Combustion, Explosion, and Shock Waves, Vol. 36, No. 6, 2000
Physical Interpretation of Explosion Welding
near Its Lower Boundary
V. G. Petushkov1 UDC 621.791.4:621.771
Translated from Fizika Goreniya i Vzryva, Vol. 36, No. 6, pp. 104 110, November December, 2000.
Original article submitted February 1, 1999; revision submitted September 23, 1999.
The paper considers existing approaches to determining conditions for the forma-
tion of a wavy profile of the joint surface of metals under explosion. Limitations of
the hydrodynamic model due to neglect of the specific properties of welded bodies
are discussed. Within the framework of an elastoplastic model, a new criterion of
wave formation under explosion welding is proposed. The criterion is based on the
assumption that the dynamic yield points of welded metals are equal upon collision.
The notion of an explosion welding region gen- to the ordinate and corresponds to regimes in which
erally implies a set of points on the (ł, vc) plane (ł is the pressure at the collision point causes plastic de-
the collision angle and vc is the velocity of the contact formation of the metal. Transition from a wavy to a
point) at which the phenomena involved in the high- waveless configuration of the joint occurs along the
rate oblique collision of metal bodies are described straight line D. The velocity of the contact point that
by the hydrodynamic model. The boundaries of the corresponds to this line is given by
welding region at which the effect of the individual
0.5
"
vc,2 = 20HV/� , (1)
properties of the metals is pronounced are the lim-
itations of the hydrodynamic model. One of these
where HV is the hardness of the metal and � is its
limitations is that it is not clear which physicome-
density.
chanical characteristics of the colliding bodies should
The curve B bounds the collision parameter val-
be used in the relations describing the boundaries
ues for which a reverse (cumulative) jet can arise.
of an explosion welding region, especially if these
The curve C is the upper bound of the explosion-
characteristics differ substantially. Originally, how-
welding region, and its position is known to depend
ever, hydrodynamic theory did not seek to take this
on the thermal and force conditions of preservation
difference into account because it was developed for
of the joints formed [1].
cumulation, which, by definition, is a geometrically
The lower boundary of the explosion-welding re-
and physically symmetric phenomenon. Therefore,
gion (curve A) is of primary interest. Within the
the hydrodynamic theory of explosion welding still
uses properties that are not associated with a specific
metal of the welded pair, for example, strength, den-
sity, melting point, heat capacity, and speed of sound,
etc., which often leads to unacceptable results.
The bodies that collide in welding are usually
treated as liquid jets. In accordance with theoretical
concepts and experimental data on explosion weld-
ing, the curves A, B, and C on the (ł, vc) plane
determine the explosion welding region (weldability
window) (Fig. 1). On the left, the welding region
is bounded by the straight line E, which is parallel
1
Fig. 1. Classification of flows on the (ł, vc) plane
Paton Institute of Electric Welding,
according to Deribas [2].
Ukrainian National Academy of Sciences, Kiev 252650.
0010-5082/00/3606-0771 $25.00 � 2000 Plenum Publishing Corporation 771
772 Petushkov
framework of the hydrodynamic model of the phe- the lower boundary. The use of the half-sum of the
nomenon, it is described by the equation of a hyper- hardnesses is, in essence, a forced compromise.
bola The concept proposed here is based on the as-
sumption that in certain collision regimes, explosion
łvc = const. (2)
welding of similar and dissimilar metals with both
close or different static strengths is accompanied by
The constant in (2) is defined as HV/� [2], and,
more or less pronounced wave formation on the joint
hence, the equation of the lower boundary becomes
face. A few theories of wave formation are known
łvc = K(HV/�)0.5. (3)
[1, 2, 10], including the hypothesis that ahead of the
contact point, peculiar plastic-strain  hillocks form
Besides HV, expression (3) can contain other
and interact [11]. The  hillocks lead the contact
strength characteristics, for example, the Hugoniot
point and are possible only if the latter moves with
elastic limit, the static yield point �y, or the static
subsonic velocity. The formation of the  hillocks is
ultimate strength �b with allowance for the empirical
likely to be due to the motion of the metal behind the
relations between them. For example, for steel, we
fronts of the rarefaction waves from the free surfaces
have [3]
of the colliding bodies, which is characterized by mass
HV H" 5�y H" 2.5�b. (4)
velocities v1 and v2. Apparently, the parameters of
wave formation on the joint surface profile must be
For better agreement with experimental data, the
determined by certain relations between v1 and v2
fitting coefficient K is introduced into formula (3).
and the elastic constants of each of the metals, i.e.,
According to [4], it varies in the range 0.6 1.8 for
by the individual physicomechanical properties of the
various combinations of welded metals and welding
metals. It is noteworthy that this mechanism forms
conditions.
the basis of the  shock-wave method of explosion
In the case of explosion welding of metals with
welding [3, 10, 12]. It is pertinent to emphasize that
equal or similar strengths, expression (3) is free of
within the framework of this theory of wave forma-
formal contradictions and agrees satisfactorily with
tion, a very strong restriction should be imposed on
experimental data. However, in the case of nonsym-
the velocity of the contact point, namely, vc c"
s
metric collision of dissimilar metals, especially those
(c" is the velocity of the surface plastic-strain wave
s
having markedly different static strength (hardness),
[13]). Then, as a first approximation, we can as-
analytical determination of the lower boundary of the
"
sume that vc,1 cs. This fact, by the way, was not
explosion welding region involves fundamental diffi-
taken into consideration in the monograph [11] and
culties associated with the choice of hardness (and
in the subsequent publications on this issue. Con-
density) of one of the metals of the welded pair to
sidering the interaction of colliding bodies within the
be introduced in Eq. (3) [5]. According to the most
framework of the elastoplastic model, Simonov [14]
widely used hypothesis on the necessity of formation
proposed the condition ł = łmin 7ć% as a criterion
of a reverse mass flux (cumulative jet) [5 7] as a cri-
of the lower boundary, which will be discussed below.
terion of explosive welding, it was suggested that this
However, regardless of the adopted concept of wave
equation contain the hardness of the stronger metal
formation, it is obvious that wave formation is noth-
[8] or, on the contrary, that of the less strong metal
ing but irreversible deformation of the near-surface
[6, 9] of a given pair, or the half-sum of these hard-
layers and mutual penetration of the metals in the
nesses [1, 2, 10] with the coefficient K varied to en-
crystalline state. This process must be determined
sure agreement between calculations and experiment.
by the physical and strength characteristics of the
The physical justifications of these variants are
metals.
not convincing. Suffice it to say that if the lower
Alternate penetration of one metal into the other
boundary is determined from the condition of forma-
is possible in at least two situations: the properties
tion of the reverse mass flux from the surface of the
reflecting the resistance of the colliding bodies to pen-
less strong (softer) metal of the pair [9], it is unclear
etration (hardness) and/or plastic strain (yield point)
how self-purification and deformation of the stronger
change periodically, thus determining the wave fre-
metal can lead to generation of symmetric sinusoidal
quency vc/, or these characteristics vary at the ini-
(in the case of equal or close densities of welded met-
tial stage of collision so that they become equal and
als [2]) waves in it. The introduction of the hardness
remain equal during the entire process of interaction
of the less strong metal into formula (3) is also un-
between the bodies. Both situations seem quite pos-
justified, since in this case, the softer metal of the
sible and, in essence, they express the fact that under
pair is obviously overloaded. This would cause its
conditions of formation of a wavy joint profile, the
melting, but this is not experimentally observed near
Physical Interpretation of Explosion Welding near Its Lower Boundary 773
strengths and, hence, the corresponding physicome-
chanical characteristics of the welded metals must
be equal identically. The main strength character-
istic is the yield point �d, which characterizes the
plastic strain resistance of a metal and is sensitive
to the plastic-strain rate and temperature conditions
of deformation [15]. The foregoing strongly suggests
that in explosion welding upon high-rate collision of
metals, a wavy joint is obtained if and only if both
metals experience plastic strain simultaneously (as
in the case of other methods of pressure welding [16
18]). In other words, the  forced interaction be-
tween explosion-welded metals, determined by the
parameters ł and vc, must cause simultaneous plas-
tic flow of the near-surface layers with strain rates
such (most likely compatible) that they correspond
u
Fig. 2. Calculated curves of �d (!
): curve 1a refers to
exactly to their equal dynamic yield points. More-
ą-iron for N0 = 1012 m-2 and B0 = 7.3 � 10-5 Pa � sec,
over, for a stationary wave to exist (regardless of
curve 1b refers to ą-iron for N0 = 1010 m-2 and
its dimensions), this state of equal plastic resistances
B0 = 7.3 � 10-5 Pa � sec, curve 2 refers to copper for
must be steady. We consider conditions which make
N0 = 1010 m-2 and B0 = 0.6 � 10-5 Pa � sec, and
this situation possible.
curve 3 refers to aluminum for N0 = 1012 m-2 and
It is well known that explosion welding is ac- B0 = 10-4 Pa � sec.
companied by high strain rates of near-contact metal
layers. The strain can be estimated, for example, by
the velocity of the submerged jet [2, 5], the width
u
shear �d versus strain rate calculated in [21] and Fig.
of a pressure plot or the time of existence of high
3 shows the experimental curves of the resistance to
pressure in the collision zone [11], and the magni-
uniaxial tensile strain (upper dynamic yield point)
tude and formation time of shear [17] and wavy [18
obtained in [13, 22]. In Fig. 2, B0 is the disloca-
20] strains. Sedykh [16] established that high-quality
tion retardation constant, determined mainly by the
joints between aluminum and St. 3 steel and between
mechanisms of phonon viscosity and phonon dissipa-
St. 3 steel and St. 3 steel are produced if the critical
tion and N0 is the initial dislocation density.
shear strain is about 10 and 50%, respectively, and
Unfortunately, for the region � 105 sec-1, reli-
Ł
the duration of welding determined from the moment
able experimental data on the quantity �d are few in
rarefaction waves arrive at the junction zone is of the
number, contradictory as a rule, or lacking because
order of a few microseconds. Hence, it follows that at
of insurmountable methodical difficulties [13, 21, 23].
a depth of several millimeters, the shear-strain rate
The available empirical data refer mainly to strain
of the near-contact layers of the welded metals is not
rates � 104 105 sec-1, which are attained in qua-
Ł
less than 105 106 sec-1. For these high rates of nor-
sistatic tests for uniaxial stress and/or strain states
mal strain (�), the dynamic yield point is many times
Ł
[23]. It was found that the relative increment in the
higher than the static value of �d, which corresponds
dynamic yield point (we call the latter the dynamic
to values of � of the order of 10-3 sec-1. Finally,
Ł
hardening) depends strongly on its static value. As
assuming that � vc/ [14], for typical values of the
Ł
early as 1946, Tailor [24] established this dependence
contact point velocity vc 103 m/sec and a wave-
for � H" 104 sec-1. Later on, this dependence was
Ł
length of  10-3 m, we obtain � 106 sec-1. Even
Ł
obtained by Ivanov, Novikov, and Sinitsyn [25] for
higher strain rates (exceeding the above-mentioned
� H" 105 sec-1 in studying the spall strength of
Ł
values by a factor of no less than 1.5 or 2) should be
iron and steel and by Petushkov [22] for strain rates
expected in the  hillocks that arise ahead of the con-
� H" 105 sec-1 in uniaxial tension of some metals.
Ł
tact point according to the wave-formation concept
Figure 4 shows these dependences and discrete
proposed in [11].
values of �d obtained in [2, 13, 21, 22] and other
The effect of strain rate on the strength and
studies for strain rates � 104 sec-1 (up to � H"
Ł Ł
plasticity of metals has been studied in many works,
107 sec-1) using various experimental techniques for
which are reviewed, for example, in [2, 13, 21, 22].
metals with substantially different strengths. One
To illustrate the character of this effect, Fig. 2 shows
can see that for high plastic-strain rates, the plas-
the curves of the upper dynamic yield point in pure
tic resistance of metals increases abruptly beginning
774 Petushkov
even for � 0.5 � 105 sec-1, the dynamic yield points
Ł
of, e.g., Armco iron (�y = 0.18 GPa), St. 3 steel
(�y = 0.24 GPa), and steel 45 (�y = 0.52 GPa) be-
come equal (�d = 1.2 GPa) [22] and approach the
value of HV for steel 45, thus  fitting into relation
(3). This regularity is also observed for other met-
als (aluminum and its alloys, copper, brass, niobium,
and steels of various hardnesses). Data on these met-
als are summarized in Fig. 4. The experiments of [26]
on high-rate penetration of a solid impactor into tar-
gets from lead, copper, aluminum, D16 alloy, and
Armco iron, which were performed as far back as
1959 and became classical, can serve as direct ver-
ification of the validity of the standpoint developed
here. Vitman and Stepanov [26] established that
when the penetrator velocity is v H" 103 m/sec (maxi-
Fig. 3. Upper dynamic yield point of steel 45 (curve
mum strain rate � H" 105 sec-1), metals with an order
Ł
1) and ą-iron (curve 2) versus strain rate for uniax-
of magnitude difference in the initial strength (Brinell
ial quasistatic extension [21] (the figures at the points
hardness is HB = 60 1100 MPa) have about the same
show the number of averaged experimental data.)
penetration resistances, i.e., the dynamic hardnesses
of interest Hd = H0 + k �v2 (H0 is the hardness for
a penetration velocity of v 10 m/sec, k is the
coefficient of shape for the penetrator tip, and � is
the density of the target) almost coincide. More-
over, according to the dislocation theory of strength,
with increase in �, the strain resistance tends to the
Ł
theoretical strength of a perfect crystal [13, 21, 22],
whose values calculated, for example, from the for-
mula �theor = Gb/2Ąa (G is the shear modulus and a
and b are the lattice parameters) are fairly close for
different metals [21].
Thus, the condition
�d,1 a" �d,2 (5)
(subscripts 1 and 2 refer to the soft and the hard
metals, respectively) can be postulated as a wave-
formation criterion for explosion welded metals, in-
cluding those having different initial strengths. It is
Fig. 4. Experimental data showing the dependence
of the relative increment in the dynamic yield point
likely that the location of the straight line E in Fig.
�d/�y on its static value: curves 1, 2, and 3 refer to
1 is determined by the condition � �min, where
Ł Ł
the data of [21], [22], and [23], respectively; curve 4 is
�min is the minimum strain rate of the near-contact
Ł
a hypothetical curve for � 0.5 � 105 sec-1; points on
Ł
layers of the welded metals, for which their dynamic
curve 4 refer to the results of tests taken from different
yield and hardness points  become equal.
sources.
At present, the direct use of this criterion in
practice involves serious difficulties, since it is not
with � H" 103 sec-1, and this increase is more intense clear how the collision parameters should be specified
Ł
and begins earlier for lower values of �y. in order to satisfy condition (5). This is due, first, to
Thus, for a certain, sufficiently high, strain rate, the lack of reliable and reasonably general empirical
the absolute values of �d for metals with different regularities in the variation of the metal flow stresses
values of �y can become equal and attain the re- with strain rate for � 105 106 sec-1, especially un-
Ł
quired level to describe experimentally determined der high hydrostatic pressure and temperature. Sec-
lower bounds of explosion welding regions for various ond, � is difficult to determine reliably in the wave-
Ł
combinations of metals using classical formulas (1) formation region. The first (and principal) difficulty
and (3) in which HV is replaced by �d. can be overcome to a known extent by calculating the
One can readily verify this, bearing in mind that temperature-velocity sensitivity of the flow stresses
Physical Interpretation of Explosion Welding near Its Lower Boundary 775
in terms of the dislocation dynamics for the region of For metals with markedly different strengths, we can,
the above-barrier motion of dislocations. Such calcu- first, set �y,1 ��,2�2 and obtain the estimation for-
Ł
lations were performed, for example, by Krasovskii mula
[21], who derived a general kinetic equation for plas-
��,1�1 �y,2, (9)
Ł
tic flow that is solved numerically and allows one to
which is apparently suitable for determining ��,1 and
predict values of �d for large � and various tempera-
Ł
�d,1 if explosion welding is performed at contact-
tures. In view of its cumbersome form, we do not give
" "
point velocities close to vc,1 or lying between vc,1 and
this equation here. Nevertheless, a comparison of the
"
vc,2, which is of independent significance. Second,
data given in Figs. 2 4 shows that this approach is
the equation of the lower boundary can be written as
promising. As for the second difficulty, for now we
0.5
can restrict ourselves to the estimates � vc/ [14]
Ł
łvc = ��,1�1/�1 . (10)
Ł
or � v /a, where v is the velocity of the strain
Ł
Using the relation �� �� 3��c2 as a first
 hillock relative to the opposite colliding surface and �
approximation, for uniaxial stress state [13, 22], we
 and a H" 0.3 are the wavelength and amplitude,
obtain
respectively. In this case, we can predict at least the
order of magnitude of the quantity �. At any rate,
Ł
łvc k1c�,1 �1!
1, (11)
there is no doubt that in the range of strain rates con-
where c� is the velocity of elastic shear waves, !
is the
"
sidered, the yield point is determined by the viscous
shear strain rate, !
= 3 �, and � is the relaxation
Ł
resistance of moving dislocations, and in the region
time of shear stresses (yield delay) [13, 21].
of their above-barrier motion, it is expressed as a first
It can be concluded that wave formation at the
approximation by the Shvedov Bingham equation of
joint interface in explosion welding involves the pro-
viscoplastic flow [21, 22]:
cess of simultaneous high-rate plastic deformation of
�d = �y + ���. (6)
Ł
the near-surface layers of the obliquely colliding met-
als, which is governed by their dynamic strengths and
Here �� is the coefficient of dynamic viscosity, whose
viscosities. The condition of wave formation is the
values for various metals (�� = 10-3 10-5) Pa � sec
equality of the dynamic yield points of the welded
can be found in [2, 5, 13].
metals for strain rates and other conditions (temper-
It is possible that the assumption of Kudinov
ature, pressure, and viscosity of metals) typical of
and Koroteev [11] expressed in the formula
explosion welding. In addition to other constraints
2
�v0/2 = G/2Ą (7) [5], if condition (5) fails, explosion welding of cer-
tain combinations of dissimilar metals is impossible.
(v0 is the collision velocity) reflects the sufficient con-
In the case of explosion welding of metals with sub-
dition of wave formation, whereas criterion (5) char-
stantially different physicomechanical properties, the
acterizes near-critical and/or subcritical regimes of
lower boundary of the explosion welding region may
collision for which mutual penetration of metals is
be determined using the dynamic strength, viscosity,
possible. In this case, formula (7) should be regarded
and density of the less strong metal in the classical
as a condition that ensures simultaneous plastic flow
equation (3) with allowance for the stress-relaxation
of metals, to which �d,1 and �d,2 from relation (5)
time typical of this process.
tend with activation of the collision regimes. This is
The above interpretation of the condition of
also supported by the Simonov s criterion ł = łmin
wave formation at the lower boundary of explosion
[14]. However, the validity of this statement should
welding in terms of the elastoplastic model can be
be verified experimentally.
useful for a better understanding of the physical na-
From (6) and (7) it follows that
ture of explosion welding and for developing new ap-
proaches to studying this process.
�y,2 - �y,1 = ��,1�1 - ��,2�2, (8)
Ł Ł
and, hence, the criterion of transition from a waveless
to a wavy joint interface is the condition that the
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