CP620, Shock Compression of Condensed Matter - 2001
edited by M. D. Furnish, N. N. Thadhani, and Y. Horie
© 2002 American Institute of Physics 0-7354-0068-7/02/$ 19.00
CORRECTING FREE SURFACE EFFECTS BY INTEGRATING THE
EQUATIONS OF MOTION BACKWARD IN SPACE*
Dennis Hayes and Clint Hall
Sandia National Laboratories, Albuquerque NM 87185-1181 USA
Abstract. Free surface and window interfaces perturb the flow in compression wave experiments. The velocity of these interfaces is routinely measured in shock-compression experiments using interferometry (i.e.,
VISAR) and the perturbations must be accounted for before meaningful material property results can be obtained. Using the VISAR results as "initial conditions" we integrate the flow fields backward in space to the
interior of the specimen where the VISAR interface has not perturbed the flow at earlier times and results can
be interpreted as if the interface had not been present. This provides a rather exact correction for free surface
perturbations. The method can also be applied to window interfaces by selecting the appropriate initial conditions. Applications include interpreting Z-accelerator ramp wave experiments. The method can be applied
to elastic-plastic and quasi-elastic materials for experiments with multiple layers and multiple reverberations.
BACKGROUND AND EXAMPLE
The backward integration technique (Backward)
was motivated by the need to analyze experimental
results from the Sandia Z-accelerator. These experiments (1) measure free-surface velocity for two or
more specimen thicknesses using VISAR(2) interferometry. The desire is to interpret these measurements by assuming the particle velocity characteristics are straight lines in space-time and thus infer the
Lagrangian wave-speed as a function of particle velocity. This is sufficient to determine the stress-strain
behavior of the specimen material through Reimann
invariants. The central problem is as follows: as
early parts of the ramp wave arrive at the free surface, they reflect and interact with the later parts of
the oncoming ramp wave. This interaction bends the
later oncoming characteristics negating the assumption of straight-line-characteristic behavior required
for conventional analysis techniques. Since ramp
waves steepen with propagation distance, perturbations are different for each specimen thickness.
For isentropic compression waves, one might erroneously expect that one-half the free surface velocity would exactly equal the in situ particle velocity (the velocity at that same location if the specimen were thick and no free surface perturbations oc-
200
400
600
time - ns
FIGURE 1. The pressure load applied to the front surface of a
0.8 mm copper specimen.
200
400
600
time - ns
FIGURE 2. The particle velocity at depth 0.8 mm in a thick plate
is compared with one-half the free-surface velocity of a 0.8 mmthick plate. Experimental and calculated free-surface results agree
almost exactly.
* This work supported by the US Department of Energy.
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although the inferred loading profile at the 0 mm location is not the one actually experienced by the copper
whenever the magnetic field has penetrated the copper during loading. However, the important results
obtained by Backward are unaffected by this uncertainty.
When a Backward-determined stress history at
some interior location is used in a subsequent forward calculation, the starting simulated VISAR history is always replicated, even if the material model
is incorrect, provided the integrations were done accurately. Because the same equations are being integrated back in space and then forward in time, the
backward/forward procedure is just a mathematical
mapping of the VISAR record onto itself. That is
why another constraint must be found to make use of
this method. For Z experiments, this constraint is usually that two specimen thicknesses must both infer the
same load. This constrains the specimen stress-strain
relation. For the spall experiment that is given as
another example, stress after spallation must remain
zero at the spall plane, etc.
At present, the method cannot be used if strong
shocks are present anywhere in the flow. Since shocks
produce entropy, which destroys information, there
are simply regions of the x-t plane that are inaccessible by information obtained at the measurement surface. As a practical matter, if a shock is present in
the VISAR record, it immediately begins to spread
into an isentropic compression wave as the integration proceeds back into the interior of the specimen.
Thus for large amplitude shocks, significant entropy
generation will be ignored by Backward. But for low
amplitude shocks like the 20 kbar shock present in the
spall experiment, the entropy generated by the shock
is completely negligible and apparently, as evidenced
by the good comparison of the starting VISAR record
with the subsequent forward calculation, solutions are
accurate enough for our purposes. Thus if we use the
method for experiments with weak shocks, a case by
case evaluation must be made to assess if errors introduced are significant.
The general method is also extended to hysteretic elastic-plastic (4) and quasielastic materials. (5) These extensions can change the governing
equations from hyperbolic to parabolic. However in
the spall example below, the growth of expected instabilities is "managed" well enough to get an accurate
solution for the stress history at the spall plane.
curred). That would be true for an isentropic step load
but is not the case for a ramp load as explained above.
For example, Fig. 1 shows a 2Mbar ramp load that is
applied to a 0.8 mm sample of copper. Figure 2 shows
a dramatic difference between u/s/2 and Uinsitu for
the ramp load of Fig. 1. Using the former as an approximation for the latter introduces errors of 10's of
ns in time and more than 10% in velocity for an experimental situation that requires more than an order
of magnitude better accuracy to be useful.
CORRECTING FOR FREE SURFACES
This paper treats the above situation. Actually, in
the above example, we started from the experimental
free-surface velocity history of Fig. 2 and the pressure history of Fig. 1 was deduced by the backward
integration technique described below. This deduced
pressure history was subsequently used as a boundary condition in a traditional forward WONDY(3)
calculation to produce the simulated in situ velocity. In order to validate the accuracy of the Backward procedure, a separate WONDY simulation used
this same pressure boundary condition to calculate
the free-surface velocity. The experimental and calculated free-surface motion agree almost exactly (see
Fig. 2) verifying the accuracy of the entire procedure.
BACKWARD METHOD
The one-dimensional Lagrangian equations of motion,
(1)
(2)
(3)
are numerically integrated from the free surface backward in space to the interior of the sample to a location where the relevant part of the wave has not been
perturbed by the free surface. The "initial" conditions
are the VISAR free surface velocity history and zero
stress.
Integration can be done backward in space to any
convenient location, provided it is far enough back
to avoid perturbations from the free surface at times
of interest. In the two-sample experiment described
in example 1 below, the same stress-strain behavior
would be obtained if the integration were carried back
to say 0.1 mm, Omm as was done in the analysis, or
even to -0.1 mm! The position x=0 is usually chosen
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200
300 T-
COPPER
150
3
- theoretical isentrope
—— Z-516
100
50
400
0.100
800
0.105
specific volume
FIGURE 3. Analysis of Z-516. Two VISAR records from copper
samples of two different thicknesses were obtained during a single
Z experiment. Each free surface record was integrated backward in
space to x = 0. The stress-strain relation was varied systematically
until the two calculated load histories at x — 0 were the same.
They agree to about 1%. The deduced stress-strain agrees with the
known behavior of copper. See Fig. (4).
0.110
3
cm /g
FIGURE 4. The deduced stress-strain behavior of copper for experiment Z-516 compared with the known behavior, our best result
to date.
tionship for the copper specimen. (Figs. 3,4)
2. Starting from two simulated LiF window velocity histories, (8) determine the stress-strain relationship for the specimen and the pressure load originally
applied in the simulations. (Fig. 5)
3. Starting from a free-surface velocity history
from a spall pullback experiment in aluminum, (5) determine the location of the spall plane and the stress
history of the failure process at that location. Also determine parameters for the time-dependent quasielastic model of plasticity. (Fig. 6)
Each of these examples is part of an ongoing experimental study in which Backward is finding application.
WINDOWED VISAR EXPERIMENTS
The method is not restricted to analyzing freesurface experiments. If the measurement interface
has a VISAR window, the initial conditions at the
measurement plane for Backward are chosen along
the time axis as: VISAR velocity, stress in the window at that particle velocity and specific volume in
the specimen. Care must be taken to ensure the window's stress-strain is calibrated for ramps and that its
optical properties (VISAR correction) is measured for
isentropic compression. (6) Furthermore, the VISAR
technique itself can give different results for ramps
than for steady waves.(7) A further complication
arises for windowed experiments in which the stressstrain relation for the specimen is being studied by
using more than one sample thickness described in
example 2 below. Backward must use the "answer" to
initialize the integration. This problem is tractable(6)
but the implicit nature of this problem magnifies the
errors in the deduced stress-strain relation for the
sample.
EXAMPLES
Backward is presently being used to analyze a
variety of experiments at Sandia, Los Alamos and
Lawrence Livermore National Labs. Below are selected examples that show the breadth of application:
1. Starting from two free surface measurements on
the Z accelerator, (6) determine the stress-strain rela1179
CONCLUSION
There are many ways to combine forward and
backward calculations. For instance, if one specimen of a two-specimen Z experiment has a shock,
the shockless result can be integrated backward and
the specimen with a shock integrated forward with
a code like WONDY. By constraining the calculated
and measured shock results to agree, the equation of
state could be determined. Or the free-surface acceleration of a high velocity flyer plate can be used
to calculate the various gradients in the flyer at impact, information needed to quantify the character
of the shock generated at impact. (9) Recent experiments have combined free-surface and window measurements in a unique way to define window optical properties.(6) Backward is an essential part of
this investigation. There is seemingly an endless vari-
FIGURE 5. Pressure load. Simulated VISAR records for two
windowed copper specimens(8) (not shown) were used by Backward to deduce the stress-strain relation used for the simulation to
^1/4% (not shown). Backward also extracts the load (shown) accurately. For this idealized example Backward makes essentially
an exact correction for the window interface perturbations.
ety of ways to put forward and backward calculations
together to tease new results from experiments, provided we can shed our habit of viewing the x-t plane
from only one direction.
ACKNOWLEDGMENTS
The authors are indebted to J. Asay, J. Fritz,
M. Knudson, and R. Menikoff for useful discussions.
Sandia is a operated by Sandia Corporation, a Lockheed Martin Company, for the Department of Energy
under contract DE-AC04-94AL85000.
REFERENCES
1. C, Hall et al., submitted to Rev. Sci. Instr., 2001.
2. L. M. Barker and R. E. Hollenbach, J. Appl. Phys. 43, 4669
(1972).
3. M. E. Kipp and R. J. Lawrence, WONDY V - a onedimensional finite-difference wave propagation code, Technical Report SAND81-0930, Sandia National Laboratories, 1982.
4. D. Hayes, Backward integration of the equations of motion
to correct for free surface perturbations, Technical Report
SAND2001-1440, Sandia National Laboratories, 2001.
5. D. Hayes, J. Vorthman, and J. Fritz, Backward integration of
a VISAR record: Free surface to the spall plane, Technical
Report LA-13830-MS, Los Alamos National Laboratory, 2001.
6. C. Hall et al., to be published.
7. D. Hayes, J. Appl. Phys. 89 (2001).
8. D. Reisman, LLNL, private communication.
9. M. Knudson, SNL, private communication.
10. J. N. Johnson, J. Phys. Chem. Solids 54, 691 (1993).
FIGURE 6. Backward integration of a spall pullback free surface
VISAR record on 6061-T6 aluminum.(5) The VISAR free surface
record (bottom) was integrated backward in space to ?^150% of
the estimated scab thickness. The 3-D graph (top) shows the calculated stress as a function of time and Lagrangian distance from
the free surface. Note the slice marked "spall plane". That is the
plane where the stress stays at zero after spall occurs. Stress at that
plane is shown in the middle graph. Backward sought best values for the two quasielastic parameters and for the position where
the RMS of the late time stress was a minimum. Two quasielastic
parameters deduced for 6061-T6 Al agree well with those determined by Johnson.(10) The position of the deduced spall plane
agrees with experiment. The two-step failure seen in the middle
graph displays secondary spall resistance behavior seen previously
in tantalum. The calculated free-surface velocity (bottom) was obtained with WONDY where the Backward-determined load (middle) was applied to an aluminum layer with thickness equal to the
Backward-determined scab thickness.
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