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
THE SPALL STRENGTH MEASUREMENT AND MODELLING OF
AQ80 IRON AND COPPER SYSTEMS
P.D. Church1, W.G. Proud2, T.D. Andrews3 and B. Goldthorpe3
1
2
DERA, Fort Halstead, Sevenoaks, Kent, TN14 7 BP, United Kingdom
Physics and Chemistry of Solids Group, Cavendish Laboratory, Madingley Road, Cambridge, CBS OHE,
United Kingdom
3
DERA Farnborough, Ively Road, Hampshire, GUM OLX, United Kingdom
Abstract A series of plate impact experiments have been conducted at a range of velocities required
to cause spall planes to grow in thiee materials. Annealed quenched iron (AQ80) and two types of
copper, high-conductivity, cold drawn (HCCD 101) and copper prepared by a DERA propriety route
(XMCu). The rear surface of the sample was observed using VISAR during impact and afterwards the
recovered targets were sectioned. The velocity-time traces were modelled using a Lagrangian
hydrocode, DYNA-2D, which incorporated the Goldthorpe Path Dependent Ductile Fracture Model.
The experimental data and modelling is presented and match closely for the iron, whereas the
agreement is much less good for the copper. The reasons for this are discussed and future work is
proposed.
INTRODUCTION
Time
The spall strength of a material is a measure of it's
high-rate tensile properties. The spall effect is
caused by the overlap of release waves as shown
in figure la. The reload signal, figure Ib, is a
measure of the spall strength (0spau).
Spall strengths have been measured in many
materials. The variation in value as a function of
the initial shock input [1] and the incipient
formation of the spall plane [2] have been studied.
Modelling of such systems is non-trivial and
requires the successful integration of several
steps; the passage of shock waves through an
uncompressed material, the dispersion of the
release waves, the interaction of the releases and
the fracture limit.
In this study three materials are tested; an
iron prepared by annealing / quenching with a
Vickers hardness of 80 (AQ80), a copper (XM
Cu) prepared by a DERA proprietary route and
HCCD 101 copper. Material parameters are
shown in table 1.
Particle Velocity /Stress
Distance
(a)
(b)
FIGURE 1, (a)Wave propagation diagram, c = compression
wave, r — release fan, (b) Velocity of rear of target plate.
EXPERIMENTAL
Samples of the materials, 6 mm thick, were
impacted by a 3 mm copper flier plate, (Cu 101)
fired from a 50 mm bore single stage light gas
gun at the Cavendish Laboratory. The rear surface
of the target impact was followed using VISAR
giving a time - velocity history with a resolution
of the order of nanoseconds.
487
Property \ Material
Density (g / cm3)
Longitudinal Wave speed (mm jas"1)
Shear Wave Speed (mm jjs"1)
Bulk Wave Speed(mm jus"1)
Poissons ratio (mm [is"1)
AQ80
7.93±0.02
5.86±0.01
3,26±0.01
4,49±0.01
0.27±0.01
CulOl
8.95±0.05
4,67±0.01
2.30±0.01
3.95±0,0l
0.347±0.01
XMCu
8.98±0.05
4.76±0.01
2.32±0.01
3.82±0.01
0.335±0.01
TABLE 1. Properties of the materials used
as a function of the stress state and the local
plastic strain increment.
SIMULATION STUDIES
The simulations were performed using the public
domain version of the Lagrangian hydrocode
DYNA2D originating from Lawrence Livermore
Laboratories. The mesh resolutions used for all
the simulations was 0.014 mm in both the flyer
and the target. The mesh was sufficient to resolve
all the wave propagation and a finer mesh gave no
difference to the result, indicating that the
solution had converged. All the experiments were
simulated and the results can be summarised for
the two materials.
dS = 0.67 exp [1,5sn-QMsn-l.5] de
(1)
Where sn = Stress triaxiality (Pressure/Flow
Stress or P/Y), de = Effective plastic strain
S = Damage.
The only measured parameter in the model is the
critical damage, which is obtained from a novel
analysis of the neck in a quasi-static tension test,
where the pressure/yield (P/Y) condition is
initially 0.33. In plate impact spallation tests, the
strain rate is about 10Vl and the P/Y value under
uniaxial strain is of the order of 400-800
depending on impact stress. For the iron the
critical value of damage was 5.4. Once the failure
condition is reached in the element, it is deleted.
The simulation results were compared with the
VISAR traces for all the experimental velocities,
figure 2. The agreement is exceptional
particularly the magnitude of the reload signal
and the subsequent 'ringing' trace. An interesting
prediction is that after the initial spall, the model
predicts many other spalls further into the
material. It is difficult to verify this
experimentally, though such phenomena have
been seen and photographed using high-speed
cameras during liquid impact studies on PMMA,
which suggests this is not unreasonable.
SIMULATIONS OF THE IRON TARGET
PLATES
The iron target plates were simulated using the
appropriate constants for modified ArmstrongZerilli model [3]. The C101 flyer plates were
simulated using the Goldthorpe Path Dependent
Deformation Model [4]. It is absolutely crucial in
the simulation of spall experiments to use the
correct elastic properties (i.e. densities, sound
speeds, moduli) for all the materials as spallation
is caused by the superposition of two release fans,
subtle differences alter the formation both
temporally and spatially. When comparing to a
VISAR trace with a 2 ns resolution, this can make
a significant difference to the result.
The ductile fracture model was based on
the Goldthorpe Path Dependent Ductile Fracture
Model [5], which expresses the damage evolution
488
600
Particle Velocity
"« 500
E
£ 400
600
500 -
^ 300
w 200
* 100
0.5
1.5
2
2.5
Time / p
3.5
FIGURE 2. The spall traces of AQ80, impacted at 256,461 and
585 m s~ l . Solid lines are experimental data, hatched lines with
markers are the model.
FIGURE 3. Effect of release from flier plate using (a) literature
values, (b) parameters measured on the flier before use.
good on the compression, Hugoniot stress, pulse
width and the start of the release. However, the
spall reload signal is captured very poorly. This
level of agreement was similar for firings of
different velocities and the C101 copper targets.
Furthermore, the model has given good results for
the fracture cylinder test where the strain rate is
similar to plate impact, but the stress system is
complex 3D and dynamically changing [6]. This
suggests the Goldthorpe model is robust with
changing boundary and stress systems.
It was crucial to perform the simulation
with accurate constitutive data for both target and
flyer plates. Whilst this is obvious for the target
plate, it is not so obvious for the flyer plate. An
illustration of the sensitivity of these results is
shown in figure 3 where the higher velocity
experiment has been simulated using an elastic
plastic model for the copper flyer and literature
data for the elastic properties. Although the
comparison is fair, there are some significant
differences relating to subtleties in the release fan
from the back of the flyer plate. This reinforces
the view that great care is required in simulating
plate impact spallation tests.
a '
£>
200
• ——Experimental
-•••Model
0.5
1
1.5
2
2.5
Ticne/jus
3
3.5
FIGURE 4. XM Cu Spall comparsion of model with
experiment.
DISCUSSION
The Goldthorpe model is giving very impressive
results for the high strain rate spallation of iron
under different impact stresses. This is
remarkable considering the model was derived
under quasi-static uniaxial stress conditions and
demonstrates the robustness of the general
approach to dynamic ductile fracture.
The discrepancy with the copper results
is disturbing and indicates that the Goldthorpe
Path Dependent Ductile Fracture model may
require some modification: the Goldthorpe model
was published as an interim and novel approach
SIMULATIONS OF COPPER SYSTEMS
The simulation methodology was identical to the
iron target plates as was the mesh resolutions. The
critical damage level for the Goldthorpe fracture
model for XM copper was 4.7. A comparison of
the simulation and the experiment at 460 m/s is
shown in figure 4. The level of agreement is very
489
CONCLUSIONS
to ductile fracture. A number of simplifying
assumptions were applied, not least of which was
the employment of the void growth law due to
Rice & Tracey [7]. This law assumes that the
voids grow in a rigid plastic material, most
materials, particularly copper exhibit significant
work hardening. The Goldthorpe approach relies
extensively on the solution of void interactions
using novel analytical techniques [8].
These solutions are complicated by the
assumption of local work hardening on the void
interactions. This hardening would tend to delay
the onset of fracture, as the local material
becomes progressively stronger as the voids
grow. The full solution to this problem is an area
of ongoing research and is attempting to calculate
the progress of damage under defined stress-states
to the boundary conditions calculated by the
hydrocode in a so-called dynamic 'unit cell'
model. This entirely novel approach will then be
able to link the damage evolution into the
constitutive model and equation of state thus
modifying the local wave propagation.
The implementation of this physically
based partnership between hydrocodes and
analytical solutions will not be trivial since the
link between them is path-dependent. Thus a very
clear appreciation of the interface between the
hydrocode numerics and the underlying model
assumptions will be required for a self-consistent
implementation.
It is clear that the plate impact spall test
is a stringent test of a fracture model and the
hydrocode numerics, as it requires the resolution
of complex wave propagation both spatially and
temporally. The VISAR trace is a demanding
comparison since its time resolution is of the
same order as the hydrocode time step, any slight
disparity in the numerical scheme will degrade
the simulation result.
1 . The Goldthorpe Path Dependent Ductile
Fracture Model gives excellent agreement
with spallation results for iron.
2. The general approach to ductile fracture has
been demonstrated as robust since the model
derived under quasi-static loading has been
successfully applied at very high strain rates.
3. The model requires some modification to
account for work hardening of the local
material around the voids.
4 . The partnership between the analytical
solutions and hydrocodes in the 'unit cell'
model shows significant promise.
ACKNOWLEDGEMENTS
D. Cross of the Cavendish Laboratory is thanked
for his technical expertise and support. A. Butler
of DERA is thanked for supplying the materials.
REFERENCES
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