Combustion, Explosion, and Shock Waves, Vol. 37, No. 2, pp. 230 235, 2001
Application of the Electromagnetic
Model for Diagnosing
Shock-Wave Processes in Metals
S. D. Gilev1 UDC 539.63:537.311.3
Translated from Fizika Goreniya i Vzryva, Vol. 37, No. 2, pp. 121 127, March April, 2001.
Original article submitted October 28, 1999.
To verify the electromagnetic model of shock compression of a conductor in an elec-
tromagnetic field, shock-wave experiments with Constantan are performed. The test
results show that the electromagnetic model gives a qualitatively correct description
of the phenomenon. Some disagreement between the numerical and experimental
data may be caused by factors ignored in the model (a finite thickness of the shock-
wave front and the nonuniformity of the shock wave and electromagnetic field in the
measurement cell). Electric conductivity of Constantan is determined experimentally
under conditions of single shock compression. These studies justify the electromag-
netic model of shock compression of metal in a magnetic field and form the basis for
development of new techniques for dynamic experiments.
INTRODUCTION possibility of using this model to describe electromag-
netic phenomena in actual conductors has to be ver-
An electromagnetic model of shock compression of
ified. The model may be tested by comparison with
a conductor in a magnetic field was formulated in [1
experimental data. In the case of agreement with the
3] under the following assumptions: 1) the geometry
experiment, a question of using the model for predict-
is one-dimensional; 2) the shock velocity is much lower
ing unknown properties of shock-compressed substances
than the velocity of light; 3) the shock wave is steady
may be posed.
and the mass velocity of the substance behind the shock-
The objective of the present work is verification of
wave front is constant; 4) the substance is not magnetic;
the electromagnetic model and its application for diag-
5) the shock front is a surface of discontinuity of hy-
nosing shock-wave processes in metal specimens. The
drodynamic parameters; 6) the electric conductivity of
electromagnetic model is discussed in Sec. 1, and a qual-
the substance is changed in a jumplike manner at the
itative description of the electromagnetic pattern for
shock-wave front. The model is applied to analyze the-
shock compression of a current-carrying conductor is
oretically the electromagnetic phenomena in a current-
given. In Sec. 2, the experiments are described, and the
carrying conductor [1], in a finite-thickness conductor
relationship between the measured electric voltage and
in an external magnetic field [2], and in a conducting
the electric field at the conductor surface is found. The
half-space in an external magnetic field [3]. In the lat-
test results are discussed and compared with model pre-
ter case, a rigorous solution of the problem could be
dictions in Sec. 3. The free parameter here is the electric
obtained, and parameters determining the structure of
conductivity of the shock-compressed metal. The qual-
current waves in metal were found. The problems in [1
itative agreement between the model and experimental
3] differ only by the form of initial and boundary con-
dependences and also the absence of disagreement with
ditions. Since the model is based on some assumptions
the known results allow us to conclude that the model
on the properties of a shock-compressed substance, the
may be effectively used in a shock-wave experiment.
1
Lavrent ev Institute of Hydrodynamics,
Siberian Division, Russian Academy of Sciences,
Novosibirsk 630090.
230 0010-5082/01/3702-0230 $25.00 © 2001 Plenum Publishing Corporation
Application of the Electromagnetic Model for Diagnosing 231
1. ELECTROMAGNETIC MODEL determined the current density ji and the electric field
OF SHOCK COMPRESSION OF Ei in the corresponding region:
A CURRENT-CARRYING CONDUCTOR
1 "Bi 1 "Bi
ji = - , Ei = - + viBi.
µ0 "x µ0Ãi "x
We consider a conductor of thickness x0 with a d.c.
Here vi is the velocity of the substance.
current I flowing through the conductor; the current is
The distributions of fields and currents in the con-
generated by an external source. At the initial time, a
ductor experience qualitative changes in the course of
planar shock wave enters the conductor: D is the shock
shock-wave motion. The electromagnetic pattern as a
velocity and U is the mass velocity of the substance
whole is not obvious and varies significantly depending
behind the shock front in the laboratory frame of refer-
on the parameters of the problem. The special features
ence. The electric conductivity of the substance changes
of shock-wave processes in the magnetic field, which
in a jumplike manner at the front from Ã1 in the incom-
were revealed by solving the problems in [2, 3], allow
pressed region to Ã2 in the compressed region. Taking
us to offer an interpretation of the electromagnetic field
into account the assumptions used, the electromagnetic
structure in a current-carrying conductor.
model of shock compression of a current-carrying con-
Shock compression of the conductor in a magnetic
ductor may be formulated in the form of the following
field generates a system of two currents identical in
boundary-value problem relative to the magnetic field
magnitude but opposite in direction. One of the cur-
B(x, t):
rents passes ahead of the shock-wave front in the incom-
"B1 1 "2B1
pressed substance and moves in space with the phase
- = 0, Dt x < x0, (1)
velocity of the wave; the oppositely directed current
"t µ0Ã1 "x2
flows in the compressed substance and diffuses from
"B2 "B2 1 "2B2 the surface inside the conductor. The absolute value of
- U - = 0,
the induction current depends on the compressibility of
"t "x µ0Ã2 "x2
the substance, on the ratio between the diffusion-layer
(2)
Ut x Dt,
thickness x" H" 1/µ0Ã1D and the conductor thickness
x0, and on time. The induction-current direction is de-
x
termined by the sign of the magnetic field and changes
B1(x, 0) = B0 1 - 2 , (3)
x0
in the course of shock-wave motion. For a greater thick-
ness of the conductor, the electromagnetic pattern may
B2(Ut, t) = B0, t > 0, (4)
be represented as a result of addition of two solutions: a
steady solution (uniform distribution of the current over
B1(x0, t) = -B0, (5)
the conductor thickness) and an unsteady solution [3].
As a result, when the shock wave enters the conduc-
B1(Dt, t) = B2(Dt, t), (6)
tor, the current density is maximum at the shock-wave
front and decreases in the incompressed substance. The
1 "B1 1 "B2
current density on the conductor surface is lower than
= - UB2 x=Dt. (7)
µ0Ã1 "x x=Dt µ0Ã2 "x
at the initial time. Approaching the shock front, the
Here (1) and (2) are the equations of magnetic diffu- current density in the compressed substance increases.
sion for the incompressed and compressed regions, re- As it follows from the numerical solution [1], the
spectively, which are written in the laboratory frame of initial and final phases of shock-wave motion may be ac-
reference, (3) is the initial condition, (4) and (5) are companied by the emergence of intense countercurrents
the boundary conditions; (6) and (7) are the conditions in the surface layers of the conductor. The physical
of continuity of the magnetic and electric fields at the meaning of this strange (at first sight) phenomenon is
shock-wave front; B0 is the magnetic field at the con- as follows. In the initial phases of shock-wave motion,
ductor boundary, which is generated by the transient an induction current passes in the incompressed sub-
current I, and µ0 is the magnetic permeability of vac- stance. The direction of the induction current coincides
uum. The problem is considered in the laboratory frame with the direction of the current generated by an exter-
of reference up to the time when the shock wave goes nal source. The closing current behind the shock front
out of the conductor x0/D. is oppositely directed. In the case of a large electric con-
System (1) (7) was solved numerically. The calcu- ductivity of the substance, the induction current may be
lation technique and some results are described in [1]. so large that the total current passing in the compressed
Based on the found magnetic field Bi(x, t), using the region becomes negative. Thus, in the surface layer of
Ampere s equation and the generalized Ohm s law, we the compressed matter, there is a current whose direc-
232 Gilev
tion is opposite to the direction of the current in the d.c. current (up to 700 A) was passing in the circuit,
remaining part of the conductor. A similar result is ob- which remained constant during the measurements. A
tained by considering the last phases of compression of planar shock wave was induced by a generator 75 mm in
the conductor, when the shock-wave front moves in the diameter with an equalizing charge 60 mm thick. The
region of a negative magnetic field, and the induction shock wave propagated in the measurement cell perpen-
current in the incompressed substance has a direction dicular to the foil plane. Air gaps in the experimental
opposite to that of the current induced by the external assembly were eliminated using epoxy resin. The volt-
source. In this case, the zone of the resultant negative age at the foil was registered by electrodes connected to
current is brought onto the surface from the depth of the foil from different sides (from the side of the enter-
the conductor. ing shock wave and on the opposite side). Hereinafter,
The electromagnetic model formulated is based on we will call them the back and front sides, respectively.
some assumptions. Therefore, the possibility of using The distance between the electrodes was H"6 mm. The
the model to describe electromagnetic phenomena in voltage on the electrodes was registered by an S1-75
shock-compressed conductors needs experimental veri- oscilloscope with a frequency band of 250 MHz. The
fication. scheme of the conducted experiments is close to that
described in [6].
We find the relation between the voltage V (t) regis-
tered by the oscilloscope and the parameters of the elec-
2. ARRANGEMENT OF EXPERIMENTS
tromagnetic field in the measurement cell. The voltage
The conductor in our experiments was Constan- on the measurement device (voltmeter) is
tan. The choice of this material was conditioned by
B
several reasons. First, Constantan has a large specific
V = ÕA - ÕB = - EÕ dl,
resistance, which facilitates signal registration. Second,
A
Constantan is a reference material in measurement of
the electric conductivity of substances in shock waves
where ÕA and ÕB are the potentials of the points A
[4]. The electric conductivity of the substance exam- and B on voltmeter clamps and EÕ is the potential
ined is determined relative to the reference specimen;
component of the electric field. The generalized Ohm s
therefore, it is important to know its behavior. Con- law for the voltmeter circuit may be written in the form
stantan has been used in shock-wave technology for a
"A
long time, but there are no detailed studies of its elec- j = Ã - "Õ - + u × B ,
"t
trophysical properties under shock compression. It is
where A is the vector-potential of the magnetic field.
known nowadays that the change in the electric con-
Thus, for the registered voltage, we may obtain
ductivity of Constantan is small. For a shock pressure
of 20 GPa, the relative increase in the resistance of Con- "Åš j
V = - - dl + (u × B) dl. (8)
stantan is H"2% [5]. This result refers to conditions of
"t Ã
multiple compression of the specimen located between
Integration in this formula is performed over the contour
the layers of a dielectric.
(see Fig. 1) consisting of the voltmeter output A C D,
The experiments were conducted in the following
voltmeter terminal B, voltmeter, and voltmeter termi-
statement (Fig. 1). MNMts 40-1.5 Constantan foil
nal A. Here Åš is the magnetic flux through this contour
(500 µm thick and 10 mm wide) was located between
and u is the mass velocity in the contour-fixed reference
thick dielectric plates (Micarta) and connected to an
system. Since the current through the cell is unchanged,
electric circuit. Before shock-wave arrival at the foil, a
we have Åš = const, and the first term in (8) vanishes.
Let us analyze the third term. If the measurement con-
tour does not move relative to the cell (u = 0), then this
term equals zero. The condition u = 0, strictly speak-
ing, is not satisfied (for instance, the connection of elec-
trodes to the measurement cable is outside the zone of
the shock-wave action). At the same time, the magnetic
field is small at a large distance from the conductor, the
angle between the vectors u and B is also small, and the
third term in (8) may most often be neglected. Thus,
Fig. 1. Measurement scheme: 1) Constantan foil; only the second term is retained in Eq. (8). Taking into
2) dielectric; 3) terminals; 4) oscilloscope.
account that the measurement device (voltmeter) does
Application of the Electromagnetic Model for Diagnosing 233
Fig. 2. Voltage records in experiments with Constantan for connections of the measurement electrodes on the side
of the incoming shock wave (a) and on the opposite side of the specimen (b); the expected moments of incoming
of the shock wave into the foil and its outgoing are marked by arrows.
not consume current, the integral over the contour re- compression and expansion waves over the specimen are
duces to the integral over the sector of the conducting reflected in test results. The registered voltage increases
specimen. In the one-dimensional case, the registered and decreases in the phases of shock-wave motion and
voltage V (t) is extremely simply related to the electric unloading, respectively. The first increase in voltage in
field at the foil boundary E(t): time corresponds to the passage of the first shock wave
over the specimen. The decrease in strength of shock
V (t) E(t)
= .
waves propagating over the specimen corresponds to the
V0 E0
decrease in deviations from the initial voltage.
Here V0 is the initial voltage at the voltmeter and E0 is
In what follows, we consider the phase of motion
the initial electric field at a given surface of the speci-
of the first shock wave over the specimen for the case
men.
where the metal is compressed one time, and its state
is known most reliably. To compare the test results
with the electromagnetic model, the experimental de-
3. TEST RESULTS AND
pendences V (t)/V0 with different connection of the mea-
COMPARISON WITH THE MODEL
surement electrodes are compared in the present work
with the numerical dependences of the electric field on
Figure 2 shows the typical voltage oscillograms ob-
time E(t)/E0 (Fig. 3).
tained in experiments with Constantan. The records
The values of E(t)/E0 are given for the reference
correspond to connection of the electrodes to the back
system connected with the substance in which the elec-
(Fig. 2a) and front (Fig. 2b) sides of the specimen. Prior
trodes are located. This is the laboratory frame of refer-
to shock loading, the voltage on the electrodes is con-
ence if the electrodes are connected to the front side of
stant. When the shock wave enters the metal speci-
the foil. If the electrodes are connected to the back side
men, the registered voltage experiences a sequence of in-
of the foil, this is the frame of reference connected with
creases and decreases, which are roughly periodic. The
the moving substance. The parameters of shock com-
arising oscillations of voltage have a decaying character;
pression of Constantan (velocity and pressure) were de-
their period corresponds to the doubled time of passing
termined by the impedance-match method. The shock
of the shock wave over the foil, which unambiguously in-
adiabat for Constantan was borrowed from [7]. The
dicates a relationship between electric signals and wave
pressure of the incident shock wave in Micarta was
motion. The shock impedance of Constantan is signifi-
20.2 GPa. The calculated pressure of the first shock
cantly greater than the impedance of Micarta; therefore,
wave in Constantan was p H" 43 GPa. The time of
the pressure of the first shock wave in Constantan is sig-
shock-wave propagation over the specimen was 89 nsec,
nificantly greater than the pressure of the incident shock
which is significantly smaller than the time of magnetic-
wave in Micarta. In the next series of waves, the pres-
field growth at the shock front to its maximum value
sure in Constantan decreases to the pressure in the am-
t" H" 10/µ0Ã2(D - U)2 [3], which is H"280 nsec for Con-
bient dielectric. The special features of propagation of
234 Gilev
Fig. 3. Comparison of experimental records of the voltage V (t)/V0 (bold solid curves) with the calculated de-
pendences of the electric field E(t)/E0 (thin solid curves) for connections of the measurement electrodes on the
side of the incoming shock wave (a) and on the opposite side of the specimen (b); the dotted curves show the
equilibrium dependences of the electric field Ee(t)/E0, which were obtained within the framework of the electrical
engineering approach.
stantan. The initial electric conductivity of Constantan nar form. Based on a comparison of these dependences,
was Ã1 H" 2·104 &!-1 ·cm-1. For the calculations plotted the estimate of the transitional-zone thickness yields a
in Fig. 3, it was assumed that the magnetic Reynolds value of H"70 µm, which is significantly greater than the
number was Rem = µ0Ã1Dx0 = 7, U/D = 0.15, and shock-front thickness for a steady shock wave in met-
the parameter s = Ã2/Ã1 was varied. For comparison, als [8]. Probably, the value obtained reflects the special
Fig. 3 shows the dependences Ee(t)/E0 obtained using features of the shock-wave formation in the conductor
the electrical engineering approach in which the electric after the incident wave passes the layer of epoxy resin
field is uniform over the specimen cross section. that fills the gap between the dielectric plate and foil.
A comparison of experimental and calculated de- As the shock wave moves along the metal, the shock-
pendences shown in Fig. 3 allows the following conclu- front thickness decreases, which improves the conditions
sions. of applicability of the electromagnetic model. For com-
The experimental and calculated dependences are parison, we note that the estimate of the thickness of the
in qualitative agreement with each other. This is obvi- current layer in unbounded Constantan yields a similar
ous if we recall a strong dependence of the electric field value x" H" 1/µ0Ã1D H" 80 µm.
on the parameter s. The experimental dependences are Apart from the initial sector, the experimental de-
within the band of the electric field values correspond- pendence in Fig. 3a as a whole may be described within
ing to a small variation of the parameter s. the framework of the model proposed for s H" 0.9 0.95.
The behavior of the time dependences of the volt- The experimental dependence of the voltage on the
age and electric field depends strongly on the position front side of the specimen (Fig. 3b) is not in complete
of the measurement electrodes. For the present model, agreement with the calculated curve. The electric field
in which the shock front has a zero thickness, shock- in this model should increase only at the late stages of
wave penetration leads to an instantaneous decrease in shock-wave motion. The further behavior of the elec-
the electric field at the back side of the specimen. The tric field depends strongly on the parameter s (increase
current density remains finite. After that, the electric or decrease). The voltage increase in the experiment
field increases monotonically. This behavior is caused begins earlier than that predicted by the model. Qual-
by the formation of a current countercurrent system in itatively, the experimental curve is more similar to the
the conductor and by spatial extension of this system in equilibrium dependence Ee(t)/E0. This behavior may
the course of shock-front motion. The experimental and be related to non-one-dimensionality of the magnetic
calculated dependences (see Fig. 3a) are significantly field. The registered signal depends strongly on the
different in the first 10 15 nsec. A smoother variation point of connection of the electrodes. Connection of
of the registered voltage may be attributed to the finite the electrodes at the foil edges is sensitive to electro-
thickness of the shock front in the experiments and also magnetic field nonuniformity. In Fig. 3b, the value of s
to a certain deviation of the shock wave from the pla- may be roughly evaluated as s H" 0.8 0.9.
Application of the Electromagnetic Model for Diagnosing 235
As a whole, the agreement between our experi- the model may be used for diagnosing shock-wave pro-
mental and numerical dependences may be estimated cesses in metals.
as satisfactory. In any case, the dependences V (t)/V0 This work was supported by the Russian Foun-
and E(t)/E0 have an identical character. The values dation for Fundamental Research (Grant No. 99-02-
s H" 0.9 0.95 found from Fig. 3a do not contradict 16807).
the known data. Extrapolation of the results of [5] to
the shock-wave pressure p = 43 GPa yields the value
s H" 1.08 1.12. The difference in these values may be
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energy densities. In particular, it seems of interest to
study the defects of the crystalline structure of metals
under conditions of a controlled loading rate.
CONCLUSIONS
Numerical [1, 2], analytical [3], and experimental
studies performed in the present work allow us to jus-
tify the electromagnetic model of shock compression of
a conductor in a magnetic field. The electromagnetic
model proposed describes experimental data on shock
compression of Constantan. With the above allowances,