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
For special copyright notice, see page 1305.
RECOVERY OF URANIUM FRAGMENTS
H. R. James, D. H. McElrue and R. E. Winter
AWE, Aldermaston, Reading, Berks, UK
Abstract. We describe a theory for calculating the penetration of fragments into foam. Comparisons
with regular projectiles show that the drag term is similar in value to the analogous term in
aerodynamics. This, plus the simple model used to describe porosity, enables the theory to be used in
predicting the levels of stress present when uranium fragments are arrested in foam catchers.
Consequently the theory can be used to assist in the design of catchers which will not distort uranium
fragments travelling at 1-3 km/s. The theory is tested against experiments using some current designs.
INTRODUCTION
Such theory, when verified, can prove a useful aid
in further developing the catcher by reducing the
number of experiments required to test various
foam configurations. Experiments have been
carried out to both validate the theory and to
provide an initial design for the catcher.
Explosive accident studies often require
experimental determination of the fragmentation
characteristics of a "donor" warhead. Information
such as fragment size, velocity and trajectory are
important when assessing the likely damage that
such an event could inflict on its surroundings. A
powerful technique for providing such
information is to arrest the donor fragments in a
medium that is sufficiently soft for the fragments
not to undergo significant erosion or break-up,
and yet sufficiently robust as to decelerate
fragments initially travelling at velocities in the
region of 1-3km/s. The design of such a catcher is
difficult, and such difficulties are increased when
dealing with high-density, pyrophoric materials
such as uranium.
THEORY
The damage inflicted upon a fragment that is
abruptly decelerated can be considered as
occurring in two separate regimes. In the first the
effect of the stresses imparted to the fragment by
the initial impact shock have to be evaluated. In
the second the deformation can be caused over a
longer timescale by the penetration of the fragment
into the target. Since in this work the target is
porous, a porous model has to be used to generate
its equation-of-state (EoS) for use in the shock and
penetration calculations. The theory developed in
this paper is an extension of that presented in
reference 1.
Current development of such a catcher at AWE
has concentrated on a design which consists of
blocks of different low density foams configured
in a manner intended to provide increasing
resistance to the fragment as its velocity
decreases. Theoretical work has been undertaken
to describe the stresses induced in the fragment by
such a design, and to attempt to identify regimes
which will cause severe erosion of the projectile.
The model of porosity used in this paper has been
referred to by a number of names such as
"Snowplough" or "locking solid" by different
authors in the past. It has the characteristics that the
pores are assumed to be massless, and so do not
contribute heat during pore collapse. The
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thermodynamic state of the porous material lies
off the EoS surface of the fully dense material, but
joins that surface once the pores have been
completely closed. A shock in the porous material
forms a smooth discontinuity and so the material
obeys the Rankine-Hugoniot conservation
equations. Any shock strength is sufficient to
collapse all the pores, and so the material jumps
from its initial porous state onto the fully dense
EoS surface.
1
1
-
-
(2)
where CTHEL is the Hugoniot Elastic Limit for the
projectile and U! is the initial projectile velocity.
Where the right side of (2) tries to drop below CTHEL
the projectile ceases to deform and the left side
becomes
In this model the foam EoS can be approximated
by the use of an incompressible hugoniot. Such a
hugoniot will exist at a specific density in the
Snowplough model, and numerical tests show that
it is a reasonable approximation for most highly
porous materials regardless of their actual density.
The only compression that then takes place in the
shock is between v0 and vos (the reciprocals of the
densities of initial and fully dense material).
(3)
where CL is the longitudinal sound velocity.
Although (2) is a steady state equation it is
assumed it can be applied to a non-steady
penetration provided the projectile deceleration is
slow compared to the wave speeds needed to adjust
the interface, shock and plastic boundaries. This
quasi-steady approach is used in conjunction with
The projectile will continue to penetrate after the
initial shock has attenuated. The penetration will
always be supersonic for this model of porosity
since any particle velocity generates a shock. A
steady state supersonic penetration has a shock
velocity that equals the interface velocity. For a
material that has strength, the stress behind the
shock, a s , equals ps+2Y/3 for material undergoing
uniaxial stress and having an elastic-perfectly
plastic strength model. Y is the dynamic yield
strength for the foam and ps=p0 uz2 /ji where ps is
the shock pressure, uz is the interface velocity and
H=pos/(pos-po).
(4)
at
where m is the projectile mass and A the projected
fragment area subjected to the stress crz. CD is
analogous to the drag coefficient used in
aerodynamics, and will be shown to have very
similar values to those obtained in that field. After
some manipulation and integration, the above
equations give
The isentropic, incompressible axisymmetric
momentum flow equation can be reduced to
= posuzduz
(5)
(1)
where Zpen is the total penetration and
in the direction of impact (the z direction) by
assuming a steady state and a negligible shear
gradient over the area of interest. Integrating in
the target between the top of the shock and the
interface, and in the projectile between the plastic
boundary and the interface gives
.
= tan
The velocity above which deformation begins (UID)
is
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Similar conclusions are reached on the lower
value of CD from experiments where a variety of
metals were impacted into strawboard3. Strawboard
has the consistency of strong cardboard and is used
in the catcher designs described below. As it
consists of cellulose and glue, pos is about 1.5g/cc,
while po is 0.64-0.70g/cc. The velocity range of
these experiments is such that GHEL should not be
exceeded for any of the metals used. Consequently
the projectile geometry should be largely unaltered
during penetration. Y was quickly established as
0.25GPa and held constant for the fits to the
complete range of experiments. It was found that
CD varied from 0.44 to 0.60 for spheres, and from
0.70 to 0.86 for cubes. This comparison also
established that both theory and experiment scale
asZpen/m1/3.
_ "HEL
Uir\ —
—
PfCL
(6).
The chief unknowns in the above analysis are CD
and Y. Comparisons with experimental data1 show
that some tuning can take place which produces
values for both terms that minimizes the
differences between theory and experiment. The
following table shows predicted penetration depths
compared to radiographic data in which copper
spheres penetrate foam for which p0=0.176 g/cc,
and pos=1.265 g/cc. CD and Y are tuned to give the
experimental penetration distances at 2.21km/s.
Thereafter Y is held constant but CD is tuned to the
final penetration distance. As the table shows,
intermediate penetrations are then well modelled
by the theory provided the sphere does not
fragment. Y=0.05GPa in the following. AZ is the
percentage difference in distance between theory
and experiment.
EXPERIMENT
Two variations of foam catcher have so far been
tested against uranium tiles. Each tile measured
10mmxlOmmx2.37mm, and had a mass of about
4.5gms. An array of either 25 or 21 tiles was driven
by an explosive plane wave lens system.
TABLE 1. Comparison of Theory with
Experimental Penetration Data.
A;EI
AZ2
CD
T 2 ^s
AZpen
TI
Ui
km/s
(IS
°/€>
%
%
-2 .43
154.1
2.21
27
-0.46 _**
0.40
0
2.9
18.5 -11.04 105.4
-0.47
0.95
107.6
3.13
1.43
0
0.81
_
_
3.27
97.3
1.83
0.90
0
100.0
27.32 -1.01
1.24*
4.28
* Sphere fragmented. ** Complete penetration in
experiment.
The peak penetration stress level is comparable
with GHEL at the 2.21 km/s impact, but exceeds it
for higher velocities indicating that projectile
deformation (and probably erosion) has occurred.
Radiographs from [1] confirm this conclusion and
show the projectile presenting an increasingly flat
surface to the foam by 2.9km/s. In aerodynamics2
the value of CD for a sphere is 0.47 (over a wide
range of Reynolds number), while that for a disc
is 1.17. It is interesting to note that as the shape
changes from a sphere to disc-like in the above,
CD goes from 0.40 to near unity (discounting the
fragmented projectile).
The first catcher design (catcher l)consisted of
1m of aqueous foam (p0=0.02g/cc), followed by
480mm of a flexible polyurethane-based sponge
foam (Intumescent dry foam, p0=0.1g/cc), and
backed with 308mm of strawboards. The impact
velocity was about 2.3mm/jis, and those tiles that
were recovered were mainly fractured and eroded.
Four mainly intact tiles were recovered from the
strawboard layer but all showed signs of severe
erosion. Large scale tumble was observed.
Figure 1 shows an improved catcher design
(catcher 2). "Termanto" foam is a rigid PVC-based
foam. The Intumescent foam has both dry and wet
layers. The latter is intended to provide quenching
for the hot uranium fragments. The calculated
impact velocity of the tiles was 1.5mm/jis and
seven intact tiles were recovered from the
penultimate layer. Two of these tiles had hit, but
not penetrated, the strawboard. The remaining tiles
had appeared to exit through the sides of the target.
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Dry Intumescent
Foam 0.1 g/cc
(480 mm)
T
Table 2 shows theoretical estimates of velocities
in km/s seen at the end of each layer in the catcher
for different initial fragment velocities. This
analysis assumes an impact in either a flat-faced or
edge-on mode. The tumble shown in the
radiographs indicate that the impact mode of the
fragments will lie between these limits. Since the
edge-on is the more effective penetration mode, the
table gives the upper and lower limits of expected
penetration as well as areas where significant
erosion can be expected.
Wet Intumescent
Foam 0.37 g/cc
(300 mm)
t
Termanto Foam 0.045
g/cc (25 & 500 mm)
Strawboard
0.68 g/cc
(304 mm)
DISCUSSION AND CONCLUSIONS
The supersonic penetration through porous media,
coupled with the simple EoS approximation, allows
the stress equation to be expressed in a readily
solvable form. Values of drag are obtained from
experiments where the projectile form and
trajectory are closely controlled. Such values are
comparable to those found from other fields. In
more complex situations where fragments tumble,
the theory gives an envelope of performance that
matches experiment. Catcher 1 is expected to give
large scale erosion for 2.3 km/s impacts, while
catcher 2 should not for 1.5 km/s fragments - both
in line with observation.
REFERENCES
FIGURE 1. Configuration of Catcher 2.
USE OF THEORY IN CATCHER DESIGN
The mechanical data for DU fragments were
taken from [4]. From this data aHEL=0.56 GPa,
and so impacts producing penetration stresses
below this level are assumed not to cause major
projectile erosion. Of course the simple theoretical
analysis does not predict pressure gradients across
the projectile, or the effect of fragment tumble,
both of which will set up bending moments
creating deformation. However, it is assumed that
normal non-tumbling impacts below the GHEL
criterion should not undergo radical deformation.
The minimum critical velocities corresponding to
this criterion for the target components are seen in
the wet intumescent foam (1.3km/s) and in the
Strawboard (0.86km/s). The initial impact shock
will exceed this criterion, but the duration of this
shock should not cause major deformation in the
fragment and so the analysis below will
concentrate on the longer-term penetration
phenomena.
1.
2.
3.
4.
TABLE 2: Theoretical Estimate of Catcher
Performance.
Catcher
Vel.
Km/s
2.3
2.3
1
1
1.5
2
2
1.5
2.3
2
2.3
2
L 77 - erosion is expected.
CD
0.81
0.63
0.81
0.63
0.81
0.63
Orientation
Flat
Edge
Flat
Edge
Flat
Edge
Trucano T.G., and Grady D.E., Int.J.Impact Engng
17,861-872(1995).
Hughes W.F., and Brighton J.A., Theory and
Problems of Fluid Dynamics, Schaum Publishing
Co., New York, 1967.
McMahon E.G., RARDE Fort Halstead, Private
Communication (1969).
Tonks D.L., Vorthman I.E., Hixon R., Kelly A. and
Zurek A.K.,"Spallation Studies on Shock Loaded
U-6 wt% Nb", in Shock Compression of Condensed
Matter -1999, ed. M.D.Furnish et.al, AIP
Conference Proceedings 505, New York, 2000, pp
329-332.
C British Crown Copyright 2001/MoD
Aq.
Foam
1.55
2.13
-
1305
Term.
Foam
1.47
1.49
2.24
2.28
Dry
Int.
0.58
1.76
0.55
1.25
0.85
1.93
Term.
Foam
0.31
1.14
0.51
1.77
Wet Int.
stopped
0.79
stopped
1.11
Straw
board
stopped
through
stopped
through