Table 8.3: Comparison of stress intensity factor calculation of CFVE composite and vinyl
ester resin matrix
Method CFVE dry CFVE soaked Vinyl ester neat matrix
K
Ic
[MPa
√
m] 4.9 3.1 1.5
The unidirectional CFVE samples exhibited a dynamic fracture toughness over 3 times
higherthanthatofitsconstituentmatrixwhentestedinneatresinform. Togeneratefracture
ontheunidirectionalCFVEsamples,theimpactspeedwasmorethandouble(34m/sfaster)
than the one used for the vinyl ester neat matrix samples. An impact of such speed would
have caused unstable cracking and crack branching in the vinyl ester neat matrix samples.
After 43 days of immersion, the vinyl ester samples gained a water content of 0.4 wt %,
doublethatoftheCVFEsampleswithawatercontentof0.2wt%. Whiletheeffectofwater
content on the dynamic fracture behavior of vinyl ester was negligible, the dynamic fracture
121
behavior of CVFE was degraded by ∼30% as a result of water sorption, in spite of CVFE
experiencing a water uptake less than that of vinyl ester. This demonstrates the complexity
of the dynamic fracture mechanics of FRPs when immersed in water. Water sorption can
introduce stress concentrations as water clusters inside of the matrix material. Furthermore,
it can also affect the bondage between fiber and matrix weakening the resistance of the
composite to overcome crack propagation at higher loads. It is recommended to complete
further experiments with CFVE samples to verify the repeatability of the results obtained
in this work by comparing to samples from different batches of CFVE laminates.
IV. Conclusions
In this study, the dynamic fracture initiation of CFVE samples conditioned in two different
environments, yielding in a water content differential of 0.2 wt %, was investigated. The
results can be concluded as follows:
• Preliminary results using PMMA samples were compared to previous results shown in
Chapter 4 for calibration and validation purposes. Results showed that the implemen-
tation of the setup was completed adequately and measurements were repeatable.
• AsignificantdeteriorationonthefracturetoughnessofCFVE,approximately30%,was
observed on samples exposed to water for 43 days. The degradation occurred despite
the low water content levels (0.2 wt %) the samples experienced after 43 days of con-
ditioning. The deterioration effect on the fracture toughness of CFVE was attributed
to debonding between fibers and matrix. It is recommended that further experiments
are completed to verify these results.
• Amatrix-compositeanalysiswasperformedbycomparingthedynamicfracturetough-
nessofCFVEwithitsconstituentmatrixinneatresinform, studiedinChapter5. The
unidirectional CFVE samples had a dynamic fracture toughness over 3 times higher
than that of vinyl ester neat resin.
122
Chapter 9
Future Direction
Multiple opportunities for future studies to extend the work presented in this thesis still
remain. A few recommendations for further work on this topic are listed as follows:
• This work only included mode-I dynamic fracture studies. Experiments can be ex-
tended to different fracture modes in which various configurations and asymmetric
loading conditions can produce pure mode-II or mixed-mode crack growth. The SIF
for different modes and crack speed measurements can be performed on CFVE and
other relevant FRPs. Previous work done by [62, 139] using unidirectional graphite
fiber-epoxy composites showed highly unstable and intersonic, shear dominated crack
propagation along the fiber direction. These cracks reached unprecedented speeds of
7400 m/s, which was more than 3 times the shear wave speed of the material tested.
Similar experiments can be completed and dynamic fracture toughness measurements
can be incorporated.
• Further studies need to be completed for various fiber orientations on unidirectional
CFVE samples to obtain a wider range of cases exemplifying the dynamic fracture
behavior of this material. Since fiber orientation determines how composite materi-
als interact with the load applied, their mechanical behavior can either be matrix-or
fiber-dominated. This means that there are possibly other mechanisms or different
outcomes for this type of experiments that have yet to be studied as fiber orientation
is varied. The long term goal for this study should focus on the establishment of a
model to describe the behavior of unidirectional or quasi-isotropic CFVE materials
as fiber orientation and water content are varied. This model could be based on em-
pirical formulations relating fracture toughness, fiber orientation and water content.
123
Additionally, it would be beneficial to attempt to improve spatial resolution used for
this experiments and develop an algorithm to calculate the exact location of a moving
crack-tip [114, 138].
• To promptly exploit the advantages offered by the method described in Appendix 9
to estimate the dynamic mechanical properties of materials, it is suggested to initially
study low-impedance materials, such as core foams. Their dynamic responses are
slower than that of other materials, and their applications are relevant as structural
components in sandwich panels in the aerospace and naval industries. This will allow
theuseoflowertemporalresolution,morefittingtothecurrentcapabilitiesavailablein
the laboratory setup, to perform accurate measurements of their dynamic mechanical
behavior, as done by [140]. Furthermore, since the method shown in Appendix 9 is
restricted to only quasi-uniaxial tests, its applicability should be extended to more
complex loading scenarios by implementing the Virtual Fields Methods [141].
124
Appendix A
I. Estimation of Dynamic Mechanical Behavior using
Displacement Fields
DIC allows the estimation of mechanical properties of materials using displacement fields.
In this thesis, an initial study was conducted to generate a platform that can be used to
further quantify the dynamic behavior of materials. This approach was initially developed
by Aloui et al. [142]. The main advantages of this procedure is that it is a non-intrusive
measurement,soitdoesnotinterferewiththebehaviorofthematerialandthatitisfull-field,
thus, allowing localized and global measurements. The main drawback is that it is highly
dependent on the experimental setup and equipment. It is practically required to use ultra
high-speedcamerasbecausehightemporalresolutionandspatialresolutionarerequired. The
higher the impedance of the material, the higher speeds required to quantify the behavior
under dynamic loading, thus, limiting the extension of applicability of this method to low
impedance materials with the technology developed and available. This method can be
applied to lower strain rates as well. However, the author believes that the main advantage
of this technique resides in dynamic events because it allows for measurements that have
never been done before with high accuracy.
Here, a preliminary study on the estimate of inertial stresses from displacement fields
was conducted. Acceleration fields were used coupled with a free-end boundary condition,
where the stress should always equal zero, to reconstruct the inertial stress in the sample. A
central scheme difference was used to calculate the acceleration fields from the displacement
components, u and v. In order to calculate the inertial stresses using the full-field data
extracted from DIC, the geometry of the sample was divided into a finite number of thin
125
sections, dz, along the specimen length. One of the main benefits of this approach is that
using the reconstructed stress and its corresponding strain averaged on the same sliced
section, local stress-strain curves can be plotted for each frame. These plots can be used to
estimate the elastic modulus, E, and Poisson’s ratio without any constitutive model of the
material [143, 144]. Figure 1 shows a free body diagram of a sample in uniaxial loading with
an isolated sliced section. Considering Newton’s law of motion in the longitudinal direction
(z in Figure 1), the following equation describes the state of the sliced section shown in
Figure 1:
F
z
(z
i
+dz
1
i,t)−F
z
(z
i
,t) = ρ
0
A
0
dz
δ
2
u
δt
2
(z
i
,t), (1)
where ρ
0
and A
0
are the initial density and cross-sectional area of the specimen respectively;
u,asintheprevioussection,isthedisplacementinthelongitudinaldirectionofthespecimen.
Therefore, Equation (1) can be rewritten as:
dF
z
(z
i
,t)
dz
i
= ρ
0
A
0
δ
2
u
δt
2
(z
i
,t). (2)
By integrating Equation (2) along the length of the sample, the force can be described as
follows:
F
z
(z
i
,t) = F
z
(0,t)+
Z
z
i
0
ρ
0
A
0
δ
2
u
δt
2
(ξ,t)dξ. (3)
As described in [142, 145], these equations allow to recover the force along the specimen’s
length as long as F
z
(0,t) is known. In other studies [140, 144, 146], various apparatus have
been used to measure this boundary condition, F
z
(0,t), such as force transducers or the
Kolsky bar method. Pierron and Forquin [147] introduced the idea of using a free-end as a
boundarycondition, inwhichF
z
(0,t) = 0. Thus, allthatisneededaretheaccelerationfields
calculated from the displacement fields measured using DIC. Therefore, the force applied at
any cross-section of the specimen can be determined by measuring the displacement field.
One of the main limitations encountered during this work is that in order to calculate
force, it is necessary to calculate the cross-sectional area as it changes with time, A(z
1
,t).
The measurement of the cross-sectional area is limited by the field of view of the camera,
126
x
z
i
i
z
z
F (z +dz, t)
F (z , t)
Projectile side Free-end side
e
F (t)=0
i
i
A
p
F (t)
z=z
z=0
i=n
i=2 i=1
dz
Figure 1: Schematic of the sample impacted and sliced into segments to calculate the inertial
stress.
which is not always the same as the height of the sample. The equation to estimate A(z
1
,t)
for any material is given by [145]:
A(z
1
,t) = A
0
(1+ǫ
22
(z
1
,t))(1+ǫ
33
(z
1
,t)), (4)
where A
0
is the original cross-sectional area, ǫ
22
is the in-plane strain on the transverse
direction of the load and ǫ
33
is the out-of-plane strain. Additionally, to calculate ǫ
33
, a
second camera and a 3D correlation software are required. For any orthotropic material, not
considered quasi-isotropic, the measurement of ǫ
33
is necessary to estimate the force within
a specimen. However, for isotropic and quasi-isotropic materials, it can be assumed that
ǫ
22
= ǫ
33
, which allows the use of only in-plane measurements done with a single camera.
This assumption yields:
A(z
1
,t) = A(z
1
,0)(1+ǫ
22
(z
1
,t))
2
. (5)
The nominal and true stress can be calculated by assuming a uniaxial stress field as follows
127
[142]:
σ
n
11
(z
1
,t) =
F
z
(z
1
,t)
A
0
,
σ
tr
11
(z
1
,t) =
F
z
(z
1
,t)
A(z
1
,t)
. (6)
Assuming that F
z
(z
1
,t)/A
0
= 0 because of the free-end condition in Equation (3). Then,
the nominal and true inertial stresses can be estimated by:
σ
n
11
(z
1
,t) =
Z
z
1
0
ρ
0
δ
2
u
δt
2
(ξ,t)dξ≈
n
X
i=n
ρ
(i)
0
s
(i)
a
(i)
,
σ
tr
11
(z
1
,t) =
ρ
0
A
0
A(z
i
,t)
Z
z
1
0
δ
2
u
δt
2
(ξ,t)dξ≈
ρ
0
A
0
A(z
i
,t)
n
X
i=1
s
(i)
a
(i)
, (7)
wheresrepresentsthethicknessoftheithslicedsection,ndenotesthetotalnumberofsliced
sections up to the location z
i
, and a is the average axial acceleration over the slice.
For this work, a second camera and a 3D correlation software were not available in our
laboratory. For this reason, inertial stresses were not calculated. For the preliminary results
shown here, the measurements performed on PMMA samples used for the calibration exper-
iments in Chapter 8 were used to calculate the acceleration fields shown in this Appendix.
Aspreviouslydescribed, accelerationfieldswerecalculatedusingthedisplacementsobtained
fromDIC.Asequenceofimagesshowingtheprogressionofthefullfieldaccelerationisshown
inFigure2. Additionally, Figure3(a)showstheaverageaccelerationexperiencedinthesam-
ple. Comparing the results obtained to the inherent noise, shown in Figure 3(b), it can be
seen that for acceleration the noise is small compared to the amplitude of the measurements.
Usingtheaccelerationfields,theinertialstressescanbecalculatedbyapplyingEquation(7).
128
(a) 0 μs (b) 6.56 μs (c) 19.68 μs
(d) 39.36 μs (e) 45.92 μs (f) 59.04 μs
(g) 65.60 μs (h) 78.72 μs (i) 91.84 μs
Figure 2: Full-field acceleration maps for a PMMA sample subjected to dynamic loading.
The transient behavior of the applied load is observed. There is an incoming compressive
pulse (travelling to the right) followed by a tensile pulse reflected (traveling to the left) from
the free end of the sample.
80 100 120 140 160 180
−6
−4
−2
0
2
4
6
x 10
5
Acceleration [m/s
2
]
Time [μs]
(a)
0 2 4 6 8 10
−5
0
5
x 10
4
Acceleration [m/s
2
]
Image #
(b)
Figure 3: (a) Average acceleration measurement from full field visualization (b) Noise calcu-
lated for axial acceleration represent less than 4% of maximum acceleration calculated.
129
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Abstract (if available)
Abstract
The useful properties that polymeric materials offer make them good candidates to be considered in naval and aircraft applications as the matrix constituent in a composite material, such as fiber reinforced polymers (FRP). Naturally, material properties are prone to degenerate as a result of exposure to their environment and the characteristics of the material. In order to confidently use these materials in outdoor environments in which, for instance, humidity levels are varied, it is important to understand how environmental conditions may affect their performance under different loading conditions. In this work, an experimental study was completed regarding the effects of varied humidity levels and sorbed water amounts on dynamic crack initiation and propagation of three polymeric materials, Poly(Methyl Methacrylate) (PMMA), vinyl ester neat resin and carbon-fiber/vinyl ester (CFVE) composite, subjected to stress pulses created by an gas-gun. Edge-on impact experiments were performed on samples conditioned in different environments, including dry specimens, specimens exposed to different relative humidity environments, and distilled water saturated specimens. Experiments varied by immersion time, and similar loading rates were applied to all sample groups. The optical, mechanical and fracture properties were investigated under different water contents. Additionally, the effect of water as a surrounding medium was studied on the dynamic fracture behavior of PMMA. High-speed photography combined with three different non-invasive visualization techniques, namely the method of transmitted caustics, a Fizeau interferometer, and digital image correlation, and simultaneous strain gauges were utilized to obtain quantitative data from the experiments depending on the material and measurements required. Among the properties measured during these experiments were the stress intensity factor, stress optical coefficient, and crack-tip speed. Results yielded repeatable responses within the same materials studied. The fracture behavior of these polymeric materials did not show a significant change due to water content or surrounding environment. This was attributed to the effect of high strain rate, which overcame the effect of water. On the other hand, the fracture behavior of CFVE was significantly affected by water content, in that the fracture toughness of the material degraded by 30%. This dissertation is part of an ongoing effort in which the experimental techniques implemented and the results obtained will be used as a baseline to study the change of dynamic fracture behavior of fiber-reinforced polymers subjected to similar conditioning and loading conditions.
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Creator
Delpino Gonzales, Orlando
(author)
Core Title
On the dynamic fracture behavior of polymeric materials subjected to extreme conditions
School
Viterbi School of Engineering
Degree
Doctor of Philosophy
Degree Program
Aerospace Engineering
Publication Date
06/21/2016
Defense Date
04/27/2016
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University of Southern California
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Tag
caustics,digital image correlation,dynamic fracture,experimental mechanics,OAI-PMH Harvest,polymers
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English
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Eliasson, Veronica (
committee chair
), Lee, Vincent W. (
committee member
), Nutt, Steven (
committee member
)
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delpinog@usc.edu,orlando_dg@hotmail.com
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Tags
caustics
digital image correlation
dynamic fracture
experimental mechanics
polymers