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 amplitude 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 aligned with the axial direction parallel to the loading. The target is designed such that on initial loading, the outer cylinder 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 field velocity interferometry (ORVIS) will also be discussed

Brown, J. L.

Ravichandran, G.

English

Justin Brown

Guruswami Ravichandran

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High pressure hugoniot measurements using mach waves

Caltech Authors - Main

High pressure hugoniot measurements using mach waves
Justin Brown and Guruswami Ravichandran 
 
Citation: AIP Conf. Proc. 1426, 485 (2012); doi: 10.1063/1.3686323 
View online: http://dx.doi.org/10.1063/1.3686323 
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HIGH PRESSURE HUGONIOT MEASUREMENTS  
USING MACH WAVES 
 
J.L. Brown and G. Ravichandran 
 
Division of Engineering and Applied Science, California Institute 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 amplitude 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 aligned with the axial direction parallel to the loading. The target is 
designed such that on initial loading, the outer cylinder 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 field 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, the 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 - 2011AIP Conf. Proc. 1426, 485-488 (2012); doi: 10.1063/1.3686323©   2012 American Institute of Physics 978-0-7354-1006-0/$0.00485
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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 inner cylinder. Under this 
condition, converging shocks 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 standard 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
Us
P (GPa)
10-1
Precursors
Incident Shock
Reflected Shock
Mach Stem
100
101
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 traces 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 thought 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 higher 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 velocities 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. 
REFERENCES 
1. Fowles, G. R. and Isbell, W. M., "Method for 
Hugoniot Equation-of-State Measurements at 
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1377-1379, 1964. 
2. Adadurov, G. A., et al., "Mach reflection 
parameters for plexiglass cylinders," ZPMTF, vol. 
10, pp. 126-128, 1969. 
3. Syono, Y., Goto, T., and Sato, T., "Pressure 
enhancement by conically convergent shocks in 
composite cylinders and its application to shock 
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7131-7135, 1982. 
4. Brown, J. L., et al., "High pressure Hugoniot 
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5. Brown, J. L., "High Pressure Hugoniot 
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7. Menikoff, R. and Plohr, B., "The Riemann problem 
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8. Barker, L. M. and Hollenbach, R. E., "Laser 
inteferometer for measuring high velocities of any 
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9. Hornung, H., "Deriving Features of Reacting 
Hypersonic Flow from Gradients at a Curved 
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10. Rice, M. H., et al., "Compression of Solids by 
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11. Marsh, S. P., LASL Shock Hugoniot Data: 
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12. Bloomquist, D. D. and Sheffield, S. A., "ORVIS, 
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System, Theory of Operation and Data Reduction 
Techniques," Sandia Report, SAND82-2918, 1982. 488
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High pressure Hugoniot measurements using Mach waves

Compression of Solids by Strong Shock Waves,&quot;

CTH: A Software Family for Multi-Dimensional Shock Physics Analysis,&quot;

Deriving Features of Reacting Hypersonic Flow from Gradients at a Curved Shock,&quot;

High Pressure Hugoniot Measurements in Solids Using Mach Reflections,&quot;

High pressure Hugoniot measurements using converging shocks,&quot;

Laser inteferometer for measuring high velocities of any reflecting surface,&quot;

LASL Shock Hugoniot Data:

Mach reflection parameters for plexiglass cylinders,&quot;

Method for Hugoniot Equation-of-State Measurements at Extreme Pressures,&quot;

ORVIS, Optically Recording Velocity Interferometer System, Theory of Operation and Data Reduction Techniques,&quot;

Pressure enhancement by conically convergent shocks in composite cylinders and its application to shock recovery experiments,&quot;

The Riemann problem for fluid flow of real materials,&quot;

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High pressure Hugoniot measurements using Mach waves

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