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Shock wave response of in situ iron-based metallic glass matrix composites

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Shock wave response of in situ iron-based metallic glass matrix composites

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Content Shock Wave Response of in situ Iron-based
Metallic Glass Matrix Composites
Gauri Khanolkar
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 Mechanical Engineering
December 2015
© 2015 by Gauri Khanolkar
To my parents Sujata and Rajendra,
for their boundless love and encouragement
i
Acknowledgments
I wish to thank my advisor Prof. Veronica Eliasson for giving me the opportunity to
work on this project and for being an exceptional mentor and teacher over the years. I
am immensely grateful to her for her guidance and support, and for always putting my
learning as a researcher first. Many thanks to Prof. Andrea Hodge for playing such an
instrumental roleinmy admission toUSC, andgiving me theopportunityofa lifetimeto
pursue my doctoral studies here, in addition to serving as my thesis committee member
and giving me invaluable advice throughout the course of my graduate studies. I would
also like to acknowledge Prof. Michael Kassner, Prof. Steve Nutt, and Prof. Yan Jin for
being on my thesis committee and for all their helpful suggestions.
I am indebted to Dr. Michael Rauls and Prof. Ravichandran of Caltech for their ex-
pert help in running my experiments. This work is due in no small part to them. I am
grateful to Prof. Y. M. Gupta, Dr. Yoshi Toyoda, Kurt Zimmerman, and Nate Argan-
brightforallthattheyhavetaughtmeaboutshockphysicsandplateimpactexperiments
during my time at Washington State University, Pullman. I would like to acknowledge
Dr. James Kelly and Prof. Olivia Graeve of the University of California, San Diego for
providing amorphous steel samples and Prof. Kathy Flores of Washington University in
St. Louis for providing Zr-based metallic glass samples. Many thanks to Dr. I-Chung
Chen and Andrew Lindo for all their help with material characterization. The Defense
Threat Reduction Agency is sincerely acknowledged for funding this work, and Program
Manager Dr. Suhithi Peiris for her support. Silvana Martinez, Samantha Graves and
Jamie Kidder of the Aerospace and Mechanical Engineering department office are grate-
ii
fully acknowledged for always being so incredibly helpful. My sincere gratitude to Gary
Kuepper, Ramon Delgadillo, Kan Lee and Don Wiggins of the USC Viterbi Machine
Shop for all their impeccable machining work for my experimental components, even at
short notice.
A big thank you to my wonderful labmates Dr. Chuanxi Wang, Orlando Delpino,
Stelios Koumlis, Shi Qui, Qian Wan, Honjoo Jeon and Jack Gross for many useful dis-
cussions and making the lab such a fun place to work. To my friends at USC and WSU-
RebeccaLee,LuciaSun,SydnieLieb,OkjooPark,JamesHumann,ShanlingYang,Angie
Lee, Aditi Bauskar, Mihindra Dunuwille, Sakun Duwal, Pritha Renganathan, Anirban
and Andrea Mandal and others - thank you for all the great times and pep talks when I
needed them, I could not have survived grad school without you.
Finally, many thanks to my family for their unconditional love and unwavering sup-
port-myparents,myin-laws,andmyyoungerbrotherAmey. TomyhusbandHrishikesh,
thank you for being my rock and inspiring me to do my best.
iii
Contents
Dedication i
Acknowledgments ii
Abstract xii
1 Introduction 1
1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.1.1 Response to quasi-static & dynamic uniaxial stress . . . . . . . . 4
1.1.2 Response to shock wave loading . . . . . . . . . . . . . . . . . . . 7
1.1.3 Iron-based bulk metallic glasses . . . . . . . . . . . . . . . . . . . 10
1.1.4 In Situ Metallic Glass Matrix Composites . . . . . . . . . . . . . 12
1.2 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
1.3 Outline. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2 Theoretical Background 18
2.1 Shock Wave Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.1.1 Shock Jump Conditions . . . . . . . . . . . . . . . . . . . . . . . 19
2.1.2 Equation of State . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.1.3 Hugoniot. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.1.4 Impedance Matching . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.2 Continuum Mechanics Theory . . . . . . . . . . . . . . . . . . . . . . . . 25
2.2.1 Elastic Response . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
iv
2.2.2 Elastic-Plastic Response to 1-D Shock Compression . . . . . . . . 29
3 Experimental Methodology 32
3.1 Material Characterization . . . . . . . . . . . . . . . . . . . . . . . . . . 33
3.2 Target and Projectile preparation . . . . . . . . . . . . . . . . . . . . . . 34
3.3 Loading Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
3.4 Velocity Interferometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
3.4.1 VISAR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
3.4.2 PDV . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
4 Experimental Results 47
4.1 Tilt-corrected elastic shock velocity . . . . . . . . . . . . . . . . . . . . . 49
4.2 Determining particle velocity at the HEL . . . . . . . . . . . . . . . . . . 53
4.3 Shock compression experiments . . . . . . . . . . . . . . . . . . . . . . . 53
4.3.1 Temperature rise due to shock waves . . . . . . . . . . . . . . . . 60
4.3.2 Elastic-Perfectly Plastic Response and the Hydrostat . . . . . . . 63
4.3.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
5 Continuum Modeling in ANSYS Autodyn 75
5.1 ANSYS Autodyn formulation . . . . . . . . . . . . . . . . . . . . . . . . 75
5.2 Autodyn model set-up . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
5.3 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
6 Dynamic Fracture Morphology of a Zr-based BMG 86
6.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
6.2 Experimental Method. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
6.3 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
7 Conclusion and Future Direction 96
Bibliography 101
v
List of Figures
1.1 Compression and tension stress-strain curves for a Zr-based alloy showing
limited plasticity in compression and catastrophic failure in tension [156]. 5
1.2 Ashby maps showing superior tensile strength and hardness of amorphous
steels as compared with other amorphous and crystalline metallic alloys [59] 11
1.3 MicrostructureandcorrespondingmechanicalstrengthsofZr-basedmetal-
lic glasses containing varying kinds of in situ nanocrystalline reinforce-
ments [61]. Here σ
f
is the fracture strength, ǫ
f
is the fracture strain, E is
the Young’s Modulus, V
f
is the volume fraction of nanoparticles, and d is
the size of the nanoparticles. . . . . . . . . . . . . . . . . . . . . . . . . . 14
1.4 Experimental techniques and real world processes across a range of strain
rates, with regime of interest highlighted in red. . . . . . . . . . . . . . . 17
2.1 Schematic of conditions ahead of and behind two shock waves . . . . . . 20
2.2 Rayleighline connecting finalstate ontheHugoniot(denoted bysubscript
1) from a given initial state (denoted by subscript 0) . . . . . . . . . . . 23
2.3 (a) Graphical representation of impedance matching using P −u
p
Hugo-
niotsshowingState1attheintersectionoftheforward-facingHugoniotfor
B and the reflected Hugoniot for A, (b) t−x plot for wave interactions in
impactor A hitting target B. States 0’ and 0 represent ambient conditions
in A and B respectively, and State 1 the shocked state in both. . . . . . . 23
vi
2.4 Representation of (a) P−u
p
Hugoniot and (b) t−x plot for wave interac-
tions in target A (backed by vacuum) with its free surface. The impactor
is not shown. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.5 Representation of (a) P−u
p
Hugoniot and (b) t−x plot for wave interac-
tions of target B backed by plate C with the boundary between the two.
The impactor is not shown. Plate C has a higher acoustic impedance than B 26
2.6 A comparison of the Hugoniot, Hydrostat and the stress-strain curve. . . 30
3.1 Schematic of target configuration at a) Caltech and b) Pullman. Further
details on materials and dimensions of target components in each set-up
are listed in Tables 4.3 and 4.4. . . . . . . . . . . . . . . . . . . . . . . . 33
3.2 Vickers hardness measurement indent . . . . . . . . . . . . . . . . . . . . 34
3.3 X-ray diffraction patterns of as-received SAM2X5 samples. . . . . . . . . 35
3.4 Pullman set-up: a) Buffer, sample and window epoxied to each other, and
b) Assembled target, viewed here on the non-impact side, with VISAR
probes at the gun-muzzle. . . . . . . . . . . . . . . . . . . . . . . . . . . 36
3.5 Caltech set-up: Picture of two fully assembled target plates - BMG sam-
ples, shortingpins, andPDVprobesareaffixedintopolycarbonateholders
- and projectiles. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
3.6 Naval powder gun at Caltech. Projectiles are loaded at the breech end on
the left and prepared targets are mounted in the chamber. . . . . . . . . 38
3.7 Schematic of projectile exiting muzzle-end of the powder gun to impact
assembled target at Caltech. T1, T2, T3, and T4 on the impact face of
the target holder denote the locations of the four shorting pins used for
tilt and time of arrival measurements. . . . . . . . . . . . . . . . . . . . . 38
3.8 The conventional VISAR . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
3.9 The push-pull VISAR. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
3.10 Schematic of the heterodyne PDV . . . . . . . . . . . . . . . . . . . . . . 46
vii
4.1 Schematic ofshorting pinlocations 2, 3, 4 andprojection ofparticle veloc-
itymeasurementlocationontotheimpactplaneat1. Dashedlinesindicate
transformed co-ordinate system obtained by rotating through angle φ. . . 50
4.2 Schematic of target plate with shorting pins and interferometry probe for
particle velocity measurement. Times of arrival in the tilt measurement
plane aredenoted ast
3
, t
4
and t
′
1
andin the particle velocity measurement
plane as t
1
. The third shorting pin is not shown in this two-dimensional
sketch. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
4.3 Relation between uncorrected shock velocity (no tilt) and velocity of the
tilted shock wave . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
4.4 Identification of HEL point at intersection of straight-line fits (black) to
the two legs of the knee in the wave profile (red). . . . . . . . . . . . . . 53
4.5 Time-distance plot showing wave interaction in flyer and target immedi-
ately following impact. Measurement from PDV probes occurs at location
denoted in orange. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
4.6 Time-distance plot showing wave interaction in impactor, buffer, target
and window immediately following impact. Measurement from VISAR
probes occurs at location denoted in orange. . . . . . . . . . . . . . . . . 57
4.7 Wave profile data for SAM2x5 showing particle velocity at interface of
sample and window as measured by Pullman VISAR. The arrow on the
plot indicates the HEL point. . . . . . . . . . . . . . . . . . . . . . . . . 58
4.8 Wave profile data for XS-1 showing free surface velocity at sample rear
surface as measured by Caltech PDV. Arrows on the plot indicate the
HEL point. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
4.9 Wave profile dataforSAM2x5 showing freesurface velocity atsample rear
surfaceasmeasuredbyCaltechPDV.ArrowsontheplotindicatetheHEL
point. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
4.10 Plastic shock velocity - particle velocity Hugoniots for XS-1 and SAM2x5.
The red circle on the y-axis denotes the ambient bulk sound speed. . . . 66
viii
4.11 Elastic shock velocity - particle velocity Hugoniots for XS-1 and SAM2x5.
The intercept on the y-axis denotes the ambient longitudinal sound speed.
The red line is a linear least squares fit through the experimental data. . 67
4.12 A comparison of calculated Hugoniot (red), hydrostat (black) and experi-
mental data (blue) for XS-1. . . . . . . . . . . . . . . . . . . . . . . . . . 68
4.13 A comparison of calculated Hugoniot (red), hydrostat (black) and experi-
mental data (blue) for SAM2x5. . . . . . . . . . . . . . . . . . . . . . . . 69
4.14 Scanning electron microscope (SEM) image of the mirror-polished surface
of a SAM2x5 sample at 3300x magnification. Fused powder grains and
the pores at their triple points are visible. . . . . . . . . . . . . . . . . . 71
4.15 Post-yield shear strength as a function of density compression for XS-1
(black),SAM2x5(blue)andVitreloy106(red)fromtheworkofTurneaure
et al. (2006) [139].. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
5.1 2D axisymmetric problem set-up in Autodyn . . . . . . . . . . . . . . . . 78
5.2 A comparison of experimentally obtained wave profiles (solid lines) and
Autodyn simulated wave profiles (dashed lines) for XS-1. Simulated pro-
files have been overlapped onto measured ones for ease of comparison. . . 82
5.3 A comparison of experimentally obtained wave profiles (solid lines) and
Autodyn simulated wave profiles (dashed lines) for SAM2x5. Simulated
profiles have been overlapped onto measured ones for ease of comparison. 83
6.1 Schematic ofexperimental set-up comprising projectile-target assembly as
mounted on gun barrel: (1) Delrin sabot, (2) Tungsten flyer plate, (3)
BMG sample, (4) Tungsten base plate and (5) Polycarbonate holder. . . 89
6.2 Scanning electron micrographs of the fracture surfaces of impacted Vit-
reloy 106a samples showing fracture surface from impact with a tungsten
flyer at (a) 110 m/s b) 150 m/s. Micrographs for each of these impact
conditions show different fracture features. . . . . . . . . . . . . . . . . . 91
ix
6.3 Scanning electron micrographs of the fracture surfaces of impacted Vit-
reloy 106a samples showing (a) Typical ductile fracture surface on sample
impacted by Aluminum, (b) Ductile fracture surface containing micro-
steps on sample impacted by stainless steel, and (c) Pattern of river and
cusp veins evenly distributed across sample impacted by tungsten. . . . . 93
6.4 Scanning electron micrographs of the fracture surfaces of impacted Vit-
reloy 106a samples showing (a) River-like veins on sample impacted at
62 m/s, (b) Cusps bordering river-like veins on surface from impact at
92 m/s with arrow indicating shearing direction, (c) Pattern of cusps in
disarray on fracture surface from impact at 110 m/s, (d) Large melted
belts on sample impacted at 150 m/s. (Inset) X-ray diffraction plots of
as-received Vitreloy 106a sample, and damaged samples impacted with
tungsten flyer at 62 m/s, 92 m/s, 110 m/s and 150 m/s. . . . . . . . . . 95
x
List of Tables
1.1 Summary of reported HEL for BMGs . . . . . . . . . . . . . . . . . . . . 8
4.1 Material characterization results . . . . . . . . . . . . . . . . . . . . . . . 48
4.2 Elastic Moduli of SAM2x5 and XS-1 . . . . . . . . . . . . . . . . . . . . 48
4.3 Summary of experimental parameters for Pullman experiments . . . . . . 55
4.4 Summary of experimental parameters for Caltech experiments. Flyers
were all made of OFHC Copper. . . . . . . . . . . . . . . . . . . . . . . . 55
4.5 Summary of calculated SAM2x5 results for Pullman experiments . . . . . 60
4.6 Summary of calculated XS-1 results for Caltech experiments . . . . . . . 64
4.7 Summary of calculated SAM2x5 results for Caltech experiments . . . . . 65
5.1 Material properties used in Autodyn Model for copper . . . . . . . . . . 79
5.2 Material properties used in Autodyn Model for XS-1 and SAM2x5 . . . . 81
5.3 Comparison of experimental and simulated shock velocities . . . . . . . . 83
5.4 Comparison of experimental and simulated particle velocities . . . . . . . 84
6.1 Summary of flyer materials used, impact velocities and loading stresses
generated in plate impact experiments on Vitreloy 106a. . . . . . . . . . 90
xi
Abstract
Bulk metallic glasses (BMG) have recently garnered interest due to superior properties
suchashigherstrength,toughnessandhardness,arisingoutoftheamorphousstructureof
these metallic alloys, as compared to their crystalline counterparts. However, BMGs are
brittle and fail catastrophically following their elastic limit, which severely restricts their
use in structural applications. To offset their brittleness, studies of various combinations
of hard nano/micro particles, in situ precipitated crystalline phases and fibers embedded
within the BMG, exist in the literature. These resulting materials are known as metallic
glass matrix composites. In this work, we study the high strain-rate response of two
novel Fe-based metallic glass matrix composites, containing varying amounts of in situ
crystalline phases, when subjected to shock compression. Shock response is determined
by making velocity measurements using interferometry techniques such as the Velocity
Interferometer System for Any Reflector (VISAR) and Photonic Doppler Velocimetry
(PDV) at the rear free surface of BMG samples, which have been subjected to impact
from a high-velocity projectile launched from a powder gun. Experiments have yielded
repeatable results indicating a Hugoniot Elastic Limit (HEL) to be 12.5 GPa and 8 GPa
respectively for the two composites. The former HEL result is higher than elastic limits
foranyBMGreportedintheliteraturethusfar. Theeffectofpartialcrystallizationinthe
amorphous matrix of BMG on the observed shock response is further explored through
a comparison of the results from both composites. In addition, the sensitivity of the
fracture morphology of a Zr-based BMG Vitreloy 106 to strain rate is examined through
a series of low-velocity impact experiments using a single-stage gas gun. Post-mortem
microscopic examination of the fracture surfaces of the retrieved failed specimens was
xii
conducted. The dynamic fracture morphology for the Zr-BMG showed a clear strain-
rate dependence in the form of various unique features at the micro-scale.
xiii
Chapter 1
Introduction
Bulk metallic glasses (BMG), also known as bulk amorphous alloys or structural amor-
phous metals, aremulti-component metallicalloys thataremetastable, amorphoussolids
i.e. their atomic arrangement lacks long-range order. Their amorphous nature, and
the consequent lack of dislocation-mediated slip, enables them to reach near theoreti-
cal strengths, which results in yield strengths that are far in excess of those of their
crystalline counterparts. The absence of crystal defects and grain boundaries in BMGs
result in attributes such as high elastic limits, hardness, toughness, corrosion and wear
resistance etc. This makes them attractive for use in a vast array of high-strength ap-
plications, such as sports equipment, satellite shields for protection against meteorite
impact, bearings for use on spacecrafts, precision tooling, and components in microelec-
tromechanical systems [10,64]. Moreover, their “self-sharpening” behavior makes them
suitable for use in kinetic energy penetrators [135]. Despite all these favorable proper-
ties of bulk metallic glasses, a major drawback yet remains and that is their inability to
undergo homogenous plastic deformation. The high strength of metallic glasses makes
them vulnerable for failure due to defects and results in a lack of work-hardening capa-
bility [43]. BMGs, therefore, are known to deform by highly localized shear banding, the
propagation of which causes them to undergo catastrophic failure immediately beyond
theirelasticlimit. Thus, despite having highvaluesofhardness andyield strengthBMGs
are essentially very brittle. In an attempt to offset this brittle behavior and raise the
1
straintofailureofBMGs,twoprocesseshavebeenexploredsofartoenhancetheirductil-
ity: causing the precipitation of in-situ crystallites in the amorphous solid and external
addition of crystalline reinforcement particles to amorphous matrix to create metallic
glass matrix composites (MGMC). Both these processes have shown to be successful in
increasing the plasticity of BMGs when subjected to compression [43].
Glasses have existed in nature since the early days of the formation of the Earth and
were formed from rapid solidification of molten rock from volcanic eruptions, lightning
strikes and meteorite impact [111]. Metals naturally exist as crystalline materials, and
metallic glasses are a very recent addition to the class of amorphous materials. While
some amorphous thin films formed frommetal deposition onto very cold surfaces existed,
it was not until 1960 that the synthesis of a Au-Si amorphous metallic alloy formed from
rapid quenching of the melt from 1570 K to room temperature was reported [76]. This
work went onto stimulate a lot of research into the synthesis of amorphous metallic al-
loys. However, as in the earlier Au-Si alloy work, the need for cooling rates as large as
10
6
/K to produce amorphicity greatly limited the thickness of the specimens that could
be produced, allowing for the formation of only ribbons and thin films. In subsequent
years, ternary Pd-based glassy metallic alloy systems with slower nucleation kinetics and
hence slower cooling rate enabled the production of samples of thickness up to a couple
millimeters [10,111]. The ability to produce bulk samples was a critical development
since it opened the possibility of using BMGs in structural applications. It was not until
the late 1980s – early 1990s that La-, Mg-, Pd-, Fe-, Zr-, Ti-based metallic glasses of
high glass forming ability with several components were synthesized by Inoue and John-
son [64,135]. This allowed for even thicker samples of up to 9 mm in thickness. The
big breakthrough came when Peker and Johnson later developed a Zr-Ti-Cu-Ni-Be alloy
with a critical casting thickness of up to 10 cm, which came to be known as Vitreloy
1 [115]and went on to become the first commercially available BMG, and thus paved the
way for the possibility of BMGs being used as structural materials.
2
Since the potential of BMGs for use in structural applications became apparent, there
arose a need for mechanically characterizing the material across a vast range of loading
rates. As a result, a number of studies focusing on the quasi-static compression and
tension response of BMGs were undertaken. A summary of these studies is presented
in Section 1.1.1. While there have been fewer studies on their response under dynamic
loading conditions, those on the shock wave compression of BMGs have been fewer still
and have been mostly limited to Zr-based BMGs. Shock waves, commonly generated
by high-speed impact, result in abrupt changes in pressure and temperature behind the
shock front. They provide an invaluable tool for probing material response under ex-
treme conditions. This dissertation explores, for the first time, the shock response of
a novel iron-based metallic glass (known to be a promising candidate for a number of
applications [34] on account of its hardness and corrosion-resistance) which also contains
in situ reinforcement additives, to assess its applicability in ballistic uses such as kinetic
energy penetrators.
1.1 Background
Since the development of specimens with critical casting thickness as large as a 1 mm or
more, and the subsequent synthesis of the first commercially available Zr-based BMG,
numerous studies exploring their mechanical response have been undertaken. These in-
cludeexperiments involving quasi-staticuniaxialtensionandcompressionaswellassome
involving dynamichighstrain-rateshockloading. Mechanicaltestinghasbeenconducted
on BMGs of varying composition with the aim of characterizing their response for use
in high-strength structural applications. An overview of work done in the mechanical
testing of BMGs under quasi-static and dynamic loading conditions will be described
next.
3
1.1.1 Response to quasi-static & dynamic uniaxial stress
Bulk metallic glasses differ vastly from crystalline metals in that they do not strain
harden, and show very limited plasticity, as seen in Figure 1.1. Their plastic deforma-
tion is influenced by both normal and shear stresses, and deformation occurs inhomoge-
neouslyatroomtemperaturethroughtheaccumulationofplasticstraininlocalizedshear
bands [114]. A number of theories have been proposed to explain deformation mecha-
nisms in metallic glasses. In general, metallic glasses can deform either homogeneously
(at low stress, high temperature) i.e. each volume element in the sample accommodates
strain or inhomegeneously (atlow temperature, high stress) i.e. strain is localized in very
narrow zones called shear bands [124]. Argon developed a model based on two deforma-
tion modes - diffuse shear transformation and dislocation loop formation - to describe
theboundarybetweenhomogeneousandinhomogeneousdeformation[5,6]. Spaepenpro-
posed a theory based on free volume created by external stress and its annihilation by
diffusion [124,125], and this has now become the most widely cited theory to describe
deformationofmetallicglasses. Accordingtothistheory,macroscopicflow,bothhomoge-
nous and inhomogeneous, occurs from individual atomic jumps to nearest neighboring
empty spaces, biased in the direction of externally applied force. If a neighboring free
space is unable to accommodate an atomic jump, the diffusing atom creates free space in
its original position, thus contributing to the creation of new free volume. The addition
of free volume results in a drop in viscosity thereby locally weakening and softening the
material through a shear band causing the material to fail along the plane of that shear
band [124,125,142]. Another explanation for the drop in viscosity within a shear band
was believed to be due to temperature rise from localized adiabatic heating [77]. As
summarized by Lund and Shuh [97], the macroscopic yield and failure of metallic glasses
consists of many collective small-scale events: a) nucleation of local shear transforma-
tion zones (STZs) in which atoms reorganize to accommodate applied shear strain, b)
propagation of shear localization (shear banding) [124], c) adiabatic heating in deformed
regions [142], d) nucleation of nanocrystallites in or near shear bands, e) nucleation of
4
nano voids in shear bands [141], and f) coalescence of voids during failure [95,156].
Figure 1.1: Compression and tension stress-strain curves for a Zr-based alloy showing limited
plasticity in compression and catastrophic failure in tension [156].
Metallic glasses exhibit varied behavior in tension and compression: linear elastic be-
havior followed by catastrophic fracture under tension whereas there is some plastic flow
following elastic response under compression [24,156,157]. The mechanistic reasons for
thisdifferenceinbehaviorareasfollows. Incompression, duetoaconfinedconfiguration,
shear bands carry large amounts of plastic strain, lending ductility to metallic glasses.
In tension, however, deformation occurs through a single shear band and the glass fails
by shear rupture through this band with very little plastic strain [102]. With the ex-
ception of some studies [19,129], failure surface orientation of metallic glasses subjected
to tension and compression show asymmetry and deviate from the 45-degree pure shear
failuresurface[20,30,55,67,78,113,150,157,158]predictedbytheVonMisescriterion[99].
Therefore, it was concluded that failure was not entirely dominated by shear stresses,
but hada contribution fromthe normalstress as well, leading to the materialdescription
given by the Mohr-Coulomb type yield criterion [20,30,35,36,57,78,97,103,105,121]
rather than the Von Mises criterion [18,129]. However, studies of the effect of im-
posed hydrostatic pressure showed it to have negligible effects on the failure strength
of metallic glasses [78,94]. It was also shown that the pressure-dependent terms in the
Mohr-Coulomb and Drucker-Prager failure criterion for metallc glasses are lower than
crystalline metals and other brittle metals (sand, concrete, granite) pointing to limited
5
pressure sensitivity in compression for the amorphous alloys studied [20]. On the other
hand, tensile mean stress was found tohave amore pronounced effect onfreevolume and
stress localization as opposed to compressive mean stress, thereby confirming the negli-
gible effect of imposed compressive hydrostatic pressure from previous studies as well as
the applicability of a Mohr-Coulomb type criterion in describing the yield behavior of
metallic glasses [36]. This is in keeping with a speculation made in an earlier work about
the sensitivity of shear band and failure surface orientation only to tensile hydrostatic
pressure, since compressive failure surfaces were seen to lie along the maximum shear
stress plane [18].
ThefracturemorphologiesofBMGsundertensionandcompression showvariedfeatures.
Acompressive fracturesurface istypically smoothwithperiodic bandsinthe directionof
fracture, with uniform vein-like structures within the bands. In contrast, a tensile frac-
ture surface shows a mixture of veins and radial cores. These differences are attributed
to the role of normal stresses during tensile loading and the dominance of shear stresses
during compressive loading [67,157]. The nature and variety of features present on frac-
ture surfaces of failed metallic glass specimens provide a rich source of information on
dominant fracture mechanisms under varying configurations and regimes of loading. Nu-
merous studies of the fracture morphology of metallic glasses under quasi-static tension
and compression, dynamic compression [19,81,83,87,113,129,131,147,148,150]and high
strain-rate impact loading [32,67,109,136,153,160,161] have been performed. Several
of these focus on the sensitivity of fracture features to the magnitude of applied strain
rate. Inone work, itwas observed thatthe fracture surfaces ofBMG specimens damaged
fromloadingatincreasing strainratesexhibitedsimilarfracturefeatures[131],whileoth-
ers observed varying fracture morphologies indicating that different damage mechanisms
dominatethefractureprocessatdifferentstrainrates[70,83,147,148,150,160],suggesting
the dependence of fracture morphology on strain rate. In addition to fracture morphol-
ogy,thestrainratesensitivity ofthestrengthofmetallicglasses hasalsobeenthesubject
of several studies. An investigation of stress-strain behavior under quasi-static uniaxial
6
stress led tothe observationthatanincrease instrain-rateandadecrease intemperature
resulted in a transition from homogenous to inhomogeneous deformation [95,96]. This
result is in keeping with the deformation map proposed by Spaepen [124]. Most stud-
ies have pointed to the negative strain-rate sensitivity of metallic glasses i.e. decreasing
strengthwithincreasingstrain-rate[44,81,90,103,113,148,160]whichhasbeenattributed
tocracknucleationimmediatelyfollowingshearbandinitiationduetorapidloadingrates
and nucleation of shear bands at increasing stresses [103]. Others have found negligible
effect of strain rate on strength [19,87,129,131], while some others have reported pos-
itive strain-rate sensitivity of metallic glasses [72,88]. It has also been suggested that
the strain rate dependence of deformation behavior in BMG varies with material com-
position, specimen shape and loading procedure [22,81,86,87,145]. Therefore, a clear
consensus onthe ratesensitivity ofthestrength andfracturebehavior ofmetallicglasses,
especially under low and intermediate strain rates, has been elusive.
1.1.2 Response to shock wave loading
The existing literature on the shock response of bulk metallic glasses, to the best of our
knowledge, is almost entirely composed of studies on Zr-based BMGs of slightly varying
composition. The first reported shock compression study of metallic glasses was con-
ducted by Zhuang et al. on Vitreloy 1 and its in situ dendritic beta-phase reinforced
composite, referred to as Vit-1 andbeta-Vit, atstrain rateof2×10
6
persecond via plate
impact experiments [161]. A surprisingly low Hugoniot Elastic Limit (HEL) of less than
0.1 GPa was reported. The HEL represents the maximum normal stress that a material
can withstand under uniaxial compressive strain without internal rearrangement taking
place at the shock front [2]. Subsequent studies on the shock compression of metallic
glasses revealed a higher HEL of about 7 GPa and a distinct two-wave structure com-
posed of an elastic precursor followed by the plastic wave [65,104,107,138,139,146,155].
Some authors observed kinks in Hugoniots calculated from data obtained from plate-
impact studies which were interpreted as evidence of phase transition [104,107,136,161].
7
In the work of Togo et al., X-ray diffraction of recovered samples, however, showed an
intactamorphousstructure identicaltotheoneobtainedbeforeimpactatambient condi-
tions, indicating that shock compression had induced a transient dense amorphous state
which subsequently reverted back to its original state [136]. In the work of Martin et al.,
a Birch-Murnaghan fit of the data obtained at large shock pressures, revealed a bulk
modulus much higher than its ambient value, which was taken as confirmation of phase
transformation [104]. On the other hand, shock compression response explored over a
similarly wide stress range as the previous works by Xi et al. showed no phase trans-
formations at all [146]. Shock wave profiles of BMGs have been simulated using Von
Mises-based constitutive models [65,139]. The applicability of the Von Mises model to
describe BMG material behavior was further confirmed by combined compression and
shear plate impact experiments which showed that hydrostatic pressure and applied nor-
mal stress have negligible influence on the yield behavior of BMGs [154]. While all of
these studies were conducted on BMGsofsimilar composition, Jaglinski et al. conducted
impact experiments on a Zr-BMG with half the zirconium substituted by hafnium, to
investigate the effects of change in composition on BMG response. Other than small
differences in sound speeds and elastic moduli resulting from the elemental substitution,
interestingly, the characteristic of shock response, including the magnitude of the HEL,
remained the same [65].
Table 1.1: Summary of reported HEL for BMGs
Author HEL (GPa) Applied Stress (GPa) Composition
Zhuang et al. [161] 0.10 5−7 Zr
41.2
Ti
13.8
Cu
12.5
Ni
10
Be
22.5
Turneaure et al. [138] 7.10 < 13 Zr
56.7
Cu
15.3
Ni
12.5
Nb
5.0
Al
10.0
Y
0.5
Mashimo et al. [107] 6.20 < 50 Zr
55
Al
10
Ni
5
Cu
30
Yuan et al. [155] 6.15 4−7 Zr
41.25
Ti
13.75
Ni
10
Cu
12.5
Be
22.5
Turneaure et al. [139] 7.00 4−16 Zr
56.7
Cu
15.3
Ni
12.5
Nb
5.0
Al
10.0
Y
0.5
Togo et al. [136] 6.00 < 45 Zr
55
Al
10
Ni
5
Cu
30
Martin et al. [104] 6.86 5−123 Zr
57
Nb
5
Cu
15.4
Ni
12.6
Al
10
Xi et al. [146] 6.90−9.60 18−110 Zr
51
Ti
5
Ni
10
Cu
25
Al
9
Arman et al. [7] 7.20 < 60 Cu
56
Zr
54
Jaglinski et al. [65] 7.40 4−16 (Hf
0.5
Zr
0.5
)
56.7
Cu
15.3
Ni
12.5
Nb
5.0
Al
10.0
Y
0.5
8
In addition to compression, several shock-induced dynamic tension or spall experiments
havealsobeenconductedonBMGs. Turneaureet al. reportedaspallstrengthof3.8GPa
for a Zr-BMG for impact stresses below the HEL. The spall strength was found to be
independent of impact stress, and much larger than that of crystalline metals which typ-
ically have spall strengths of 1 GPa or less [137]. Yuan et al. reported a similar value
for spall strength but found that it decreased with increasing levels of shock stress below
the HEL and remained steady with changing stress above the HEL. They attributed
this phenomenon to the accumulation and dominance of brittle deformation below the
HEL, and the onset of ductile inelastic processes above the HEL [153,155]. On the other
hand, the simulations of tensile fracture by Turneaure et al. showed that the response
becomes increasingly brittle with increased stress [137]. Astroshenko et al. reported a
spall strength of 2.9 GPa for a Ti-based BMG below its HEL [8]. Examination of recov-
ered samples shows cracks formed from the nucleation of microvoids [8,161]. Escobedo
et al. conducted microscopic examination of spall-recovered samples and observed three
main attributes of the damaged microstructure: at the macroscale, damage becomes in-
creasinglylocalizedwithincreasingimpactstress; atthemicroscale, fracturesurfacewent
from being rough to smooth with increasing compressive stress – both tensile and shear
stresses are present at the lower stresses, while at the higher stresses, a depletion of free
volume causes a dominance of mean tensile stress, which also causes a localized increase
in temperature resulting in melting and droplets; at the nanoscale, there are bumps and
corrugations which are explained by the Shear Transformation Zone and Tensile Trans-
formationZone theories as resulting from a sequence of arrest and propagationof mode I
cracks. The authors concluded this work with the result that the BMG shows attributes
of both a metal and a glass, in that its fracture tends to seem brittle at the macroscopic
level, and ductile at the microscopic level [32]. Huang et al. conducted spall experiments
at a stress below the HEL while varying the duration of the stress pulse and observed
evidence of ductile fracture. A micro void nucleation model they proposed to describe
theirobservations ofBMG fractureindicated thediffusion andcoalescence offree volume
results in microvoid nucleation and that mean tensile stresses play a dominant role in
9
this process [54], as also shown by Escobedo et al. in their work.
TheexistingliteratureontheshockresponseofBMGs,mostlylimitedtoZr-basedBMGs,
reveals that they have HELs of approximately 7 – 9 GPa. This is an order of magnitude
higher than that of crystalline materials, for e.g. the HEL of Aluminum 7075 is about
0.7 GPa. In all the studies, the response is typically characterized by elastic-plastic
deformation and a tendency to lose post-yield strength. While the study of the shock
response of an in situ MGMC designed to enhance ductility and the monolithic BMG
from which it was made indicated differences in damage mechanisms when spalled, no
observable difference was seen in the shock compression response or strength. This dis-
sertation aims to further explore the shock response of BMGs in addition to determining
the effect of in situ matrix additives, if any.
1.1.3 Iron-based bulk metallic glasses
Fe-based bulk metallic glasses, or amorphous steels, are known to have hardness as
high as 12 GPa [23,50,62,63,92,116,122], high fracture strengths of 4 GPa or more
[23,45,62,63,92,143,152], good corrosion resistance and magnetic properties [132], with
the added advantage over other metallic glasses of lower material cost. Figure 1.2 con-
tains Ashby maps showing the superior strength and hardness of amorphous steel in
comparison with other metallic glass compositions. Several compositions of Fe-based
BMGs have been explored in order to optimize glass forming ability as well as ductility
and strength. Even though Fe-BMGs have several attractive attributes, most of them
have fracture toughness as low as 5 MPa-m
1/2
and are consequently extremely brittle
and undergo catastrophic failure [23]. The low toughness of Fe-BMGs is associated with
high shear modulus and high glass transition temperature which result in high activation
barriers for shear flow thereby limiting plastic deformation [28,48]. However, lowering
the glass transition temperature has resulted in reduced glass forming ability, and there-
fore composition variations which result in reduced shear modulus and hence greater
10
ductility have been explored [28,47,82]. It has also been shown that brittle amorphous
steels are associated with a low Poisson’s ratio and a high shear modulus to bulk mod-
ulus ratio [45,79]. Developing different compositions by employing varying chemistry
and alloying strategy that systematically changes elastic constants has lead to improved
ductility and toughness [80]. Experimenting with different iron-based compositions has
also shed light on the effect of metal-metal and metal-metalloid bonding on the overall
ductility of the amorphous specimen, and has indicated that while the addition or sub-
stitution ofmetalloids enhances the glass forming ability, a higher metallicity ofthe alloy
system favors plasticity [48]. Further evidence for this is given by a new composition
which possesses unprecedented room-temperature plasticity of over 20 percent, resulting
from the unique clusters of metal-metal bonds which accommodate shear strain that the
alloy contains [151].
Figure 1.2: Ashby maps showing superior tensile strength and hardness of amorphous steels as
compared with other amorphous and crystalline metallic alloys [59]
In addition, iron-based metallic glass matrix composites containing precipitated crys-
talline phases through varying degrees of annealing or micro-addition of nano crys-
tallization inducing elements, have also been studied and proven to enhance ductil-
11
ity [46,85,123], with one such study, for example, resulting in significantly improved
plasticity of over 30 percent plastic strain due to the in situ α-Fe dendritic phase as
opposed to only 3 percent plastic strain in monolithic samples [46]. The presence of
either precipitated α-Fe phase or iron-metalloid compounds or a combination of both
in the amorphous matrix is seen to have resulted in increased plasticity or even higher
hardness [112].
1.1.4 In Situ Metallic Glass Matrix Composites
In order to exploit the favorable properties of metallic glasses while overcoming their
inherent limitation of lack of plasticity, metallic glass composites have been fabricated
to generate materials with enhanced properties that combine those of both the matrix
as well as the reinforcements. The fabrication and mechanical properties of two types of
metallic glass matrix composites (MGMCs) have been explored thus far – ones in which
reinforcement phases are formed in situ through controlled devitrification and others
where reinforcements are added externally or ex situ in the form of particle, rods, and
wires. Since this thesis deals with the mechanical response of two in situ composites, a
brief background only on these type of MGMCs is presented here.
The following factors have been identified as being conducive to nanocrystallization
within an amorphous metallic matrix: a) the occurrence of crystallization through mul-
tiple steps, b) the availability of nucleation sites within the amorphous phase, c) a de-
celeration in the growth of the nanocrystallites, and d) high thermal stability of the
amorphous matrix phase [60]. Figure 1.3 summarizes the effect of microstructure on
the mechanical properties of various in situ MGMCs and shows that in general while
the composites show better strength and plastic strain to failure in comparison with
the pristine monolithic BMG, the one containing 20 - 30 percent volume fraction of
5 nanometer sized nanocrystallites achieved the highest strength [61]. The composites
studied in the work of this thesis also contain nanocrystallites of a comparable size, how-
ever their volume fraction within the matrix is significantly smaller. Fracture strength,
12
Vickers hardness and Young’s modulus have been shown to have a linear increase with
volume fraction of nanocrystallization products [60,61]. The efficacy of precipitated
crystalline phases in improving fracture strength, ductility (plastic strain to failure), and
elastic moduli of composites with varying compositions has been demonstrated in sev-
eral works [14,21,51,60,61,117,144]. A similar work involving a Ti-based MGMC with
dendritic phases with high tensile ductility also resulted in large fracture toughness K
IC
of 170 MPa-m
1/2
and fracture energy G
IC
of 340 kJ/m
2
, surpassing those of even tough
fullycrystalline titaniumandsteel alloys, thusachieving theranksofthetoughestknown
materials [53].
The interplay between the precipitated particulates and deformation due to shear local-
ization results in the confinement and arrest of shear bands as well as the promotion of
the growth of networks of multiple shear bands, both resulting in higher strength and
enhanced ductility [51]. The hardness and strength of nanocrystalline phases themselves
as well as interfacial effects due to the high interface-to-volume ratio of nanoparticles
are also both attributable factors for improved mechanical properties [33]. In a work
involving a Zr-based MGMC, Inoue identified the following additional reasons for their
superior mechanical properties: a) large free volume in the amorphous phase formed
during annealing in the supercooled liquid regime, which aids in accommodation of large
strains, b) highly dense packed atomic configuration at the nanocrystallite-amorphous
phase interface due to lower interface energy of liquid/solid interface as compared to
solid/solid interface, and c) small size of nanocrystallites and their homogeneous and
isolated dispersion within the glassy matrix [58].
Whilethepresentliteratureindicatesthattheadditionofin situ reinforcementstoBMGs
enhances their mechanical properties, the type and quantity of additives required to
achieve this effect seems to be unique for each BMG composition. Also, thus far, the
mechanical response of MGMCs has been mostly studied under quasi-static conditions.
The only published work on the shock response of an MCMC shows that the in situ den-
13
Figure 1.3: Microstructure and corresponding mechanical strengths of Zr-based metallic glasses
containing varying kinds of in situ nanocrystalline reinforcements [61]. Here σ
f
is the fracture
strength, ǫ
f
is the fracture strain, E is the Young’s Modulus, V
f
is the volume fraction of
nanoparticles, and d is the size of the nanoparticles.
dritic phase has no effect on shock compression behavior as compared with the pristine,
monolithic BMG [161]. Therefore, it is not entirely clear yet whether matrix additions
are effective in altering mechanical response at high strain rates.
14
1.2 Motivation
As described in the preceding sections, bulk metallic glasses are a novel class of high-
strength, high-toughness materials which present the possibility for use in wide-ranging
engineering applications. While several different compositions and their ease of forming
an amorphous alloy have been explored, the focus of studies on their mechanical prop-
erties has largely been restricted to that of Zr-based compositions. Amorphous steels
present an untapped resource since their quasi-static strengths are almost twice as large
as those of other metallic glass compositions, in addition to there being other advantages
to their use, such as their superior wear, corrosive, and magnetic properties, and more
importantlytheirlowmaterialcost. InordertoutilizeBMGsinengineering applications,
such as structural uses, it is necessary to characterize their mechanical response over a
wide range of strain-rates. While studies on the quasi-static and dynamic uniaxial stress
response have been fairly comprehensive, those on high strain-rate shock response have
been limited. Figure 1.4 shows a schematic of experimental techniques and processes
across a range of strain rates, with the regime of interest for this work highlighted in red.
This dissertation attempts at filling these two deficiencies in the current state of under-
standing, that of the mechanical response of an amorphous steel, and furthering that
of the shock behavior of amorphous metals, through an experimental examination of
the shock compression behavior of an iron-based metallic glass. In addition, this work
deals with another important aspect that remains to be fully understood. Since metallic
glasses are inherently brittle, attempts at enhancing their ductility by fabricating com-
posites have been made. Several of these have been shown to be successful, especially
thoseonnanocrystallinecomposites, however apartfromonestudy[161],therehavebeen
no attempts as yet to understand the effects of composite reinforcements on the overall
mechanical response at very high strain rates.
Further, it has been theorized, based on high strain rate experimental work on one slight
15
variation of the Zr-based composition, that all strong, brittle metallic glasses can be de-
scribedbythesamegeneralmechanical response[65]. This workprovidesanopportunity
to further test and explore this observation under comparable strain rates. The study
of material response necessitates the establishment of equations of state and constitu-
tive strength relations which describe the variation of hydrostatic pressure and strength
respectively with changes in loading states such as strain, strain rate, temperature etc.
These can then set the framework for building material models for use in numerical
simulations of mechanical response, which in turn can help inform choice of experimen-
tal parameters, thereby furthering the understanding of the material across previously
unexplored regimes. Moreover, in our quest to design and synthesize materials with tun-
able properties, it is important to understand the link between structure and mechanical
response. Given all of these larger goals, the objectives of this work are as follows:
1. To study the shock wave response of a novel high-hardness iron-based metallic
glass by subjecting it to shock compression experiments, and to construct its high
pressure equation of state.
2. To study the effect of partial devitrification in the amorphous alloy on the high
strain rate dynamic response by studying and comparing the response of an amor-
phous sample and partially crystalline sample of the same iron-based composition,
referred to as XS-1 and SAM2x5 respectively. Even though the extent of devitrifi-
cation in both alloy systems is on the order of only 1 – 2 percent, the term in situ
MGMC has been used to describe both, in order to reflect the existence of more
than one phase in the specimens.
3. To simulate shock wave profiles using experimental equations of state through con-
tinuum mechanical modeling in commercial hydrocode ANSYS Autodyn.
4. To study the effect ofstrainrateonthe dynamic fracturemorphology ofa Zr-based
metallic glass after it has been subjected to low velocity plate impact experiments.
16
10
8
10
6
10
4
10
2
10
0
10
-2
10
-4
Strain rate (s
-1
)
Figure1.4: Experimental techniques and real world processes across a range of strain rates, with
regime of interest highlighted in red.
1.3 Outline
This dissertation has been organized as follows: Chapter 1 contains a survey of the rele-
vant literature and background on the topic of research undertaken here, and presents a
motivationforthiswork. Chapter2providesthetheoreticalframeworkforthemethodsof
analysis and concepts used inthis work. Chapter 3 contains details ofall relevant experi-
mental methods and facilities used including techniques for sample preparation, material
characterization, velocity interferometry and high-velocity loading systems. Chapter 4
presents asummary ofresults andanalysis oftheexperimental work, including details on
calculationsandmeasuredquantities. Chapter5presentsresultsofsimulationsperformed
using the equation of state constructed from experiments, and provides a comparison of
simulated and experimental results. Chapter 6 presents work done on the post-mortem
microscopic examination of failed Zr-based samples after they had been subjected to low
velocity plate impact experiments, in order to study the sensitivity of its fracture mor-
phology to strain rate. Chapter 7 summarizes the conclusions and contributions of this
work, and provides recommendations for future directions for this research.
17
Chapter 2
Theoretical Background
Since this work involves experiments in shock compression, a brief background on shock
waves and their propagation through solids is presented here. Shock waves are formed
whenadisturbancepropagatesthroughamediumfasterthanthespeedwithwhichparti-
cles can move away. They are treated as mathematic discontinuities which cause sudden
changes in pressure, temperature, internal energy and density. However, in reality shock
waves have a finite rise time, i.e. the time during which the material is taken from its
initial state to the high-pressure shock-compressed state, due to material dissipative pro-
cesses such as thermal conduction and viscosity which cause smearing of the shock front.
Shock waves also have a finite thickness which is determined by the competition between
nonlinear shock-strengthening effects and dissipation, however very narrow shocks can
be treated as discontinuities [49]. When a material is subjected to a large amplitude
disturbance, it gets stiffer from undergoing compression. The speed of sound in the com-
pressed material increases as it gets stiffer, and as the disturbance front moves faster it
steepens to form a shock wave. The propagation of shocks in solids gives rise to a state
of uniaxial strain resulting from inertial confinement thus creating the opportunity to
interrogate material response at high strain rates and develop high-pressure equations of
state and constitutive relationships. There are several ways of introducing a controlled
plane one-dimensional shock wave into a material of interest, the most common of which
isthehigh-velocity plateimpactexperiment [9],which isdescribedindetailinChapter4.
18
2.1 Shock Wave Theory
2.1.1 Shock Jump Conditions
The feature that mainly distinguishes solids fromfluids is that solid materials have rigid-
ity and aretherefore able to support shear stresses. The mechanical response ofa solid is
dividedintotwoparts-thebulkorhydrostaticresponse, whichisdescribedbyamaterial
equation ofstate which defines a relation between its thermodynamic properties, and the
deviatoric response which is described by a constitutive material strength relationship.
When the stress produced by a shock wave in a material greatly exceeds its strength
i.e. the maximum shear stress that the material can sustain, the difference between the
normal and lateral stress (the shear stress) is very small as compared to the average
of the normal and lateral stresses (the pressure). The fact that the shear stresses can
be ignored reduces the response of the solid to that of a fluid. This consideration of a
solid that flows like a fluid at a very large build-up of hydrostatic pressure is called the
hydrodynamic approximation [25,26,110].
The calculation of shock wave parameters is performed by applying conservation laws
between the material behind and ahead of the shock. As a shock wave travels through
the material at shock velocity U
s
, the material ahead of and behind it may be described
by quantities such as pressure P, internal energy E, and density ρ (often expressed as its
inverse whichisthespecific volume v). Inaddition, thematerialbehindtheshocktravels
withparticlevelocityu
p
. Ifthematerialaheadoftheshockisatrest,theparticlevelocity
in the undisturbed state is zero, and the pressure, density and internal energy are those
correspondingtotheambientstate. ThefollowingEulerianjumpconditions[9,26,37,120]
prove useful in computing a certain shock compression state (denoted by subscript 2),
given that the conditions of the pre-state (denoted by subscript 1) are already known,
19
P
2
,E
2
,ρ
2
, u
p2
P
1
,E
1
,ρ
1
, u
p1
U
1
U
2
P
0
,E
0
,ρ
0
Figure 2.1: Schematic of conditions ahead of and behind two shock waves
and are commonly used for calculations based on data from plate impact experiments.
The equations (2.1), (2.2)and(2.3)describe the relationship between the material ahead
of and behind the shock traveling with speed U
2
, as seen in Figure 2.1. The underlying
assumptions oftheshock jump conditions are: 1)theexistence ofa steady, discontinuous
shock front so as to be able to assign a distinctive wave velocity U
s
, and 2) the states
ahead of and behind the shock wave are in thermodynamic equilibrium, so as to be able
to identify the thermodynamic pressure P.
Mass Conservation
ρ
1
(U
2
−u
p1
) = ρ
2
(U
2
−u
p2
) (2.1)
Momentum Conservation
P
2
−P
1
=ρ
1
(U
2
−u
p1
)(u
p2
−u
p1
) (2.2)
Energy Conservation
E
2
−E
1
=
1
2
(P
2
+P
1
)(v
2
−v
1
) (2.3)
20
2.1.2 Equation of State
The conservation equations above (2.1), (2.2), (2.3) contain five parameters - pressure
P, shock velocity U
s
, particle velocity u
p
, internal energy E, and specific volume v, but
there are only three equations. Therefore, an additional equation is required if all the
parameters are to be expressed as a function of one parameter. This equation can be
expressed in terms of the kinematic parameters measured in a plate impact experiment
U
s
and u
p
and can be empirically described by a polynomial equation with parameters
C
0
, S
1
, S
2
, S
3
and so on as:
U
s
=C
0
+S
1
u
p
+S
2
u
p
2
+... (2.4)
The equation (2.4) is known as the equation of state (EOS) of a material. Here, S
1
and
S
2
are empirical parameters and C
0
is the ambient pressure bulk sound velocity, which is
given by
p
K
S
/ρ
0
where K
S
isthe isentropic bulk modulus. FormostmetalsS
2
= 0, and
therefore equation (2.4) is reduced to a linear relationship which is valid for describing
the shock response of metallic materials not undergoing a phase transformation:
U
s
=C
0
+S
1
u
p
(2.5)
2.1.3 Hugoniot
If a material is subjected to shocks of varying strength, an EOS describing the U
s
-u
p
relationship can be obtained, and in conjunction with the jump conditions, one can
determine a pair of data points (such as P-u
p
, P-v)corresponding to each incident shock
strength. These data points can be used to construct a curve describing the locus of end
states that are reached by a shock transition from a given initial state. This curve is
known as the Hugoniot curve or simply the Hugoniot. The Hugoniot is also referred to
as the shock adiabat since the transition to the shocked state from an initial state is an
adiabatic process, whereas release from the shocked state occurs along the isentrope. A
Hugoniot is centered around a specific initial condition and each point on the curve is
21
a final state reached through shock transition from that given initial condition. Unlike
an isotherm or an isentrope, since the Hugoniot curve is not a thermodynamic path but
rather a locus ofend states, each point onthe curve corresponds to an experiment with a
given incident shock strength for a material at a given initial condition. Hugoniots for a
widevarietyofmaterialscontainedintheLosAlamoscompendiumwereconstructedfrom
peak state data meticulously determined through hundreds of plate impact experiments
[101]. As seen in Figure 2.2, the line connecting the initial state to a final state on the
Hugoniot is called the Rayleigh line. Its slope for a P-u
p
Hugoniot is a direct result from
the jump condition for the conservation of momentum, and is given as follows [110]:
P −P
0
U
p
=ρ
0
U
s
(2.6)
For Hugoniots that contain kinks or discontinuities in slope from phase transformations
or elastic-plastic transitions, the initial and final states cannot be connected by a single
Rayleigh line and therefore, the final state is reached by a series of two or more Rayleigh
lines each with a slope in the P-u
p
plane proportional to the shock velocity of the corre-
sponding shock wave [9]. The area under the Rayleigh line for a P-v Hugoniot gives the
specific energy of the material behind the shock front. Entropy always increases along
the Hugoniot and it can be shown that this leads to the Hugoniot curve being concave
upwards [49,120]. For materials with concave upward Hugoniots, the shock velocity
increases with increasing shock pressure i.e the material can sustain the formation of a
shock wave, andthisis animportantcondition forthestability ofashock wave. Material
Hugoniots are also used in calculating conditions in the shocked state, as outlined in the
next section.
2.1.4 Impedance Matching
The product of a material’s density ρ and shock speed U
s
is defined as the acoustic
impedance of the material. When a projectile moving at a certain velocity, impacts
a stationary target material, there is a backward-facing wave that goes through the
22
P
v
v
0
(P
1
,v
1
)
Rayleigh Line
Hugoniot
Figure 2.2: Rayleigh line connecting final state on the Hugoniot (denoted by subscript 1) from
a given initial state (denoted by subscript 0)
A B
P
u
p
0
0'
1
u
A
u
B
u
A
(P
1
,u
p1
)
(a)
A
B
1 1
0'
0
t
x
(b)
Figure2.3: (a) Graphical representation of impedance matching using P−u
p
Hugoniots showing
State 1 at the intersection of the forward-facing Hugoniot for B and the reflected Hugoniot for
A, (b) t−x plot for wave interactions in impactor A hitting target B. States 0’ and 0 represent
ambient conditions in A and B respectively, and State 1 the shocked state in both.
23
projectile,andaforward-facingwavethatgoesthroughthetarget. Usingthetractionand
velocitycontinuityboundarycondition,andtheshockjumpequations,onecandetermine
all thermodynamic parameters resulting from the impact, as shown in Figure 2.3. This
approach is known as impedance matching. To graphically solve for the final state, the
point of intersection of the two material Hugoniots can be determined. This can be
expressed by the following. Here the subscripts I and T refer to the impactor/projectile
and target states respectively:
P
I
=ρ
0I
U
sI
(u
p
−u
pI
) =ρ
0T
U
sT
u
p
(2.7)
While the impedance matching technique is a very useful approach to determine shocked
material states, it is an approximation when multiple waves are involved since only the
principal Hugoniots (which represent a shock with a specific initial state ahead of it)
are used. For a rarefaction wave also the principal Hugoniot is still used instead of the
release isentrope. For most condensed matter materials, this error is smaller than can be
measured and is therefore ignored [37].
In plate impact experiments, velocity measurements areusually made ata free surface or
at a material interface. It is then necessary to deduce the in-material particle velocities
usingthisdatatodeterminetheresponseofthetargetmaterial. Thegraphicalimpedance
matching approach is used to do this and is depicted in Figures 2.4 and 2.5 for the free
surface and material interface measurement cases respectively. When a forward-facing
shockwaveimpingesonafreesurface,abackward-facingrarefactionwaveisreflectedinto
the material, relieving the pressure and accelerating the material in the direction of the
shock wave. The intersection of the incident and reflected Hugoniots in the P-u
p
space
leads to the in-material particle velocity being half of that of the free surface velocity.
This is known as the free surface approximation and is equivalent to the small-strain
approximation thatallows fortherelease isentrope to be replaced by a reflected principal
Hugoniot [9,120]. For the case of a forward-facing wave being transmitted across an
interface between two materials, the in-material conditions in the material on the left are
24
found at the intersection of the reflected principal Hugoniot (of the material on the left)
and the principal Hugoniot (of the material on the right). In Figures 2.3, 2.4 and 2.5,
the vertical solid line in the t−x plot represents the material interface, while the dashed
line represents a moving boundary and the slope of the lines representing the traveling
shock waves is the inverse of the speed of the shock. Through impedance matching it
can be shown that if the material on the left has a lower impedance than the material
on the right, a shock wave is reflected back into the material on the left. If the converse
if true, a rarefaction wave is reflected into the material on the left.
A
P
u
p
0
1
(P
1
,u
p1
)
A
0'
A
(0,2u
p1
)
(a)
A
Vacuum
1
0'
0
t
0'
x
(b)
Figure 2.4: Representation of (a) P −u
p
Hugoniot and (b) t−x plot for wave interactions in
target A (backed by vacuum) with its free surface. The impactor is not shown.
2.2 Continuum Mechanics Theory
A comparison of the expressions for Hooke’s law in the elastic regime for stress load-
ing states of uniaxial stress, hydrostatic pressure and uniaxial strain configurations are
presented in the section below. These will then be used to derive the relations forelastic-
plastic material response under one-dimensional shock loading using isotropic Hooke’s
law in the section that follows.
25
B
B
P
u
p
0
1
(P
1
,u
p1
)
B
2
(P
2
,u
p2
)
(a)
B
1
2
0 0
t
2
x
(b)
Figure 2.5: Representation of (a) P − u
p
Hugoniot and (b) t− x plot for wave interactions
of target B backed by plate C with the boundary between the two. The impactor is not shown.
Plate C has a higher acoustic impedance than B
2.2.1 Elastic Response
For an elastic isotropic solid, Hooke’s law is given by [99]:
σ
ij
=2µǫ
ij
+λǫ
kk
δ
ij
(2.8)
Here,µ (alsotheshearmodulusG)andλaretheLameconstantsandδ
ij
istheKronecker
delta function. The linear elastic Hooke’s law foreach ofthethree types ofloading states
mentioned above will now be explored. For a state of uniaxial stress (as produced by a
uniaxial tension or compression test), the Cauchy stress tensor is given by:
σ =





σ
11
0 0
0 0 0
0 0 0





(2.9)
Using equation (2.8), the principal stresses σ
11
, as well as σ
22
and σ
33
which are both
zero, can be expressed as follows:
26
σ
11
=2µǫ
11
+λǫ
kk
(2.10)
σ
22
=2µǫ
22
+λǫ
kk
= 0 (2.11)
σ
33
=2µǫ
33
+λǫ
kk
= 0 (2.12)
Since,thePoisson’sratioν isdefinedasthatbetweenstraininthelateralandlongitudinal
directions, i.e.−ǫ
22
and ǫ
11
respectively, it follows from the above relations between the
Lame constants that:
ν =
λ
2(µ +λ)
(2.13)
Also, using the above equations (2.10), (2.11), (2.12), σ
11
for the elastic regime may
also be expressed as:
σ
11
=

µ (2µ +3λ)
µ +λ

ǫ
11
(2.14)
In equation (2.13), the constant of proportionality between σ
11
and ǫ
11
is the Young’s
Modulus E. For a state in which hydrostatic pressure is exerted on a solid, the Cauchy
stress is given by:
σ =





P 0 0
0 P 0
0 0 P





(2.15)
Each of the principal strains and stresses are equal to each other. Therefore:
ǫ
kk
=3ǫ
11
(2.16)
It follows from equation (2.8) that each of the principal stresses are given by:
27
σ
11
= (3λ+2µ )ǫ
11
(2.17)
Here, the constant of proportionality between σ
11
and ǫ
11
is the isentropic bulk modulus
K
s
. Forastateofuniaxialstrain(asproducedbyaplateimpactexperiment),theCauchy
stress is given by the following. Here, the two lateral stresses σ
22
and σ
33
are equal to
each other by symmetry of an isotropic material are denoted by σ
lateral
:
σ =





σ
11
0 0
0 σ
lateral
0
0 0 σ
lateral





(2.18)
In the case of uniaxial strain, the only non-zero principal strain is in the direction of
loading i.e. ǫ
11
. Therefore ǫ
kk
is equal to ǫ
11
. Hooke’s law can then be expressed as
follows:
σ
11
=(2µ +λ)ǫ
11
(2.19)
Using the definition of the isentropic bulk modulus from equation (2.17), an alternate
expression for Hooke’s law is:
σ
11
=

K
s
+
4
3
µ
ǫ
11
(2.20)
σ
lateral
=λǫ
11
(2.21)
The mean stress is given by:
P =
σ
11
+2σ
lateral
3
(2.22)
Therefore, the Hooke’s law for the elastic uniaxial strain loading configuration in terms
of the mean stress and shear stress is:
28
σ
11
=−P +
2
3
(σ
11
−σ
lateral
) (2.23)
2.2.2 Elastic-Plastic Response to 1-D Shock Compression
The relations for elastic-plastic material response under shock compression are derived
here using Hooke’s Law for isotropic solids [37,40]. Consider a loading and release path
containing both elastic and plastic components. Here, for simplicity of notations, the
principal stress in the direction of shock loading σ
11
is denoted as σ
x
and the lateral
stress σ
lateral
as σ
y
. The shear stress τ, which acts in plane oriented 45 degrees to the
direction of propagation of the shock front, is expressed in terms of the difference in
longitudinal and lateral stress acting on the material:
τ =
(σ
x
−σ
y
)
2
(2.24)
The mean stress, using equations (2.22), (2.24) can then be expressed in terms of the
shear stress as:
P =σ
x
−
4
3
τ (2.25)
The elastic response along the loading path is given by equation (2.19). An alternate
expression for the elastic loading using equations (2.13), (2.22) is:
σ
x
= 3P

1−ν
1+ν

(2.26)
It can be shown that elastic release is given by the same expression with a change in
sign. When the material reaches the end of the elastic portion of its loading path, it
becomes plastic and a yield condition is required. The yield point on a loading path of
uniaxial strain is known as the Hugoniot Elastic Limit (HEL). Here, the Von Mises yield
criterion, the simplest yield model and one which describes the yield behavior of a lot of
materials, is applied. A general form of the yield criterion is given by:
29
(σ
x
−σ
y
)
2
+(σ
y
−σ
z
)
2
+(σ
z
−σ
x
)
2
=2Y (2.27)
Here, Y is the yield strength of the material under uniaxial stress. For an isotropic
material under uniaxial strain, the Von Mises yield criterion reduces to:
|σ
x
−σ
y
|≤Y (2.28)
σ
x
x
2Y/3
HEL
Y
1-D Stress
Hydrostat
1-D Strain
Figure 2.6: A comparison of the Hugoniot, Hydrostat and the stress-strain curve.
Using equations (2.24),(2.28), the yield strength under uniaxial stress maybe expressed
in terms of shear stress as:
Y = σ
x
−σ
y
=2τ (2.29)
At the yield point Y, using equations (2.22), (2.26), (2.29), the HEL stress can be ex-
pressed as [40]:
σ
HEL
x
= Y

1−ν
1−2ν

(2.30)
This equation relates the yield strength under uniaxial stress to that under uniaxial
strain for a material exhibiting elastic-plastic response and is useful for estimating the
30
HEL stress using data from uniaxial stress experiments. Using equations (2.25), (2.29),
the Hooke’s law along the path of plastic compression is given by:
σ
x
=P +
2
3
Y (2.31)
Similarly it can be shown for the path of plastic release that:
σ
x
= P −
2
3
Y (2.32)
The mean pressure is therefore obtained by subtracting (2/3)Y from the Hugoniot stress
σ
x
for a plastic shock wave for various values of strain, as is shown in Figure 2.6.
31
Chapter 3
Experimental Methodology
The high strain rate shock response of amorphous steels was explored via plate impact
experiments. This involves launching impactors mounted on projectiles at velocities as
high as 2000 m/s from a gun onto the sample of interest, thus generating a normal shock
wavepropagatingintothesample. Theprimarydiagnosticinplateimpactexperimentsis
velocity interferometry, which enables the measurement of particle velocity. In addition,
velocity of the shock is also measured by monitoring shock arrival times at the front and
rear of the sample. Experiments for this work were conducted at two facilities: Institute
for Shock Physics (ISP), Washington State University Pullman and Graduate Aerospace
Laboratories (GALCIT) of the California Institute of Technology. The velocity interfer-
ometrytechniquesusedatPullmanandCaltechweretheVISAR(VelocityInterferometer
System forAny Reflector) and PDV(Photonic Doppler Velocimetry) respectively. While
both these techniques result in the same measurement, the configuration of the target
set-up varied at each facility. In the Pullman facility, impactors were launched onto a
target assembly comprising a buffer plate backed by the BMG sample, which was in turn
backed by an optical window resulting in a sample-window interface particle velocity
measurement at the sample rear surface. At the Caltech facility, impactors were directly
launched onto the BMG sample resulting in a free surface velocity measurement at the
sample back surface. A schematic of both target set-ups is shown in Figure 3.1.
32
Flyer
Sample
1 mm
1.5 mm
(a)
Flyer Buffer
Sample Window
2 mm 2 mm
1.5 mm 5 mm
(b)
Figure 3.1: Schematic of target configuration at a) Caltech and b) Pullman. Further details on
materials and dimensions of target components in each set-up are listed in Tables 4.3 and 4.4.
3.1 Material Characterization
All BMG samples as well as impactor, buffer and window materials were characterized
using density, Vickers hardness and longitudinal and shear sound speeds. Densities were
determined using the Archimedean method (Ohaus Solids Density Determination Kit)
whereas sound speeds were measured using ultrasonic transducers and the pulse-echo
technique (Olympus 38Dl Plus Ultrasonic Thickness Gauge). Vickers hardness measure-
ments were made using a 300 gram-force held for 10 seconds (Leco LM100). An image of
anindentonaSAM2x5sample, withcracksemanatingfromtheindentindicating typical
ofabrittlematerial, canbeseeninFigure3.2. Incaseswhere single-crystals wereused in
the experiment, crystalline orientations were confirmed using Laue X-ray diffraction im-
ages. X-ray diffraction (XRD) of as-received BMG samples confirmed their amorphous
nature. XRD patterns of a few SAM2x5 samples are shown in Figure 3.3. A narrow
primary peak as well as presence of secondary peaks indicates partial devitrification [73].
33
Figure 3.2: Vickers hardness measurement indent
3.2 Target and Projectile preparation
At the Pullman facility, all target components i.e. sample, impactor and buffer, were
lapped flat and parallel to within 2 μm and were given a final polish using a 3 μm
diamond-gritoilsuspension tocreatea specularreflective surface forinterferometry mea-
surements. These were then epoxied together using a 5:1 ratio of warmed Epon Resin
815C and Hysol Loctite HD3475 hardener and cured for 24 hours while ensuring that
bondthicknesses were nomorethan2μm. Forallexperiments, aVelocityInterferometer
System for Any Reflector (VISAR) was used to measure the particle velocity history of
the rear surface using a 200 μm laser spot size. In cases where the polished rear surface
of the sample was not interrogated for velocity measurements and an optical window
epoxied to the sample was used instead, a thin layer of aluminum was vapor-deposited
onto the sample-window interface in order to obtain a reflective surface for laser interfer-
ometry. Prepared samples were then attached to pre-machined aluminum target plates
which could be bolted onto fixtures on the muzzle-end of the gun for the impact ex-
periment. Pictures of epoxied target components and a shot-ready target affixed to the
gun are shown in Figure 3.4. The design of the projectile, at ISP, Pullman, varied with
34
the range of velocities of the experiments - 200 mm long aluminum 6061-T6 projectiles
were used for the low velocity experiments, and 86 mm long Lexan projectiles were used
for the higher velocities. Impactors were either epoxied directly onto the projectiles or
using an impactor mount. Tilt of the projectile face was measured after attachment
of impactor with projectile, and only those projectiles which had impact tilt with their
axisofroughlyatenthofthetypicallyexpectedimpacttiltofthegunorbetterwereused.
30 35 40 45 50 55 60 65 70 75 80
2Theta (°)
0
25
100
225
400
625
900
Intensity (counts)
SAM-F
SAM-C
SAM-G
SAM-H
SAM-K
Figure 3.3: X-ray diffraction patterns of as-received SAM2X5 samples.
At the Caltech facility, target samples were lapped flat and then epoxied into a polycar-
bonatetargetholder,witharidgeforsampleplacementatthecenterandholeplacements
for shorting pins (Dynasen CA-1038) and interferometry probes. Lapped samples were
diffusely reflective for ease of interferometry free surface measurements. Pictures of as-
sembled Caltech targets and projectiles can be seen in Figure 3.5. All BMG samples
used in this work were nominally 18 mm in diameter and 2 mm in thickness. In all of
the experiments dimensions of impactors and target components were chosen to ensure a
35
Window
Sample
Buffer
(a)
Muzzle End
VISAR probes
Target Holder
(b)
Figure 3.4: Pullman set-up: a) Buffer, sample and window epoxied to each other, and b)
Assembled target, viewed here on the non-impact side, with VISAR probes at the gun-muzzle.
state of uniaxial strain at the point of measurement for the duration of the experiment.
In order to do this, the speed of the fastest wave was used in calculations, as is the
practice in experimental shock physics, to determine appropriate diameter to thickness
ratioswhich allowfordatatoberecorded beforethearrivalofedgewaves. The projectile
used at the Caltech facility comprised of a 45 mm long Nylatron sabot with a cavity at
the front end to accommodate a 34 mm diameter, 1 mm thick OFHC copper flyer plate.
Flyer plates were epoxied into the sabot using Hysol Loctite E-20 HP epoxy and were
cured under a flat steel weight overnight.
3.3 Loading Systems
Impact facilities at ISP, Pullman that were used for experiments in this work comprised
of a 102 mm (4-inch) smooth bore single-stage light gas gun with a 12 meter long barrel
and a 64 mm (2.5-inch) smooth bore powder gun with a 1.5 meter long barrel. The
muzzle-end of both guns contained a fixture for target attachment that was set flat to
within a fraction of a milliradian using a depth gauge. The catch tanks were filled with
36
Target Holder
Projectile
Figure3.5: Caltech set-up: Picture of two fully assembled target plates - BMG samples, shorting
pins, and PDV probes are affixed into polycarbonate holders - and projectiles.
tightly packed fabric to slow down the projectiles and were pressured down to create
vacuum for experiments. The light gas gun was capable of launching projectiles between
400 and 1000 m/s while the powder gun could attain velocities up to 2000 m/s. The
typical impact tilts on the guns were about 0.5 mrad and 2 mrad respectively.
The facility at Caltech has a 36 mm smooth bore naval powder gun with a 3 meter long
barrel, capable of launching projectiles at velocities between 400 m/s and 2000 m/s. A
picture of the powder gun can be seen in Figure 3.6. Polycarbonate target plates were
attached to a custom-built target fixture in the chamber that consisted of three screws
to adjust for planarity. A charge assembled with up to 50 grams of H4198 smokeless rifle
powder and 3 grams of Aliiant 2400 pistol powder was ignited using a 120 V solenoid
(McMaster Carr) [127]. Typical impact tilts for experiments on this gun were on the
order of 5 mrad. A schematic of the projectile exiting the muzzle-end of the gun right
before impact is seen in Figure 3.7.
37
Figure 3.6: Naval powder gun at Caltech. Projectiles are loaded at the breech end on the left
and prepared targets are mounted in the chamber.
Figure 3.7: Schematic of projectile exiting muzzle-end of the powder gun to impact assembled
target at Caltech. T1, T2, T3, and T4 on the impact face of the target holder denote the
locations of the four shorting pins used for tilt and time of arrival measurements.
38
3.4 Velocity Interferometry
In order to solve for thermodynamic quantities in the shocked state using the Rankine-
Hugoniot equations and thereby determine a material’s EOS, the measurement of atleast
two quantities must be made. These are commonly the shock velocity, determined from
timeofarrivaloftheshockatthefrontandbackofthesample,andparticlevelocitywhich
is measured using velocity interferometry techniques. Velocity interferometry makes use
of the basic concept of the Doppler Effect since velocity is inferred from fringes produced
from the mixing of shifted light collected from a moving surface and unshifted light.
The two commonly used laser interferometry systems used to make particle velocity
measurements in shock experiments are the VISAR (used in Pullman) and PDV (used
at Caltech), which are both described in detail in the following sections.
3.4.1 VISAR
The VISAR (Velocity Interferometer System forAny Reflector) was developed by Barker
andHollenbachin1972[12]. Althoughvelocityinterferometrysystemsexistedbeforethis
one, theyrequired specimens thatwere polishedtoamirrorfinish andwere verysensitive
to tilt of the surface of interest during motion. The VISAR, on the other hand, is able to
monitor the motion of surfaces that are specular or diffusely reflective to varying degrees
while being relatively insensitive to tilting of the sample surface as well as maintaining
the 1 - 2 percent accuracies of previous techniques. The VISAR, among other optical
components, consists of the two legs of a Michelson interferometer that are of the same
optical length, however a time delay is introduced in one of the legs using etalons. A
schematic of a conventional VISAR is shown in Figure 3.8. Laser light reflected from
the moving surface of interest and which has thus become Doppler shifted due to the
motion, is split among these two legs, one of which is time-delayed, and then mixed back
to obtain fringes. The following equation relates the fringe count F(t) to the surface
velocity u(t−τ/2):
39
u(t−τ/2)=
λF(t)
2τ (1+Δν/ν
0
)
(3.1)
Here, t is time, λ is the wavelength of the laser light, τ is the interferometer delay time,
and Δν/ν
0
is an index of refraction correction factor which is zero unless the specimen
is backed by a window material through which laser interferometry measurements are
made. In Figure 3.8, a third of the light reflected from the sample is sent to a beam
intensity monitor (BIM), in order to check changes in intensity of reflected light which
are mainly caused by shock-induced changed in surface reflectivity, while the rest is split
50-50 through the two legs of the interferometer. Light that passes through one of the
legs,istime-delayedfrompassagethroughtheetalonsandisalsophase-shifted90degrees
due to a 1/8 wave plate which effectively acts as a quarter wave plate since light passes
through it twice. Light coming from the two legs is then passed through a polarizing
beamsplitter(PBS)which splitsthelightintohalf-Pandhalf-Spolarizedlightwhich are
90 degrees out of phase and result in interferometry fringes when recombined. Obtaining
signals in quadrature (90 degrees out of phase) in this manner allows one to distinguish
between acceleration and deceleration in the specimen’s motion [11].
BIM
Target
Probe
Mirror
Etalon
Detectors
Beam Splitter
BS
1/8 Wave
Plate
Figure 3.8: The conventional VISAR
40
A modification to the conventional VISAR was proposed by Hemsing [52] in order to
make better use of light in the interferometer system. The VISAR set-up described
above, results in loss of half the light initially entering the system due to reflection and
transmission to the other side of the main beam splitter. In the modified VISAR, also
known as the push-pull VISAR, half the light that is lost in the original set-up is used to
drive a second set of detectors resulting in 2 sets of signals that are in quadrature, and
therefore, 4 optical signals. A schematic of the push-pull VISAR is shown in Figure 3.9.
The instrumentation of the plate impact experiments in Pullman consisted of an in-
house,custom-builtpush-pullmultipointVISARcapableofmakingsimultaneousvelocity
measurements at 4 points. Data was recorded using Tektronix DPO7254C oscilloscopes
with 8 GHz bandwidth at a sampling rate of 50 ps/point. The surfaces of interest were
typically polished to a mirror finish to have the samples be specularly reflective. The
ISP VISAR made use of a 6 W, 532 nm laser (Coherent, Inc.) and has a variable delay
time capability. The VISAR produces fringes that are proportional to the velocity of the
surface of interest, and the number of fringes can be determined through an analysis of
the fringe record. Multiplying the number of fringes by the VISAR’s Velocity Per Fringe
(VPF) constant yields the velocity of the surface. For a variable delay time VISAR, the
VPF is set in advance ofthe experiment and is calculated based on the expected velocity
(which may be determined beforehand using an impedance matching calculation). Using
Equation (3.1), the VPF is expressed as [11]:
VPF =
u(1+Δν/ν
0
)
F(t)
=
λ
2τ
(3.2)
Hereuis theexpected velocity and(1+Δν/ν
0
)isthe window correction factor(equal to
1 if a free surface measurement is being made). Typically no more than 2 or 3 fringes are
recorded. For example, if the expected velocity is about 250 m/s, the window material
is c-axis sapphire (window correction factor of 1.783 [69]), and the number of fringes is
3, the VPF constant is then 154 m/s/fringe with a delay time of 1.58 ns. Interferometer
delay times for all experiments conducted were typically 1 - 2 nanoseconds, and this is
41
BIM
Target
Probe
Mirror
Etalon
Detectors
Beam Splitter
BS
1/8 Wave
Plate
PBS
Figure 3.9: The push-pull VISAR
also equal to the rise time of the shock waves measured. Once the VPF for a given
experiment is calculated, using Equation (3.2), the delay time may also be calculated.
Thelengthofetalonsrequiredforadesiredinterferometerdelayτ maythenbecalculated
using the following relation:
τ =
2h
c
(n−1/n) (3.3)
Here, h is the etalon length, n is the refractive index of the etalon material, and c is
the speed of light in vacuum. The ISP VISAR typically used fused silica etalons with a
refractive index of 1.795. Finally, the velocity is calculated from recorded fringe data as
follows [11]:
v(t) =VPF xF(t)=
λ
2τ
F(t) (3.4)
Oftenjumpsinfringescancauseambiguityinthedata. Iftheexpectedvelocityisknown,
fringes can be added to the record after the experiment during data reduction to match
the known velocity. A more feasible solution is to use more than one push-pull VISAR
system in conjunction with one another. In this manner, the leg in each VISAR system
can be set to a different delay time, and therefore different VPFs which are not multiples
42
of each other. This is known as the dual delay VISAR. Reduced data signals are related
to the optical phase difference by a tangent function, and due to the periodicity of this
function, each value for the data signal could correspond to an infinite number of values
for the change in phase, if all of them are a multiple of 2π [29]. Dual delay VISAR
systems thus help in eliminating this so called 2π ambiguity by employing more than
one VPF thereby resulting in an unambiguous fringe count and subsequently velocity
measurement. The uncertainty in the VISAR velocity measurement is given by [29]:

Δv
v

2
=

VPF
v
ΔF

2
+

ΔVPF
VPF

2
(3.5)
Here, Δv/v, ΔF, and ΔVPF/VPF is the uncertainty in velocity, fringe count, and VPF
constant respectively. ΔF is typically on the order of 1 percent and ΔVPF/VPF is
typically 0.1 percent. For velocities that are almost equal in magnitude to the VPF, the
velocity uncertainty is almost equal to the uncertainty in VPF [29].
3.4.2 PDV
APhotonicDopplerVelocimetry(PDV)system isafiber-based Michelson interferometer
which uses recently developed detector technology for laser wavelengths in the infra-red
region (1550 nm wavelength) and fast digitizers to record very high beat frequencies [3].
In addition to being compact and simple in its operation, the PDV is also simpler to
build than a VISAR and is composed of off-the-shelf components manufactured by the
telecommunications industry. In the PDV, fibers are used to send 1550 nm wavelength
light through a probe that focuses this light onto the target surface. Part of this light
from the laser source is sent to a detector, while the remainder undergoes Doppler shift
due to surface motion. The two are then recombined resulting in beat frequencies which
are captured at a digitizer. Thus, in contrast to the VISAR, wherein Doppler-shifted
light collected from a moving surface is split into two legs, one of which is delayed in
time, and then remixed to obtain fringes, the PDV uses the heterodyne method wherein
lightoftwodifferentfrequencies –theoriginalnon-Dopplershiftedreferencelightandthe
43
Doppler-shifted light from the moving surface - are mixed to produce beat frequencies.
The first example of a PDV system was described by Strand et al [128]. Unlike the
VISAR, the PDV is a displacement interferometer wherein displacement of the target by
a distance equal to λ/2, λ being the wavelength of laser light, produces a single fringe.
It is assumed that the velocity of the moving surface over a very small period of interest
τ remains nearly constant, and the instantaneous position of the target surface, x(t), is
then given by [4]:
x(t)≈x

t

+v

t

t−t

(3.6)
Here, v is the average velocity over the time interval τ, and t is the center of that time
interval. The optical phase difference φt over this time interval may then be given by:
φ(t)≈φ(t
1
)−ω

t

t−t

(3.7)
Here,ωistheradialbeatfrequencywithinthetimeintervalwhichisdirectlyproportional
tothevelocityofthemovingtargetandisgivenby4πv/λ,andφ(t
1
)isthephasedifference
from the previous time interval. The output signal at the detector s(t) is then given in
terms of the phase difference as follows:
s(t) =Acos(φ−ω

t−t

) (3.8)
The frequency content of the signal is determined by performing a short-time Fourier
transform (STFT) to calculate the power spectrum:
S(ω,t) =
Z
∞
−∞
s(t)w(t)e
−iωt
dt (3.9)
Here w(t) is the window function used in the STFT (typically a Hamming window).
Thus, at each t within the period of interest τ, the beat frequency is extracted from the
peak in the power spectrum. The velocity within each τ is directly proportional to the
frequency f and is given by:
44
v =
λf
2
=
λω
4π
(3.10)
The relationship between the time interval τ and frequency peak with Δf results in the
following uncertainty product:
(Δf)(τ)≥
1
4π
(3.11)
For example, uncertainty in velocity Δv of 10 m/s would require an STFT analysis win-
dowof6nanoseconds. Forvelocities greaterthan1000m/s, thisresults inanuncertainty
Δv/v ofless than 1 percent, which is reasonable forthe calculation ofthe material Hugo-
niot [4]. However, for low velocities this presents a problem since time-scale of the beat
frequencies containing features in the velocity profile may be larger than the analysis
window itself. The period of the analysis window could be increased, but this would
result in poor time resolution. In order to overcome this issue, frequency conversion is
often adopted. In this method, the reference unshifted light and target light are set at
different frequencies. This is done using an acousto-optic (AO) frequency shifter to mod-
ify the frequency of the laser light, as shown in Figure 3.10. Alternatively, separate laser
sources may be used for the reference light and target light, as is done in the Caltech
PDV system. The beat frequency, using frequency conversion, is related to the target
velocity by [119]:
f
beat
=f
mod
+
2v
λ
(3.12)
Here, f
mod
is the frequency modification in the target light due to the AO frequency
modulator. The Caltech PDV, which is a heterodyne PDV like the one depicted in the
schematic in Figure 3.10 (separate reference laser used instead of the AO modulator
shown), allows for four simultaneous velocity measurements by splitting the laser light
(NKT Photonic 2 W drive laser with a 20mW reference laser) four ways into four PDV
probes (AC Photonics part number: 1CL15P006LCC01)corresponding to Channels 1, 2,
3 and 4. Since this PDV system only makes use of two detectors (Miteq photosensors)
45
Attenuator
Laser
1x2 Splitter
AO Modulator
Circulator
Circulator
1x2 Splitter
Attenuator
Sensor
Probe
Target
Figure 3.10: Schematic of the heterodyne PDV
with each detector for two probe channels, Channels 2 and 4 are delayed in time by a
4 km long single mode fiber (Timbercon TM-902-00278-008KM) which results in a 20
microsecond delay in the signals collected by Channels 1 and 2, and 3 and 4. However,
for experiments for this work, only Channels 1 and 3 were used, one focused on the flyer
plate for projectile velocity measurement, and the other focused at nominally the center
of the metallic glass sample for particle velocity measurement, and none of the time
multiplexed channels were used. Data acquisition is carried out throughan Agilent MSO
9104Aoscilloscope with a sampling rate of 20 GS/s. As partof target preparation before
eachexperiment, targetsurfacesarelappedandPDVprobesaresetintotheprobeholder
ata height thatresults in theleast possible return loss, monitored by a hand-held optical
return loss meter (JDSU SmartClass ORL-55). Once targets were completely assembled,
the probes were affixed to the PDV system while making sure that the fiber ends were
clean by examining through a fiber scope. Attenuators on the front panel of the Caltech
PDV (OZ Optics BB-700-11-1550-9/125-S-60-3A3A-1-0.5-LL)were adjusted to tune the
magnitude of the laser return from the target. Finally, recorded fringe data were reduced
to particle velocity wave profiles using the PlotData software fromSandia National Labs.
46
Chapter 4
Experimental Results
This chapterdescribes theresults ofshockcompression experiments ontwo typesofiron-
based in situ metallic glass matrix composites (MGMC), each with the same nominal
composition of Fe
49.7
Cr
17.7
Mn
1.9
Mo
7.4
W
1.6
B
15.2
C
3.8
Si
2.4
and designated as SAM2x5 and
XS-1. Both samples are synthesized by the method of Spark Plasma Sintering (SPS)
of amorphous powders and have been each sintered at different temperatures in the su-
percooled liquid regime (temperature between the glass transition and crystallization
temperature of a metallic glass), resulting in slight variations in the microstructure from
varying extent of devitrification between the two sample types [73]. SAM2x5 samples
were produced from sintering powders at 900 K which resulted in the precipitation of
sub-micron sized (Fe,Cr,Mn,W)
23
(B,C,Si)
6
crystallized phase scattered throughout the
amorphous matrix. XS-1 was produced from sintering powders at 870 K which is below
the onset temperature of any crystallization products. While high-resolution microscopy
stillshows speckling inXS-1indicating somecrystallite precipitation, thesearebelow the
X-ray detection limit, and the samples are as such X-ray amorphous. The difference in
extent of devitrification in the two sample types allows a systematic study ofthe effect of
partialdevitrificationonthemechanicalresponseofthesecompositesathighstrainrates.
Physical properties of both SAM2x5 and XS-1 samples measured using the methods de-
scribed in Section 3.1 are listed in Table 4.1. A summary of elastic moduli of SAM2x5
47
and XS-1 and their comparison with a Zr-based metallic glass Vitreloy 106 [65] is pre-
sented in Table 4.2. The elastic moduli have been calculated using the properties listed
in Table 4.1. All ofthe elastic moduli of the iron-based composite of this work are higher
than those of the Zr-based alloy, except for the Poisson’s ratio. The lower Poisson’s ratio
of SAM2x5 and XS-1 indicates a more brittle deformation response as compared with
that of Vitreloy 106 [45,79]. In general, the elastic moduli of XS-1 are lower than those
of SAM2x5. This has been seen previously while comparing moduli of fully amorphous
metallic glasses vs their crystalline composites and has been attributed to the so called
elastic softening effect due to the larger free volume in amorphous composition owing to
its glassy structure [38].
Table 4.1: Material characterization results
Sample Density (g/cc) Longitudinal Shear Vickers
Sound Speed (km/s) Sound Speed (km/s) Hardness (GPa)
SAM2X5 7.87±0.02 6.61±0.04 3.68±0.02 16.34±0.50
XS-1 7.75±0.04 6.12±0.04 3.42±0.06 16.16±0.54
Table 4.2: Elastic Moduli of SAM2x5 and XS-1
Sample Longitudinal Young’s Shear Bulk Poisson’s
Modulus Modulus Modulus Modulus Ratio
(GPa) (GPa) (GPa) (GPa)
SAM2X5 343.67±3.85 273.86±5.32 107.68±2.56 200.09±3.32 0.27±0.01
XS-1 289.94±4.29 231.02±6.89 90.86±3.45 168.78±4.57 0.27±0.01
Vitreloy 106 152.0 83.4 30.3 111.6 0.3755
This study undertakes an examination of the response of these two Fe-based MGMCs to
shock compression through the determination of their high-pressure equations of state.
The technique employed is the plate impact experiment with velocity interferometry
as the primary diagnostic. The resulting measurements are those of particle velocity
and shock velocity, with all the other quantities then being accessible by the Rankine-
Hugoniot conservations equations. The methods of calculation, the experimental results
andadiscussionontheimplicationofthoseresultsarepresentedinthefollowingsections.
48
4.1 Tilt-corrected elastic shock velocity
Plate impact experiments rarely ever involve perfectly planar impacts onto the target
plate and inevitably result in a finite tilt, θ
tilt
, between the flyer plate and target plate.
The tilt in the projectile, θ
tilt,pr
, leads to an even larger tilt in the shock wave generated
in the target, θ
tilt,sh
, and may result in significant miscalculation of the shock velocity
if not properly accounted for. This could in turn result in large errors in the calculated
Hugoniot since quantities such as stress, volume compression, and internal energy are
derived from the shock velocity measurement. This section describes the calculation of
elastic shock velocity from known thickness of the target and time of shock arrival at
front and back of target while accounting for the tilt of the shock wave. The tilt of the
projectile is measured by placing three to four shorting pins in a specific arrangement
aroundthe targetplate. Alternatively, impact tiltmay be inferred fromtime ofarrivalof
the tilted shock wave on a target plate by velocity measurement at three or more points.
When the flyer plate sweeps across the impact surface, the pins are sequentially shorted
thus marking the time of arrival of the impact at each location. The tilt is then solved
for using the geometric co-ordinate locations of each pin as well as its time of impact in
the manner outlined below.
Consider location of three shorting pins or alternatively points of interferometry probe
measurement at points 2, 3 and 4 in the xy-plane as shown in Figure 4.1. The point
at which particle velocity measurement is made lies directly behind point 1. The line
of closure across which the flyer sweeps the impact face makes an angle φ with the
horizontal axis. Therefore, one can consider a transformed co-ordinate system in which
the originalaxes are rotated by the azimuthal angle ofthe line ofcontact φ such that the
transformed y-axis y
′
lies parallel to the line of contact. The transformed co-ordinates
may be expressed in terms of the original co-ordinates as follows:
49
y
x
2
,y
2
)
3
,y
3
)
4
,y
4
)
1
1
,y
1
)
Figure 4.1: Schematic of shorting pin locations 2, 3, 4 and projection of particle velocity mea-
surement location onto the impact plane at 1. Dashed lines indicate transformed co-ordinate
system obtained by rotating through angle φ.


x
′
y
′


=


cosφ sinφ
−sinφ cosφ




x
y


(4.1)
The velocity with which the flyer sweeps across the impact-facing target surface is the
velocity of the line of closure V
closure
, which may be calculated using the time separation
ofshorting between anytwo pins andtheknown distance between them, andisexpressed
as follows:
V
closure
=
y
,
2
−y
,
4
t
2
−t
4
=
y
,
3
−y
,
2
t
3
−t
2
(4.2)
Once the closure velocity has been determined, the impact tilt of the projectile θ
tilt,pr
may be solved for as follows:
θ
tilt,pr
=
V
projectile
V
closure
(4.3)
Here, V
projectile
is the velocity with which the flyer plate-sabot assembly is launched from
50
the gun. In most plate impact configurations, including the ones used in this work, the
location of the shorting pins and thus the tilt measurement is laterally separated from
the location of particle velocity measurement at the sample rear surface, as shown in
Figure 4.2. Therefore, in order to get the transit time of the traversal of the shock across
a straight-line path, the time of arrival of the wave at a point on the tilt measurement
plane that is directly collinear with the point of velocity measurement, t
′
1
needs to be
calculated as follows:
V
closure
=
y
,
1
−y
,
4
t
,
1
−t
4
(4.4)
Target plate
V
t
t
t
t
Figure 4.2: Schematic of target plate with shorting pins and interferometry probe for particle
velocity measurement. Times of arrival in the tilt measurement plane are denoted as t
3
, t
4
and
t
′
1
and in the particle velocity measurement plane as t
1
. The third shorting pin is not shown in
this two-dimensional sketch.
Theshockvelocity, whilenotaccountingforeffectsfromtilt,canthenbecalculatedgiven
the target thickness d and the traversal time given by the difference between measured
arrivaltime fromthe particle velocity probe and the arrivaltime calculated ata collinear
point in the same plane as the shorting pins:
51
sh,el,uncorrected
U
sh,el
tilt,sh
Figure 4.3: Relation between uncorrected shock velocity (no tilt) and velocity of the tilted shock
wave
U
sh,el,uncorrected
=
d
t
1
−t
′
1
(4.5)
Since the tilt of the shock wave itself scales in relation to the speed of the shock wave,
the relation for the tilt of the shock wave relative to the tilt of the flyer plate is given by
the following relation:
V
closure
=
V
projectile
θ
tilt,pr
=
U
sh,el
θ
tilt,sh
(4.6)
Here, U
sh,el
is the calculated elastic shock velocity which incorporates the effect of the
tiltoftheshock. The velocity ofthetilted shock wave is related to theuncorrected shock
velocity (assuming zero tilt) by the cosine of the angle of the shock wave tilt, as shown
in Figure 4.3.
U
sh,el,uncorrected
=U
sh,el
cosθ
tilt,sh
(4.7)
From equations (4.6) and (4.7), we obtain the following relation in order to solve for the
shock tilt θ
tilt,sh
and consequently the tilt-corrected shock velocity U
sh,el
:
θ
tilt,sh
cosθ
tilt,sh
=
U
sh,el,uncorrected
V
closure
(4.8)
For small tilts of the flyer plate, the correction to the shock traversal time is usually of
the order of a few picoseconds.
52
4.2 Determining particle velocity at the HEL
The convention followed in identifying the particle velocity at the HEL varies by author.
In several cases, including the ones in this work, the HEL results in a gradual knee
in the wave profile, signifying the onset of inelastic deformation processes. Therefore,
the identification of a distinct HEL point can become ambiguous. In order to maintain
consistency in calculations, the method for determining the HEL is also kept consistent.
As shown in Figure 4.4, lines are fit to the two legs of the knee in the wave profile, and
the HEL is identified as the point at the intersection of these two lines.
0.01 0.02 0.03 0.04 0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
Time (µs)
Particle Velocity (mm/µs)
HEL
Figure 4.4: Identification of HEL point at intersection of straight-line fits (black) to the two legs
of the knee in the wave profile (red).
4.3 Shock compression experiments
Aspreviouslyintroduced, plateimpactexperimentswereconductedattwofacilities: ISP,
WSU Pullman and GALCIT, Caltech. The experiments conducted at ISP consisted of
of a flyer plate impacting a target composed of a buffer plate, sample and window, while
53
thoseatCaltechconsistedofaflyerimpactingasample,asseeninFigure3.1. Asummary
of experimental parameters are listed in Tables 4.3 and 4.4. Figures 4.5 and 4.6 depict
the wave interaction in the flyer and target components immediately following impact.
Backward facingandforwardfacing shocks propagateintheflyerandtargetrespectively.
Incaseofimpactstresses abovetheHEL,twowaves propagateintothetarget,theelastic
precursor wave and a slower moving deformation shock wave. When these waves arrive
at the measurement interface (denoted in orange in Figures 4.5 and 4.6), it sets off a
sharp spike in velocity followed by features corresponding to the materials response in
what is called the measured wave profile. In both of these experimental configurations,
in-material particle velocity has to be calculated from measured velocity which is the
sample-window interface velocity in the Pullman set-up, and the free surface velocity
in the Caltech set-up. The difference between the in-material and measured states is
indicated by States 0 and 1 in the free surface configuration in Figure 2.4 and by States
2 and 2
′
in the window configuration in Figure 2.5 in Section 2.1.4. For the former,
the in-material state is calculated from the measured state from impedance matching
between the sample and window as follows:
u
in-material
=
u
measured
(ρ
S
U
S
+ρ
W
U
W
)
2ρ
S
U
S
(4.9)
Inequation(4.9)above,thesubscriptsS andW indicatesampleandwindowrespectively.
For the free surface configuration, the in-material particle velocity is half of the free
surface velocity by the free surface approximation [9,120].
u
in-material
=
u
free surface
2
(4.10)
Using the elastic shock velocity and particle velocity atHEL determined using the meth-
ods described in sections 4.1 and 4.2, the HEL stress may be computed as follows:
σ
HEL
= ρ
0
U
sh,el
u
HEL
(4.11)
Here, ρ
0
is the ambient density of the sample. The density of the sample after passage
54
Table 4.3: Summary of experimental parameters for Pullman experiments
Sample Impact Flyer/ Buffer Window Window Flyer Buffer Sample
Velocity Material Material thickness thickness thickness thickness
(km/s) (mm) (mm) (mm) (mm)
SAM-F 0.467 C-sapphire C-sapphire 4.998 2.019 2.010 1.795
SAM-C 0.465 C-sapphire C-sapphire 5.020 2.010 2.006 1.816
SAM-H 1.468 OFHC Copper LiF 6.360 2.225 2.059 1.591
Table 4.4: Summary of experimental parameters for Caltech experiments. Flyers were all made
of OFHC Copper.
Sample Impact Flyer Sample
Velocity (km/s) thickness (mm) thickness (mm)
XS1-J 1.300±0.002 1.020 1.665
XS1-H 1.104±0.002 1.016 1.659
XS1-M 0.584±0.001 1.015 1.841
SAM-O 0.705±0.001 1.015 1.722
SAM-R 0.918±0.002 1.017 1.666
SAM-V 1.001±0.002 1.016 1.572
of the elastic precursor is determined from the first Rankine-Hugoniot jump condition as
follows:
ρ
HEL
=
ρ
0
U
sh,el
U
sh,el
−u
HEL
(4.12)
The density in the peak shocked state may be determined in a similar manner:
ρ
peak
=
ρ
HEL
U
sh,pl
U
sh,pl
−(u
peak
−u
HEL
)
(4.13)
The Eulerian plastic shock speed is calculated after determining the transit time of the
plastic wave after precursor arrival to the mid-point of the plastic wave rise δt as follows:
U
sh,pl
=
d(ρ
0
/ρ
HEL
)
d/U
sh,el
+δt
(4.14)
Here, d is the original thickness of the sample. Finally, the stress in the peak shocked
state of the sample is calculated as follows:
55
Figure 4.5: Time-distance plot showing wave interaction in flyer and target immediately follow-
ing impact. Measurement from PDV probes occurs at location denoted in orange.
σ
peak
=σ
HEL
+ρ
HEL
U
sh,pl
(u
peak
−u
HEL
) (4.15)
A summary of calculations performed using this method for both Pullman and Caltech
experiments are summarized in Tables 4.5, 4.6 and 4.7.
Initial experiments conducted on SAM2x5 at peak stresses of about 11 GPa showed a
single elastic wave, indicating purely elastic response. However, even though the sample
was shocked below its elastic limit, a slight decay was apparent in the peak particle ve-
locity in the flat-top section of the wave profile, as seen in Figure 4.7 indicating stress
relaxation and consequently, the onset of some inelastic processes. This suggests that
the impact stress the sample was subjected to was approaching the elastic limit of the
material. A high yield strength for this material is expected based on measurement of
indentation hardness H
V
. The yield strength of bulk metallic glasses is known to be
predicted by the H
V
/3 relation [62]. As seen in Table 4.1, the measured Vickers hard-
ness for SAM2x5 is 16.34 GPa, and therefore it follows that the predicted quasi-static
56
Figure 4.6: Time-distance plot showing wave interaction in impactor, buffer, target and window
immediately following impact. Measurement from VISAR probes occurs at location denoted in
orange.
yield strength would be 5.45 GPa. This is even higher than the yield strength of over
4 GPa previously reported forsuper-high-strength amorphous steels [62,63]. Using equa-
tion (2.30), the HEL can be predicted as well, since the Poisson’s ratio ν as well as
other elastic moduli are known through calculations from ultrasonic measurements. A
value of 0.27 for the Poisson’s ratio thus gives us an expected HEL of 8.65 GPa. The
HEL represents the onset of yielding under uniaxial strain. An experiment at a peak
stress of about 32 GPa revealed a two-wave elastic-plastic structure with an HEL of
12.89 GPa corresponding to a large elastic strain of 3.73 percent. The amount of elas-
tic strain incurred by the specimen is comparable to that of Zr-based metallic glass in
previous shock compression studies [65,139]. The discrepancy between calculated and
57
0 0.1 0.2 0.3 0.4
0
0.2
0.4
0.6
0.8
1
Time (µs)
Particle Velocity (mm/µs)
SAM C
SAM F
SAM H
Figure 4.7: Wave profile data for SAM2x5 showing particle velocity at interface of sample and
window as measured by Pullman VISAR. The arrow on the plot indicates the HEL point.
measured HEL may have arisen since the strengthening effect and mechanical properties
of in situ precipitated phases are not taken into account in the calculation. On the other
hand, the HEL calculation is in very good agreement with the measured HEL for the
XS-1 samples. The more homogeneously amorphous structure of XS-1 as compared to
SAM2x5 with barely detectable nanocrystallites is likely responsible for this agreement.
Discrepancies in measured HEL and the one calculated from quasi-static yield strength
have also been observed previously and have thought to have arisen due to strain-rate
effects [139].
A repeat of the experiment on SAM2x5 at 11 GPa peak stress yielded highly repeatable
wave profiles thereby establishing the reproducibility of the sample response. Agreement
58
betweenresultsobtainedusingbothWSUPullmanandCaltechset-upsfurtherconfirmed
reproducibility of the results. Figures 4.8 and 4.9 show wave profiles obtained at Caltech
for XS-1 and SAM2x5 respectively with each of these containing a two-wave elastic-
plastic structure with the HEL at the kink or knee. Strain rates for all the experiments
ranged from 6×10
5
– 2.5×10
6
per second. These experiments on XS-1 and SAM2x5
revealed HELsof8–9GPaand10–12.5GParespectively. These arehigherthanelastic
limits under shock loading previously reported for metallic glasses of other compositions
[65,104,107,136–139]. Thelatterespecially isconsiderably higher, andisnearly1.5times
that of previous results for the HEL of a metallic glass. The elastic limit measured for
SAM2x5 is comparable to that of other high-strength hard, brittle ceramics such as SiC
and TiBr
2
[106].
0 0.5 1 1.5
0
200
400
600
800
1000
1200
Time (µs)
Free surface velocity (m/s)
XS1−J
XS1−H
XS1−M
Figure 4.8: Wave profile data for XS-1 showing free surface velocity at sample rear surface as
measured by Caltech PDV. Arrows on the plot indicate the HEL point.
59
Table 4.5: Summary of calculated SAM2x5 results for Pullman experiments
SAM-F SAM-C SAM-H
Elastic Shock Velocity (km/s) – 6.75±0.02 6.74±0.02
Plastic Shock Velocity (kms) – – 5.71±0.12
HEL Particle Velocity (km/s) – – 0.242±0.001
HEL (GPa) – – 12.89±0.19
Peak Particle Velocity (km/s) 0.219±0.001 0.222±0.001 0.657±0.033
Peak Density Compression 0.0341±0.003 0.0343±0.003 0.118±0.012
Peak Stress (GPa) 11.42±0.39 11.74±0.41 32.31±1.13
−0.5 0 0.5 1 1.5
0
200
400
Time (µs)
Free surface velocity (m/s)
−
−
−
Figure 4.9: Wave profile data for SAM2x5 showing free surface velocity at sample rear surface
as measured by Caltech PDV. Arrows on the plot indicate the HEL point.
4.3.1 Temperature rise due to shock waves
The compression of a material from the propagation of a shock wave through it results
in large temperature rises. The experimental measurement of shock temperatures can be
challenging and therefore estimations for temperature rise are generally calculated using
the material equation of state [108,110]. The temperature along the Hugoniot may be
calculated using the following relationship [66]:
60
T
H
= T
S
+
Z
P
H
P
S
VdP
ΓC
V
(4.16)
Here, T
H
, T
S
, P
H
andP
S
aretemperature andpressure alongthe Hugoniotandisentrope
respectively, while V, C
V
and Γ are the specific volume, specific heat and Gruneisen
parameter respectively. The temperature along the isentrope is calculated using the
Gruneisen parameter at ambient conditions, γ
0
, as follows [66,133]:
T
S
= T
0
exp

γ
0

1−
V
V
0

(4.17)
Here, T
0
is the ambient temperature and V and V
0
represent the final and initial specific
volume onthe Hugoniot fora given compression state. Equation (4.16)can bere-written
as follows, after evaluation of the integral [133]:
T
H
= T
S
−
Δe
S
C
V
+
1
2C
V
(V
0
−V)P
H
(4.18)
Here, Δe
S
represents the change in specific internal energy along the isentrope, and is
generally expressed as a function of the ratio of initial and final specific volumes for a
given compression state V
0
/V, isentropic bulk modulus K
S
and the pressure derivative
of the isentropic bulk modulus dK
S
/dP written as K
S
′
. The isentropic bulk modulus
is given by ρC
0
2
while its pressure derivative K
S
′
is estimated by 4s− 1 where s is
the slope of the U
S
− u
p
Hugoniot. Δe
S
is given by the integration of the isentropic
Birch-Murnaghan equation of state [15] and is expressed as follows:
Δe
S
=
9V
0
K
S
16



"

V
0
V

2/3
−1
#
2
"
6−4

V
0
V

2/3
#
+K
S
′
"

V
0
V

2/3
−1
#
3



(4.19)
As can be seen from Equations (4.17), (4.18) and (4.19), the calculation of the shock
temperature requires knowledge of the thermal properties of the material. Since thermal
measurements were notmade onXS-1 andSAM2x5, these were estimated frommeasure-
ments previously made on other iron-based metallic glasses. Γ
0
is calculated using the
61
following relation:
Γ
0
=
3αK
C
V
ρ
(4.20)
InEquation(4.20),K isthebulkmodulus,αisthecoefficientoflinearthermalexpansion,
C
V
isthespecificheatcapacity,andρistheambientdensity. Valuesforheatcapacityand
coefficientforthermalexpansionwereobtainedfrompreviousworksonthemeasurements
ofthermalpropertiesofamorphoussteels[93,134,159]. Thefollowingassumptionismade
throughout the calculation:
Γ
0
V
0
=
Γ
V
=constant (4.21)
It is also assumed that the specific heat C
V
remains constant. In order to calculate the
isentropic bulk modulus K
S
and the pressure derivative of the isentropic bulk modulus
K
S
′
, C
0
and s must be determined from the plastic U
S
−u
p
Hugoniot, which has been
plotted in Figure 4.10. The vertical error bars in Figure 4.10 represent the uncertainty
in measured plastic shock velocity arising out of the uncertainty in time of arrival and
sample thickness measurements. The horizontal error bars represent uncertainty in par-
ticle velocity measurement by the PDV which is calculated using Equation (3.11). The
velocity of the plastic shock, which is the main deformation-causing shock front, is a
measure of the strength of the shock and by the very definition of a steadily-sustained
shock, generally increases (linearly, for most materials) with increasing impact stress.
The plastic shock velocity–particle velocity Hugoniot for XS-1, with the exception of
one data point at the intermediate loading stress, displays this general trend. On the
other hand, this trend does not clearly present itself in the data for SAM2x5. In ad-
dition, the velocities of the shock lie below the ambient bulk sound speed of SAM2x5
indicating severe lossofstrength. Thisisinkeeping withprevious observationsforelastic
strain-softening solids in which the plastic shock is often slower than the ambient bulk
wave speed of the material [71]. A linear fit to the data in Figure 4.10 results in U
S
−u
p
relationsofU
S
=4.61+0.69u
p
andU
S
= 5.05−1.16u
p
forXS-1andSAM2x5respectively.
62
Finally, all the gathered material properties are substituted into Equation (4.18) to cal-
culated temperature rise within the shock front. Temperatures are only calculated for
the maximum impact stresses in this dissertation, since these will result in the largest
possible temperature rise for this set of experiments. It is found that a shock stress of
25.13 GPa results in a final temperature of 438 K (165 °C) in XS-1, and an incident
stress of 19.83 GPa results in a temperature of 423 K (150°C) in SAM2x5, assuming an
ambient temperature of300 K(27°C).Bothofthese temperatures aresignificantly lower
than the glass transition temperature of 850 K (577°C) for the chemical composition of
XS-1 and SAM2x5, as well as the temperatures associated with long-range ordering and
devitrification [73]. The shock response of both composites is therefore not influenced by
any potential shock-induced phase transitions since these do not occur under the range
of impact stresses examined here.
4.3.2 Elastic-Perfectly Plastic Response and the Hydrostat
A high HEL is indicative of the extent of energy-absorbing ability and therefore the
shock-resistance of a material and is seen to be a benchmark for ballistic performance,
however, the elastic limit alone may not be a good indicator of material strength and
performance. The offset between static isothermal data and shock data is, therefore,
commonly used to assess the shear strength achieved in the shock compressed state [42].
For this purpose, the static data may be obtained from diamond-anvil cell experiments,
or it may be computed from data from shock experiments if lateral stress measurements
aremadeinadditionto theusuallongitudinalstress measurements. Intheabsence ofex-
perimental hydrostatic data however, the hydrostat may be analytically calculated while
making some assumptions. Such an approach has been adopted in this work.
Elastic shock velocity from each experiment, calculated from pin arrival times and the
method outlined in Section 4.1, is plotted against particle velocity corresponding to the
elastic limit, as seen in Figure 4.11. A linear fit through this data results in an equation
63
Table 4.6: Summary of calculated XS-1 results for Caltech experiments
XS1-J XS1-H XS1-M
Elastic Shock Velocity (km/s) 6.20±0.02 6.20±0.02 6.40±0.02
Plastic Shock Velocity (kms) 5.09±0.10 4.72±0.10 5.05±0.10
HEL Particle Velocity (km/s) 0.167±0.009 0.182±0.009 0.181±0.009
HEL (GPa) 7.99±0.12 8.73±0.13 9.02±0.14
Elastic Density Compression 0.0276±0.003 0.0301±0.003 0.0292±0.003
Peak Particle Velocity (km/s) 0.590±0.009 0.485±0.009 0.251±0.009
Peak Density Compression 0.1209±0.012 0.1007±0.010 0.0436±0.004
Peak Stress (GPa) 25.13±0.89 20.15±0.71 11.84±0.41
of the form:
U
sh,el
=c
L,0
+su
p
(4.22)
Here, c
L,0
is the ambient longitudinal sound speed measured using the ultrasonic pulse
echo technique and the parameter s is the slope of the linear fit. This equation is
converted to the stress - density compression σ−µ plane using the Rankine-Hugoniot
jump conditions [108] and is expressed as [139]:
σ
x
= L
0

µ +(2s−1)µ 2
+(1−4s+3s
2
)µ 3

(4.23)
Here, L
0
is the ambient longitudinal modulus. Equation (4.23) represents the longitudi-
nal elastic stress as a function of density compression. In order to make inferences about
the strength of the material, a comparison of the Hugoniot and the hydrostat is neces-
sary. The hydrostat represents the average of stresses in the three principal directions
i.e. the longitudinal stress (measured here) and two lateral stresses. Since no lateral
stress measurements were made in any of the experiments in this work, an assumption
pertaining to the elastic moduli of the material is needed in order to calculate the mean
stress or the hydrostatic stress. Following the framework for this calculation as laid out
by Fowles in 1961 [40], the shear modulus G is assumed to remain constant. The mean
stress σ
m
is then calculated from the longitudinal stress as follows:
64
Table 4.7: Summary of calculated SAM2x5 results for Caltech experiments
SAM-O SAM-R SAM-V
Elastic Shock Velocity (km/s) 6.53±0.02 6.59±0.02 6.88±0.02
Plastic Shock Velocity (kms) 4.15±0.10 4.82±0.10 4.69±0.10
HEL Particle Velocity (km/s) 0.242±0.009 0.242±0.009 0.189±0.009
HEL (GPa) 12.43±0.19 12.54±0.19 10.30±0.15
Elastic Density Compression 0.0385±0.004 0.0382±0.004 0.0283±0.003
Peak Particle Velocity (km/s) 0.306±0.009 0.409±0.009 0.439±0.009
Peak Density Compression 0.0546±0.005 0.0753±0.008 0.0861±0.009
Peak Stress (GPa) 14.57±0.51 19.08±0.67 19.83±0.69
σ
m
=σ
x
−
4G
3

µ −
µ 2
2
+
µ 3
9

(4.24)
Themeanstress calculated fromEquation(4.24)thusrepresents theanalyticalhydrostat
for the material, whereas Equation (4.23) represents the elastic Hugoniot. The Hugo-
niot beyond the elastic limit is constructed assuming elastic-perfectly plastic material
response as described in Section 2.2.2 by adding the constant offset value of 2Y/3 to the
calculated hydrostat. The experimentally obtainedstresses and density compressions are
then plotted alongside the calculated Hugoniots and hydrostat to evaluate the strength
of the material, as seen in Figures 4.12 and 4.13.
Although the spread in elastic shock velocity data is within 5 percent for both sample
types, as seen in Figure 4.11, a linear fit from the ambient sound speed on the y-axis to
either extreme in value of shock velocity results in significant variation in the s parame-
ter, which is the slope of the line. Since the construction of the Hugoniot and hydrostat
entirely depend on the value of s, this in turn has a direct effect on the results obtained
for material strength. For the sake of consistency, the results and following interpreta-
tions have been drawn from a linear fit through three representative experimental data
points. ThefitresultsinaU
sh,el
−u
p
relationgivenbyU
sh,el
= 6.61+0.176u
p
forSAM2x5
and U
sh,el
= 6.12+0.833u
p
for XS-1. As in Figure 4.10, the vertical and horizontal error
bars in Figure 4.11 represent uncertainties in elastic shock velocity (arising out of un-
65
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
4
4.2
4.4
4.6
4.8
5
5.2
Particle Velocity (km/s)
Shock Velocity (km/s)
(a) XS-1
0 0.1 0.2 0.3 0.4 0.5
4
4.2
4.4
4.6
4.8
5
5.2
Particle Velocity (km/s)
Shock Velocity (km/s)
(b) SAM2x5
Figure 4.10: Plastic shock velocity - particle velocity Hugoniots for XS-1 and SAM2x5. The red
circle on the y-axis denotes the ambient bulk sound speed.
certainty in time of arrival, sample thickness and elastic shock velocity measurements)
and particle velocity respectively. While the measured elastic shock velocity in a shocked
material is generally higher than its ambient longitudinal sound speed as is apparent for
XS-1, two of the data points plotted for SAM2x5 seem to lie slightly below the ambient
sound speed. A lower than ambient velocity which could potentially result in a Hugo-
niot with a negative slope was also observed in the work of Martin et al. on a Zr-based
metallic glass [104]. This was explained to be a consequence of considerable dispersion
of the shock front in the observed stress traces, which is a direct result of the negative
first pressure derivative of the elastic modulus of amorphous metals [27]. However, since
dispersed shock fronts were not observed in the wave profiles measured in this work, it
is likely that the measured elastic shock velocities appear to be lower than the ambient
longitudinal sound speed as a result of the uncertainty in the measurement. Values for
the elastic shock velocity that are in the upper extreme of the uncertainty bounds could
result in a positive sloped Hugoniot, and this lends further evidence for this possibility.
Mashimo describes a system of classification for the yielding of shocked solids based
66
0 0.05 0.1 0.15 0.2
6
6.1
6.2
6.3
6.4
6.5
6.6
Particle Velocity (km/s)
Elastic Shock Velocity (km/s)

(a) XS-1
0 0.05 0.1 0.15 0.2 0.25 0.3
6
6.2
6.4
6.6
6.8
7
Particle Velocity (km/s)
Elastic Shock Velocity (km/s)

(b) SAM2x5
Figure 4.11: Elastic shock velocity - particle velocity Hugoniots for XS-1 and SAM2x5. The
intercept on the y-axis denotes the ambient longitudinal sound speed. The red line is a linear
least squares fit through the experimental data.
on the offset of the Hugoniot from the hydrostat or the isotropic loading state beyond
the elastic limit [27,106]. Following this system, materials may be classified into three
main categories based on the comparison of their response to shock and isotropic load-
ing when considered in the stress - density compression space: toughening solids whose
shearstrengthincreases withappliedstress, perfectelasto-plasticsolidswhich maintaina
constant offset between their Hugoniot and hydrostat, and perfect elasto-isotropic solids
which catastrophically lose shear strength above the HEL with the stress collapsing onto
the isotropic state. However, in reality, the perfect elasto-plastic and perfect elasto-
isotropic solids do not exist, since the strength of a material is transformed by plastic
deformation and therefore does not remain constant, and under dynamic loading even
liquids preserve some shear strength by virtue of viscosity and therefore a complete col-
lapse onto the isotropic stress state is unrealistic for a solid. Material behavior often lies
between these two idealized extreme scenarios, and this is seen in the material response
of XS-1 and SAM2x5 as is apparent in Figures 4.12 and 4.13.
For both materials, the data points clustered around the inflection in the Hugoniot cor-
67
0 0.02 0.04 0.06 0.08 0.1 0.12
0
5
10
15
20
25
30
Density Compression
Stress (GPa)

XS−1 Hydrostat
XS−1 Hugoniot
Experiment
Figure 4.12: A comparison of calculated Hugoniot (red), hydrostat (black) and experimental
data (blue) for XS-1.
respond to the HEL. For the XS-1 samples, the data point corresponding to the peak
state at the lowest impact stress seems to be in accordance with the elasto-plastic cal-
culation. However at the higher impact stresses, both data points lie very close to the
hydrostat, thus suggesting that there exists a certain threshold between 12 and 20 GPa
beyond which XS-1 catastrophically loses all strength with the Hugoniot consequently
nearly collapsing onto the isotropic stress state. Such catastrophic loss of shear strength
hasalsobeenobservedinhigh-HEL,brittleandhardmaterialssuchasboroncarbideand
silicon nitride [71]. On the other hand, peak state measurements for SAM2x5 compared
to its calculated Hugoniot suggest that although the material strain-softens it seems
to retain its strength even at impact stresses nearly one and half times its HEL. This
observed loss of post-yield strength in SAM2x5 and XS-1 is likely a result of yielding
phenomena arising out of slip from propagation of cracks, cleavage and partial melting.
68
0 0.02 0.04 0.06 0.08 0.1 0.12
0
5
10
15
20
25
30
Density Compression
Stress (GPa)

SAM2x5 Hydrostat
SAM2x5 Hugoniot
Experiment
Figure 4.13: A comparison of calculated Hugoniot (red), hydrostat (black) and experimental
data (blue) for SAM2x5.
Under intense loading scenarios such as the ones shock compression experiments present,
the free volume in the amorphous matrix is quickly depleted and large stress concen-
trations are accumulated at those sites. Micro-voids coalesce into shear bands, which
then provide pathways for the propagation of micro-cracks arising out of areas of intense
shear localization. Evidence for strength loss beyond the elastic limit has also been seen
in previous shock studies of various Zr-based metallic glasses [65,104,139,153,155]. This
has manifested itself in the wave profiles - in the steeply rising elastic wave front fol-
lowed by stress relaxation ahead of plastic wave arrival [104,155], which is a signature
of an elasto-isotropic solid undergoing catastrophic strength loss [27] - as well as in the
examination of post-yield strength [65,139]. This signature of strength loss in the shock
wave trace was absent in all of the measured wave profiles in this work, except for two
69
measurements obtained for SAM2x5, as seen in Figure 4.9.
4.3.3 Discussion
Whilethemechanisticphenomenabehindtheelasticlimitandyieldingofcrystallinemet-
als under shock loading is fairly well understood in terms of dislocation-mediated slip,
thephysical interpretation ofthe elasticlimit inbrittleamorphous solids remained some-
what obscure until recently. Subsequently, it has been hypothesized that shock-yielding
in brittle solids corresponds to the onset of relieving of shear strains from fracture by
the joining and interaction of damaged zones and subsequent flow of the material [16].
In general, various mechanisms by which slip, and thereby yielding, may occur to ac-
commodate shear strains have been proposed; these include plastic-brittle deformation
comprising of slip zones with cracks, micro-cracks and dislocations, brittle destruction
deformation consisting of cracks and cleavages, deformation from partial-melting zones,
and deformation from lattice destruction [106]. Of these, deformation from cracks and
cleavage, partial melting, and lattice destruction are known to result in catastrophic loss
of shear strength beyond the elastic limit.
The HEL in brittle materials, particularly in polycrystalline ceramics, has been seen to
be heavily influenced by ceramic density/porosity, impurity content and material pro-
cessing [71,106]. This is because complex heterogenous stress states are created at the
microscale by the presence of impurities and irregularities even though a macroscopi-
cally clean, uniform shock wave is initially introduced into the material. In a study on
Ti-based in situ metallic glass matrix composites synthesized from sintering of powders,
yield strength was found to decrease with increasing porosity ofthe microstructure [140].
Slight variations in sintering parameters, the extent and size of defects and their dis-
tribution, the degree of powder compaction and subsequent porosity could, in a similar
manner, explain the variation in elastic limits seen in both XS-1 and SAM2x5. A scan-
ning electron microscope (SEM) image of the polished surface of a SAM2x5 sample,
as seen in Figure 4.14, contains evidence of some of these material irregularities in the
70
form of boundaries indicating fused sintered powder grains as well as porosity at grain
triple-points. These were also observed for XS-1 samples. Passage of the shock over a
heterogeneity or large pore could result in significant dissipation and a large localized
temperature rise causing melting in a manner similar to that seen in porous granular
solids, thus resulting in an observed smaller elastic limit for that specimen.
Figure 4.14: Scanning electron microscope (SEM) image of the mirror-polished surface of a
SAM2x5 sample at 3300x magnification. Fused powder grains and the pores at their triple
points are visible.
Figure 4.15 shows a comparison of the post-yield shear strength in SAM2x5 and XS-
1 compared with that calculated for a Zr-based metallic glass Vitreloy in a previous
work [139]. The strength in both this work and the cited work is calculated using the
constant shear modulus assumption and this enables a meaningful comparison of the
three cases. Like Vitreloy, SAM2x5 and XS-1 also how a general trend of decreasing
strength at increasing stresses and compressions. A comparison of the fully amorphous
Vitreloy with the X-ray amorphous XS-1 shows that the iron-based composition XS-1
presents a significantly larger strength to begin with than Vitreloy, lending further ev-
idence to the superior strength of amorphous steels compared to other compositions of
amorphous metallic alloys. However both catastrophically lose strength at volume com-
71
pressions ofabout10 percent and higher. Onthe otherhand, SAM2x5 presents a further
improvement in initial strength over XS-1, and retains significant strength at volume
compressions of nearly 10 percent, on account of the reinforcing nanocrystalline phase it
contains. Therefore, the extent of devitrification, however small, and nature of devitrifi-
cation products is seen to have a significant effect on the shock response of the metallic
glasses studied here, unlike the results of Zhuang et al. where a comparison of the shock
response of Vitreloy 1 and its dendritic phase reinforced in situ composite (containing
25 percent volume fraction of crystallinity [51] as opposed to merely 1 - 2 percent in
this work) yielded no observable differences [161]. Controlled devitrification is thus a po-
tentially viable adjustable parameter for synthesis of amorphous metallic materials with
properties that can be tailored as desired.
0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 0.11 0.12
0
1
2
3
4
5
6
Density Compression
Strength (GPa)

XS−1 (this work)
SAM2x5 (this work)
Vitreloy (Turneaure et al)
Figure 4.15: Post-yield shear strength as a function of density compression for XS-1 (black),
SAM2x5 (blue) and Vitreloy 106 (red) from the work of Turneaure et al. (2006) [139].
72
Since themaximum shock stress thatSAM2x5 was subjected toisalittleshy ofthemax-
imum impact stress for XS-1, a further loss of strength for SAM2x5 than is suggested by
the current data cannot be completely ruled out. Even then, such dramatically differ-
ent responses at high Hugoniot pressures within a comparable range of impact stresses
for samples that are nearly identical in their make-up, except for a very small amount
of crystallinity (on the order of 1 or 2 percent), cannot be ignored. Therefore, a fur-
ther examination of the crystalline phases present in SAM2x5 that distinguish it from
XS-1 and their effect on overall mechanical response is needed. In synthesizing sam-
ples of SAM2x5, the amorphous powders were sintered at 900 K, which is above the
temperature at which structural transformation corresponding to the crystallization of
(Fe,Cr,Mn,W)
23
(B,C,Si)
6
occurs. On the other hand, samples of XS-1 were made by
sintering powders at 870 K, just below the onset of this crystallization, but above the
temperature at which structural relaxation and some long-range ordering occurs [73].
Metal carbides and borides are known to have high hardnesses, and the presence of their
nanocrystallites dispersedintheamorphousmatrixwithinSAM2x5arelikely responsible
for strengthening this material – in both increasing its elastic limit compared with XS-1,
as well as retention of post-yield strength – by acting as barriers to the propagation
of failure fronts in the form of shear bands and cracks. In a previous work, annealing
of an Fe-Co-B-Si-Nb metallic glass resulted in enhanced hardness and Young’s modulus,
withmechanical hardeningbeing attributedtotheprecipitationofthemetastableFe
23
B
6
phase [38], which is believed to be present in SAM2x5 as well. In another work, the pre-
cipitation of the Fe
23
B
6
phase in a Fe-Dy-B-Nb metallic glass was also seen to enhance
fracture strength and Vickers hardness, in addition to improving the thermal stability
and glass forming ability of the composition [84].
While a higher Young’s modulus is observed in SAM2x5 as compared to XS-1, no signif-
icant difference in Vickers hardness is observed between the two sample types, as seen in
Tables 4.1 and 4.2. Therefore, while a small proportion of crystalline precipitate is not
largeenough tocause anyobservable distinction in theambient andquasi-static mechan-
73
ical response of SAM2x5 and XS-1, it proves to be significant in influencing the response
of the shocked samples at very high strain rates (of the order of 10
6
per second). This
is in stark contrast with results from previous works on Vitreloy 1 and its in situ com-
posite wherein quasi-static compression yielded a markedly different response between
the two, with the composite showing enhanced ducility (up to 8 percent plastic strain
to failure) [51], whereas the high strain rate shock compression response of the two was
nearly identical [161]. The exact manner in which the highly dispersed, sub-micron sized
isolated as well as clusters of crystallites in XS-1 and SAM2x5 aid in strengthening the
material is not fully understood, and a more detailed characterization of the size, distri-
bution, nature and mechanical properties of the precipitates will have to be performed
to establish further correlation with the bulk mechanical response at high strain rates.
74
Chapter 5
Continuum Modeling in ANSYS
Autodyn
5.1 ANSYS Autodyn formulation
The response of the metallic glasses explored in this work was modeled in ANSYS Au-
todyn by using material equations of state from experimentally measured data as an
input. Autodyn is designed to model non-linear, highly transient problems and has been
successfully used to simulate impact, shock propagation, blast waves in solids andliquids
etc. Autodyn has also been used in the constitutive modeling of a Zr-based BMG and
its composite when subjected to modified Taylor tests in a previous work [105]. As part
of the explicit dynamics formulation of the Autodyn solver, the conservation equations
for mass, momentum and energy are solved for each element of the Lagrangian mesh
in the model using results from the previous time step as an input. For a Lagrangian
description, the mesh distorts and moves along with the material, and the new den-
sity is calculated using the current volume of the element and its initial mass using the
conservation of mass as follows [1]:
ρ
0
V
0
V
=
m
V
(5.1)
75
The conservation of momentum relates the acceleration to the stress tensor σ
ij
as fol-
lows (the partial differential equation describing momentum conservation only in the
x-direction is given here):
ρ¨ x = b
x
+
∂σ
xx
∂x
+
∂σ
xy
∂y
+
∂σ
xz
∂z
(5.2)
The conservation of energy is given by the following:
˙ e =
1
ρ
(σ
xx
˙ ǫ
xx
+σ
yy
˙ ǫ
yy
+σ
zz
˙ ǫ
zz
+2σ
xy
˙ ǫ
xy
+2σ
yz
˙ ǫ
yz
+2σ
zx
˙ ǫ
zx
) (5.3)
A central difference scheme - the Leapfrog scheme - is applied to the integration in time.
After forces (arising from internal stresses, contact, boundary conditions etc.) at the
nodes are calculated, the nodal accelerations are derived as follows:
¨ x
i
=
F
i
m
+b
i
(5.4)
The acceleration thus calculated by dividing nodal forces by nodal mass undergoes ex-
plicit integration in time to obtain new nodal velocity:
˙ x
i
n+
1
2
= ˙ x
i
n−
1
2
+ ¨ x
i
n
δt
n
(5.5)
Finally, nodal positions for the next time step are calculated by integrating calculated
nodal velocities as follows:
x
i
n+1
= x
i
n
+ ˙ x
i
n+
1
2
δt
n+
1
2
(5.6)
The advantages of the explicit time integration for solving a set of nonlinear equations
include the direct solving of the equations without the need for iteration or checks for
convergence. In addition, an inversion of the stiffness matrix is not required [1]. The size
ofthetimestepusedinexplicittimeintegrationislimitedbytheCourant-Fredrichs-Levy
(CFL) condition by which the stress wave or disturbance cannot travel farther than the
smallest characteristic element dimension in the mesh in a single time step:
76
δt≤ f

h
c

min
(5.7)
Here f is the stability time step factor (set to a default value of 0.66), h is the charac-
teristic dimension of an element, and c is the local material sound speed in the element.
5.2 Autodyn model set-up
The experimental set-up corresponding to the one from Caltech consisting of a cop-
per flyer plate impacting a BMG sample is replicated in Autodyn in a 2-dimensional
axisymmetric set-up, as shown in Figure 5.1. Two rectangular parts with dimensions
corresponding to actual experimental dimensions of the flyer and sample are created and
filled with a Lagrangian mesh with a grid size of 2 microns. Finer meshes are able to
better resolve the abrupt nanosecond-timescale rise of a shock wave and avoid smeared
shock fronts, however they are computationally expensive. It was found that a grid size
of 2 microns was able to adequately capture the shock wave profiles in a reasonable com-
putation time, and hence this has been used for all simulations in this work. A total of
90,000 grid points were used in the computation for each of the components. The flyer
plate part was given an initial condition for velocity in the x-direction corresponding to
the actual impact velocity in the experiment. To simulate a state of uniaxial strain in
the sample, a boundary condition of zero velocity in the y-direction was imposed on the
sample part. The model was runfora totalsimulation duration of450 - 600nanoseconds
which corresponds to the interferometry recording time in the experiment.
In order to simulate the mechanical response of a material, it is necessary to define three
aspects of its properties: the equation of state (EOS), a constitutive strength model,
and a failure criterion. Each of the parts in the model was filled with the appropriate
material. The properties for copper in the flyer plate were chosen from the Autodyn
material library. These comprised of a shock EOS, which is expressed in the form of
77
y
x
Flyer
Sample
1 mm
1.5 mm
Figure 5.1: 2D axisymmetric problem set-up in Autodyn
Equation (2.5), and Steinberg-Guinan strength model which expresses the change in
shear modulus G and yield strength Y, while capping the change in yield strength to a
certain limit strength Y
max
, and is expressed by the following [126]:
G =G
0

1+

(dG/dP
G
0

P
η
1/3
+

dG/dT
G
0

(T −300)

(5.8)
Y =Y
0
[1+β(ǫ+ǫ
i
)]
n

1+

dY/dP
Y
0

P
η
1/3
+

dG/dT
G
0

(T −300)

(5.9)
Y
0
[1+β(ǫ+ǫ
i
)]
n
≤Y
max
(5.10)
Amaximum tensile principalstress failurecriterion, as well asanerosion model, was also
specified. Material parameters for copper have been summarized in Table 5.1.
The sample part was filled with material properties for the iron-based metallic glasses.
The sample’s equation of state was obtained from experimental data, and the Von Mises
strengthandPrincipalStressfailurecriterionwereused. Followingthemethodofanalysis
78
Table 5.1: Material properties used in Autodyn Model for copper
Equation of State Shock
Ambient bulk sound speed 3670 m/s
S 1.489
Gruneisen Parameter 2.02
Reference Temperature 300 K
Specific Heat 383 J/kgK
Strength Steinberg-Guinan
Shear Modulus 47.7 GPa
Yield Stress 0.12 GPa
Maximum Yield Stress 0.64 GPa
Hardening Constant 36
Hardening Exponent 0.45
dG/dP 1.35
dG/dT -0.01798 GPa/K
dY/dP 0.003396
Melting Temperature 1790 K
Failure Principal Stress
Principle Tensile Stress 3 GPa
Erosion Geometric Strain
Erosion Strain 2.00
Type of Geometric Strain Instantaneous
79
described in Section 4.3, the Hugoniot corresponding to the ideal elastoplastic response
was constructed by adding a constant offset of 2Y/3 from the hydrostat calculated from
the experimental data. This ideal elastoplastic Hugoniot was used as the EOS input in
the Autodyn model. While ANSYS does not make the source code for the finite element
calculation available for modification, one can create user-defined subroutines to define
an EOS, strength model or failure criterion. In order to simulate ideal elastoplastic wave
profilesforcomparisonwithexperimentalones,thecalculatedHugoniotwasincorporated
as the material EOS through a modification of the polynomial EOS in Autodyn. The
polynomial EOS in Autodyn is a general form of the Mie-Gruneisen EOS and defines the
pressure as the following [1]:
p = A
1
µ +A
2
µ 2
+A
3
µ 3
+Γ
0
p
0
e (5.11)
In Equation (5.11) above, µ is the density compression defined by ρ/ρ
0
− 1 where ρ
0
is the ambient density and ρ is the density in the shock compressed state, Γ
0
is the
Gruneisen parameter at ambient conditions, and e is the specific internal energy. Here,
it is assumed that the product Γρ remains a constant which is equal to the product of
the Gruniesen parameter and density at ambient conditions, Γ
0
ρ
0
. The user-subroutine
that was created as part of this work, defines the polynomial EOS in the following form
of the Mie-Gruneisen EOS:
p =p
H

1−
Γ
0
µ 2

+Γ
0
ρE (5.12)
In Equation (5.12) above, p
H
is the pressure on the Hugoniot, and p is the pressure in
an off-Hugoniot state. Here we substitute into p
H
the calculated Hugoniot obtained in
Section 4.3, which we define as:
p
H
=p
hydrostat
+
2Y
3
(5.13)
Here, we define p
hydrostat
in terms of the density compression, up to third-order terms in
density compression, as follows:
80
Table 5.2: Material properties used in Autodyn Model for XS-1 and SAM2x5
XS-1 SAM2x5
Equation of State User-Defined Shock User-Defined Shock
A
1
168.78 GPa 200.07 GPa
A
2
253.68 GPa -151.04 GPa
A
3
-112.87 GPa 85.98 GPa
Gruneisen Parameter 1.52 1.77
Strength Von Mises Von Mises
Shear Modulus 90.86 GPa 106.88 GPa
Yield Stress 5 GPa 7.8 GPa
Failure Principal Stress Principal Stress
Principle Tensile Stress 1 GPa 1 GPa
p
hydrostat
= A
1
µ +A
2
µ 2
+A
3
µ 3
(5.14)
In Equation (5.14), the terms A
1
, A
2
, and A
3
are the constants obtained following the
calculation of the mean stress from Equation (4.24). A
1
is the bulk modulus of the
material. The mean stress or hydrostat thus obtained is substituted in Equation (5.13)
which is in turn substituted into Equation (5.12) in the user-defined subroutine. The
yield strength specified in the Von Mises model is that calculated from the measured
HEL using Equation (2.30). An estimated value of the tensile strength is defined as
the failure limit in the failure criterion. A summary of the material parameters used to
simulate SAM2x5 and XS-1 is presented in Table 5.2. Each of the experiments presented
in Tables 4.4 were simulated using the model parameters presented in this section.
5.3 Simulation Results
Figures 5.2 and 5.3 show a comparison of simulated shock wave profiles (dashed) and
experimental wave profiles (solid). In addition, Tables 5.3 and 5.4 present a comparison
of simulated and experimental elastic and plastic shock velocities U
sh,el
and U
sh,pl
, and
HEL andpeak state particle velocities u
HEL
andu
peak
respectively. It canbeen seen that
there is good overall qualitative agreement in the structure of the waves, as well as in
81
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35
0
200
400
600
800
1000
1200
Time(μs)

Free surface velocity (m/s)
1300−exp
1100−exp
600−exp
1100−sim
1300−sim
600−sim
Figure 5.2: A comparison of experimentally obtained wave profiles (solid lines) and Autodyn
simulated wave profiles (dashed lines) for XS-1. Simulated profiles have been overlapped onto
measured ones for ease of comparison.
the HEL and peak states.
SimulatedwaveprofilesinFigures5.2and5.3havebeenshiftedintimesuchthatthatthe
foot of the steeply rising part of the elastic wave coincides with that in the experimental
wave profiles for ease of comparison. While the simulations capture the two-wave struc-
ture of the profiles, certain finer signatures in the wave profiles such as stress relaxation
following the elastic limit in the SAM2x5 samples, and the apparent hardening follow-
ing yield in the XS-1 samples, have not been captured. Moreover, a lack of consistency
in the appearance of these features amongst various samples added to the challenge of
incorporating these behaviors in the Autodyn model in order to simulate them. Further
characterization of the samples, through both quasi-static and dynamic mechanical ex-
perimentation, will help in the determination of parameters that govern these features.
In addition, characterization of the defects within the samples would yield parameters
82
0 0.1 0.2 0.3 0.4 0.5
0
200
400
600
800
1000
Time(μs)

Free surface velocity (m/s)
900−exp
700−exp
1000−exp
700−sim
900−sim
1000−sim
Figure 5.3: A comparison of experimentally obtained wave profiles (solid lines) and Autodyn
simulated wave profiles (dashed lines) for SAM2x5. Simulated profiles have been overlapped
onto measured ones for ease of comparison.
Table 5.3: Comparison of experimental and simulated shock velocities
Experiment Simulation Experiment Simulation
Sample U
sh,el
(km/s) U
sh,el
(km/s) U
sh,pl
(km/s) U
sh,pl
(km/s)
XS1-J 6.20 6.44 5.09 5.35
XS1-H 6.20 6.49 4.72 5.32
XS1-M 6.40 6.44 5.05 5.06
SAM-O 6.53 6.73 4.15 4.83
SAM-R 6.59 6.73 4.82 4.91
SAM-V 6.88 6.74 4.69 4.92
83
Table 5.4: Comparison of experimental and simulated particle velocities
Experiment Simulation Experiment Simulation
Sample u
HEL
(m/s) u
HEL
(m/s) u
peak
(m/s) u
peak
(m/s)
XS1-J 167 171 590 612
XS1-H 182 171 485 511
XS1-M 181 171 251 262
SAM-O 242 239 306 308
SAM-R 242 239 409 408
SAM-V 189 239 439 448
relevant for simulating variations in elastic limit, such as the one seen for SAM-V. A
discussion of the effect of defects on the HEL is presented in Section 4.3.3.
In some cases, simulated wave profiles showed a slightly higher peak state than the ex-
perimental ones, as is especially apparent in the simulated cases for XS1-J and XS1-H.
Although these were within 5 % of each other, the higher peak state could possibly be
a result of the slightly higher simulated elastic shock velocities, which create a higher
acoustic impedance for the simulated material. However, in general, the elastic shock
velocities obtained from simulations are consistent to within 5 % of ultrasonic measure-
ments of ambient sound speeds, and the elastic moduli of the samples, as well as most
experimental data points obtained at both Pullman and Caltech. In the cases where
the simulated elastic shock speed is closer to the experimental one, the peak states also
seem to have better agreement, as in XS1-M and SAM2x5 samples. In addition, the sim-
ulated wave profiles assume homogeneous, isotropic material response and do not take
into account defects or heterogeneities which result in dissipation and therefore lower
peak Hugoniot states, and this may likely account for the discrepancy in elastic shock
speeds. In general, the simulated elastic shock speeds lie within the range of measured
elastic shock velocities.
In general, the simulated plastic shock velocities, which are a result of the ideal elasto-
plasticprediction, arehigherthanexperimentallymeasuredones. Thisisexpected, based
84
on the results presented in Figures 4.12 and 4.13 in Section 4.3, since both sample types
undergo post-yield strain-softening. However, the difference in simulated and experi-
mental plastic shock speeds is especially higher for samples XS1-J and XS1-H, which are
seen to catastrophically lose shear strength and approach the isotropic stress state on
the hydrostat in Figure 4.12. Therefore, simulations based on elasto-plastic predictions
provide shock wave profiles for ideal elasto-plastic response, and are in keeping with ob-
servations of experimental results, and are further evidence of the post-yield strength
loss trends shown by each of the amorphous steel composites. Moreover, the agreement
between simulated and experimental wave profiles also demonstrates the appropriateness
of the Von Mises strength model in describing the yield behavior of these metallic glass
composites, thereby eliminating theneed toincorporatetheeffect ofhydrostatic pressure
on strength as observed in other works which are discussed in Section 1.1.1.
85
Chapter 6
Dynamic Fracture Morphology of a
Zr-based BMG
This chapter describes the results of a study on the effect of varying strain rate on the
fracture morphology of a commonly studied Zr-based metallic glass known as Vitreloy
106, in a previously unexplored range of dynamic strain rates of approximately 10
3
−10
4
s
−1
using the planar plate impact technique [74]. Studies of the morphology were per-
formed through a post-mortem microscopic examination of fracture surfaces of recovered
failed samples.
6.1 Motivation
Bulk metallic glasses (BMG) are a novel class of materials with mechanical properties
that are superior compared to their crystalline counterparts. Their high strength, hard-
ness, corrosion and wear resistance make them attractive fora wide range ofapplications
from athletic equipment to kinetic energy penetrators [68]. As such, there is a need to
characterize their mechanical response over a broad range of loading rates. In general,
three types of testing regimes, each requiring experimental techniques unique to that
regime, are needed in order to achieve loading rates between 10
5
− 10
6
s
−1
; in order
of increasing loading rates those include: quasi-static (servo-hydraulic machines), dy-
86
namic (Taylor anvil tests, Hopkinson bar) and shock-loading tests (normal plate impact,
pulsed laser, explosives). To date, most studies on the mechanical response of BMG
have been performed in the 10
−5
−10
3
s
−1
range of loading rates by using either quasi-
static tension/compression tests or dynamic tests using the Split Hopkinson Pressure
Bar (SHPB) [19,44,56,70,72,75,81,83,87,89,90,100,129–131,147–150]. Additionally,
shock loading experiments using the plate impact technique have been performed in the
range of 10
5
−10
6
s
−1
[32,67,109,136,153,160,161]. However, findings on the strain rate
dependence of the strength and fracture morphologies of BMG have been conflicting and
to the best ofourknowledge no such study has been performedin the intermediate range
of 10
3
−10
4
s
−1
loading rate using the plate impact configuration.
While several authors reported that the effect of strain rate on the strength of BMG un-
der dynamic loading conditions is negligible, [19,89,91,129,131] some observed positive
strain rate sensitivity [72,87] and others reported observing negative strain rate sensitiv-
ity of BMG [44,81,90,100,130,148,149,160]. Among those who reported a lack of strain
rate sensitivity, some observed that the fracture surfaces of BMG specimens damaged
fromloadingatincreasing strainratesexhibitedsimilarfracturefeatures[131],whileoth-
ers observed varying fracture morphologies indicating that different damage mechanisms
dominate the fracture process at different strain rates [70,83,147,148,150,160], suggest-
ing the dependence of fracture morphology on strain rate. It has also been suggested
that the strain rate dependence of deformation behavior in BMG varies with material
composition, specimen shape and loading procedure [22,81,87].
In this work, Zr-based BMG samples with nominal composition
Zr
58.5
Cu
15.6
Ni
12.8
Al
10.3
Nb
2.8
, also known as Vitreloy 106a, were subjected to plate im-
pact experiments to determine the effect of varying loading rate and stress on observed
fracturefeatures. Inordertoelucidate onthedynamicbehavior atanintermediate range
of loading rates, the impact velocities from the single-stage gas-gun tests ranged from 60
to 150 m/s which are equivalent to loading rates of the order 10
3
−10
4
s
−1
. Specifically,
87
these tests are performed in a regime not previously explored using the plate impact
experimental technique.
6.2 Experimental Method
The Vitreloy 106a sample was prepared by vacuum arc melting a mixture of high-purity
metals inanargonatmosphere andthencasting into coppermolds to make a rectangular
slab ofthe BMG [13]. Cylindrical samples, 18.75 mmin diameter, and3 mm inthickness
were cut using electric discharge machining. Samples were polished using 400-grit SiC
paper in order to remove a thin crystalline surface layer formed due to interaction with
the mold. X-ray diffraction of as-received sample confirmed its amorphous nature. The
samples, which were backed by a 3 mm thick tungsten base plate, were then impacted
by projectiles - comprising of a tungsten flyer 22 mm in diameter, 13 mm in thickness
embedded in a Delrin sabot 50 mm in diameter, 25 mm in thickness - launched from a
single stage gas gun. A schematic of the experimental set-up is shown in Figure 6.1. Ve-
locity ofthe projectile is measured by two receivers (Avago Technologies HFBR-2505CZ)
mounted 10 cm apart on the gun barrel.
A total of six cylindrical samples were impacted in two sets of experiments: the first in
which three samples were impacted with flyers of different materials at the same impact
velocity to determine the effect of varying stress amplitude, and the second in which
three additional samples were impacted using a tungsten flyer with increasing velocities
to ascertain the effect of varying impact velocity, and therefore loading rate. For all the
experiments performed, theinitialimpactstress intheBMGsamplewascalculatedusing
the impedance matching technique, which has been derived elsewhere [118]. Using the
second Rankine-Hugoniot jump condition, the stress in the flyer may be expressed as
follows
P
F
= ρ
0F
(C
0F
+s
F
u
pF
+V)(V −u
pF
). (6.1)
88
1 2
3
4
5
Figure 6.1: Schematic of experimental set-up comprising projectile-target assembly as mounted
on gun barrel: (1) Delrin sabot, (2) Tungsten flyer plate, (3) BMG sample, (4) Tungsten base
plate and (5) Polycarbonate holder.
Here,P isthestress, ρ
0
isthedensityandC
0
thebulksoundspeedatambientconditions,
u
p
is the particle velocity, s is the parameter used to express the shock velocity in terms
oftheparticlevelocity asC
0
+su
p
, andV istheimpactvelocity. Thestress inthesample
may also be expressed in a similar manner.
P
S
= ρ
0S
C
0S
u
pS
+ρ
0S
C
0S
u
pS
2
(6.2)
The subscripts F and S denote flyer and sample respectively. At the moment of impact,
traction and velocity continuity require that P
F
and P
S
are equal and u
pF
and u
pS
are
equal to each other. Equations 6.1 and 6.2 are therefore set equal to each other, and by
solving a quadratic equation in u
p
, the stress at the flyer-sample interface at impact is
obtained. The initial impact stress is calculated in this manner and is summarized in
Table 6.1.
89
Flyer Material Impact Velocity (m/s) Initial Impact Stress (GPa)
W 62± 0.4 1.37
W 92± 1.1 2.10
W 110± 1.4 2.51
W 150± 2.0 3.43
Al 2024 60± 0.3 0.65
SS 304 60± 0.1 1.05
Table 6.1: Summary of flyer materials used, impact velocities and loading stresses generated in
plate impact experiments on Vitreloy 106a.
6.3 Results and Discussion
The impacted samples and their fragments were recovered from the target chamber fol-
lowing impact, and their fracture surfaces were examined using JEOL JSM-7001 SEM.
Forall the samples impacted, bright flashes of visible light were seen upon fracture. This
wasalsoobservedbyothersandwasnotedasbeingsuggestiveoflocalmeltingonfracture
of the BMG [41,56,81].
ThemicrographsinFigure6.3showthefracturesurfacesfromsamplessubjected tovary-
ing amplitudes ofloading stress frombeing impacted by flyer plates ofdifferent materials
-aluminum 2024, stainless steel 304 and tungsten- at impact velocities 60.67±0.26 m/s.
For the sample impacted with aluminum, as seen in Figure 6.3(a), the fracture surface
shows dimpled patterntypical ofductilefracture. The sample impacted bystainless steel
shows a similar fracture surface; however, there also seems to be a dense accumulation of
melted droplets and numerous cracks. As seen in Figure 6.3(c), the sample impacted by
tungsten, however, seemstohaveafracturesurfacethatlooksverydifferentfromthepre-
vioustwo i.e. nodimple patterns. Insteadthereisanevenly distributed patternofrivers,
as well as some cusps and smooth areastypical ofbrittle, BMG failure. This is indicative
ofthenatureofdeformationturning brittlefromductileastheimpactstress isincreased.
Fracture surfaces of samples impacted by tungsten flyers at varying loading rates show a
90
Figure 6.2: Scanning electron micrographs of the fracture surfaces of impacted Vitreloy 106a
samples showing fracture surface from impact with a tungsten flyer at (a) 110 m/s b) 150 m/s.
Micrographs for each of these impact conditions show different fracture features.
combination of different fracture patterns (rivers, veins, smooth areas, droplets etc), as
is apparent from Figure 6.3(c), since final fracture likely occurs due to different modes at
different locations [98]. This occurrence of various fracture patterns for a given impact
condition is also apparent in Figure 6.2. However, at each loading rate, one dominant
fracture pattern which covered most of the fracture surface was observed, while other
features were distributed over smaller areas. The dominant fracture pattern correspond-
ing to each loading rate was studied further at higher magnification and is discussed in
the following paragraph.
Themicrographs inFigure6.4depict fracturesurfaces fromsamples subjected atvarying
loading rates from being impacted by a tungsten flyer at velocities of 62 m/s, 92 m/s,
110 m/s and 150 m/s. For the sample impacted at 62 m/s, as seen in Figure 6.4(a),
river-like patterns cover most of the fracture surface. This is typical of fracture surfaces
of BMG [75,149,150]. It is observed that rivers are also interspersed with occasional
smooth regions i.e. areas of fast catastrophic failure, and cell-like veins or cusps. River
patterns arise out of the flow of low viscosity BMG from localized adiabatic heating.
When the low-viscosity material separates in the shear slip bands as the BMG deforms,
pores are opened up, resulting in the formation of cell-like vein morphology [19,98]. The
fracturesurfaceofthesampleimpactedat92m/sshowed beltsofmeltedBMG,river-like
91
veins and arrays of cusps along with resolidified melted droplet particles amidst them.
Cusps, as seen in Figure 6.4(b), form in the direction of shear force and are evidence of
plastic flow in the direction of shear. The sample impacted at 110 m/s depicts arrays
of cusps, in Figure 6.4(c), with the overall fracture surface being more inhomogeneous
than previous ones. The cusps are lined by coarser cell walls and contain resolidified
droplets. Regions of cusps bordered by melted bands indicate areas of crack growth
and propagation. BMG impacted at 150 m/s shows patterns of rivers and cusps that
are flanked by large melted belts, as seen in Figure 6.4(d). The observed morphology
fromthese samples damaged atfourdifferent loading rates shows varied fracturefeatures
indicating that damage mechanisms are strain rate dependent. Increasing amounts of
melted droplets and zones on fracture surfaces of samples impacted at higher velocities
is evidence of higher temperatures being reached at high strain rates [160]. The X-ray
diffraction patterns of the as-received and fractured samples, as seen in the inset in Fig-
ure 6.4, confirm the amorphous structure of the as-received sample as well as that of the
impacted sample fragments.
Micrographsofsamples impactedwithtungstenatincreasing velocities, andthereforein-
creasing loading rates, show that the deformation becomes increasingly inhomogeneous,
suggesting a dependence of fracture mechanisms and morphology on strain rate. There
also seems to be an increased level of melting, suggesting that there is a greater release
of fracture energy resulting in higher temperatures, with each subsequent velocity, as
seen from increasing amounts of resolidified particles and viscous, melted belts. Frac-
ture morphologies of samples impacted at the lower loading rates (corresponding to the
lower impact velocities of 62 m/s and 92 m/s), show evidence of shear fracture being
the dominant mode of failure through the presence of cleavage river and veins, which
are formed from the pulling apart of low viscosity material. As the loading rate is in-
creased (corresponding to increased impact velocities of 110 m/s and 150 m/s), there is
not enough time for heat generated from friction effects during the shear fracture process
to be sufficiently conducted through the material. Therefore, there is a large agglomera-
92
Figure 6.3: Scanning electron micrographs of the fracture surfaces of impacted Vitreloy 106a
samples showing (a) Typical ductile fracture surface on sample impacted by Aluminum, (b)
Ductile fracture surface containing micro-steps on sample impacted by stainless steel, and (c)
Pattern of river and cusp veins evenly distributed across sample impacted by tungsten.
93
tion of melted droplets and bands. At the higher loading rates, temperature and melting
effects are more pronounced in the failure process than at low loading rates where shear
fracture is the main mode. Micrographs of samples impacted at 60.67±0.26 m/s with
different flyer materials, resulting in varying shock amplitudes, show that with increas-
ing amplitude, the nature of damage from compression changes from ductile to brittle.
The morphology of samples impacted at lower stress exhibit ductility due to failure from
multiple shear bands. As the shear band formed along the principal shear plane propa-
gates, secondary shear bands are formedfromgrowth andcoalescence ofmicrovoids [160]
resulting in the accumulation of plastic strain thus lending ductility to the material. At
the higher stresses, and therefore higher loading rates, there is not enough time for voids
to coalesce and form new shear bands, and therefore failure occurs along a single shear
band resulting in brittle behavior.
The presented results, obtained under low-velocity impact loading conditions using the
plate impact technique at previously unexplored intermediate levels of strain rate (10
3
−
10
4
s
−1
), show a dependence on the applied loading rate resulting from the occur-
rence of different fracture mechanisms at varying levels of loading rate. This work in-
dicates that there is a strain rate sensitivity in Vitreloy 106 at intermediate loading
rates, in agreement with some of the previous observations at lower and higher strain
rates [70,83,147,148,150,160]. Therefore, the strain rate sensitivity in this particular
BMG appears to be independent of loading procedure. Further investigations of other
BMGs over a wide range of loading rates and stresses will allow the study of possible
transitions regions for the mechanical deformation, thus opening the possibility for new
applications.
94
Figure 6.4: Scanning electron micrographs of the fracture surfaces of impacted Vitreloy 106a
samples showing (a) River-like veins on sample impacted at 62 m/s, (b) Cusps bordering river-
like veins on surface from impact at 92 m/s with arrow indicating shearing direction, (c) Pattern
ofcusps indisarray onfracture surface from impact at110 m/s, (d) Large meltedbelts on sample
impacted at 150 m/s. (Inset) X-ray diffraction plots of as-received Vitreloy 106a sample, and
damaged samples impacted with tungsten flyer at 62 m/s, 92 m/s, 110 m/s and 150 m/s.
95
Chapter 7
Conclusion and Future Direction
Through this dissertation, the shock wave response of two novel iron-based metallic glass
matrix composites has been explored. The main contributions and conclusions of this
work are listed as follows:
• Shock compression experiments have been conducted for the first time on an amor-
phous steel, and the measurement of particle velocity and shock velocity have been
made using velocity interferometry techniques. These enabled obtaining not only
the wave profiles, which contain several distinct signatures of the high strain rate
mechanical response of the metallic glass specimens, but also the calculation and
construction of Hugoniots for the materials examined.
• A comparison of the shock response of an iron-based metallic glass and its in
situ matrix composite in order to investigate the role of partial devitrification on
mechanical response under uniaxial strain loading has been conducted for the first
time.
• The Hugoniot Elastic Limit (HEL) was calculated from measured wave profiles,
and while that of XS-1 (X-ray amorphous samples), in the range of 8 - 9 GPa, is
comparableto onesobtainedpreviously forZr-basedamorphousalloys [65,104,107,
136–139,146],SAM2x5 (partiallydevitrified samples) hasanHELof10 -12.5GPa,
which is nearly 1.5 times that of previously studied Zr-based compositions. A high
96
elastic limit indicates a large ability for energy absorption and makes this material
an attractive candidate for ballistic and armor applications.
• Despite being significantly different in composition from metallic glasses that have
been studied until now under shock compression, the iron-based alloys of this work
demonstrateverysimilarshockbehavior, namelylargeamplitudeelasticwaves (the
exact magnitude of the elastic wave amplitudes differs) and significant loss of post-
yield strength. As was suggested in a previous work [65], there appears to be a
universality in the shock compression response of metallic glasses that show strong
but brittle behavior under quasi-static conditions. Further research will have to
be undertaken to understand the underlying mechanisms responsible for seemingly
unifying metallic glasses in their shock wave response.
• A comparison of the shock response of each of the sample types to their calculated
ideal elasto-plastic response (using the constant shear modulus assumption) shows
that both XS-1 and SAM2x5 possess higher post-yield strength than a Zr-BMG as
well as a Hf-Zr-BMG from previous studies [65,139]. However, at higher loading
stresses, XS-1 seems to lose most of its strength reducing to a hydrostatic stress
state, as do previously studied alloys. However, SAM2x5 samples seem to retain
large post-yield strength even up to density compressions of nearly 10 percent.
Therefore, the presence of products of devitrification, although small in volume
fraction (on the order of 1 - 2 percent), plays a significant role in strengthening
the material (as evidenced by its larger HEL) as well as lending it more ductility
(from retaining shear strength beyond the HEL). This is likely a result of the high
hardnessandstrengthofdevitrificationproductssuchasFe
23
B
6
aswellasthearrest
of shear bands by clusters of nanocrystallites. Therefore, it has been demonstrated
that the extent of devitrification is an important adjustable parameter to tune the
mechanical response of the material as desired.
• Lastly, numericalsimulationsusing acommercialhydrocodewerealsoperformedto
generate ideal elasto-plastic wave profiles for comparison with measured ones. The
97
applicability of the Von Mises strength criterion for predicting yield under shock
compression has been successfully shown. Further evidence for post-yield strength
loss was obtained from simulated wave profiles in the faster shock speeds of the
simulations compared to experiments.
In conclusion, the amorphous steel studied here is seen to show similar overall shock
response as previously studied amorphous Zr-based compositions. However a minor ad-
dition of nanocrystallinity results in a significant improvement in yield strength and
post-yield shear strength retention. These amorphous steels have also been shown to be
superior in their elastic limit and strength compared to stainless steel 304 which has an
HEL of 0.35 GPa with material strength only accounting for 2 - 3 percent of the total
stress at stresses above 25 GPa [31]. In contrast, material strength of SAM2x5 is nearly
15 percent of the total stress at an impact stress of 20 GPa as can be seen in Figure 4.13
and 4.15. Also, unlike a previous work on Zr-BMG Vitreloy 1 and its in situ dendritic
phase composite, where no difference was seen in the shock response of the monolithic
BMG and its composite [161], the amorphous steels of this dissertation show a signifi-
cant difference in response based on the extent of devitrification. This work therefore
reveals in situ reinforced metallic glass composites as promising candidates for use in ap-
plications such as: armor, turbine blades, protective shields for electronic devices, sports
equipment such as clubs and racquets, precision tooling, and biomedical devices among
many other potential uses.
However, several avenues of future research still remain. A few recommendations for
further work on this topic are listed as follows:
• This work only involved shock compression experiments, and the response of the
material to dynamic tension or spall was not explored. Spall experiments are de-
signed such that the shock reflecting off the free surface of the sample and the
rarefaction wave from the back surface of the impactor collide at roughly the cen-
terofthesamplecreatingdynamictension. Whencoupledwithaset-upthatallows
98
for recovery of deformed specimens in order to conduct post-mortem examination
of failed samples, it is known as a spall recovery experiment. Several such works
on Zr and Ti-based BMGs have been performed [8,32,54,136,161]. A next step in
this work would be to conduct spall recovery experiments in a similar manner to
ascertain damage mechanisms from a microscopic examination of the spalled sur-
face morphology. This would help shed light unambiguously on exactly what the
operating mechanisms are through which in situ reinforcements behave within the
amorphous matrix in a metallic glass composite when subjected to shock loading.
• While a thorough characterization of the ambient mechanical properties of both
XS-1 and SAM2x5 was performed, no quantification or characterization of the de-
fects or porosity of the samples was conducted. Brittle solids, such as the metallic
glassesstudiedhere,inherentlycontainflawssuchascavities, inclusionsandvarious
other inhomogeneities [71]. These play a very important role in the shock behavior
as well as overall mechanical response of the material, and are often responsible for
variation in response among samples. Cavities or voids disperse shock fronts, and
act as sites where the energy of the shock is focused resulting in localized heating
andsubsequent melting. Localtensilestresses mayalso arisefromstress concentra-
tions around material heterogeneities. Randomly oriented defects coalesce under
applied shear stresses to form bands of intense localized deformation. Continuum
mechanical modeling, such as the one described in Section 5.3 is inadequate in
capturing all these aspects of the mechanical response (such as stress relaxation
following yield, apparent hardening behind the precursor wave, strain softening
and material degradation etc) and therefore, a close examination of the processes
occurring at the mesoscale is necessary. Thorough microscopic examination of the
microstructure and subsequent reproduction of the features of the microstructure
in a finite element model, along the lines of the mesoscale simulations which cap-
ture the role of plasticity of grains and collapse of pores in the deformation of
alumina [17], would be useful in determining mechanistic phenomena behind the
shock response of MGMCs.
99
• A systematic study of composites subjected to controlled devitrification resulting
in varying volume fractions of nanocrystallinity would be useful to determine the
microstructure that is optimal for high strain rate mechanical performance. In a
previous work, increasing devitrification of an amorphous steel was found to result
in embrittlement [21]. In another work, only a minor addition of 4 percent volume
fraction Niobium to a Ti-based BMG composition, which was found to promote
nanocrystallization,resultedinsignificantenhancementinplasticity,hardness,elas-
tic moduli and corrosion properties [39]. Therefore, samples with varying levels of
nanocrystallization must be further experimented with to determine best available
mechanical properties.
100
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Abstract (if available)

Abstract Bulk metallic glasses (BMG) have recently garnered interest due to superior properties such as higher strength, toughness and hardness, arising out of the amorphous struc- ture of these metallic alloys, as compared to their crystalline counterparts. However, BMGs are brittle and fail catastrophically following their elastic limit, which severely restricts their use in structural applications. To offset their brittleness, studies of various combinations of hard nano/micro particles, in situ precipitated crystalline phases and fibers embedded within the BMG, exist in the literature. These resulting materials are known as metallic glass matrix composites. In this work, we study the high strain-rate response of two novel Fe-based metallic glass matrix composites, both of the same composition Fe49.7Cr17.7Mn1.9Mo7.4W1.6B15.2C3.8Si2.4, containing varying amounts of in situ crystalline phases, when subjected to shock compression. Shock response is determined by making velocity measurements using interferometry techniques such as the Velocity Interferometer System for Any Reflector (VISAR) and Photonic Doppler Velocimetry (PDV) at the rear free surface of BMG samples, which have been subjected to impact from a high-velocity projectile launched from a powder gun. Plate impact experiments have yielded repeatable results indicating a Hugoniot Elastic Limit (HEL) to be 12.5 GPa and 8 GPa respectively for the two composites. The former HEL result is higher than elastic limits for any BMG reported in the literature thus far. The effect of partial crystallization in the amorphous matrix of BMG on the observed shock response is fur- ther explored through a comparison of the results from both composites. It was found that the presence of products of devitrification, although small in volume fraction, plays a significant role in strengthening the material (as evidenced by its larger HEL) as well as lending it more ductility (from retaining shear strength beyond the HEL). This is likely a result of the high hardness and strength of devitrification products such as Fe23B6 as well as the arrest of shear bands by clusters of nanocrystallites. Therefore, it has been demonstrated that the extent of devitrification is an important adjustable parameter to tune the mechanical response of the material as desired. ❧ In addition, the sensitivity of the fracture morphology of a Zr-based BMG Vitreloy 106 to strain rate is examined through a series of low-velocity impact experiments using a single-stage gas gun. Post-mortem microscopic examination of the fracture surfaces of the retrieved failed specimens was conducted. The dynamic fracture morphology for the Zr-BMG showed a clear strain-rate dependence in the form of various unique features at the micro-scale resulting from the occurrence of different fracture mechanisms at varying levels of loading rate.

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Creator Khanolkar, Gauri Rajendra (author)

Core Title Shock wave response of in situ iron-based metallic glass matrix composites

School Viterbi School of Engineering

Degree Doctor of Philosophy

Degree Program Mechanical Engineering

Publication Date 09/15/2015

Defense Date 08/07/2015

Publisher University of Southern California (original), University of Southern California. Libraries (digital)

Tag amorphous metals,bulk metallic glasses,high strain rate loading,mechanics of materials,OAI-PMH Harvest,plate impact experiments,shock physics,shock wave compression

Format application/pdf (imt)

Language English

Contributor Electronically uploaded by the author (provenance)

Advisor Eliasson, Veronica (committee chair), Hodge, Andrea M. (committee member), Kassner, Michael E. (committee member)

Creator Email gaurikhanolkar@gmail.com,khanolka@usc.edu

Permanent Link (DOI) https://doi.org/10.25549/usctheses-c40-177897

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