Shock and Release Temperatures in Molybdenum
Damian C. Swift∗ and Achim Seifter
Group P-24, Physics Division, Los Alamos National Laboratory,
MS E526, Los Alamos, New Mexico 87545, USA
David B. Holtkamp and David A. Clark
Group P-22, Physics Division, Los Alamos National Laboratory,
MS D410, Los Alamos, New Mexico 87545, USA
(Dated: March 12, 2007 – LA-UR-07-1660)
Shock and release temperatures in Mo were calculated, taking account of heating from plastic flow
predicted using the Steinberg-Guinan model. Plastic flow was calculated self-consistently with the
shock jump conditions: this is necessary for a rigorous estimate of the locus of shock states accessible.
The temperatures obtained were significantly higher than predicted assuming ideal hydrodynamic
loading. The temperatures were compared with surface emission spectrometry measurements for Mo
shocked to around 60GPa and then released into vacuum or into a LiF window. Shock loading was
induced by the impact of a planar projectile, accelerated by high explosive or in a gas gun. Surface
velocimetry showed an elastic wave at the start of release from the shocked state; the amplitude of
the elastic wave matched the prediction to around 10%, indicating that the predicted flow stress in
the shocked state was reasonable. The measured temperatures were consistent with the simulations,
indicating that the fraction of plastic work converted to heat was in the range 70-100% for these
loading conditions.
PACS numbers: 62.50.+p, 62.20.Fe, 65.40.-b, 07.20.Ka
Keywords: shock physics, shock temperature, plasticity, molybdenum
I. INTRODUCTION
The behavior of matter subjected to extreme condi-
tions through dynamic loading is of interest from a direct
physical standpoint, as dynamic loading is often the only
practical way to induce extreme conditions, and because
important engineering problems in hypervelocity impact
and weaponry involve dynamic loading [1]. Temperature
is a key parameter in understanding the properties of
matter, as it governs the population of vibrational pro-
cesses and excitations past energy barriers. Temperature
is thus important for a physical understanding of many
types of behavior and the associated models. The equa-
tion of state (EOS) includes contributions from the ex-
citation of atomic vibrations and electronic excitations.
Plastic flow is mediated by the excitation of dislocations
and twin boundaries past Peierls barriers. Phase changes
depend on the thermodynamic state’s location in the
phase diagram. The kinetics of phase changes are de-
scribed by the nucleation and growth of the daughter
phase in the matrix of the parent phase, requiring the
excitation of atoms past barriers. Chemical reactions
are governed by the excitation of atoms or electrons over
barriers. Diffusion in condensed matter is the motion of
atoms past the barriers formed by their neighbors. Con-
ductivities include scattering contributions from thermal
excitations.
Temperature is notoriously difficult to measure dur-
∗Electronic address: dswift@lanl.gov
ing dynamic loading, in particular for opaque materials
[2]. Extreme states of matter are often hidden within a
sheath of matter in a different state – this is the usual
situation in shock loading experiments. Probes made of
matter generally disrupt the state of interest, e.g. by pre-
senting an impedance mismatch to compression waves.
Much interesting physics in condensed matter occurs at
compressions of a few tens of percent, where the heating
may be in the range of a few hundred kelvin and the re-
sulting thermal emissions are small. Most temperature
measurements of shock-loaded systems have been made
using photon emission spectroscopy, commonly called py-
rometry [3, 4]. However, many materials of interest (e.g.
metals) are opaque in the relevant region of the spectrum:
infra-red through visible for shocks up to the terapascal
regime. Emission from an opaque material comes from
the surface, which cannot be maintained at the pressure
of the initial shock for long enough to allow useful emis-
sion spectra to be collected. A transparent window can
be placed in contact with the sample to maintain an el-
evated pressure, but the mismatch in shock impedance
must be taken into account, along with the effect of heat
conduction. Accurate temperature data have been ob-
tained from transparent sample materials, where ther-
mal radiation from the shocked state can escape from
the sample [5, 6].
Neutron resonance spectroscopy (NRS) has been inves-
tigated as a fundamentally different technique for mea-
suring the temperature inside a dynamically-loaded spec-
imen, which can be used on opaque materials [7]. Trial
measurements of NRS temperatures were performed on
shock-loadedMo, as a standard material for high pressure
work; the shock temperature was found to lie significantly
http://arxiv.org/abs/0704.1850v1
mailto:dswift@lanl.gov
above the temperature predicted using reasonable EOS
for Mo [7]. Measurements were also made using pyrome-
try of the temperature of Mo which was shocked and then
released into a LiF window or into vacuum; these exper-
iments also yielded temperatures which lay significantly
above EOS predictions [8].
Here we consider the effect of plastic flow on the Mo
pyrometry experiments. Plastic flow was neglected in
previous comparisons of predictions with the tempera-
ture data. The contribution of plastic work to heating
has been mentioned in studies of other metals [9] but has
not been quantified in detail or consistently. The contri-
bution to the total internal energy from plastic heating
has been estimated in order to extract the scalar EOS
from shock data [10], which involves a similar analysis of
shock heating, though less general.
II. CORRECTIONS AND SYSTEMATIC
UNCERTAINTIES IN PYROMETRY
Polycrystalline materials, such as the Mo for which
the discrepancy in temperature measurements was re-
ported, are heterogeneous in that they are composed of
an aggregate of grains of different crystallographic orien-
tation. The Mo samples were machined from material
which had been prepared by pressing from powder, so
there was a population of voids and there were impuri-
ties concentrated along grain boundaries. On shock load-
ing and subsequent release, different regions of the sam-
ple would thus respond differently, producing a variation
in local temperature. Given enough time, temperature
variations disperse through thermal conduction, but this
typically takes of order microseconds for grains tens of
micrometers in size, which is long on the time scale of
the experiments. We wish to compare pyrometry mea-
surements of temperature in Mo with predictions using
continuum models, so the temperature of interest is the
mean, bulk value. Thermal radiation is described by the
Stefan-Boltzmann relation, where the total power is pro-
portional to the fourth power of temperature. Pyrometry
measurements are often more accurate at shorter wave-
lengths where the power varies with higher powers of tem-
perature [11]. Thus unquantified temperature variations
(spatial or temporal, within the respective resolutions of
the detectors) lead to an overestimate of the mean tem-
perature. Spatial variation of brightness temperature has
been observed in shock-loaded Sn [12], which has a much
lower flow stress than Mo.
Pyrometry measurements from metals probe the sur-
face temperature. The surface is prone to increased
plastic work around surface features such as machining
marks: when a rough surface in contact with material
of lower impedance (or vacuum, in the extreme case) is
shocked, flow may be exaggerated in pits and grooves. At
sufficiently high shock pressures, localized jetting may
occur. Any such localized increase in plastic flow will
produce a higher local temperature, which may appear
as a higher mean temperature as discussed above.
For experiments in which the sample is observed
through a transparent window to maintain an elevated
pressure, there may be additional radiation from com-
pression of any gas or glue in the gap between the sample
and the window, or from the shocked window material it-
self. These effects were considered and corrected for the
Mo data [13]. After the shock passes from the sample
into a window, the temperature of the shock state is gen-
erally different to that in the sample, so heat conduction
can alter the temperature of the sample surface.
In multiple channel pyrometry measurements, thermal
emission from the sample is recorded using a set of de-
tectors responding to different ranges of photon wave-
length. In the simplest case, a grey body spectrum can
be fitted to the signals, and the mean emissivity and
temperature deduced from the shape of the spectrum.
In general, the emissivity of a material varies with state
and wavelength, and also with surface roughness, which
may change during dynamic loading. The emissivity may
be measured directly, for example by ellipsometry. It is
more common to assume that the multiple pyrometry
channels over-sample the wavelength variation of the ra-
diance and emissivity, at least over part of the spectrum,
allowing both to be deduced. Again, these effects were
taken into account for the Mo measurements [13].
Aside from gaps and glues between the sample and
any window, it is common for components of the shocked
target assembly to include sharp corners and low-density
materials such as plastics, foams, and glues, as part of
the engineering structure. As with glued windows, low
density materials in general may be shock-loaded to a
higher temperature than the sample. Sharp internal cor-
ners, when shocked, may form jets with large amounts
of plastic heating. Thermal emission from any of these
sources may be present as a background against which
the emission from the sample must be distinguished.
The net effect of the complications associated with py-
rometry measurements on metals is that it is possible to
over-estimate the temperature.
III. HEATING FROM PLASTIC WORK IN
SHOCK AND RELEASE
Shock compression involves the transit of a shock wave
through each element of material. Subsequently, axial
and lateral release and recompression waves reverberate
through the sample until it ultimately comes to rest at
zero pressure. In the simplest case, which many shock
physics experiments are designed to achieve, the shock
is steady and planar and the initial release is planar:
the material is compressed and released uniaxially. In
general, the shock and release may be converging or di-
verging, but locally the compression and decompression
induced is close to uniaxial. Specifically, the strains are
not not isotropic. If a crystalline solid is subjected to
non-isotropic strains then shear stresses must be induced,
leading to plastic flow if the flow stress is exceeded.
In continuum dynamics situations such as shock load-
ing, simulations and analysis may be performed accu-
rately by a scalar solution of the shock jump and isen-
tropic expansion relations if the material is represented
by an EOS, i.e. if the effects of elasticity and plastic
flow are ignored. It is common practice to use spatially-
resolving numerical simulations if the material is to be
represented with any greater complexity. However, sim-
ulations of shock waves are complicated by the need to
ensure that the shock – discontinuous at the continuum
level – is captured accurately without inducing numerical
artifacts such as oscillations or excess heating. However,
numerical methods have been developed to allow shock
compression and ramp decompression to be simulated by
a scalar solution for general material models including
elasticity and plastic flow [14]. These numerical solutions
were used to interpret the temperature measurements on
Spatially-resolved simulations were also performed for
comparison with velocity history measurements. These
simulations used a Lagrangian representation of the
shock experiments, integrated in time with a predictor-
corrector numerical scheme employing artificial viscosity
to stabilize the shock wave [15]. The material models
were identical with those used in the scalar solution.
The conservation equations for shock and release states
in material dynamics are usually formulated in terms of
compression and pressure. An order to take account of
elastic-plastic effects, the equations were formulated in
terms of the stress and strain tensors. Thus the Rankine-
Hugoniot equations [16] for conservation across the shock
were expressed in terms of the total stress normal to the
shock, τn, rather than the pressure p:
u2s = −v
τn − τn0
v0 − v
, (1)
∆up =
−(τn − τn0)(v0 − v), (2)
e = e0 −
(τn + τn0)(v0 − v), (3)
where v is specific volume (the reciprocal of the mass
density ρ), e is specific internal energy, us is the speed
of the shock wave with respect to the material, ∆up is
the change in material speed normal to the shock wave
(i.e., parallel to its direction of propagation), and sub-
script 0 refers to the initial state. The specific internal
energy was defined to exclude elastic strain energy, so
the energy equation above included only the volumetric
and plastic strain contributions to the volume change.
The relation for adiabatic compression and release was
expressed similarly:
pdiv ~u
: elastic
(||σgrad~u|| − pdiv ~u) : plastic
where τ is the stress tensor, σ the deviatoric stress
σ ≡ τ −
Tr τI = τ + pI, (5)
grad~u the velocity gradient tensor, and div ~u its trace.
For uniaxial compression, ||σgrad~u|| = σn∂~un/∂rn i.e.
the product of the components in the direction normal
to the wave, all others being zero. In the non-spatially-
resolved calculations, the velocity gradient was simply
the assumed or imposed strain rate.
The state of the material was expressed in terms of ρ
and e (allowing a mean pressure p(ρ, e) to be calculated
from the EOS), a deviatoric elastic strain tensor ǫ (allow-
ing the deviatoric stress contributions σ to be calculated),
and a scalar equivalent plastic strain ǫ̃p, used to calculate
work hardening. As discussed elsewhere [14], a hypere-
lastic formulation using strain rather than a hypoelastic
formulation using stress was preferred for consistency and
accuracy in situations where shear strains are applied at
different compressions. Thus the stress deviator σ was
calculated from the instantaneous strain,
σ = 2Gǫ, (6)
where G(ρ, T ) is the shear modulus. Plastic flow was
taken to occur using a von Mises yield surface [17]. De-
formation was plastic rather than elastic if the scalar ef-
fective shear stress
fσ||σ2|| (7)
exceeded the yield stress Y (ρ, e, ǫ̃p), in which case plastic
strain for work hardening was accumulated at a rate
˙̃ǫp =
||ǫ̇ǫ||+ ||ǫǫ̇||
, (8)
where ǫ̇ is the deviatoric part of the symmetric part of
the velocity gradient,
ǫ̇ ≡ Ė −
Tr ĖI : Ė ≡
(U + UT ), U ≡ grad~u.
If σ̃ < Y , the elastic deformation was simply ǫ̇. For
uniaxial compression along the x-direction, the only non-
zero component of U is [U ]100.
If plastic flow occurs, then the material is always
heated to some degree. Plastic flow occurs through the
motion and generation of defects in the crystal lattice,
such as dislocations. Usually in polycrystalline materials,
defects accumulate during plastic deformation. Heating
generally represents less than the total plastic work as
some potential energy is absorbed in the structure of de-
fects. The fraction of plastic work converted to heat fp is
thought to be 0.85-0.95. It was assumed here to be 0.9.
Thus the contribution of plastic work to heating was
ėp = fp
||σė||
if σ̃ > Y and zero otherwise.
When a metal is deformed, shear strains result in the
accumulation of elastic energy until the flow stress is
reached. Continued deformation results in plastic work.
If the material work-hardens, the rate of plastic working
increases. If thermal softening occurs, the rate of plastic
working decreases but the stored elastic energy is also
converted to plastic work. In an idealized material ex-
hibiting a constant flow stress (elastic-perfectly plastic),
arbitrarily large amounts of plastic work may be accu-
mulated by large deformations – uniaxial as well as pure
shear – beyond the flow stress. Ceramics may behave dif-
ferently, the flow stress decreasing to a small fraction of
its initial value as deformation continues beyond the elas-
tic limit [9], presumably as interatomic bonds are broken
and brittle damage occurs. Many transparent materials
are ceramic; this reduction of the flow stress may explain
why good agreement has been obtained between shock
temperatures and predictions neglecting heating from the
constitutive response [6].
Plastic flow is largely irreversible. If a sample of mate-
rial is shock loaded and then released [24], the pressure
reduces on release but further plastic work is done.
Mo was represented by an empirical EOS fitted to
shock compression data [18, 19, 20], with a deviatoric
strength model developed and calibrated to data on the
amplitude and shape of elastic waves running ahead of
shocks [18]. The EOS was likely to be accurate to a
few percent in temperature for the shock pressures of a
few tens of gigapascals considered here. The Steinberg-
Guinan strength model includes a prediction of the flow
stress at elevated pressures. The flow stress, and hence
the heating from plastic flow, was uncertain at the level of
a few tens of percent. As discussed below, measurements
of surface velocity provided an independent measure of
the flow stress.
Material models for continuum dynamics are often im-
plemented in varying ways in different computer pro-
grams. The results may depend on details such as the
way in which numerical limits, e.g. on flow stress, are
applied. In our simulations, the EOS was represented by
an expression for pressure p in terms of mass density ρ
and specific internal energy e. This is sufficient to allow
the dynamical equations for the continuum to be inte-
grated in time. Two different EOS were used, a tabular
form from the ‘SESAME’ library [19], and an analytic
form of the Grüneisen type, using the principal Hugoniot
as the reference curve [18]: shock speed us in terms of
particle speed up,
us = c0 + s1up, (11)
together with a relation for the Grüneisen parameter
γ(ρ) = γ0 + b(ρ/ρ0 − 1). (12)
The shear modulus G and flow stress Y followed the
Steinberg-Guinan model [21], which includes explicit de-
pendence on temperature T and accumulated plastic
strain ǫp:
G(p, T ) = G0
1 +Ap(ρ/ρ0)
−1/3 −B(T − T0)
Y (p, T ) = Y0f(ǫp)G(p, T )/G0 (14)
f(ǫp) = min [(1 + βǫ̃p)
n, Ymax/Y0] . (15)
TABLE I: Grüneisen equation of state and Steinberg-Guinan
strength parameters for Mo.
equation of state strength
ρ0 10.2 g/cm
G0 125 GPa
c0 5.143 km/s Y0 1.6 GPa
s1 1.255 A 1.14 × 10
−2 GPa−1
γ0 1.59 B 1.52 × 10
−4 K−1
b 0.30 β 20
cp 2.43 × 10
−4 MJ/kg.K n 0.15
a 1.3 Ymax 2.8 GPa
Source: [18] with unit conversions.
Because of the scaling of flow stress by shear modulus,
the maximum flow stress at high pressures was not lim-
ited by the ‘maximum’ flow stress Ymax – this allowed
the flow stress to be significantly greater than Ymax in
the Mo impact experiments. The usual definition of the
Steinberg-Guinan model [18] includes an explicit initial
plastic strain from manufacture; we treated ǫp as a local
material parameter in addition to ρ, e, and the elastic
strain, and set ǫ̃p to a non-zero value in the initial condi-
tions if required. The factors fǫ and fσ used in calculat-
ing the scalar effective magnitudes of the corresponding
tensors were chosen for consistency with the definitions
of stress and strain used in deducing strength parameters
for Mo from experiments: fǫ = fσ = 3/2.
The SESAME EOS were defined as a pair of tables
{p, e}(ρ, T ), so the p(ρ, e) relation was obtained by nu-
merical inversion and the temperature was readily calcu-
lated. Temperatures were calculated from the Grüneisen
EOS with reference to a compression curve along which
the temperature and specific internal energy were known,
{Tr, er}(ρ), and using a specific heat capacity defined as a
function of density cv(ρ) (constant in practice). The ref-
erence curve chosen was the zero kelvin isotherm (‘cold
curve’), Tr = 0K. This was calculated from the principal
isentrope e(ρ)|s0 using the estimated density variation of
Grüneisen parameter:
er(ρ) = e(ρ)|s0 − T0cpe
a(1−ρ0/ρ)
)γ0−a
. (16)
The isentrope was calculated by numerical integration of
the second law of thermodynamics,
= −p(1/v, e). (17)
Mechanical properties and temperatures calculated by ei-
ther EOS gave the same result to o(1%), which consti-
tutes good agreement for models in material dynamics.
The Grüneisen EOS have slightly smoother loci, so the
results presented below are from this EOS.
Simulations were performed in units of millimeters,
gigapascals, microseconds, kelvin, and Mg/m3=g/cm3.
Parameters for Mo in these units are listed in Table I.
projectile
thermal
radiation
collecting lens;
to spectrometer
reflected laser light
for Doppler velocimetryro
window
sample
FIG. 1: Schematic of impact-induced shock experiments with
surface temperature measurements. Aspect ratios are repre-
sentative of the experiments discussed here. If the window
is omitted, the experiment measures the free surface (zero
normal stress) temperature.
IV. PYROMETRY EXPERIMENTS ON
MOLYBDENUM
Pyrometry measurements of the temperature in
shocked and released Mo have been made using two types
of experiment. In both cases, the shock was induced by
the impact of a flat projectile. The projectiles were ac-
celerated using a high explosive launcher, as in the NRS
experiments, and by a gas gun. The pyrometry measure-
ment was performed at the surface opposite the impact,
the shocked state releasing either to vacuum or into a
LiF window to sustain an elevated pressure (Figs 1 and
2). In each case, the shock state was calculated using the
published EOS and strength properties for the projectile
and Mo target.
In all cases, the impact conditions were calculated us-
ing the scalar solution, and were repeated with and with-
out strength in all components of the impact experiment.
For experiments with a LiF window, the temperature in
the window was also predicted; a high temperature would
signal an increased possibility of thermal radiation from
the window obscuring the emissions from the Mo sam-
ple. Where an uncertainty in impact velocity was re-
ported, the calculations were repeated for velocities at
the extremes of uncertainty, giving an estimate of the
uncertainty in pressure and temperature.
Similar calculations were performed with and without
strength in each component separately. The Steinberg-
Guinan model is least appropriate for LiF, so this is the
only component where it would be useful to make such
additional comparisons. However, the effect on states
releasing into LiF were dominated by the strength of the
Mo, so the additional comparisons are omitted for clarity.
An indication that the contribution of strength in the LiF
is a small effect in the simulations is that the predicted
shock temperature in LiF varied much less as a function
of strength than did the temperature of any of the Mo
states.
Taking strength into account, on release into LiF, the
 0  0.5  1  1.5  2  2.5  3  3.5  4
particle speed (km/s)
Al-6061
projectile
impact shock
elastic release
window release
free surface release
FIG. 2: Shock and release states induced in impact experi-
ments with and without a window. Solid lines: shock Hugo-
niots; dashed line: release adiabat. Example calculation for
Al-6061 projectile traveling at 3.6 km/s, impacting stationary
Mo target, releasing into LiF window or into vacuum. The
initial states of the Mo and LiF are at the origin; the initial
state of the projectile is at zero normal stress and 3.6 km/s.
On impact, the shock states in the projectile and sample are
at the elevated pressure intersection marked ‘impact shock.’
When the shocked state in the Mo releases into the LiF win-
dow, the resulting state is the intersection marked ‘window
release.’ When the shocked state in the Mo is released at a
free surface, the resulting state is the zero normal stress state
marked ‘free surface release.’ Release from the shocked state
shows an inflexion when plastic flow occurs.
normal stress in the Mo was lower than the in-plane stress
because the elastic strain is a distension in the axial di-
rection. For this reason, the calculations with strength
have a lower normal stress: a result which may be coun-
terintuitive.
Various improvements could be made in future pyrom-
etry measurements to reduce the temperature uncertain-
ties. Some optimization could be performed by repeating
experiments multiple times, adjusting detector gains and
digitization ranges for best accuracy. However, the diffi-
culty and cost of each projectile impact experiment can
make multiple repeats impractical.
A. Gas gun
The projectile was Ta, 3mm thick, accelerated to
1.70 km/s using a two stage gas gun. The target was
Mo, 5mm thick. Thermal emission was measured on re-
lease into a LiF window, using a 7 channel pyrometer.
The measured release temperature was 683± 41K.
The shocked state in the Mo was calculated to be
58.7GPa and 645K, of which 51K was from plastic work.
The state on release into LiF was thus calculated to be
24.8GPa and 614K, of which 82K was from plastic work.
 200  300  400  500  600  700  800
temperature (K)
Hugoniot
release
adiabat
elastic
LiF release
solid: Steinberg-Guinan
dashed: no strength
FIG. 3: Temperature measurement from a shock of 59GPa,
on release to 25GPa into LiF, compared with predictions
based on the Steinberg-Guinan strength model and with
strength neglected. The crosses on the release adiabats show
where release pauses when a shock is transmitted into the
LiF. When strength is included, the first portion of release
is elastic; the elastic portion of the adiabat is marked; flow
becomes plastic below the inflexion.
The measured surface temperature was just 1.5 standard
deviations above the temperature predicted using the
Steinberg-Guinan strength model, and more than three
standard deviations above the temperature predicted ig-
noring material strength. (Table II and Fig. 3.)
B. Forest Flyer
The high explosive launcher used the Forest Flyer de-
sign [22]. With this system, the projectile was slightly
dished on impact, though this should not affect the py-
rometry measurement significantly. The projectile was
accelerating slightly, reverberating elastically from the
acceleration process, and possibly had a porous region
through its thickness as a result of tensile stresses dur-
ing acceleration. The relatively strong reverberations in
the projectile affect its effective speed on impact, and
contributed to the uncertainty in material states.
The projectile was Al-6061 alloy, 6mm thick, acceler-
ated to 3.6 ± 0.1 km/s. The target was Mo, also 6mm
thick. Six experiments were performed, four for release
into a LiF window and two into vacuum. Surface emis-
sion was measured with a 5 channel visible-near infrared
pyrometer or a 4 channel near infrared pyrometer. The
free surface temperature had a relatively large uncer-
tainty, and the signals on release into LiF showed ev-
idence of thermal emission from the LiF itself with a
temperature of around 580K [13]. The measured release
temperature was 762 ± 40K into LiF, and 568 ± 100K
from the free surface.
 200  300  400  500  600  700  800  900
temperature (K)
Hugoniot
release
adiabat
LiF release
free surface
solid: Steinberg-Guinan
dashed: no strength
FIG. 4: Temperature measurements from a shock of 64GPa,
on release to 27GPa into LiF and to zero pressure, compared
with predictions based on the Steinberg-Guinan strength
model and with strength neglected. The release adiabat from
the mean shock pressure is shown, along with adiabats re-
flecting the uncertainty in shock pressure. The lines across
the release adiabats show where release pauses when a shock
is transmitted into the LiF.
The shocked state in the Mo was calculated to be 63.9±
2.4GPa and 707±31K, of which 53±3K was from plastic
work. The state on release into LiF was thus calculated
to be 27.1 ± 1GPa and 670 ± 25K, of which 89 ± 1K
was from plastic work. The state on release into vacuum
was calculated to be 635± 23K, of which 126± 4K was
from plastic work. The uncertainties are correlated: the
smallest, mean, and largest of each go together.
The surface temperature on release into LiF was 1.5-2.5
standard deviations of the temperature predicted using
the Steinberg-Guinan strength model, and 3.5-4.5 stan-
dard deviations from the temperature predicted with-
out strength. The uncertainty in the free surface re-
lease temperature was too large to discriminate between
a purely hydrodynamic calculation (no strength) and the
Steinberg-Guinan model – both lay within one standard
deviation of the measurement. The predicted temper-
ature of the LiF itself also matched the measurement
to within the experimental uncertainties. (Table II and
Fig. 4.)
The velocity history of the surface of the sample was
measured by laser Doppler velocimetry of the ‘VISAR’
type [23]. General features of the velocity history in-
cluded a rapid acceleration when the shock reached the
surface, a roughly constant peak velocity corresponding
to the sustained pressure behind the shock, deceleration
caused by the release wave from the rear of the projectile,
and a slight re-acceleration as the sample was subjected
to tensile stress causing spall type damage (Fig. 5). The
epoch of peak velocity was not perfectly constant, but
showed some acceleration. This was probably caused by
the compression gradient in the projectile from the resid-
 0  0.5  1  1.5  2  2.5  3  3.5  4
time after impact (µs)
shock
release
FIG. 5: Surface velocity histories measured in Forest Flyer
impact experiments with and without a LiF window. Separate
lines are from different experiments. The upper two traces are
from free surface release; the lower three are from release into
a LiF window.
ual accelerating pressure at impact, and any regions of
porosity resulting from tensile damage as the projectile
was accelerated by the relatively strong pressures induced
by the detonating high explosive. The onset of release
showed a clear elastic precursor (Fig. 6).
The measured velocity histories were compared with
spatially-resolved one-dimensional continuum dynamics
simulations. The projectile was modeled as ideal, i.e.
at uniform STP conditions and traveling at a constant
3.6 km/s with no reverberations. As a result, the peak ve-
locity epoch was flatter than measured, but was in good
agreement for amplitude and duration. Release into the
LiF was also reproduced well overall. The shape of the
elastic precursor to release was not reproduced perfectly
using the Steinberg-Guinan strength model, but its am-
plitude was reproduced to within around 10% and the
time of arrival was in good agreement with the exper-
iment. The difference in shape could be caused by in-
adequacy in the Steinberg-Guinan model – for example,
in the detailed work-hardening history in the shocked
state – but is more likely to reflect density variations in
the projectile as discussed above. The uniaxial strains
greatly exceeded the elastic limit on release as well as on
compression, so the plastic work should be dominated by
the flow stress rather than the precise path before plastic
flow occurred. Thus the agreement between calculated
and observed amplitudes suggests that the plastic work
should be correct to around 10%.
Spallation did not affect the shock and release states
of interest for the temperature measurements considered
here. The simulations used a crude spall model of the
minimum pressure type, with a minimum pressure of -
1.5GPa [18], meaning that the maximum tensile stress
induced by the Mo as it was distended was 1.5GPa.
 0  0.5  1  1.5  2  2.5  3  3.5  4
time after impact (µs)
experiment
experiment
strength
strength
FIG. 6: Surface velocity history in Forest Flyer impact ex-
periment with a free surface (upper traces) and a LiF window
(lower traces). Each experimental records is compared with
two continuum dynamics simulations, with (solid lines) and
without strength (dashed lines) in the Mo. The elastic precur-
sor to the release wave is evident where the experimental ve-
locity histories deviate from the simulations without strength.
No treatment of accumulating porosity was included, so
the Mo as simulated continued to exert a tensile stress
when in reality voids or cracks would open, reducing the
stress. Thus the simulations of velocity history did not
show a re-acceleration after the deceleration associated
with the release wave. Tensile damage and spall can de-
pend strongly on the strain rate and loading history. The
simulated and observed release deceleration matched to
within a few percent, suggesting that the published spall
strength applies well to the loading history induced by
these projectile impact experiments.
V. CONCLUSIONS
Shock and release temperatures were calculated self-
consistently using the equation of state and a published
constitutive model for Mo. Strength was calculated to
make a significant difference to states in experiments ex-
ploring pressures of tens of gigapascals. The high pres-
sure flow stress predicted using the Steinberg-Guinan
strength model matched the elastic release precursor ob-
served using surface Doppler velocimetry, suggesting that
the flow stress was correct to around 10%. The predicted
temperatures were consistent with pyrometry measure-
ments for shocks of around 60GPa, releasing into a LiF
window or into vacuum. The LiF release temperatures
were clearly more consistent with plastic work as pre-
dicted using the Steinberg-Guinan model than with hy-
drodynamic flow (no strength). The uncertainties in tem-
perature were however too large to discriminate between
strength models to better than several tens of percent in
TABLE II: Shock and release states.
no strength strength measured
particle speed normal stress temperature particle speed normal stress temperature temperature
(km/s) (GPa) (K) (km/s) (GPa) (K) (K)
gas gun
Mo shock 0.905 57.9 594 0.902 58.7 645
Mo release into LiF 1.374 25.4 532 1.337 24.8 614 683 ± 41
LiF shock 1.374 25.4 535 1.337 24.8 532
Forest Flyer
Mo shock 0.97 ± 0.03 63.3 ± 2.4 654± 28 0.97± 0.03 63.9 ± 2.4 707± 31
Mo release into vacuum 1.95 ± 0.07 0 509± 19 1.91± 0.07 0 635± 23 566± 100
Mo release into LiF 1.48 ± 0.04 27.8 ± 1 581± 24 1.44± 0.04 27.1± 1 670± 25 762 ± 40
LiF shock 1.48 ± 0.04 27.8 ± 1 570± 17 1.44± 0.04 27.1± 1 566± 16 624± 100
flow stress.
Heating from plastic work was calculated to be around
50K for shock pressures around 60GPa, 90K on sub-
sequent release into LiF, and 125K on release at a free
surface. The fraction of plastic work converted to heat
was assumed to be 90% – the heating would have been
about 10% greater if all the plastic work appeared as
heat. Taking plastic flow into account, there was no dis-
crepancy between predictions and measured release tem-
peratures for Mo. This is a validation of the models of
EOS and strength for Mo, and of the use of pyrometry
to measure release temperatures in metals – though the
pyrometry measurements obtained in these experiments
were not precise enough to discriminate between models
calibrated against similar mechanical data such as veloc-
ity histories. The fraction of plastic work converted to
heat was most likely close to 100%, though the uncer-
tainty in the temperature measurements means that this
figure cannot be justified statistically to better than a
few tens of percent.
Plastic flow makes a significant contribution to recon-
ciling the temperature discrepancy observed in the neu-
tron resonance spectrometry experiments on shocked Mo,
although the complete explanation is more complicated
and will be reported separately.
Acknowledgments
We would like to acknowledge the contribution of Carl
Greeff for assistance and advice on equations of state
and their uncertainties or certainties for Mo, of Ron Ra-
bie, David Funk, Rob Hixson, and Chuck Forest for de-
tailed information on the design and testing of the Forest
Flyer loading system, and of Sheng-Nian Luo for gen-
eral advice and comments on pyrometry and material
dynamics. The gas gun experiments were performed by
D.B. Holtkamp, P. Paulsen, P. Fiske, D. DeVore, J. Gar-
cia, and L. Tabaka at Lawrence Livermore National Lab-
oratory in 1999. The work was performed under the aus-
pices of the U.S. Department of Energy under contracts
W-7405-ENG-36 and DE-AC52-06NA25396.
[1] M. Eremets, “High Pressure Experimental Methods”
(Oxford University Press, 1996, New York).
[2] S.-N. Luo and D.C. Swift, Physica B 388, 139 (2007).
[3] S.B. Kormer, Sov. Phys. Usp. 21, 689700 (1965) – trans-
lation.
[4] M.B. Boslough and T.J. Ahrens, Rev. Sci. Instrum. 60,
3711-6 (1989).
[5] S.-N. Luo, J.A. Akins, T.J. Ahrens, and P.D. Asimow,
J. Geophys. Res. 109, B05205 (2004).
[6] S.-N. Luo, D.C. Swift, R.N. Mulford, N.D. Drummond,
and G.J. Ackland, J. Phys.: Cond. Matt., 16, 30, 5435 -
5442 (2004).
[7] V.W. Yuan, J.D. Bowman, D.J. Funk, G.L. Morgan,
R.L. Rabie, C.E. Ragan, J.P. Quintana, and H.L. Stacy,
Phys. Rev. Lett. 94, 125504 (2005).
[8] A. Seifter, K. Boborides, D.A. Clark, R.B. Corrow,
D.B. Holtkamp, G.L. Morgan, J.R. Payton, P. Quintana,
C.E. Ragan, P. Rodriguez, H.L. Stacey, W.S. Vogan,
V.W. Yuan, and A.W. Obst, Temperature measurements
of shock-loaded molybdenum (Los Alamos National Labo-
ratory report LA-UR-04-2561), Proc. TEMPMEKO 2004
conference, held in Dubrovnik, Croatia, 22-27 June 2004.
[9] S.A. Raikes and T.J. Ahrens, Geophys. J. of the Roy.
Astron. Soc. 58, pp 717-747 (1979).
[10] C.E. Morris and J.N. Fritz, J. Appl. Phys. 51, 2, pp 1244-
1246 (1980).
[11] A. Seifter and A. Obst, About the proper wavelength for
pyrometry on shock physics experiments, submitted to
Int. J. of Thermophysics.
[12] A. Seifter (Los Alamos National Laboratory), infra red
camera measurements, in preparation.
[13] A. Seifter (Los Alamos National Laboratory), detailed
pyrometry analysis, in preparation.
[14] D.C. Swift, Numerical solution of shock and ramp load-
ing relations for general material properties, submitted,
arXiv:cond-mat/0704.0008.
[15] D. Benson, Computer Methods in Appl. Mechanics and
Eng. 99, 235 (1992).
[16] For a recent review and introduction, see e.g.
M.R. Boslough and J.R. Asay, in J.R. Asay and
M. Shahinpoor (Eds), “High-Pressure Shock Compres-
sion of Solids” (Springer-Verlag, New York, 1992).
[17] R. Hill, “The Mathematical Theory of Plasticity”
(Clarendon Press, Oxford, 1950).
[18] D.J. Steinberg, Equation of state and strength parame-
ters for selected materials, Lawrence Livermore National
Laboratory report UCRL-MA-106439 change 1 (1996).
[19] K.S. Holian (Ed.), Los Alamos National Laboratory re-
port LA-10160-MS (1984).
[20] C. Greeff (Los Alamos National Laboratory), unpub-
lished work and private communications.
[21] D.J. Steinberg, S.G. Cochran, and M.W. Guinan,
J. Appl. Phys. 51, 1498 (1980).
[22] D.C. Swift, C.A. Forest, D.A. Clark, W.T. Buttler,
M. Marr-Lyon, and P. Rightley, On High Explosive
Launching of Projectiles for Shock Physics Experiments,
submitted, arXiv:cond-mat/0702693.
[23] L.M. Barker and R.E. Hollenbach, J. Appl. Phys. 43, 11,
pp 4669-4675 (1972).
[24] In purely hydrodynamic analyses, where the effect of
plastic flow is ignored, release from a shocked state fol-
lows an isentrope. This is no longer true when additional
dissipative processes occur, such as plastic flow and vis-
cosity, which lead to heating with an increase in entropy.
The term ‘quasi-isentropic’ is sometimes used in this con-
text, particularly for shockless compression; here we pre-
fer to refer to the release loci as adiabats since this is a
more specific term.
http://arxiv.org/abs/cond-mat/0702693

  Shock and release temperatures in Mo were calculated, taking account of
heating from plastic flow predicted using the Steinberg-Guinan model. Plastic
flow was calculated self-consistently with the shock jump conditions: this is
necessary for a rigorous estimate of the locus of shock states accessible. The
temperatures obtained were significantly higher than predicted assuming ideal
hydrodynamic loading. The temperatures were compared with surface emission
spectrometry measurements for Mo shocked to around 60GPa and then released into
vacuum or into a LiF window. Shock loading was induced by the impact of a
planar projectile, accelerated by high explosive or in a gas gun. Surface
velocimetry showed an elastic wave at the start of release from the shocked
state; the amplitude of the elastic wave matched the prediction to around 10%,
indicating that the predicted flow stress in the shocked state was reasonable.
The measured temperatures were consistent with the simulations, indicating that
the fraction of plastic work converted to heat was in the range 70-100% for
these loading conditions.

Shock and Release Temperatures in Molybdenum
Damian C. Swift∗ and Achim Seifter
Group P-24, Physics Division, Los Alamos National Laboratory,
MS E526, Los Alamos, New Mexico 87545, USA
David B. Holtkamp and David A. Clark
Group P-22, Physics Division, Los Alamos National Laboratory,
MS D410, Los Alamos, New Mexico 87545, USA
(Dated: March 12, 2007 – LA-UR-07-1660)
Shock and release temperatures in Mo were calculated, taking account of heating from plastic flow
predicted using the Steinberg-Guinan model. Plastic flow was calculated self-consistently with the
shock jump conditions: this is necessary for a rigorous estimate of the locus of shock states accessible.
The temperatures obtained were significantly higher than predicted assuming ideal hydrodynamic
loading. The temperatures were compared with surface emission spectrometry measurements for Mo
shocked to around 60GPa and then released into vacuum or into a LiF window. Shock loading was
induced by the impact of a planar projectile, accelerated by high explosive or in a gas gun. Surface
velocimetry showed an elastic wave at the start of release from the shocked state; the amplitude of
the elastic wave matched the prediction to around 10%, indicating that the predicted flow stress in
the shocked state was reasonable. The measured temperatures were consistent with the simulations,
indicating that the fraction of plastic work converted to heat was in the range 70-100% for these
loading conditions.
PACS numbers: 62.50.+p, 62.20.Fe, 65.40.-b, 07.20.Ka
Keywords: shock physics, shock temperature, plasticity, molybdenum
I. INTRODUCTION
The behavior of matter subjected to extreme condi-
tions through dynamic loading is of interest from a direct
physical standpoint, as dynamic loading is often the only
practical way to induce extreme conditions, and because
important engineering problems in hypervelocity impact
and weaponry involve dynamic loading [1]. Temperature
is a key parameter in understanding the properties of
matter, as it governs the population of vibrational pro-
cesses and excitations past energy barriers. Temperature
is thus important for a physical understanding of many
types of behavior and the associated models. The equa-
tion of state (EOS) includes contributions from the ex-
citation of atomic vibrations and electronic excitations.
Plastic flow is mediated by the excitation of dislocations
and twin boundaries past Peierls barriers. Phase changes
depend on the thermodynamic state’s location in the
phase diagram. The kinetics of phase changes are de-
scribed by the nucleation and growth of the daughter
phase in the matrix of the parent phase, requiring the
excitation of atoms past barriers. Chemical reactions
are governed by the excitation of atoms or electrons over
barriers. Diffusion in condensed matter is the motion of
atoms past the barriers formed by their neighbors. Con-
ductivities include scattering contributions from thermal
excitations.
Temperature is notoriously difficult to measure dur-
∗Electronic address: dswift@lanl.gov
ing dynamic loading, in particular for opaque materials
[2]. Extreme states of matter are often hidden within a
sheath of matter in a different state – this is the usual
situation in shock loading experiments. Probes made of
matter generally disrupt the state of interest, e.g. by pre-
senting an impedance mismatch to compression waves.
Much interesting physics in condensed matter occurs at
compressions of a few tens of percent, where the heating
may be in the range of a few hundred kelvin and the re-
sulting thermal emissions are small. Most temperature
measurements of shock-loaded systems have been made
using photon emission spectroscopy, commonly called py-
rometry [3, 4]. However, many materials of interest (e.g.
metals) are opaque in the relevant region of the spectrum:
infra-red through visible for shocks up to the terapascal
regime. Emission from an opaque material comes from
the surface, which cannot be maintained at the pressure
of the initial shock for long enough to allow useful emis-
sion spectra to be collected. A transparent window can
be placed in contact with the sample to maintain an el-
evated pressure, but the mismatch in shock impedance
must be taken into account, along with the effect of heat
conduction. Accurate temperature data have been ob-
tained from transparent sample materials, where ther-
mal radiation from the shocked state can escape from
the sample [5, 6].
Neutron resonance spectroscopy (NRS) has been inves-
tigated as a fundamentally different technique for mea-
suring the temperature inside a dynamically-loaded spec-
imen, which can be used on opaque materials [7]. Trial
measurements of NRS temperatures were performed on
shock-loadedMo, as a standard material for high pressure
work; the shock temperature was found to lie significantly
http://arxiv.org/abs/0704.1850v1
mailto:dswift@lanl.gov
above the temperature predicted using reasonable EOS
for Mo [7]. Measurements were also made using pyrome-
try of the temperature of Mo which was shocked and then
released into a LiF window or into vacuum; these exper-
iments also yielded temperatures which lay significantly
above EOS predictions [8].
Here we consider the effect of plastic flow on the Mo
pyrometry experiments. Plastic flow was neglected in
previous comparisons of predictions with the tempera-
ture data. The contribution of plastic work to heating
has been mentioned in studies of other metals [9] but has
not been quantified in detail or consistently. The contri-
bution to the total internal energy from plastic heating
has been estimated in order to extract the scalar EOS
from shock data [10], which involves a similar analysis of
shock heating, though less general.
II. CORRECTIONS AND SYSTEMATIC
UNCERTAINTIES IN PYROMETRY
Polycrystalline materials, such as the Mo for which
the discrepancy in temperature measurements was re-
ported, are heterogeneous in that they are composed of
an aggregate of grains of different crystallographic orien-
tation. The Mo samples were machined from material
which had been prepared by pressing from powder, so
there was a population of voids and there were impuri-
ties concentrated along grain boundaries. On shock load-
ing and subsequent release, different regions of the sam-
ple would thus respond differently, producing a variation
in local temperature. Given enough time, temperature
variations disperse through thermal conduction, but this
typically takes of order microseconds for grains tens of
micrometers in size, which is long on the time scale of
the experiments. We wish to compare pyrometry mea-
surements of temperature in Mo with predictions using
continuum models, so the temperature of interest is the
mean, bulk value. Thermal radiation is described by the
Stefan-Boltzmann relation, where the total power is pro-
portional to the fourth power of temperature. Pyrometry
measurements are often more accurate at shorter wave-
lengths where the power varies with higher powers of tem-
perature [11]. Thus unquantified temperature variations
(spatial or temporal, within the respective resolutions of
the detectors) lead to an overestimate of the mean tem-
perature. Spatial variation of brightness temperature has
been observed in shock-loaded Sn [12], which has a much
lower flow stress than Mo.
Pyrometry measurements from metals probe the sur-
face temperature. The surface is prone to increased
plastic work around surface features such as machining
marks: when a rough surface in contact with material
of lower impedance (or vacuum, in the extreme case) is
shocked, flow may be exaggerated in pits and grooves. At
sufficiently high shock pressures, localized jetting may
occur. Any such localized increase in plastic flow will
produce a higher local temperature, which may appear
as a higher mean temperature as discussed above.
For experiments in which the sample is observed
through a transparent window to maintain an elevated
pressure, there may be additional radiation from com-
pression of any gas or glue in the gap between the sample
and the window, or from the shocked window material it-
self. These effects were considered and corrected for the
Mo data [13]. After the shock passes from the sample
into a window, the temperature of the shock state is gen-
erally different to that in the sample, so heat conduction
can alter the temperature of the sample surface.
In multiple channel pyrometry measurements, thermal
emission from the sample is recorded using a set of de-
tectors responding to different ranges of photon wave-
length. In the simplest case, a grey body spectrum can
be fitted to the signals, and the mean emissivity and
temperature deduced from the shape of the spectrum.
In general, the emissivity of a material varies with state
and wavelength, and also with surface roughness, which
may change during dynamic loading. The emissivity may
be measured directly, for example by ellipsometry. It is
more common to assume that the multiple pyrometry
channels over-sample the wavelength variation of the ra-
diance and emissivity, at least over part of the spectrum,
allowing both to be deduced. Again, these effects were
taken into account for the Mo measurements [13].
Aside from gaps and glues between the sample and
any window, it is common for components of the shocked
target assembly to include sharp corners and low-density
materials such as plastics, foams, and glues, as part of
the engineering structure. As with glued windows, low
density materials in general may be shock-loaded to a
higher temperature than the sample. Sharp internal cor-
ners, when shocked, may form jets with large amounts
of plastic heating. Thermal emission from any of these
sources may be present as a background against which
the emission from the sample must be distinguished.
The net effect of the complications associated with py-
rometry measurements on metals is that it is possible to
over-estimate the temperature.
III. HEATING FROM PLASTIC WORK IN
SHOCK AND RELEASE
Shock compression involves the transit of a shock wave
through each element of material. Subsequently, axial
and lateral release and recompression waves reverberate
through the sample until it ultimately comes to rest at
zero pressure. In the simplest case, which many shock
physics experiments are designed to achieve, the shock
is steady and planar and the initial release is planar:
the material is compressed and released uniaxially. In
general, the shock and release may be converging or di-
verging, but locally the compression and decompression
induced is close to uniaxial. Specifically, the strains are
not not isotropic. If a crystalline solid is subjected to
non-isotropic strains then shear stresses must be induced,
leading to plastic flow if the flow stress is exceeded.
In continuum dynamics situations such as shock load-
ing, simulations and analysis may be performed accu-
rately by a scalar solution of the shock jump and isen-
tropic expansion relations if the material is represented
by an EOS, i.e. if the effects of elasticity and plastic
flow are ignored. It is common practice to use spatially-
resolving numerical simulations if the material is to be
represented with any greater complexity. However, sim-
ulations of shock waves are complicated by the need to
ensure that the shock – discontinuous at the continuum
level – is captured accurately without inducing numerical
artifacts such as oscillations or excess heating. However,
numerical methods have been developed to allow shock
compression and ramp decompression to be simulated by
a scalar solution for general material models including
elasticity and plastic flow [14]. These numerical solutions
were used to interpret the temperature measurements on
Spatially-resolved simulations were also performed for
comparison with velocity history measurements. These
simulations used a Lagrangian representation of the
shock experiments, integrated in time with a predictor-
corrector numerical scheme employing artificial viscosity
to stabilize the shock wave [15]. The material models
were identical with those used in the scalar solution.
The conservation equations for shock and release states
in material dynamics are usually formulated in terms of
compression and pressure. An order to take account of
elastic-plastic effects, the equations were formulated in
terms of the stress and strain tensors. Thus the Rankine-
Hugoniot equations [16] for conservation across the shock
were expressed in terms of the total stress normal to the
shock, τn, rather than the pressure p:
u2s = −v
τn − τn0
v0 − v
, (1)
∆up =
−(τn − τn0)(v0 − v), (2)
e = e0 −
(τn + τn0)(v0 − v), (3)
where v is specific volume (the reciprocal of the mass
density ρ), e is specific internal energy, us is the speed
of the shock wave with respect to the material, ∆up is
the change in material speed normal to the shock wave
(i.e., parallel to its direction of propagation), and sub-
script 0 refers to the initial state. The specific internal
energy was defined to exclude elastic strain energy, so
the energy equation above included only the volumetric
and plastic strain contributions to the volume change.
The relation for adiabatic compression and release was
expressed similarly:
pdiv ~u
: elastic
(||σgrad~u|| − pdiv ~u) : plastic
where τ is the stress tensor, σ the deviatoric stress
σ ≡ τ −
Tr τI = τ + pI, (5)
grad~u the velocity gradient tensor, and div ~u its trace.
For uniaxial compression, ||σgrad~u|| = σn∂~un/∂rn i.e.
the product of the components in the direction normal
to the wave, all others being zero. In the non-spatially-
resolved calculations, the velocity gradient was simply
the assumed or imposed strain rate.
The state of the material was expressed in terms of ρ
and e (allowing a mean pressure p(ρ, e) to be calculated
from the EOS), a deviatoric elastic strain tensor ǫ (allow-
ing the deviatoric stress contributions σ to be calculated),
and a scalar equivalent plastic strain ǫ̃p, used to calculate
work hardening. As discussed elsewhere [14], a hypere-
lastic formulation using strain rather than a hypoelastic
formulation using stress was preferred for consistency and
accuracy in situations where shear strains are applied at
different compressions. Thus the stress deviator σ was
calculated from the instantaneous strain,
σ = 2Gǫ, (6)
where G(ρ, T ) is the shear modulus. Plastic flow was
taken to occur using a von Mises yield surface [17]. De-
formation was plastic rather than elastic if the scalar ef-
fective shear stress
fσ||σ2|| (7)
exceeded the yield stress Y (ρ, e, ǫ̃p), in which case plastic
strain for work hardening was accumulated at a rate
˙̃ǫp =
||ǫ̇ǫ||+ ||ǫǫ̇||
, (8)
where ǫ̇ is the deviatoric part of the symmetric part of
the velocity gradient,
ǫ̇ ≡ Ė −
Tr ĖI : Ė ≡
(U + UT ), U ≡ grad~u.
If σ̃ < Y , the elastic deformation was simply ǫ̇. For
uniaxial compression along the x-direction, the only non-
zero component of U is [U ]100.
If plastic flow occurs, then the material is always
heated to some degree. Plastic flow occurs through the
motion and generation of defects in the crystal lattice,
such as dislocations. Usually in polycrystalline materials,
defects accumulate during plastic deformation. Heating
generally represents less than the total plastic work as
some potential energy is absorbed in the structure of de-
fects. The fraction of plastic work converted to heat fp is
thought to be 0.85-0.95. It was assumed here to be 0.9.
Thus the contribution of plastic work to heating was
ėp = fp
||σė||
if σ̃ > Y and zero otherwise.
When a metal is deformed, shear strains result in the
accumulation of elastic energy until the flow stress is
reached. Continued deformation results in plastic work.
If the material work-hardens, the rate of plastic working
increases. If thermal softening occurs, the rate of plastic
working decreases but the stored elastic energy is also
converted to plastic work. In an idealized material ex-
hibiting a constant flow stress (elastic-perfectly plastic),
arbitrarily large amounts of plastic work may be accu-
mulated by large deformations – uniaxial as well as pure
shear – beyond the flow stress. Ceramics may behave dif-
ferently, the flow stress decreasing to a small fraction of
its initial value as deformation continues beyond the elas-
tic limit [9], presumably as interatomic bonds are broken
and brittle damage occurs. Many transparent materials
are ceramic; this reduction of the flow stress may explain
why good agreement has been obtained between shock
temperatures and predictions neglecting heating from the
constitutive response [6].
Plastic flow is largely irreversible. If a sample of mate-
rial is shock loaded and then released [24], the pressure
reduces on release but further plastic work is done.
Mo was represented by an empirical EOS fitted to
shock compression data [18, 19, 20], with a deviatoric
strength model developed and calibrated to data on the
amplitude and shape of elastic waves running ahead of
shocks [18]. The EOS was likely to be accurate to a
few percent in temperature for the shock pressures of a
few tens of gigapascals considered here. The Steinberg-
Guinan strength model includes a prediction of the flow
stress at elevated pressures. The flow stress, and hence
the heating from plastic flow, was uncertain at the level of
a few tens of percent. As discussed below, measurements
of surface velocity provided an independent measure of
the flow stress.
Material models for continuum dynamics are often im-
plemented in varying ways in different computer pro-
grams. The results may depend on details such as the
way in which numerical limits, e.g. on flow stress, are
applied. In our simulations, the EOS was represented by
an expression for pressure p in terms of mass density ρ
and specific internal energy e. This is sufficient to allow
the dynamical equations for the continuum to be inte-
grated in time. Two different EOS were used, a tabular
form from the ‘SESAME’ library [19], and an analytic
form of the Grüneisen type, using the principal Hugoniot
as the reference curve [18]: shock speed us in terms of
particle speed up,
us = c0 + s1up, (11)
together with a relation for the Grüneisen parameter
γ(ρ) = γ0 + b(ρ/ρ0 − 1). (12)
The shear modulus G and flow stress Y followed the
Steinberg-Guinan model [21], which includes explicit de-
pendence on temperature T and accumulated plastic
strain ǫp:
G(p, T ) = G0
1 +Ap(ρ/ρ0)
−1/3 −B(T − T0)
Y (p, T ) = Y0f(ǫp)G(p, T )/G0 (14)
f(ǫp) = min [(1 + βǫ̃p)
n, Ymax/Y0] . (15)
TABLE I: Grüneisen equation of state and Steinberg-Guinan
strength parameters for Mo.
equation of state strength
ρ0 10.2 g/cm
G0 125 GPa
c0 5.143 km/s Y0 1.6 GPa
s1 1.255 A 1.14 × 10
−2 GPa−1
γ0 1.59 B 1.52 × 10
−4 K−1
b 0.30 β 20
cp 2.43 × 10
−4 MJ/kg.K n 0.15
a 1.3 Ymax 2.8 GPa
Source: [18] with unit conversions.
Because of the scaling of flow stress by shear modulus,
the maximum flow stress at high pressures was not lim-
ited by the ‘maximum’ flow stress Ymax – this allowed
the flow stress to be significantly greater than Ymax in
the Mo impact experiments. The usual definition of the
Steinberg-Guinan model [18] includes an explicit initial
plastic strain from manufacture; we treated ǫp as a local
material parameter in addition to ρ, e, and the elastic
strain, and set ǫ̃p to a non-zero value in the initial condi-
tions if required. The factors fǫ and fσ used in calculat-
ing the scalar effective magnitudes of the corresponding
tensors were chosen for consistency with the definitions
of stress and strain used in deducing strength parameters
for Mo from experiments: fǫ = fσ = 3/2.
The SESAME EOS were defined as a pair of tables
{p, e}(ρ, T ), so the p(ρ, e) relation was obtained by nu-
merical inversion and the temperature was readily calcu-
lated. Temperatures were calculated from the Grüneisen
EOS with reference to a compression curve along which
the temperature and specific internal energy were known,
{Tr, er}(ρ), and using a specific heat capacity defined as a
function of density cv(ρ) (constant in practice). The ref-
erence curve chosen was the zero kelvin isotherm (‘cold
curve’), Tr = 0K. This was calculated from the principal
isentrope e(ρ)|s0 using the estimated density variation of
Grüneisen parameter:
er(ρ) = e(ρ)|s0 − T0cpe
a(1−ρ0/ρ)
)γ0−a
. (16)
The isentrope was calculated by numerical integration of
the second law of thermodynamics,
= −p(1/v, e). (17)
Mechanical properties and temperatures calculated by ei-
ther EOS gave the same result to o(1%), which consti-
tutes good agreement for models in material dynamics.
The Grüneisen EOS have slightly smoother loci, so the
results presented below are from this EOS.
Simulations were performed in units of millimeters,
gigapascals, microseconds, kelvin, and Mg/m3=g/cm3.
Parameters for Mo in these units are listed in Table I.
projectile
thermal
radiation
collecting lens;
to spectrometer
reflected laser light
for Doppler velocimetryro
window
sample
FIG. 1: Schematic of impact-induced shock experiments with
surface temperature measurements. Aspect ratios are repre-
sentative of the experiments discussed here. If the window
is omitted, the experiment measures the free surface (zero
normal stress) temperature.
IV. PYROMETRY EXPERIMENTS ON
MOLYBDENUM
Pyrometry measurements of the temperature in
shocked and released Mo have been made using two types
of experiment. In both cases, the shock was induced by
the impact of a flat projectile. The projectiles were ac-
celerated using a high explosive launcher, as in the NRS
experiments, and by a gas gun. The pyrometry measure-
ment was performed at the surface opposite the impact,
the shocked state releasing either to vacuum or into a
LiF window to sustain an elevated pressure (Figs 1 and
2). In each case, the shock state was calculated using the
published EOS and strength properties for the projectile
and Mo target.
In all cases, the impact conditions were calculated us-
ing the scalar solution, and were repeated with and with-
out strength in all components of the impact experiment.
For experiments with a LiF window, the temperature in
the window was also predicted; a high temperature would
signal an increased possibility of thermal radiation from
the window obscuring the emissions from the Mo sam-
ple. Where an uncertainty in impact velocity was re-
ported, the calculations were repeated for velocities at
the extremes of uncertainty, giving an estimate of the
uncertainty in pressure and temperature.
Similar calculations were performed with and without
strength in each component separately. The Steinberg-
Guinan model is least appropriate for LiF, so this is the
only component where it would be useful to make such
additional comparisons. However, the effect on states
releasing into LiF were dominated by the strength of the
Mo, so the additional comparisons are omitted for clarity.
An indication that the contribution of strength in the LiF
is a small effect in the simulations is that the predicted
shock temperature in LiF varied much less as a function
of strength than did the temperature of any of the Mo
states.
Taking strength into account, on release into LiF, the
 0  0.5  1  1.5  2  2.5  3  3.5  4
particle speed (km/s)
Al-6061
projectile
impact shock
elastic release
window release
free surface release
FIG. 2: Shock and release states induced in impact experi-
ments with and without a window. Solid lines: shock Hugo-
niots; dashed line: release adiabat. Example calculation for
Al-6061 projectile traveling at 3.6 km/s, impacting stationary
Mo target, releasing into LiF window or into vacuum. The
initial states of the Mo and LiF are at the origin; the initial
state of the projectile is at zero normal stress and 3.6 km/s.
On impact, the shock states in the projectile and sample are
at the elevated pressure intersection marked ‘impact shock.’
When the shocked state in the Mo releases into the LiF win-
dow, the resulting state is the intersection marked ‘window
release.’ When the shocked state in the Mo is released at a
free surface, the resulting state is the zero normal stress state
marked ‘free surface release.’ Release from the shocked state
shows an inflexion when plastic flow occurs.
normal stress in the Mo was lower than the in-plane stress
because the elastic strain is a distension in the axial di-
rection. For this reason, the calculations with strength
have a lower normal stress: a result which may be coun-
terintuitive.
Various improvements could be made in future pyrom-
etry measurements to reduce the temperature uncertain-
ties. Some optimization could be performed by repeating
experiments multiple times, adjusting detector gains and
digitization ranges for best accuracy. However, the diffi-
culty and cost of each projectile impact experiment can
make multiple repeats impractical.
A. Gas gun
The projectile was Ta, 3mm thick, accelerated to
1.70 km/s using a two stage gas gun. The target was
Mo, 5mm thick. Thermal emission was measured on re-
lease into a LiF window, using a 7 channel pyrometer.
The measured release temperature was 683± 41K.
The shocked state in the Mo was calculated to be
58.7GPa and 645K, of which 51K was from plastic work.
The state on release into LiF was thus calculated to be
24.8GPa and 614K, of which 82K was from plastic work.
 200  300  400  500  600  700  800
temperature (K)
Hugoniot
release
adiabat
elastic
LiF release
solid: Steinberg-Guinan
dashed: no strength
FIG. 3: Temperature measurement from a shock of 59GPa,
on release to 25GPa into LiF, compared with predictions
based on the Steinberg-Guinan strength model and with
strength neglected. The crosses on the release adiabats show
where release pauses when a shock is transmitted into the
LiF. When strength is included, the first portion of release
is elastic; the elastic portion of the adiabat is marked; flow
becomes plastic below the inflexion.
The measured surface temperature was just 1.5 standard
deviations above the temperature predicted using the
Steinberg-Guinan strength model, and more than three
standard deviations above the temperature predicted ig-
noring material strength. (Table II and Fig. 3.)
B. Forest Flyer
The high explosive launcher used the Forest Flyer de-
sign [22]. With this system, the projectile was slightly
dished on impact, though this should not affect the py-
rometry measurement significantly. The projectile was
accelerating slightly, reverberating elastically from the
acceleration process, and possibly had a porous region
through its thickness as a result of tensile stresses dur-
ing acceleration. The relatively strong reverberations in
the projectile affect its effective speed on impact, and
contributed to the uncertainty in material states.
The projectile was Al-6061 alloy, 6mm thick, acceler-
ated to 3.6 ± 0.1 km/s. The target was Mo, also 6mm
thick. Six experiments were performed, four for release
into a LiF window and two into vacuum. Surface emis-
sion was measured with a 5 channel visible-near infrared
pyrometer or a 4 channel near infrared pyrometer. The
free surface temperature had a relatively large uncer-
tainty, and the signals on release into LiF showed ev-
idence of thermal emission from the LiF itself with a
temperature of around 580K [13]. The measured release
temperature was 762 ± 40K into LiF, and 568 ± 100K
from the free surface.
 200  300  400  500  600  700  800  900
temperature (K)
Hugoniot
release
adiabat
LiF release
free surface
solid: Steinberg-Guinan
dashed: no strength
FIG. 4: Temperature measurements from a shock of 64GPa,
on release to 27GPa into LiF and to zero pressure, compared
with predictions based on the Steinberg-Guinan strength
model and with strength neglected. The release adiabat from
the mean shock pressure is shown, along with adiabats re-
flecting the uncertainty in shock pressure. The lines across
the release adiabats show where release pauses when a shock
is transmitted into the LiF.
The shocked state in the Mo was calculated to be 63.9±
2.4GPa and 707±31K, of which 53±3K was from plastic
work. The state on release into LiF was thus calculated
to be 27.1 ± 1GPa and 670 ± 25K, of which 89 ± 1K
was from plastic work. The state on release into vacuum
was calculated to be 635± 23K, of which 126± 4K was
from plastic work. The uncertainties are correlated: the
smallest, mean, and largest of each go together.
The surface temperature on release into LiF was 1.5-2.5
standard deviations of the temperature predicted using
the Steinberg-Guinan strength model, and 3.5-4.5 stan-
dard deviations from the temperature predicted with-
out strength. The uncertainty in the free surface re-
lease temperature was too large to discriminate between
a purely hydrodynamic calculation (no strength) and the
Steinberg-Guinan model – both lay within one standard
deviation of the measurement. The predicted temper-
ature of the LiF itself also matched the measurement
to within the experimental uncertainties. (Table II and
Fig. 4.)
The velocity history of the surface of the sample was
measured by laser Doppler velocimetry of the ‘VISAR’
type [23]. General features of the velocity history in-
cluded a rapid acceleration when the shock reached the
surface, a roughly constant peak velocity corresponding
to the sustained pressure behind the shock, deceleration
caused by the release wave from the rear of the projectile,
and a slight re-acceleration as the sample was subjected
to tensile stress causing spall type damage (Fig. 5). The
epoch of peak velocity was not perfectly constant, but
showed some acceleration. This was probably caused by
the compression gradient in the projectile from the resid-
 0  0.5  1  1.5  2  2.5  3  3.5  4
time after impact (µs)
shock
release
FIG. 5: Surface velocity histories measured in Forest Flyer
impact experiments with and without a LiF window. Separate
lines are from different experiments. The upper two traces are
from free surface release; the lower three are from release into
a LiF window.
ual accelerating pressure at impact, and any regions of
porosity resulting from tensile damage as the projectile
was accelerated by the relatively strong pressures induced
by the detonating high explosive. The onset of release
showed a clear elastic precursor (Fig. 6).
The measured velocity histories were compared with
spatially-resolved one-dimensional continuum dynamics
simulations. The projectile was modeled as ideal, i.e.
at uniform STP conditions and traveling at a constant
3.6 km/s with no reverberations. As a result, the peak ve-
locity epoch was flatter than measured, but was in good
agreement for amplitude and duration. Release into the
LiF was also reproduced well overall. The shape of the
elastic precursor to release was not reproduced perfectly
using the Steinberg-Guinan strength model, but its am-
plitude was reproduced to within around 10% and the
time of arrival was in good agreement with the exper-
iment. The difference in shape could be caused by in-
adequacy in the Steinberg-Guinan model – for example,
in the detailed work-hardening history in the shocked
state – but is more likely to reflect density variations in
the projectile as discussed above. The uniaxial strains
greatly exceeded the elastic limit on release as well as on
compression, so the plastic work should be dominated by
the flow stress rather than the precise path before plastic
flow occurred. Thus the agreement between calculated
and observed amplitudes suggests that the plastic work
should be correct to around 10%.
Spallation did not affect the shock and release states
of interest for the temperature measurements considered
here. The simulations used a crude spall model of the
minimum pressure type, with a minimum pressure of -
1.5GPa [18], meaning that the maximum tensile stress
induced by the Mo as it was distended was 1.5GPa.
 0  0.5  1  1.5  2  2.5  3  3.5  4
time after impact (µs)
experiment
experiment
strength
strength
FIG. 6: Surface velocity history in Forest Flyer impact ex-
periment with a free surface (upper traces) and a LiF window
(lower traces). Each experimental records is compared with
two continuum dynamics simulations, with (solid lines) and
without strength (dashed lines) in the Mo. The elastic precur-
sor to the release wave is evident where the experimental ve-
locity histories deviate from the simulations without strength.
No treatment of accumulating porosity was included, so
the Mo as simulated continued to exert a tensile stress
when in reality voids or cracks would open, reducing the
stress. Thus the simulations of velocity history did not
show a re-acceleration after the deceleration associated
with the release wave. Tensile damage and spall can de-
pend strongly on the strain rate and loading history. The
simulated and observed release deceleration matched to
within a few percent, suggesting that the published spall
strength applies well to the loading history induced by
these projectile impact experiments.
V. CONCLUSIONS
Shock and release temperatures were calculated self-
consistently using the equation of state and a published
constitutive model for Mo. Strength was calculated to
make a significant difference to states in experiments ex-
ploring pressures of tens of gigapascals. The high pres-
sure flow stress predicted using the Steinberg-Guinan
strength model matched the elastic release precursor ob-
served using surface Doppler velocimetry, suggesting that
the flow stress was correct to around 10%. The predicted
temperatures were consistent with pyrometry measure-
ments for shocks of around 60GPa, releasing into a LiF
window or into vacuum. The LiF release temperatures
were clearly more consistent with plastic work as pre-
dicted using the Steinberg-Guinan model than with hy-
drodynamic flow (no strength). The uncertainties in tem-
perature were however too large to discriminate between
strength models to better than several tens of percent in
TABLE II: Shock and release states.
no strength strength measured
particle speed normal stress temperature particle speed normal stress temperature temperature
(km/s) (GPa) (K) (km/s) (GPa) (K) (K)
gas gun
Mo shock 0.905 57.9 594 0.902 58.7 645
Mo release into LiF 1.374 25.4 532 1.337 24.8 614 683 ± 41
LiF shock 1.374 25.4 535 1.337 24.8 532
Forest Flyer
Mo shock 0.97 ± 0.03 63.3 ± 2.4 654± 28 0.97± 0.03 63.9 ± 2.4 707± 31
Mo release into vacuum 1.95 ± 0.07 0 509± 19 1.91± 0.07 0 635± 23 566± 100
Mo release into LiF 1.48 ± 0.04 27.8 ± 1 581± 24 1.44± 0.04 27.1± 1 670± 25 762 ± 40
LiF shock 1.48 ± 0.04 27.8 ± 1 570± 17 1.44± 0.04 27.1± 1 566± 16 624± 100
flow stress.
Heating from plastic work was calculated to be around
50K for shock pressures around 60GPa, 90K on sub-
sequent release into LiF, and 125K on release at a free
surface. The fraction of plastic work converted to heat
was assumed to be 90% – the heating would have been
about 10% greater if all the plastic work appeared as
heat. Taking plastic flow into account, there was no dis-
crepancy between predictions and measured release tem-
peratures for Mo. This is a validation of the models of
EOS and strength for Mo, and of the use of pyrometry
to measure release temperatures in metals – though the
pyrometry measurements obtained in these experiments
were not precise enough to discriminate between models
calibrated against similar mechanical data such as veloc-
ity histories. The fraction of plastic work converted to
heat was most likely close to 100%, though the uncer-
tainty in the temperature measurements means that this
figure cannot be justified statistically to better than a
few tens of percent.
Plastic flow makes a significant contribution to recon-
ciling the temperature discrepancy observed in the neu-
tron resonance spectrometry experiments on shocked Mo,
although the complete explanation is more complicated
and will be reported separately.
Acknowledgments
We would like to acknowledge the contribution of Carl
Greeff for assistance and advice on equations of state
and their uncertainties or certainties for Mo, of Ron Ra-
bie, David Funk, Rob Hixson, and Chuck Forest for de-
tailed information on the design and testing of the Forest
Flyer loading system, and of Sheng-Nian Luo for gen-
eral advice and comments on pyrometry and material
dynamics. The gas gun experiments were performed by
D.B. Holtkamp, P. Paulsen, P. Fiske, D. DeVore, J. Gar-
cia, and L. Tabaka at Lawrence Livermore National Lab-
oratory in 1999. The work was performed under the aus-
pices of the U.S. Department of Energy under contracts
W-7405-ENG-36 and DE-AC52-06NA25396.
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[24] In purely hydrodynamic analyses, where the effect of
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The term ‘quasi-isentropic’ is sometimes used in this con-
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more specific term.
http://arxiv.org/abs/cond-mat/0702693
