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
A NEW TEMPERATURE-DEPENDENT EQUATION OF STATE
FOR INERT, REACTIVE AND COMPOSITE MATERIALS
O. Heuze* , J.C Goutelle**, G. Baudin **
** CEA Bruyeres, B.P.I2, 91680 Bruyeres-le-Chatel, France
** DGA/DCE/Centre d'Etudes de Gramat, 46500 Gramat, France
e-mail: [email protected]
Abstract. The improvement of numerical simulations of transition to detonation processes is dependent
on the temperature law of the equation of state of the initial explosive composition. But in most cases,
only a few physical data are available for these materials. We propose a new equation of state based on
Mie-Griineisen model, which gives a D-u linear law or more sophisticated, available for single
components (metals...) or mixtures (alloys, reactive mixtures) and requiring short computational time.
INTRODUCTION
where F is the Griineisen coefficient. The complex
analytical solution cannot be used in numerical
codes. We propose a development using a Pade
approximate. This gives a good agreement for shock
waves obtained in metal up to 1 Mbar. Then we
propose a description for mixtures: alloys, and
explosive-binders.
There is an increasing requirement for Equations
of State (E.O.S.) in computer codes that describe
properties of matter in a wide range of validity,
together with a short computation time [1].
For most metals, shock velocity/particle velocity
obeys a linear law. Then it is possible to deduce
mechanical properties (e.g. pressure, density, and
velocity..).
The improvement of our knowledge also requires
also access to thermal properties (Temperature,
entropy...), although measurements of these
properties are difficult or impossible, and often not
reliable in shock waves. We obtain such models with
an assumption for heat capacities. This leads to the
description of material by a complete E.O.S.
It is convenient to use a Mie-Griineisen (MG)
E.O.S. in computer codes because it gives a realistic
description of matter together with short
computation time [2]: this linear P-E law can be
solved analytically with the energy equation. The
properties are defined by a sum of a cold part and a
thermal part.
In the present paper, we show that the exact
agreement with the D-u linear law is obtained with a
first order linear differential equation for the cold
part [3,4], with the assumption of constant F/V,
MATHEMATICAL FORMULATION
Assuming constant heat capacity Cv = —
and Griineisen coefficient F = V\ —
= r(v), the
\°E )v
E.O.S. are described by the following MG laws:
(i)
(2)
dEc(V)
(3)
dV
where V is the specific volume, T the temperature, P
the pressure and E the internal energy. PC(V) and
EC(V) represent the 0 Kelvin isotherm curves. This is
a minimum model for Cv, valid for temperatures
higher than the Debye temperature. The RankineP C (V) = -
169
Hugoniot (RH) equation for energy is written in the
following form:
(10)
Po
EH-E,=(PH+P^yo-y)/2 (4)
where the subscripts H and 0 represent the shock
and initial states. According to the RH equations for
the mass and the momentum, the shock velocity D
and the particle velocity u are given by:
V)
dY
(H)
dx
Elimination of temperature gives the P(V,E) E.O.S.:
(5)
APPROXIMATED AND GENERALIZED
MODEL
a) Approximated HZ model
This exact solution should lead to expensive
computer time. Then we tried to develop Y(x) in a
Pade approximate. It gives the following result:
u = JPH(V»-V}
(6)
Using the classical law D=CQ+SU where CQ is the
Bulk sound speed and solving equations (5) and (6),
the pressure PH(V) can be obtained:
P0c0 x
where
x=l-p 0 V (7)
Y(x)= ^l
Elimination of EH and Pc in the former equations
(l)-(5) gives the differential equation:
^ + -(E c -E -lAjcWH flJl
2
nc
I
2
PoC j
° **x (; i r° ,\p °
2
1
1+sx 3 s r
/ - ( °-s)x2/6
2
(]2)
1-sx
Then
E =£ \c2*2
° C° 2
-JC-1
dV
V(
° 2Po J
(2p^
y
Using equation (7) for PH(x), this differential
equation can be written in the following form
1 dF
F (F - F
P
^
*
_ ±^_Jzl__£o_v
c
-c 0 2 Y--^x
Po
dx
1+ s x
/3-s(F 0 -s)x 2 /6
1-sx
(13)
Po
For V/Vo=0.7 (i.e. ~1 Mbar for aluminum and ~3
Mbar for copper) , this potential provides a pressure
discrepancy (compared to the linear D-u law) less
than 1 %. Such a model is linked to our assumptions :
constant pF, and linear D-u law.
(8)
V-H'^T0
The 0 Kelvin isotherm EC(V) can be obtained by
integration of this equation. We choose to solve
equation in the case : r(V)/V= 1
Let us introduce:
b) Generalize H-G model
Let us try now to generalize our result to more
realistic material properties :
(14)
2
3(u/D)
Integration of equation (8) gives :
u
(15)
This D-u law lead to the following PH(V):
P0c0 x
P —
r
—
H
(9)
~~
°
/t
II
C
V
C
U — O j A — v *2
We suggest to use for E C (V):
Ec(v)=E0+c02Y(x)
V
2
—
C
3
V
3 \ I2
(16)
/
(17)
where :
a^^I±
_2s-F0
2s
Then, the model is defined by :
with Y(x) = -
4F 0 s-r o 2 -2s 2
2s
— sx The CyT0 exp(Fox) term has been included in the
Fade approximate Y(x). The coefficients zj are
4
170
For each component, a minimum volume for
which the shock pressure becomes infinite can be
deduced from D^CJ+SI.U:
VLi=VQi(\-\/Si)
(22)
We define a minimum volume for the mixture:
V L (P) = 2>jV u (P)
(23)
determined by use of the derivatives of the Hugoniot
pressure in the reference state (x=0), or with the
power series of D. We obtain :
=
z, =-
Sifci~ro)
i
12cA2
+3s
Thus, a global parameter s is determined from this
minimum volume:
—
s = ———-———
(24)
I-VL/VO
Our model is now defined for the whole mixture:
As expected, only z5 depends on b, j, s2 and 83.
Notice that : ——— =
(25)
dE c (v)
-C V T
dv
Following our former result obtained for onecomponent material, we suggest to use for EC(V)
Ec=EQ+cQ2Y(x)-CvTQerx
EXTENSION TO MIXTURES: MHG MODEL
In former sections, we have defined a onecomponent model. For many applications, metals are
generally used as alloys and explosive molecules
are used with binders and/or additives.
In this section, our aim is to extend to mixtures
the model associated with a linear D-u law. For that
purpose, we need to determine V0, c0, s, F0, Cv for
the mixture. Let jii denote the mass fraction of the ith
component in the mixture.
We consider equilibrium pressure for all the
components of the mixture. Since energy and
volume are extensive properties, we deduce :
E-X^Ei
(18)
with Y(x) =
k=0
The zk parameters are determined with use of the
mixture Hugoniot pressure and its derivatives in the
reference state (x=0). The Hugoniot is given by the
equation
'ni(P)
(26>
The derivatives of PH in the reference state can be
calculated using the VH(P) derivatives.
i
v = £mVi
(19)
RESULTS
i
In order to show the validity of our model and our
assumptions, we have made several comparisons
with theoretical and experimental results.
At first, we compare the H-G model with the
exact solution for the Hugoniot pressure. Figures 1,
2, and 3 show the result for copper, HMX pressed
explosive and Teflon. In both cases, it is not possible
to make a
Assuming a MG equation of state for both the
components and the mixture, the three last
hypothesis for the pressure, internal energy and
specific volume lead to the following form for the
Griineisen coefficient and Cv:
£ = SH,£
<*»
c
(21)
=
-Vi
The model is limited to a constant Fp law for the
mixture and each component.
171
400
800
1200
1600
400
Particle velocity (m/s)
FIGURE 1 COPPER
HUGONIOT PRESSURE.
800
1200
1600
2000
0
Particle velocity (n^s)
FIGURE 2. HMX
HUGONIOT PRESSURE.
800
1200 1600 2000 2400
200
Particle velocity (m/s)
FIGURE 3: TEFLON
HUGONIOT PRESSURE.
400
600
800
800
400
1200
Particle velocity (m/s)
FIGURES. RESULTS
FORCOMP-B3.
difference between the approximated and exact
solutions.
Figure 4, 5, 6 show the comparison of our model
for mixtures with experimental results respectively
forTa+10%W, Comp-B3, and PBXW-115. These
figures represent Hugoniot pressure versus particle
velocity. The results obtained for Ta+10%W exhibit
a significant divergence between experiment and
calculation for pressure greater than 40 Gpa. For
Comp-B3, we see no difference between our model
and experiment.
400
400
800
1200
1600
Particle velocity (nVs)
FIGURE 6. RESULTS
FOR PBXW-115.
parameters. Its potential exactly provides a linear Du mainly for material for which only a few data
exist. In order to minimize computational time, we
have deduced an approximated E.O.S. which gives a
very good agreement with the linear D-u law in the
range of detonation pressures (up to 1 Mbar).
A generalized model has been determined to
extend the former model to a more sophisticated D-u
law. Our complete MG equation of state for solid
single-component or mixtures is a good compromise
between realistic physics, the requirements of a few
physical data, and short computational time.
We have also extended this model to mixtures.
Mixing laws have been established to deduce the
parameters of the mixture from the parameters of its
individual components. This extension is useful for
alloys, explosive compositions, or solid mixtures
whose composition changes with time. Comparison
of our mixing model with experimental results shows
a good agreement in some cases.
1000
Patide velocity (iris)
Figure 4. RESULTS FOR
REFERENCES
TA+10%W.
[1] W.C. Davis, «Equations of State for
Detonation Products», this symposium.
[2] D.J. Steinberg, «Equation of state and
strength properties of selected materials», LLNL
report UCRL-MA-106439, 1991.
[3] O. Heuze «Une equation d'etat pour milieux
sous choc », unpublished report.
[4]C. Renero, F.E Prieto, De Icaza M.
"Comparison of two universal equations of state for
solids" J. Phys. Cond. Matter 2 (1990) 295-3.
For PBXW-115 (figure 6), we can see a difference
between our model and experiment, increasing with
particle velocity. It comes from the poor accuracy of
data of the different components. This shows the
limits of our model.
CONCLUSION
We have developed a complete Mie-Griineisen
equation of state using a minimum set of physical
law. Such a model is useful in many cases,
172