Shock Waves (1997) 7:49 54
Revisiting shock waves in metals
S.M. Chitanvis
Theoretical Division, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA
Received 25 March 1996 / Accepted 20 August 1996
Abstract. We cast Wallace s theory of thermoplastic flow In order to perform our analysis, we found it useful
in conservative form. We point out the difference between to cast the equations of thermoplastic flow in conservative
our formulation, which accounts for contact with an external form. In this formulation, we felt it essential to include the
energy reservoir, and previous formulations of thermoplastic fact that in cases of interest (where a metal is being  pushed
flow. The theory is exploited to show that the experiments by an external agent), the system under consideration is in
of Johnson and Barker on 6062-T6 Al can be interpreted contact with an energy reservoir. This leads us to invoke
as a weak shock wave that splits into an infinite sequence the Helmholtz potential minimum principle in order to write
of  infinitesimal shocks, caused by increasing plasticity, down the correct energy conservation equation.
leading to the observed smooth temporal velocity profile (a The notion of shock-splitting in fluids has received con-
dispersed wave). We predict that overdriven shock waves siderable attention over the years [4]. It is well-known that
in metals will split as well. We also re-examine the need shock waves in fluids near a phase transition can split due
for invoking a heat dissipation mechanism for overdriven the loss of convexity of the equation of state. Intuitively it
shocks. would appear that a similar phenomenon should ensue as
It is briefly pointed out that our approach of casting the well in metals that are beginning to flow in a plastic fash-
theory of thermoplastic flow in divergence form can be gen- ion under an applied stress. In fact, it was the qualitative
eralized easily to account for heat release in energetic solids. similarity [4] in the split-shock profiles in liquids, and the
double-structure of weak shock waves in 6062-T6 Al that
Key words: Shock splitting, Metals, Strong shocks
caused us to think along this line. We will argue later in this
paper that this is indeed the case. However, the plastic wave
is a fully dispersed wave which is associated with a finite
rate of development of plasticity, while in the case of fluids,
1 Introduction we have an equilibrium discontinuity near a phase transi-
tion. There is only a cursory discussion of shock splitting in
Wallace developed a theory of irreversible thermodynamics
an elastic-plastic solid in Zeldovich and Raizer [5], where
and flow in metals many years ago [1]. The cornerstone of
it is implied that an initial shock in a ductile solid can split
his theory is the experimental observation that virtually all
simultaneously into a fast component (elastic precursor) and
the plastic work done in deforming a metal is dissipated
a slower plastic component. We were unable to find in the
[2]. This energy goes into increasing the entropy. This ap-
literature an explicit interpretation of experimental results
proach was applied to construct plastic constitutive equations
on weak shocks in metals in terms of shock splitting. Hence
from experiments involving propagating weak shock waves
we thought it would be useful, for the record, to perform
in 6062-T6 Al [3]. In this fashion one circumvents the ne-
a straightforward analysis of what some would consider an
cessity for detailed theories of microstructural deformation
obvious truth. Our observations do not affect the practical
for the construction of the desired constitutive relations. On
aspects of constructing plastic constitutive relations from ex-
the other hand, it is clear that detailed theories of structural
perimental data using Wallace s method [1].
deformation will lead to fundamental insights.
We remark that the double-wave structure that Johnson
We will argue in this paper that one can go a bit further
and Barker observed (Johnson and Barker (1967)) may ap-
in interpreting the two-wave structure in metals using ther-
pear to be anomalous to the uninitiated, since we see a rel-
modynamics and conservation laws alone, without having
atively rapid rise in the velocity (or equivalently, the pres-
to invoke detailed theories of microstructural deformation.
sure) behind the elastic precursor, even as plasticity devel-
We will show, using conventional Hugoniot curve analysis,
ops, which would imply a certain softening of the metal. A
that it is possible for the elastic precursor (weak shock) in
physical reason for the pressure going up behind the pre-
6062-T6 Al to split in a time-dependent manner, as plasticity
cursor is that increased plasticity in metals is due to the
develops in the metal, forming a dispersed wave.
50
generation of dislocations whose interactions then cause a (3) Energy conservation:
certain amount of  stiffening [6]. It will be shown later
Wallace s basic tenet is:
that the splitting of the elastic precursor is consistent with
developing plasticity in the metal. Indeed, it is the develop-
dU = dWelastic + TdS ,
ment of the plasticity that is essential for the splitting of the
elastic precursor to occur. In atomic terms, one can imagine TdS = dWplastic , (2.3)
that the elastic precursor is described by an energy change
2
that is harmonic (quadratic) in the displacements from their
dWplastic = ÄdÈ
equilibrium positions of the affected atoms. Further displace- Á
ment of the atoms vitiates the initial ground state structural
From this, Wallace [1] writes down the energy conser-
arrangement even more.
vation equation as:
The major portion of our paper considers the behavior
of weak shock waves in metals. Obviously, a good reason
"(ÁE) - - "!

)
+ "· (Á- - "· ( ·- =0 ,
uE) Ä u
for this is the existence of well-resolved experimental re-
"t
sults in this regime. The temporal resolution available in
1-
2
experiments is not sufficient to determine whether the shock-
E= u +U , (2.3a)
2
splitting, which occurs for weak shocks, takes place for over-
driven shocks as well. This should not be construed to mean
"!
Ä "!
that in the case of overdriven shocks, the plastic wave def-
U = - : d µ
Á
initely overtakes the elastic precursor. It could be that the
splitting is simply too small to be observable. Indeed, we
where U is the specific internal energy. The last of
will argue towards the end of section III, that overdriven
Eqs. (2.3a) is an  equation of state , which helps us
shocks in metals do split. We will provide in this paper an
close our set of equations. It has been defined thusly on
alternate view of overdriven shocks, where the need for a
physical grounds, viz., that the internal energy changes
heat dissipation mechanism will be re-examined.
are wrought in the material by a displacement of atoms
from their equilibrium positions. For fluids on the other
hand, one assumes that the energy change is given by the
2 Formulation of thermoplastic flow in divergence form
mean-square deviations of the atomic velocities from the
The equations of motion for thermoplastic flow in divergence
average velocity, and leads to a pressure change, as per
form are:
sta ndard statistical mechanics treatments. In this discus-
sion, we speak entirely in terms of the total stress, and
(1) Mass conservation:
we have not separated out the pressure contribution.

"Á -
)
+ "· (- = 0 (2.1)
u
"t Alternatively, the kind of system we wish to consider
- in this paper is nominally a slab of metal, which is be-

where Á is the density and u is the particle velocity.
ing  pushed on one side by an external agent. In other
(2) Momentum conservation:
words, our system is not isolated, since it is in contact with
)
an energy reservoir, which supplies it energy required to
"(Á- - "!
u
-

+ "· (Á- " u - Ä ) = 0 (2.2)
u
cause plastic deformations. Then, thermodynamic consid-
"t
erations tell us [7] that the Helmholtz potential minimum
"!
Ä
The tensor is the stress tensor, and its usual decompo- principle holds, which states that d(U - TS) = 0, consis-
sition in terms of elastic and plastic strains, in accordance tent with Eq. 2.3. Therefore, the energy per unit volume
with the Prandtl-Reuss law is as follows [1]: 1 -
2
which is conserved is not the usual Á u + U , but rather,
2
2
1 -
Á u + U - TS . Thus, the energy conservation equa-
Äij = -P´ij + sij ; 2
tion for a metal in contact with an external energy reservoir
(2.2a)

may be written down as follows:
1
P = - Äij
3
"(ÁEp) - - "!

)
"!
+ "· (Á- - "· ( ·- =0 ,
uEp) Ä u
s
where the traceless stress deviator tensor is defined by
"t
Eq. (2.2a). The stress tensor is defined in terms of the
"!
1-
2
µ
total strain tensor , which in turn is assumed to be the
Ep = u +Up , (2.4)
2
sum of an elastic component and a plastic component
"! "! "!
Up = U - Wplastic
µ µ µ
(d = d +d ). A measure È of the plastic strain is
e p
defined terms of an effective shear stress Ä by:
Note that Wplastic is the entropy term, as defined in Eq. (2.3).

"! "!
3 We could interpret Eq. (2.4) as one which conserves the
s s
Ä2 = : , Ä e" 0 ,
8
reversible portion of the energy. Or, one which which per-
(2.2b)
mits entropy production in a continuous fashion in space and
"!

"!
3 s
time, in addition to entropy production across shocks. This
µ
d = dÈ
4 Ä
p
is best seen by transferring the terms depending on plastic
51
work to the right hand side of Eq. (2.4). In particular, it al- are self-sustaining, while plastic deformations in metals are
lows entropy production to occur, in principle, behind the caused by an external agent supplying the energy to cause
elastic precursor shock in a metal. We propose that the plas- plastic deformations. In either case, however, entropy can be
tic work Wplastic is governed by the following rate equation: produced in a continuous fashion in space and time, in addi-
tion to entropy production across shocks. It is this fact which
"(ÁWplastic) - "!


leads to a certain formal similarity in the description of these
+ "· (Á- µ
uWplastic) =Rplastic(Á, , Wplastic) ,
"t
two phenomena. In general, we expect plastic work to lower
"! "!
the state of stress in the material (if we hold the total strain
µ µ
Rplastic(Á, , Wplastic) > 0" e"µ" , (2.5)
fixed), thereby affecting the rate of energy release. It would
"! "! therefore be reasonable to assume that the rate of energy
µ µ
Rplastic(Á, , Wplastic) =0" <µ"
release is tied to the plastic work performed on the material,
perhaps through a dependence on the stress. We therefore
where . indicates a norm. The inequality in Eq. (2.5) indi-
begin to see the interplay (difference) between plastic flow
cates the irreversible nature of the plastic deformation pro-
and energy release. In the case of metals, plasticity is caused
cess. For metals the value of µ" is non zero, indicating vis-
by dislocations, etc., while in solid explosives, plasticity is
coplastic behavior. If µ" were to be zero, we would say the
caused by irreversible atomic/molecular rearrangements in
material is viscoelastic. From a physical viewpoint, it seems
the ductile polymeric binder, as well as brittle behavior in
adequate to use Eqs. (2.1), (2.2), (2.4) and (2.5) to describe
the grains of high explosive. Since Wallace s theory of ther-
thermoplastic flow. For a deeper mathematical discussion,
moplasticity is based almost exclusively on thermodynamic
we refer the reader to Plohr and Sharp [8]. The conservation
grounds, we speculate that it would be reasonable to extend
of energy in their consideration is given by Eq. (2.3a). For
this approach to studying high explosives as well. In other
the record, we note that Eq. (2.5) and (2.4) could be com-
words, it may prove possible to extract constitutive relations
bined into just one equation. For the purposes of analysis,
from experimental data via this approach. We are not aware
however, we shall retain Eq. (2.5) as a separate entity, as
of extended discussions along this line. In this paper, how-
done by Plohr and Sharp.
ever, we will restrict ourselves to studying shocks in ductile,
We note that instead of Eq. (2.5), we could have chosen
non-energetic solids.
the following rate equation:
"(Wplastic) - "!


+ "· (- µ
uWplastic) =Rplastic(Á, , Wplastic) ,
"t 3 One-dimensional analysis
"! "!
µ µ
Rplastic(Á, , Wplastic) =0" e"µ" , (2.5a)
Our basic idea here is to argue that the experimentally ob-
served velocity profile is formed in two stages. First, a weak
"! "!
µ µ
Rplastic(Á, , Wplastic) =0" <µ"
shock (the elastic precursor) is sent propagating down the
sample by the applied external pressure. This shock is con-
It is easy to see that Eq. (2.5), in conjunction with Eq. (2.1)
sidered to be weak because its velocity is quite close to the
allows Wplastic to be continuous across a shock, while
sound velocity in the elastically shocked material. Secondly,
Eq. (2.5a) permits a jump in Wplastic. If we assume the metal
as the plasticity in the metal develops behind the shock in
is in a pristine condition before being shocked, then both
time, the material properties change to an extent where it is
forms of the rate equation give us identical results across
possible for the precursor to split into a sequence of shocks
the elastic precursor. If the rate in either Eq. (2.5) or (2.5a)
in a continuous fashion, giving rise to the smooth observed
- -

profile i.e., a dispersed wave. A plausible assumption we will
contains the divergence of a vector function (i.e. "· F ),
make in our analysis is that, given the viscoplastic nature of
then it is easy to see that this term will lead to a jump in the
metals (see Eq. (2.5)), the splitting of the precursor does not
plastic work across a shock. The existence of such a term
begin until after a shock has formed, under the application
may be determined via experiments. We will assume in this
of an external pressure pulse.
paper that for the case of weak shocks in metals, this term
In the one-dimensional case, where a shock wave prop-
is absent. We will consider this term only briefly in section
agates through a thick slab of ductile solid, we have two
III, when we discuss overdriven shocks in metals.
stresses, the normal stress (Ã) and the transverse stress
In our approach, we will assume for simplicity that work
(Ã
hardening is accounted for implicitly by a combination of - 2Ä). We will follow Wallace in taking the total strain
and the plastic strain È as independent variables. Using the
Eq. (2.5) and the stress-strain/plasticity relation (Eq. (3.4))
usual Hugoniot conditions, we obtain the following equa-
used in the next section.
tions for the Rayleigh line (mass and momentum conserva-
Notice that Eqs. (2.1), (2.2), (2.4) and (2.5) are formally
tion):
analogous to those in the theory of detonations (ZND model)
in explosives [9]. Indeed, energy release in the solids we are
(Ã
"Ã - Ãi)
= = Á0(us - ui)2 ,
considering could be accounted for by the following trans-
(µ - µi) (µ - µi)
formation Ep E - Wplastic - Wenergy-released, and one ad-
ditional equation governing the release of this energy. The 1
V = , (3.1)
equation of state will also need to have an additional con-
Á
tribution which reflects the effect of temperature rising as
V
energy is released. A major difference between energetic
µ = 1 -
solids and metals is of course that detonations in explosives
V0
52
where us is the shock velocity, the subscript i denotes the
initial state ahead of the shock, and the subscript 0 denotes
the original unshocked state. The Hugoniot line (energy con-
servation) is given by:
1
Up(Wplastic) - Up(Wplastic ) - (Ã + Ãi)(µ - µi)Vi =0 ,
i
2
µ,È µi,Èi
Up -Up = Ã(µ , È )V dµ - Ã(µ , È )V dµ
i
0 0
È,µ Èi,µi
-2 Ä(µ , È )V dÈ - Ä(µ , È )V dÈ
0 0
(3.2)
where the subscript i refers to the initial state of the system.
We obtain from Eq. (3.2),

2
à = Up(Wplastic) - Up(Wplastic ) - Ãi (3.3)
i
V0(µ - µi)
where Wplastic = 0. From Eq. (2.5), it is clear that the cor-
i
responding jump condition for plastic work, in conjunction
with mass conservation implies that plasticity is continu-
ous across the shock. And furthermore, we assume that the
material was pristine (zero plasticity) to begin with. This as-
sumption is consistent with our analysis, as shown below,
in our discussion of the first Hugoniot curve.
Fig. 1. a This is the first Hugoniot which takes the initial state (0) to the
We get the following relations, applied to 6062-T6 Al
elastically shocked state (1). The solid line is the Hugoniot Ho, and the
from Wallace [1], which relate explicitly the stress à to the dashed line Ro is the Rayleigh line. Notice how shallow is the convexity
of Ho. The shock velocity is 0.6405 cm/µs. For a better perspective, see b;
plastic variable È, and also includes the entropy contribution
b To alleviate the shallow curvature of H0 we have plotted here (R0 H0).
from the plastic work given by Eq. (2.3):
Given that R0 is a straight line, the curvature in this figure is entirely due

3
to H0. Notice the scale on the y-axis is much smaller than the one in a
à =(+2µ)µ-2µÈ-  +3µ+Å› +2¾ µ2+(4+10µ+4¾)µÈ
2
µ,È
3 3 v
to occur. Since we are working in a one-dimensional geom-
-  +6µ+ ¾ + È2 +2Å‚a Ä(µ , È )dÈ ,
2 2 4
0 etry, we do not have to consider a separate Hugoniot for the
(3.4) shear stress. Eqs. (3.2) and (3.4) are the only way that the
shear stress enters our analysis.
3 3
The first Hugoniot curve H0, which takes the system
Ä = µ µ - È - ( + µ + ¾)µ2
2 2
from an initial state (0) (with zero stress and strain) to the
elastic precursor state (1) is shown in Figure 1a. R0 de-
3 9 3 v 9 3
notes the corresponding Rayleigh line. The elastic shocked
+  + µ + ¾ + µÈ - µ + v È2
2 2 2 4 4 8
state is given by the intersection R0 and H0. To alleviate
the shallowness of the curvature of H0, we have plotted
 =0.544 Mbar, µ =0.267 Mbar, Å› = -1.40 Mbars,
R0 H0 in Fig. 1b. This gives us an improved perspective.
(3.5) Given the quadratic dependence of the Rayleigh line on the
shock velocity, and the sharply rising nature of the Hugoniot,
¾ = -2.82 Mbars, v = -4.69 Mbars, Å‚a =2.16
the strain in the shocked state (1) turns out to be a surpris-
where Å‚a is the Gruneisen coefficient for aluminum, and the ingly sensitive function of the shock speed. Now the deduced
other constants are the Lamé and Murnaghan elastic con- value of the strain right behind the shock is about 0.0037
stants for aluminum. [1]. This value is obtained from the particle velocity behind
Equation (3.4) represents a consistent perturbative ex- the shock, while the particle velocity is in turn inferred from
pansion of the stress and the total shear stress in powers of the free surface velocity. In order to get a strain consistent
the total strain and the plastic strain. Notice that both the with the above value, we needed to adjust the shock velocity
total strain and the total shear stress are convex functions of slightly, to 0.6405 cm/µs, which gave us a stra in at state (1)
the total strain. It is permissible to perform the integration in of 0.00358, and a stress of just under 4 kbars. The shock ve-
Eq. (3.4) holding the total strain fixed. Similar considerations locity we used is fairly close (99.15%) to the value of 0.646
apply to Eq. (3.2). In Eq. (3.4), the irreversible nature of the cm/µs deduced by Wallace [1], and may be compared with
plastic flow is captured through the plastic flow variable and 0.637 cm/µs which is the sound speed in the unshocked re-
the total shear stress which is responsible for plastic work gion, using finite strain theory. With this slight adjustment
53
ing only the mass and momentum conservation equations.
However, for a correct interpretation of shock-splitting, our
formulation of the energy equation, which is appropriate for
a system in contact with an energy reservoir, does make a
difference.
The initial (infinitesimal) relaxation of stress from state
(1) which triggers the subsequent dispersed wave is a tran-
sient analog of the Taylor (rarefaction) wave in detonation
theory [13]. We are therefore tempted to think of the subse-
quent dispersed wave a sort of a rarefaction shock wave.
3.1 Overdriven shocks
Fig. 2. This figure displays secondary Hugoniots H1 (È = 0) (short dashed
Our focus in this paper so far has been entirely on weak
line) and H1 (È = 3.610-4) (solid line), in addition to the original Hugo-
shocks in metals, where the shock velocity is rather close to
niot Ho and the Rayleigh line Ro (long dashes). It is clearly impossible
the sound velocity. Wallace [14] defines overdriven shocks
energetically for the state (1) to make a transition to H1. But a transition
to H1 is certainly possible, as it lies below Ro
as those with a velocity greater than the sound velocity.
Within our approach, which represents a generalization of
the formalism of Wallace, it is easy to see that all the re-
to achieve consistency with experimental observations, we marks we made about weak shocks apply equally well to
proceed to look at shock splitting. strong shocks in metals, in the sense that a strong shock will
The question now is whether the system can make a split as well, as the metal develops an increasing amount of
transition from (1) to another state via a second shock. To plasticity (we will assume for the moment that plasticity is
determine this, we must consider the Hugoniot that passes continuous across the shock). The experimental verification
through the point (1) (Fig. 2). The second Hugoniot indicates of this statement is difficult, given the tremendous amount
the possible states that the precursor state (1) can reach via of temporal resolution that will be required for this purpose.
a shock. If the Hugoniot is such that it lies (at least for We have performed Hugoniot analyses for overdriven shocks
some range of the strain) below the Rayeigh line R0, then using shock velocities greater than the speed of sound. As an
it is clear that a shock wave with a lower velocity than the example, we find for a shock velocity of 0.646 cm/µs, the
initial elastic precursor will develop. Else, the velocity of strain achieved by the elastic precursor is about 0.008. This
the second shock will be higher than the velocity of the first value is about twice that for the weak shock case. The stress
shock, so that it will merge with and overtake the first shock, achieved is also about twice the value attained in the weak
indicating the sense in which the first shock is stable against shock case. We have not bothered to include figures for this
such an event occurring [4]. case, as they are not any more instructive than Figs. 1 and 2.
H1 indicates the Hugoniot for a state of zero plasticity It is difficult to perform this kind of analysis for higher shock
in Fig. 2. As we just indicated, the system is stable against a speeds, since it seems reasonable to assume that the stress-
transition to this Hugoniot. On the other hand, in an infinites- strain relations given by Eq. (3.4) will become increasingly
imal amount of time a small amount of plasticity develops inaccurate as the strain increases. As evidence for this , we
(while the total strain remains constant), and the subsequent point out that we get inconsistent results for shock speeds
softening of the metal will cause the Hugoniot to literally considerably larger than the speed of sound, viz., the line
drop, as the overall stress in the system is reduced. When H1 is no longer co-terminus with the intersection of R0 and
this occurs (see e.g. H1 in Fig. 2), the Hugoniot will lie H0 (for contrast, see Fig. 2 for the weak shock case).
below the Raleigh line, indicating a transition via a slower We remark that we did not have to invoke a heat dissipa-
shock can indeed take place to this Hugoniot. A second tion mechanism to discuss overdriven shocks [14]. The basic
splitting will now occur on the second Hugoniot as well, as reason for this is the way we formulated the rate equation
more plasticity develops in time. This sequence will continue for the development of plasticity (Eq. (2.5)), which guaran-
smoothly, until the plasticity becomes saturated, [10 12] and tees in conjunction with mass conservation that the plasticity
can develop no further, as indicated by the experimental ve- parameter is continuous across the shock (as long as there
locity profiles [3]. Indeed, the central point of Wallace s is no divergence term in the rate function). And since we
method is to develop a method to extract th e finite rate assumed that the material was pristine (zero plasticity) to
of development of plasticity from weak shock wave data in begin with, plasticity is also zero immediately behind the
metals. precursor (initial shock). This contrasts with the assumption
It is a curious feature of the type of theory we are study- made in earlier papers that plasticity changes along the first
ing, that because of the way the internal energy is defined Hugoniot for the case of overdriven shocks [14].
(see Eq. (2.3a)), the mass conservation and momentum con- As we have seen earlier, in order for the plasticity to
servation equations may be solved independently of the en- change along the first Hugoniot, we have to assume the
ergy equation. Thus, even though our energy equation is existence of a divergence term in the rate function which
defined differently than Wallace s, the constitutive relations describes the development of plasticity. This would clearly
deduced from Wallace s original method remain unaltered. lead to the lowering of the stress, which would in turn cause
This is because the constitutive relations are obtained us- the Hugoniot to  sag below the Hugoniot for zero plasticity.
54
However, this does not imply that a shocked state reached Acknowledgements. I would like to acknowledge useful discussions with
John Bdzil, Brad Holian, Ralph Menikoff and Duane Wallace.
along this Hugoniot occurs for a lower state of stress [14]. In
order to see this, we point out that the shocked state will be
given by the intersection of a Rayleigh line (corresponding
References
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to visualize that the  sagged Hugoniot will intersect the
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Rayleigh line at a higher strain and stress than the original
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Hugoniot (with zero plasticity). Thus, we have provided an
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alternate viewpoint in which one does not have to invoke
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heat dissipation in order to understand overdriven shocks in
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metals. Finally, even in this case, the overdriven sh ock will Zeldovich Ya B and Raizer Yu P (1967) Physics of Shock waves and high-
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so strong that the initial shocked state somehow takes the
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4 Conclusion
We developed a conservative form of Wallace s theory of
thermoplastic flow. Our formulation differs from previous
ones in that we account explicitly for the fact that in cases
of interest, the system is in contact with an external energy
reservoir which supplies it energy to cause deformations.
Our approach allowed us to perform a wave-curve analysis
of Johnson and Barker s experimental data on weak shocks
in 6062-T6 Al, and show that the two-shock-wave structure
they observed is actually due the phenomenon of shock split-
ting. We pointed out that finite rate effects cause this split to
smooth out the wave profile into a sequence of  infinitesi-
mal shocks. We predict, using our approach, that overdriven
shocks will split as well. Difficulties with the experimental
observation of this phenomenon for the case of overdriven
shocks remain unsolved.