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
GROWTH OF PERTURBATIONS ON METALS
INTERFACE AT OBLIQUE COLLISSION WITH
SUPERSONIC VELOCITY OF CONTACT POINT MOTION
O.B. Drennov, A.L. Mikhaylov, P.N. Nizovtsev, V.A. Raevskii
RFNC-VNIIEF, IPhE, Sarov, Nizhni Novgorod region, 37 Mira ave.
Russia, 607190
Abstract. By now the subsonic mode of collision is studied in detail. In the supersonic mode, when
shock waves arrive to the contact point, jet formation is impossible. It is assumed that perturbation
growth at metal interfaces is also impossible. It is obtained in our experiments that perturbations are
formed in the mode of supersonic jetless oblique collision at the metals interface. Numerical
calculations with use of the two-dimensional Lagrange technique showed presence of an area with large
gradient of velocity and high intensity of strains near the contact point.
also impossible. The basic generator, a shaped jet is
not present.
INTRODUCTION
At oblique collision of metal layers in the contact
zone the intensive shift strains are growing, nearboundary layers of metal are strongly heated, and
shaped jets are formed [1,2,3]. The specified effects
cause distortion of the metal interface profile after
collision. There are regular waves, asymmetrical
distorted waves, and layers of melt of intermixed
components.
The subsonic regime of oblique collision has been
studied in detail to the present time: oc < C0 (uc is
the velocity of contact point; C0 is the sound velocity
in this material).
Collision of samples with velocity uc > C0 is
described by analogy to the description of
supersonic flow flowing around the wedge [2,4].
The critical value of velocity uc exists at a constant
angle of collision y=const (y is the analog of
expansion angle of wedge). When uc > ucr at the
contact point the attached oblique shock waves
arrive. The oblique shock waves turn the flows for
angles equal in the total to y. No jet formation is
observed in this case. The authors [1,5] believe that
growth of perturbations on the metal interfaces is
EXPERIMENTAL PROCEDURE AND
ANALYSIS
We performed a series of experiments using the
basic experimental set-up for oblique impact study.
After dynamic loading the plates were caught by
a layer of porous material. Fragments were cut out
of the samples' middles for manufacture of
microsections. In fig.l one can see the dependence
of perturbation amplitude, a, on interface between
two metals (aluminum alloy AMTs) witch Mach
number uc /C0
[6]. The amplitude of the
perturbation, a, (but not the perturbation wave length
A) is chosen as one of independent variables, since
as approaching the maximum value of a the
interface can achieve a form of turbulent mixing of
melts of both materials (see further fig.2 for
aluminum alloy AMTs), where A, is just not
recorded.
The uprising branch of dependence #=f(M) is
determined by the regime of jet formation at the
contact point (subsonic area of flow).
595
, mm
AMTs,
M=1.5magn.x50
FIGURE 2. Samples interface after high-velocity oblique
collision
instability growth due to the tangential break of
velocities at the interface between the initiallystationary and launched plates, which are in various
thermodynamic states.
Relative sliding of the materials occurs behind the
front of an oblique shock wave. Sample temperature
increases at the wave front. During relative slide on
the interface, the intensive shift strains occur, and
the contact plane is melted. The Kelvin-Helmholtz
instability grows, and it results in perturbations on
the interface between two similar metals. The source
of the oscillating initial perturbation is the front of
oblique shock wave.
M
FIGURE 1. Dependence of perturbations amplitude a at interface
of samples made of AMTs alloy (Al; 1/5% Mn) on the Mach
number M=uc /C0 at y «13° = const.
Growth of velocity uc is accompanied by increase
in loading pressure and intensity of plastic shift
strains in the contact zone. A large mass of metal is
involved in jet flow, the perturbations amplitude
grows. Values where uc « ucr (attachment of shock
waves to the contact point), which describes the
transition from the jet formation regime to the jetless
regime of oblique collision, the amplitude of the
perturbations reaches the maximum value.
The transition to jetless regime of plate collision
assumes practically instantaneous termination of the
perturbation growth process (their basic generator
disappears). However, it was obtained in our
experiments that the perturbation amplitude
decreases monotonically with further increase in the
contact point velocity (uc > ucr).
Similar experiments were performed with other
materials (copper, Armco iron). The same results
were obtained.
In fig.2 one can see micro sections of samples
interfaces, which illustrate geometrical shape of the
interface after loading with various velocities.
One can see the scheme of flow configuration in
thefig.3.
In this regime the perturbations are formed on
interface, probably, at Kelvin-Helmholtz
AMTs,
ANALYTICAL SOLUTIONS
The problem concerning perturbation growth on
metal interfaces at oblique collision is rather
complicated for theoretical analysis. Presence of
high strains and a zone of high heating near the
interface impedes application of the method of small
perturbations and simple models of shear strength. It
is of interest to consider the process of growth of
small perturbations for the case when an ideal liquid
slides on the surface of a material having strength.
FIGURE 3. Scheme of flow occurred near to collision zone in
the coordinate system related to the contact point O in regime of
supersonic oblique collision with attached shock waves. Ui, U2 is
the launcher and stationary plates, respectively; SW is the shock
wave; RW is the rarefaction wave; co is the angle of turn of
stationary plate.
M=1.3 magn.xSO
596
boundary is growing in the same way as in the ideal
liquid. The amplitude of the running wave at first
grows and reaches the maximum value, where stress
intensity is comparable to hydrodynamic pressure.
According to the results of the numerical
calculations and experiments, a zone with elevated
temperature is formed at oblique collision of plates,
even in case of the absence of jet formation, as a
result of high strains near the boundary. Besides, for
short time an intensive shift flow with velocity
gradient dependent on the velocity and angle of the
plate collision occurs. In this zone, the shear
modulus and the yield strength are significantly
lower than their values at normal conditions.
Therefore, in a certain zone the conditions for
instability growth can be fulfilled. Thus the greatest
growth should be observed for perturbations with
wavelengths a little longer than the critical one.
This formulation of the problem corresponds to
the variant, where one metal slides on another metal
where strength is lost as a result of thermal softening
in a layer adjacent to the contact surface.
Ideally elastic medium
Let us assume that in the upper half-plane (y>0)
there is a layer of ideal liquid having thickness H
and density p. This layer is moving in direction x
with velocity U0. In the area y<0, there is an ideally
elastic, incompressible material with velocity of the
transverse waves, C. There are harmonic small
perturbations on the interface
$(x) = 40exp(i£x),
(1)
where K is the wave number.
Analysis of analytical solution shows that for
each value of KH^nH/X there is a maximum value
\i=Uo/C9 corresponding to a stable solution
describing wave propagation on the surface of the
elastic half-space. At \i>\icr(KH) there are only
exponentially growing solutions, i.e. the interface
becomes unstable.
NUMERICAL MODELING
The configuration of the system considered in the
calculations is identical to the basic experimental
set-up. The calculations were performed with the use
of the two-dimensional Lagrange technique.
Two variants are considered: absolute slippage,
and absolute friction ("adhesion" of plate surfaces
during contact).
It is necessary to note that the first variant of the
modeling corresponds to the situation, where, as a
result of impact, a local area is formed with high
levels of strain and, therefore, heating, causing
softening of the material.
The plates' material behavior is described with
use of equation of state in the Mie-Gruneisen form
and equations of elastic-plastic flow. The expression
for dynamic yield strength Y in the elastic-plastic
model is accepted in the form typical for the
maj ority of metals
Elastic-plastic medium
The elastic-plastic approximation is close to
reality, but it is impossible, probably, to obtain an
analytical solution in this case. It is clear that
presence of the strength limit will result in
displacement of the instability boundary to smaller
Mach numbers in comparison with the case of ideal
elasticity.
Intensity of stresses in the propagating wave in
elastic medium reaches the value
G—
(2)
Y=(Y0+a.-P)(l-V-ET/Em)
where £, is the amplitude of perturbations on
boundary, G is the shear modulus, Si is the strain. At
a,->7, where Y is the dynamic yield strength, there
is the beginning of plastic deformation of the
material and transition to the unstable regime. If the
initial perturbation of boundary has the form
a(x)=a0cos(Kx) at the undeformed initial state (s/=0,
a,=0), during the initial stage the perturbation on the
(3)
where Y0 , a, (3 is the numerical constants, P is the
loading pressure, ET is the thermal energy of
material, Em is the energy of material melting that is
function of the material state parameters. It is
determined according to Lindemann law.
Of special interest are calculations of the area
behind the critical one, when contact point motion
597
The width of the zone of high levels of strain
decreases as material strength increases. This is in
agreement with the experimental results.
It should be noted that the presence of a zone with
high strain and a considerable gradient in velocity at
certain velocities of collision is observed both in
variants with absolute sliding of contacting surfaces
and in calculations with absolute friction .
Due to intensive deformation, heating in local
zone results in significant softening of the material.
Thus, at collision velocities W0 < 2mm/jj,s the
conditions for Kelvin-Helmholtz instability growth
are implemented.
velocity, uc, significantly exceeds sound velocity,
C0, and exceeds the critical value ucr.
As calculations revealed, in this regime a rather
complicated character of material deformation in the
zone adjacent to the contact surface is implemented.
An area with a large gradient of velocity and,
correspondingly, with a high strain intensity is
formed near the contact point. In fig.4 the
dependence of material velocity in the OX direction
on the initial velocity of collision is presented. As
one can see, the velocity gradient decreases in
accordance with collision velocity growth.
With the increase of collision velocity and the
reduction of gradient of relative velocity between
contacting surfaces, there is decrease in zone width
with high level of strains.
According to calculation results, the character of
strain distribution depends also on the materials
strength.
Wo=1.5mm/|as
CONCLUSIONS
In oblique collisions of metal plates the
perturbation growth is observed in the regime behind
the critical regime, where the contact point velocity
is significantly higher than the sound velocity, and a
shaped jet is not formed.
Perturbation growth stops at rather high velocity
of collision.
The reasons for perturbation growth are the
significant gradient of velocity occurring near the
contact point for a short time, and deformation
heating in a narrow zone near the interface.
Approximated analytical solutions show that at a
sufficient gradient of velocity the Kelvin-Helmholtz
instability occurs at the boundary between a liquid
and elastic or elastic-plastic material.
I
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x(mm)
REFERENCES
1.
W0=2 mm/jos
2.
3
3.
4.
-1 2
6,
5.
D
1
_^A0
10
20
30
6.
4(
x(mm)
FIGURE 4. Distribution of material velocity in the OX
direction at various initial velocities of collision
598
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