Shock compression of water and
solutions of ammonium nitrate
Michael James Morley
Trinity College
&
Fracture and Shock Physics, Cavendish Laboratory
University of Cambridge
A Thesis Submitted for the Degree of Doctor of Philosophy
June 2011

SHOCK COMPRESSION OF WATER AND SOLUTIONS OF
AMMONIUM NITRATE
Michael James Morley
Modern mining explosives employ solutions of ammonium nitrate, where
the solution is the oxidising component of a fuel/oxidiser mixture. This the-
sis is primarily concerned with the shock response of water and of aqueous
solutions of ammonium nitrate. Of particular interest are the temperatures
induced through shock compression. An experimental facility, using a single-
stage gas gun in the ‘plate impact’ configuration, is described, along with asso-
ciated experimental diagnostics. Measurements of, and improvements to, the
tilt at impact are reported. The problem of shock temperature is discussed,
including a brief review of the relevant literature. It is demonstrated that di-
rect measurement of shock temperature is a complex issue that is not yet fully
understood, whereas determination of temperature from an equation of state
is an established technique.
In a series of experiments, plate impact techniques were utilised to deter-
mine the Hugoniot and, through shock/reload experiments, the equation of
state of water and aqueous solutions of ammonium nitrate. In-situ manganin
gauges were used to measure stresses in the liquids and, from the arrival
times of the shock wave, determine the shock velocity. Linear shock velocity–
particle velocity Hugoniots for the liquids were determined, up to particle
velocities of 1 km s−1, with uncertainties on the intercept and slope of these
Hugoniots of 5 %. A Mie-Gru¨neisen equation of state was used to describe the
shock/reload experiments. Approximate calculations of shock temperature
are reported. Increasing ammonium nitrate concentrations resulted in greater
calculated temperatures. It was demonstrated that the liquids investigated
in this thesis show a temperature dependence of the Gru¨neisen parameter, Γ,
which cannot be accommodated in the model. The present work is believed to
be the first demonstration of this effect in shock compressed liquids. The data
presented provide constraints on future theoretical development of equations
of state.

RELATED PUBLICATIONS
Some of the results presented in Chapter 4 were presented at the Sixteenth
Biennial International Conference of the American Physical Society Topical
Group on Shock Compression of Condensed Matter:
M. Morley, D. J. Chapman, and W. G. Proud. Temperature mea-
surements at material interfaces with thin-foil gauges. In Elert,
M. L. and Butler, W. T. and Furnish, M. D. and Anderson, W. W.
and Proud, W. G., editors, Shock Compression of Condensed Matter —
2009, pages 599–6014. American Institute of Physics, 2009.
Some of the results presented in Chapter 5 have been used in the development
of models of shock wave structure:
K. B. Jordan Direct Numeric Simulation of Shock Wave Structure with-
out the Use of Artificial Viscosity. PhD thesis. Marquette University,
2011.
Some of the results presented in Chapters 5 and 6 were presented at the Six-
teenth Biennial International Conference of the American Physical Society
Topical Group on Shock Compression of Condensed Matter:
M. Morley, D. J. Chapman, and W. G. Proud. Equation of state data
from off-Hugoniot measurements in multiply-shocked liquids. In
Elert, M. L. and Butler, W. T. and Furnish, M. D. and Anderson, W.
W. and Proud, W. G., editors, Shock Compression of Condensed Matter
— 2009, pages 560–563. American Institute of Physics, 2009.
Some of the results presented in Chapters 5 and 6 will be presented at the Sev-
enteenth Biennial International Conference of the American Physical Society
Topical Group on Shock Compression of Condensed Matter:
M. Morley and D. M. Williamson. Shock/reload response of water
and aqueous solutions of ammonium nitrate. In Shock Compres-
sion of Condensed Matter — 2011. American Institute of Physics, in
preparation.

DECLARATION
This thesis is the result of my own work and includes nothing
which is the outcome of work done in collaboration, except where
noted in the text. It is not substantially the same as any that I have
submitted or will be submitting for a degree or diploma or other
qualification at this or any other University and satisfies the word
limit prescribed by the Faculty Board of Physics and Chemistry.

ACKNOWLEDGEMENTS
This work has been supported financially by Orica Mining Services and I
would like to extend my thanks to Richard Goodridge and the team at Kurri
Kurri for their assistance and hospitality. In particular, I am grateful to Drs.
Jim Chan, Ian Kirby, and Alan Minchinton for their advice and words of wis-
dom. I am grateful for the supervision that has been afforded to me by Drs.
Bill Allison and David Williamson, and to Dr. Bill Proud for arranging the
funding of this project.
I must acknowledge Dr. David Chapman, now at the Institute for Shock
Physics, Imperial College, who early on pointed me in interesting directions.
I should also thank Dr. Chris Braithwaite, for his advice and assistance with
VISAR measurements, and David Bell, for his assistance with interferometeric
and optical diagnostics. To the others of the Fracture and Shock Physics Group
- Hannah Brierley, Adam Collins, Dr. Dan Drodge, Tuuli Gro¨hn, Simon Kirk,
Nick Taylor and Dr. Stephen Walley - who were always on hand for useful
advice or a pint, often simultaneously, thank you. I am particularly grateful
to Amy Adams, of the University of Florida, who helped me stay sane during
a difficult summer, and believed in me even when I didn’t.
Without the invaluable mechanical skills of Rik Balsod, Ray Flaxman, Dave
Johnson, Peter Norman, Nigel Palfrey, Chris Summerfield, and the other staff
of the Mott and Main Workshops, I doubt whether any of the experiments I
conducted would have progressed beyond a roughly-drawn sketch. I learnt
much from their advice and expertise, and owe them a debt of gratitude. In
the wider Cavendish, Chris, Jacqui, Jess, Lisa, and Susan - amongst others
- have made my time here far more enjoyable than it might otherwise have
been. Further thanks should be given to Drs. Bill Allison, Chris Braithwaite,
David Chapman, and Daniel Eakins (of the Institute of Shock Physics, Impe-
rial College) who kindly gave up their time to proof-read my thesis, and who
offered valuable and useful insights.
It would be remiss of me not to thank my parents who, years ago, started
me on the road that lead to this document and to whom I shall remain eter-
nally indebted.
Finally, and most importantly, I must thank Rachael, who has helped in
ways she can’t imagine, and who proof-read much of what follows. Without
her love and support, I doubt I could have found the motivation to finish this
thesis.
CONTENTS
Contents ix
1 Introduction 1
1.1 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2 Fundamentals of shock wave propagation 9
2.1 Planar longitudinal shock waves . . . . . . . . . . . . . . . . . . 10
2.2 Graphical representations . . . . . . . . . . . . . . . . . . . . . . 15
2.3 Ideal Elastic-Plastic materials . . . . . . . . . . . . . . . . . . . . 20
2.4 The Mie-Gru¨neisen Equation of State . . . . . . . . . . . . . . . . 21
2.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.6 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
3 Experimental techniques 29
3.1 The Cavendish plate-impact facility . . . . . . . . . . . . . . . . 29
3.2 In-situ stress measurements . . . . . . . . . . . . . . . . . . . . . 33
3.3 Particle velocity diagnostics . . . . . . . . . . . . . . . . . . . . . 39
3.4 Sample preparation and design . . . . . . . . . . . . . . . . . . . 40
3.5 Measurement of impact tilt in plate-impact experiments . . . . . 47
3.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
3.7 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
4 Shock temperature 61
4.1 Direct temperature measurements . . . . . . . . . . . . . . . . . 61
4.2 Indirect temperature measurements . . . . . . . . . . . . . . . . 70
4.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
4.4 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
5 Shock compression of water 95
5.1 Cell development . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
5.2 Pure water, ambient temperature . . . . . . . . . . . . . . . . . . 115
5.3 Pure water, elevated temperature . . . . . . . . . . . . . . . . . . 122
ix
5.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
5.5 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
6 Shock compression of ammonium nitrate solutions 137
6.1 Ammonium nitrate solutions . . . . . . . . . . . . . . . . . . . . 138
6.2 Principal Hugoniots . . . . . . . . . . . . . . . . . . . . . . . . . 139
6.3 Reload experiments . . . . . . . . . . . . . . . . . . . . . . . . . . 148
6.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156
6.5 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
7 Analysis and discussion 159
7.1 Relationships amongst the Hugoniots . . . . . . . . . . . . . . . 159
7.2 Functional dependencies of Gru¨neissen’s parameter . . . . . . . 166
7.3 Pathological behaviour of water . . . . . . . . . . . . . . . . . . . 171
7.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176
7.5 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177
8 Conclusions and future work 181
8.1 Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184
8.2 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186
A Selected gauge traces 189
A.1 50 % ammonium nitrate, 14± 1 ◦C . . . . . . . . . . . . . . . . . 191
A.2 50 % ammonium nitrate, 65± 2 ◦C . . . . . . . . . . . . . . . . . 193
A.3 80 % ammonium nitrate, 64± 2 ◦C . . . . . . . . . . . . . . . . . 195
CHAPTER 1
INTRODUCTION
Jones, Scott and Co. was formed over 130 years ago as a supplier of explosives
to the newly-discovered Victorian gold fields. Since then, first as part of No-
bel, then ICI, and now as Orica, the company has become one of the largest
producers of commercial explosives anywhere in the world [Orica, 2004] with
dozens of major manufacturing locations across the world (Figure 1.1).
In 2010, Orica Mining Services recorded annual earnings in excess of half
a billion pounds, with nearly two-thirds of this being generated from Aus-
tralian operations [Orica, 2010]. A large proportion of this revenue is gen-
erated through sales to coal mines in New South Wales, where the port of
Newcastle sees coal exports of over 90 million tons per annum [Port Waratah
Coal Services, 2011]. Affecting the underlying profitability of all mining op-
erations is, of course, the price of raw materials, i.e., of the explosives them-
selves. The production of ammonium nitrate produces significant amounts of
carbon dioxide; for environmental reasons, and under the continued threat of
the introduction of a carbon tax [Commonwealth of Australia, 2011], it would
be desirable to reduce Orica’s carbon footprint. One way to achieve this would
be simply to reduce the amount of ammonium nitrate incorporated in emul-
sion explosives. It is not immediately clear how much ammonium nitrate can
be removed whilst retaining the necessary explosive output. The experimen-
tal development and testing of a new explosive formulation is, however, an
expensive and time-consuming processs. A theoretical framework, enabling
prediction of the behaviour of novel explosives, is therefore desirable. In-
deed, accompanying the development of new explosives has been a develop-
ment of computational models - first in one-dimensional models [Chan, 1982;
Kirby and Chan, 1986], and then two-dimensionally [Chan, 2007; Kirby and
Chan, 2002]. Orica’s present model is implemented in the ‘Commerical Prop-
erties of Explosives’ (CPeX) code [Chan, 2011]. More recently, there has been a
move towards developing holistic models, that model the entirety of the blast-
ing process, from explosive behaviour, through explosive/rock interaction, to
fragmentation behaviour of the rock. One such approach is the De Beers-led
‘Hybrid Stress Blasting Model’ [Braithwaite, 2009]. Orica’s ‘Mechanistic Blast-
1
Figure
1.1:O
rica’s
m
ajorm
anufacturing
locations,adapted
from
O
rica
[2010].The
m
ap
does
notshow
sm
allerregionalm
anufacturing
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such
as
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R
oburite
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igan.
2
ing Model’, when combined with the CPeX model of detonation behaviour,
fulfils much the same role [Chan, 2011]. Predicting the behaviour of novel
emulsion compositions requires developing knowledge of the behaviour of
the constituent ingredients at the pressures, and over the time scales, that
would be found in a mine. Of particular interest are the temperature rises that
occur due to the passage of the shock, as reaction in ammonium nitrate is, to
a large extent, thermally activated. This thesis considers aqueous solutions
of ammonium nitrate. Using established shock compression techniques, the
high pressure (several giga-Pascal), dynamic (few micro-seconds) behaviour
of these solutions is investigated, in well-defined laboratory conditions. That
ammonium nitrate solutions can be considered explosives is a fact largely ne-
glected by the present work, which focusses on the behaviour of the unreacted
solutions.
It is both interesting and instructive to consider, from an historical view-
point, the development of the explosives that are used in the mining industry.
Gunpowder was discovered in antiquity in China, although it was not un-
til the seventeenth century, in Hungary, that it saw widespread (albeit rather
dangerous) use in mining [Dolan and Langer, 1997]. The development of dy-
namites - the first high explosive - by Alfred Nobel in the 1860s, together with
his invention of the blasting cap, or detonator, led to marked improvements
in safety. The first use of ammonium nitrate as a component in explosives
came at this time with its inclusion in dynamite formulations [Cook, 1974]. By
the 1960s, dynamites were being replaced in the mining industry by cheaper
explosives based around the mixture of granular ammonium nitrate with fuel
oil (ANFO). The properties of the granular ammonium nitrate used in ANFO
have recently been studied in this laboratory [Taylor, 2011]. ANFO delivers
a large amount of blasting energy whilst being cheap to produce, however, it
suffers a major drawback. Mines, by their very nature - as large holes in the
ground - tend to be damp environments, and often bore holes contain a large
amount of water. The addition of more than 10 % water to ANFO causes it to
fail to detonate. Ammonium nitrate is highly hygroscopic, and the addition of
fuel oil only enhances this unfortunate property [Hustrulid, 1999], preventing
the use of ANFO in damp bore holes. A solution to this problem came with the
development, first of ‘slurry’, and then, of ‘emulsion’ explosives [Cook, 1974;
Sudweeks, 1985]. In an ammonium nitrate emulsion explosive, a solution of
ammonium nitrate is mixed with a fuel to create a water-in-oil type emulsion.
Micrographs of two such emulsions are shown in Figure 1.2. Droplets of am-
3
monium nitrate solution are coated entirely in fuel, providing an intimate mix-
ing of fuel and oxidiser, and effectively water-proofing the explosive, through
the immiscibility of the oil phase with any surrounding water. Ammonium
nitrate emulsions are now widely used in the industry. Indeed, Orica are cur-
rently seeking to create a new emulsion manufacturing plant at their technical
centre in Kurri Kurri, near Newcastle [Orica, 2009].
Figure 1.2: Micrographs of two ammonium nitrate emulsions, from Sudweeks [1985].
The circular droplets are the ammonium nitrate solution, which is sur-
rounded by a continuous oil phase.
Galbraith [1997] reported four experiments conducted in the ‘ring-up’ con-
figuration to measure the principal Hugoniot of a variety of ammonium ni-
trate solutions, and one similar experiment on pure water. In this geometry,
the sample is confined between two high-impedance anvils and is thus subject
to the action of a number of shocks which ‘ring-up’ the stress in the the sample.
These reflected shocks result in states that do not lie on the principal Hugo-
niot of the sample. A given stress in the ‘ring-up’ geometry is reached for a
lower entropy rise than would be the case for a single shock wave. The lower
entropy rise results in a lower thermal pressure and, ultimately means that a
4
given stress in the ‘ring up’ geometry is reached at higher density and, hence,
lower particle velocity than the same stress on the principal Hugoniot. If an
equation of state is available, then the exact difference between the principal
Hugoniot and multiple shock state can be calculated. Applying an equation of
state for water [Gurtman et al., 1971], it can be shown that, for the experiment
on water reported by Galbraith, the first reflected shock state is at a stress ap-
proximately 2 % below the Hugoniot. It can be assumed that the ammonium
nitrate solutions show a similar magnitude of displacement between the prin-
cipal Hugoniot and first reflected shock state. Galbraith comments that this
displacement is within the experimental uncertainty of the diagnostics and,
thus, can be neglected. Galbraith assumed that subsequent states in the ‘ring-
up’ lie similarly below the Hugoniot. However, it can also be shown from the
equation of state that these states lie progressively further below the Hugo-
niot, with the difference in some of the stresses reported by Galbraith being as
much as 10 %. Hence, the Hugoniots reported by Galbraith underestimate the
Hugoniot stress in the liquid by as much as 10 %. An additional complication
arises from the specific diagnostic employed by Galbraith. In order to mea-
sure the longitudinal stress in the liquid, manganin gauges were employed.
The particular gauges in use at the time had ‘legs’, which extend out of the
target enabling electrical contacts to be made, that are also made from man-
ganin. The measured resistance change is, therefore, dependent not only on
the stress experience by the gauge grid, but also by the pressure experienced
by the legs, which may have a significant deviatoric component (i.e., a lateral
stress). Such effects have recently been reported on for lateral stress gauges by
others in this Laboratory [Chapman et al., 2009]. The effect of the manganin
legs on measured stress was not understood at the time of Galbraith’s work,
but generally appears to result in stresses that, when reduced from the mea-
sured resistance change, are around 2 % lower than the stress that was present
in the experiment. Because of these two effects, whilst Galbraith’s data com-
prise the best results on the Hugoniot of ammonium nitrate solutions that has
been reported in the open literature, advances that have been made in the past
decade now enable measurements of a greater accuracy to be made.
Chapter 2 of this thesis develops a mathematical formulation of the propa-
gation of shock waves and introduces the Mie-Gru¨neisen equation of state, an
equation that is widely used in the literature and believed by Orica to provide
an appropriate description of emulsion explosives [Chan, 2011]. In Chapter 3,
the experimental technique of ‘plate impact’ is described; particular attention
5
is paid to diagnostics for measuring stress in dynamic experiments. The ex-
isting experimental facilities in the Cavendish Laboratory are described, and
reductions of the tilt at impact reported on. Chapter 4 discusses techniques for
measuring shock temperature that have been reported in the literature and
presents new experimental results on the use of thin-foil thermistors. From
these, it is concluded that direct measurement of temperature is an expen-
sive and complex issue that remains largely unsolved. The calculation of
temperature from equations of state is, however, an established technique,
and a method for so doing is described. In Chapters 5 and 6, measurements
are made of the Hugoniots and equations of state of water and ammonium
nitrate solutions, investigating the effects of initial temperature and ammo-
nium nitrate concentration on the resultant behaviour. The results reported
enable approximate calculations of shock temperatures to be made. Finally,
Chapter 7 critically assesses the present data and discusses the applicability
of the Mie-Gru¨neisen equation of state to the liquids studied in this thesis. It
is demonstrated that the assumptions underpinning Mie-Gru¨neisen equation
are unlikely to apply to water and ammonium nitrate solutions in the range of
stresses considered in the present work. Complexities in the theoretical mod-
eling of water are discussed, and it is demonstrated that whilst surprisingly
little is well understood about the shock response of water and aqueous solu-
tions at low stresses, the results presented in this thesis are an important test
of the common assumptions made when dealing with such materials.
1.1 Bibliography
C. H. Braithwaite. High Strain Rate Properties of Geological Materials. PhD thesis,
University of Cambridge, 2009.
J. Chan. Personal communication, 2011. Orica Mining Services.
S. K. Chan. A theory to predict the velocity-diameter relation of explosives.
In Seventh Symposium (International) on Detonation — 1981, pages 589–601.
Office of Naval Research, 1982.
S. K. Chan. Prediction of confinement effects on detonation with an analytical
two-dimensional model. In Thirteenth Symposium (International) on Detona-
tion — 2006, pages 224–232. Office of Naval Research, 2007.
6
D. J. Chapman, C. H. Braithwaite, and W. G. Proud. Calibration of wire-like
manganin gauges for use in planar shock-wave experiments. In M. L. Elert,
W. T. Butler, M. D. Furnish, W. W. Anderson, and W. G. Proud, editors, Shock
Compression of Condensed Matter — 2009, pages 603–606. American Institute
of Physics, 2009.
Commonwealth of Australia. Department of Climate Change and Energy Ef-
ficiency, 2011. URL http://www.climatechange.gov.au/.
M. A. Cook. The Science of Industrial Explosives. Ireco Chemicals, 1974.
J. E. Dolan and S. S. Langer, editors. Explosives in the Service of Man. The Royal
Society of Chemistry, 1997.
S. Galbraith. The responses of potassium chloride, ammonium nitrate solutions, and
emulsion explosives to plate impact loading. PhD thesis, University of Cam-
bridge, 1997.
G. A. Gurtman, J. W. Kirsch, and C. R. Hastings. Analytical equation of state
for water compressed to 300 kbar. Journal of Applied Physics, 42(2):851–857,
1971.
W. Hustrulid. Blasting principles for open pit mining, volume 1 - General design
concepts. A. A. Balkema, 1999.
I. J. Kirby and S. K. Chan. A small divergent detonation theory for inter-
molecular explosives. In Eighth Symposium (International) on Detonation —
1985, pages 176–186. Office of Naval Research, 1986.
I. J. Kirby and S. K. Chan. Analysis of VOD-diameter data using an analytical
two-dimensional non-ideal detonation model. In M. D. Furnish, M. Elert,
T. P. Russell, and C. T. White, editors, Shock Compression of Condensed Matter
— 2005. American Institute of Physics, 2002.
Orica. Orica Limited — History, 2004. URL http://www.orica.com/
BUSINESS/COR/orica/COR00254.nsf/Page/About_OricaHistory.
Orica. Proposed Ammonium Nitrate Emulsion (ANE) Production Facility and
Continued Operations of Orica Mining Services Technology Centre, 2009.
URL http://www.oricaminingservices.com/download/file_id_8072/.
Orica. On Solid Ground, 2010. URL http://www.orica.com/BUSINESS/COR/
orica/COR00254.nsf/Page/InvestorsReports. Orica Annual Report.
7
Port Waratah Coal Services. PWCS Operating Statistics, 2011. URL http:
//www.pwcs.com.au/display/assets/download.php?id=498.
W. B. Sudweeks. Physical and chemical properties of industrial slurry explo-
sives. Industrial & Engineering Chemistry Product Research and Development,
24(3):432–436, 1985.
N. Taylor. Hot spots in Ammonium Nitrate. PhD thesis, University of Cam-
bridge, 2011.
8
CHAPTER 2
FUNDAMENTALS OF SHOCK WAVE PROPAGATION
The equations governing the propagation of shocks (i.e., discontinuous flows)
were derived in the latter half of the 1800s [Hugoniot, 1887, 1889; Rankine,
1869], but it was only in the 1940s - as part of the War effort - that the exper-
imental study of shock waves in condensed matter, and the development of
a theoretical framework to support such experiments, began. An historical
overview of this early experimental programme is given by Taylor [1984].
Shock waves are the necessary and unavoidable result of matter moving
at velocities greater than those at which the surrounding media can move out
of the way: a shock wave, therefore, is supersonic with respect to the material
into which it propagates.
Figure 2.1: A rigid piston, moving at up, impinges on a column of fluid. The resultant
shock wave propagates through the fluid at US , taking the material from its
initial state, 0, to a state of elevated pressure, density, and internal energy,
labelled 1.
Figure 2.1 shows a piston driving a shock wave into column of fluid. In this
simplified system, it is assumed that the resultant flow is steady, body forces
may be neglected, and the initial and final states are in mechanical equilib-
rium. Additionally, it is assumed that the material has zero strength, is invis-
cid, and that no reactions or phase changes occur. Following the derivations
9
presented by, amongst others, Asay and Shahinpoor [1993, Chapter 2] and
Davison [2008, Chapter 3], this chapter considers such a simplified system,
and shows it is possible to derive a series of conservation equations and the
mathematical description of a planar longitudinial shock.
2.1 Planar longitudinal shock waves
Considering the piston in Figure 2.1, per unit time, the discontinuity moves
through a mass of material ρ0USA, where ρ0 is the initial density of the fluid,
US is the velocity of the discontinuity andA is its area. The material behind the
discontinuity is at some higher density ρ1 and the mass of material between
the piston and the discontinuity is ρ1(US−up)Awhere up is the particle veloc-
ity of the material and is identical to the velocity of the piston. Conservation
of mass is expressed by equating these terms and dividing by the area A:
ρ0US = ρ1(US − up). (2.1)
Conservation of momentum can be derived by considering a tube of unit
area parallel to the plane normal of the discontinuity and with one face on
either side of the discontinuity. The difference in force over this unit area,
then, is identical to the change in pressure P1 − P0 between the two faces,
and is equal to the change in momentum due to the acceleration of the mass
element ρ0US to the particle velocity up. Thus, conservation of momentum is
expressed as
P1 − P0 = ρ0USup, (2.2)
where P0 is the initial pressure of the material and P1 the pressure behind the
discontinuity.
In a time interval ∆t, the work done per unit area by the piston is P1up∆t
and is equal to the increase in the kinetic and internal energies of the material.
The mass accelerated in this time interval is ρ0US∆t, and it is accelerated to the
particle velocity up; the change in internal energy is the difference in specific
internal energy,E1−E0, multiplied by this mass element. Hence, conservation
of energy gives the expression
P1up =
1
2
(ρ0USu
2
p) + ρ0US(E1 − E0), (2.3)
as the time interval ∆t cancels from both sides.
10
These three equations combined are termed the ‘jump conditions’ and ex-
press the conservation of mass, momentum, and energy between an initial
and final state. It is not necessary that there is a discontinuity between the
two states, as this was not a requirement of the derivation, and so the equa-
tions hold as true for steady compressions as they do for shock discontinuities.
What is required is that the initial and final states are uniform and in mechan-
ical equilibrium. This latter condition merely requires that the state behind
the shock is changing slowly with respect to rate of change at the shock itself:
thus the jump conditions, whilst not strictly valid, are a useful approximation
for decompression waves or in the case of chemical reaction or some rate-
dependent behaviour (e.g., a viscoelastic response) occurring, provided the
final state only varies slowly with time.
Equations 2.1, 2.2, and 2.3 may be combined to eliminate the velocities
to give the Rankine-Hugoniot equation, which relates the change in internal
energy to the changes in specific volume and pressure:
E1 − E0 = 1
2
(V0 − V1)(P + P0), (2.4)
where the densities have been replaced by the specific volumes (V ≡ 1/ρ).
The jump conditions can be generalised to cover the transition from one
steady state to another, where the material in the initial state moves with some
velocity u1 with respect to an arbitrary reference frame:
ρ1/ρ = (up − u1)/(US − u1), (2.5)
P − P1 = ρ1(US − u1)(up − u1), (2.6)
E − E1 = 1
2
(P + P1)(V1 − V ). (2.7)
The jump conditions, then, give a set of three equations in five variables
(P ,V ,E,US ,up); an additional equation - the ‘equation of state’ E(P, V ) - is
required to close the system. This equation describes the behaviour of the ma-
terial, and is not derivable from the idealised system described above. One
such equation of state is the Mie-Gru¨neisen equation, discussed in §2.4. Thus,
after the addition of an equation of state to the jump conditions, a single in-
dependent variable exists defining a curve in a five-dimensional parameter
space: the ‘Hugoniot.’ (This curve, as discussed in §2.2 below, may be rep-
resented in any two-dimensional plane). The Hugoniot does not represent a
thermodynamic path but is merely the locus of possible states, centered on
11
some initial state, achievable through the action of a single shock on a mate-
rial. The term ‘principal Hugoniot’ is used to describe the Hugoniot centered
on the zero pressure state (i.e., the ambient state of the material).
By equating the total differential of the Rankine-Hugoniot Equation (2.4)
to the differential form of the first law of thermodynamics, dE = TdS − PdV ,
an expression for the entropy, S, on the Hugoniot may be obtained:
TdS =
1
2
(V0 − V1)dP + 1
2
(P + P0)dV. (2.8)
Expanding the entropy as a Taylor series along the Hugoniot gives:
S = S0 +
dS
dV
∣∣∣∣
0
(V −V0) + 1
2!
d2S
dV 2
∣∣∣∣
0
(V −V0)2 + 1
3!
d3S
dV 3
∣∣∣∣
0
(V −V0)3 + · · · . (2.9)
From Equation 2.8 it is clear that the first order derivative in the expansion
is zero:
dS
dV
∣∣∣∣
0
=
1
2T
[
P − P0 + (V0 − V )dP
dV
]
P=P0,V=V0
. (2.10)
Similarly, the second order derivative is
d2S
dV 2
∣∣∣∣
0
=
1
2T
[
dP
dV
− 2(V0 − V )2 dP
dV
+ (V0 − V )d
2P
dV 2
]
P=P0,V=V0
, (2.11)
which is also zero. The third order derivative is non-zero, and hence the en-
tropy along the Hugoniot is
S − S0 = 1
12T0
d2P
dV 2
(V0 − V )3 +O(V0 − V )4. (2.12)
The entropy, then, is constant along the Hugoniot to third order in strain.
It is therefore, for weak shocks, often reasonable to approximate the Hugoniot
to a line of constant entropy: the isentrope. As the shock process is sufficiently
rapid that no heat is transferred to the surrounding material, the second law
of thermodynamics requires the entropy change, S − S0, to be positive. Thus,
the shock is thermodynamically stable if
d2P
dV 2
∣∣∣∣
S
> 0. (2.13)
This is equivalent to requiring the P–V Hugoniot to be concave upwards
everywhere. It is important to recall that the Hugoniot is not a thermodynamic
12
curve and does not represent the loading path of the material; this point is
discussed further in §2.2 below.
Temperature is defined thermodynamically as the reciprocal of the partial
derivative of entropy with respect to energy:
T ≡
(
∂S
∂E
)−1
. (2.14)
Thus, the entropy rise associated with the passage of a shock wave leads to
a temperature rise in the shocked material. This temperature rise is discussed
further in Chapter 4.
Figure 2.2: A compressive wave propagating into a material whose sound speed in-
creases with pressure, shown as a function of distance and pressure, at three
different times. The sound speed, cb, in the higher pressure state is greater
than that at lower pressures, ca, and hence the higher pressure portions of
the wave front will catch the lower pressure sections. This results in the
formation of a shock discontinuity moving at a single velocity US in, and
imparting a particle velocity up to, the shocked material.
In most materials, sound speed is an increasing function of pressure. As
shown in Figure 2.2, it is this property which allows a compressive wave to
steepen into a shock discontinuity. The sound speed in the higher pressure
state is greater than that in the lower pressure state, and thus compressional
waves propagating in the higher pressure region travel faster than those in
the lower pressure region. Of course, in the laboratory frame, both propagate
at a velocity equal to the local sound speed plus the material’s particle veloc-
ity. Thus, waves from the higher pressure region catch those from the lower
pressure region, resulting in the steepening of the wave front until it becomes
13
a single discontinuity travelling into unshocked material at the shock veloc-
ity, which is itself supersonic with respect to the unshocked material. Some
materials do not exhibit this behaviour. For example, at low pressures, the
sound speed in fused silica decreases with pressure and, hence, shocks are
not supported [Asay and Shahinpoor, 1993].
It follows that, in addition to being supersonic with respect to the un-
shocked material, the shock must be subsonic with respect to the material
behind it: if this were not the case, then compressional disturbances behind
the shock would not be able to catch it. Thence, the shock would not be stable
with respect to trailing perturbations. Therefore, it is these three conditions -
that sound speed increases with pressure, the shock is supersonic with respect
to the material ahead of it and subsonic with respect to that behind it - that are
required to ensure that a shock is stable.
In a real material, especially at moderate to low pressures, a shock is not
a true discontinuity and its sharpness is limited by dissipative processes in-
cluding heat transport and viscosity. For example, in a viscous material, an
infinite velocity gradient cannot be supported. Hence, in experimental mea-
surements, shock rise times of several tens to hundreds of nanoseconds are
not uncommon, depending on the material. A final complication occurs when
the sound speed is not a continuous function of pressure: for example, in the
vicinity of a phase change or in an elastic-plastic material, the discontinuity in
sound speed will result in the emergence of a two-wave structure.
A material, once in an elevated pressure state, must, necessarily, return
to ambient pressure: the mechanism through which this is achieved is that
of the release, or rarefaction. Release waves spread as they propagate, and
are often termed ‘release fans’ as a consequence. Such a spreading is shown in
Figure 2.3; it should be obvious that the mechanism responsible for steepening
a shock is similarly responsible for the spreading of a release fan: the decrease
of sound speed with decreasing pressure results in the front portion of the
release out-running the trailing portions. The degree of ‘fanning’ depends
on the material and in many cases it is more useful to consider only the lead
characteristic of the release.
Whereas a shock, which is limited by dissipative processes, is irreversible
and results in an increase in entropy, a release is (at least in an inviscid fluid) a
nearly isentropic process. Therefore, an irreversible rise in entropy occurs in a
medium that is shocked to some state and then returned to ambient conditions
by the action of release waves.
14
Figure 2.3: The same mechanism that results in the sharpening of the shock shown in
Figure 2.2 is also responsible for the spreading of a release wave. Portions
of the release wave in the lower pressure regions travel slower than those
in the higher pressure regions and hence trail behind. This results in the
spreading of the wave into a ‘release fan’.
2.2 Graphical representations
The principal Hugoniot of a material is often represented in US–up space, as
the experimental determination of this relationship is common. It is observed
empirically that, for the vast majority of materials, the velocity of the shock
wave generated under plate-impact type loading is linearly proportional to
the particle velocity imparted at impact. This result, sometimes termed the
shock equation of state, is conventionally expressed in the form
US = c0 + Sup, (2.15)
where c0 and S are material constants. c0 is usually found to be approxi-
mately equal to the bulk soundspeed at zero pressure and S is positive for
most materials. The values of c0 and S are tabulated for a wide range of ma-
terials in Asay and Shahinpoor [1993, Appendix C] and Marsh [1980]. Such a
linear relationship between shock and particle velocities gives a Hugoniot in
pressure–particle velocity space that is curved upwards. In some materials,
such as nylon, there is a weak quadratic dependence of shock velocity on par-
ticle velocity, and an extra term may be added to Equation 2.15 to capture any
such behaviour.
The shock impedance, Z ≡ ρ0US , indicates the ability of a material to
support pressure under the action of a shock wave. For a given particle ve-
15
locity up a material of higher impedance will support a higher pressure than
a lower impedance one. It is often convenient to consider impedance in the
limit of a weak shock, where Z0 is given by ρ0cL, where cL is the longitudi-
nal sound speed at zero pressure; this quantity allows for easy comparison
between materials. In general, materials with a high impedance have a high
density, although this is not always the case: sapphire, for example, has a high
impedance due to its very high sound speed.
It is both useful and instructive to consider the graphical representations of
the results derived above. Three representations are commonly encountered
in the design and analysis of shock experiments: x–t diagrams, P–v, and P–up
diagrams.
x-t diagrams are instructive in showing the propagation of shocks and
releases through a series of materials, and are helpful in determining what
interactions may occur at boundaries or between various waves. Figure 2.4
shows an example x-t for a symmetric plate impact experiment. A flyer plate
is fired at some velocity into a stationary target of the same material, produc-
ing shocks in both. These shocks, upon reaching the rear surfaces of both flyer
and target, are reflected as release waves that propagate back into each plate.
Figure 2.4: An x-t diagram for a symmetric plate impact experiment. Shocks are gener-
ated at impact that propagate into the target and flyer at identical velocities;
upon reaching the free surfaces, these shocks are reflected as release fans.
If, as is shown in Figure 2.5, the target plate was not backed by a free sur-
face but by a material of higher impedance then, upon reaching the interface,
16
there is a shock transmitted into the rear anvil, and one reflected into the target
plate. It should be noted that this reflected shock acts to increase the pressure
in the target plate, whilst simultaneously lowering its particle velocity (as it
propagates in the opposite direction).
Figure 2.5: An x-t diagram for a symmetric plate impact experiment backed by a ma-
terial of higher impedance. Upon reaching this interface, the shock is re-
flected back into the target material; in addition, a shock propogates into
the rear material.
The jump conditions, Equations 2.1 - 2.3, can be solved for the shock and
particle velocities:
US = V0
√
(P − P0)/(V0 − V ), (2.16)
up =
√
(P − P0)(V0 − V ). (2.17)
Figure 2.6 shows a Hugoniot in P–V space. The chord connecting the ini-
tial and final state, which has slope (US/V0)2, is termed the ‘Rayleigh line’ and
represents the loading path between the initial and final states. From Equa-
tion 2.17, it is clear that specific kinetic energy is (P − P0)(V0 − V )/2 and is
the triangle below the Rayleigh line. The increase in specific internal energy is
given by the Rankine-Hugoniot Equation (Equation 2.4) and is the total area
below the Rayleigh line. In the case that P0 = 0, which is approximately true
in the case when the initial state is at ambient pressure, then the internal and
kinetic energy generated are equal.
As shown in Equation 2.12, the Hugoniot is isentropic to third order in
strain. Therefore, it is reasonable to approximate the release by the Hugoniot.
17
Figure 2.6: In P–v space, the chord between the initial and final states - the Rayleigh
line - gives the loading path of a material under single-shock loading. The
specific kinetic energy imparted by the shock wave is given by the area
of the triangle below the Rayleigh line; the specific internal energy rise is
given by the total area under the Rayleigh line. As the Hugoniot is isen-
tropic to third order in strain, and hence a good approximation of the re-
lease isentrope, the area between the Hugoniot and the Rayleigh line repre-
sents approximately the ‘waste heat’ irreversibly gained by a material that
is shocked to some state and then released to ambient conditions.
18
Thus, the internal energy returned to the surroundings upon release may be
approximated by the area under the Hugoniot. Hence, the irreversible nature
of the shock process leads to ‘waste heat’ which is the internal energy retained
and is approximately given by the area between the Hugoniot and Rayleigh
line.
P–up diagrams enable determination of the pressure and particle velocity
states resulting from a given interaction. In order that an interface does not
separate, it is necessary that particle velocity be conserved across it; similarly
the pressures must be identical. As the equations derived above are invariant
under transformation of the reference frame, it should be obvious that Hugo-
niots in pressure-particle velocity space may simply be translated along the
particle velocity axis or rotated through the pressure axis to match the partic-
ular shock under consideration. As is shown in Figure 2.7, which corresponds
to the experiment represented in Figure 2.4, the Hugoniot of the flyer is trans-
lated such that it is centered on the impact velocity and reflected - for the shock
in the flyer travels in the opposite direction. It is easy to see, therefore, that
the particle velocity in such a symmetric impact is half the impact velocity.
Figure 2.7: A P–up diagram for a symmetric plate impact. The flyer Hugoniot is re-
flected, as it travels in the opposite direction, and is translated so that it is
centered on the impact velocity. The Hugoniot state is given by the inter-
section of the two Hugoniots; in the symmetric case, the particle velocity is
simply half the impact velocity.
If, instead of the higher impedance material of Figure 2.5, the target plate
19
was backed by a material of lower impedance, then, whilst a shock would
still propagate into the rear material, a release would propagate back into the
target plate. The corresponding P–up diagram is shown in Figure 2.8. The tar-
get material is as expected, shocked to the same state as in Figure 2.7. How-
ever, the release process then takes it to a state lying on the rear material’s
Hugoniot. As, in most materials, it is reasonable to consider that releases are
nearly isentropic, and the Hugoniot and isentrope differ only by terms that
are third-order in strain, it is conventional to approximate the release path by
the isentrope and, hence, by the Hugoniot. Hence, this final state lies on the
intersection between the rear material’s Hugoniot and that of the flyer plate.
Figure 2.8: For a target material backed, not by a free surface, but by a material of
lower impedance, the P–up diagram enables both the Hugoniot and re-
lease state to be found. As in Figure 2.7, the Hugoniot state is given by the
intersection of the target and flyer Hugoniots. Approximating the release
isentrope of the target by its Hugoniot, the release state is the intersection
of the rear material’s Hugoniot with a Hugoniot for the target material that
passes through the previously-calculated Hugoniot state; in the case of a
symmetric impact, this is identical to the flyer Hugoniot.
2.3 Ideal Elastic-Plastic materials
The preceding discussions have explicitly neglected the effect of material strength
on the response of materials to shocks. At high pressures, or for weak ma-
terials, this assumption seems justified, as the effect of strength is consid-
20
ered to be negligible and the material behaves as an inviscid fluid. However,
at moderate shock pressures, many materials exhibit a ‘deviatoric’ response
[Chartagnac, 1982]. Whilst it is not necessary, in the context of the present
discussion, to consider the full mathematical treatment of such materials, the
qualitative response of one such material - the ideal elastic-plastic material -
is discussed below.
The hydrodynamic pressure P is equal to the mean stress and under con-
ditions of uniaxial strain, is 13 (σx + 2σy), where σx is the longitudinal stress
and σy the deviatoric strain. Combining this with the Tresca (or, equivalently,
von Mises) yield criteria, 2τ = σx−σy , the forward yield condition is satisfied
[Kanel et al., 2004] when
|σx − P | = 2
3
Y, (2.18)
where the yield strength Y is simply twice the shear strength τ at the onset of
yield, giving the offset loading cycle shown in Figure 2.9.
For stresses below the ‘Hugoniot elastic limit’ (HEL) of the material, the
response is entirely elastic and consists of a single elastic wave. The response
above the HEL is shown schematically in Figure 2.10 and exhibits a so-called
‘two wave’ response. The first wave is an elastic wave, of stress-amplitude
equal to the HEL, and runs ahead of the plastic shock: for this reason, it is
conventially refered to as the ‘elastic precursor’. The release behaviour is sim-
ilar: the initial unloading is elastic, and occurs before plastic unloading. The
gradual nature of both unloading segments is due to the dispersive nature of
release waves, shown in Figure 2.3.
2.4 The Mie-Gru¨neisen Equation of State
The thermodynamic Gru¨neisen parameter, Γ, is defined [McQueen, 1989] as
Γ = v
∂P
∂E
∣∣∣∣
v
. (2.19)
This equation can be used [e.g., Davison, 2008] to relate the energy on the
principal Hugoniot, EH , along an isochor, to an energy state not on the prin-
cipal Hugoniot:
P − PH = Γ
v
(E − EH), (2.20)
21
Figure 2.9: Loading and unloading, in P–v space for an ideal elastic-plastic material.
The longitudinal stress increases elastically until the forward yield criterion
is reached. The plastic portion is given by a curve identical to the Hugoniot
(the dashed line, often termed the ‘hydrostat’) that is displaced above it by
two-thirds the yield strength Y . The unloading is similar: first, the unload-
ing is elastic continuing until the reverse yield criterion is reached. Plastic
unloading occurs along a line parallel to the Hugoniot, displaced below the
hydrostat by two-thirds the yield strength.
where PH is the pressure on the principal Hugoniot. Equation 2.20 is often
refered to as the ‘Mie-Gru¨neisen equation’ [Davison, 2008] is a P–V –E equa-
tion of state. A ‘Mie-Gru¨neisen material’ is one in which Γ is a function of
volume alone.
It is instructive to consider the origin of Equation 2.20. The following
derivation follows those given by Rice et al. [1958] and [Eliezer et al., 2002].
Consider a solid ofN atoms to be composed ofN simple harmonic oscillators.
Neglecting any phonon-electron interactions, the thermodynamic functions of
the system can be considered as a superposition of terms [Eliezer et al., 2002].
The pressure is expressed as P = Pc + PTa + PTe, where the subscript c de-
notes the pressure at 0 K. PTa is the contribution to pressure due to vibration
22
Figure 2.10: Representative wave profile for the ideal elastic-plastic material shown in
Figure 2.9. A similar profile would be found in both stress and free surface
velocity and consists of an initial elastic response (the ‘elastic precursor’)
followed by the plastic shock. The release process is similar and consists
of elastic unloading followed by plastic unloading.
of the atoms. PTe is the electronic contribution, and can be neglected over the
pressure and temperature range relevant to the present work. Similarly, the
internal energy can be considered as E = Ec + ETa + ETe. Following the
derivation given by Rice et al. [1958], the energy and pressure of such a sys-
tem is calculated below. The energy levels of the oscillators are [Ashcroft and
Mermin, 1976]
Ej =
(
n+
1
2
)
hνj , (2.21)
where the eigenfrequencies νj are functions of volume alone, j = 1, 2, · · · , 3N
for the 3N independent degrees of freedom, n = 0, 1, · · · ,∞ is the excitation
number of the particular mode, and h is Planck’s constant. The partition func-
tion for the oscillator system is then
23
Z =
3N∏
j=1
( ∞∑
n=0
(
e−(n+
1
2hνj)/kT
))
, (2.22)
where k is the Boltzman constant. The sum over the excitation numbers is a
convergent geometric series, and so the partition function becomes
Z =
3N∏
j=1
(
e−hνj/2kT
1− e−hνj/kT
)
. (2.23)
The Helmholtz free energy, F , is defined as
F = −kT ln(Z), (2.24)
and so the Helmholtz free energy of the solid is
F = φ(v) +
3N∑
j=1
hνj
2
+ kT
3N∑
j=1
ln
(
1− e−hνj/kT
)
, (2.25)
where the term φ(v) is the potential energy of the atoms at rest. The pressure
is defined as the negative of the volume derivate of F at constant T
P = − ∂F
∂V
∣∣∣∣
T
= −dφ(v)
dV
+
1
V
3N∑
j=1
[
Γj
(
hνj
2
+
hνj(
ehνj/kT − 1)
)]
, (2.26)
where Γj is defined as
Γj ≡ −d ln(νj)
d ln(V )
. (2.27)
The internal energy can be determined from the Helmholtz free energy
through the relation
E = F − T ∂F
∂T
∣∣∣∣
V
, (2.28)
and so the internal energy is
E = φ(v) +
3N∑
j=1
[
hνj
2
+
hνj(
ehνj/kT − 1)
]
. (2.29)
Considering Einstein’s model of solids [Eliezer et al., 2002], the eigenfre-
quencies of the oscillators are identical: νj ≡ ν for all values of j. Hence,
Γj ≡ Γ for all j and Equation 2.27 becomes
24
Γ ≡ −V
ν
dν
dV
, (2.30)
and as, in a real solid, the eigenfrequencies are functions of volume alone, so
too is Γ. The sums over j in Equations 2.26 and 2.29 are then trivially solved
and the pressure (Equation 2.26) becomes
P = −dφ(v)
dV
+
3NΓ
V
[
hνj
2
+
hνj(
ehνj/kT − 1)
]
, (2.31)
and Equation 2.29 is
E = φ(v) + 3N
[
hνj
2
+
hνj(
ehνj/kT − 1)
]
. (2.32)
Combining these equations, we have the Mie-Gru¨neisen equation of state
referenced to the ‘cold compression curve’
P − Pc = Γ
v
(E − Ec), (2.33)
as φ(v) ≡ Ec and −dφ(v)/dV ≡ Pc are the energy and pressure at 0 K. Differ-
entiating Equation 2.33, we recover the definition of Γ given in Equation 2.19.
As Γ is a function solely of volume (Equation 2.30), then Equation 2.33 can be
referenced to pressures and energies at any temperature. Replacing the cold
compression curve with the Hugoniot, the familar Mie-Gru¨neisen equation
(Equation 2.20) is recovered:
P − PH = Γ
v
(E − EH). (2.34)
It should be stressed that this final step is valid if, and only if, Γ is a func-
tion of volume alone. The Mie-Gru¨neisen equation can similarly be derived
in the context of Debye theory, where there is a spectrum of eigenfrequencies
νj . As noted by [Ashcroft and Mermin, 1976], all the eigenfrequencies scale
linearly with the Debye frequency, ωD:
Γj = −d ln(ωD)
d ln(V )
, (2.35)
and, therefore, Γj is independent of j and therefore the same for all the eigen-
frequencies. Consequently, Γ ≡ Γj , which is now a function of volume alone,
can be taken outside the sum over j in Equation 2.26, and by substituting the
energy (Equation 2.29), Equations 2.33 and 2.34 can be recovered.
25
The physics underlying the foregoing discussion is generally neglected
and the Mie-Gru¨neisen equation is often applied to materials that would not
conventionally be described as a collection of harmonic oscillators. In these
cases, it should be regarded as simply a P–V –E equation of state that can
be used to describe shock experiments. It has been experimentally demon-
strated that the Mie-Gru¨neisen equation provides an appropriate description
of the behaviour of many materials, including mixtures of different materials
[McQueen, 1989], porous solids [Meyers, 1994], and liquids (including water)
[Davis, 1973; Gurtman et al., 1971]. Further, the equation can be used to calcu-
late Hugoniots that are not centered on the ambient conditions (i.e., those due
to multiple shock compression) [McQueen, 1989]. For a material that is first
shocked to a state (P1, V1), the locus of second shock states (P2, V2) is given
by:
P2 =
PH − (Γ/V2)[(PH − P1)(V0 − V2)/2]
1− (Γ/V2)(V1 − V2)/2 , (2.36)
where PH is the pressure on the principal Hugoniot at the volume V2. Thus,
the loci of reflected shocks may be calculated given knowledge of Γ; alter-
natively, Γ may be determined from experimental measurement of reflected
shock states [e.g., Mitchell and Nellis, 1982].
2.5 Summary
• A theoretical description of ideal planar shock waves has been described.
• Two equations of state governing material response have been presented.
• It has been shown that the passage of a shock wave through a material
leads to an irreversible rise in entropy and temperature.
2.6 Bibliography
J. R. Asay and M. Shahinpoor, editors. High-pressure shock compression of solids.
Springer, 1993.
N. Ashcroft and N. Mermin. Solid state physics. Holt, Rinehart and Winston,
1976.
P. F. Chartagnac. Determination of mean and deviatoric stresses in shock
loaded solids. Journal of Applied Physics, 53(2):948–953, 1982.
26
R. O. Davis. Further comments on thermodynamic response of Mie-Gruneisen
materials. Zeitschrift fu¨r Physik B Condensed Matter, 17:63–70, 1973.
L. Davison. Fundamentals of shock wave propagation in solids. Springer, 2008.
S. Eliezer, A. Ghatak, and H. Hora. Fundamentals of equations of state. World
Scientific, 2002.
G. A. Gurtman, J. W. Kirsch, and C. R. Hastings. Analytical equation of state
for water compressed to 300 kbar. Journal of Applied Physics, 42(2):851–857,
1971.
H. Hugoniot. On the propagation of motion in bodies and in perfect gases in
particular - I. Journal de l’E´cole Polytechnique, 57:3–97, 1887.
H. Hugoniot. On the propagation of motion in bodies and in perfect gases in
particular - II. Journal de l’E´cole Polytechnique, 58:1–125, 1889.
G. I. Kanel, S. V. Razorenov, and V. E. Fortov. Shock-Wave Phenomena and the
properties of condensed matter. Springer, 2004.
S. P. Marsh, editor. LASL Shock Hugoniot Data. University of California Press,
1980.
R. G. McQueen. Shock-waves in condensed media: Their properties and the
equation of state of materials derived from them. In Enrico Fermi summer
course CX11: nuclear collisions from the mean field into the fragmentation regime,
1989.
M. A. Meyers. Dynamic Behavior of Materials. John Wiley & Sons, 1994.
A. C. Mitchell and W. J. Nellis. Equation of state and electrical conductivity
of water and ammonia shocked to the 100 GPa (1 Mbar) pressure range.
Journal of Chemical Physics, 76:6273–6281, 1982.
W. J. M. Rankine. On the thermodynamic theory of waves of finite longitudi-
nal disturbance. Proceedings of the Royal Society of London, 18:80–83, 1869.
M. Rice, R. McQueen, and J. M. Walsh. Compression of solids by strong shock
waves. In F. Seitz and D. Turnbull, editors, Advances in Research and Applica-
tions, volume 6 of Solid State Physics, pages 1–63. Academic Press, 1958.
27
J. W. Taylor. Thunder in the mountains. In J. R. Asay, R. A. Graham, and
G. K. Straub, editors, Shock Waves in Condensed Matter — 1983, pages 3–15.
American Institute of Physics, 1984.
28
CHAPTER 3
EXPERIMENTAL TECHNIQUES
In Chapter 2, the equations governing the propagation of planar longitudinal
shocks in condensed media were derived. The practicalities of generating a
planar longitudinal shock and measuring the resultant longitudinal stress are
now considered. A number of experimental techniques for the production of
well-defined planar shock waves in laboratory conditions are available [Asay
and Shahinpoor, 1993, §3.2 – §3.4]. In the present thesis, a single-stage gas gun
is used in the so-called ‘plate-impact’ geometry to produce shock stresses to
30 GPa.
3.1 The Cavendish plate-impact facility
The experiments described in this thesis were performed on the Cavendish
plate-impact facility. The facility was originally designed to investigate the
behaviour of hard ceramics above their elastic limits [Bourne et al., 1995], al-
though it has subsequently been used to investigate a wide range of phenom-
ena under uniaxial plate-impact and ballistic loading [e.g., Braithwaite, 2009;
Chapman, 2009; Forde, 2000; Hammond, 2004; Murray, 1997; Willmott, 2004].
The facility, shown in Figure 3.1, consists of a single-stage light-gas gun, with
a 5 m barrel with 50.8 mm interior bore, and associated diagnostics.
The gun employs a ‘wrap-around’ breech, where the projectile seals the
openings of the gas reservoirs [Fowles et al., 1970]. The reservoirs have a
volume of 0.02 m3; release of the gas in these reservoirs - either compressed
air or helium - accelerates the projectile along the barrel. In order to avoid
an air-blast ahead of the projectile disturbing the target, and to ensure that
the gun is effectively silenced, the target area is part of a sealed expansion
chamber, of total volume 2.2 m3, which is evacuated to below 15 mbar prior to
firing.
The details of the operation of the plate impact facility are adequately
described elsewhere (e.g., Bourne et al. [1995], Murray [1997], Braithwaite
[2009]). However, aspects relevant to the experiments described in this the-
sis are discussed in detail below.
29
Figure 3.1: The single-stage light-gas gun in the Cavendish plate-impact facility. The
gun utilises a ‘wrap-around’ breech, with the projectile forming the seal of
the high-pressure breech reservoir. A sabot is inserted in the breech and
held in place by a vacuum drawn behind it. The central section of the tar-
get area is bolted in place before an experiment is performed, and the en-
tire barrel and expansion chamber are evacuated. The breech reservoirs are
filled with air or helium, either directly or via a gas pump to obtain pres-
sures as high as 350 bar. When the vacuum behind the projectile is released,
the projectile drifts forward through the breech until the reservoirs are no
longer sealed. The projectile is then accelerated along the barrel and into
the target area. Various diagnostics are available, including a laser-based
interferometer system (VISAR). From Braithwaite [2009].
30
Figure 3.2 shows the target area of the plate impact facility. The target
plate, holding the specimen mount, is attached by three adjustable threaded
rods to a collar mounted on the muzzle of the gun. It is by the adjustment of
these three rods that the sample may be aligned perpendicular to the barrel of
the gun.
Figure 3.2: Target area of the plate impact facility. The specimen mount is held within
an adjustable target plate attached to the muzzle of the gun. Three threaded
rods, of which one is labelled, are used to adjust the tilt of the the target
plate (and, hence, the target itself) with respect to the axis of the barrel.
A number of BNC connections enable diagnostic outputs from within the
target chamber. In addition, a number of windows for high speed photog-
raphy are available. When not in use, these are covered with aluminium
blanking plates.
A brass plug, with a hardened steel face, is inserted into the muzzle of the
gun. The face of the plug is machined so that it is perpendicular to the axis
of the barrel. An electronic dial indicator, the probe of which is mounted in a
disc that can be held against the sample mount, is used to measure the posi-
tion of the mount with respect to the face of the alignment plug. By repeated
measurement of the displacement at three points, and careful adjustment of
the threaded rods, it is possible to bring the specimen mount into alignment.
The mount is considered aligned when the displacements measured at three
31
points differ by no more than ±1µm.
Figure 3.3: Schematic representation of the target area of the plate impact facility. For
clarity, the adjustable rods on which the target plate is mounted (Figure 3.2)
are not shown. On exiting the muzzle, the flyer makes contact with a series
of pairs of thin conducting pins. The pins form an electrical circuit, which
is completed at impact, producing a pulse on an oscilloscope. The velocity
of the projectile is determined from the average time between pulses.
The target area is shown schematically in Figure 3.3. For clarity, the ad-
justable rods on which the target plate is mounted are not shown. Projec-
tile velocity is measured immediately prior to impact. A series of eight thin
(0.3 mm diameter) pins, of either graphite or brass, are arranged in pairs above
the path of the projectile and protrude approximately 5 mm vertically down-
wards in front of the flyer plate. As the flyer plate impacts each pair of pins,
an electrical circuit is completed and a corresponding pulse recorded on an
oscilloscope. The time between each pulse enables the velocity to be deter-
mined, and the impact velocity is given by the average velocity recorded from
the four pairs of pins (three timings).
Experiments E081026A (flyer velocity 0.697 mmµs−1, brass pins) and E081027A
(flyer velocity 0.394 mmµs−1, graphite pins) enabled an independent measure-
32
ment of projectile velocity by the use of streak photography [Courtney-Pratt,
1949]. Streak records, obtained with a DRS Lightning Digistreak camera, were
compared to the measured velocity; an extract from such a streak record is
shown in Figure 3.4. In each case, the velocities determined by the pins and
by the streak record agree to within 1 %. It is therefore concluded that the
error on the measured impact velocity is 1 %.
Figure 3.4: Extract from the streak image of E081026A. The black object is the leading
edge of a flyer, travelling from left to right at 0.697 mmµs−1. The time axis
is increasing vertically downwards, with each pixel representing approx-
imately 1.8µs. The vertical lines are spaced 1 mm apart, and are from a
calibration image that has been artificially added to the image.
3.2 In-situ stress measurements
In-situ measurements of longitudinal stress are a key diagnostic in the experi-
ments described in this thesis. One such diagnostic is the thin-foil piezoresis-
tive gauge [Asay and Shahinpoor, 1993, §3.9.1]: the pressure-dependent resis-
tance of a thin metallic foil can be used to measure the applied pressure.
The pioneering work of Bridgman [1917] measured the effect of static pres-
sure on the resistance of a range of metals. Lawson [1956] discusses the use of
pressure-dependent resistance to construct pressure gauges for high-pressure
experiments. For such a gauge to be of use, it must be sensitive to the applied
pressure, insensitive to temperature changes, and, preferably, must exhibit lin-
ear behaviour over the pressure range of interest. For these reasons manganin
(an alloy of copper, manganese, and nickel), which has a low temperature co-
efficient of resistance and approximately linear pressure coefficient, has been
used as piezoresistive pressure gauge. As the resistance change is dependent
on the mode of the applied strain [Lawson, 1956], the resistance change due
to an applied hydrostatic pressure is not the same as that due to an equivalent
33
uniaxial dynamic strain. Considering manganin as a perfect elastic-plastic
material, Barsis et al. [1970] were able to measure the piezo-resistive coeffi-
cients in manganin under shock loading.
Figure 3.5 shows a commercial manganin foil gauge. It is a ‘grid-like’
gauge (LM-SS-125CH-048) and is supplied by Vishay Measurements Group
U.K. Ltd. The nominal resistance of the gauge grid is 48 Ω ± 1 % and it is
mounted on a glass-fibre-reinforced epoxy-phenolic substrate.
Figure 3.5: Thin-foil manganin gauges used in this work were supplied by Vishay
Micro-Measurements, and consist of 48 Ω gauge (LM-SS-125CH-048) on a
glass-fibre-reinforced epoxy-henolic substrate. As targets are larger than
the gauges, in order to extract a signal from a gauge, ‘legs’ of copper shim
are attached using a low-melting point solder.
The gauges pictured are embedded in experiments using a slow-setting
low-viscosity epoxy. As the gauges are much smaller than the targets, in or-
der to extract signals from the gauges ‘legs’ of 25µm-thick copper shim are
attached to the gauge with a low-melting-point Indalloy solder. When embed-
ding gauges between conducting surfaces, 25µm-thick sheets of either mylar
or teflon (PTFE) are used to insulate the gauge. Samples are clamped between
glass plates whilst the epoxy dries. The low-viscosity and long curing time
ensure that, during the clamping process, any bubbles in the epoxy flow out-
wards, and ensures a uniform and thin glue layer. Total overall thicknesses of
gauge packages are around 100µm.
A Wheatstone bridge [Horowitz and Hill, 1989] is a measuring device ca-
pable of accurately determining, through a differential measurement, an un-
known electrical resistance. For the bridge circuit shown in Figure 3.6, the
output voltage is proportional to the size of the unknown resistance Rx and,
hence, recording this voltage when the unknown resistance is a manganin
gauge allows determination of the resistance of the gauge element. Using Kir-
choff’s Laws, the output voltage in an Wheatstone bridge may be calculated:
34
Vout =
(
Rx
R3 +Rx
− R2
R1 +R2
)
Vsupply, (3.1)
where Vsupply is termed the ‘excitation voltage’ of the bridge. The output volt-
age is zero, and the bridge described as ‘balanced’ when
Rx
R3
=
R2
R1
. (3.2)
Figure 3.6: A Wheatstone bridge. The bridge is ‘balanced’, and Vout is zero, when
RX =
R2
R1
R3.
In order to be sensitive to small resistance changes, the excitation volt-
age of the bridge must be large (of order 100 V). For the 48 Ω gauges used
in this work, the resultant current in the gauge (around 2 A) would damage
the gauge if applied continuously. Pulsed power supplies (Dynasen CK1-
50/0.050-300 and Dynasen CK2-50/0.050-300) are used in this Laboratory to
measure resistance histories of piezoresistive gauges. The power supplies
have been modified in-house [Bourne and Rosenberg, 1997] in order to more
accurately capture the fast transient resistance changes in gauges under shock
compression.
Whilst the output of the Dynasen power supplies is roughly linear with
respect to changes in the gauge’s resistance, the output voltage changes with
both the excitation voltage of the bridge and the resistance value at which
it is balanced. In order to calibrate the bridge, and to enable the calculation
of stress, the output voltage as a function of gauge resistance is measured
before every experiment. The power supplies have an adjustable ‘balance’
resistance. Once a gauge has been connected to the power supply, the balance
35
resistance is adjusted such that the output voltage of the bridge is zero. After
the bridge has been balanced, a number of well-defined resistances are placed
in series with the gauge. These ‘shunt’ resistors mimic the effect of a changing
resistance in the gauge. A typical shunt calibration is shown in Figure 3.7.
A simple quadratic relation is used to fit the output voltage to a resistance
change.
0 0.5 1 1.5 2 2.5
0
5
10
15
20
25
30
35
Voltage/V
R
es
is
ta
n
ce
/Ω
 
 
Figure 3.7: Gauge calibration, from experiment E090624B, to ∆R
R0
≈ 0.5. The calibration
curve is described by the equation R = aV + bV 2, with a = 9.1699 ΩV−1
and b = 1.3072 ΩV−2.
The calibration, relating the change in resistance of the gauge to the stress
in the gauge, is dependent only on the geometry of the gauge element and,
so, the calibration of foil-like manganin gauges to 18.1 GPa in plate impact
experiments, reported by Rosenberg et al. [1980], can be applied to the gauges
used in the present work. They found that the response of the gauge is elastic-
plastic. Below the yield strength of the gauge, the gauge behaves elastically,
and the resistance change varies linearly with stress:
σ =
1
k
∆R
R0
, (3.3)
where k = 1.95 × 10−2 GPa−1 is the one-dimensional piezoresistive constant
36
given by Rosenberg and Partom [1985]. The onset of plastic behaviour and,
hence, the HEL of the gauge, is at a stress of 1.5 GPa [Rosenberg et al., 1980]
Above the HEL, Rosenberg et al. [1980] fitted a fourth-order polynomial func-
tion to their data:
σ = A0 +A1
(
∆R
R0
)
+A2
(
∆R
R0
)2
+A3
(
∆R
R0
)3
+A4
(
∆R
R0
)4
, (3.4)
whereA0 = 0.572 GPa,A1 = 29.59 GPa,A2 = 95.20 GPa,A3 = −312.74 GPa,
and A4 = 331.77 GPa. The fit, shown in Figure 3.8, passes within 2 % of all
data points, with a standard deviation of 0.1 GPa. Within error, this fit applies
to multiple-shock loading as well as to single shocks [Rosenberg and Partom,
1985].
Figure 3.8: Calibration for manganin gauges, from Rosenberg et al. [1980]; the differ-
ent data points represent different target materials. The fit is described by
Equations 3.3 and 3.4 and passes within 2 % of all data points.
Due to the elastic-plastic nature of manganin, resistive hysteresis is ob-
served on unloading. This hysteresis has been calibrated in the range 1.5–
15 GPa by Yaziv et al. [1980]. The calibration relates the in-material stress on
unloading to the peak stress the gauge has been subjected to and the ratio of
37
measured resistance changes at each point. The normalised data obtained by
Yaziv et al. [1980], relating the ratio of stresses to the ratio of resistances, is
shown in Figure 3.9.
0 0.2 0.4 0.6 0.8 1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Ratio of resistance changes
R
at
io
of
st
re
ss
es
 
 
Figure 3.9: Manganin exhibits a strong resistive hysteresis upon unloading. Data re-
lating the resistance changes and stress in the partially unloaded states to
those at the peak stress are from from Yaziv et al. [1980]. The unloading
data is described, in the plastic region, by Equation 3.5. For stresses above
80 % of the peak stress, the fit is undefined and ideal reverse elastic yielding
has been assumed.
A non-linear least-squares method was used in Matlab to fit a fifth-order
polynomial to the data:
σrelease
σpeak
= M0 + M1
(
∆Rrelease
∆Rpeak
)
(3.5)
+ M2
(
∆Rrelease
∆Rpeak
)2
+M3
(
∆Rrelease
∆Rpeak
)3
+ M4
(
∆Rrelease
∆Rpeak
)4
+M5
(
∆Rrelease
∆Rpeak
)5
,
where M0 = −0.1375, M1 = 1.3995, M2 = −0.3768, M3 = −1.1112, M4 =
2.5171, and M5 = −1.3515.
38
The fit, which passes within 1 % of the data, is valid to
∆Rrelease
∆Rpeak
= 0.9, and
σrelease
σpeak
= 0.8. (3.6)
As noted by Yaziv et al. [1980], the calibration above this region is uncer-
tain and difficult to determine experimentally. In the analysis discussed be-
low, the resistive hysteresis is taken to be linear in this region, as the gauge is
assumed to undergo reverse yielding and hence, initially, unloads elastically.
The data reduction procedure, then, for manganin gauges combines the
calibration results described above. The initial resistance of the gauge, R0, is
measured at multiple stages during the construction of a target to ensure that
no damage (indicated by a resistance that differs greatly from the nominal
gauge resistance) has occurred. A four-terminal digital multimeter (Keith-
ley 2100) enables high-precision measurements of the initial resistance to be
made. Simple Matlab routines have been written to calculate the fitting pa-
rameters necessary to convert bridge output voltages into resistance change
(Figure 3.7) and to reduce the data to a stress–time history using Equations
3.3, 3.4, and 3.5.
3.3 Particle velocity diagnostics
In addition to measurements of in-material longitudinal stresses, direct mea-
surements of particle velocity have been accomplished through the use of in-
terferometric techniques. One diagnostic that has been used on the Cavendish
plate-impact facility is the VISAR interferometer. The Velocity Interferometer
System for Any Reflector (VISAR) was developed by Barker and Hollenbach
[1972]. A detailed overview of the development of VISAR is not relevant to
the present discussion, and the interested reader is directed to the historical
overview given by Barker [2000] or the technical descriptions given by Dolan
[2006]. The basic principal of all VISAR systems is to interfere laser light that
has been reflected from a target at two different times. Due to the relativis-
tic Doppler effect, light reflected from a moving surface undergoes a change
in phase [Barker and Hollenbach, 1972]. The fringe pattern that results from
interfering light reflected from the moving surface at two different times is re-
lated to the relative changes in phase and, hence, to the velocity of the surface.
The VISAR system used with the plate impact facility uses the ‘modified
quadrature’ or ‘push-pull’ design proposed by Hemsing [1979]. Figure 3.10
39
shows schematically Hemsing’s VISAR interferometer. The particular system
available in the Cavendish (VMBV-04-1027-532, produced by Valyn V.I.P. Ltd.)
is capable of simultaneous measurements at up to three distinct points, using
a neodymium-doped yttrium orthovanadate (Nd:YVO4) solid-state laser, fre-
quency doubled by a LiB3O5 crystal to 532 nm (Coherent Verdi V5). Light for
each beam is carried to the target in a 50µm-core optical fibre, and returned to
the interferometer in a 300µm-core fibre. The interferometer is schematically
identical to that shown in Figure 3.10. After passing through a collimator,
an ultranarrow bandpass filter (UBNP) prevents light not at the laser wave-
length from entering the interferometer. Light from each beam is split by the
principal beam splitter, BS, with half traveling through the delay leg. The un-
compensated components that make up the delay leg are the remaining glass
of the beam splitter, up to four etalons, E, and the waveplate, WP. The wave
plate can be rotated axially to ensure the phase difference between the two
polarisations, β, is pi/2. This splitting of polarisations ensures that whenever
the interference in one polarisation results in a minimum of the detected sig-
nal, the other polarisation gives a maximum, resulting in no loss of sensitivity
near the minima. Upon reflection at the mirror M2, this delayed light passes
back to the beam splitter. The remaining light travels through free space to
the mirror, M1, in the reference leg before returning to the beam splitter. M1
is mounted on three piezoelectric elements, the voltage supplied to which can
be adjusted to align the interferometer. Finally, this recombined light trav-
els to the two polarising beam splitters. The four output beams associated
with each input beam are collected into optical fibres, that take the outputs
to a separate ‘beam module’. The beam modules contain the necessary opto-
electronic components to allow for velocity histories to be reduced from the
ouptut beams of the interferometer.
The details of the analysis are not relevant to the present discussion and
have been described adequately elsewhere [Dolan, 2006; Hemsing, 1979]. A
detailed analysis of the performance of the VISAR system used in the Cam-
bridge plate impact facility is reported by Chapman [2009].
3.4 Sample preparation and design
In addition to requiring that the target is well-aligned, as discussed above, it
is important that the flyer plate and target plates are machined both flat and
parallel, as any mis-machining will have the same effect as a poorly-aligned
40
Figure 3.10: Hemsing’s modified quadrature VISAR interferometer, the ‘push-pull’
VISAR after Dolan [2006]. The interferometer consists of a principal beam
splitter (BS), a number of glass etalons (E), a one-eighth waveplate (WP),
two mirrors (M1, M2), polarising beam splitters (PBS) and photomultipli-
ers (D1a, D1b, D2a, D2b). Light not at the laser wavelength is rejected by
an ultra-narrow bandpass filter (UNBP).
target.
A vacuum chuck has been designed in-house to enable the machining of
target plates to the required degree of planarity. The chuck is fixed to the bed
of a CNC mill, and skimmed flat. Target plates are then skimmed to near
the final thickness by placing them on the chuck and skimming one face. The
plates are turned over and the reverse face is then skimmed to reduce the plate
to the required thickness. This also ensures that both faces are parallel. With
this process, plates as thin as 1 mm are skimmed flat, and parallel, to, typically,
better than ±2µm.
For reasons discussed in Section 3.5 below, two main sabot designs were
used in the experiments reported in this thesis: the older ‘Mark II’ polycar-
bonate design (Figure 3.11), and the new ‘Mark III’ nylon design (Figure 3.12).
Both designs have three variants: the ‘standard’, which has a solid poly-
mer body; the ‘lightweight’, which has material drilled from it, and allows
41
Figure 3.11: The standard ‘Mark II’ sabot, machined from a solid polycarbonate cylin-
der to an outer diameter of 50.60+0.00−0.10 mm, is shown with a copper flyer
plate. The ‘o’-rings are placed on the sabot prior to firing, and form the
seal of the firing reservoirs. The front face of the flyer plate is skimmed
and/or lapped after being attached to the sabot to ensure the front face is
perpendicular to the sabot’s axis.
for higher velocities to be achieved for a given gas pressure; and the ‘heavy-
weight’, which has a brass cylinder inserted in it. Lower firing pressures tend
to result in less reproducible projectile velocities; the increased mass of the
heavyweight projectiles necessitates an increase in firing pressure and, hence,
results in more reproducible velocities. In all variants there is a recess behind
the flyer plate. A small hole is drilled through the sabot wall into this recess.
Thus, the space behind the flyer is evacuated with the expansion chamber and
barrel. In this way, the rear surface of the flyer releases fully, as a free surface,
rather than partially, from the interface between the flyer and the polymer.
Figure 3.13 superposes a sabot in its pre-firing position in the breech of the
gun. The two ‘o’-rings seal the openings of the firing reservoirs whilst they are
being filled. In order to ensure that the sabot remains in place prior to firing,
a vacuum is drawn behind it. When the gun is to be fired, a small amount
of air is allowed in behind the sabot. The sabot then begins to drift along
42
Figure 3.12: The standard ‘Mark III’ sabot, constructed from a nylon 6,6 rod that is pre-
ground to an outer diameter of 50.75+0.00−0.05 mm, is shown with a PMMA
flyer plate. An aluminium ring is placed on non-conducting flyers to en-
able velocity measurements. The grooves in which the ‘o’-rings sit are
clearly visible.
the barrel until the rear ‘o’-ring no longer seals the firing reservoirs. At this
point, the full pressure of the firing gas expands into the barrel, accelerating
the projectile towards the target.
The diagnostic techniques described above, and the equations derived in
Chapter 2, assume the propagation of a steady planar shock. In order, there-
fore, that the analysis remains valid, diagnostics must be employed only where
the strain is uniaxial and the wave steady. Longitudinal and lateral releases
will erode these conditions and, therefore, the effect of release waves must be
considered when designing plate impact targets.
At impact, in addition the forward-moving shock which propagates into
the target, a backward-moving shock propagates into the flyer plate. As shown
schematically in Figure 3.14, interaction of this shock with the rear of the flyer
plate results in a forward-moving release wave propagating into the target.
As noted in Chapter 2, this release propagates as a fan across a range of de-
43
Figure 3.13: The breech of the Cavendish gun, with a sabot superposed in its pre-firing
position. A vacuum is drawn behind the projectile to hold it in place,
and the ‘o’-rings form a seal against the openings of the firing reservoirs.
When the vacuum behind the projectile is released, the projectile drifts
forward and ceases to seal the firing reservoirs. At this point, the firing
gas is released and accelerates the projectile along the length of the barrel.
From Braithwaite [2009].
44
creasing velocities. The first, fastest, characteristic of the fan is an elastic wave
propagating at C+up, where C is the local sound speed, and may be approxi-
mated by the ambient pressure longitudinal sound speed. The point at which
the shock ceases to be steady, defined by the interaction of the shock with the
first release characteristic, can be calculated by simply considering the propa-
gation of the forward-moving shock and the backward-moving shock, and the
propagation of the release wave through the partially-compressed medium.
Figure 3.14: Schematic representation of longitudinal release wave propagation from
the rear of a flyer plate. The shock propagating into the target remains
steady until is caught by the first release characteristic.
In the conventional plate impact scenario, the flyer plate is generally of
a smaller diameter than the target plate. Immediately, then, upon impact,
release waves propagate into both the target and flyer from material outside
the original impact area. The analysis below, following that of Swift and Kraus
[2008], derives an equation governing the propagation of lateral release waves
into the target. It is only necessary to consider the leading characteristic of
the release fan, as it is this characteristic which first erodes the condition of
uniaxial compression in the target.
Considering the frame co-moving with the shocked material, as shown in
Figure 3.15, the lateral release propagates as a spherical wave moving at the
bulk sound speed of the shocked state, cr.
At a time ∆t, the shock will, in this frame, have moved a distance (US −
up)∆t. The radius encompassed by the lead lateral release characteristic at
this time is cr∆t, and, hence, the angle φ′ along which the 1D strain becomes
45
Figure 3.15: Schematic representation of lateral release wave propagation into a tar-
get in a frame co-moving with the shocked material. The vertical axis is
distance from the axis of symmetry.
(in cylindrical polar co-ordinates) 2D is given, in the co-moving frame, by
φ′ = cos−1
(
US − up
cr
)
. (3.7)
So this relation may be of use, it is necessary to find the angle φ along
which the shock becomes two dimensional in the laboratory frame. This is ac-
complished by a Gallilean transformation from the shock frame, which moves
at up with respect to the laboratory frame. Thus,
φ = tan−1
(√
c2r − (US − up)2
US
)
. (3.8)
A similar expression holds for the release in the flyer itself. It is often
convenient to assume that the φ is approximately 45 ◦, which - in most cases
- gives an over-estimate of the angle at which the releases propagate. The
angle of release may be used to calculate the diameter over which the shock
46
remains uniaxial. For an Lagrangian distance l0 into the target, a shock of
radius r0 leaves a radius r1 under uniaxial compression:
r1 = r0 − 2l0φ. (3.9)
In designing shock experiments, it is important to ensure that the diag-
nostics are within a region that has not been laterally released and, therefore,
Equation 3.9 is used to determine where such diagnostics may be placed.
3.5 Measurement of impact tilt in plate-impact experiments
Consider a target at some small angle, Θ, to an impactor, travelling at velocity
uimpact, as shown schematically in Figure 3.16. A shock is produced in the
target and travels perpendicular to the impact face at US ; the point of contact
moves across the impact face at ucontact.
Figure 3.16: For an impactor incident on a target at some small angle Θ, the shock front
moves at some greater angle, Φ, to the front surface of the target.
By the small-angle approximation, the velocity of the contact point is
ucontact =
uimpact
sin(Θ)
≈ uimpact
Θ
. (3.10)
The angle made by the shock to the impact face, Φ, is given by
sin(Φ) ≈ Φ = Ushock
ucontact
, (3.11)
and, hence, from Equation 3.10,
47
Φ ≈ Ushock
uimpact
Θ. (3.12)
As the shock velocity is, generally, greater than the impact velocity, the tilt
of the shock is larger than the physical tilt. This multiplication of the physi-
cal misalignment can have a significant effect on any measurements. For ex-
ample, the tilt results in straining of manganin gauges, leading to erroneous
measurements of stress.
An unknown Hugoniot is most efficiently determined by measurement
of the relationship of shock velocity to particle velocity. The simplest such
method is to determine the ‘time of flight’ of a shock wave through a known
thickness of the material under consideration. Measurement of the arrival of
the shock wave at two longitudinally separate points enables the transit time,
and hence, shock velocity, to be determined.
Figure 3.17: Schematic representation of the target used in Experiment E100211A. Two
manganin gauges are separated longitudinally by a 3.185 mm-thick cop-
per plate and laterally spaced 20 mm apart. The measured shock velocity
recorded between the two gauges will be sensitive to any impact tilt.
In the case where these two measurement points are separated laterally
and longitudinally, then tilt of the shock wave will distort the measured shock
velocity. Experiment E100211A, shown schematically in Figure 3.17, demon-
strates the effect of tilt on the determination of shock velocity in copper. Com-
parison of the shock velocity predicted from the linear US–up Hugoniot given
by Asay and Shahinpoor [1993], and that measured in E100211A, enables a
measure of the tilt across the diameter on which the gauges lie. After the
48
impact of a copper flyer at 0.389± 0.004 mmµs−1, experiment E100211A indi-
cated a shock velocity of 4.224 ± 0.022 mmµs−1, compared to the calculated
value of 4.519±0.006 mmµs−1. The impact tilt required to account for this dis-
crepancy is approximately 30 mRad. However, as E100211A measured shock
arrival times at only two points, the tilt measurement is under-constrained,
and a greater tilt could exist in some other direction. Additionally, a tilted
shock introduces an uncertainty in the stress value measured by a manganin
gauge, as the loading is no longer perpendicular to the gauge face and the cal-
ibration no longer applies. Therefore, in measuring impact tilt, it is desirable
to use more than two points and to use a point-like diagnostic, rather than a
gauge or other diagnostic with significant lateral extent.
Experiment E100312A employed two diagnostics to measure impact tilt:
VISAR and optic fibres; the experimental geometry is shown schematically in
Figure 3.19. As a gas is compressed adiabatically, it is strongly heated and will,
ultimately, emit light. Whilst plate impact experiments are conducted under
vacuum, there is sufficient residual gas between the target and flyer plate to
generate an ‘impact flash’ as the gap between the two surfaces is closed. An
optical fibre flush to front surface of a target plate is able to detect light output
as the flyer impacts the target. A series of such fibres is therefore able to re-
solve impact tilt. Figure 3.18 shows the normalised output of a photodiode for
a 400µm silica core optic fibre flush to the front surface of an aluminium target
impacted by an aluminium flyer plate at an impact velocity of 0.391 mmµs−1.
The first peak, at around 1µs, is the impact flash, the tail features of this peak
depend partially on how well the fibre was polished. In around half of the
photodiode traces a secondary peak, smaller in amplitude, is seen. This peak
occurs at around 14µs in Figure 3.18. This peak is attributed to tribolumines-
cence [Smiel and Fisher, 1982]. As the silica core is fractured by the passage
of the shock wave, light is generated. This process occurs on a much longer
time-scale than impact flash, and the light output is observed well after the
target is no longer in a state of uniaxial strain. In the determination of impact
tilt, then, it is the impact flash which is considered.
Figure 3.20 shows the impact flash measured by the three silica core optic
fibres, shown in Figure 3.19. Three different photodiodes were used, each with
a different sensitivity; the detector outputs have been normalised for ease of
comparison. One detector produced a larger-than-expected output voltage,
resulting in the trace being ‘clipped’.
For a series of detectors i, placed at angles φi around a circle of radius R,
49
0 2 4 6 8 10 12 14 16 18 20
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Time/µs
P
h
o
to
d
io
d
e
 o
u
tp
u
t/
a
rb
it
ra
ry
 u
n
it
s
Figure 3.18: Normalised photodiode output from experiment E100312A. The initial
peak, at 1µs is due to impact flash; the secondary peak, at 14µs is tri-
boluminescence caused by fracture of the fibre itself.
Figure 3.19: Schematic representation of the target used in experiment E100312A. The
target, a 3 mm thick aluminium plate, held three 400µm silica core optic
fibres normal to the impact face and flush with the front surface, spaced
equally around a circle of radius 23 mm. The rear surface was illuminated
with three beams from the Cavendish’s multibeam VISAR system, equally
spaced around a circle of radius 10 mm, offset by 60 ◦ with respect to the
silica fibres.
50
Mitchell and Nellis [1981] relate the deviation between the arrival time at any
detector, ti and the average arrival time, t, to the tilt through the relationship
ti − t = −R
uimpact
tan(θ) cos(α0 + φi), (3.13)
where θ is the amplitude of the tilt; this maximal tilt occurs along a diameter
at α0 to the first diagnostic to respond (φ0 = 0). By determining the average
arrival time at all three detectors and fitting the arrival time data to Equation
3.13 the maximal tilt, as measured by the three optic fibres, was found to be
9.4± 0.1 mRad at an angle of 77 ◦ to the vertical.
 0.5  0.4  0.3  0.2  0.1 0 0.1 0.2 0.3 0.4 0.5
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Time/µs
P
h
o
to
d
io
d
e
 o
u
tp
u
t/
a
rb
it
ra
ry
 u
n
it
s
Figure 3.20: Normalised photodiode outputs, from experiment E100312A; times, in mi-
croseconds, are relative to the average arrival time at all three detectors.
Deviations from the average arrival time are −0.53, +0.12, and +0.40µs
indicating a tilt of 9.4± 0.1 mRad at an angle of 77 ◦ to the vertical.
The rear surface of the target was illuminated with three probes from the
Cavendish’s multi-beam VISAR system, arranged around a circle of radius
10 mm. The arrival time of the elastic precursor, as measured by the three
probes, was used to calculate the maximal tilt. The tilt, as measured by the
VISAR, was 9.1± 0.1 mRad at an angle of 123 ◦ to the vertical. That the angle
of maximal tilt deduced from the VISAR data is not the same as that deduced
from the photodiode data is attributed to difficulties in ensuring the mount
that holds the VISAR probes is accurately positioned, and that the rear surface
of the target is parallel to the front surface.
Experiment E100312A demonstrated that impact flash, as measured by
embedded silica-core fibres and photodiodes, could be used to measure dy-
51
 0.4  0.3  0.2  0.1 0 0.1 0.2 0.3 0.4
 0.05
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
Time/µs
F
re
e
 s
u
rf
a
c
e
 v
e
lo
c
it
y
/m
m
µ
s
 1
Figure 3.21: VISAR traces, from experiment E100312A. The fringe constant,KVPF, was
187.5 mmµs−1. The deviations from the average arrival time are −0.23,
+0.11, and +0.13µs indicating a tilt of 9.1± 0.1 mRad of 123 ◦ to the ver-
tical.
namic tilt in plate impact experiments. Tilt measured in this way agreed with
the tilt measured by multi-beam VISAR. However, in order to accurately and
reliably resolve both the magnitude and phase of the tilt, more than three
measurement points are required; for this reason, the Cavendish’s multi-beam
VISAR system is not suited to measuring dynamic tilt. In addition, as VISAR
measures shock arrival at the rear of the target, in order that lateral releases
have not affected the shock, the thickness of the target would be limited. It
was concluded that measuring tilt with a number of embedded silica-core fi-
bres would be the simplest and most effective method.
Three principal contributions to the dynamic tilt measured in a plate im-
pact experiment, shown schematically in Figure 3.22, are (i) misalignment of
the target itself, (ii) misalignment of the flyer plate, and (iii) balloting of the
sabot. In order to reduce the tilt at impact, each of these contributions were
considered.
The adjustable alignment system, described above, is designed to allow
targets to be aligned to significantly better than 1 mRad. Over time, the threads
that hold the alignment plate in place become worn and allow the alignment
plate to ‘slip’ after it has been aligned. The system was rebuilt with reduced
clearances allowed between the moving threads in order to allow for finer
adjustment and reduce any ‘slip’ in the system. In addition, the ‘alignment
plug,’ was re-machined to ensure its front face was perpendicular to the bore
of the barrel. The conventional alignment system, described above, relies on a
dial indicator that is held in place by manually applying pressure. Variations
52
Figure 3.22: Dynamic tilt at impact, in the conventional plate impact geometry, can be
considered as the sum of three independent effects: misalignment of the
target plate with respect to the normal to the barrel axis; misalignment of
the flyer plate with respect to the normal to the sabot axis; and balloting
of the sabot.
in the force with which the indicator is held in place can result in changes in
the measured displacement, thus affecting the alignment by an unquantifiable
amount. A new dial indicator system was manufactured in order to eliminate
this uncertainty. The dial indicator probe is mounted in a brass disc and the
disc is held in place by three stiff spring steel clips; the clips provide a strong
and constant force holding the indicator in place.
A Michelson interferometer has been used as an alternative method of
aligning targets perpendicular to the barrel. The interferometer, shown schemat-
ically in Figure 3.23, was constructed by Dr. Daniel Eakins of the Institute of
Shock Physics, Imperial College. In order to test the accuracy of the dial in-
dicator system, the interferometer was used to align a mirrored target. The
mirror was removed, and the dial indicator used to measure the position of
the alignment plate. The plate was found to be perpendicular to the axis of
the barrel to within 25µm. By providing an independent measurement, the
interferometer demonstrates that the improved dial indicator system allows
for targets to be aligned to within 0.5 mRad. The accuracy of inteferometer
system is limited by the the position of the laser beam with respect to the axis
of the barrel. If the beam is off-axis, or the two ends of the barrel are not paral-
53
lel (i.e., the barrel is not straight), then a misalignment with respect to the dial
indicator will be observed. It is therefore concluded that the dial indicator is
accurate to, at worst, 0.5 mRad.
Figure 3.23: The Michelson inteferometer used for alignment in the plate impact facil-
ity. Incident laser light is directed centrally through the barrel of the gun
by adjusting the mirrors A and B. An iris is inserted at each of the barrel
to allow for more precise positioning of the beam. A beam splitter, BS, is
used to split half the incident light; this light is reflected back to the beam
splitter by mirror 1 and forms one ‘leg’ of the interferometer. Mirror 2 is
mounted in place of a conventional target and reflects light back down
the barrel to the beam splitter. If the target is not aligned correctly, the
reflected light (dotted line) will not co-incide with the incident light, and
two spots will be visible on the viewing screen. For light of a large enough
coherence length, Fizeau fringes can be observed near perfect alignment;
if mirror 2 is perfectly aligned, Haidinger fringes will be visible.
Whilst the sabot itself is machined in a single operation on a lathe, en-
suring that, to within machining tolerances, the front face is perpendicular to
both the sabot axis and the outer diameter, flyer plates are attached in a sepa-
rate process as discussed above. Two distinct types of misalignment can exist
in a completed sabot, as shown in Figure 3.24: a pure tilt of the front surface
with respect to the normal to the sabot axis, and a concave (or convex) bow-
ing of the front surface. This latter type may be introduced due to tolerances
in the lathe used to ‘skim’ the front of the flyer. In a typical lathe, the per-
mitted tolerance of the mechanisms that could contribute to this bowing are
such that the machined surface is planar to significantly better than 5µm. The
introduction of a pure tilt should not be possible in a well-maintained lathe,
as the lathe’s axis is perpendicular to the tool’s direction of travel to within a
high degree of tolerance.
In order to measure both tilt and bowing, an accurate dial indicator was
54
Figure 3.24: Misalignment of the flyer plate, with respect to the normal to the sabot
axis, takes two forms: pure tilt (a), and concave or convex bowing (b).
placed in the toolpost of a well-aligned lathe whilst a finished sabot was ro-
tated in the chuck. Typical measurements indicated a bowing of less than 5µm
and that flyer plates were perpendicular to the axis to within 5µm; both these
measurements are at the maximum attainable measurement accuracy for the
lathe used, and correspond to a pure tilt of order 1 mRad.
Experiment E100415A employed six photodiodes, equally spaced around
a radius of 20 mm, to measure the tilt of an aluminium flyer plate, flat to
±4µm, incident on an aluminium target plate (3.056±0.010 mm) at 0.396 mmµs−1.
The target was aligned to within 1 mRad. Two of the detectors recorded no
output; the remaining photodiode outputs are shown in Figure 3.25, with out-
put voltages normalised. By fitting Equation 3.13 to the data, the resultant tilt
at impact was found to be 4.4± 0.8 mRad. at an angle of −40◦ to the vertical.
Balloting, the in-bore yaw of a projectile, is a well-known phenomenon in
the study of interior ballistics [Ansari and Baugh, 1986]. Simplistically, the
maximum possible balloting angle can be calculated by considering the angle
of yaw that can be introduced due to a projectile of diameter dsabot in a barrel
of (larger) bore dbore, as shown schematically in Figure 3.26.
For a sabot of length Lsabot, this maximum balloting angle occurs when
dbore = Lsabot sin(Φballoting) + dsabot cos(Φballoting). (3.14)
For small angles, however, the balloting angle may be expressed as
Φballoting ≈ dbore − dsabot
Lsabot
. (3.15)
The gun of the Cavendish plate impact facility has an internal bore of
50.80 mm near the muzzle, as measured by a dial bore gauge. The bore is ap-
proximately 50.85 mm near the breech, where repeated insertion of the breech
plug has widened the internal diameter somewhat.
55
 0.2  0.1 0 0.1 0.2 0.3 0.4
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Time/ µs
P
h
o
to
d
io
d
e
 o
u
tp
u
t/
a
rb
it
ra
ry
 u
n
it
s
Figure 3.25: Photodiode response, from experiment E100415A. Detector output has
been normalised to the range [0 : 1] and times, in microseconds, are
relative to the average shock arrival time of all four traces. One detec-
tor (shown in black) produced a larger than expected output voltage, re-
sulting in ‘clipping’ of that trace. The deviations from the average ar-
rival times were −0.017,−0.015,+0.010, and +0.025µs indicating a tilt of
4.4± 0.8 mRad at an angle of −40◦ to the vertical.
Figure 3.26: The principal contribution to balloting, or in-bore yaw, is due to the sabot
necessarily having a smaller diameter than the bore through which it trav-
els. The maximum balloting angle occurs when the leading (or, equally,
trailing) edge of the sabot touches one side of the bore and the opposite
end touches the opposite side of the bore.
The standard, ‘Mark 2,’ polycarbonate sabot has an external diameter of
50.60+0.00−0.10 mm, and hence Φballoting = 2
+1
−0 mRad.As the experiments discussed
above indicated a tilt at impact that was not attributable to misalignment of
the target or the flyer, it was concluded that the magnitude of the yaw possible
with a ‘Mark 2’ sabot was the principal contribution to tilt.
56
Therefore, a larger ‘Mark 3’ sabot was developed in conjunction with Peter
Norman and Chris Summerfield of the Cavendish Laboratory’s workshops.
The ‘Mark 3’ sabots have an external diameter of 50.75+0.00−0.05 mm, allowing for
a maximum in-bore yaw of Φballoting = 0.5+0.5−0.0 mRad. In addition to enlarg-
ing the diameter of the sabot, it was also decided to manufacture the ‘Mark
3’ sabots from nylon 6,6; nylon was considered to have improved long-term
storage characteristics and to be easier to machine than polycarbonate.
Experiment E100429A was a replication of experiment E100415A, using
a ‘Mark 3’ sabot at an impact velocity of 0.395 mmµs−1. Figure 3.27 shows
the signal associated with impact flash for the four detectors that functioned
correctly. Fitting Equation 3.13 to these data, the tilt was found to be 1.1 ±
0.2 mRad at an angle of −5◦ to the vertical. The coefficient of determination
was 0.92.
 0.06  0.04  0.02 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Time/µs
P
h
o
to
d
io
d
e
 o
u
tp
u
t/
a
rb
it
ra
ry
 u
n
it
s
Figure 3.27: Photodiode response, from experiment E100429A. Detector output has
been normalised to the range [0 : 1] and times, in microseconds, are rel-
ative to the average shock arrival time of all four traces. The deviations
from the average arrival times were−0.051,−0.017,−0.009, and +0.026µs
indicating a tilt of 1.1± 0.2 mRad at an angle of −5◦ to the vertical.
As further confirmation, Experiment E100528A repeated previous exper-
iments, using a ‘Mark 3’ sabot at an impact velocity of 0.392 mmµs−1. Fig-
ure 3.28 shows the signal associated with impact flash for the four detectors
that functioned correctly. As before, Equation 3.13 was fitted to these data,
and the tilt was found to be 1.6 ± 0.1 mRad at an angle of +6◦ to the verti-
cal. The coefficient of determination was 0.98. It is, therefore, concluded that
57
a ‘Mark 3’ sabot will reproducibly give tilts, at impact, of less than 2 mRad.
With this in mind, any experiments must be designed such that they are not
sensitive to errors introduced by tilts of order 2 mRad.
 0.1  0.05 0 0.05 0.1 0.15
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Time/µs
P
h
o
to
d
io
d
e
 I
n
te
n
s
it
y
/a
rb
it
ra
ry
 u
n
it
s
Figure 3.28: Photodiode response, from experiment E100528A. Detector output has
been normalised to the range [0 : 1] and times, in microseconds, are rel-
ative to the average shock arrival time of all four traces. The deviations
from the average arrival times were−0.076,−0.030,+0.047, and +0.058µs
indicating a tilt of 1.6± 0.1 mRad at an angle of +6◦ to the vertical.
3.6 Summary
• The Cavendish Plate Impact Facility, a versatile experimental facility for
gathering shock wave data, has been described.
• The principal diagnostics used in this thesis, the in-situ manganin stress
gauge, and the VISAR interferometer, have been described.
• The reduction of data from these diagnostics has been described and
evaluated.
• Impact tilt has been investigated in experiments conducted with the
Plate Impact Facility, and significantly reduced through improvements
to aspects of the facility’s design.
58
3.7 Bibliography
K. A. Ansari and J. W. Baugh, Jr. Dynamics of a balloting projectile in a mov-
ing gun tube. Technical Report BRL-CR-605, Ballistic Research Laboratory,
Aberdeen Proving Ground, 1986.
J. R. Asay and M. Shahinpoor, editors. High-pressure shock compression of solids.
Springer, 1993.
L. Barker and R. E. Hollenbach. Laser interferometer for measuring high ve-
locities of any reflecting surface. Journal of Applied Physics, 43(11):4669–4675,
1972.
L. M. Barker. The development of the VISAR, and its use in shock compression
science. In M. D. Furnish, L. Chhabildas, and R. S. Hixson, editors, Shock
Compression of Condensed Matter - 1999, pages 11–17, 2000.
E. Barsis, E. Williams, and C. Skoog. Piezoresistivity coefficients in manganin.
Journal of Applied Physics, 41(13):5155–5162, 1970.
N. K. Bourne and Z. Rosenberg. On the ringing observed in shock-loaded
piezoresistive stress gauges. Measurement Science and Technology, 8:570–573,
1997.
N. K. Bourne, Z. Rosenberg, D. J. Johnson, J. E. Field, A. E. Timbs, and R. P.
Flaxman. Design and construction of the UK plate impact facility. Measure-
ment Science and Technology, 6(10):1462–1470, 1995.
C. H. Braithwaite. High Strain Rate Properties of Geological Materials. PhD thesis,
University of Cambridge, 2009.
P. Bridgman. The electrical resistance of metals under pressure. Proceedings of
the American Academy of Arts and Sciences, 52(9):573–646, 1917.
D. J. Chapman. Shock-compression of porous materials and diagnostic development.
PhD thesis, University of Cambridge, 2009.
J. S. Courtney-Pratt. Photography of fast Transients. Interferometric study of sur-
faces. PhD thesis, University of Cambridge, 1949.
D. H. Dolan. Foundations of VISAR analysis. Technical Report SAND2006-
1960, Sandia National Laboratories, 2006.
59
L. C. Forde. Ballistic Impact of Rods. PhD thesis, University of Cambridge, 2000.
G. R. Fowles, G. E. Duvall, J. Asay, P. Bellamy, F. Feistmann, D. Grady,
T. Michaels, and R. Mitchell. Gas gun for impact studies. Review of Scientific
Instruments, 41(7):984–996, 1970.
R. Hammond. Shock and ballistic properties of bainitic steels and tungsten alloys.
PhD thesis, University of Cambridge, 2004.
W. F. Hemsing. Velocity sensing interferometer (VISAR) modification. Review
of Scientific Instruments, 50(1):73–78, 1979.
P. Horowitz and W. Hill. The Art of Electronics. Cambridge University, 2nd
edition, 1989.
A. W. Lawson. The effect of hydrostatic pressure on the electrical resistivity of
metals. Progress in Metal Physics, 6(1):1–44, 1956.
A. C. Mitchell and W. J. Nellis. Shock compression of aluminum, copper, and
tantalum. Journal of Applied Physics, 52(5):3363–3374, 1981.
N. H. Murray. The response of alumina ceramics to plate impact loading. PhD
thesis, University of Cambridge, 1997.
Z. Rosenberg and Y. Partom. Longitudinal dynamic stress measurements with
in-material piezoresistive gauges. Journal of Applied Physics, 58(5):1814–1818,
1985.
Z. Rosenberg, D. Yaziv, and Y. Partom. Calibration of foil-like manganin
gauges in planar shockwave experiments. Journal of Applied Physics, 51(7):
3702–3705, 1980.
A. J. Smiel and T. J. Fisher. Broadband triboluminescence in silica core fiber
optic waveguides. Applied Physics Letters, 41(4):324–326, 1982.
D. C. Swift and R. G. Kraus. Properties of plastic ablators in laser-driven ma-
terial dynamics experiments. Physical Review E, 77(6):066402, 2008.
G. R. Willmott. Shock studies of kimberlite, diamond, and brittle embedded particles.
PhD thesis, University of Cambridge, 2004.
D. Yaziv, Z. Rosenberg, and Y. Partom. Release wave calibration of manganin
gauges. Journal of Applied Physics, 51(12):6055–6057, 1980.
60
CHAPTER 4
SHOCK TEMPERATURE
It was demonstrated in Chapter 2 that a shock wave irreversibly increases the
entropy, and hence temperature, of a material. The measurement of shock
temperature is an open problem: in their 1979 review of the field of shock
compression, Davison and Graham [1979] noted that
“The problem of measuring the temperature of shock-compressed
solids has a long history, but measurements of accuracy sufficient
for the development of an equation of state have yet to be made.
Such a measurement is of the highest value and any advance in
this area most welcome.”
In the subsequent three decades, much progress in the measurement of
shock temperature has been reported. However, despite significant advances,
a general-purpose diagnostic technique, capable of measuring temperatures
to within 1 % has yet to be proposed. A review of the literature regarding these
techniques is presented below, together with a discussion of new experimental
results on the measurement of shock temperature using foil thermistors.
4.1 Direct temperature measurements
Thermocouples
The Seebeck effect, first reported by Seebeck [1826], is a thermoelectric phe-
nomenon and refers to the electric field produced by a conductor in a tem-
perature gradient. If, as shown in Figure 4.1, two different conductors are
joined at two junctions and a temperature difference is maintained between
the junctions then, in the open circuit configuration, a potential difference is
measured. The Seebeck coefficient, S, or thermopower, of such a configura-
tion is defined as dV/dT .
A number of authors [Crosnier et al., 1966; Palmer and Turner, 1964; Yuan
and Billingsley, 1971] have fielded thermocouples as shock temperature diag-
nostics. It was found experimentally that the output of the thermocouple ex-
61
Figure 4.1: A thermocouple, after MacDonald [1962]. Material 2 is attached to material
1, and the junctions are held at different temperatures. In the open circuit
configuration shown, an potential difference, dV , is measured; the ratio of
dV to dT is the Seebeck coefficient, or thermopower, S, of the thermocouple
materials.
ceeded, by as much as a factor of six, that which would be expected given the
temperature rise in the material, as predicted by its equation of state. Whilst
the effect of static pressure on thermocouple emf are known [Bundy, 1961], the
dynamically measured emfs indicate that a further, dynamic, pressure effect
is at work.
Bloomquist et al. [1979] attempted to address some of the previously re-
ported uncertainties. By careful preparation of the thermocouple junction,
minimising any porosity which would lead to heating of the junction itself,
they were able to measure temperatures, in copper, that were within 20 % of
those predicted for stresses in the range 14.5–36.0 GPa, as shown in Figure 4.2.
Whilst it is unclear whether this discrepancy is due to inaccuracies in the mea-
surement or the equation of state, the authors attributed it to locally higher
temperatures at the junction, when compared to the bulk.
Further work [Bloomquist and Sheffield, 1980] attempted to measure the
temperature rises in PMMA, epoxy, fused silica, and sapphire. As shown in
Figure 4.3, the measured temperatures in PMMA do not fit well with theo-
retical predictions above 2 GPa. The authors note that in PMMA there is a
distinct change in the slope of the Hugoniot at this stress [Schuler, 1970], and
take this as evidence of some previously-unobserved exothermic reaction. It
was argued that, in the inelastic regime, an increase in the thermal diffusivity
by a factor of 103 would allow for the thermocouples to reach equilibrium, as
62
Figure 4.2: Measured thermocouple emfs in copper as a function of shock pressure,
from Bloomquist et al. [1979]. Temperature rises predicted from the equa-
tion of state have been converted to emfs, and the correction due to the
equivalent static pressure applied.
was suggested by the emf–time histories, and thus the measured temperatures
were truly the temperature of the material under investigation. In sapphire
and fused silica, which remain elastic over the stress range investigated, the
temperature rises were found to more strongly correlate with the expected
temperature rise in the copper of the thermocouple, and the thermocouples
were therefore concluded not to be in equilibrium.
Figure 4.3: Measured temperatures in PMMA as a function of shock pressure, from
Bloomquist et al. [1979], compared with predicted values from the equa-
tions of state of McQueen et al. [1970] and Nunziato and Walsh [1973].
63
However, Rosenberg and Partom [1982] disagree with this interpretation,
suggesting that the observed response is more likely due to a strain depen-
dence of the thermocouple emf, as the large lateral extent of the thermocou-
ples meant they would experience significant strain. This alternative inter-
pretation was not accepted by Bloomquist and Sheffield [1982]. It is difficult
to accept the unsupported conclusion that some hitherto unknown processes
are at work in PMMA, and further study is perhaps required. Whilst there
have been some attempts [Ishutkin et al., 1986] to measure the the effect of dy-
namic pressure on thermocouple emf, the high uncertainty (20 %) associated
with these measurements do not provide a reasonable constraint on the effect.
In addition, the lingering questions as to what is actually being measured -
a purely thermal effect, or some complex mix of heat, pressure, and strain -
render thermocouples rather ineffective shock temperature diagnostics.
Radiative techniques
All hot objects radiate energy, the spectrum of which depends on their tem-
perature. The theoretical formulations for such spectra have been discussed
at length [e.g., Longair, 2003]. A pyrometer is a device capable of measuring
the intensity of radiation from a body at one or more wavelengths [Russell
et al., 1940]. Assuming the material acts as a ‘grey body’ (i.e., the emissivity
is independent of frequency) it is possible to determine the temperature from
the ratio of these radiative intensities.
Early work on optical determination of residual temperature, where the
test material has been released from a free surface, in metals [Taylor, 1963],
and of shock temperature in transparent media, where radiation propagates
ahead of the shock through the unshocked material [Kormer, 1968; Kormer
et al., 1965] found agreement with predicted temperature rises [Clark, 1959;
Walsh and Christian, 1955]. However, the sensitivities of the detectors em-
ployed limited the measurements to temperatures of order 10000 K (i.e., pres-
sures of order 100 GPa. The temperature sensitivity is partly determined by
the detector wavelength: the ‘grey body’ spectrum at lower temperatures does
not vary greatly in the visible wavelength range, increasing the experimental
uncertainty; the use of infrared detectors is required to decrease uncertainties
at these lower temperatures.
Subsequent improvements in detector technology allowed shock temper-
ature measurements to be extended to lower regimes. Von Holle and Trimble
64
[1976] studied copper in the 30 – 50 GPa range using two-wavelength infrared
pyrometry. Measured temperatures (Figure 4.4) were higher than those pre-
dicted [McQueen and Marsh, 1960]. This discrepancy was attributed to irre-
versible effects in the surface layer which are not counteracted by heat con-
duction to the bulk over the timescale of the experiment.
Figure 4.4: Residual temperature rises in shock-compressed copper, from Von Holle
and Trimble [1976]. Experimental data are compared to predicted temper-
atures from McQueen and Marsh [1960]. The highest experimental data
point is from Taylor [1963].
Swift et al. [2007] demonstrate that, whilst it is possible to take into ac-
count effects such as plastic work at surfaces, detailed prior knowledge of
the equation of state of the material is required. In their studies of shocked
molybdenum, Swift et al. [2007], were, however, unable to discriminate be-
tween various models for the temperature, as their experimental uncertain-
ties (of order 10 % on temperatures of 600–700 K) were too large to facili-
tate this. Furthermore, a number of other challenges to pyrometric measure-
ments have been reported. These include absorption and/or luminescence
in window materials [Nellis and Yoo, 1990]; the effect of surface preparation
[Shenton-Taylor et al., 2009]; surface jetting (localised plastic flow) [Swift et al.,
65
2007]; and hotspots/spatial inhomogeneities in the temperature, wavelength-
specific emissions, and light that is non-thermal in origin (e.g., impact flash
or nitrogen luminesecence at the shocked surface) [Seifter and Obst, 2007].
One further challenge is the determination of spectral emissivity; emissivity
is poorly constrained by pyrometric measurements [Nellis and Yoo, 1990] and
its determination would enable the uncertainty in measured temperature to
be reduced. Whilst some progress has been made in this area [e.g., Bakshi
et al., 2009; Partouche-Sebban and Pe´lisser, 2003; Pottlacher and Seifter, 2002;
Shenton-Taylor et al., 2009], and a number of techniques have been proposed,
dynamic emissivity measurements are still in their infancy and have not, as
yet, proved a useful constraint on temperature measurements.
Raman scattering [Raman and Krishnan, 1928] describes the inelastic scat-
tering of photons. Incident light, at a frequency ν0 is scattered to a lower
frequency, ν0 − νi, (i.e., energy is absorbed) — so-called Raman Stokes scat-
tering — or, to a higher frequency ν0 + νi (Raman anti-Stokes scattering). νi
is the Raman frequency. ‘Incoherent spontaneous Raman scattering’ [Moore
and Schmidt, 1989; Schmidt et al., 1984] can be used to measure temperature.
For a particular Raman frequency, the ratio of intensities of Stokes to anti-
Stokes scattering is directly proportional to the temperature. As spontaneous
scattering does not affect the vibrational thermal equilibrium significantly, the
temperature of the test material may be determined from this intensity ratio
[Holmes et al., 1984]. Raman spectroscopy has been used to measure shock
temperatures in a variety of liquids [Pangilinan and Gupta, 1994, 1997a; Winey
et al., 2000]. Uncertainties in the experimentally determined temperatures
were of order 15 %, which is comparable to pyrometric techniques. Raman
spectroscopy does not suffer from the same wavelength-based limitation that
pyrometry does: the Raman frequencies are independent of temperature, as it
is the ratio of Stokes to anti-Stokes scattering that is temperature dependent.
This suggests that Raman spectroscopy may be of use at much lower tem-
peratures than pyrometry, although it is only applicable to systems in which
active Raman modes exist. In addition, as noted by Pangilinan and Gupta
[1997b], Raman-based temperature measurements, whilst conceptually better
than many other techniques, suffer from practical difficulties relating to poor
signal-to-noise ratios and difficulties in data analysis. Significant advances in
these areas have yet to be reported in the open literature [e.g., Sato et al., 2007].
66
Thermistors
The temperature-dependence of resistance in metals has been known for some
time [Siemens, 1871]. As shock temperature diagnostics, thin foils of thermo-
resisitive materials can be considered an extension of the standard thin-foil
stress measurement technique, discussed in Chapter 3. A thermo-resistive
material, or thermistor, subject to an incident shock wave undergoes a change
in resistance due to some combination of the shock heating and its piezo-
resistive response. The pressure dependence of resistance has been measured
for a large number of materials by Bridgman [1964]; these results may be em-
ployed to correct for the effect of stress on resistance.
The stress can be calculated for standard materials, or a manganin gauge
can employed to measure the longitudinal stress in the conventional way.
Bridgman’s data, tabulated in Lawson [1956], gives the relationship between
resistance change and volumetric strain under hydrostatic loading conditions.
To calculate the resistance change due to pressure under dynamic uniaxial
loading, these values must be corrected. As reported by Barsis et al. [1970]
the dynamic resistance change is increased at a given pressure by two-thirds
the absolute change in volume. Once the resistance change due to the applied
stress is calculated, any remaining resistance change in the thermistor must
be due to a change in its temperature. Bridgman [1917a,b] demonstrated that
the temperature coefficient of resistance for a number of metals is virtually
independent of hydrostatic pressure. Therefore, in reducing the measured
resistance change to a temperature, it is sufficient to use the temperature co-
efficient of resistance measured at room temperature. Nickel has a high tem-
perature coefficient of resistance, and is one of the materials listed in Lawson
[1956] in which the pressure dependence of resistance has been calculated.
Therefore, it is well suited for use a thermistor. Kormer [1968] and Dremin
et al. [1973] demonstrated the feasibility of using a thin-foil nickel thermis-
tor as shock temperature diagnostic by showing that a change in resistance is
measured after account has been taken of the pressure effect.
In order to further investigate the behaviour of thin nickel gauges, a se-
ries of experiments were conducted using commercially available, 5µm thick,
nickel foil gauges (ETG-50B/E, Vishay Measurements Group U.K. Ltd) to mea-
sure temperature rises in PMMA. A constant-excitation Wheatstone bridge
amplifier (SGA400, Pentagon Instruments) was used with a bridge voltage of
0.2 V. As the resistance of nickel varies strongly with temperature, a lower ex-
67
citation voltage can be used than with manganin gauges. This has the advan-
tage of producing no significant self-heating of the gauge over periods of sev-
eral hours, much longer than the time required to set-up and perform a plate
impact experiment. Initial experiments, replicating the geometry employed,
were found to be in agreement with measurements reported by Rosenberg
and Partom [1981].
In order to investigate the temporal response of the nickel gauge, an exper-
iment was conducted to measure the residual temperature in PMMA. Man-
ganin and nickel gauges were sandwiched between two 2 mm thick PMMA
plates. When impacted by a copper flyer at 0.672 kms−1, this results in a
2.6 GPa shock that persists for around 1µs before released longitudinally from
the rear of the target. The stress in this loading cycle remains entirely uniaxial.
The reduced stress and temperature traces from this experiment are shown in
Figure 4.5.
After the gauge plane has been released, thermal conduction should bring
the nickel gauge foil into equilibrium with the surrounding PMMA. The tem-
perature trace in Figure 4.5, however, is clearly still rising, indicating that equi-
librium has not been reached. This seems to confirm the suggestion made by
Rosenberg and Partom [1981] that thinner foils are required.
Producing thinner nickel foils is, however, challenging. Whilst sputter
coating/vapour deposition techniques allow for thin films to be deposited,
it is difficult to produce films of sufficiently large initial resistance to enable
measurements to be made. The necessity to deposit such films on to some sub-
strate, and to electrically insulate them from conducting media, will increase
the time required for them to reach thermal equilibrium. Indeed, in liquids,
or any material that is not identical to the substrate, understanding the gauge
response becomes complex. If the substrate and test material can be consid-
ered identical, the gauge behaves as a thin layer in an infinite bulk. In the
case of a gauge in a more complicated environment, it is not clear how the re-
sponse of the gauge would depend on the properties of the substrate and the
material it is embedded in. Whilst these gauges show some promise as shock
temperature diagnositcs, the development of sub-micron thickness nickel foil
temperature gauges is considered to be beyond the scope of the present work.
68
Figure 4.5: Stress-time and temperature-time histories, from Experiment E090603A. A
PMMA target was impacted by a copper flyer at 0.672 km s−1. Manganin
and nickel gauges were embedded in a plane 2 mm from the impact face;
the total target thickness was 4 mm. The plane is released longitudinally
approximately 1µs after the shock first enters the gauges. The temperature
trace (red) is still rising at this point, indicating that thermal equilibrium
has not been reached.
Conclusions
Radiative techniques - either pyrometry or Raman spectroscopy - show promise
for the future, but at present are hampered by a number of significant experi-
mental challenges limiting their accuracy and usefulness. Indeed, these tech-
niques have generally only been deployed in materials where a pre-existing
equation of state is available, in order to verify the measured temperature;
at present the experimental uncertainties have proved too great to allow for
choices between different material models, and currently provide only the
loosest of bounds on equations of state. The present research confirms that
in-material techniques, based around thin-foil thermistors, show promise as
probes of temperature in shock experiments. However, experiments demon-
strate that these foils must be significantly thinner than those currently avail-
69
able in order to reach thermal equilibrium. The development of such foils is
beyond the scope of the present work.
4.2 Indirect temperature measurements
With the limitations of direct measurements of shock temperature in mind,
the present work focuses on determination of shock temperature through in-
direct means, and specifically through determination of the equation of state.
Walsh and Christian [1955] note that the temperature, T , on the Hugoniot of a
material may be calculated by integration of the thermodynamic identity
TdS = CvdT + T
∂P
∂T
∣∣∣∣
V
dV, (4.1)
where S is the entropy and Cv is the heat capacity at constant volume
Cv =
∂E
∂T
∣∣∣∣
V
, (4.2)
provided Cv and ∂P/∂T |V are known. Thus, the measurement of a thermo-
dynamically consistent equation of state enables calculation of shock tempera-
tures. By assuming constant values for Cv and ∂P/∂T |V , Walsh and Christian
[1955] calculated temperature rises in copper, aluminium, and zinc from their
pressure and volume Hugoniot measurements in these materials. Walsh et al.
[1957] calculated temperature rises in twenty-seven metals, having assumed
a Mie-Gru¨neissen equation of state, where the Gru¨neissen parameter,
Γ =
V
Cv
∂P
∂T
∣∣∣∣
V
, (4.3)
was calculated from theoretical consideration of the vibrational modes of the
metals. The validity of these equations of state was generally determined
by comparing isothermal pressure-volume relationships, calculated using the
equation of state and Hugoniot, with isotherms measured directly under static
compression. Cowperthwaite and Shaw [1970] performed similar calculations
and verified their results by direct comparison with measured temperatures
in carbon tetrachloride. The accuracy of these calculations is of order 2 %.
Gurtman et al. [1971] applied a Mie-Gru¨neissen equation of state to wa-
ter under the assumption that the heat capacity Cv is constant. For a Mie-
Gru¨neissen equation of state, Equation 4.1 becomes
70
dT =
(V0 − V )
2Cv
dP +
[
(P − P0)
2Cv
− T Γ
V
]
dV. (4.4)
Thus, temperatures along the Hugoniot may be determined by integration
of Equation 4.4, providing a continous form of Γ(v) is available. The standard
assumption made in the literature, and which is made in this thesis, is that the
heat capacity is constant. This assumption that the heat capacity is constant,
which may or may not be justified in any given material, and which greatly
simplifies the model, introduces a certain level of uncertainty to the calcula-
tion of temprature. As is discussed later in this thesis, it is the uncertainity
in the heat capacity which becomes the dominant contribution to the uncer-
tainty in calculated temperatures. By combining the experimental results of
Walsh and Rice [1957] with isothermal compression data [Bridgman, 1942]
for water, they constructed a functional form for Γ(v) from v = 1 cm3 g−1 to
v = 0.49 cm3 g−1, which is reproduced in Figure 4.6. The validity and con-
sistency of the equation of state was checked by comparison with published
adiabats and isotherms. Figure 4.7 shows the calculated shock temperatures
in water from the constant Cv model proposed by Gurtman et al. [1971] to-
gether with those determined by Walsh and Rice [1957] and Cowperthwaite
and Shaw [1970]. The disagreement with the model proposed by Cowperth-
waite and Shaw [1970] was attributed to that model neglecting the compres-
sional contribution to internal energy. The close agreement with Walsh and
Rice [1957] was taken as evidence that constant Cv is an appropriate model
for water; uncertainties were of order 2 %. Hence, it is concluded that when
Γ(V ), the Hugoniot, and heat capacity Cv of a material have been determined,
temperatures along the Hugoniot may be determined in a consistent manner
and with uncertainties of a few percent.
The remainder of this chapter describes the experimental techniques that
will enable the calculation of shock temperatures in liquids. First, methods of
determining the liquid Hugoniot are described, and their effectiveness com-
pared. The equation of state is determined from shock/reshock experiments,
and the analysis of these experiments is discussed with reference to standard
materials.
Determination of the Hugoniot of an unknown material
As discussed in Chapter 2, the principal Hugoniot of a material is the locus
of states attainable from ambient conditions due to compression by a single
71
Figure 4.6: Γ(v) for water, from the fit given by Gurtman et al. [1971]. The fit
is a seventh-order polynomial through, at high volumes, the isothermal
compression data given by Bridgman [1942] and, at low volumes, the
shock/reload experiments reported by Walsh and Rice [1957]. These lat-
ter data were quoted without errors. It can be seen that Γ initially increases
with decreasing volume, before reaching a maximum and beginning to de-
crease.
shock wave. Conventionally, Hugoniots are presented in terms of the shock
equation of state, US = c0 +Sup, and therefore both the shock velocity US and
particle velocity up must be determined from experiment.
Trivially, the shock velocity may be inferred from the difference in arrival
time of the shock at two points separated longitudinally within a sample by
some known distance. In a real sample, such a measurement is complicated
by a number of factors, chief amongst which is the the passage of the shock
through materials other than the sample. Where the arrival time is measured
with manganin gauges, the gauges must be insulated as discussed in §3.2 and,
hence, the shock propagates through the gauge package at a velocity deter-
mined by the Hugoniot of the insulating material. Furthermore, it is often
72
Figure 4.7: Shock temperatures in water, assumingCV is constant, from Gurtman et al.
[1971]. Their formulation was compared to other models in the literature.
The disagreement with the model proposed by Cowperthwaite and Shaw
[1970] was attributed the latter model neglecting the compressional contri-
bution to internal energy. The close agreement with the calculations made
by Walsh and Rice [1957] was taken as evidence that a constant CV model
is appropriate for water.
73
necessary to include a buffer material between the gauge package and sample
and, as before, the shock velocity in this medium must be accounted for when
deriving the shock velocity in the sample
For the liquid impact cell shown schematically in Figure 4.8, the total time-
of-flight of the shock comprises passage through gauge packages at the front
and rear, a buffer ahead of the sample, and the sample itself. The gauges are
considered to be located in the middle of each package and therefore it is half
the thickness of each gauge package that must be considered.
Figure 4.8: Schematic representation of a typical liquid impact cell. In this represen-
tation, the projectile would impact the left-most thin plate generating a
shock that propagates through the front buffer plate and liquid sample be-
fore reaching the rear anvil. The two gauges are embedded in glue and,
where necessary, insulating layers. The time-of-flight of the shock between
the two gauges is a function of the thickness of the buffer and sample and
the half-thicknesses of both gauge packages.
The glue, insulating material, and gauge substrate are, typically, three dif-
ferent materials; however, the shock response of each material is similar to the
shock response of PMMA. As the gauge packages are very thin, it can be as-
sumed, without the introduction of significant error, that the shock response
of the entire package is identical to that of PMMA1. The shock velocity in
1For a gauge embedded between PMMA plates, the lack of effects due to impedance mis-
matches show that, within the diagnostic sensitivity of the gauge, this assumption is valid.
74
the front gauge package can be inferred from conventional impedance match-
ing; whilst the shock velocity in the rear gauge package will differ (and will
depend on the shock response of the sample) from that in the front gauge
package, the error associated with taking the shock velocity in both packages
to be identical is again insignificant. As with the gauge package, impedance
matching arguments can be employed to infer the shock velocity in the buffer
material. Obviously, it is important to use as a buffer a material with a well-
known Hugoniot.
The thicknesses of the various relevant parts of one such cell are listed in
Table 4.1. In experiment E110211B, the front buffer material was PMMA and
the sample material was water at 66 ± 2 ◦C. Figure 4.9 shows the front (red)
and rear (black) gauge traces from experiment E110211B, due to the impact of
a PMMA flyer at 0.589± 0.006 km s−1.
Cell part Thickness /mm
Front buffer (PMMA) 2.003 ± 0.003
Front gauge package (PMMA) 0.089 ± 0.015
Sample thickness (Water) 3.601 ± 0.001
Rear gauge package (Copper) 0.220 ± 0.039
Table 4.1: Thickness of the various parts of the cell used in experiment E110211B. The
front buffer material was PMMA and the sample material was water at 66±
2 ◦C.
As the impact is symmetric, the Hugoniot state of the buffer can simply
be determined from the impact velocity, as the particle velocity in the PMMA
is half the impact velocity. The initial plateau in the front gauge trace shown
in Figure 4.9 corresponds to the resultant Hugoniot state in PMMA. Thus,
the shock velocity and, hence, transit time through the front buffer may be
determined. The liquid has a lower impedance than the PMMA and, hence,
the shock which propagates through the liquid is also reflected as a release
fan into the buffer. After the release process is complete, the front gauge
equilibrates at the Hugoniot stress of the liquid, corresponding to the sec-
ond plateau in the front gauge trace in Figure 4.9. In experiment E110211B,
this stress was measured as 0.753 ± 0.010 GPa. The difference in arrival time
of the shock between the front and rear gauges gives the total transit time of
the shock: in experiment E110211B, this was 2.470 ± 0.014µs. A single rever-
beration is visible in the rear gauge profile; this reverberation is due to the
impedance mismatch between the gauge package and the rear copper anvil.
75
The effect is small, due to the similarity in impedance between the gauge pack-
age and water, which can be deduced from the fact that the reverberation is at
a similar stress to the Hugoniot stress of the water. The exact effect can be cal-
culated using standard impedance matching techniques; it can be shown that
the effect of the reverberation is less than 0.1 % on the measured stress. This
means that the reverberation can be neglected as it has practically no effect on
the measured reload stress.
Figure 4.9: Front (red) and rear (black) stress gauge traces from experiment E110211B.
The front gauge is first shocked to the Hugoniot state of PMMA, before
being released to the Hugoniot state of the liquid at a stress of 0.753 ±
0.010 GPa; the shock reaches the rear gauge 2.470 ± 0.014µs after first ar-
riving at the front gauge. A reverberation is visible in the rear gauge pack-
age, which is caused by the impedance mismatch between the water, gauge
package, and rear copper anvil. The effect of this reverberation is suffi-
ciently small that it may be neglected, as the impedance of the gauge pack-
age is virtually identical to that of the water and so it does not significantly
perturb the reloading of the water.
From the measured stress and transit time, there are three methods, dis-
cussed below, which enable calculation of the particle velocity in the sample.
76
These methods are valid regardless of the relative impedances of the materi-
als involved as, in all cases, a shock will propagate between the front and rear
gauges.
Method 1: from σ and US
The first method assumes the propagation of a steady shock wave through
the sample. As such, the loading path is a Rayleigh line of slope ρ0US . This
Rayleigh line intersects the principal Hugoniot of the sample at the measured
longitudinal stress σ1b, as shown in Figure 4.10.
Figure 4.10: Illustration, in σ − up space, of Hugoniot determination by method 1: the
Rayleigh line, of slope ρ0US , intersects the principal Hugoniot of the sam-
ple at the measured longitudinal stress σ1b.
The uncertainty in the measured shock velocity, typically 2 %, and the un-
certainty in the measured stress, typically 2 %, combine to give an uncertainty
in particle velocity of 2 %. It should be noted that, as shown in Figure 4.15, if
the front anvil and buffer of higher impedance than the liquid sample, then
the stress σ1b is a release state in the anvil and, hence, in the gauge. For stresses
77
above the HEL of manganin, consideration must be given to the behaviour of
such gauges in release, as discussed in §3.2. The net result is to increase the
uncertainty associated with the stress, and this is especially true for releases
that are small in amplitude (i.e., when the release propagates from an interface
between two materials of similar impedance.)
In experiment E110211B, the shock velocity, determined from the transit
time shown in Figure 4.9 was 2.040± 0.022 km s−1. Using method 1, this gives
a particle velocity of 0.377± 0.007 km s−1.
Method 2: from σ and vimpact
The second method constrains the measured stress to lie on the release curve
of the anvil material, as shown in Figure 4.11. Hence, the method requires
knowledge of the release curve of the anvil material, but neglects the mea-
sured shock velocity. The release curve of the anvil is typically approximated
by reflecting its Hugoniot through the state given by the intersection of the
anvil Hugoniot with that of the flyer.
Uncertainties in the particle velocity and shock velocity determined2 by
this method are a function of the uncertainties in impact velocity (1 %), stress
(2 %), and the Hugoniot of the flyer and/or release characteristics of the anvil
(1 %). Hence, the total uncertainties are typically 2 % on particle velocity and
4 % on shock velocity.
Method 2 gives a particle velocity for experiment E110211B of 0.372 ±
0.007 km s−1 and, assuming a steady shock, a shock velocity of 2.068±0.056 km s−1.
Method 3: from vimpact and US
The final method again constrains the Hugoniot state of the sample to the re-
lease state of the anvil, and similar arguments regarding uncertainty in that
state apply here. However, as with the first method, the shock is assumed to
be steady and the measured shock velocity used to define a Rayleigh line of
slope ρ0US . The intersection of this line with the release curve, as shown in
Figure 4.12, defines both the particle velocity and stress on the sample Hugo-
niot.
2The shock velocity may be determined by assuming the shock is steady and thus that stress–
particle velocity Hugoniot is intersected by a Rayleigh line of slope ρ0US at the measured stress.
although this is not a necessary requirement of the method.
78
Figure 4.11: Illustration, in σ − up space, of Hugoniot determination by method 2: the
measured longitudinal stress, σ1b, is constrained by measurement of the
impact velocity vimpact to lie on the release curve of the flyer which, for
symmetric impact is taken to be identical to the Hugoniot of the flyer. The
particle velocity is therefore inferred from a single point of measurement
and, hence, it is not necessary to assume that a steady shock propagates
through the sample.
The uncertainties in impact velocity (1 %), shock velocity (2 %), and the
Hugoniot of the flyer and/or release characteristics of the anvil (1 %) give rise
to typical uncertainties of 2 % on particle velocity and 2 % on stress.
The shock velocity for experiment E110211B, is calculated from the transit
time, as it was for method 1, and is 2.068±0.056 km s−1 at a particle velocity of
0.374±0.005 km s−1; the stress calculated using method 3 is 0.746±0.010 GPa.
The three methods, as they rely on somewhat different assumptions, yield
slightly different values for the shock and particle velocity state. For exper-
iment E110211B, these states are plotted in Figure 4.13, and are found to lie
within error of each other. This may not always be the case. Figure 4.14 shows
the Hugoniot states in water compressed from room temperature as deter-
79
Figure 4.12: Illustration, in σ − up space, of Hugoniot determination by method 3: the
Rayleigh line, of slope ρ0US , intersects the release curve of the anvil. The
particle velocity in the sample is therefore inferred from measurements of
the shock velocity and impact velocity.
mined by the three methods. Method 2 is clearly at odds with the other two
methods. This is due to the method relying solely on the measured stress,
which is unreliable for gauges in partial release, as is the case in the experi-
ments reported in the present work. Methods 1 and 3, which are reliant on the
transit time, are considered more accurate, as there is greater confidence in the
measurement of transit time than of stress. Due to its unreliability, method 2 is
not subsequently used in this thesis. The systematic uncertainties in methods
1 and 3 are discussed further in Chapter 5. By conducting a series of experi-
ments at different impact velocities, it is possible to generate a range of shock
velocity/particle velocity pairs and thus determine the previously unknown
Hugoniot of the test liquid.
80
Figure 4.13: Hugoniot state, in shock velocity–particle velocity space, of experiment
E110211B, as determined by the three methods described above. In this
case, the three methods agree with each other within their experimental
uncertainties.
Shock/reload experiments
If the rear anvil of the liquid cell shown schematically in Figure 4.8 is of a
higher impedance than the liquid sample, then a shock is reflected from the
interface back in to the sample. The propagation of shocks and releases in
such a sample in x—t space is represented in Figure 4.15.
Assuming that the flyer and front anvil/buffer materials are identical (i.e.,
that the impact is symmetric) and are of higher impedance than the sample
liquid then the initial shock that propagates into the sample is reflected back
through the buffer towards the front gauge. Thus, the front gauge is shocked
first to a state on the principal Hugoniot of the buffer material before being
released to the Hugoniot state of the liquid. The shock that is transmitted into
the liquid sample is reflected at the interface with the rear anvil, as the anvil is
of higher impedance than the sample. The reflected (backward-propagating)
81
Figure 4.14: Hugoniot state, in shock velocity – particle velocity space, determined by
the three methods described above in water compressed from room tem-
perature. The unreliability of method 2 is evident, and it is not subse-
quently used in this thesis.
shock acts to increase the pressure and density of the sample whilst decreasing
its particle velocity.
It is possible, graphically, to solve, approximately, for the states obtained
in the liquid, as demonstrated in Figure 4.16. Initially, the gauge in the front
anvil/buffer is shocked to the state, 1a, defined by the intersection of the
Hugoniot of the anvil/buffer with that of the flyer. The gauge is subsequently
released to the Hugoniot state, 1b, in the liquid. A separate release curve has
been drawn for the front anvil material; as has been noted above, this release
curve is conventionally approximated by the reflected Hugoniot of the anvil.
The Hugoniot state can be determined using the methods described above.
The reload state, 2, is constrained in P—up space to lie on the Hugoniot of the
rear anvil. Idealised responses for the front and rear gauges, demonstrating
the stress levels corresponding to the three states attained, are shown in Fig-
82
Figure 4.15: x—t diagram for shock and release propagation in the liquid impact cell
(Figure 4.8) subject to a symmetric impact. Gauges are placed between
the front anvil and buffer (dotted line) and at the interface between the
liquid and the rear anvil. Both the front buffer and rear anvil are higher in
impedance than the sample. The liquid sample is taken to a state on the
principal Hugoniot by the forward-propagating shock entering from the
buffer; the backward-propagating shock that is reflected from the interface
with the rear anvil takes the liquid to a reloaded state that is not on the
principal Hugoniot. Secondary interactions of release fans are irrelevant
to the present discussion and not shown.
83
ure 4.17 and should be compared to the real traces shown above in Figure 4.9.
Analysis of gauge traces, in particular when the measured stress is a state in
partial release, is discussed in §3.2.
Figure 4.16: Representative graphical solution, in P—up space, for shock/reload ex-
periments. Labelled states - 1a, 1b, and 2 - correspond to those that would
be measured in the front and rear gauges and are, respectively, the Hugo-
niot state in the front anvil/buffer, the Hugoniot state in the liquid, and
the reload state in the liquid. The general case, where the anvil’s release
curve differs from the Hugoniot, is shown; the effect of neglecting this is
discussed in §4.2.
Figure 4.18 shows the relationship between the shock (i.e., Hugoniot) and
reload states for a typical experiment in (a) P—up and (b) P—V space. In the
limit that the material was loaded to a given stress by an infinite number of in-
finitesimally small compression waves, the loading would be quasi-isentropic.
The shock/reload experiment is, hence, the first step towards such a quasi-
isentropic compression, and the reload state, therefore, has a lower entropy
84
Figure 4.17: Idealised response, in P—t space for the two gauges indicated in Fig-
ure 4.8; the labelled pressures correspond to those in Figure 4.16. The
gauge responses are shown for the case where the materials behave per-
fectly plastically: i.e., there is no elastic component to either shocks or re-
leases. The idealised response should be compared to the real experimen-
tal data shown in Figure 4.9.
and, hence, a lower pressure and temperature, for a given specific volume,
and, thence, lies below the principal Hugoniot in P—V space. It is this rela-
tion, between Hugoniot and reload states along an isochor, that is required in
determining a material’s equation of state.
Figure 4.18: Representations of the states attained in a a typical shock/reload experi-
ment in (a) P—up and (b)P—v space. In the pressure—volume represen-
tation, the reload state lies below the corresponding isochoric Hugoniot
state.
85
Data reduction for shock/reload experiments
Hence, it is necessary to determine the specific volume corresponding to the
stress measured in the reload state. The reduction of data for shock/reload
experiments presented below follows the method described by Brown et al.
[2007] in their experiments on the shock compression of dry sand.
The inferred Lagrangian shock velocity associated with the reloaded state,
US2 is
US2 =
σ2 − σ1
ρ0(up1 − up2) , (4.5)
where the subscripts 1 and 2 refer to the the measured Hugoniot and reload
states respectively.
The Hugoniot state in shock/reload experiments can be determined using
the methods described above for constructing the Hugoniot of an unknown
material. The particle velocity in the reloaded state is constrained to lie on the
Hugoniot of the rear anvil and may be inferred from the measured stress; the
same stress corresponding to the reloaded state of the liquid is an Hugoniot
state for the anvil.
The volume, V2 in the reloaded state is thence
V2 = V0
(
ρ0
ρ1
− up1 − up1
US2
)
. (4.6)
It should be recalled that the Gru¨neissen parameter was defined in Chap-
ter 2 as the thermodynamic derivative of pressure with respect to internal
energy at constant volume:
Γ = V
∂P
∂E
∣∣∣
v
(4.7)
and therefore, in the reloaded state, Γ is approximately given by the rela-
tion
Γ ≈ VR PH − PR
EH − ER (4.8)
where subscript R denotes the measured reload state and H the Hugoniot
state on the isochor through VR. Furthermore, in an inviscid fluid, the hydro-
dynamic pressure and longitudinal stress in uniaxial compression are identi-
86
cal3 and, therefore, Equation 4.8 may be rewritten in terms of the measured
stresses viz.,
Γ ≈ VR σH − σR
eH − eR . (4.9)
Hence, from the previously-measured principal Hugoniot, the measured
stress in the reload state, and the calculated specific volume on reload, it is
possible to determine the equation of state of the liquid.
Equation 4.9 may be rewritten, by substitution of Equations 4.5, 4.6 and the
known Hugoniots (in the form of the shock equation of state) of the sample
and rear anvil, in terms of experimentally measured quantities. The resultant
equation is a function of the measured stresses in the Hugoniot and reload
states, the measured shock velocity in the Hugoniot state, and the parameters
ρ0, c0, and S for both the sample and rear anvil. Assuming the uncertainties
in these nine quantities are independent, the uncertainty in Γ is given by the
sum, in quadrature, of these errors.
The above analysis was applied to experiment E081105C. In this exper-
iment, a PMMA sample plate was placed between two copper anvils. The
target was impacted by a copper flyer plate at 0.458±0.005 km s−1. The resul-
tant front and rear gauge traces are shown in Figure 4.19, and show the form
expected from Figure 4.17.
Experiment E081105C was not originally designed for the determination
of the equation of state of PMMA and the reload state is at comparatively
low stress where differences between the Hugoniot and reload curve are min-
imal. However, the measured state does lie below the Hugoniot in stress–
specific volume space, as is evident in the inset graph in Figure 4.20. The
principal Hugoniot plotted in Figure 4.20 is from Asay and Shahinpoor [1993,
Appendix C]. Applying Equation 4.9 gives a value of Γ = 1.1 ± 0.8; the large
uncertainty is primarily due to the close proximity of the reload state to the
principal Hugoniot, in that the error bars in stress overlap with the principal
Hugoniot. This measured value is within 15 % of the literature value at zero
pressure, Γ0 = 0.97 [Asay and Shahinpoor, 1993, Appendix C].
3This can be seen by noting that pressure is defined as one-third the trace of the stress tensor,
and is equal to the longitudinal stress when the shear strength is zero [e.g., Davison, 2008].
87
Figure 4.19: Front (red) and rear (black) stresses in experiment E081105C. Experiment
E081105C consisted of a PMMA block between two copper anvils, sub-
jected to the impact of a copper flyer plate at 0.458± 0.005 km s−1.
4.3 Summary
• A number of techniques for direct measurement of shock temperature
have been presented in the literature.
• The uncertainties in temperatures measured with these methods are gen-
erally of order 10 %.
• Experiments performed with foil thermistors demonstrate that the tech-
nique shows promise, but difficulties in creating foils that are thin enough
to reach thermal equilibrium over the time-scale of a shock experiment
prevent their use in this thesis.
• Temperatures calculated from thermodynamically consistent equations
of state generally have uncertainties of order 2 %. One such equation of
state is the Mie-Gru¨neisen equation.
88
Figure 4.20: Experiment E081105C represented in stress – specific volume space. The
solid black curve is the literature Hugoniot for PMMA [Asay and Shahin-
poor, 1993, Appendix C]. The Hugoniot and reload states have been la-
beled. The inset shows a more detailed view of the stress–specific volume
space near the reload state, demonstrating that it lies below the principal
Hugoniot, although due to the close proximity of reload and Hugoniot
stresses at low compressions, the uncertainities in stress overlap the prin-
cipal Hugoniot.
• Determination of the equation of state requires characterisation of the
principal Hugoniot. Experimental techniques to allow the measurement
of the principal Hugoniot and determination of the Gru¨neisen parame-
ter, Γ, from shock/reload experiments have been discussed and evalu-
ated.
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94
CHAPTER 5
SHOCK COMPRESSION OF WATER
In addition to the usual considerations of flatness and planarity discussed in
Chapter 3, shock experiments involving liquids also require an effective vac-
uum seal around the liquid, and the chemical compatibility of the liquid with
the materials surrounding it must be considered. The term ‘cell’ is used to de-
scribe the liquid-containing target. Careful consideration must also be given
to the propagation of lateral release waves, as there will exist an impedance
mismatch between the liquid and the material of the cell’s wall. The first part
of this chapter concerns the design of cells for shocking liquid samples. The
cell must be vacuum-tight, to prevent the liquid being drawn out of the cell
as the gun is evacuated. The liquid itself must be confined in such a way that
it has a well-defined thickness and parallel, planar faces. The development of
such a cell is discussed below.
5.1 Cell development
In order to accurately determine the Hugoniot and equation of state of the
liquids considered in this thesis, several quantities must be determined from
each experiment: the Hugoniot and reload stresses, and the transit time of
the shock through the liquid. Thus, an experimental design that produces
two stress histories, from the front and rear gauges, that have well-defined
rises (associated with the arrival of the shock at each gauge) and reach steady
plateaus reflecting the appropriate stresses, is required. The final cell used in
this thesis was developed in several stages, which are described below. At
each stage, small modifications were made to the cell design in order to ob-
tain the desired gauge traces. A number of unexpected experimental difficul-
ties were encountered during the development process. The first part of this
chapter considers these challenges and describes their solutions.
The initial liquid cell design is shown schematically in Figure 5.1 and was
constructed from three copper plates. The front and rear anvil are simple cop-
per plates; the central plate has a 2 mm deep pocket machined into it forming
the sample volume. The front gauge is then glued in place between the front
95
anvil and the central plate and the rear gauge is glued to the rear anvil. Teflon
was used to insulate the gauges, as it has been suggested that it results in
more robust gauge packages [Chapman, 2009b]. An ‘o’-ring groove machined
around the central plate allows the cell to be sealed when it is screwed to-
gether.
Figure 5.1: Schematic representation of the initial liquid cell; the horizontal scale is ex-
aggerated by a factor of 10. The front anvil is a 1 mm copper plate. The
sample, indicated by the shaded region, is contained within a ‘cup’ ma-
chined from a 4 mm copper plate: a 2 mm deep pocket is machined into
this plate, leaving 2 mm of copper to form the front buffer. The front gauge
is glued in place between these two copper sections, and the rear gauge is
bonded to the 10 mm copper rear anvil. A groove suitable for an ‘o’-ring is
machined into the copper ‘cup’ forming a seal outside the sample volume.
The cell was filled from the rear and the filling holes sealed with a bolt and
PTFE sealing tape.
Figure 5.2 shows the front and rear gauge traces from experiment E090417A
which resulted from the impact of a copper flyer at 0.702±0.007 km s−1 on the
cell described in Figure 5.1 and filled with water. A number of reverberations
can be seen in the front gauge trace; partly, these reverberations are due to the
impedance difference between the gauge package and copper and the thick-
ness of the gauge package. The large amplitude, and long time-scale, of the
96
reverberations indicate that the gauge package is of an unsatisfactory thick-
ness. The relatively thick gauge package is a result of the use of 125µm-thick
Teflon insulation which was the thinest Teflon available at the time; in order
to continue to use Teflon, it must be replaced with sheets that are consider-
ably thinner. The large reverberations may in part explain the lack of a single
steady plateau in the reload state observed on the rear gauge trace. In addi-
tion, the Hugoniot state of the water is ill-defined in the front gauge trace, as
the tail end of the longitudinal release from the water into the copper overlaps
with what appears to be a partial release. The source of this release is unclear,
and was not anticipated from consideration of the design, where the diameter
of the liquid sample is greater than that of the flyer plate.
In addition, it is difficult to ensure that the central copper plate is machined
in a way that ensures that its front surface is parallel to the front surface of the
‘pocket’. In order to ameliorate these problems, a modified design (Figure 5.3)
was implemented. The modified cell replaces the ‘one piece’ front buffer and
sample confinement with two parts - a simple buffer plate and an aluminium
annulus. An ‘o’-ring is supported against the internal diameter of this annu-
lus in order to provide a seal. This construction allows for greater control over
thickness and planarity of the various parts of the target and the gauge pack-
ages themselves. Additionally, a stock of thinner ( 30µm) Teflon was obtained.
The gauge traces from an impact on a water-filled cell by a copper flyer
at 0.298 ± 0.003 km s−1 are shown in Figure 5.4, and are representative of ex-
periments conducted with this cell design. After reaching the Hugoniot stress
of copper, the front gauge is released to the Hugoniot stress of the liquid. It
was found in over half of the experiments conducted with this cell design that
failure of the front gauge occured shortly after the arrival of the release wave.
The electrical noise associated with the front gauge failure is ‘picked up’ by
the rear gauge, as it acts as an antenna. The noise is such that it obscures not
only the arrival time of the shock at the rear gauge, but also the stress level
reached. The high failure rate makes it impractical to use this cell design.
Whilst it would be possible to eliminate the front gauge and still determine
the reload stress in the liquid, without the front gauge no determination of
the Hugoniot is possible as neither the Hugoniot stress of the liquid, nor the
transit time of the shock, would be measured. The ultimate cause of failure
of the front gauge package is believed to be due to the large magnitude of the
release wave that propagates into it across a significant impedance mismatch
(the copper/Teflon/copper boundaries). This was not a problem in the initial
97
Figure 5.2: Front (red) and rear (black) gauge traces from experiment E090417A, in
which the cell described in Figure 5.1 was impacted by a copper flyer at
0.702 ± 0.007 km s−1. A series of reverberations in the front gauge pack-
age are evident; in part this is due to the impedance mismatch between
the gauge package and the surrounding copper. The large amplitude and
time-scale of the reveberations are caused by the relatively thick (125µm)
Teflon that was available to insulate the gauge at the time. These reserva-
tions suggest that there would not be a single well-defined shock entering
the liquid, which may be responsible for the the rear gauge not reaching a
steady plateau at the reload stress of the liquid. In addition, the Hugoniot
stress in the liquid is poorly measured, as partial lateral releases affect the
front gauge shortly after the longitudinal release from the water/copper
interface has occurred.
98
Figure 5.3: Schematic representation of the second iteration of the liquid cell design;
the horizontal scale is exaggerated by a factor of 10. The front gauge is
glued between two thin copper plates, and the sample (shaded region) is
confined by a 2 mm aluminium annulus which supports an ‘o’-ring against
its internal diameter. The rear gauge and rear anvil are identical to those of
the initial cell design (Figure 5.1.) The cell was filled from the rear and the
filling holes sealed with a bolt and PTFE sealing tape.
design, as the relatively thick gauge package ‘smoothed out’ the effect of the
impedance mismatch. That the gauge package in the new cell design is sig-
nigicantly thinner can be observed by noting the lack of reverberations in the
shock rise in Figure 5.4 when compared to Figure 5.2.
In order to avoid these problems, a cell, shown schematically in Figure 5.5,
was developed that employed ‘off-axis’ gauges. In this cell, which is not ax-
isymmetric, the front and rear gauges do not share a common axis. The front
gauge is embedded between two copper plates, and thus not subject to a re-
lease from the liquid although, for this reason, it does not measure the Hugo-
niot stress in the liquid which, in addition to the measured shock velocity, is
used in the determination of the liquid Hugoniot.
Experiment E100623A subjected such a cell, filled with water, to the im-
pact of a copper flyer at 0.491 ± 0.005 km s−1. The resultant gauge traces are
99
Figure 5.4: Front (red) and rear (black) gauge traces from experiment E090611B. The
trace, which is representative of those conducted using the cell described
in Figure 5.3 demonstrates the failure of the front gauge during the release
process. The electrical noise associated with this failure is evident on the
rear gauge trace, and acts to obscure both the arrival time of the shock at
the rear gauge and the stress level that results. The cell was impacted by a
copper flyer at 0.298± 0.003 km s−1.
shown in Figure 5.6. Lateral release waves, in addition to propagating from
the edges of the cell are also generated at the copper/water interface at the
centre of the cell. The release that propagates from this interface is evident
in the front gauge trace; reloading of the front gauge occurs when the shock
in the water is transmitted into the copper at the copper/water interface. In
addition, the rear gauge is partially released before it reaches an equilibrium
value; again, it seems likely that releases at the copper/water interface are
responsible for this. These release waves were not anticipated to affect the
gauges until significantly later in the experiment. That they do suggests that
the releases propagate much faster than was initially anticipated
The shock velocity determined from the transit time measured in experi-
100
Figure 5.5: Schematic representation of the third iteration of the liquid cell design; the
horizontal scale is exaggerated by a factor of 10. The entire cell is con-
structed from copper, ‘o’-rings are placed in grooves machined in the front
anvil and confining ring in order to provide a seal. The thickness of these
two parts are partly defined by the available sizes of ‘o’-rings. The front
gauge is placed between the front anvil and copper confiner and thus does
not experience a significant longitudinal release. The rear gauge is placed
at the liquid/copper interface and the two gauges are placed 15 mm later-
ally from the central axis of the cell. The cell was filled from the rear and
the filling holes sealed with a bolt and PTFE sealing tape.
ment E100623A is some 10 % slower than the Hugoniot of water [e.g., Asay
and Shahinpoor, 1993, Appendix C] would imply. In order to bring the ex-
perimental and expected shock velocities into agreement, a tilt of the target
with respect to the flyer of around 100 mRad is required, which is an order of
magnitude greater than the maximum tilt observed in Chapter 3 and would
lead to much greater rise times than are observed. It seems therefore that
some property of the cell is causing this discrepancy between measured and
expected shock velocities. It is not clear, however, what the precise cause of
the discrepancy is, as any significant non-uniformity in the thickness of the
gauge packages would have been detected when measuring the cell before
assembly. It is not possible to measure impact tilt in this cell design, as the
relatively thick front plate is not suitable for drilling the necessary holes for
optic fibres and it would not be possible to sufficiently reduce the size of the
101
Figure 5.6: Front (red) and rear (black) gauge traces from experiment E100623A. The
cell shown schematically in Figure 5.5 was impacted by a copper flyer at
0.491 ± 0.005 km s−1. The front gauge is shocked to the Hugoniot state
of copper and experiences a number of partial lateral releases, the greatest
in magnitude of which comes from the copper/water interface. The front
gauge is also partially reloaded as the shock propagating through the water
is also transmitted at the copper/water interface. The rear gauge is partially
released, again due to wave interactions at the copper/water interface be-
fore it has reached an equilibrium value.
sample in order to allow for placement of the fibres. The lack of measurement
of the Hugoniot stress of the liquid mean that the transit time must be relied
upon to determine the shock velocity; without measurement of the tilt, relying
solely on the measured transit time is inaccurate. Further, the thicknesses of
the various plates can not be greatly reduced due to the need to include seal-
ing ‘o’-rings - rendering it difficult, if not impossible to measure the reload
stress in the liquid before releases have affected the gauges. For these rea-
sons, it was concluded that the ‘off-axis’ style of cell was not suitable for the
determination of either the Hugoniot of the liquid or its reload behaviour.
The final iteration of the liquid cell design returned to the on-axis design
102
shown in Figure 5.3, but replaced the front anvil, which had previously been
made from copper, with PMMA. As the impedance of PMMA is much closer
to that of the liquid, the release that propagates back into the anvil is small
in magnitude. In addition, the impedance of the glue in which the gauge is
embedded is conventionally taken as identical to the impedance in PMMA
and thus, there no significant impedance mismatch is presented by the gauge
package. This modification of the front anvil material reduces the particle
velocity that can be imparted through symmetric impact, due to the lower
impedance of PMMA; however, impacting the cell with a copper flyer will
allow for the same particle velocities to be reached at the expense of a slight
increase in uncertainty due to the use of two standard materials.
The final cell design is shown schematically in Figure 5.7. The sample
thickness is defined by the thickness of the PMMA annulus, which was cho-
sen as either 3.6 mm or 3.8 mm, depending on the exact experiment, in order
to maximise the transit time of the shock (and, hence, reduce the uncertainty
on the measured shock velocity) whilst ensuring the rear gauge remains in a
state of uniaxial strain. As the plate impact facility is fired into a rough vac-
uum, the cell must be vacuum tight. An ‘o’-ring is used to seal the sample,
with the outer diameter of the ‘o’-ring being identical to the inner diameter of
the PMMA annulus. To prevent bowing of the front plates, a soft (50 shore
hardness) nitrile ‘o’-ring was used; the ‘o’-ring has a diameter of 4 mm and is
therefore only lightly compressed. No bowing of the front plates was measur-
able. This ‘o’-ring can clearly be seen in Figure 5.8, which is a photograph of
one such cell. In order to fill the cell with a liquid, two holes were tapped in
the rear anvil. A syringe was used to fill from the lower cell, with air being
allowed to escape through the top hole. The transparent front plates enabled
each cell to be visually inspected prior to impact to ensure that no air was left
in the sample. The holes were sealed with by inserting a bolt that that had
been wrapped in PTFE thread sealing tape. Whilst the insertion of a bolt into
the filling hole compresses the water slightly, the effect is minimal - result-
ing in an increase in pressure of around 25 MPa - compared to the several-
gigaPascal shock pressure obtained during experiments. The exact value of
the pressure increase depends on how the bolt seals the hole, and the value
given above is a maximum value, assuming that the bolt fills the entirety of
the threaded hole; the actual pressure increase is likely to be much less than
this.
103
Figure 5.7: The final liquid cell design; the horizontal scale is exaggerated by a factor
of 10. The front of the cell consists of two PMMA plates with a manganin
gauge between them. The rear anvil is a 10 mm copper plate to which a
manganin gauge, insulated with Teflon, has been bonded. The sample is
confined by a PMMA annulus of thickness either 3.6 mm or 3.8 mm. An
‘o’-ring is used to seal the cell, which is sccrewed to a standard aluminium
or tufnol mounting ring. Two threaded holes in the copper plate enable the
cell to be filled from the rear. These holes are then sealed with a bolt used
to compress a small ‘o’-ring. The internal diameter of the PMMA annulus
is 40 mm, enabling the placement of optic fibres, to measure tilt, around the
edge of the sample.
Release-wave propagation
The propagation of release waves in plate impact targets was discussed in §3.4.
Whilst Equations 3.8 and 3.9 enable determination, approximately, of when
lateral releases erode the uniaxial stress state, the more complicated propaga-
tion of releases in the liquid impact cell are not amenable to this simple ana-
lytical treatment, as has been evidenced by the unexpected arrival of release
waves in the earlier cell designs. A hydrocode is a computational code capable
of solving numerically the Euler equations for fluid flow, and the jump con-
ditions for a given experimental scenario [Benson, 1992]. Hydrocode simula-
104
Figure 5.8: Typical cell used in studying the shock response of liquids, shown schemat-
ically in Figure 5.7. The cell is shown after optic fibres, used to measure tilt,
have been inserted but before final assembly and filling. The rear gauge is
insulated with Teflon, whilst the front gauge is simply glued between two
PMMA plates. The central PMMA annulus holds an ‘o’-ring and defines
the thickness of the liquid sample.
tions of the cell described in Figure 5.7 were implemented in the C-AMR hy-
drocode in collaboration with Stefan Schoch [Schoch, 2011]. Figure 5.9 shows
the model of the liquid cell implemented in th hydrocode simulations. The
simulations are axisymmetric about y = 0, and the liquid is modelled as wa-
ter. The buffer, anvil, and annulus shown in Figure 5.7 are considered as single
continous piece of PMMA in the hydrocode. The rear anvil is modeled as a
block of copper. The flyer is placed in contact with the front surface of the
cell, and given an appropriate initial velocity. The red rectangles, which are
not part of the simulation, indicate the approximate Lagrangian location, and
lateral extent, of the front and rear gauges. Two simulations, shown below,
were conducted. In the first simulation (Figure 5.10), the cell was impacted by
a PMMA flyer at 0.500 km s−1; the second simulation (Figure 5.11) used a cop-
per flyer at 1.000 km s−1. The simulation outputs have been plotted in a way
105
that highlights variations in density such that shocks and releases can visu-
ally be tracked. The approximate Eulerian position and sizes of the embedded
manganin gauges have been overlayed on the simulation results. Both sim-
ulations show similar features: releases propagate from the edge of the flyer
and from the interface between the water and PMMA as anticipated. These
lateral releases can be seen as the expanding circular dark lines in the sim-
ulation results. In both cases, lateral releases do not reach the gauges until
some time after the shock has reached the rear surface of the target. It is con-
cluded, therefore, that the gauges experience uniaxial loading over the entire
time-scale of the experiment.
PMMA
flyer
Front gauge Rear gauge
Liquid
sample
PMMA
annulus
PMMA
anvil/buffer
Copper
rear plate
Figure 5.9: Model of the liquid cell shown in Figure 5.7 as implemented in the C-AMR
hydrocode. The model is axisymmetric about y = 0. The appriximate La-
grangian positions of the front and rear manganin gauges is indicated by
the red rectangles, which are not a part of the simulation, but have been
added graphically afterwards. The PMMA parts of the cell are treated as
a single piece of PMMA, with the rear anvil treated as a separate piece of
copper. The liquid is modeled as water. The flyer is placed in contact with
the front face of the cell, and assigned an initial velocity equal to the desired
impact velocity.
106
AB
C
D
Figure 5.10: Axisymmetric hydrocode simulation of a PMMA flyer impacting the liq-
uid cell shown in Figure 5.7 at a velocity of 0.500 km s−1 showing varia-
tions in density. Immediately after impact, lateral releases (A) start to en-
croach on the target. After 1.5 mus, the shock has reached the liquid and
the resultant release has propagated back to the front gauge. Secondary
releases (B) are formed at the edge of the liquid. At around 2.8 mus, the
shock reaches the rear gauge package and releases (C) have travelled a
short distance into the liquid. It is not until 7.0 mus after impact that re-
leases (D) begin to approach the gauges. Therefore, the gauges can be
considered to undergo unixial loading for the entire experiment.
107
Figure 5.11: Axisymmetric hydrocode simulation of a copper flyer impacting the liquid
cell shown in Figure 5.7 at a velocity of 1.000 km s−1 showing variations
in density. Similar to those observed in Figure 5.10 are observed, although
the greater shock velocity shortens the timescale over which these events
occur. In the case of this high velocity impact, the gauges remain in uniax-
ial strain until at least 4.5 mus when the simulation ended due to numeri-
cal instabilities associated with the longitudinal release from the rear free
surface.
108
Considerations regarding anvil materials
A number of different anvil materials have employed in the research described
in this thesis; whilst the rear anvil has, by virtue of its relatively high impedance,
been machined from copper, front anvils and buffers, and flyer plates, have
been constructed from both copper and poly(methyl methacrylate) (PMMA).
The analysis discussed above, and in particular the wave diagram pre-
sented in Figure 4.15, neglects deviations from an assumed shock response
that results in the emergence of only a single wave. From Chapter 2, it should
be recalled that, for example, an elastic-plastic material shocked above its HEL
exhibits a two-wave response that is not catered for in the present analysis.
Such an approximation, of only a single-wave response, greatly simplifies the
reduction of experimental data yet unavoidably introduces certain errors. The
magnitude of such errors is, however, smaller than the experimental uncer-
tainties discussed above but is presented in detail below.
Figure 5.12 shows a typical shock profile in PMMA, as recorded by an em-
bedded manganin gauge. PMMA does not exhibit the precursor associated
with an elastic-plastic material, nor does it exhibit a sharp rise to a final state;
instead the longitudinal stress in PMMA jumps to an instantaneous stress
(point A) around two-thirds that of the equilibrium state before ramping to
the final state (point B).
By modelling PMMA as a thermoviscoelastic material, Nunziato and Walsh
[1973] were able to account for these features. Barker and Hollenbach [1970]
have demonstrated that, to moderate particle velocities, the behaviour of PMMA,
in the shock velocity – particle velocity plane, differs markedly from the an-
ticipated linear Hugoniot. Their data, which were quoted without errors, are
reproduced in Figure 5.13.
The best, third-order, polynomial fit to these data is described by Equation
5.1 and is valid to the maximum reported particle velocity: up = 0.6250 km s−1.
US = 2.745 + 3.534up − 8.834u2p + 8.361u3p. (5.1)
Above this stress, the Hugoniot is adequately described by that reported
in the literature [Asay and Shahinpoor, 1993, Appendix C]:
US = 2.598 + 1.516up. (5.2)
Further, it has been reported [Barker and Hollenbach, 1970; Chapman,
2009a] that the release path in PMMA differs markedly from the crude ap-
109
Figure 5.12: Stress history recorded by a manganin gauge embedded in PMMA, from
experiment E110225A, due to symmetric impact at 0.494 km s−1. The trace
shows an initial sharp rise, of approximately 50 ns duration, indicating a
well-aligned shock, to the point A, which is approximately two-thirds the
expected stress. The stress then gradually rises to the peak stress, B, which
co-incides with the expected stress.
proximation that releases occur along a path close to the Hugoniot. Chap-
man [2009a] suggests that the crude approach yields, in impedance matching,
a stress and particle velocity some 3 % higher than the stress that would be
found by using the actual release curve. Indeed, this appears to be a worst-
case scenario: for materials whose impedances are close to that of PMMA, as
is the case for the liquids studied in Chapters 5 and 6, the uncertainity intro-
duced by approximating the release by the Hugoniot is negligible, and thus
ignored. Additionally, without an accurate model for the release path from a
given state, it is impossible to analytically take this difference into account.
The analysis described above is greatly simplified in the case where a sin-
gle, linear, Hugoniot describes the anvil’s response in the US–up plane for all
up. The experiments reported in Chapters 5 and 6 consider particle velocities
110
Figure 5.13: Shock velocity – particle velocity Hugoniot for PMMA, after Barker and
Hollenbach [1970]. The solid line, a third-order polynomial providing the
best fit to the experimental data, passes within 1 % of the data, and is de-
scribed by Equation 5.1
in PMMA both above and below the particle velocity, up = 0.6250 km s−1, at
which deviations from the linear Hugoniot become significant. However, the
effect of considering the more complex relation of Equation 5.1 on the deter-
mination of the liquid Hugoniots is minimal. Indeed, the more complicated
analysis shifts the resultant Hugoniot by considerably less than 1 %: far less
than the uncertainity in the Hugoniots reported, and within the uncertainties
between the different analysis methods. This latter point is discussed further
below.
In addition to simplifying the analysis, relying on the linear Hugoniot in-
troduces fewer assumptions. Without direct measurements of the shock- and
particle-velocities in the specific PMMA used in this thesis, there can exist
some doubt as to the validity of Equation 5.1. Whilst there are, undeniably, er-
rors associated with this simplified approach, it is believed that they are small
111
in magnitude and insignificant when considered alongside other experimen-
tal uncertainties, such as the behaviour of the manganin gauges in release,
impedance mismatches between the gauge package and surrounding PMMA,
or variations in sample thickness. Indeed, the variation in thicknesses of target
plates and gauge packages, which vary randomly between experiments, rep-
resent the dominant contribution to the uncertainty in the transit time through
anvils. In dealing with cells that have been pre-heated, the room temperature
Hugoniot of PMMA has been used. As PMMA is relatively well characterised,
its published equation of state [e.g. Asay and Shahinpoor, 1993] can be used
to calculate the effect of temperature, using the method decribed by Dai et al.
[2006]. The PMMA Hugoniot at 65 ◦C lies within error of the ambient tem-
perature Hugoniot; thus the assumption that the effect of pre-heating on the
PMMA Hugoniot can be neglected is justified, as it is substantially smaller
than the other experimental uncertainties.
Strength effects in Copper
Copper is widely used in shock experiments, as it has a high impedance, low
elastic limit, does not exhibit any phase transitions, and is both easy to ma-
chine and available as a reproducible stock material [Davison and Graham,
1979].
Figure 5.14 shows experimentally-determined stress–particle velocity states
in copper (Cu101), as measured by embedded manganin gauges under sym-
metric impact. The data are collected from previous work conducted in this
laboratory, and have been re-analysed for the present work. The experimen-
tal configurations are reported elsewhere [Chapman, 2009c]. The re-analysed
data are shown together with the linear shock equation of state for copper,
Equation 5.3, given by Asay and Shahinpoor [1993, Appendix C]:
US = 3.940 + 1.489up. (5.3)
The data of Figure 5.14, which lie within experimental error of Equation
5.3, would suggest that there are no appreciable strength effects in copper.
However, the existence of an HEL in Cu101 has been demonstrated experi-
mentally by Chapman [2009a]. Figure 5.15 (ibid.) shows the free surface ve-
locity, determined by VISAR, in Cu101 under low-velocity symmetric impact.
The two-wave structure consists of an elastic precursor, of stress amplitude
σHEL = 0.3 ± 0.1 GPa, and a second wave which increases the longitudinal
112
Figure 5.14: Copper Hugoniot states, measured with embedded manganin gauges, un-
der symmetric impact, re-analysed from experiments reported in Chap-
man [2009c]. The solid line is the linear shock velocity–particle velocity
Hugoniot given by Equation 5.3.
stress to that of the shock; the yield strength at the HEL is found by applying
the Tresca or von Mises yield criterion and is YHEL = 0.14± 0.05 GPa.
The elastic precursor becomes overdriven, leaving a single wave response,
when the elastic loading line, of slope ρ0Ce (where Ce is the elastic wave
speed) intersects the linear shock velocity Hugoniot in the stress–particle ve-
locity plane. For the elastic wave speed in copper, 4.76 km s−1 [Chapman,
2009a], this occurs at a stress of approximately 23.5 GPa.
Figure 5.16 considers the deviation of longitudinal stress from hydrody-
namic pressure in a material with finite strength. The low yield strength of
copper, however, means that there is little difference between these curves.
Indeed, it is unlikely that a manganin gauge would be sensitive to this slight
offset, and, furthermore, the impedance mismatch between the gauge package
and copper would ‘smooth out’ the elastic precursor wave. The close agree-
113
Figure 5.15: From Chapman [2009a]. The two-wave structure measured by VISAR at
the free surface of Cu101 during symmetric impact at two different impact
velocities.
ment between longitudinal stresses measured by manganin gauges and the
linear shock velocity equation derived from over-driven experiments (Equa-
tion 5.3), shown in Figure 5.14 indicate that, within the diagnostic sensitivity
of manganin gauges, this effect can be ignored.
Figure 5.17 shows the wave profile that would be observed at the interface
between an ideal elastic-plastic material and a lower impedance window. The
unloading portion would occur along the lower line of 5.16.
The ideal elastic-plastic material provides the ‘worst case’ scenario: in a
material which is not perfectly elastic-plastic, the longitudinal stress lies closer
to the hydrostat. Johnson et al. [1992] report measurements of the release wave
structure in oxygen-free electronic copper (OFEC), interpreting the observed
‘quasi-elastic’ behaviour in terms of internal stresses and dislocation motion.
Figure 5.18, from Johnson et al. [1992], is representative of the release wave
profile in OFEC.
The release wave structure of OFEC, whilst more ideally elastic-plastic
than in other materials, still deviates from the ideal case, and the release state
lies closer to the Hugoniot than Figure 5.16 implies. Coupling this with the
low yield strength measured in C101, it is concluded that the error introduced
by approximating the release of copper by its Hugoniot is negligible and may
114
Figure 5.16: Ideal elastic-plastic unloading, in P–up space. The longitudinal stress in
release is displaced below the pressure, P (calculated using the, strength-
less, linear shock-velocity Hugoniot, by − 2
3
CL
CL+CB
Y where CL and CB
are the ambient pressure longitudinal and bulk sound speeds respectively.
The additional factor of CL
CL+CB
when compared to the stress-volume
curve of Figure 2.9 reflects the fact that the displacement is smaller in
stress–particle velocity space, and is derived by Davison [1998].
be ignored.
5.2 Pure water, ambient temperature
The water used in experiments in this thesis was de-ionised and vacuum de-
gassed prior to performing each experiment. The shock response of pure wa-
ter has been widely reported in the literature [Gurtman et al., 1971; Mitchell
and Nellis, 1982; Nagayama et al., 2002], and thus a small number of experi-
ments on pure water at ambient temperature were perfomed to demonstrate
the validity of the experimental design described above and of the data anal-
ysis methods described in Chapter 4. The gauge traces resulting from such
an experiment, at an impact velocity of 0.787 ± 0.008 km s−1 with a PMMA
115
Figure 5.17: From Johnson et al. [1992]. An experimental geometry to measure the
release wave structure in a test material. The particle velocity-time history
for an ideal elastic-plastic material in this geometry is shown.
Figure 5.18: From Johnson et al. [1992]. Measured interface velocity for OFCE shocked
to 10 GPa in the geometry shown in Figure 5.17. The quasi-elastic nature
of the release wave is evident.
116
flyer are shown in Figure 5.19. The trace is representative of those obtained
in the experiments discussed in the remainder of this chapter and in Chap-
ter 6. The tilt in this experiment was measured as 1.8±1.1 mRad. Comparison
should be made between these gauge traces, those shown in Figure 5.2 ob-
tained with the initial cell design, and with the idealised responses shown in
Figure 4.17. Clearly, the final design produces siginificantly better stress his-
tories from both gauges; the arrival of the shock at each is clearly visible, and
identifiable stress levels, associated with the Hugoniot and reload state of the
liquid, persist for significant periods of time. The reverberation visible on the
rear gauge trace is due to the impedance mismatch between water and Teflon;
the effect of this mismatch was discussed in §4.2 and is sufficiently small that
it may be neglected.
Principal Hugoniot
In order to demonstrate that the reduction of the Hugoniot data yields ap-
propriate values, two experiments were conducted on water at an initial tem-
perature of at 18 ± 2 ◦C. Figure 5.20 shows the experimentally determined
Hugoniot states in water, analysed using the first and third analysis methods
discussed in §4.2. The solid line is the linear Hugoniot of water given by Na-
gayama et al. [2002] as US = 1.45 + 1.99up, and the dashed lines represent the
quoted uncertainties in the intercept and slope of the Hugoniot. Both meth-
ods of analysis yield shock and particle velocities that lie within error of this
Hugoniot, indicating that both methods produce sensible results; the slight
differences between the two methods are discussed further in §5.3.
Rybakov [1996] note that there is a slight break in the gradient of the Hugo-
niot of water at P = 2.86 GPa, suggesting this is due to the onset of a partial
phase change, 0.8 % by mass, to ice VII. Dolan and Gupta [2003] report on
the quasi-isentropic compression of water. Their pressure–temperature phase
diagram for water is reproduced in Figure 5.21; the temperatures on the Hugo-
niot are calculated using the model proposed by Gurtman et al. [1971]. The
temperatures calculated in this way suggest that the principal Hugoniot of
water does not cross the ice VII boundary; Dolan and Gupta [2003] note that
the freezing of water under shock compression is an open question. In consid-
ering the relevant literature, Duvall and Graham [1977] had previously con-
cluded similarly, citing the work of Kormer and Krishkevich [1968]: the freez-
ing of water into ice VII only occurs in isentropic loading. It seems, therefore,
117
Figure 5.19: Front (red) and rear (black) stress gauge traces from experiment E100927A.
A liquid cell, containing water at 18 ± 2 ◦C, was impacted by a PMMA
flyer at 0.787±0.008 km s−1. The front gauge is first shocked to the Hugo-
niot state of PMMA, before being released to the Hugoniot state of wa-
ter at a stress of 1.179 ± 0.013 GPa; the shock reaches the rear gauge
2.250±0.014µs after first arriving at the front gauge. The stress in the rear
gauge, corresponding to the reload state of water, is 2.987 ± 0.022 GPa.
Comparing these gauge traces with those obtained with the initial cell de-
sign (Figure 5.2), it is clear that the final cell design represents a significant
improvement. The reverberation visible on the rear gauge trace is due
to the impedance mismatch between water and Teflon; the effect of this
mismatch was discussed in §4.2 and is sufficiently small that it may be
neglected.
118
Figure 5.20: Measured Hugoniot states in water at 18 ± 2 ◦C, compared to the lin-
ear Hugoniot, US = 1.45 + 1.99up, given by Nagayama et al. [2002]; the
dashed lines reflect the quoted 3 % uncertainty in the fitting constants. The
data analysed by methods 1 and 3 lie within error of the fit, demonstrating
the validity of these analysis methods.
that the gradient change in the water Hugoniot reported by Rybakov [1996],
which has not been reported elsewhere in the literature, is not due to to a
freezing transition and the results presented may be in error.
Dolan [2003] has noted that the Hugniot given by Nagayama et al. [2002]
does not extrapolate well to particle velocities outside the experimental range.
In order to examine the reload behaviour of water, a Hugoniot which can be
extrapolated to higher particle velocities is required. Figure 5.22 presents the
Hugoniot states in water determined by method 3, and the data for water
given by Marsh [1980] for particle velocities below 1.5 km s−1. The fitted line
is the best-fit linear Hugoniot through these data and is given by US = 1.50±
0.04 + 1.85± 0.04up. The data are well-described by this fit, and it is therefore
concluded that a single linear Hugoniot represents the best fit to the data to
up = 1.5 km s
−1, and it is this Hugoniot which will be used in the analysing
119
Figure 5.21: Pressure–temperature phase diagram for water, from Dolan and Gupta
[2003]. The two dashed curves represent the limiting temperature curves -
the Hugoniot at the upper temperature range, calculated using the model
propsed by Gurtman et al. [1971], and the isentrope.
reload experiments.
Reload experiments
The Mie-Gru¨neisen equation of state has been applied to the shock/reload be-
haviour of water by a number of authors [e.g., Gurtman et al., 1971; Mitchell
and Nellis, 1982; Nagayama et al., 2002]. Indeed, Gurtman et al. [1971] pro-
vide a form for the volume-dependent Gru¨neisen parameter, Γ, against which
it is possible to compare experimental results. Using the Hugoniot shown
in Figure 5.22, and the methodology described in §4.2, it is possible to ob-
tain a value for Γ from experiment E100927A (Figure 5.19). At a volume in the
reloaded state of 0.707±0.010 cm3g−1, the value of Γ determined in this way is
1.07± 0.57, compared to the value given by Gurtman et al. [1971] of 1.05. The
uncertainty in Γ is determined by analysis, in quadrature, of the uncertain-
ties in the measured stresses, shock velocity, and the parameters defining the
Hugoniots of both the water and the rear copper anvil; it is believed that this
method gives rise to a larger uncertainty than would be suggested by the data,
as it neglects covariance in the uncertainties on these quantities. The covari-
ance arises as the Hugoniot of the liquid has been generated from the experi-
mental data, and these data are used in conjunction with the Hugoniot when
120
Figure 5.22: Hugoniot of water below up = 1.5 km s−1. The data comprise the present
research, as determined by method 3, together with that of Nagayama
et al. [2002] and Marsh [1980]. The line fitted through these data is given
by US = 1.50 ± 0.04 + 1.85 ± 0.04up and is shown by the solid line, with
the dashed lines representing the upper and lower bounds of this fit. The
coefficient of determination of the fit is 0.99. It is therefore concluded that
the Hugoniot of water over this particle velocity range is best described
by the present fit.
calculating Γ. It is not apparent how this covariance may be accounted for in
the present analysis. It is felt that the agreement demonstrated between the
determined value of Γ, and that given by Gurtman et al. [1971], demonstrates
the validity of the experimental techniques presented. Experiment E100816A,
which had a reload stress of 1.678 ± 0.040 GPa yields a value of Γ with an
uncertainty of around 500 %. As Γ is, ultimately, dependent on the difference
between the Hugoniot and reload stresses, when this difference is very small
(as is the case at low reload stresses), the proportionate uncertainties are large.
Γ can only be generally determined from the experiments in this thesis when
the reload stress is at least 3 % below the Hugoniot; this criterion is satisfied
121
when the reload stress is around 3 GPa.
5.3 Pure water, elevated temperature
The method adopted in this thesis to determine the shock temperature rise in
a material requires first the measurement of the principal Hugoniot of the ma-
terial. Thence, shock/reload experiments enable an empirical fit to the Mie-
Gru¨neisen equation of state to be made. Combining these two measurements
with a suitable model for the heat capacity, it is possible to calculate shock tem-
peratures along the Hugoniot. The remainder of this chapter discusses a series
of experiments conducted on water, at an initial temperature of 66 ± 2 ◦C, to
measure the Hugoniot and volume dependence of the Gru¨neisen parameter,
Γ. These measurements will then be combined to calculate the shock temper-
atures in water.
In order to investigate liquids at different initial temperatures, it was nec-
essary to pre-heat the cell described above. Heating was achieved by attach-
ing two metal clad resistors to the rear copper anvil, and applying around 1 A
at 30 V. During plate impact experiments, the initial temperature was deter-
mined by a single thermocouple mounted on the rear anvil. Prior to conduct-
ing any plate impact experiments on the heated cells, the effects of heating
the cell were investigated by placing thermocouples on the front anvil and
within the liquid itself . It was found that if the cell was heated over a period
of several hours, and then filled with the test liquid, which had been sepa-
rately heated, no significant thermal gradients were introduced. A series of
experiments were conducted on de-ionised water at an initial temperature of
66 ± 2 ◦C. The initial density, ρ0, of water at this temperature is 0.878 g cm−3
[Washburn, 2003].
Principal Hugoniot
Figure 5.23 shows the experimentally determined Hugoniot states, as deter-
mined by the three methods given in §4.2, in shock velocity–particle velocity
space.
In addition to the experimental uncertainties discussed earlier, various sys-
tematic errors are associated with the two methods. Principally, these can be
considered to come from two sources: firstly, the uncertainty associated with
stress measurements made in release and, secondly, the precise nature of the
122
Figure 5.23: Measured Hugoniot states for water at 66± 2 ◦C, determined using using
two of the methods described in §4.2. The two methods yield somewhat
different results for the Hugoniot state for each experiment. Method 3 is
considered to give the most accurate determination of the Hugoniot. The
systematic uncertainties relating to each method are discussed in the text.
release path of the front anvil material. As the water has a lower impedance
than PMMA, the Hugoniot state lies on the release curve of PMMA. The stress
measured by a gauge embedded in PMMA, therefore, also lies on the release
curve of the PMMA with the gauge itself in a state of partial release.
Considering first the determination of stress from the resistance change of
the manganin gauge, it should be recalled that the resistance of manganin is
subject to hysteresis on unloading. It is assumed that, when the peak stress
experienced by the gauge is below 1.5 GPa, the gauge behaves elastically.
For peak stresses above this elastic limit, a second calibration curve [Yaziv
et al., 1980] can be applied, which gives stresses in the release state of the
gauge. This calibration is, however, somewhat uncertain for stresses that
are above 80 % of the peak stress. Figure 5.24 shows the front stress gauge
trace, from experiment E101126A, which resulted from the impact of a cop-
123
per flyer at 0.590 ± 0.006 km s−1. The peak stress (PMMA Hugoniot) was
2.034 ± 0.011 GPa, and the release stress (water Hugoniot/PMMA release
curve) was 1.964 ± 0.015 GPa. This latter figure is around 97 % of the peak
stress, and is therefore in the uncertain region of the calibration.
Figure 5.24: Front stress gauge trace from experiment E101126A. The peak stress,
2.034± 0.011 GPa, corresponds to the Hugoniot stress of PMMA. The sec-
ond plateau, at 1.964±0.015 GPa, is the Hugoniot stress of the liquid, and
lies on the release curve of PMMA. It is also, evidently, a state in release
of the manganin gauge and thus subject to the release calibration given by
[Yaziv et al., 1980].
§5.1 discussed the behaviour of PMMA in release. It was concluded that,
in general, the widely-used approximation that the release curve of a material
may be approximated by the Hugoniot, was inappropriate for PMMA, which
demonstrates a complex, non-linear, thermo-visco-elastic, rate-dependent re-
sponse. For states in partial release that lie close to the Hugoniot state from
which they are released, the uncertainty introduced by neglecting the exact
nature of the release curve of PMMA is negligble, as the strain involved is
small. Further, as there is no analytical description of the release behaviour of
124
PMMA, without detailed experimental determination of the release path for
each experiment, it is not clear how this effect could be applied to the present
analysis.
The uncertainty introduced by not including the exact release behaviour
of PMMA in the data analysis is, as has been noted, considered to be neglig-
ble. Further evidence for this is provided by Figure 5.25, which shows the
Hugoniot states determined by methods 1 and 3 in stress–particle velocity
space. The close agreement demonstrated by the two methods is considered
evidence that the uncertainties in the release behaviour of the gauges and of
PMMA are small. It is concluded that the uncertainity in the gauge behaviour
in states of partial release is, however, significantly larger than the uncertainity
introduced by neglecting the true release behaviour of PMMA. For these rea-
sons, it is concluded that method 3 gives the most accurate determination of
the Hugoniot states of the water in the shock velocity–particle velocity plane,
and these data will be used to construct a relationship for the Hugoniot. Mea-
surements of the shock response of low-density granular systems by Chap-
man [2009a] yielded similar conclusions.
The resultant Hugoniot for water at 66± 2 ◦C is shown in Figure 5.26. The
fit, which is weighted by the errors on the data, has a coefficient of determina-
tion of 0.98, and is given by US = 1.44±0.05 +1.53±0.09up. The dashed lines
represent confidence bounds at 95 %. That the data are well-fitted by a single
linear relationship is taken as evidence that no phase changes occur. The qual-
ity of fit also suggests that a linear relationship is the appropriate description
of the present data.
Reload experiments
In addition to measuring the principal Hugoniot, the reported experiments
measured the reload stress in water at 66 ± 2 ◦C. These stress measurements
allow for the determination of the volume of the reload state in the way de-
scribed in the preceeding chapter, and are presented as the open symbols in
Figure 5.27 together with the Hugoniot states (blue symbols) and fitted Hugo-
niot (blue curve). By using the Mie-Gru¨neisen equation of state to describe
these reload states, we will be able to calculate temperature rises along the
Hugoniot. The fitted Hugoniot, and its uncertainties (dashed curves) were
calculated from the US–up relationship using the equation
125
Figure 5.25: Measured Hugoniot states for water at 66± 2 ◦C. In P–up, the three anal-
ysis methods yield similar Hugoniots, indicating that the approximations
made yield only small uncertainties. Method 2, which neglects the mea-
sured transit time of the shock wave, in which there is greatest confidence,
is considered unreliable. Methods 1 and 3 are in close agreement, indicat-
ing that neglecting the exact release behaviour of PMMA introduces only
a small uncertainty in the calculations.
P =
ρ0c
2
0∆
(1− S∆)2 , (5.4)
where ∆ is the Lagrangian compression, ρ0(v0 − v1) and c0 and S are, respec-
tively, the intercept and slope of the US–up Hugoniot.
The arrows in Figure 5.27, linking the Hugoniot and reload stress for the
experiments with reload stresses above 3.5 GPa, correspond in each case to the
Rayleigh line describing the loading path taken in these experiments. These
states lie below the Hugoniot on isochors through their respective reload vol-
umes. The lower stress experiments do not differ markedly from the Hugo-
niot, with two experiments found to be exactly on the Hugoniot. It is not clear
why the remaining two reload states lie above the Hugoniot. It is unlikely that
126
Figure 5.26: Hugoniot of water at 66 ± 2 ◦C. The data points are determined from
the experimental values using the third analysis method described in §4.2.
The solid line is the weighted best fit to the data and is given by US =
1.44 ± 0.05 + 1.53 ± 0.09up; the dashed lines are confidence bounds at
95 %.
this is due to a phase change, as this would require the states to be at a lower
temperature than the isentrope centered on ambient conditions shown in Fig-
ure 5.21, which is clearly unphysical. It is likely that the deviation in these
states is an artefact of the particular experiments involved, as at low stresses,
a small offset in the measured reload stress would manifest as proportionately
larger offset in the calculated volume.
For those states which lie below the Hugoniot, it is possible to determine
a value of the Gru¨neissen parameter, Γ,
Γ ≈ vR pH − pR
eH − eR , (5.5)
where the subscripts R and H refer to the reload state and the Hugoniot
state on an isochor through the reload volume. The energies are calculated
127
Figure 5.27: Reload states in water at 66 ± 2 ◦C. The solid points are the previously
determined Hugoniot states, and the blue curve is given by the Hugoniot
US = 1.44 ± 0.05 + 1.53 ± 0.09up, where the dashed lines represent the
uncertainities in this relationship. The open symbols are the measured
reload states. Arrows indicating the Rayleigh line between the Hugoniot
and reload state have been added for three of the experiments.
using the generalised jump condition for energy between the states (V, P ) and
(V1, P1)
E − E1 = 1
2
(P + P1)(V1 − V ). (5.6)
These values for Γ are shown in Figure 5.28, where the uncertainties in Γ
have been determined in quadrature. The calculation of Γ relies on the mea-
sured US–up Hugoniot and the measured stress and volume in the Hugoniot
state. However, the US–up Hugoniot is determined from the stress measure-
ment, and the volume is, in turn, determined from the stress measurement
and the US–up Hugoniot. The errors in these quantities are, therefore, to an
extent, covariant and the determination of errors in quadrature will lead to an
128
over-estimate of the error.
Figure 5.28: Experimentally determined values of the Gru¨neissen parameter, Γ, in wa-
ter at 66±2 ◦C. The uncertainties in Γ have been determined in quadrature
from Equation 5.5 and are believed to be an over-estimate, as the quadra-
ture analysis neglects covariance of the errors in the measured Hugoniot
stresses and the fitted US–up Hugoniot.
The data presented in Figure 5.28 suggest that Γ decreases with increasing
density in the range investigated. Dolan [2003] suggests that, in water com-
pressed from ambient conditions, a maximum is observed in Γ at a specfic
volume of around 0.675 cm3 g−1, likely due to the collapse of the hydrogen
bonding which would allow for greater compaction.
In order to calculate temperatures along the Hugoniot, a form of Γ(v) is
required that is defined for all v; following the methodology given by Gurt-
man et al. [1971], a polynomial fit to Γ is found. In order to ensure correct
behaviour at low compressions, the 60 ◦C and 70 ◦C isotherms of water given
by Grindley and Lind [1971] are used in the equation
129
CvΓ(V )
V
=
Pi(V )− Pj(V )
Ti − Tj . (5.7)
Gurtman et al. [1971] suggest that the heat capacity, Cv , of water shock
compressed from an initial temperature of 20 ◦C is within 10 % of 3.2 J g−1 K−1.
This is in agreement with the value of 3 J g−1 K−1 used by Dolan [2003], and is
lower than the heat capacity at ambient pressure (4 J g−1 K−1). The following
analysis assumes that the value given by Gurtman et al. [1971] is approxi-
mately correct for water at an initial temperature of 66 ± 2 ◦C. The isotherms
are described by a Murnaghan equation
P (v) =
K
g
[(
v0
v
)g
− 1
]
, (5.8)
where K was chosen to match the bulk isothermal modulus at ambient pres-
sure [Kell, 1975] and the temperature of the isotherm, leaving g, which is the
rate that the isothermal modulus varies with pressure, as the only adjustable
parameter. The resultant values for Γ are plotted in Figure 5.29 together with
a fifth-order polynomial fit to the data
Γ(v cm3 g−1) = − 253 (5.9)
+ (1510 g cm−3)v
− (3521 g2 cm−9)v2
+ (4057 g3 cm−27)v3
− (2324 g4 cm−81)v4
+ (531 g5 cm−243)v5,
which is the lowest-order polynomial that can describe the behaviour of Γ at
high and low specific volumes.
It is not possible, generally, to integrate the the equation for Hugoniot tem-
peratures, Equation 4.4, in closed form. McQueen [1989] gives a centered dif-
ference equation that enables the temperature at some point on the Hugoniot,
(Pi, Vi, Ti), to be calculated from some other point (Pi−1, Vi−1, Ti−1):
Ti =
Ti−1[1 + ρΓ∆V/2] + [(V0 − V )∆P + (P − P0)∆V ]/2Cv
1 + ρΓ∆V/2
, (5.10)
130
Figure 5.29: Volume-dependent Gru¨neissen parameter, Γ, in water at 66 ± 2 ◦C from
shock/reload experiments (open squares) and isotherms (closed squares,
Equation 5.7). The fit is a fifth-order polynomial described by Equation 5.9
and is the lowest-order polynomial that can describe the behaviour of Γ at
high and low specific volumes.
where the quantities ρ, Γ, Cv , V , and P refer to the density, Gru¨neissen pa-
rameter, heat capacity, volume, and pressure averaged between the two states
and ∆V , ∆P the differences in volume and pressure between the two states.
By specifying the temperature T0 at the initial volume V0 and pressure P =
P0 = 0, it is possible construct a full temperature Hugoniot by iterative ap-
plication of Equation 5.10 using the polynomial form for Γ. The calculated
shock temperatures, as a function of pressure, are shown in Figure 5.30. The
present analysis has been conducted for three values of the specific heat - the
expected value of Cv = 3.2 J g−1 K−1, and the upper and lower bounds that
are within 10 % of this value, as suggested by Gurtman et al. [1971]. This vari-
ation in Cv yields uncertainties in the shock temperature of around 7 %, and
represents the dominant contribution the uncertainty in the temperature cal-
131
culation. As is standard in the literature, the heat capacity has been assumed
to be constant; as is discussed in Chapter 7 this approximation may not be
valid, and Cv is likely to increase slightly over the stress range considered.
The magnitude of the variation is uncertain, and it is believed the present 10 %
variation is an appropriate description; a 50 % variation in the heat capacity
would increase the uncertainty in the calculated temperature rise to approxi-
mately 25 %. Combining the 10 % variation in CV with the uncertainties in the
experimentally determined Hugoniot and the polynomial fit for Γ, it is felt
that the present results enable calculation of the shock temperature in water,
at an initial temperature of 66 ◦C, to within ±10 % for stresses to 8 GPa. The
assumptions underpinning these calculations of temperature are discussed in
detail in Chapter 7.
5.4 Summary
• A cell has been developed which allows for well-defined shocks to be
generated in liquids, and the ability to pre-heat the liquid has been demon-
strated.
• Manganin gauges at the front and rear of the liquid show sharp rises, at
the arrival of the shock, and steady plateaus in the Hugoniot and reload
states.
• Comparison with literature values for the Hugoniot and Gru¨neisen pa-
rameter, Γ, for water compressed from room temperature, demonstrate
the cell and analysis techniques produce consistent and reliable data.
• The Hugoniot of water for water compressed from 66 ± 2 ◦C has been
determined experimentally.
• The Gru¨neisen parameter has been determined from shock/reload ex-
periments, and combined with appropriate isothermal compression data
to produce a polynomial fit describing the volume-dependence of Γ.
• Shock temperatures have been calculated along the principal Hugoniot
of water compressed from 66± 2 ◦C to 8 GPa with uncertainties of 10 %.
132
Figure 5.30: Calculated shock temperatures in water at 66 ± 2 ◦C, using the mea-
sured Hugoniot US = 1.44 + 1.53up and polynomial expression for Γ
(Equation 5.9), assuming a constant heat capacity Cv = 3.2 J g−1 K−1.
The dashed lines are calculated for heat capacities 10 % above and below
Cv = 3.2 J g
−1 K−1. The variation in heat capacities uncertainties in cal-
culated temperature of approximately 7 %. To account for uncertainties in
the experimentally determined Hugoniot and the polynomial fit for Γ, it
is concluded that the present results give the shock temperature in water
at 66± 2 ◦C for pressures to 8 GPa to within ±10 %.
5.5 Bibliography
J. R. Asay and M. Shahinpoor, editors. High-pressure shock compression of solids.
Springer, 1993.
L. M. Barker and R. E. Hollenbach. Shock-wave studies of pmma, fused silica,
and sapphire. Journal of Applied Physics, 41(10):4208–4226, 1970.
D. Benson. Computational methods in Lagrangian and Eulerian hydrocodes.
Computer Methods in Applied Mechanics and Engineering, 99:235–394, 1992.
133
D. J. Chapman. Shock-compression of porous materials and diagnostic development.
PhD thesis, University of Cambridge, 2009a.
D. J. Chapman. Personal communication, 2009b. Fracture and Shock Physics,
Cavendish Laboratory.
D. J. Chapman. Standard materials for use with the Cavendish plate-impact
facility. Technical report, University of Cambridge, 2009c.
C. Dai, H. Tan, J. Hu, Y. Yu, and J. Zheng. Hugoniot evaluation of the pre-
heated metal from its principal hugoniot. Journal of Applied Physics, 99(5):
056102, 2006.
L. Davison. Attenuation of longitudinal elastoplastic pulses. In L. Davi-
son and M. Shahinpoor, editors, High-pressure shock compression of solids III,
pages 277–327. Springer, 1998.
L. Davison and R. A. Graham. Shock compression of solids. Physics Reports,
55:255–379, 1979.
D. H. Dolan. Time dependent freezing of water under multiple shock wave compres-
sion. PhD thesis, Washington State University, 2003.
D. H. Dolan and Y. M. Gupta. Time-dependent freezing of water under dy-
namic compression. Chemical Physics Letters, 374(5-6):608 – 612, 2003.
G. E. Duvall and R. A. Graham. Phase transitions under shock-wave loading.
Reviews of Modern Physics, 49(3):523–579, 1977.
T. Grindley and J. E. Lind. PVT properties of water and mercury. The Journal
of Chemical Physics, 54(9):3983–3989, 1971.
G. A. Gurtman, J. W. Kirsch, and C. R. Hastings. Analytical equation of state
for water compressed to 300 kbar. Journal of Applied Physics, 42(2):851–857,
1971.
J. N. Johnson, R. S. Hixson, G. T. Gray, III, and C. E. Morris. Quasielastic
release in shock-compressed solids. Journal of Applied Physics, 7(2):429–441,
1992.
G. S. Kell. Density, thermal expansivity, and compressibility of liquid water
from 0 ◦C to 150 ◦C correlations and tables for atmospheric pressure and
134
saturation reviewed and expressed on 1968 temperature scale. Journal of
Chemical and Engineering Data, 20(1):97–105, 1975.
K. B. Kormer, S. B. Yushko and G. V. Krishkevich. Phase transformation of
water into ice VII by shock compression. Soviet Physics JETP, 27:879–881,
1968.
S. P. Marsh, editor. LASL Shock Hugoniot Data. University of California Press,
1980.
R. G. McQueen. Shock-waves in condensed media: Their properties and the
equation of state of materials derived from them. In Enrico Fermi summer
course CX11: nuclear collisions from the mean field into the fragmentation regime,
1989.
A. C. Mitchell and W. J. Nellis. Equation of state and electrical conductivity
of water and ammonia shocked to the 100 GPa (1 Mbar) pressure range.
Journal of Chemical Physics, 76:6273–6281, 1982.
K. Nagayama, Y. Mori, K. Shimada, and M. Nakahara. Shock hugoniot com-
pression curve for water up to 1 GPa by using a compressed gas gun. Journal
of Applied Physics, 91(1):476–482, 2002.
J. W. Nunziato and E. K. Walsh. Propagation of steady shock waves in non-
linear thermoviscoelastic solids. Journal of the Mechanics and Physics of Solids,
21:317–335, 1973.
A. Rybakov. Phase transformation of water under shock compression. Journal
of Applied Mechanics and Technical Physics, 37:629–633, 1996.
S. Schoch. Personal communication, 2011. Laboratory for Scientific Comput-
ing, University of Cambridge.
E. Washburn, editor. International Critical Tables of Numerical Data, Physics,
Chemistry and Technology. Knovel, 1st electronic edition, 2003.
D. Yaziv, Z. Rosenberg, and Y. Partom. Release wave calibration of manganin
gauges. Journal of Applied Physics, 51(12):6055–6057, 1980.
135

CHAPTER 6
SHOCK COMPRESSION OF AMMONIUM NITRATE SOLUTIONS
The liquid impact cell developed in previous chapters was applied to the
study of aqueous solutions of ammonium nitrate. Specifically, measurements
were made of the principal Hugoniot and shock/reload response of a series
of ammonium nitrate solutions. From these measurements, an empirical fit
describing the volume-dependence of the Gru¨neisen parameter, Γ in each so-
lution is determined. Shock temperatures can then be calculated along the
Hugoniot, as has been discussed in the preceeding chapters.
The ‘solubility limit’ of a solution is defined as the temperature below
which, for a given concentration (i.e., weight-percentage) of the solute, crystal-
isation would begin to occur. The solubility limit for the ammonium nitrate–
water system, as a function of the weight-percentage of ammonium nitrate, is
shown in Figure 6.1.
The emulsion explosives used in the mining industry were discussed in
Chapter 1. The ‘water’ phase of these emulsions is an aqueous solution of
ammonium nitrate, typically containing 70 % or more ammonium nitrate by
weight [Sudweeks, 1985]. From Figure 6.1, it is clear that such solutions have
solubility limits above room temperature. Indeed, the manufacture of emul-
sion explosives occurs at elevated temperatures, and it is only after the emul-
sion matrix has been formed that the resultant explosive may be cooled to
room temperature [Hustrulid, 1999].
Thus, it is clear that the ammonium nitrate solutions of interest are satu-
rated at room temperature and experiments on them must be conducted at an
elevated initial temperature. It is not immediately obvious what effect raising
the initial temperature will have on the behaviour of the solution. A second
solution, which has a solubility limit below room temperature, can be used to
investigate any effects by conducting experiments on it at room temperature
and at the temperature of the more concentrated solution.
137
Figure 6.1: Phase diagram for the ammonium nitrate/water system as a function of
weight-percentage of ammonium nitrate and temperature [data supplied
by Chan, 2011]. Stable solutions exist for temperatures above the ‘solubility
limit’, shown by the solid curve. For a given concentration, crystallisation
will occur at temperatures below this line.
6.1 Ammonium nitrate solutions
The ‘o’-ring used in the liquid cell ceases to provide an effective seal for tem-
peratures above 70 ◦C, thus limiting the maximum concentration which may
be experimented upon. From Figure 6.1, it can be seen that this limiting con-
centration is approximately 82 %. To provide some margin of error, both in
terms of the efficacy of the ‘o’-ring seal and the solubility limit data, the max-
imum concentration was set at 80 % ammonium nitrate by weight. The sol-
ubility limit for this solution is 57 ◦C. A solution of 50 % ammonium nitrate
by weight, which has a solubility limit below room temperature, was used to
investigate what effect changing the initial solution temperature has on the
calculated shock temperatures.
Solutions were prepared using vacuum-de-gassed and de-ionised water,
and reagant grade ammonium nitrate. The ammonium nitrate was > 98 %
138
purity, and purchased in granular form from Sigma Aldrich. Appropriate
quantities of water and ammonium nitrate were weighed using a balance ac-
curate to ±0.005 g and mixed in a sealed container at 70 ◦C in an oil bath for
several hours until the ammonium nitrate had fully dissolved. As the analy-
sis of shock wave data require knowledge of the initial density of the material
being shocked, it was necessary to determine the density of the solution. At
ambient temperature, this can be accomplished by using a density bottle. A
density bottle is a glass vessel whose volume is accurately known. Measuring
the mass of fluid the bottle contains allows for the density of the fluid to be
determined. At elevated temperatures, the volume of the density bottle is no
longer precisely known, and attempting to compensate for this would intro-
duce unnecessarily large uncertainties into subsequent calculations. Densities
of the solutions at elevated temperature were determined from the literature
[Washburn, 2003]; it is not clear what method was used to measure these den-
sities. The physical properties of the solutions discussed in this chapter are
summarised in Table 6.1.
6.2 Principal Hugoniots
Hugoniots were determined for the three solutions described above using the
methods already presented. As with the pure water experiments described
in Chapter 5, different methods of analysing the data yield subtly different
results. These differences are attributed to the systematic uncertainties associ-
ated with the assumptions used by each method. As in the previous chapter, it
is felt that the measured Hugoniot stresses are less reliable, as they lie in states
of partial release of the gauge. Method 1 combines the measured stress with
the shock velocity deduced from the transit time to calculate the Hugoniot
particle velocity in the solution. It is therefore considered unreliable, whereas
method 3, which constrains the Rayleigh line defined by the measured shock
velocity to lie on the release curve of PMMA, is believed to provide a more
accurate determination of the Hugoniot state. That both these uncertainties
are small is evidenced by the generally close agreement between both meth-
ods. As before, however, method 3 is considered the more reliable method,
and has hence been used to fit the shock velocity–particle velocity Hugoniots.
The Hugoniot of the 50 % ammonium nitrate solution at a temperature of
14 ± 1 ◦C is presented in shock velocity–particle velocity space in Figure 6.2,
and in stress–particle velocity space in Figure 6.3, and the Hugoniot is given
139
C
oncentration
/
W
eigh
t
%
am
m
o
n
iu
m
n
itra
te
Initialtem
perature
/
K
D
ensity
/
g
cm
−
3
c
v /
J
g −
1
K
−
1
Γ
0
5
0
1
4±
1
1.2
2
0±
0
.0
0
5
2.572
1.13±
0.05
5
0
6
5±
2
1.1
9
0±
0
.0
0
5
2.590
0.57±
0.05
8
0
6
4±
2
1.3
7
0±
0
.0
0
5
1.706
0.68±
0.05
Table
6.1:Properties
of
the
am
m
onium
nitrate
solutions
used
in
the
present
w
ork.
D
ensities
at
elevated
tem
peratures
are
from
W
ashburn
[2003],
the
other
data
w
ere
provided
by
C
han
[2011].
c
v
is
the
isochoric
heat
capacity
at
zero
pressure.
Γ
0
is
the
G
ru¨neisen
param
eter,also
atzero
pressure.
140
Figure 6.2: Measured Hugoniot states for a solution of 50 % ammonium nitrate by
weight at 14±1 ◦C. The methods described in §4.2 yield somewhat different
results for the Hugoniot state for each experiment. The blue line is the fit
through the states determined by method 3, which is considered to provide
the most accurate determination of the Hugoniot states, and is given by the
equation US = 1.86 ± 0.03 + 1.80 ± 0.06up. The dashed lines are the 95 %
confidence bounds of the fits, and suggest that data is well represented by
a linear relationship. The large discrepancy between the two methods for
the highest particle velocity datum are likely due to mismeasurement of the
stress, potentially due to a fault in the construction of the gauge package.
There is no evidence from the data for a phase change over the range of
particle velocities investigated.
by the linear relationshipUS = 1.86±0.03+1.80±0.06up. It is not clear why the
two methods give markedly different results for the highest particle velocity
experiment. Such a discrepancy could be due to the stress being recorded
incorrectly in this particular experiment, possibly due to a fault, such as a gas
bubble in the glue layer, introduced when the cell was constructed.
The Hugoniot of the 50 % ammonium nitrate solution at a temperature of
65 ± 2 ◦C is presented in shock velocity–particle velocity space in Figure 6.4,
141
Figure 6.3: Stress–particle velocity Hugoniot for a solution of 50 % ammonium nitrate
by weight at 14±1 ◦C, determined by two analysis methods, and presented
with the Hugoniot given by US = 1.86 ± 0.03 + 1.80 ± 0.06up. That the
highest datum analysed by method 1 is not well-described by the fit is con-
sidered further evidence that the Hugoniot stress in the liquid was mis-
measured in this experiment.
and in stress–particle velocity space in Figure 6.5. The fitted Hugoniot is given
by the equation US = 1.73 ± 0.05 + 1.49 ± 0.10up, and therefore lies below
the ambient temperature Hugoniot in both spaces. Chapman [2009] reported
the Hugoniots for samples of silica powder at various initial densities; the
lower initial density samples were found to lie below those of higher initial
densities. Similar results can be seen in the Hugoniot data given by Marsh
[1980] for a number of different materials. The present results act to confirm
the conclusion that lower density forms of a given material in general will
have lower Hugoniots. From a phenomenological view point, the elevated
temperature solution can be considered to be a less dense modification of the
ambient temperature solution, and therefore has a lower shock impedance.
By definition of the impedance, it must, therefore, have a lower Hugoniot, as
142
Figure 6.4: Measured Hugoniot states for a solution of 50 % ammonium nitrate by
weight at 65±2 ◦C. The methods described in §4.2 yield somewhat different
results for the Hugoniot state for each experiment. The blue line is the fit
through the states determined by method 3, which is considered to provide
the most accurate determination of the Hugoniot states, and is given by the
equation US = 1.73 ± 0.05 + 1.49 ± 0.10up. The dashed lines are the 95 %
confidence bounds of the fits, and suggest that data is well represented by
a linear relationship. As with the ambient temperature experiments, there
is no evidence of any phase changes occurring over the range of particle
velocities investigated.
the sound speed is unlikely to increase with decreasing density.
The Hugoniot of the 80 % ammonium nitrate solution at a temperature of
64 ± 2 ◦C is presented in shock velocity–particle velocity space in Figure 6.6,
and in stress–particle velocity space in Figure 6.7. The Hugoniot is given by
the equation US = 1.83± 0.04 + 1.73± 0.09up. There is no evidence in any of
the presented Hugoniots to suggest that any phase changes occur in the so-
lutions over the stress/particle velocity ranges investigated, nor is there any
suggestion from the data of non-linearity of the shock velocity–particle veloc-
ity Hugoniot.
143
Figure 6.5: Stress–particle velocity Hugoniot for a solution of 50 % ammonium nitrate
by weight at 65±2 ◦C, determined by two analysis methods, and presented
with the Hugoniot given by US = 1.73± 0.05 + 1.49± 0.10up. The stress–
particle velocity data determined by both methods are well-described by
the present fit.
The Hugoniots of the three solutions, in stress–particle velocity space, are
presented in Figure 6.8 together with their associated uncertainties. These
Hugoniots have been extrapolated to particle velocities approximately twice
the maximum particle velocity in the experiments (≈ 1 km s−1). As has al-
ready been discussed, the elevated temperature Hugoniot of the 50 % ammo-
nium nitrate solution lies below the ambient temperature Hugoniot due to the
decreased density. The 80 % ammonium nitrate solution Hugoniot lies above
the two 50 % ammonium nitrate solution Hugoniots, principally due to its in-
creased density.
144
Figure 6.6: Measured Hugoniot states for a solution of 80 % ammonium nitrate by
weight at 64 ± 2 ◦C. The three described in §4.2 yield somewhat different
results for the Hugoniot state for each experiment. The blue line is the fit
through the states determined by method 3, which is considered to provide
the most accurate determination of the Hugoniot states, and is given by the
equation US = 1.83 ± 0.04 + 1.73 ± 0.09up. The dashed lines are the 95 %
confidence bounds of the fits, and suggest that data is well represented by
a linear relationship. As with the 50 % ammonium nitrate solutions, there
is no evidence that any phase changes have taken place.
145
Figure 6.7: Stress–particle velocity Hugoniot for a solution of 80 % ammonium nitrate
by weight at 64±2 ◦C, determined by two analysis methods, and presented
with the Hugoniot given by US = 1.83± 0.04 + 1.73± 0.09up. As with the
50 % ammonium nitrate solutions, the data reduced by both methods are
well-described by the fitted linear Hugoniot.
146
Figure 6.8: Stress–particle velocity Hugoniots for the ammonium nitrate solutions; the
solid curves are given by the fitted Hugoniots, and the dashed lines repre-
sent the uncertainties in the intercept and slope of the linear shock velocity–
particle velocity Hugoniots. The elevated temperature 50 % ammonium
nitrate solution can be considered as a less dense version of the ambient
temperature 50 % solution, and it is expected that this should result in a de-
crease in the impedance of the material. The 80 % ammonium nitrate solu-
tion, which has a higher density than the other solutions, consequently has
a higher impedance. The experimental Hugoniot data are limited to parti-
cle velocities of around 1 km s−1; the lack of any evidence for non-linearity
in the present data allow for the Hugoniots to be extended to higher par-
ticle velocities with ease, as the linear Hugoniots are well-behaved at all
particle velocities, whereas a quadratic relationship between particle veloc-
ity and shock velocity would not behave well in stress–volume space when
extrapolated.
147
6.3 Reload experiments
The reported experiments, in addition to measuring the principal Hugoniots
of the solutions, enabled determination of the reload state of the solutions.
These reload states are plotted for the various solutions in Figures 6.9, 6.10,
and 6.11. In some of the experiments, the rear gauge failed before reaching an
equilibrium value; these experiments have been neglected in the analysis of
reload states, and are indicated by the circled points on the principal Hugo-
niots in the figures below. Rayleigh lines, connecting the Hugoniot and reload
states, have been added to the graphs. As expected, the reload states gener-
ally lie below the principal Hugoniot; one exception is the reload state with
lowest stress in Figure 6.10, which lies somewhat above the Hugoniot. This is
likely due to the inherently larger proportional uncertainties at lower stresses
causing the reload volume to be miscalculated slightly.
The measured reload states can be used to obtain values of the Gru¨neisen
parameter, Γ. Generally, the method for determining values of Γ discussed in
the preceeding chapters is only reliable for reload stresses above 3 GPa. Be-
low this reload stress, when the Hugoniot stress is of order 1 GPa, the relative
uncertainties in the measured stresses are sufficient to prevent an accurate de-
termination of Γ, and the calculation of Γ produces unphysically large values.
The calculation of temperature on the Hugoniot requires a form for Γ that can
be integrated from the initial specific volume, v0. Whereas, in water, accurate
compressional isotherms are reported in the literature enabling determination
of the behaviour of Γ at low compressions, for the solutions reported, the low
compression behaviour is limited to a single value of Γ at the initial density
[Chan, 2011]. In analysing the volume-dependence of Γ, a linear form has
been assumed. Whilst the data in water suggest that a more complicated form
might be appropriate, without isothermal compression data, it is not possible
to ensure appropriate low compression behaviour for higher-order polyno-
mials. Thus, as values of Γ are only available above 3 GPa and at the initial
density, a linear dependence on volume is the simplest admissible form.
Figure 6.12 shows the measured values of Γ in a solution of 50 % ammo-
nium nitrate by weight at 14± 1 ◦C together with the zero-pressure value and
the linear fit, Γ = 3.65 − 3.62v, through the data. The fact that Γ decreases
with decreasing volume in the reload experiments suggests that there is a
maximum in Γ near v = 0.6 cm3 g−1, as was demonstrated in the water exper-
iments. However, without further measurements of Γ at low compressions, it
148
Figure 6.9: Hugoniot and reload states, in stress–specific volume space, for a solution
of 50 % ammonium nitrate by weight at 14 ± 1 ◦C. The triangular sym-
bols are the measured Hugoniot states; the error bars, which are generally
smaller than the symbols, have been omitted for clarity. The solid straight
lines are the Rayleigh lines linking each Hugoniot state to its corresponding
reload state (square symbols). The solid black curve is the fitted Hugoniot
for the solution. The two circled points on the Hugoniot do not have mea-
sured reload states, as the rear gauges in these experiments failed before
reaching equilibrium stress values.
149
Figure 6.10: Hugoniot and reload states, in stress–specific volume space, for a solu-
tion of 50 % ammonium nitrate by weight at 65 ± 2 ◦C. The triangular
symbols are the measured Hugoniot states; the error bars, which are gen-
erally smaller than the symbols, have been omitted for clarity. The solid
straight lines are the Rayleigh lines linking each Hugoniot state to its cor-
responding reload state (square symbols). The solid black curve is the fit-
ted Hugoniot for the solution. The two circled points on the Hugoniot do
not have measured reload states, as the rear gauges in these experiments
failed before reaching equilibrium stress values. The lowest stress reload
state appears to lie above the Hugoniot, probably due to the reload stress
being incorrectly measured in this experiment.
150
Figure 6.11: Hugoniot and reload states, in stress–specific volume space, for a solution
of 80 % ammonium nitrate by weight at 64 ± 2 ◦C. The triangular sym-
bols are the measured Hugoniot states; the error bars, which are generally
smaller than the symbols, have been omitted for clarity. The solid straight
lines are the Rayleigh lines linking each Hugoniot state to its correspond-
ing reload state (square symbols). The solid black curve is the fitted Hugo-
niot for the solution. The circled point on the Hugoniot does not have a
measured reload state, as the rear gauge in this experiment failed before
reaching an equilibrium stress value.
151
Figure 6.12: Γ in a solution of 50 % ammonium nitrate by weight at 14 ± 1 ◦C de-
termined from shock/reload experiments, together with the calculated
value of Γ at v0 [Chan, 2011]. The linear fit through these data is given
by Γ = 3.65 − 3.62v. The data derived from shock/reload experiments
suggest that Γ reaches a maximum near a specific volume of 0.6 cm3 g−1;
without further data near v0, however, it is not possible to fit a higher-
order polynomial through the data that behaves physically for all vol-
umes.
is not possible to fit a more complicated function whilst maintaining realistic
behaviour as v tends to v0. These results imply that there is a temperature-
dependence of Γ at constant volume; this point is discussed further in Chap-
ter 7.
Increasing the initial temperature of the solution to 65±2 ◦C has little effect
on the measured values of Γ, which are presented in Figure 6.13. Again, the
data seem to imply that there is a maximum in Γ near v = 0.6 cm3 g−1, but
the lack of data in the low compression region do not enable a higher-order
polynomial to be fitted to the data. The linear fit is given by Γ = 3.93− 4.01v.
The variation of Γ with volume in a solution of 80 % ammonium nitrate by
152
Figure 6.13: Γ in a solution of 50 % ammonium nitrate by weight at 65 ± 2 ◦C de-
termined from shock/reload experiments, together with the calculated
value of Γ at v0 [Chan, 2011]. The linear fit through these data is given
by Γ = 3.93 − 4.01v. The data derived from shock/reload experiments
suggest that Γ reaches a maximum near a specific volume of 0.6 cm3 g−1;
without further data near v0, however, it is not possible to fit a higher-
order polynomial through the data that behaves physically for all vol-
umes.
weight at 64± 2 ◦C is shown in Figure 6.14. In contrast to the experiments on
the 50 % ammonium nitrate solution, there is no evidence for a maximum in Γ
over the range of volumes investigated, and the linear relationship Γ = 3.87−
3.76v provides a good fit through the data from shock/reload experiments
and the calculated value of Γ at v0.
The linear fits for the volume dependence of Γ in each solution have simi-
lar gradients, and the reported values of Γ lie close to each other. This suggests
that the response of the three solutions is quite similar, a suggestion supported
by the similarity of the solution Hugoniots reported above.
In order to calculate temperatures along the Hugoniot, in addition to de-
153
Figure 6.14: Γ in a solution of 80 % ammonium nitrate by weight at 64 ± 2 ◦C deter-
mined from shock/reload experiments, together with the calculated value
of Γ at v0 [Chan, 2011]. The linear fit through these data is given by
Γ = 3.87 − 3.76v. In contrast to the the other experiments reported in
this thesis, there is no evidence that Γ reaches a maximum value over the
range of specific volumes investigated.
termining the Hugoniot and the volume-dependence of Γ, a model for the
heat capacity at constant volume, cv , is required. In the present analysis, the
value of cv has been taken to be identical to the value at zero pressure given by
Chan [2011]. That cv is constant is the standard assumption made in the litera-
ture, and is usually justified by assuming the material is well above its Debye
temperature; without detailed data on the behaviour of cv under dynamic
compression, constant cv is also the simplest assumption. These heat capaci-
ties are given in Table 6.1. The calculated temperature rises in the solutions are
presented in Figure 6.15, along with the temperatures in water determined in
Chapter 5, as a function of the Lagrangian compression, ρ0(v0 − v). The data
are presented as a function of compression, rather than pressure or density, as
154
the liquids have different initial densities and different impedances. Giving
the temperatures as a function of compression allows for easier comparison
between the materials. Errors have been calculated by considering the effect
of uncertainties in the measured Hugoniots and the heat capacity used, and
are of order 10 % on the temperature rise.
Given the similarities in the Hugoniots and values of Γ, the variations
in calculated shock temperatures in the solutions can chiefly be explained
through their different heat capacities. As lower heat capacities would give
rise to larger calculated temperature rises, the lower heat capacity in the 80 %
ammonium nitrate solution, compared to the 50 % solution at the same initial
temperature, explains the majority of the difference in calculated shock tem-
peratures for these experiments. As with the water, the effect of increasing
the initial temperature for the 50 % solution is predominantly to increase the
shock temperatures by the difference in initial temperature.
Changes in the functional form or values of Γ are manifested by changes
in the curvature of the temperature-compression curves shown in Figure 6.15.
The present temperature calculations are relatively insensitive to the exact
form of Γ. McQueen [1989] has commented that Γ cannot be taken as con-
stant, however, as this results in unphysical behaviour being predicted. For
the 50 % ammonium nitrate solutions, replacing the single linear fit with a fit
that is piecewise linear between the three values changes the calculated tem-
peratures by only one or two percent, and is thus significantly smaller than
the other experimental uncertainties. Variations in the value of cv change the
magnitude of the temperature rise but not the curvature. The assumptions
that the heat capacity is constant and within 10 % of that at zero pressure, and
that Γ takes the linear form described above, yield an uncertainty in the cal-
culated shock temperature rises of 10 %. As with the previous results in this
thesis, variation of the heat capacity along the Hugoniot has been neglected,
as is generally the case in the literature. It is unclear what form such variation
may take in these solutions. It is felt that, over the compression range of in-
terest, the present 10 % uncertainty in the calculated temperatures is sufficient
to account for any such variations. The calculated temperature rises are only
valid over the compression range measured along the Hugoniot, where the
maximum compression was approximately 0.25. However, as is discussed in
the following chapter, these calculations must be considered as only approxi-
mations.
155
Figure 6.15: Calculated temperature rises, assuming a constant value of cv equal to
the zero pressure heat capacity, in the ammonium nitrate solutions, and
temperatures in water determined in Chapter 5, as a function of the La-
grangian compression, ρ0(v0 − v). The calculated temperatures are only
valid over the compression range of the Hugoniots measured above. The
uncertainty on the temperatures is 10 %.
6.4 Summary
• The principal Hugoniots of three ammonium nitrate solutions have been
determined experimentally.
• The shock velocity–particle velocities are linear over the range of particle
velocities studied.
• The Hugoniots are similar to each other, and the variations seem most
strongly correlated with the variations in initial density.
• Stress and specific volume in states off the principal Hugoniots have
been measured through shock/reload experiments.
• Extrapolation of the principal Hugoniots to compressions enabled de-
termination of the Gru¨neisen parameter, Γ.
156
• By assuming cv was constant and equal to the zero pressure value, tem-
peratures along the Hugoniot were calculated for each solution.
• To Lagrangian compressions of 0.25, these temperatures have uncertain-
ties of 10 %.
• Within the present experimental uncertainties, for a given solution, the
shock temperatures at a higher initial temperature can be found by adding
the increase in initial temperature to the calculated shock temperatures
for the lower initial temperature.
6.5 Bibliography
J. Chan. Personal communication, 2011. Orica Mining Services.
D. J. Chapman. Shock-compression of porous materials and diagnostic development.
PhD thesis, University of Cambridge, 2009.
W. Hustrulid. Blasting principles for open pit mining, volume 1 - General design
concepts. A. A. Balkema, 1999.
S. P. Marsh, editor. LASL Shock Hugoniot Data. University of California Press,
1980.
R. G. McQueen. Shock-waves in condensed media: Their properties and the
equation of state of materials derived from them. In Enrico Fermi summer
course CX11: nuclear collisions from the mean field into the fragmentation regime,
1989.
W. B. Sudweeks. Physical and chemical properties of industrial slurry explo-
sives. Industrial & Engineering Chemistry Product Research and Development,
24(3):432–436, 1985.
E. Washburn, editor. International Critical Tables of Numerical Data, Physics,
Chemistry and Technology. Knovel, 1st electronic edition, 2003.
157

CHAPTER 7
ANALYSIS AND DISCUSSION
The preceding chapters have described a series of experiments that were con-
ducted to investigate the response of water and of ammonium nitrate solu-
tions to shock loading. The Mie-Gru¨neisen equation of state has previously
been used to describe the thermodynamic behaviour of water under shock
loading and, under the assumption that the heat capacity Cv is constant, has
been used to calculate shock temperature rises [Gurtman et al., 1971]. The
present work applied this model to water at an elevated initial temperature
(Chapter 5) and to solutions of ammonium nitrate (Chapter 6). Attention is
now turned to the validity of the assumptions made in the present model
and, hence, to the applicability of the Mie-Gru¨neisen equation of state to the
behaviour of water and ammonium nitrate solutions. Firstly, the effect of
preheating on a material’s Hugoniot is considered and comparison between
the experimentally-determined and theoretically-predicted Hugoniots made.
Secondly, the functional dependencies of Γ are examined, and it is demon-
strated that the present results show a dependence on temperature in addition
to volume. Finally, the theoretical understanding of the shock compression of
water is examined, with particular reference to the behaviour of Γ and the
heat capacity CV .
7.1 Relationships amongst the Hugoniots
The experimentally-determined Hugoniots for water at initial temperatures of
18±2 ◦C and 66±2 ◦C are shown in stress–particle velocity space in Figure 7.1.
The elevated temperature Hugoniot lies significantly below the ambient tem-
perature Hugoniot, suggesting that a significant softening has occurred. Soft-
ening has been experimentally observed in preheated metal samples [e.g., Dai
et al., 2006], although the level of softening is typically much smaller than that
shown in Figure 7.1.
The Mie-Gru¨neisen equation can be used to calculate Hugoniots at ele-
vated temperatures. Using the principal Hugoniot as the reference curve, the
Mie-Gru¨neisen equation becomes
159
Figure 7.1: Pressure–particle velocity Hugoniots of water at 18 ± 2 ◦C (blue) and
66± 2 ◦C (red), from the results described in Chapter 5. The elevated tem-
perature Hugoniot lies below the ambient temperature Hugoniot, indicat-
ing a reduction in shock impedance with temperature over this range of
temperatures and particle velocities.
P
′
H − PH =
Γ
v
(E
′
H − EH). (7.1)
whereE
′
H and P
′
H are the respectively the energy and pressure on the elevated
temperature Hugoniot. Considering the generalised energy jump condition
(Equation 2.7), we have, on the principal Hugoniot
EH − E0 = 1
2
PH(V0 − V ), (7.2)
and, similarly, on the elevated temperature Hugoniot
E
′
H − E
′
0 =
1
2
P
′
H(V
′
0 − V ), (7.3)
where E
′
0 is the specific internal energy at ambient pressure at the elevated
temperature and is simply E0 + Cp∆T , where Cp is the heat capacity at con-
160
stant pressure, which is typically assumed to be constant - in water, over the
temperature range of interest, the variation ofCp with temperature is less than
one part in a thousand [Washburn, 2003]. Combining this result with Equa-
tions 7.1, 7.2, and 7.3, the pressure–volume Hugoniot at elevated temperature
is
P
′
H = PH
 1− Γv (v0−v2 )
1− Γv
(
v
′
0−v
2
)
+ Γv cp∆T
1− Γv
(
v
′
0−v
2
) , (7.4)
and thus may be calculated from the principal Hugoniot provided Γ is known.
Using the principal Hugoniot for water determined experimentally in Chap-
ter 5 and the polynomial form for Γ(v) given by Gurtman et al. [1971], the
elevated temperature Hugoniot for water in pressure–volume space has been
calculated; the heat capacity was taken to be 4.2 J g−1 K−1 [Washburn, 2003].
This calculation is presented in Figure 7.2, together with the experimentally
determined principal and elevated temperature Hugoniots.
The calculation suggests that the elevated temperature Hugoniot should,
in pressure-volume space, lie to the right of the ambient temperature Hugo-
niot, with the shift being partly due to the change in initial volume and partly
due to the increase in specific internal energy at zero pressure. The calculated
elevated temperature Hugoniot crosses the measured ambient Hugoniot at a
stress of around 7 GPa. The implication of this crossing over is that the spe-
cific internal energy at a given volume for the elevated temperature material
is lower than that of the ambient temperature material; it should however be
recalled that an additional term Cp∆T must be added to the energy of the
elevated temperature material to reflect the energy added by pre-heating it.
The measured elevated temperature Hugoniot does not agree well with the
calculated Hugoniot, as it crosses the ambient Hugoniot at around 0.5 GPa
and thence lies significantly below it at most stresses. A similar discrepancy
between the calculated and measured elevated temperature Hugoniots occurs
in pressure–particle velocity space, as is shown in Figure 7.3. As expected, the
calculation shows a softening of the water, which results in the calculated el-
evated temperature Hugoniot lying below the principal Hugoniot. However,
the calculation is, within error, the same as the ambient temperature Hugo-
niot. The measured elevated temperature Hugoniot shows a much greater de-
gree of softening. It has been noted by a number of authors [Jonas et al., 1976;
Postorino et al., 1993; Tromp et al., 1994] that, as the pressure and temperature
161
Figure 7.2: Pressure–volume Hugoniots of water at initial temperatures of 18 ± 2 ◦C
(blue) and 66 ± 2 ◦C (red), from the results described in Chapter 5. The
green curve has been calculated using Equation 7.4, the experimentally de-
termined water Hugoniot, and the form of Γ for ambient temperature water
given by Gurtman et al. [1971] under the assumption that the heat capacity
cp at ambient pressure is independent of temperature. The calculations sug-
gest that, in pressure–volume space, the increase in specific internal energy
at zero pressure, together with the reduction in initial density, dominate
the behaviour of the higher temperature Hugoniot. The calculations indi-
cate that Hugoniots will cross over at a stress of around 7 GPa. The experi-
mentally determined elevated temperature Hugoniots crosses the ambient
Hugoniot at 0.5 GPa, the the errors cease to overlap at around 1.5 GPa.
162
of water are increased, significant structural changes occur and there appears
to be a transition from behaviour that is dominated by hydrogen bonds to a
more ‘normal’ liquid response. The implication of these studies is that, for a
given pressure, the elevated temperature water would be significantly denser;
Bradley and Pitzer [1979] note that these structural changes begin to become
important at temperatures above 50 ◦C. Due to these structural changes, it
would appear that water at elevated temperature is a fundamentally differ-
ent material than water at ambient temperature. The increase in temperature
appears to increase markedly the compressibility of water and reduce signifi-
cantly its impedance.
Figure 7.3: Pressure–particle velocity Hugoniots of water at initial temperatures of
18 ± 2 ◦C (blue) and 66 ± 2 ◦C (red), from the fits described in Chapter 5.
Particle velocities corresponding to the calculated elevated temperature
Hugoniot shown in Figure 7.2 have been calculated using the relationship
u
′
p =
√
P ′(v
′
0 − v′), where the primes indicate that the values are on the
elevated temperature Hugoniot. The calculation suggests that the elevated
temperature should lie below, but within error of, the ambient temperature
Hugoniot. The measured elevated temperature Hugoniot is significantly
softer than the ambient temperature Hugoniot and is not within error of it.
163
A similar behaviour, of softening (reduction in impedance) is observed in
the compression of a solution of 50 % ammonium nitrate by weight at 14±1 ◦C
and 65 ± 2 ◦C. The Hugoniots determined in Chapter 6 for this solution are
presented in Figure 7.4. Such behaviour is to be expected: whilst the weight
percentages of ammonium nitrate and water are equal, an ammonium ni-
trate molecule is four-times heavier than a water molecule and, thus, there
are many more water molecules in the solution. The solution can be consid-
ered as ammonium (NH+4 ) and nitrate (NO
−
3 ) ions completely surrounded
by water and therefore the response of the solution will be dominated by the
response of water.
Figure 7.4: Pressure–particle velocity Hugoniots of a solution of 50 % ammonium ni-
trate by weight at 14 ± 1 ◦C (blue) and 65 ± 2 ◦C (red), from the fits de-
scribed in Chapter 6. As with the water Hugoniots shown in Figure 7.1, the
elevated temperature Hugoniot lies below the ambient temperature Hugo-
niot, indicating a reduction in shock impedance with temperature over this
range of temperatures and particle velocities.
Figure 7.5 shows the pressure–particle velocity data given by Galbraith
[1997] for a solution of 54 % ammonium nitrate by weight at 20±1 ◦C and 80±
164
1 ◦C. The uncertainties associated with this data were discussed in Chapter 1.
For ease of comparison, a quadratic (constrained through the origin) has been
fitted to Galbraith’s data. As with the liquids discussed above, it is clear that
the elevated temperature solution has a lower impedance than the ambient
temperature solution.
Figure 7.5: Hugoniots, in pressure–particle velocity space, of a solution of 54 % ammo-
nium nitrate by weight at 20 ± 1 ◦C (blue) and 80 ± 1 ◦C (red). The data
are from Galbraith [1997], and have been re-analysed to give the present
Hugoniots, which are taken as quadratic relationships between pressure
and particle velocity, and have been forced through the origin. Again, the
higher temperature Hugoniot lies below that at lower temperature, indicat-
ing a softening of the liquid and effective reduction in impedance.
The present experimental results, together with those of Galbraith [1997]
and the Mie-Gru¨neisen calculations described above, demonstrate that water,
and water-dominated solutions, become significantly softer as they are pre-
heated to an extent that is not described by the Mie-Gru¨neisen equation of
state. In water, it is not possible to achieve agreement between the measured
and calculated elevated temperature Hugoniots for any physically plausible
165
values of Γ or the heat capacityCp. As has been noted in the literature, it seems
that there is a significant reduction in the effect of the peculiar structural prop-
erties of water as temperatures increase. It is, perhaps, therefore unwise to
consider ambient and elevated temperature water as being the same material.
7.2 Functional dependencies of Gru¨neissen’s parameter
Throughout this thesis, it has been assumed that Γ is a function of volume
alone; indeed this is required if one is to use the Hugoniot, and not the cold
compression curve, as the reference pressure-energy curve. In discussing the
field of shock compression and of equations of state, McQueen [1989] stated
that
“One of the basic results from these studies is that Γ is a func-
tion of volume alone. To our knowledge this has never really been
tested, since Γ has never been measured at two temperatures at the
same density.”
Whilst it is true that such measurements of Γ from shock compression ex-
periments have not been reported in the literature, other experimental mea-
surements of Γ have been reported, but are largely neglected by the shock
physics community. Knopoff and Shapiro [1969] discuss the applicability of
the Mie-Gru¨neisen equation of state to shock wave data, noting the logical
contradiction between the derivation of the Mie-Gru¨neisen equation of state
(which assumes a crystalline solid) and the jump conditions (which assume
an inviscid fluid). From the thermodynamic definition of Γ (Equation 2.19),
we have
Γ = V
∂P
∂E
∣∣∣∣
V
(7.5)
= V
∂P
∂T
∣∣∣∣
V
/
∂E
∂T
∣∣∣∣
V
(7.6)
=
V
CV
(
− ∂V
∂T
∣∣∣∣
P
/
∂P
∂V
∣∣∣∣
T
)
(7.7)
=
αKTV
CV
(7.8)
where CV = ∂E∂T
∣∣
V
is the heat capacity at constant volume, KT = −V ∂P∂V
∣∣
T
is
the isothermal bulk modulus, and α = 1V
∂V
∂T
∣∣
P
is the volumetric coefficient of
166
thermal expansion. These quantities have been measured at a variety of tem-
peratures under static loading for a number of materials, and as Knopoff and
Shapiro [1969] note, the resultant values of Γ determined in a variety of solids
are independent of temperature. Knopoff and Shapiro [1969] determined val-
ues of Γ as a function of both temperature and pressure in water from litera-
ture data of the heat capacity, bulk modulus, and thermal expansivity. Their
results are shown in Figure 7.6.
Figure 7.6: Variation of Γ with volume at different temperatures in water, from Knopoff
and Shapiro [1969]. The values were calculated from published isostatic
compression data, and indicate a dependence of Γ on temperature. The
variation with temperature appears to decrease with increasing pressure
(decreasing volumes).
One interesting feature is that Γ increases with decreasing V , which is the
opposite to the behaviour found in sold metals [McQueen, 1989]. This be-
haviour, of Γ increasing with decreasing V , has also been observed in mercury
(Figure 7.7) which, whilst metallic, is also a liquid; there is also a temperature
dependence of Γ at constant volume. It is obvious from Figure 7.6 that in water
there is a significant dependence of Γ on temperature, and at larger tempera-
tures Γ is larger. However, as the pressure increases and the volume decreases,
167
the temperature dependence decreases. Knopoff and Shapiro [1969] conclude
that at high pressures the Mie-Gru¨neisen equation of state provides a valid
description of liquids, suggesting that the effect of compression is to increase
the spatial correlation of the water molecules to a point at which the lattice dy-
namic description becomes more appropriate. They note that further experi-
mentation is required, at higher compressions, to investigate the temperature
dependence of Γ in liquids. The present study, therefore, represents a test of
the validity of the assumption that Γ is independent of temperature.
Figure 7.7: Variation of Γ at different temperatures in mercury as a function of the rel-
ative volume V/V0 (where V0 is the volume at room teperature and pres-
sure), from Knopoff and Shapiro [1969]. The values were calculated from
published isostatic compression data, and indicate a dependence of Γ on
temperature. Γ is observed to increase with decreasing volume, in contrast
to the standard behaviour in solid metals [McQueen, 1989].
Figure 7.8 shows the values of Γ measured in water at 66 ± 2 ◦C, from
the experiments described in Chapter 5. These data, and the fit derived from
them, are compared to the fit for Γ in water at ambient temperature given by
Gurtman et al. [1971]. It is clear from the values of Γ determined from the
shock/reload experiments conducted that even a moderate degree of heating
(48 ± 3 ◦C) significantly alters the value of Γ. The present results, therefore,
support the trend suggested by the results of Knopoff and Shapiro [1969]. In
producing their fit, Gurtman et al. [1971] constructed the low-compression
data from isothermal compression curves, choosing the isotherms at 75 ◦C
and 175 ◦C to provide the best fit to the shock data; that different pairs of
168
Figure 7.8: Variation of Γ with volume for water at 66 ± 2 ◦C, from the experiments
described in Chapter 5. The fit given by Gurtman et al. [1971] for water at
ambient temperature is shown for comparison. The present results indicate
that Γ has a temperature dependence over a wide range of specific volumes.
isotherms produce different values of Γ is a clear suggestion that there is a
temperature dependence of Γ. It is surprising that the isothermal data sug-
gest that Γ is lower in the higher temperature water, as the opposite result is
found from the shock data. This could be due to uncertainties in the value
used for CV when calculating Γ from isotherms, which has been discussed in
the preceeding chapters.
Similar results were found for a solution of 50 % ammonium nitrate by
weight at 14±1 ◦C and 65±2 ◦C. Values of Γ determined from the experiments
described in Chapter 6 are shown in Figure 7.9. As with water, the ambient
pressure values of Γ (calculated by Chan [2011] using Equation 7.8) suggest
that, initially, Γ is smaller in the higher temperature liquid; again, this trend is
reversed in the dynamic experiments.
169
Figure 7.9: Variation of Γ with volume for a solution of 50 % ammonium nitrate by
weight at 14 ± 1 ◦C (blue) and 65 ± 2 ◦C (red), from the experiments de-
scribed in Chapter 6. The present results indicate that Γ has a temperature
dependence over a wide range of specific volumes.
The implication, then, of the present experimental results is that in wa-
ter and ammonium nitrate solutions, at pressures below 10 GPa, there is sig-
nificant dependence of Γ on temperature and, hence, as noted by Anderson
[2000], the Mie-Gru¨neisen equation of state is a poor model of the thermal
pressure. In light of this conclusion, the temperature calculations presented in
the preceeding chapters should be considered only as rough approximations.
Further, Anderson [2000] suggests that the Mie-Gru¨neisen equation of state
is only applicable to systems that have a high level of coordination; it seems
likely then that water, and ammonium nitrate solutions, will not be amenable
to the simple treatment that is generally applied in the literature. It should be
stressed that the standard form of the Mie-Gru¨neisen equation of state, which
uses the principal Hugoniot as a reference curve, is only valid when Γ is a
function of volume alone.
170
7.3 Pathological behaviour of water
The foregoing sections have discussed the results presented in this thesis in
the context of the Mie-Gru¨neisen equation of state. It has been shown the Mie-
Gru¨neisen equation provides a poor description of the effect of preheating on
the water Hugoniot, and that the present results indicate a significant varia-
tion of Γ with temperature. There is some suggestion from the literature that
structural effects in water may be responsible for both of these phenomena. It
has also been noted in earlier chapters that uncertainties in the heat capacity
CV limit the ability to calculate Γ from isothermal compression data, and have
an effect on calculated shock temperatures. The remainder of this chapter con-
cerns the theoretical understanding of the behaviour of water, with particular
reference to the functional form and dependencies of CV and Γ.
Water is a pathologically complex molecule, and its properties vary dra-
matically with temperature and pressure [Nellis, 2008]. In the liquid form, wa-
ter molecules auto-dissociate into hydronium (H3O+) and hydroxide (OH−)
ions:
2H2OEGGGGGCH3O
+ +OH−. (7.9)
Under shock compression, single water molecules also dissociate:
H2OEGGGGGCH
+ +OH−, (7.10)
and the proportions of the various ions and molecules varies strongly with
both pressure and temperature [Goldman et al., 2009a; Holzapfel, 1969]. The
complex interaction between these various ionic species, and the non-spherical
potentional of the water molecule itself, have hampered ab initio modelling
efforts. Ree [1982] opted to focus on modeling higher pressures and temper-
atures in order that the complex interactions could be neglected. Their calcu-
lated value of Γ as a function of temperature and density on the Hugoniot is
reproduced in Figure 7.10.
The calculated values of Γ are markedly lower than the experimentally
determined values [e.g. Gurtman et al., 1971], which Ree attributed to uncer-
tainties in the experimental measurements, having noted that the molecular
potential used was able to accurately reproduce the principal Hugoniot, and
also commented that the finite-difference approximation used to calculate Γ
leads to an over-estimate of the value of Γ. The dashed portion of the curve
171
Figure 7.10: Ab initio calculations of the variation of Γ with (a) temperature and (b)
density on the Hugoniot, from Ree [1982]. The dashed portions of the
curves represent the low temperature region where complex interactions
neglected in the calculations become important. The calculated values of Γ
are significantly lower than those measured experimentally; this discrep-
ancy was attributed to errors in the experiments.
represents the lower temperature region where the calculations become uncer-
tain. This is the region of densities and temperatures of interest to the present
work. The decrease in Γ at high pressures is attributed to an increase in the
compressibility due to the collapse of the hydrogen bonding structure of wa-
ter. In contrast Nellis [2008] suggests this behaviour indicates that the energy
is being absorbed internally as well as thermally. Nigmatulin and Bolotnova
[2008] note that experiments suggest Γ in water must have a temperature de-
pendence, and explicity included this in their construction of an equation of
state for water and steam. Their calculation of Γ as a function of volume at
different temperatures is reproduced in Figure 7.11.
Whilst not directly comparable to the form for Γ given by Gurtman et al.
[1971] or the calculations made by Ree [1982] (Figure 7.10), comparison can be
made to the values of Γ deduced from isothermal compression data discussed
in Chapter 5, some of which are shown in Figure 7.8 above. The calculated
values of Γ shown in Figure 7.11 do not agree with those determined from
isothermal compression, both in terms of their value and the variation with
volume. The calculations made by Nigmatulin and Bolotnova [2008] appear
172
Figure 7.11: Variation of Γ with volume at different temperatures, from Nigmatulin
and Bolotnova [2008]. In the figure, V00 corresponds to the initial vol-
ume at room temperature, which has been referred to as v0 elsewhere in
this thesis. Whilst not directly comparable to the preceeding results, and
the calculations given by Ree [1982], it is reasonably clear that the func-
tional form given by Nigmatulin and Bolotnova [2008] does not match
well with the experimentally determined values of Γ. For example, at all
temperatures, the figure shows Γ increasing with decreasing volume be-
low V/V00 ≈ 0.6, whereas the experiments show a decrease in Γ at these
volumes.
also to contradict the calculations made by Ree [1982]. It is unclear what the
origin of this discrepancy is, and no explanation is offered in the paper. In a
subsequent paper [Nigmatulin and Bolotnova, 2011], the authors offer a form
for Γ that neglects any temperature dependence, arguing that it applies to
states where the anomalous behaviour of water is unimportant. It is not clear
what pressure and temperature range this applies to, although it is implied
this is the gaseous phase. In general, recent modeling efforts [e.g., French and
Redmer, 2009; Goldman et al., 2009a,b] have been concerned with the temper-
atures and pressures that are relevant to water in planetary interiors, which
are much higher than those dealt with in the present thesis, and the tempera-
ture dependence of Γ is generally neglected.
The heat capacity of water at constant volume, Cv , is often, for simplicity,
assumed to be constant Dolan [2003]; Gurtman et al. [1971]. This assump-
tion can be justified by arguing that effective Debye temperature of water is
173
lower than the actual temperature. Kor and Tripathi [1974] applied a quasi-
crystaline model to water, assuming that over short time-scales, the molecules
behaved in a similar way to those in solids, and calculated effective Debye
temperatures as a function of both temperature and pressure. From their cal-
culations, it was concluded that the Debye temperature remained constant, at
around 170 K, as the temperature was varied at ambient pressure. However,
it was found that the Debye temperature increases rapidly with pressure at
constant temperature, rising to over 400 K at a pressure of 0.3 GPa and tem-
perature of 30 ◦C. It seems likely, therefore, that the Debye temperature will be
close to the Hugoniot temperature and the approximation that the heat capac-
ity is constant may not be valid. Ree [1982] calculated the variation of the heat
capacity along the Hugoniot of water, noting it decreased from a maximum of
around 11.5R to a near-constant 7.6R, as is shown in Figure 7.12.
Figure 7.12: Variation of Cv/R as a function of Hugoniot temperature in water, from
Ree [1982]. The dashed portion of the curve represents the low temper-
ature region where complex interactions neglected in the calculations be-
come important. Significant variation of the heat capacity with tempera-
ture is predicted.
Ree [1982] comments that the maximum greatly exceeds the value of 6.94R
that would be expected from summing the ideal-gas translation, rotational,
and bond vibration contributions (3.94R) and the potential energy contribu-
tions from translational and librational modes for each molecule. The high
temperature value is close to the ideal gas contribution of 6R and the the po-
tential energy contribution of 1.5R. The variation is attributed to the complex
interplay of the increasing density and temperature along the Hugoniot - the
174
shock compression tends to increase Cv through enhanced librational motion,
whereas the shock temperature tends to promote bond vibration (increasing
Cv) and molecular rotation over libration (decreasing Cv). Ree [1982] suggests
in light of these calculations, the constant-Cv model of Gurtman et al. [1971]
may require modification; no explanation as to the apparent agreement be-
tween that model and experimental measurements of temperature is offered.
Nigmatulin and Bolotnova [2008] also calculated the variation in heat capac-
ity; these calculations as functions of volume at different temperatures and
temperature at different volumes are shown in Figure 7.13. As with the Γ
calculations discussed above, a direct comparison between Figure 7.13 and
Figure 7.12 is not possible; however, it should be noted that values of Cv cal-
culated by Nigmatulin and Bolotnova [2008] are wildly different from those
calculated by Ree [1982], which themselves differ markedly from the constant
values used by Gurtman et al. [1971] and Dolan [2003].
Figure 7.13: Variation of heat capacity with (a) relative volume at different temper-
atures and (b) temperature at different relative volumes, from Nigmat-
ulin and Bolotnova [2008]. Whilst not directly comparable to calculations
shown in Figure 7.12, it is reasonably clear that the calculations made by
Ree [1982] and by Nigmatulin and Bolotnova [2008] give fundamentally
different forms and values for the heat capacity as a function of both vol-
ume and temperature.
The picture, then, which emerges from these modeling results is an unclear
one. The variation, value, and even functional dependencies of Γ are still not
175
agreed upon; the same is true of the heat capacity Cv . It is puzzling that the
molecular models suggest a constant value for Cv is inappropriate and yet
empirical models, such as that proposed by Gurtman et al. [1971] are able to
accurately calculate shock temperatures in water using a constant value for
Cv determined from experiments. There are no comparable calculations in
the open literature for ammonium nitrate solutions. In considering the soft-
ening of the Hugoniot that occurs when a material is pre-heated, it has been
experimentally demonstrated that a solution of 50 % ammonium nitrate be-
haves in a similar way to water, with both undergoing significant softening
on pre-heating. This is taken as evidence that the behaviour of the solution is
dominated by the behaviour of water. In both the solution and in water, it has
been shown that pre-heating results in an increase in the value of Γ, demon-
strating that the standard assumption that Γ is a function of volume alone, is
invalid over the stress range investigated. The present results are believed to
be the first test of this assumption in shock-compressed liquids at low stresses.
It follows from this result that the Mie-Gru¨neisen equation of state is a poor
model for the thermal pressure in water and ammonium nitrate solutions at
low stresses and, therefore, further work on the theoretical description of these
materials is required.
7.4 Summary
• Measured Hugoniots in water and ammonium nitrate for different ini-
tial temperatures indicate a significant softening (reduction in impedance)
with increasing temperature.
• Calculations using the Mie-Gru¨neisen equation of state are unable to
correctly account for this softening.
• From the literature, it is believed that water (and hence the solutions)
undergo significant structural changes affecting their behaviour as tem-
perature increases. These changes would lead to a marked softening,
and it is believed this may explain the observed behaviour.
• It is arguable that upon heating, water and ammonium nitrate solutions
change sufficiently as to effectively be different materials.
• The variation of Γ with temperature, which has been observed experi-
mentally, is largely ignored in the literature.
176
• The present work suggest that the variation of Γ with temperature is
significant under shock loading.
• There is no widely accepted method for incorporating a temperature-
dependent Γ into equations of state, as it invalidates the standard form
of the Mie-Gru¨neisen equation of state.
• Water is a pathologically complex material that has not proved amenable
to modeling efforts. Published models of water show different, and con-
tradictory, variations in Γ and Cv .
• In lieu of more concrete modeling on the behaviour of Cv the simplest
approximation is to assume that it remains constant as a function of pres-
sure and temperature.
• Most modeling efforts are concerned with the pressures and tempera-
tures that may exist in planetary interiors, and thus are not relevant to
the present work.
• A surprisingly small amount is known for certain about the behaviour
of water and aqueous solutions under shock compression, and it is be-
lieved that the present work represents the first published test of many
of the commonly used assumptions.
7.5 Bibliography
O. L. Anderson. The Gru¨neisen ratio for the last 30 years. Geophysical Journal
International, 143(2):279–294, 2000.
D. J. Bradley and K. S. Pitzer. Thermodynamics of electrolytes. 12. Dielectric
properties of water and Debye-Hu¨ckel parameters to 350◦C and 1 kbar. The
Journal of Physical Chemistry, 83(12):1599–1603, 1979.
J. Chan. Personal communication, 2011. Orica Mining Services.
C. Dai, H. Tan, J. Hu, Y. Yu, and J. Zheng. Hugoniot evaluation of the pre-
heated metal from its principal hugoniot. Journal of Applied Physics, 99(5):
056102, 2006.
D. H. Dolan. Time dependent freezing of water under multiple shock wave compres-
sion. PhD thesis, Washington State University, 2003.
177
M. French and R. Redmer. Estimating the quantum effects from molecular vi-
brations of water under high pressures and temperatures. Journal of Physics:
Condensed Matter, 21(37):375101, 2009.
S. Galbraith. The responses of potassium chloride, ammonium nitrate solutions, and
emulsion explosives to plate impact loading. PhD thesis, University of Cam-
bridge, 1997.
N. Goldman, E. Reed, I.-F. W. Kuo, L. E. Fried, C. J. Mundy, and A. Curioni.
Ab initio simulation of the equation of state and kinetics of shocked water.
The Journal of Chemical Physics, 130(12):124517, 2009a.
N. Goldman, E. J. Reed, and L. E. Fried. Quantum mechanical corrections to
simulated shock hugoniot temperatures. The Journal of Chemical Physics, 131
(20):204103, 2009b.
G. A. Gurtman, J. W. Kirsch, and C. R. Hastings. Analytical equation of state
for water compressed to 300 kbar. Journal of Applied Physics, 42(2):851–857,
1971.
W. B. Holzapfel. Effect of pressure and temperature on the conductivity and
ionic dissociation of water up to 100 kbar and 1000[degree]c. The Journal of
Chemical Physics, 50(10):4424–4428, 1969.
J. Jonas, T. DeFries, and D. J. Wilbur. Molecular motions in compressed liquid
water. The Journal of Chemical Physics, 65(2):582–588, 1976.
L. Knopoff and J. N. Shapiro. Comments on the interrelationships between
Gru¨neisen’s parameter and shock and isothermal equations of state. Journal
of Geophysical Research, 74(6):1439–1450, 1969.
S. K. Kor and N. D. Tripathi. Temperature and pressure dependence of ef-
fective debye temperature in associated liquids based on quasi crystalline
model. Journal of the Physical Society of Japan, 36(2):552–554, 1974.
R. G. McQueen. Shock-waves in condensed media: Their properties and the
equation of state of materials derived from them. In Enrico Fermi summer
course CX11: nuclear collisions from the mean field into the fragmentation regime,
1989.
W. J. Nellis. Water at high dynamic pressures and temperatures. Zeitschrift fu¨r
Naturforschung, 63b(6):605–607, 2008.
178
R. Nigmatulin and R. Bolotnova. Wide-range equation of state for water and
steam: Calculation results. High Temperature, 46:325–336, 2008.
R. Nigmatulin and R. Bolotnova. Wide-range equation of state for water and
steam: simplified form. High Temperature, 49:303–306, 2011.
P. Postorino, R. H. Tromp, M.-A. Ricci, A. K. Soper, and G. W. Neilson. The
interatomic structure of water at supercritical temperatures. Nature, 366:
668–670, 1993.
F. Ree. Molecular interaction of dense water at high temperature. The Journal
of Chemical Physics, 76(12):6287–6302, 1982.
R. H. Tromp, P. Postorino, G. W. Neilson, M. A. Ricci, and A. K. Soper. Neu-
tron diffraction studies of H2O/D2O at supercritical temperatures. A direct
determination of gHH (r), gOH (r), and gOO(r). The Journal of Chemical Physics,
101(7):6210–6215, 1994.
E. Washburn, editor. International Critical Tables of Numerical Data, Physics,
Chemistry and Technology. Knovel, 1st electronic edition, 2003.
179

CHAPTER 8
CONCLUSIONS AND FUTURE WORK
The aim of this thesis has to been to further develop an understanding of
the behaviour at high dynamic pressures of materials relevant to the mining
industry. The specific materials considered have been aqueous solutions of
ammonium nitrate; such solutions are employed as the oxidising component
of the ‘emulsion’ explosives that are widely used in the industry.
The experiments reported in this thesis were performed on the Cavendish’s
Plate Impact Facility, a versatile experimental apparatus constructed sixteen
years ago [Bourne et al., 1995], which has not been significantly altered or
improved upon. Worn parts, particularly of the alignment mechanism, were
rebuilt at the author’s instigation. Minimisation of the impact tilt in such an
apparatus has been identified as a key requirement in ensuring high fidelity
data are obtained. Measurements of impact tilt on the Cavendish’s Plate Im-
pact Facility have not previously been reported. As such, a technique for char-
acterising impact tilt, using optic fibres, was developed. Using this technique,
impact tilt was measured in a number of experiments, and found to be of or-
der 10 mRad. This has subsequently been reduced to less than 2 mRad, chiefly
through the design and use of a new projectile.
A survey of the literature indicated that a number of experimental tech-
niques have been proposed to measure the temperature rise associated with
shock compression. A brief investigation of the use of thin-foil thermistors
demonstrated that their response is chiefly limited by their thickness. The
technical challenges in making ultra-thin foils prevented their use in the present
investigation. Other methods, including Raman spectroscopy, show promise
for future research, but represent a significant capital investment beyond the
scope of this investigation.
Accurate shock temperatures may be calculated from a thermodynami-
cally consistent equation of state. Historically, such equations have not been
used, with simplified analytic expressions being used instead. The simplest
such equation, which has been applied widely in the shock compression field,
is the Mie-Gru¨neisen equation of state, which relates the energy in a given
state to the energy on some reference curve (typically the Hugoniot) through
181
the Gru¨neisen parameter, Γ. A ‘Mie-Gru¨neisen’ material is one in which Γ
is assumed to be a function of volume alone. If this parameter can be deter-
mined then, under assumptions regarding the heat capacity of the material,
shock temperatures can in general be calculated to accuracies of a few per-
cent.
To this end, it was necessary to develop an experimental configuration and
technique that would enable characterisation of the principal Hugoniot and
equation of state of liquids under shock loading. The study of shocks in liq-
uids presented several experimental challenges, as generating well-defined
shocks in liquids of precise thickness is markedly more complex than do-
ing the same in a solid sample. A cell was designed that would enable the
measurement of the shock response of liquids. A number of unanticipated
problems were encountered during the testing of the cell, requiring significant
modifications to be made to the design. An iterative design process resulted
in the cell employed in the present work. Shock velocities in the liquids were
determined from measurement of the transit time through a known thickness.
Embedded manganin gauges allowed accurate measurement of the Hugoniot
and reshock stresses in shock compressed liquids. The measured Hugoniot
states were fitted by linear shock velocity–particle velocity relations and, ex-
trapolated to higher compressions. Thus, the reshock stresses enabled param-
eterisation of the Mie-Gru¨neisen equation of state. The validity and consis-
tency of the cell and analysis techniques were demonstrated by studying the
behaviour of water compressed from room temperature, which has previously
been reported in the literature [Gurtman et al., 1971].
A series of experiments were then conducted on a number of liquids: wa-
ter, at an initial temperature of 66±2 ◦C, a solution of 50 % ammonium nitrate
by weight at initial temperatures of 14 ± 1 ◦C and 65 ± 2 ◦C, and a solution
of 80 % ammonium nitrate by weight at an initial temperature of 64 ± 2 ◦C.
Shock velocities of these materials were studied to a maximum particle veloc-
ity of around 1 km s−1, enabling determination of the Hugoniot. The reported
Hugoniots show no evidence of phase changes or, in the ammonium nitrate
solutions, shock-induced reaction, and have uncertainties in the intercept and
slope of less than 5 %. The Hugoniots were shown to lie close to each other in
both shock velocity–particle velocity space and stress–particle velocity space.
As might be expected, increasing the initial density of the liquid (i.e., increas-
ing the weight-percentage of ammonium nitrate), resulted in increased shock
impedances. For this reason, the Hugoniot of the 50 % ammonium nitrate so-
182
lution at an initial temperature of 65 ± 2 ◦C lies below the Hugoniot for the
same solution at an initial temperature of 14 ± 1 ◦C. In water, the softening
that occurs on heating exceeds greatly that which is predicted from the equa-
tion of state; from a review of the literature, it is believed that this softening is
the result of structural changes in the liquid as it is heated.
Shock/reload experiments on water at an initial temperature of 66 ± 2 ◦C
enabled the determination of the Gru¨neisen parameter, Γ, at specific volumes
between 0.64 and 0.54 cm3 g−1. Combining these data with reported compres-
sional isotherms [Grindley and Lind, 1971], a fifth-order polynomial was used
to describe the volume dependence of Γ, as this is the lowest order polynomial
which can correctly describe the isothermal compression data and dynami-
cally measured values. Shock/reload experiments also enabled determina-
tion of Γ in the ammonium nitrate solutions. In water and the solution of 50 %
ammonium nitrate by weight, a temperature dependence of Γ at constant vol-
ume was found. There have been some reports in the literature concering such
a temperature varation [e.g. Knopoff and Shapiro, 1969], but such investiga-
tions have been generally neglected [e.g. McQueen, 1989]. There is no widely
accepted method for incorporating a temperature-dependent Γ into equations
of state, as it invalidates the standard form of the Mie-Gru¨neisen equation of
state. There is no consensus in published models of the behaviour of wa-
ter [e.g. Nigmatulin and Bolotnova, 2008, 2011; Ree, 1982] on the functional
form or values of Γ that might expected, and most recent modeling efforts
[e.g., French and Redmer, 2009; Goldman et al., 2009a,b] have been devoted to
pressures in planetary interiors, which far exceed those relevant to this study.
Neglecting these concerns, and under the assumption that the heat ca-
pacity cv is constant, and, for water, within 10 % of 3.2 J g−1 K−1 [Gurtman
et al., 1971], temperatures on the Hugoniot, which exceeded 1000 K at 8 GPa,
were calculated with an uncertainity of 10 %. Temperatures were determined
in the ammonium nitrate solutions, by assuming that cv was constant and
equal to the zero pressure value, and combining the dynamic data with the
calculated zero pressure value of Γ [Chan, 2011]. The calculated tempera-
tures, which have uncertainties of 10 % are valid to Lagrangian compressions
of 0.25 (equivalent to stresses of approximately 4.5 GPa). It was found that
increasing the initial temperature for a given concentration resulted in higher
temperatures on shock loading, the difference in shock temperature being ap-
proximately equal to difference in initial temperatures. At the same initial
temperature, higher concentrations yielded higher shock temperatures. The
183
lowest calculated temperature rise, of around 100 K at a compression of 0.25,
was found for the 50 % ammonium nitrate by weight at initial temperatures
of 14±1 ◦C. A temperature rise of 250 K at the same level of compression was
calculated for the 80 % ammonium nitrate by weight at an initial temperature
of 64 ± 2 ◦C. Even though these calculations must be considered only as ap-
proximations, it is believed that these results represent the first calculations
of shock temperatures in ammonium nitrate solutions in the open literature.
The assumption of constant cv is justified if the Debye temperature is signifi-
cantly lower than the temperature of the material. There is some suggestion
[Kor and Tripathi, 1974] that this may not be the case as water is compressed.
Again, however, literature models [e.g. Nigmatulin and Bolotnova, 2008; Ree,
1982] present a confused and contradictory picture.
8.1 Future work
In order to reduce uncertainties in the calculated temperatures, four areas
must be addressed in future work. Firstly, the measurement of the principal
Hugoniots should be extended to higher particle velocities. This would re-
duce the need to extrapolate the Hugoniot beyond its measured region when
analysing shock/reload experiments, and would reduce the uncertainties in
the intercept and slope of the fitted linear shock velocity–particle velocity rela-
tionship. Such measurements are not possible with the present experimental
facilities. The highest particle velocities in the liquid were obtained by us-
ing copper flyer plates at near the maximum velocity of the Cavendish’s Plate
Impact Facility. No significant increase in velocity is possible, as the wrap-
around breech design imposes certain constraints on the wall thickness, and
hence mass, of the sabot. Some increase in particle velocity could be achieved
through the use of higher-impedance flyer materials, such as tungsten or tan-
talum. A well-characterised stock of these materials is not available at the
present time, and characterising the shock response of either of these mate-
rials to a suitable degree of accuracy is itself a significant experimental pro-
gramme. To achieve particle velocities that would cover the full compression
range of the reshock measurements, experiments would need to be conducted
at around twice the maximum particle velocities reported in the present work.
Such particle velocities would require projectile velocities that are not physi-
cally possible with a single-stage gas gun, requiring instead a two-stage gas
gun, powder gun, or explosive plane-wave generator [Asay and Shahinpoor,
184
1993, §3.2 – §3.4].
Accurate measurement of the compressional isotherms of the ammonium
nitrate solutions would provide a stronger constraint on the behavior of Γ at
volumes close to the initial specific volume. To be of use, measurements of the
specific volume to pressures below 1 GPa, with an accuracy of 0.01 %, compa-
rable to the isotherms reported by Grindley and Lind [1971] for water, would
be required. This would allow for an improved fit describing the volume-
dependence of Γ to be determined by constraining the behaviour at low com-
pressions, thus increasing the accuracy of the calculated temperatures.
The calculations of shock temperature have assumed that cv is constant
and, in the ammonium nitrate solutions, equal to the zero-pressure value. An
accurate experimental technique for measuring cv in shock experiments has
not been reported in the open literature. Some success has been had in apply-
ing density functional theory to the calculation of heat capacities during com-
pression. Ree [1982] calculated cv along the Hugoniot of water, but report that
the calculations are only approximate for temperatures near 1000 K, yielding
uncertainties larger than those reported by Gurtman et al. [1971]. More recent
calculations for water [French and Redmer, 2009; Goldman et al., 2009b] do
not significantly improve on these uncertainties in the pressure/temperature
region relevant to the present work. More accurate calculations of the heat
capacity of shock compressed minerals have been reported [de Koker, 2010;
de Koker et al., 2008], although the applicability of the calculations to the
temperatures and pressures in the liquids studied in the present work is not
clear. Future experimental work should also attempt the direct measurement
of shock temperature in these liquids, which could then be used to improve
the calculations by constraining the value of cv .
Finally, it is evident from the present work that the behaviour of water at
low stresses is poorly understood from a theoretical viewpoint, as published
models give contradictory results. The application of the Mie-Gru¨neisen equa-
tion to describe the thermal pressure appears to be invalid, as there is a strong
variation of Γ with temperature at constant volume. Therefore, it must be con-
cluded that a better model of thermal pressure is needed; the results presented
in this thesis provide a constraint on the behaviour of such a model and thus
act as a useful constraint on future developments. The mining industry must,
therefore, in developing the desired holistic models of the blasting process,
attempt to capture the underlying physical processes that occur, at a level that
is not currently considered.
185
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187

APPENDIX A
SELECTED GAUGE TRACES
In Chapter 6 it was noted that in some experiments, the stress in the reshock
state was not measured, due to failure of the rear gauge. These rear gauge
traces are presented in this appendix. Figure A.1 shows the front and rear
gauge traces from experiment E100927A, where the plateau associated with
the stress in the reshock state is clearly visible. The trace is typical of those
used in the analyses in Chapters 5, 6, and 7. The front gauge trace shows
two plateaus, corresponding to first the Hugoniot stress in PMMA and then
to the Hugoniot stress in the test liquid. The rear trace rises to the reload
stress of the test liquid. The reverberation evident on the rear gauge rise is
due to the impedance mismatch presented by the Teflon used to insulate the
gauge. As noted in Chapter 4, this reverberation has no significant effect on
the measured reload stress and can be neglected. Comparing the following
traces to Figure A.1, the failure of the rear gauge is evident.
189
Figure A.1: Front (red) and rear (black) stress gauge traces from experiment E100927A.
A liquid cell, containing water at 18±2 ◦C, was impacted by a PMMA flyer
at 0.787 ± 0.008 km s−1. The front gauge is first shocked to the Hugoniot
state of PMMA, before being released to the Hugoniot state of water at
a stress of 1.179 ± 0.013 GPa; the shock reaches the rear gauge 2.250 ±
0.014µs after first arriving at the front gauge. The stress in the rear gauge,
corresponding to the reload state of water, is 2.987± 0.022 GPa.
190
A.1 50% ammonium nitrate, 14± 1 ◦C
Figure A.2: Front (red) and rear (black) stress gauge traces from experiment E101118A.
A liquid cell, containing 50 % ammonium nitrate at 14±1 ◦C, was impacted
by a PMMA flyer at 0.591 ± 0.006 km s−1. The failure of the rear gauge at
around 2µs is evident. Electrical noise associated with this failure appears
on the front gauge trace at around 2.5µs.
191
Figure A.3: Front (red) and rear (black) stress gauge traces from experiment E101130A.
A liquid cell, containing 50 % ammonium nitrate at 14±1 ◦C, was impacted
by a copper flyer at 0.799 ± 0.008 km s−1. The failure of the rear gauge at
around 1.5µs is evident. Electrical noise associated with this failure ap-
pears on the front gauge trace at around 1.7µs.
192
A.2 50% ammonium nitrate, 65± 2 ◦C
Figure A.4: Front (red) and rear (black) stress gauge traces from experiment E110305A.
A liquid cell, containing 50 % ammonium nitrate at 65±2 ◦C, was impacted
by a PMMA flyer at 0.592 ± 0.006 km s−1. The failure of the rear gauge at
around 2.1µs is evident.
193
Figure A.5: Front (red) and rear (black) stress gauge traces from experiment E110308B.
A liquid cell, containing 50 % ammonium nitrate at 65 ± 2 ◦C, was im-
pacted by a PMMA flyer at 1.014 ± 0.010 km s−1. The failure of the rear
gauge at around 1.7µs is evident. Electrical noise associated with this fail-
ure appears on the front gauge trace at around 1.8µs
194
A.3 80% ammonium nitrate, 64± 2 ◦C
Figure A.6: Front (red) and rear (black) stress gauge traces from experiment E110228B.
A liquid cell, containing 80 % ammonium nitrate at 65±2 ◦C, was impacted
by a PMMA flyer at 0.598 ± 0.006 km s−1. The failure of the rear gauge
at around 1.8µs is evident. It is not clear why the rear gauge seems to
then equilibrate at around 1.6 GPa, although this may indicate mechanical
failure (e.g., the gauge de-bonding) rather than the electrical failure seen in
the other figures.
195