Shock Waves (1999) 9: 193 200
Shock waves in solids: an evolutionary perspective
J.N. Johnson1, R. Ch�ret2
1
Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM 87545, USA (e-mail: [e-mail])
2
Commissariat ą l Energie Atomique, Paris, France
Received 28 December 1998 / Accepted 31 December 1998
Abstract. A historical summary of the early classic papers in shock compression science is presented with
the purpose of establishing the foundations for more recent studies of shock waves in solids. The major
(largely theoretical) works of the period from 1808 to 1949 have been previously brought together into a
single collection [Johnson and Ch�ret, Springer-Verlag, New York, 1998]. Important papers on shock waves
in condensed matter from 1948 to the present were not included in the collection, largely because of their
relatively easy accessibility. This paper provides a supplement to the original collection allowing research
workers and historians to trace the major developments leading to the establishment of the unique field of
shock compression in solids.
Key words: History, Solids, Polymorphic phase transition, Elastic-plastic transition, Equation of state
1 Introduction Also, with the exception of the aforementioned report
by Bethe, the classic papers of this collection refer mainly
The study of nonlinear acoustics and shock-wave compres- to nonlinear and shock-wave behavior of gases, and ideal
sion in gases, liquids and solids has had a long and var- gases at that. Immediately following World War II there
ied evolution. This includes connections with the related
was an acceleration of interest in the high-pressure prop-
fields of high-speed gas dynamics, detonation science and
erties of solids and liquids, particularly dynamic impact
static high-pressure research. The genesis of this field was
behavior related to Hugoniot states and, in the case of
in studies of infinitesimal acoustic waves in air, which took
solids, polymorphic phase transformations and material
place in the 18th Century and early 19th Century. It then
strength.
evolved to include effects of nonlinear convective terms in
It is the purpose of the present article to outline the
the equations of motion, and on to fully nonlinear behavior
main results of early classic papers, the transition pe-
arising from realistic equations of state, dissipation, and,
riod (1910 1949) and to trace the further evolution of the
in the case of solids, material constitutive properties re- field of shock waves in solids from 1948 to the present
lated to strength and polymorphic phase transformations.
by means of the most significant technical literature. It
A collection (Johnson and Ch�ret, 1998) of the early
is hoped from this outline that the evolution of the field
papers (up to 1910) on the subject of shock compression
of shock compression of solids can be viewed with a more
and two significant theoretical works from the 1940s has
objective perspective, particularly in comparison to static
been published by the authors of the present article. Spe- high-pressure (Bridgman, 1958), gas dynamics (Glass and
cific early papers in the published collection were chosen
Sislian, 1994) and detonation (Ch�ret, 1993). The stan-
with several criteria in mind. Some of these papers were
dard reference work for shock waves (with particular ap-
difficult to locate or were published in French or German:
plication to a polytropic gas) is Courant and Friedrichs
translation of the latter papers into English was deemed
(1948).
a positive contribution to the historical and technical lit-
erature. Certain other papers, although readily accessible,
in English, were of such importance they needed to be in-
2 Early classic papers
cluded in a comprehensive collection of this kind. Once we
reach the post-World-War II period, almost all the signif-
2.1 Poisson, Stokes and Earnshaw
icant papers can be found in English and are available to
anyone with access to a good university library. Except
We begin with the work of Poisson (1808) in which he
for the work of Weyl (1949), added to provide comparison
wrote the equations for mass and momentum conservation
with the government report by Bethe (1942), these papers
as
were not included.
"�
+ "� (�"�) =0 (1)
Correspondence to: J.N. Johnson
"t
194 J.N. Johnson, R. Ch�ret: Shock waves in solids: an evolutionary perspective
dp "� 1
proportional to density. Under the condition that p " �,
q - = + "2� (2)
� "t 2
Eq. (4) can be obtained by equating the discontinuous
jump in internal energy (what later became known as the
shown here in modern notation, where p is the pressure, �
Rankine-Hugoniot relation) to the negative integral of pdv
is the material density, q is the potential function for body
across the discontinuity:
forces, and � is the velocity potential. The early papers on
sound propagation made no distinction between a partial
1/�
derivative and a total derivative; the reader must judge
1 1 1
(p + p ) - = - pdv (5)
the meaning from the context in which the equations are
2 � �
initially written. Equation (2) is a rather unusual form of
1/�
Newton s second law, and is formally correct only for the
case in which pressure p is a function of density � alone. By equating the discontinuous jump in internal energy to
Poisson then went on to obtain a solution to Eqs. (1) the value appropriate for continuous motion, Stokes failed
and (2) in various simple geometrical configurations, as- to allow for the irreversible nature of the shock process.
suming first homogeneity and then heterogeneity of air This may not be surprising since the basic concepts of
as to density and temperature. In all these situations he irreversibility were being developed and debated at this
extensively followed a fruitful procedure, searching for so- time. But it is interesting historically that the  Rankine-
lutions in the form Hugoniot jump condition for energy might have carried
the name of  Stokes had he been aware of the discon-
"� "�
tinuous change in a second (independent) state variable
= f x " at - t (3)
"x "x
together with density. This is no criticism of Stokes, since
the common assumption of 1848 until Hugoniot (1887,
with appropriate choice of x, and appropriate approxima-
1889) was that the pressure could be considered a func-
tions to (1/a)"�/"x. Equation (3) represents planar waves
tion of density alone.
propagating in the positive and negative x directions with
Earnshaw (1860), a Sheffield clergyman educated at
speeds ąa + u, respectively, where u is the local parti-
Cambridge, began with the observation that if the parti-
cle velocity. In his derivations, Poisson generally escaped
cle velocity were an arbitrary function of strain, this pro-
the difficulty of incorrect sound velocity (at that time the
vided a first integral of a rather general one-dimensional
only calculated sound velocity was for isothermal condi-
wave equation. Thus, if y is the displacement and X is the
tions), by taking into account, as suggested by Laplace,
Largangian position variable,
 the production of heat associated with the compression
of air. He was thus able to evaluate the additional pres- "y "y
= F (6)
sure due to adiabatic compression in comparison to the
"t "X
isothermal constraint. However, unaware as he and his
contemporaries were of the concept of entropy, his calcu- can be differentiated with respect to time to give
lations remain without further physical meaning or inter-
2
pretation.
"2y "y "2y
= F (7)
Approximately forty years later Stokes (1848) exam-
"t2 "X "X2
ined the results of Poisson as prompted by discussions
between Professor Challis and the Astronomer Royal. For
Equation (7) is the equation for wave motion along the
finite-amplitude waves of the type considered by Poisson,
x axis with a rather general wave speed as a function
one would expect a sinusoidal disturbance to steepen in
of the strain. Earnshaw then went on to examine wave
the direction of propagation and, at some point down-
propagation under isothermal and adiabatic conditions for
stream, achieve an infinite negative gradient. Stokes then
condensation and rarefaction. Of interest to the theory of
considered the possibility of the formation of a propagat-
shock propagation is his investigation regarding the prop-
ing discontinuity in fluid motion. Following publication of
agation of steady waves (i.e., finite-amplitude waves that
this speculation, discussions with Sir William Thomson
do not change shape as they propagate) under the con-
and Lord Rayleigh prompted Stokes to withdraw his as-
dition that the pressure is a function of density alone.
sumption (Stokes, 1883).
He found the necessary relationship between pressure and
The jump conditions for mass and momentum conser-
density in this case to be
vation across a propagating discontinuity were well estab-
B
lished and not in question at this time. Stokes did not
p = A - (8)
set forth the energy jump relationship in the initial paper
�
(1848), but did give the following expression for conser-
and concluded such a condition could not be realized for
vation of energy across a propagating discontinuity in his
any real material. Thus Eq. (8) was considered inappli-
collected papers (1883):
cable to the problem of finite-amplitude wave propaga-
tion. Today we recognize Eq. (8) as an expression for the
2�� log (�/� ) =�2 - � 2 (4)
Rayleigh line. Rayleigh (1910) and Taylor (1910) later ex-
where � is the initial density (ahead of the discontinuity), amined conditions under which steady waves, or waves of
� is the final density, and the pressure is assumed to be permanent regime, could be realized.
J.N. Johnson, R. Ch�ret: Shock waves in solids: an evolutionary perspective 195
2.2 Riemann wave and lower case symbols indicate conditions far be-
hind a steady propagating disturbance. Reciprocal density
Riemann, the German mathematician, occupies a unique
is termed the bulkiness, m is the mass-velocity or somatic
place in the development of shock-wave concepts because
velocity, and U is the shock velocity in modern notation
of his high level of mathematical rigor and sophistication.
(not used by Rankine).
Riemann (1860) began, as his contemporaries, with the
The additional thermodynamic constraint imposed by
assumption that the pressure is a unique function of den- Rankine is that  from the front to the back of the wave,
sity, p = �(�), and sought solutions to the equations for
the integral transfer of heat received must be nothing; . . . .
mass and momentum conservation:
Rankine expressed this condition as
"� "(�u) �2
+ = 0 (9)
� d� = 0 (16)
"t "x
�1
"u "u " log �
+ u = -� (�) (10)
where � is the absolute temperature and � is the  ther-
"t "x "x
modynamic function, i.e., what we would now call the
These expressions can be combined to give the familiar
entropy. Rankine s solution to the problem of finite-ampli-
characteristic form, with
tude wave propagation consisted of using Earnshaw s rela-
tion, Eq. (8), in conjunction with the condition of global
c(�) = � (�) (11)
null heat transfer, Eq. (16), and knowledge of the ther-
mal conductivity, to obtain an expression for pressure as
as
a function of position through the continuous transition.
"u "u
In Rankine s approach the wave is steady and has finite
+(u ą c(�))
"t "x
rise time.
" log � " log � It is Hugoniot (1887, 1889) who gave the correct anal-
ą c(�) +(u ą c(�)) = 0 (12)
ysis of a propagating discontinuity for the specific case of
"t "x
an ideal gas, whose equation of state is given by
or
pv
E = (17)
dr =0 on dx =(u + c(�)) dt (13a)
ł - 1
ds =0 on dx =(u - c(�)) dt (13b)
where p is the pressure, � is the specific volume and ł is the
ratio of the heat capacities. In ż 154 Hugoniot presented
where (here in modern notation)
1
dx
r a" c(�)d ln � + u (14a)
�0(u1 - u) = p1 - p (18)
2
dt
1
s a" c(�)d ln � - u (14b)
dx
2
�0(v1 - v) =(u1 - u) (19)
dt

1 p1v1 - pv dx
These expressions provided a very powerful mathematical
�0 u2 - u2 + = p1u1 - pu (20)
1
technique for the analysis of finite-amplitude waves. How-
2 ł - 1 dt
ever, the formation of discontinuities were still not permit-
where dx/dt is the relative upstream velocity and sub-
ted; all that could be done with Riemann s analysis was
script 1 (in p1, u1, v1) refers to the downstream state of the
to estimate when and where they might form.
discontinuous transition, while no subscript (in p, u, v =
1/�0) refers to the upstream state.
2.3 Rankine and Hugoniot
Rankine (1870) wrote down the correct expression for con- 3 Transition period (1910 1949)
servation of energy in a plane steady propagating wave of
either discontinuous or finite extent. He called the kinetic
3.1 Rayleigh and Taylor
energy the  actual energy of the disturbance, and the
internal energy the  work done in altering bulkiness. He
Rayleigh (J. W. Strutt, 1910) reviewed the literature pre-
thus correctly obtained the equation for conservation of
ceding his study and then examined the joint effects of
energy as
viscosity and thermal conduction on waves of permanent
regime; that is, waves which propagate without change of
m
pu = u2 +(p + P )(S - s) (15)
shape in the sense of Rankine.
2
Taylor s (1910) work paralleled that of Rayleigh, but
where m = �0U, S = 1/�0, and s = 1/� in modern no- without the careful review of the previous papers of Stokes,
tation. Capital symbols indicate conditions ahead of the Earnshaw, Riemann, Rankine and Hugoniot. Rayleigh and
196 J.N. Johnson, R. Ch�ret: Shock waves in solids: an evolutionary perspective
Taylor, in effect, resolved the issues raised by Stokes in his 4 Shock waves in solids (1948 1958)
discussions with Sir William Thomson and Lord Rayleigh.
One could say that with Rayleigh and Taylor analy- We mark the beginning of research in shock wave com-
pression of solids with the work of Pack, Evans and James
sis regarding the correct interpretation of finite-amplitude
(1948). This overlaps slightly with the transition period
wave propagation in ideal gases had reached maturity.
(1910 1949) discussed in the previous section, but the last
of those works (Weyl, 1949) was exclusively theoretical
and applied to fluids. It might also be argued that the
3.2 Bethe and Weyl
study of solids began with the work of Bethe (1942) be-
cause of his mention of phase transformations and consid-
The report of Bethe (1942) for the National Defense Re- erations of the thermodynamic properties of solids. The
following discussion presents the major works in experi-
search Council is the first important work to examine the
mental and theoretical research from 1948 to 1958 related
theory of shock waves for fluids with arbitrary equations of
state and to discuss possible effects related to shock com- to solids undergoing impact and explosive loading.
pression of solids. The main results of this report are the
three conditions which guarantee the stable propagation
4.1 Pack, Evans and James
of shock waves:
(elastic-plastic transition)
"2p
> 0 (21)
2
"V
S
Pack, Evans and James (1948) measured the shock wave
velocity of steel and lead in contact with a high explosive
"p
(unnamed) which produced peak target pressures of 28.3
V > -2 (22)
and 27.1 GPa, respectively. Shock speeds were measured
"E
V
by means of an ionization detector at the front surface of
the sample, and an electrical ball contact on the back sur-
"p
face. The time difference between the activation of these
< 0 (23)
"V
E two detectors then gave the transit time through the sam-
ple. Theoretical shock velocities are obtained from the ex-
Materials considered with regard to these conditions were
pression for what appears to be isothermal compression:
solids, liquids and gases at various extreme and unusual
states. 1/3 2/3
1/3
V0 - V V
A very useful result presented by Bethe is the deriva- p = ą exp � - 1 (25)
1/3
V0
V0
tion of the entropy increase for low-amplitude shocks in
fluids. The result is given by
where ą and � are two parameters related to the two-body
interatomic potential; they are determined from empirical
"2p ("V )3
information on compressibilities (Bridgman, 1931). The
"S = - + � � � (24)
2
"V 12T
dominance of specific volume on the solid equation of state
S
at modest temperatures is discussed in Sect. 4.6, below. In
and thus we see from Eqs. (21) and (24) that the entropy spite of the neglect of thermal contributions to the equa-
change in normal fluids is third order in the volume change tion of state, Pack et al. were able to deduce the shock
and positive for compression ("V < 0, and T is the abso- speed in steel (5240 m/s) to be less than its elastic planar
lute temperature). wave speed, and for lead (3020 m/s) it is greater.
The particular interest in shock stability apparent in This is reflected in the experimental data, where the
this report comes from discussions Bethe had with Von velocity of the leading disturbance reaching the back sur-
Neumann regarding whether it is possible for materials face of the target sample is unchanged in the case of steel,
to form shocks in rarefaction. This is indeed the case for and is decreasing in the case of lead. In the former case the
solids which undergo reverse phase transformation upon leading disturbance is comprised of an elastic wave at the
release from the shocked state, and has been observed ex- plastic yield point of steel, and the latter case the leading
perimentally, as we shall discuss in connection with the disturbance is the decaying shock wave.
work of Barker and Hollenbach (1974) on shock waves in While these measurements do not contain a great deal
iron at pressures above 13 GPa. of information on the details of the dynamic plastic flow
Weyl (1949) also presented analysis of the shock wave process, and the equation of state lacks functional de-
in arbitrary fluids and conditions for the existence of pendence on internal energy or temperature, this is the
shocks and the variation of entropy along the Hugoniot first real experimental observation and interpretation of
curve. In addition, he considered the problem of calculat- the unique elastic-plastic nature of solids subject to shock
ing the structure of the shock layer. Weyl s work is con- wave compression.
siderably more mathematical than Bethe s, in the same An important related study is that of Taylor (1948) in
way Riemann s paper was more mathematical than those which right-circular cylinders are impacted end-on with
of his contemporaries in the field of finite-amplitude wave a nearly rigid wall. Recovered samples exhibit the effects
propagation. of plastic wave propagation and permanent deformation
J.N. Johnson, R. Ch�ret: Shock waves in solids: an evolutionary perspective 197
in measurement of sample diameter as a function of dis- the computational zoning and choice of the constant had
tance from the impacted end. Conditions in the Taylor negligible influence on the flow outside of the shock layers
impact test are considerably different from those in the and other important physical processes, was an important
Pack, Evans and James experiment, particularly the lack positive step in numerical computation.
of a large isotropic mean stress component in the former A possible negative aspect of this numerical technique
case. Hence, we tend to place the Taylor impact test in the is that, as time passes and large-scale computation be-
category of high rate, continuous deformation rather than comes such a dominant part of the research into shocks in
in the field of shock compression. Nevertheless, the work solids, the assumptions under which this method provides
of Taylor has had an important influence on the study of good physical solutions are forgotten and the method may
dynamic plasticity from 1948 to the present and deserves be applied inappropriately. It is worthwhile to read, or re-
mention in the context of the research work described here. read, this important paper to see its proper applicability.
4.2 Von Neumann and Richtmyer
4.3 Walsh and Christian
(numerical solution)
(thermodynamic properties)
Another important contribution to the field of shock wave
The work of Walsh and Christian (1955) consists of the
compression is the theoretical paper by Von Neumann and
determination of Hugoniot curves for aluminum, copper
Richtmyer (1950) on a computational method for the so-
and zinc at peak shock pressures of approximately 15 to
lution of hydrodynamical problems involving shocks. The
50 GPa. Shock and free-surface velocities were measured
initial application considered by the authors was to ideal
using a high-speed sweep camera. Mass, momentum and
gases, but it is this development which allowed the com-
energy conservation jump conditions provided necessary
putation of complex impact phenomena in gases, liquids
constraints for the determination of the thermodynamic
and solids of almost any variety. Its essential feature is
state behind the shock, along with the relationship
that it relaxed the requirement of actually following in-
1/2
p1
dividual shocks in numerical computation. In the case of
dV
"ur = - dP (30)
homogeneous gases and fluids with numerous wave reflec-
dP
0
tions and interactions, this could be complicated enough,
adi
but in the case of solids undergoing elastic-plastic defor-
for the particle velocity change due to the centered simple
mation and phase transformations in addition to the usual
wave produced when the shock reaches the stress-free rear
pressure-volume nonlinearity this would be prohibitive.
surface. In Eq. (30) p1 refers to the pressure behind the
The well-known method of Von Neumann and Richt-
shock (before reflection) and the subscript adi refers to
myer consists of the addition of an artificial dissipative
integration along a path of constant entropy (reversible
term q to the conservation laws, here written in Lagran-
process in continuous flow regime).
gian coordinates:
In this way Walsh and Christian were able to estab-
"V "u lish the Hugoniot states as well as determine the errors
�0 - = 0 (26)
made in using the common assumption that particle ve-
"t "X
locity doubles upon reflection from a free surface. The full
"u "(p + q)
expression for the free-surface velocity is
�0 + = 0 (27)
"t "X
ufs = up + "ur (31)
"E "V
+(p + q) = 0 (28)
"t "t
maximum values of the ratio "ur/up are determined for
The authors demonstrate that, for an ideal gas, a form
aluminum, copper and zinc and applied to the experimen-
for the artificial dissipative stress q giving a numerically
tal data, with the result that overall uncertainties in com-
computed shock thickness of the same order of the spatial
pression for a given temperature are approximately 1 per-
increment "X in finite-difference solutions is
cent for aluminum and approximately 2 percent copper
and zinc.
(�0c"X)2 "V "V
Thermodynamic considerations were employed to es-
q = - � (29)
V "t "t
timate temperatures along Hugoniot curves and adiabats.
Specific assumptions were that Cv and ("p/"T )V remained
where c is a dimensionless constant near unity.
constant, as suggested by the insensitivity of both param-
Equation (29) is used currently in numerical calcula-
eters to pressure and temperature in static compressibility
tions not only for gases and fluids, but also for solids under
work. The solution for the temperature along the Hugo-
almost universal conditions. Other forms have been pro-
niot was given by
posed and used in conjunction with the quadratic form,
Eq. (29), particularly a linear form that damps numerical
T1(V1) =T0 exp [b(V0 - V1)]
oscillations behind steep shocks (Wilkins, 1964).
V1
f(V )ebV
The recognition that an artificial viscosity could elim-
+ exp(-bV1) dV (32)
inate the need for detailed shock tracking, provided that Cv Hug
V0
198 J.N. Johnson, R. Ch�ret: Shock waves in solids: an evolutionary perspective
where Graham (1993) has discussed the importance of these
("p/"T )V
measurements in connection with those of the static high-
b = (33)
pressure community, and the usefulness of shock data as a
Cv
reliable source of calibration data for static high-pressure
1 dp p
experiments.
f(V ) = (V0 - V ) + (34)
2 dV 2
Calculated isotherms were compared with low-pressure
isothermal compression data. 4.6 Walsh, Rice, McQueen and Yarger
This work represents the first serious consideration of (twenty-seven metals)
thermodynamic properties of solids and the comparison
between isotherms, adiabats and Hugoniot curves.
Based on the rapidly developing experimental techniques
unique to studies of planar shock wave compression in
solids, Walsh et al. (1957), systematically measured the
4.4 Minshall
Hugoniot curves for twenty-seven metals to several tens
(elastic-plastic transition)
of GPa. The experimental technique employed in these
studies consisted of argon flash gaps to optically record
shock arrival times. The measured Hugoniot curves were
The second major paper on the effects elastic and plastic
waves in solids is that of Minshall (1955). Steel and tung- then used in conjunction with the Mie-Gr�neisen equa-
tion of state and the Dugdale-McDonald formula (for the
sten samples were loaded explosively with composition B
(40 percent TNT and 60 percent RDX). The experimen- Gr�neisen coefficient) to calculate complete thermody-
namic information for states neighboring the experimen-
tal procedure used here was very direct. Electrical contact
tal curves. Comparison to static measurements were also
pins were placed at various depths into the back surfaces of
performed, with the results showing the strong comple-
the samples. Those pins at the greatest depth recorded the
mentary nature of these two fields. This, and other work
arrival of the shock and those nearer the surface recorded
was summarized in the very important review article by
the arrival of an elastic wave, once the shock had decayed
Rice, McQueen and Walsh (1958).
to an amplitude below which its speed was less than the
The Mie-Gr�neisen equation of state is specific to sol-
elastic wave speed.
The experimental records showed the elastic-wave ve- ids. It can be derived from the assumption that the ther-
locities in both materials to be equal to their correspond- mal energy E of a metallic crystal is adequately described
by a set of simple harmonic oscillators whose frequencies,
ing sound-wave speeds. The  pressure in the elastic waves
of the steel samples was nominally 1.0 to 1.2 GPa, as deter- �i, i = 1, 2, � � � 3N, are functions of volume alone. If the
zero-temperature potential energy of the system is Ś , the
mined by measured closing times for the elastic-wave pins.
internal energy is then given by
This quantity would later be termed the Hugoniot Elastic
Limit, or HEL, and is the longitudinal stress component
1 h�i
at the elastic-plastic yield point. The HEL for tungsten
E = Ś + h�i + (35)
was determined to be 1.3 to 1.6 GPa.
2 exp(h�i/kT ) - 1
i
Also observed in these experiments was the time evolu-
tion of the elastic wave amplitude. One of the experiments
and the Helmholtz free energy becomes
on SAE 1040 steel exhibited a distinct change in slope of
arrival time vs. distance for the elastic wave. This would
1
A = Ś + h�i + kT ln (1 - exp(-h�i/kT )) (36)
correspond to a qualitative observation of elastic precursor
2
i i
decay.
The pressure is thus given by
4.5 Bancroft, Peterson and Minshall
"A dŚ
(polymorphic phase transition)
P = - = -
"V dV
T
1 1 h�i
Perhaps the single most important experimental paper on
+ łi h�i + (37)
shock waves in solids is that of Bancroft, Peterson and
V 2 exp(h�i/kT ) - 1
i
Minshall (1956) on polymorphism of iron under shock-
wave compression. As in the previous paper, electrical con-
where łi is a dimensionless variable defined by
tact pin techniques were used to determine arrival times
at differing depths in the target sample.
d ln �i
łi a"- (38)
These data provided evidence for a three-wave struc-
d ln V
ture consisting of an elastic disturbance followed by two
shocks. The first shock had an amplitude of 13 GPa and
When all of the łi are equal, Eq. (37) can be simplified to
was interpreted as the onset of a polymorphic phase trans-
formation from the initial solid phase to a second phase
dŚ ł
P = - + Evib (39)
with a volume collapse of approximately 9.4 percent.
dV V
J.N. Johnson, R. Ch�ret: Shock waves in solids: an evolutionary perspective 199
where Evib is the vibrational contribution to the internal guns for flat-plate-impact work, electromagnetic particle
energy and ł is the logarithmic derivative of any eigenfre- velocity gauges, optical methods, piezo-resistive gauges,
quency. Equation (39) can be written in the form and piezo-electric gauges. The optical methods include
standard displacement interferometry, velocity interferom-
ł
etry, and the widely used VISAR (Velocity Interferometer
P = PH(V ) + [E - EH(V )] (40)
V
for Surfaces of Any Reflectivity) technique (Barker and
Hollenbach, 1972).
where PH and EH are the pressure and internal energy
An important application of the VISAR technique was
on the Hugoniot (or any other reference curve if it is more
the time- resolved study of the ą � phase transition in
convenient).
iron (Barker and Hollenbach, 1974). These measurements
Of particular interest with regard to the difference be-
provided new information concerning rate effects associ-
tween shock waves in gases and solids is a comparison of
ated with the transformation at 13 GPa (Bancroft, Peter-
Eq. (40) with that for the ideal gas, Eq. (17). For gases
son and Minshall, 1956) and the stress level at which iron
the dependence of the pressure on specific volume is equiv-
reverts to the ą phase upon unloading. This particular
alent to its dependence on internal energy. For solids at
work demonstrates the strength of modern experimental
normal pressures and modest departures from isentropic
methods in resolving physical phenomena in shock-loaded
(or Hugoniot) conditions, the dependence of pressure on
solids on a time scale approaching nanoseconds.
specific volume is much stronger than on the internal en-
The collection by Asay and Shahinpoor (1993) presents
ergy. It is for this reason that purely mechanical theories
review articles concerning various aspects of high-pressure
can be used in the case of solids at moderate compressions
shock compression of solids. This includes the introduction
with reasonable quantitative success (see also Sect. 4.1,
to experimental measurement mentioned above (Barker,
this paper).
Shahinpoor and Chhabildas, 1993) as well discussions of
Another area of investigation with regard to solid equa-
equation of state, plastic deformation, fracture and shock
tions of state is the dependence of ł (the Gr�neisen co-
recovery. A more advanced discussion of experimental
efficient) on specific volume. Rice, McQueen and Walsh
measurement, particularly piezo-electric gauges and their
(1958) investigate this in some detail. For example, the
historical development (along with development of other
Dugdale-McDonald relationship was employed in early
major measuring techniques) is given by Graham (1993).
data analysis:
This work outlines the important developments of the me-
chanical, physical and chemical studies of shock compres-
2/3 2
V "2(PV )/"V 1
ł = - - (41) sion in solids. Additional information on the details of
2/3
2 "(PV )/"V 3
shock compression chemistry of solids can be found in the
work of Horie and Sawaoka (1993).
where the derivatives in Eq. (41) are to be taken along
the 0ć%K isotherm. Hugoniot data on solids provide con-
siderable information on thermodynamic properties that
6 Conclusion
are difficult to obtain by other means.
From a mainly historical perspective J. W. Taylor
In this article we have attempted to connect the early
(1984) documents the pioneering work on the shock com-
evolution of the field of nonlinear acoustics and finite-
pression of solids carried out at Los Alamos from about
amplitude wave propagation in gases with the field of
1944 until 1958. In his Conclusion to this article he writes:
shock compression in solids. The collection of Johnson
 This history begins with a few hastily constructed demon-
and Ch�ret (1998) provides considerable detail concern-
stration experiments and ends with a new field of scientific
ing theoretical research from 1808 to 1949. In the interest
inquiry. It is significant to (note) that at about the same
of providing a supplement to this collection and to illumi-
time there began to appear numerous papers on the sub-
nate the unique nature of the subject of shocks in solids,
ject in the Soviet journals. Undoubtedly there is a similar
important experimental work from 1948 to 1958 is summa-
story to be told from their viewpoint.
rized along with a single paper on numerical computation.
The period from 1958 to the present is adequately covered
in numerous review articles and reference books, three of
5 Progress since 1958
which are listed in Sect. 5.
The period from 1958 to the present has been very active
with respect to the development of new experimental tech-
References
niques to record longitudinal stress, pressure and particle
velocity in shock compressed solids. It has also been a pe-
Asay JR and Shahinpoor M (1993) High-pressure shock com-
riod of reasonably faithful documentation of methods and
pression of solids. Springer-Verlag, New York
results. We present here references to a number of sources
Bancroft D, Peterson EL, Minshall S (1956) Polymorphism of
which give details of this work.
iron at high pressure. J. Appl. Phys. 27: 291 298
A useful introductory summary of experimental tech- Barker LM, Hollenbach RE (1972) Laser interferometer for
niques is given by Barker, Shahinpoor and Chhabildas
measuring high velocities of any reflecting surface. J. Appl.
(1993): this includes the design and performance of gas Phys. 43: 4669 4675
200 J.N. Johnson, R. Ch�ret: Shock waves in solids: an evolutionary perspective
Barker LM, Hollenbach RE (1974) Shock wave study of the Rankine WJM (1870) On the thermodynamic theory of waves
ą �! � phase transition in iron. J. Appl. Phys. 45: 4872 of finite longitudinal disturbance. Philosophical Transac-
4887 tions 160: 277 288 [Johnson and Ch�ret (1998), pp. 133
Barker LM, Shahinpoor M, Chhabildas LC (1993) Experimen- 147]
tal and diagnostic techniques. In: Asay, J. R. and Shahin- Rice MH, McQueen RG, Walsh JM (1958) Compression of
poor, M. (eds) High-pressure shock compression of solids. solids by strong shock waves. In: Seitz F, Turnbull D (eds)
Springer-Verlag, New York, pp. 43 73 Solid state physics 6. Academic Press, New York, pp. 1 63
Bethe H (1942) On the theory of shock waves for an arbi- Riemann B (1860) Die Fortplanzung ebener Luftwellen
trary equation of state. Report No. 545, Office of Scien- von endlicher Schwingungsweite. Abhandlungen der
tific Research and Development, NDRC-B-237 [Johnson Gesellschaft der Wissenschaften zu Gottingen 8: 43 65 [in
and Ch�ret (1998), pp. 421 - 492] English, Johnson and Ch�ret (1998), pp. 109 128]
Bridgman PW (1931 and 1958) The physics of high pressure. Stokes GG (1848) On a difficulty in the theory of sound. Phil.
G. Bell and Sons, Ltd., London Mag. XXXIII: 349 356 [Johnson and Ch�ret (1998), pp.
Ch�ret R (1993) Detonation of condensed explosives. Springer- 71 79]
Verlag, New York Stokes GG (1883) On a difficulty in the theory of sound. Col-
Courant R, Friedrichs KO (1948) Supersonic flow and shock lected papers II, Cambridge University Press, Cambridge,
waves. Interscience Publishers, Inc., New York pp. 51 55
Earnshaw S (1860) On the mathematical theory of sound. Strutt JW, Third Baron Rayleigh (1910) Aerial plane waves
Philosophical Transactions 150: 133 148 [Johnson and of finite amplitude. Proc. Roy. Soc. A84: 247 284 [Johnson
Ch�ret (1998), pp. 83 104] and Ch�ret (1998), pp. 361 404]
Glass II, Sislian JP (1994) Nonstationary flows and shock Taylor GI (1910) The conditions necessary for discontinuous
waves. Clarendon Press, Oxford motion in gases. Proc. Roy. Soc. A84: 371 377 [Johnson
Graham RA (1993) Solids under high-pressure shock compres- and Ch�ret (1998), pp. 409 416]
sion. Springer-Verlag, New York Taylor GI (1948) The use of flat-ended projectiles for determin-
Horie Y, Sawaoka AB (1993) Shock compression chemistry of ing dynamic yield stress. Proc. Roy. Soc. A194: 289 299
materials. KTK Scientific Publishers, Tokyo Taylor JW (1984) Thunder in the mountains. In: Asay JR,
Hugoniot H (1887) Sur la propagation du mouvment dans les Graham RA, Straub GK (eds) Shock waves in condensed
corps et specialement dans les gaz parfaits  premiŁre par- matter  1983. Elsevier Science Publishers B. V., pp. 3 15
tie. J. de l Ecole Polytechnique 57: 3 97 [in English, John- Von Neumann J, Richtmyer RD (1950) J. Appl. Phys. 21: 232
son and Ch�ret (1998), pp. 161 243] 237
Hugoniot H (1889) Sur la propagation du mouvment dans les Walsh JM, Christian RH (1955) Equation of state of metals
corps et specialement dans les gaz parfaits  deuxiŁme par- from shock wave measurements. Phys. Rev. 97: 1544 1556
tie. J. de l Ecole Polytechnique 58: 1 125 [in English, John- Walsh JM, Rice MH, McQueen RG, Yarger FL (1957) Shock-
son and Ch�ret (1998), pp. 245 358] wave compression of twenty-seven metals. Equations of
Johnson JN, Ch�ret R (1998) Classic papers in shock compres- state of metals. Phys. Rev. 108: 196 216
sion science. Springer-Verlag, New York Weyl H (1949) Shock waves in arbitrary fluids. Comm. on
Minshall S (1955) Properties of elastic and plastic waves de- Pure and Appl. Math. II (Nos. 2 3): 103 122 [Johnson and
termined by pin contactors and crystals. J. Appl. Phys. 26: Ch�ret (1998), pp. 497 519]
463 469 Wilkins M (1964) Calculation of elastic-plastic flow. In: Alder
Pack DC, Evans WM, James HJ (1948) The propagation of B, Fernbach S, Rotenberg M (eds) Methods in computa-
shock waves in steel and lead. Proc. Phys. Soc. 60: 1 8 tional physics (3). Academic Press, New York, pp. 211 263
Poisson S-D (1808) Sur la th�orie du son. J. de l Ecole Poly-
technique 14: 319 392 [in English, Johnson and Ch�ret
(1998), pp. 3 65]