High pressure hugoniot measurements using mach waves
Justin Brown and Guruswami Ravichandran
Citation: AIP Conf. Proc. 1426, 485 (2012); doi: 10.1063/1.3686323
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HIGH PRESSURE HUGONIOT MEASUREMENTS
USING MACH WAVES
J.L. Brown and G. Ravichandran
Division of Engineer
ing and Applied Science, California Institu
te of Technology,
Pasadena CA 91125
Abstract.
Traditionally, most dynamic shock compression experiments are conducted using a plane
one-dimensional wave of uniaxial strain. In this case, the evaluation of the equation of state is
simplified due to the geometry, but the amplitud
e of the induced shock wave is limited by the
magnitude of the input load. In an effort to dramatically increase the range of pressures that can be
accessed by traditional loading methods, a composite target assembly is examined. The target consists
of two concentric cylinders aligne
d with the axial direction parallel to the loading. The target is
designed such that on initial loading, the outer cylind
er will have a higher shock velocity than the inner
material of interest. Conically converging shocks will be generated at the interface between the two
materials due to the impedance mismatch. Upon convergence, an irregular reflection occurs and the
conical analog of a Mach reflection develops. The Mach reflection will grow until it reaches a steady
state, at which point the wave configuration becomes self similar. The resulting high pressure
Hugoniot state can then be measured using velocity interferometry and impedance matching. The
technique is demonstrated using a planar mechanical
impact generated by a powder gun to study the
shock response of copper. Two systems are examined which utilize either a low impedance (6061-T6
aluminum) or a high impedance (molybdenum) outer cylinder. A multipoint VISAR experiment will
be presented to validate the technique, and will be compared to numerical simulations. The feasibility
of measuring an entire Hugoniot curve using full fiel
d velocity interferometry (ORVIS) will also be
discussed.
Keywords:
Hugoniot, Mach reflection, converging shocks.
PACS:
47.40.Nm, 07.35.+k, 62.50.-p.
INTRODUCTION
Generally, Hugoniot measurements are made
using well-controlled one-dimensional shocks.
Plate impact experiments, for example, provide
avenues for extremely accurate equation of state
(EOS) measurements. The maximum achievable
impact velocity, however, limits the magnitude of
the induced shock. In an effort to increase the
pressures accessible with a given impact velocity,
converging shock waves are generated through a
composite target consisting of two concentric
cylinders. This configuration was originally
examined for a solid material confined by a high
explosive [1, 2]. Later, th
e method was extended to
mechanical impact testing, where the outer
explosive was replaced by a solid material [3] and
shocks are generated by a standard plate impact.
Recently, this target assembly has been shown to
produce a high-pressure planar shock at the center
of the inner cylinder for which Hugoniot
measurements can be made [4].
Shock Compression of Condensed Matter - 2011
AIP Conf. Proc. 1426, 485-488 (2012); doi: 10.1063/1.3686323
© 2012 American Institute of Physics 978-0-7354-1006-0/$0.00
485
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EXPERIMENTAL SETUP
The Mach lens target configuration is shown
in Fig.1. A normal plate impact generates a plane
shock on one surface of the composite target
assembly. The target materials are selected such
that the shock speed in the outer cylinder is higher
than that in the inne
r cylinder. Under this
condition, converging shoc
ks are generated at the
material interface due to the impedance mismatch.
Upon convergence, the geometry forces an
irregular reflection, and the conical analog of a
Mach reflection occurs. After some transient build
up, the Mach reflection becomes steady in time and
the Mach reflection approaches a self-similar
solution. The axisymmetric
nature of the target
results in a normal Mach stem at the center of the
inner cylinder. Thus, in the steady state, the
configuration essentially produces a plane shock
traveling at the shock speed of the outer cylinder,
and a single measurement of the particle velocity
results in an estimate of the shocked state behind
the Mach stem.
Figure 1
. Mach lens target assembly. A plane shock is
generated at the left of the target with a normal plate
impact.
NUMERICAL SIMULATIONS
Numerical simulations, the details of which can be
found elsewhere [4, 5], were performed with the
CTH hyrdrocode [6] to properly design the
experiments. The results of a typical simulation are
shown in Fig. 2. In this simulation a thick
aluminum flyer impacts an aluminum outer
cylinder and copper inner cylinder 6.4 mm in
diameter at 1.6 km/s. Fig. 2(a) – (d) illustrate the
initial converging shocks in the inner cylinder,
iirregular reflection, growth of the Mach wave, and
finally the steady state Mach reflection. A more
quantitative view of the simulation is taken in Fig.
2(e) where equally spaced Lagrangian tracers along
the center of the inner
cylinder illustrate the
behavior of the interacting shocks. At early times
the axial particle velocity trace represents what
would be typical of a stan
dard plate impact. At ~1
s, the converging shocks arrive at the centerline,
and the particle velocity begins to increase. The
transient build up of the Mach reflection is
captured until at ~4.5
s, the wave profile becomes
steady in time. As shown, the corresponding
length-to-diameter ratio (L/D) associated with this
point in time is ~4. This provides the effective
design criteria for the experiment. Thus, in this
case, the length of the assembly must be at least 4
times larger than the inner cylinder diameter while
the outer cylinder diameter must be large enough
such that any edge effects can be neglected.
6061-T6 Al
OFHC Cu
U
s
P (GPa)
10
-1
Precursors
Incident Shock
Reflected Shock
Mach Stem
10
0
10
1
a) t = 0.75
s
b) t = 1.25
s
c) t = 1.75
s
d) t = 3.00
s
L/D = 3
4
5
e)
Figure 2
. Pressure contours for a typical simulation at
various stages of the Mach reflection (time from impact)
are shown in (a)–(d). Equally spaced axial velocity traces
along the center of the target are shown in (e).
It should be noted that simple analytic methods can
be used to determine the nature of the steady state
Mach reflection. The methods are based on the
shock polar framework commonly used in gas
dynamics but generalized for an arbitrary EOS [7].
486
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These shock polar solutions can be used to provide
useful physical insights and have been shown to be
in excellent agreement with the numerical
simulations [4, 5].
EXPERIMENTAL RESULTS AND
DISCUSSION
In an attempt to validate the experimental
method, an experiment which utilized multiple
VISAR [8] measurements will be discussed in
detail. In this experiment, an aluminum flyer 13
mm thick and 89 mm in diameter impacted an
aluminum outer cylinder and copper inner cylinder
at 1.59 km/s. The outer diameters of the inner and
outer cylinders were 76 mm and 6.4 mm,
respectively. The length of the target was 22 mm.
As shown in the insert in Fig. 3, probes were used
to monitor the rear free surface of the target at radii
of 0, 1.35mm, 2.69 mm, and 7.21 mm.
Interface
Free Surface
0
7.21
2.69
1.35
Figure 3
. Experimental free surface velocities measured
in an aluminum / copper Mach lens. The tr
aces are color
coded to the probe locations shown in the inset. The
dashed gray traces are the corresponding numerical
profiles.
Results of the experiment, along with the
corresponding simulated waveforms, are shown in
Fig. 3. As shown, the experiment agrees well with
the expected Mach wave profile shown in the inset.
The outer most probe corresponds to the plane
wave corresponding to the shock in the outer
cylinder. The discrepancy between the simulation
and experiment is though
t to be a result of the
impactor tilt which was ~2 mrad. The probe near
the cylinder’s interface, at 2.69 mm, illustrates the
two-shock structure corresponding to the incident
shock followed by the reflected shock. The
gradient in the measured particle velocity between
these two shocks is thought to be a result of the
curvature of the wave front [9]. The probe at 1.35
mm represents a point close to half the radius of
the inner cylinder and exhibits a similar double
shock structure. The hi
gher initial particle
illustrates the increase in
pressure due to the
gradient along the Mach wave while the shorter
spacing between the two shocks is simply a
function of the geometry of the reflection. Finally,
at the center of the inner cylinder, the Mach stem is
monitored where the single shock exhibits the
dramatic pressure increase in this portion of the
Mach reflection.
As alluded to previously, a shocked state can be
calculated easily for the profile measured at the
center of the target. Given the measured impact
velocity, impedance matching [10] can be used to
determine the shocked state in the outer cylinder as
long as the impactor and outer cylinder material
Hugoniots are well known. Thus, assuming the
Mach wave has reached a steady state, the axial
wave velocity of the entire Mach reflection is
known. Making a free surface approximation, the
axial particle velocity can be estimated to be half of
the measured free surface velocity. The
conservation equations can
then be used to
calculate the rest of the mechanical variables
associated with the shocked state behind the Mach
stem.
Technically, the Mach reflection provides a
continuous pressure gradient between the state at
the interface to the center of the inner cylinder.
Hence, a full field measurement of the particle
velocity distribution along the entire reflection can
theoretically be used to calculate the entire
Hugoniot curve between these two points. As a
rough illustration of the idea, the two profiles
monitoring the two-shock regime can be used to
estimate an average shocked state. Given the radial
position of each measurement and the time of
arrival of the initial shock, the average incident
shock angle can be calculated through the known
axial wave speed for which the average normal
shock and particle velocitie
s can then be calculated
[4]. The calculated shocked states are shown in
Fig. 4 along with the copper Hugoniot measured
using conventional methods [11]. As shown, the
487
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measured points are in good agreement with the
literature. Further, the high pressure point
represents a gain in pressure of approximately 4.4
when compared to the equivalent plate impact
experiment.
Figure 4
. Shocked states calculated from the
experimental profiles shown in Fig. 3 along with the
copper Hugoniot from [11].
CONCLUSIONS
In the present study, the feasibility of using a
simple composite target to generate a steady Mach
reflection is examined. Numerical simulations are
used to gain insights into the problem and design
the experiments. The experiment presented here
illustrates the ability to calculate a shocked state
using a single measurement of the free surface
velocity at the center of the cylinder. In this case,
the Mach stem is monitored for which an
extremely high-pressure state exists. Further
experiments have shown repeatability [4, 5] and
suggest this target configuration may be used to
greatly extend the capabilities of existing shock
compression techniques. The experiment also
demonstrates the ability to calculate multiple
Hugoniot points in the same experiment. Given
quality data with both a high temporal and spatial
resolution, such as that provided by ORVIS [12], it
should be possible to calculate a significant portion
of the Hugoniot in a single experiment.
ACKNOWLEDGEMENTS
The research support provided by the Caltech
Center for the Predictive Modeling and Simulation
of High-Energy Density Dynamic Response of
Materials through the U.S. Department of Energy
National Nuclear Security Administration under
Award Number DE-FC52-08NA28613 is
gratefully acknowledged.
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