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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
ON THE HEL AND THE "RAMPING" ABOVE HEL
E. Bar-on1, Y. Partom1, M. B. Rubin2 and D. Z. Yankelevsky3
1
2
RafaelBallistics Center, P.O. Box 2250, Haifa 31021, Israel.
Faculty of Mechanical Engineering, Technion-Israel Institute of Technology, Haifa 32000, Israel.
3
Faculty of Civil Engineering, Technion-Israel Institute of Technology, Haifa 32000, Israel.
Abstract. Unlike in metals, stress and particle velocity histories of shock waves in ceramic materials
show a typical "ramping" above the Hugoniot Elastic Limit (HEL). Under the assumption that the HEL
signifies the beginning of failure of the material, this "ramping" has been described by viscoplasticity or
by moduli degradation. However, there is ample experimental evidence to rule out plastic flow at the
HEL level of stress, and it seems improbable that the moduli would degrade significantly during the
compressive phase of a plate-impact. The proposed micro-mechanical mechanism for the HEL, and for
the ramping beyond the HEL, is based on the process of porous compaction due to pressure above a
threshold pressure pcri4sh - Although the pore volume of high-grade ceramics is quite small, it is
sufficient to cause the ramping observed in Hugoniot measurements from the seventies. Additional
evidence for the effect of porosity on the HEL stress has been given in recent years. The experimental
evidence is supported by simulations, in which a simple model for porous compaction was used. These
simulations suggest that porous compaction is the main micro-mechanical mechanism causing the
"ramping" and other features related to the HEL.
INTRODUCTION
<a>
The concept of the HEL (Hugoniot Elastic
Limit) is used extensively in high velocity impact
dynamics. For metals under impact the material
undergoes a compression phase, which at first
causes elastic deformations and at a specified value
of stress causes inelastic deformations to evolve.
This point on the Hugoniot curve is denoted as the
HEL and for the case of metals it is related to the
yield stress and to the onset of plastic deformations.
The HEL in the compressive phase curve of metals
is recognized by the sharp break of the stress-strain
curve, which continues to increase to higher values
of stress as a concave function (Fig. la).
In brittle materials behavior, the concept of the
HEL is not so clear. For brittle materials the
transition from the elastic phase to the inelastic
phase occurs gradually, and the compressive phase
curve describes a convex function (Fig. Ib).
1
V
V*»
«J
__
"***
""""*
FIGURE 1. Typical compressive phase:
(a) In metals;
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(b) In ceramics
Many researchers (e.g. works by Longy and
Cagnoux [1], Beachump et al [2], Lankford [3], [4]
and Lankford et al. [5]) suggest that micro-plasticity
plays a major role as a limiting factor in the
compressive failure of high strength ceramics. It is
possible, that under high hydrostatic pressures due
to confinement, or near crack tips due to highpressure singularity points, one can find
dislocations, which are associated with plasticity.
However, the difference in the behavior at the
transition between the elastic phase and the inelastic
phase in ceramics, as compared with metals,
suggests that there is a different physical
mechanism acting in ceramics. Specifically, in the
present work, the HEL characterizes a transition
phase between elastic phase and the phase of
inelastic strains due to pore crushing.
This conclusion is based on the following
experimental observations. However, it is also
consistent with the fact that the proposed model,
which assumes pore collapse but no plasticity in the
regular sense, produces results that are in good
agreement with experimental data.
about 300 kbar, the value obtained for u 2 from
impedance matching was consistently greater than
Ufs2/2. This is the opposite of results obtained for
metals and is consistent with a model in which the
porosity is permanently crushed out of the material
by the shock compaction".
ALUMINUM OXIDE
(A1203)
Co.-HOT Pness U,,/2
IMP. MATCH
X
W E S G O -995
OIAMONITC
"
U,,/2
P-3J42-1
.
Q COOHS AD-85
.
U,,/2
.
l(-|>
MATCH
UM//2
IMP. MATCH
V SINGLE CRYSTAL.MSauEEN and MAMSH
EXPERIMENTAL PROOF FOR THE
POROSITY EFFECT
Hugoniot data for several ceramics were
measured by Gust and Royce [6] who applied shock
waves using explosives. The tested aluminas were
distinguished according to their initial porosity,
starting with 6.6% porosity AD-85 and ending with
0.2% porosity Lucalox. In Fig. 2, the measured
steady-state Hugoniot stress and the best-fit curves
for four of these ceramics (from [6]) are plotted.
Studying this figure, one can conclude that as the
value of porosity decreases (going from right to the
left in the graph) - (1) the HEL value increases, and
(2) the HEL cusp becomes indistinguishable and the
curve converges to the Hugoniot of a single crystal
A12O3 (sapphire). Another conclusion from these
results is that the measured Hugoniots do not follow
one curve in the stress-volume plane (like in a
p-a model), but there is a family of curves,
where each curve is associated with the initial
porosity of a specific ceramic.
When they analyzed their data, Gust and Royce
noted that (in [6], bottom of page 289) - "For the
more porous materials,...it was noted that in the
compaction regime, i.e., from the yield point to
020
O.Z1
0.2Z
O.M
O.t*
0.23
0.2«
O.JT
O.2«
0.»
FIGURE 2: Experimental Hugoniots for A12 O3 ceramics [6]
20
o
*
AD-85 [1]
B4C[7]
£
015
.ts:
5
10
15
20
Porosity <|> (%)
FIGURE 3: The effect of porosity on the HEL stress magnitude
of AD-85 alumina [1] and Boron Carbide [7]
740
O.SO
The effect of porosity on the magnitude of the
HEL stress was studied over the years by different
researchers. Rosenberg reported the effect of
porosity on the HEL stress in Boron Carbide (B 4 C)
ceramics [7], and Longy and Cagnoux showed
similar behavior in AD-85 alumina [1]. Both works
have shown that increasing porosity decreases the
magnitude of the HEL stress (Fig. 3).
Figure 7 demonstrates the good agreement, which
can be achieved in simulating the stress curve in a
plate-impact experiment [8], using this model.
2.5
POROSITY IN THE MODEL
In many constitutive models, a Representative
Volume Element dv of brittle material in the
present configuration (at time / ) is decomposed
into a solid part whose volume is dvs and a pore
volume dv p , such that
dv = dvs + dvp,
0.5
1.2
1.4
TIME (jiSec.)
dV = d\
FIGURE 4: The effect of varying pcnish in Eq. (4)
where dV,dVs,dVp are the values of dv,dvs,dvp
in a fixed reference configuration, and the indexes
s, p denote the solid and pores, respectively. The
porosity <j> and its reference value O are then
defined by
1
P
J-
crush
! —— Pcrush=2-8GPa
t4
V p
(k = —-, O = —dv
dV
__
-
2.5r
v
(2a,b)
7
In the compressive phase, the porosity </> is
determined through integration of the following
evolution equations:
Uto-
1.2
1.4
TIME (uSec.)
for
p > pcrush
far
pcrush > p
(3a,b)
FIGURE 5: The effect of varying C} in Eq. (4)
2.5r
Here pcrush is a threshold pressure to begin
pore collapse in compaction, v is the current
element volume and /3 (p) is given by
-P™^
Pcrush
)
V
Pcrush
(4)
)
Pcrush9 ^i 9 Q are material parameters to be
determined, and are unique for each ceramic. Here
they are taken to be constants. However, they are
probably functions of the initial porosity and/or
other variables. Figs. 4, 5 and 6 show the way that
the calculated stress curve is affected according to
the varied values of these material parameters.
FIGURE 6: The effect of varying C2 in Eq. (4)
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1.8
100
2.5
AD995-model
Dandekar and Bartkowsky
Grady and Moody
AD85-model
Gust and Royce
80
60
-.1.5
f
Q_
o
CD
40
20
0.5
1.2
1.4
TIME (>iSec.)
1.6
0.22
1.8
0.24
0.26
Volume (Cm3/gm.)
0.28
0.3
FIGURE 8: Experimental data and model simulations, proofs of
the porosity effect on the HEL value and the "ramping" in the
Hugoniots of two ceramics
FIGURE 7: Comparison of the measured compressive phase in a
plate-impact experiment [8] with the model calculation
DISCUSSION
This work shows that porous compaction is the
main mechanism active during the compressive
phase of shock loading of ceramics. This conclusion
has been based on the analysis of experimental data
that has been collected over the last 30 years. The
proposed model of porous compaction correctly
predicted the ramping observed in the compression
wave, without causing precursor decay or wave
separation, as observed in pseudo-plasticity models.
It is important to emphasize that here attention
has been focused on the compression phase, and the
unloading or tension phases have been omitted.
Consequently, it has been possible to consider only
a simple model for porous compaction, without the
necessity to go into detail of a model to simulate the
response of brittle materials to impact loads
(compressive and tensile phases). For general
material response it is necessary to consider a much
more complicated phenomena associated with
micro-cracking, and possibly a pseudo "plasticflow" (related to the amount of crushed pores).
[2] Beachump E. K., Carr M. J. and Graham R. A.,
"Plastic Deformation in Alumina by Explosive
Shock Loading". J. of Am. Ceramics Soc., Vol. 68,
No. 12. PP. 696-699. (1985).
[3] Lankford
J.,
"Compressive
Strength
and
Microplasticity in Polycrystalline Alumina". J. of
Materials Science, Vol. 12. PP. 791-796. (1990).
[4] Lankford J., "High Strain Rate Compression and
Plastic Flow of Ceramics". J. of Materials Science,
Vol. 15. PP. 745-750. (1996).
[5] Lankford J., Predebon W. W., Staehler J. M.,
Subhash, Pletka B. J. and Anderson C. E., "The Role
of Plasticity as a Limiting Factor in the Compressive
Failure of High Strength Ceramics". Mechanics of
Materials, Vol. 29. PP. 205-218. (1998).
[6] Gust W.H., and Royce E.B., "Dynamic Yield
Strength of B4 C, Be O and A12 03 Ceramics". J.
of Appl. Phys., Vol. 42, No. 1, PP. 276-295. (January
1971).
[7] Rosenberg Z., "The Response of Ceramic Materials
to Shock Loading". Proceedings of the APS Shock
Waves in Condensed Matter Conference, PP. 439446. Williamsburg, VA. (1991).
[8] Rajendran A. M. and Grove D. J., "Modeling the
Impact Behavior of AD-85 Ceramic". 24th Int.
SAMPE Technical Conf. T925-T934. (1992).
REFERENCES
[1] Longy F., Cagnoux J., "Plasticity and Microcracking
in Shock-Loaded Alumina". J. Am. Ceram. Soc.,
Vol. 72(6), PP. 971-979. (1989).
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