CP620, Shock Compression of Condensed Matter - 2001
edited by M. D. Furnish, N. N. Thadhani, and Y. Hone
© 2002 American Institute of Physics 0-7354-0068-7/02/$ 19.00
APPROXIMATE BLAST THEORY: APPLICATION TO SOLIDS
Gregory J. Hutchens
P. O. Box 5578, Aiken SC 29804
Abstract. A method for analyzing strong shock waves in solids is developed for one-dimensional
geometries. An approximation to classical Taylor-Sedov theory is applied to materials described by a
Mie-Grueneisen equation of state. This methodology is then extended to the near-field case where
source mass is not negligible. Example solution results are given.
INTRODUCTION
where p is the material density, P is the pressure, e
is the specific internal energy, F(p) is the
Grueneisen coefficient, and Pref and eref are the
pressure and specific internal energy in some
conveniently chosen reference state.
First, an approximation to the Taylor-Sedov
approach5 will be reviewed and applied to a MieGrueneisen solid for an arbitrary compression, (3
To analyze the propagation of a strong shock
wave it is generally assumed that the initial energy
driving the shock is released in an infinitesimal
volume, i.e., the approach developed by Taylor and
Sedov. This assumption is valid at large distances
from the location of initial energy release so that
the mass of the ambient medium enclosed by the
expanding shock, mo, is much larger than the initial
source mass, ms, that is, m0»ms.
However, near the location of energy release the
assumption of negligible source mass is invalid and
the presence of the source mass must be included.
Previous analyses1"3 have addressed shock
propagation in the near-field where the source
mass is not negligible. These analyses assumed
that both the ambient medium and the source
debris could be described by the ideal gas equation
of state with identical y (specific heat ratio, CP/CV).
Further work4 extended these results to account for
chemical differences between the ambient medium
and source debris by allowing different y for each,
yg for the ambient medium and yd for the source
debris. The objective here is to extend this work to
solids described by a Mie-Grueneisen equation of
state,
P(p,e)=P ref (p)+pr(pXe-e ref (p)) (
= p/
/Po
where p0 is the density of the ambient
medium (assumed constant).
Next, this
approximation will be extended to the near-field
region by explicitly including the source mass.
Finally, example calculations will be performed to
demonstrate the methodology.
APPROXIMATE TAYLOR-SEDOV THEORY
To develop an approximate expression for the
shock position as a function of time, R(t), begin
with the integral expressing conservation of energy
R(t
0=CN
J
Vi- u
oA
2
>
+ e b r N d r , N = 0 , l , 2 (2)
/
where E0 is the initial energy release, p, u, and e are
the mass density, flow speed, and specific internal
(1)
442
energy, respectively, in the region behind the shock,
and CN and N are geometrical constants. For
Cartesian geometry (N=0), CN = A, where A is the
flow channel cross sectional area; for cylindrical
geometry (N=l), CN = 2ftL, where L is the source
length; and for spherical geometry (N=2), CN = 4n.
For the shock wave to be considered "strong" the
initial energy release, E0, must be much larger than
the internal energy of the ambient medium. This
allows the Rankine-Hugoniot relations to be written
in the strong shock form
E 0 -K 3 R N + 1
K 4 R N+l
dR
~d7
where K$ and IQ are given by
K
CN
3 = 7Tr~APo
(e
refW-
(J3-1)2
N
Po
~~
(3)
Eq. (8) is an ordinary differential equation for the
shock position as a function of time, in the absence
of source mass. Note that when the reference state
is such that Pref and eref vanish or K3=0 (e.g., on the
Hugoniot), then the solution of Eq. (8) is the
expression for self-similar expansion
(4)
(5)
(6)
R self
LK4J
where subscript "1" denotes quantities behind the
shock front, F(P) is the Grueneisen coefficient, and
R is the shock speed. Furthermore it is assumed
that the shock is sufficiently strong to completely
pulverize the solid so that behind the shock the
material behaves as an ideal fluid. Hence, material
strength effects are neglected.
The basic approach to approximating the energy
integral, Eq. (2), is to evaluate the integral over a
small (relative to shock position, R) mass shell of
thickness
AT-
(8)
APPROXIMATE NEAR-FIELD BLAST
THEORY
An approximate expression for the shock
position near the source of energy release may be
obtained from the energy integral, Eq. (2), modified
to include the source debris
f 1
"\
(A
(1)
•*
)
l
}(
2
\
(9)
N^
-N J - u a + e a P a r *
Now, begin by eliminating the specific internal
energy, e, using the equation of state, Eq. (1). Next,
assume that u and P are approximately constant
across the mass shell and equal to their values at the
shock front. With this assumption, the energy
integral may be evaluated as the product of the
energy per unit volume and the volume of the mass
shell. Finally, use the Rankine-Hugoniot relations,
Eqs. (4) and (5), to eliminate Ui and PI to find
R \
y
where, again, N = 0, 1,2, subscript "d" denotes the
source debris, subscript "a" denotes ambient
medium set in motion by the shock, and R<> is the
location of the contact discontinuity between the
source debris and the post-shock ambient medium,
defined by
443
R(t)=R c +Ar.
t,
The approximation of the second integral in Eq.
(9) was developed above and is given by Eq. (8).
The first integral in Eq. (9) is approximated in
exactly the same manner although it is assumed that
the source debris is described by the ideal gas
equation of state,
while for large R (Ki becomes negligible), Eq. (11)
becomes the expression for self-similar expansion
given above.
EXAMPLES
(10)
Example calculations, using two different forms
of the Mie-Grueneisen equation of state have been
performed. The objective here is not to assess or
discuss the relative merits of a particular form of the
equation of state but to demonstrate implementation
of the methodology. The results from approximate
blast theory (in the near-field case) will also be
compared with exact solutions for free-expansion
and self-similar expansion. This will provide a
consistency check with the limiting cases.
The first example uses the Hugoniot as the
reference state and is derived from Us-up data. In
this case the constant K3 vanishes since PH and eH
are related by the expression for the Hugoniot curve.
Hence, previously published formulas4 apply,
except with the constants K! and K2 defined as
above. Note that all example calculations use the
same input data as Ref. 4, specifically, Cartesian
geometry and E0 = 2.28xl06 J/m2.
With Us-up data the compression and Grueneisen
coefficient are given by
Following the same methodology as given above
for approximate Taylor-Sedov theory and the using
the result from shock tube theory that u and P are
continuous across the contact surface,6' 1 the first
integral in Eq. (9) may be evaluated as the product
of the energy per unit volume of source debris and
the volume enclosed by the contact surface. Finally,
using the Rankine-Hugoniot relations to eliminate
Ui and PI and combining the resulting expression
with Eq. (8) gives the approximation of the energy
integral for near-field blast theory,
dR = E 0 -K 3 R N + 1
dt
v
(U)
• v r> N+l
where Kj and K^ are given by
K, =-1) 2p
2
-P S R
N+1
s-1
K, =
(N+l)
(12)
where s is the slope of the straight-line fit given by
Us=c0+sup.
where subscript "s" identifies the source density and
radius (or thickness), and K3 is given above. Eq.
(11) is a first-order ordinary differential equation for
the shock position as a function of time including
the effects of source mass. Note that when the
reference state is such that Pref and ercf vanish or
K3=0 (e.g., on the Hugoniot), then for small R Eq.
(11) reduces to the equation for free-expansion
(13)
Figure 1 gives the solution results for iron8
(p0=7.586 g/cm3, s= 1.624).
444
state begins to diverge from that for self-similar
expansion (that is, with K3=0). This is expected
since, as the shock propagates, the product K3RN+1
will increase until it is of the same order of
magnitude as the initial energy release, E0. This
also indicates the distance at which self-similar
behavior no longer occurs.
Free-expansion
Modified theory
CONCLUSION
The second example uses a general form for the
equation of state with Pref given by the BirchMurnaghan equation. With this form of the
equation of state, Pref and eref are related by
By extending previous work, an approximate
method for determining the shock position as a
function of time for materials described by a MieGrueneisen equation of state has been developed.
This methodology may be used to support a variety
of materials science studies by providing a scoping
type analysis. Using the formulas derived here, data
may be quickly analyzed to determine under what
conditions further measurements or computational
simulations are required.
(P)=|Pref(pQ dp'.
ACKNOWLEDGEMENTS
0.100
1.000
Time(microsec)
FIGURE 1. Shock position in iron as a function of time
(Hugoniot equation of state)
e
ref
(14)
The author gratefully acknowledges review and
comments from W. G. Winn of the University of
South Carolina at Aiken and the unwavering love
and support of his wife, Martha.
In this case the constant K3 does not vanish. Figure
2 gives a comparison of the results for shock
position as a function of time using the general form
of equation of state for iron (p0 = 7.969 g/cm3, see
Table 1 of Ref. 9) with, and without, the constant
K3.
REFERENCES
1. D. A. Freiwald, J. Appl. Phys. 43, 2224-2226 (1972).
2. G. J. Hutchens, J. Appl Phys. 77, 2912-2915 (1995).
3. D. A. Freiwald and R. A. Axford, J. Appl. Phys. 46,
1171-1174 (1975); 46, 3697 (1975).
4. G. J. Hutchens, J. Appl. Phys., 88, 3654-3658 (2000).
5. E. G. Harris, Naval Research Laboratory Report No.
4858,1956.
6. F. H. Harlow and A. A. Amsden, Fluid Dynamics: A
LASL Monograph, LA4700,1971, pp. 58-60.
7. Y. B. Zeldovich and Y. P. Raizer, Physics of Shock
Waves and High Temperature Hydrodynamic
Phenomena, edited by W. D. Hayes and R. F.
Probstein, Academic, New York, 1966, pp. 233-239.
8. LASL Shock Hugoniot Data, edited by Stanley P.
March, University of California Press, Berkeley,
1980, pp. 89-92.
9. G. I. Kerley, SAND93-0027, February 15,1993
Without Kg
Time (mlcrosec)
FIGURE 2. Shock position in iron as a function of time (general
form of equation of state)
As seen in Figure 2, in less than 10 jisec the
shock position described by the general equation of
445